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A.S. Argon E. A n t H. K. D. H. Bhadeshia H. Biloni J. L Bocquet W. J. Boettinger G. Brebec R. W. Cahn G.Y. Chin? T.W. Clyne R.D. Doherty H.E. Exner R. Ferro D.R. Gaskell H. Gleiter A.L. Greer P. Haasen? J.P. Hirth S. Hofmann E.D. Hondros E. Hornbogen G. Kostorz
C. Laird P. LejEek W.C. Leslie Y. Limoge J. D. Livingston F.E. Luborsky T. B. Massalski J. R. Nicholls AD. Pelton D.G. Pettifor D.P. Pope M. Riihle A. Saccone S. R. J. Saunders M.P. Seah W. Steurer J.-L. Strudel R.M. Thomson C.M. Wayman M. Wilkens A.H. Windle H. J. Wollenberger
PHYSICAL METALLURGY Fourth, revised and enhanced edition
ELSEVIER SCIENCEB.V. Sara Burgemartstraat 25 P.O. Box 211,1000 AE Amsterdam, The Netherlands
ISBN 0 444 89875 1 0 1996 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced,stored in a retrieval system or transmitted in any form of by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, lo00 Ah4 Amsterdam, The Netherlands.
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Printed in The Netherlands
Regretfully unnoticed, in the final printing process a layout error has occurred on the original page v, due to which the authors’ names of chapters 15-19 are not correctly aligned with their chapter titles. Please use this corrected page instead.
SYNOPSIS OF CONTENTS Volume 1 1. 2. 3. 4.
5. 6. 7. 8. 9.
Crystal structure of the metallic elements Electron theory of metals Structure and stability of alloys Structure of intermetallic compounds and phases Appendix: Quasicxystals Metallurgical thermodynamics Phase diagrams Diffusion in metals and alloys Solidification Microstructure
Volume 2 10. Surface microscopy, qualitative and quantitative 11. Transmission electron microscopy 12. X-ray and neutron scattering 13. Interfacial and surface microchemistry 14. Oxidation, hot corrosion and protection of metallic materials 15. Diffusive phase transformations in the solid state
16. Nondifisive phase transformations 17. Physical metallurgy of steels 18. Point defects 19. Metastable states of alloys
Volume 3 20. Dislocations 21. Mechanical properties of single-phase crystalline media: deformation at low temperatures 22. Mechanical properties of single-phase crystalline media: deformation in the presence of diffusion 23. Mechanical properties of solid solutions 24. Mechanical properties of intermetallic compounds 25. Mechanical properties of multiphase alloys 26. Fracture 27. Fatigue 28. Recovery and recrystallization 29. Magnetic properties of metals and alloys 30. Metallic composite materials 31. Sintering processes 32. A metallurgist’s guide to polymers
Hirth Argon Aqon Haasen? Pope Stnrdel Thornson Laird Cahn Livingston,Luborsky, Chin? Clyne h e r ;A c t W d e
SYNOPSIS OF CONTENTS Volume 1 1. 2. 3. 4.
5. 6. 7. 8. 9.
Crystal structure of the metallic elements Electron theory of metals Structureand stability of alloys Structure of intermetallic compounds and phases Appendix: Quasicrystals Metallurgical thermodynamics Phasediagrams Diffusion in metals and alloys Solidification mcrosmhlre
Volume 2 10. SurFace microscopy, qualitative and quantitative 11. Transmission electron microscopy 12. X-ray and neutron scattering 13. Interfacial and surface microchemistry 14. Oxidation, hot corrosion and protection of metallic materials 15. Diffusive phase transformations in the solid state 16. Nondiffusive phase transformations 17.Physical metallurgy of steels 18. Point d e f m 19. Metastable states of alloys
h e r RWe, wilkepas Kostorz Hondms,Seah,Hojham, LejEek Saunders, Nicholls
Volume 3 20. Dislocations 21. Mechanical properties of single-phase crystalline media: deformation at low temperatures 22.Mechanical properties of single-phase crystalline media: deformation in the presence of diffusion 23. Mechanical properties of solid solutions 24. Mechanical properties of intermetallic compounds 25. Mechanical properties of multiphase alloys 26. Fracture 27. Fatigue 28. Recovery and recrystallization 29. Magnetic properties of metals and alloys 30. Metallic composite materials 31. Sintering processes 32. A metalIurgist’sguide to polymers V
Hirth Argon Argon Haasen? Pope Strudel Thornon
Laird Cahn Livingston, M o r s @ , Chin? c2yne .Erne 2.37 cF4 Cu 8' > 4.22 cF4 Cu Y >4.27 t14 6 >IO E >72 cF4?
Cu
a-Po
Sr 38 V =56.35A3 1s22s2p63s2p6~'o4s2p65s2 ff
p
>504
y
>896 or >3.5 >26 > 35
6
t14
>728 or > 19.5 > 32
E
cF4 hP2 cI2
Cu Mg
w
1.636
Ba
56 V =63.36A3 ls~s2p63s2p6~'04s2p6d105s2p66s2 ff cI2 w P >5.33 hP2 Mg Y >7.5 6 > 12.6
1.581
Ra 88 Va=68.22A3 Fr 87 ls~s~63s~6d'04s2p6d10f145~2p6d106s$~s' 1s22s2p63s2p6d'04s2p6d10f'45s2p6d106s2p67~2 ff cI2 w
increasing number of inner electron shells, is shown by the example of Cs (fig. 13). With increasing pressure, the valence electrons change from s to d character, giving rise to a large number of pressure-induced phase transitions at ambient temperature (YOUNG[ 199I]): 2.37 GPa 4.22 GPa 4.27 GPa 10 GPa 72 GPa a-Cs c3 p-cs e P'-CS e y-cs e 8-cs c3 E-CS References: p . 45.
18
u! Steurer
Ch. 1, $ 3
Fig. 11. Relationship between body-centered cubic (bcc) a-Fe, cI2-W type, space group I m h , No. 229, la: 0 0 0, and face-centered cubic (fcc) y-Fe, cF4-Cu type, space group F m h , No. 225, 4a: 0 0 0. The facecentered tetragonal unit cell drawn into an array of four bcc unit cells transforms by shrinking its faces to fcc.
The alkaline earth metals behave quite similarly to the alkali metals. They crystallize under ambient conditions in one of the two closest-packed structures (ccp or hcp) or in the body-centered cubic (bcc) structure type and also show several allotropic forms (fig. 14). The large deviation c/u = 1.56 from the ideal value of 1.633 for beryllium indicates covalent bonding contributions. For alkali and alkaline earth metals, the pressure-induced phase transitions from cI2-W to cF4-Cu occur with increasing atomic number at decreasing pressures.
3.3. Groups 3 to 10, transition metals The elements of groups 3 to 10 are typical metals which have in common that their d-orbitals are partially occupied. These orbitals are only slightly screened by the outer s-electrons, leading to significantly different chemical properties of the transition elements going from left to right in the periodic system. The atomic volumes decrease rapidly with increasing number of electrons in bonding d-orbitals, because of cohesion, and increase as the anti-bonding d-orbitals become filled (fig. 15). The anomalous behavior of the 3d-transition metals, Mn, Fe and Co, may be explained by the existence of non-bonding d-electrons (PEARSON[19721). Scandium, yttrium, lanthanum and actinium (table 6) are expected to behave quite
Fig. 12. Unit cell of the body-centered cubic structure type cI2-W, space group I m h , No. 229, la: 0 0 0.
Ch. 1, $ 3
19
Crystal structure of the metallic elements
PRESSURE, GPa
Fig. 13. The variation of the atomic volume of cesium with pressure (after DONOHUE [1974]).
similarly. Indeed they show similar phase sequences: the high-pressure phases of light elements occur as the ambient-pressure phases of the heavy homologues. The hP4 phase of lanthanum, with the sequence ..ACAB.., is one of the simpler closest-packed polytypic structures common for the lanthanides (fig. 16 and fig. 10). Another typical polytype for lanthanides is the hR3 phase of yttrium with stacking sequence ..ABABCBCAC.. (fig. 17 and fig. 10). Titanium, zirconium and hafnium (table 6) crystallize in a slightly compressed hcp structure type and transform to bcc at higher temperatures. At higher pressures the w-Ti phase is obtained (fig. 18). The packing density of the hP3-Ti structure with -0.57 is slightly larger than that of the simple cubic a-Po structure (-0.52) but substantially lower than for bcc (-0.68) or ccp and hcp (-0.74) type structures. Calculations have shown that the w-Ti phase: is stable owing to covalent bonding contributions from s-d electron transfer. At even higher pressures, zirconium and hafnium transform to the cI2-W type, while titanium remains in the hp3-Ti phase up to at least 87 GPa. By theoretical considerations it is also expected that titanium performs this transformation at sufficiently high pressures (AHUJAet al. [19931). A general theoretical phase diagram for Ti, Zr and Hf is shown in fig. 19. Vanadium, niobium, tantalum, molybdenum and tungsten have only simple bcc References: p. 45.
20
N? Steurer
Ch. 1, 9 3
Fig. 14. Illustration of the bcc-to-hcp phase transition of Ba. (a) bcc unit cell with (110) plane marked. (b) Projection of the bcc structure upon the (110) plane. Atomic displacements necessary for the transformation are indicated by arrows.
structures (table 7). Up to pressures of 170 to 364 GPa no further allotropes could be found, in agreement with theoretical calculations. Chromium shows two antiferromagnetic phase transitions, which modify the structure only very slightly (YOUNG [ 19911). The high-temperature phases of manganese (table 8), y-Mn, cF4-Cu type, and 8-Mn, cI2-W type, are typical metal structures, whereas a-Mn and p-Mn form very complicated structures, possibly caused by their antiferromagnetism. Thus, the a-Mn structure can be described as a 3 x 3 x 3 superstructure of bcc unit cells, with 20 atoms slightly shifted and 4 atoms added resulting in 58 atoms over all (fig. 20). The structure of p-Mn (fig. 21) is also governed by the valence electron concentration (“electron compound” or Hume-Rothery-type phase). The variation of the atomic volume of manganese with temperature is illustrated in fig. 22. For technetium, rhenium, ruthenium and osmium, only simple hcp structures are known. The technically most important element and the main constituent of the Earth’s core, iron (table 8) shows five allotropic forms (fig. 23): ferromagnetic bcc a-Fe transforms to paramagnetic isostructural p-Fe with a Curie temperature of 1043 K; at 1185 K fcc y-Fe forms while at 1667 K a bcc phase, now called &Fe, appears again. For the variation of
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21
Ctystal structure of the metallic elements
c
lo
"5 w
I 3 J
0
> 0 -
I 0 IQ
Y Zr N b M o T c Ru RhPd Ag Cd I n Sn Hf To W Re Os I r Pt Au H g T 1
Fig. 15. Atomic volumes of the transition metals. A means cF4-Cu type, v hm-Mg,0 cI2-W, (after PEARSON [1972]).
other types
the atomic volume with temperature see fig. 24. High-pressure nonmagnetic E-Fe, existing above 13 GPa, bas a slightly compressed hcp structure. Cobalt (table 9) is dimorphous, hcp at ambient conditions and ccp at higher temperatures. By annealing it in a special way, stacking disorder can be generated: the hcp sequence ..ABAB.. is statistically disturbed by a ccp sequence ..ABCABC.. like ..ABABABABCBCBCBC.. with a frequency of about one ..ABC.. among ten ..AB... Rhodium, iridium, nickel, palladium and platinum all crystallize in simple cubic closestpacked structures. 3.4. Groups 11 and 12, copper and zinc group metals
The "mint metals", copper, silver and gold (table 10) are typical metals with ccp structure type (fig. 25). Their single ns electron is less shielded by the filled d-orbitals than the ns electron of the alkali metals by the filled noble gas shell. The d-electrons also contribute to the metallic bond. These factors are responsible for the more noble References: p . 45.
22
W Steurer
Ch. 1, $ 3
Fig. 16. One unit cell of the hP4-La structure type, space group P6,/mmc, No. 194, 2a: 0 0 0, 2c: % Y3 %.
character of these metals than of the alkali metals and that these elements sometimes are grouped to the transition elements. For zinc, cadmium and mercury (table 10) covalent bonding contributions (filled dband) lead to deviations from hexagonal closest packing (hcp), with its ideal axial ratio c/u= 1.633, to values of 1.856 (Zn) and 1.886 (Cd), respectively. The bonds in the hcp layers are shorter and stronger, consequently, than between the layers. With increasing pressure, clu approximates the ideal value 1.633: for Cd clu = 1.68 was observed at 30 GPa (DONOHUE [ 1974]), and for Hg,c/u = 1.76 at 46.8 GPa (SCHULTE and HOLZAPFEL [19931). The rhombohedral structure of a-Hg may be derived from a ccp structure by compression along the threefold axis (fig. 26). In contrast to zinc and cadmium, the ratio c/u= 1.457 for a hypothetical distorted hcp structure is smaller than the ideal value. There also exist several high-pressure allotropes (fig. 27).
3.5. Groups 13 to 16, metallic and semi-metallic elements Only aluminum, thallium and lead crystallize in the closest-packed structures characteristic for typical metals (table 11). The s-d transfer effects, important for alkaliand alkaline-earth metals, do not appear for the heavier group 13 elements owing to their filled d-bands. Orthorhombic gallium forms a 63 network of distorted hexagons parallel to (100) at heights x=O and 1/2 (fig. 28). The bonds between the layers are considerably
Ch. 1, 8 3
Crystal structure of ihe meiallic elements
23
JA
C >A
C B
C B
)A B >A Fig. 17. One unit cell of the hR3-Sm Structure type, space group R h , No. 166, 3a: 0 0 0, 6c: 0 0 0.22.
References: p . 45.
24
Ch. 1, $ 3
U? Steurer
Fig. 18. The hP3-Ti structure type, space group P6/mmm, No. 191, la: 0 0 0, 2d: % % %.
weaker than within. At higher pressure gallium transforms to a bcc phase, cI12-Ga, and additionally increasing the temperature leads to the tetragonal indium structure type t12-In (fig. 29). In an alternative description based on a face-centered tetragonal unit cell with a’ = fia, the resemblance to a slightly distorted cubic close-packed structure with c/a = 1.08 becomes clear.
/
T
mechanically I
unstable at
bcc
; I
T=O
I
stable at
T=O
I
I I I I
I I I I I
I I I
Fig. 19. Schematic calculated phase diagram for Ti, Zr and Hf (from AHUJAet al. [1993]).
Ch. 1, $ 3
25
Crystal structure of the metallic elements
Table 6 Structure information for the elements of groups 3 and 4. In the first line of each box the chemical symbol, atomic number Z, and the atomic volume V, under ambient conditions is listed. In the second line the electronic ground state configuration is given. For each phase there is tabulated limiting temperature T[K] and pressure P[GPa], Pearson symbol PS,prototype structure PT, and, if applicable, the lattice parameter ratio clu. T[K]
P[GPa]
21 V,=24.97A3 ls22s2p63s2p6d'4s2 a p >1610 Y > 19
PS
PT
hP2 cI2 tP4?
Mg
w
39 Vat=33.01A3 1s22s~63sZp6d'04s~6d'5s2 a hP2 Mg p >1751 cI2 w > 10 hR3 Sm Y 6 > 26 hP4? E > 39 cF4 Cu
1.592
a
p
PS
PT
cla
hP2 cI2 hP3
w
Mg
1.587
>2
w-Ti
Zr 40 V,=23.28A3 1s22s~63s2p6d104s2p6d25s2 1.571
hP2 cI2 hP3 cI2
a
p
>1136
>2 > 30
6J
w'
Hf
V,=37.17A3
>1155
w
Y
57
P[GPa]
Ti 22 Vat=17.65A3 ls22s2p63s2p6d24s2
Se
La
T[K]
cla
72
V,=22.31
Mg
w
w-Ti
w
A3
1s22s2p63s~6d'04s~6d'05s2p6d16s2 1s22sZp63s~6d'04s2p6d'of145s~6d26~2 a hP4 a - h 2~1.61 a hP2 Mg ~3 >583or>2.3 cF4 Cu p >2016 cI2 w y >1138 cI2 w w > 38 hP3 w-Ti
6
> 7.0
hP6
Ac 89 V,=37.45 A3 at 293 K ...3s~6d'04s2p6d'Of145s2p6d106s2p6d17s2 ff cF4 Cu
>71
w'
Ku
1.593
cI2
1.581
w
104
...3szp6d'04s2p6d'of145s2p6d10f146s2p6d27s2
Silicon and germanium (table 11) under ambient conditions crystallize in the diamond structure, owing to strong covalent bonding. At higher pressures they transform to the metallic white-tin (tI4-Sn) structure. This structure type consists of a body-centered tetragonal lattice which can be regarded as being intermediate between the diamond structure of semiconducting a-Sn and ccp lead (fig. 30). For an ideal ratio of c/u = 0.528 one atom is sixfold coordinated. The high-pressure phase hP1-BiIn has a quasi-eightfold coordination, the ideal ratio for CN = 8 would be clu = 1. At higher pressures, closestpacked structures with twelvefold coordinations are obtained. Thus with increasing pressure silicon runs through phases with coordination numbers 4, 6, 8 and 12. The effective radius of tin in p-Sn and of lead in a-Pb is large compared with that of other typical metals with large atomic number due to uncomplete ionization of the single ns electron. This means that in a-Sn, for instance, the electron configuration is ...5s'5p3, allowing sp3-hybridization and covalent tetrahedrally coordinated bonding, whereas in p-Sn with ...5s25p2only two p-orbitals are available for covalent and one further p-orbital for metallic bonding. The structure of arsenic, antimony and bismuth (isotypic under ambient conditions) References: p. 45.
26
Ch. 1, 0 3
ISteurer
T[K]
P[GPal
PS
FT
V
23 V,=13.82A3 ls?2s%63szpdd34s2 cI2
PS
FT
cI2
w
1s~s~63s2p6d'04s2p6d55s1 cn
w
T[KI
c/a
P[GW
Cr 24 v,=12.00A3 ls22s%63s%6d54d
w
Nb 41 Vas=17.98A3
c/o
MO 42 V,=15.58A3
1s22s~63s~6d'04s~6d45s' cI2
w
Ta
73 Vat=18.02A3 1s~s~63s~6d'04s~6d10f145s2p6d36s2 cI2 w
74 Vat=15.85 A3 1s~s~63s~6d'04s~6d10f~45s2p6d46sz cI2 w
W
(table 12) consists of puckered layers of covalently bonded atoms stacked along the hexagonal axis (fig. 31). The structure can be regarded as a distorted primitive cubic structure (a-Po) in which the atomic distance d, in the layer equals that between the layers 4. The metallic character of these elements increases for d,/d, approximating to 1 (table 13). The helical structures of isotypic a-Se and a-Te may also be derived from the Table 8 Structure information for the elements of groups 7 and 8. In the first line of each box the chemical symbol, atomic number Z, and the atomic volume V, under ambient conditions is listed. In the second line the electronic ground state configuration is given. For each phase there is tabulated: liiting temperatureT[K] and pressure P[GPa], Pearson symbol PS, prototype structure PT, and, if applicable, the lattice parameter ratio c/o. T[K]
P[GPal
~n 25 va,=12.21A3 i S 2 2 s % ~ p ~ ~ 2 a p >loo0 y >1373 S >1411
PS
PT
T[K]
c/a
P[GPa]
PS
FT
cr2 cF4 cI2
w
hP2
Mg
1.603
Mg
1.582
c/a
Fe
26 Va,=11.78A3 ls%%63s~6d64s2 a y 21185 6 >1667 E > 13
cI58 a-Mn cP20 p-Mn cF4 Cu cI2 w
TC 43 Vat=14.26A3 ls~~63s~6d'04s~6d65s' hP2
Cu
w
Ru 44 Vat=13.57A3 1s ~ s ~ 6 3 s ~ 6 d 1 0 4 s ~ 6 d 7 5 s 1
Mg
1.604
Re 75 V =14.71A3 1s~s~63s~~'04s~6dp6d10f145SZp6dS6SZ hP2 Mg 1.615
hP2
OS 76 V,=13.99A3 1s"2s~~s~6d'o~~6d4s"Of'45s2p6d66sZ hP2 Mg 1.580
Ch. 1, $ 3
Crystal structure of the metallic elements
27
Fig. 20. One unit cell of cI58-Mn, space group Id3m, No. 217, with four different types of Mn atoms in 2a: 0 0 0, 8c: 0.316 0.316 0.316, 24g: 0.356 0.356 0.034, 24g: 0.089 0.089 0.282, shown (a) in perspective view and (b) in projection. 'Avo types of Mn atoms are coordinated by CN 16 Friauf polyhedra, one by a CN 14 Frank-Kasper polyhedron and one by an icosahedron.
primitive cubic a-Po structure (fig. 32). The infinite helices run along the trigonal axes, and have three atoms per turn. The interhelix bonding distance d, plays a comparable References: p . 15.
28
W Steurer
Ch. 1, 1 3
Fig. 21. One unit cell of cPZCkMn, space group P4,32, No 213, with two types of Mn atoms: 8c: 0.063 0.063 0.063, 12d: 0.125 0.202 0.452, shown (a) in perspective view and (b) in projection. The atoms in 8c are coordinated by 12 atoms in a distorted icosahedron, the Mn atoms in 12d by 14 atoms in a distorted FrankKasper CN 14 type polyhedron.
role for the metallic character of these elements as does the interlayer distance in the case of the group 15 elements. Wih increasing pressure, the transition to the metallic p-Te phase takes place.
3.6. Lanthanides and actinides Lanthanides and actinides (table 14) are characterized by the fact that their valence electrons occupying the f-orbitals are shielded by filled outer s- and p-orbitals. The chemical properties of the lanthanides are rather uniform since the 4f-orbitals are largely screened by the 5s- and 5p-electrons. The chemical behavior of the actinides, however, is somelike in between that of the 3d transition metals and the lanthanides since the 5forbitals are screened to a much smaller amount by the 6s- and 6p-electrons. With the exception of Sm and Eu, all lanthanides under ambient conditions show either a simple hcp structure with the standard stacking sequence ..AB.. or a twofold superstructure with a stacking sequence ..ACAB ... Samarium has, with ..ABABCBCAC.., an even more
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Crystal structure of the metallic elements 15.0
I
I
I
I
I
T[K]
P[GPa]
I
14.5 -
14.0-
w'
I
/'
0 13.5-
>
u
/ /
I
/
0
5 13.012.5
-
T[K] P[GPal 27 V,=11.08A3 ls~s2p63s2p6d74s2 e LY >695
PS
PT
cla
PT
cF4
Cu
cF4
Cu
Ni
Co
Rh 45
PS
28 V =10.94A3 ls22s2p63s2pG84s2
hP2 cF4
Mg Cu
1.623
pa 46 v =i4.72A3 1s%$63s2p6~'04s2p6d10
V,=1?.75A3
1
cF4
Cu
Ir 77 V =14..15%r3 1s~s~63s2p6~'04s2p6d10f14Ss2p6d76~2 cF4 Cu
Pt
78
V =15.10A3
1s~s~63s2p6~'04sZp6dlaf14SS2p6d96S1
cF4
Cu
cla
30
W
1000
Ch. 1, 9 3
Steurer
1
Y(fCC1
0
50.
Fig. 23. Phase diagram of iron (from VAINSHTEIN e?aZ. [1982]). Table 10 Structure information for the elements of groups 11 and 12. In the first line of each box the chemical symbol, atomic number Z and the atomic volume V,, under ambient conditions is listed. In the second line the electronic ground state configuration is given. For each phase there is tabulated: limiting temperatureT[K] and ' , and, if applicable, the lattice parameter ratio c/u. pressure P[GPa], Pearson symbol PS, prototype structure FT T[K]
P[GPa]
PS
PT
c/a
CU 29 Va,=11.81A3
I
T[K]
P[GPa]
PS
PT
c/u
hF2
Mg
1.856
Mg
1.886
Zn 30 Va=15.20A3 ls22s2p63sZp6d'04s2 cF4
Cu
Ag
47 Va,=17.05A3 ls22s~63s~6d'04s~6dp6dlD5S1 cF4
Cd 48 V =21.60A3 1s22s~63s2p6~'04s~6d10Ss2
Cu
AU 79 V,=16.96A3
hF2 Hg 80 V =23.13A3 at 80K
1s22s~63s~6d'04s2p6d'of14S~2~6d106s1 cF4
Cu
1s~s2p63szp6"do4s2p6d'of14Ss2p6d'06s2 a ~234.3 hR1 a-Hg P >3.7 t12 a-Pa Y > 12 0p4 8 > 37 hP2 Mg 1.76
Ch. 1, 9 3
31
Crystal structure of the metallic elements
TEMPERATURE ,OC
Fig. 24. The variation of atomic volume of iron with temperature (from DONOHUE [1974]).
complicated stacking order with 4.5-fold superperiod. For all lanthanides the ratio clu is near the ideal value of n x 1.633. It is interesting that with increasing pressure and decreasing atomic number the sequence of closest-packed phases hP2-Mg (..AB..) 3 hR3-Sm (..ABABCBCAC..) hP4-La (..ACAB..) jcF4-Cu (..ABC..) 3 hP6-Pr appears (cf. figs. 10, 17 and 33). Cerium undergoes a transformation from the y to the a-phase at pressures >0.7 GPa:
Fig. 25. The structure of c F 4 - c ~ .space group Fmgm, No. 225, 4a 0 0 0.
References: p . 45.
32
Ch. 1, $ 3
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Fig. 26. The structure of hR1-Hg, space group R3m, No. 166, 3a 0 0 0.
the ccp structure is preserved but the lattice constant decreases drastically from 5.14 to 4.84 A owing to a transition of one 4f-electron to the 5d-level (fig. 34). This isostructural transition is terminated in a critical point near 550K and 1.75 GPa (YOUNG[1991]). Further compression gives the transformation at 5.1 GPa to the a’-phase, and finally at 12.2 GPa to the &-phase. Europium shows a completely different behavior, as do the other lanthanides, owing to the stability of its half filled 4f-orbitals. Thus, it has more similarities to the alkaline earth metals; its phase diagram is comparable to that of barium
Table 11 Structure information for the elements of groups 13 and 14. In the first line of each box the chemical symbol, atomic number Z, and the atomic volume Vat under ambient conditions is listed. In the second line the electronic ground state configurationis given. For each phase there is tabulated: limiting temperature T[K]and pressure P[GPa], Pearson symbol PS, prototype structure PT, and, if applicable, the lattice parameter ratio cla. T[K]
P[GPa]
PS
PT
cla
T[K]
P[GPa]
PS
PT
cF8 t14 hP1
p-Sn 0.552
cia
Si 14 Va=Z0.02A3 1s22s2p63s2p2 LY
B Y
S E
r Ga 31 Va,=19.58A3 ls22s2p63sZp6d'04s~' LY
p y
e330 >330
>1.2 >3.0
> 12 > 13.2 > 36 >43 >78
Cd BiIn
0.92
hP2 cF4
Mg Cu
1.699
cF8 t14 hP1 hP4
C p-Sn 0.551 BiIn 0.92
O?
Ge 32 V,,=22.63A3 1s?2s~63s2p6d104s2p2
oC8 c112 t12
a-Ga In
a 1.588
In
49 Vat=2Cj.16A3 ls'2s'p63s2p6d'04s~6d'05s2p1
P
> 11
Y S
> 75 > 106
Sn
tI2
In
1.521
TI
81 Vat=28.59A3 1 s'2szp63s2p6d104~z~6d'of'45~2p6d'06szp' LY hP2 Mg p >503 cI2 w Y >3.7 cF4 Cu
50 Vat=34.16A3 at 285K 1s ~ s ~ 6 3 s 2 p 6 d ~ 0 4 ~ z p 6 d 1 0 5 s ~ 2 LY 9.2 t12 Pa 0.91 S >40 cI2 w
Pb 1.598
82 Va1=30.32A3 1s22s~~s2p6d'04s2p6d10f145s2p6d106s~2 LY cF4 Cu 1.650 B > 13.7 hP2 Mg Y > 109 cI2 w
rather than to the other lanthanides. A similar behavior is observed for ytterbium which is divalent owing to the stability of the completely filled 4f-orbitals; its phase diagram resembles that of strontium. The c-lattice parameter of gadolinium exhibits an anomalous expansion when cooled below 298 K (fig. 35) due to a change in the magnetic properties of the metal. Several other lanthanides show a similar behavior. According to their electronic properties, the actinides (table 14) can be divided into two subgroups: the elements from thorium to plutonium have itinerant 5f-electrons contributing to the metallic bond, whereas the elements from americium onwards have more localized Sf-electrons. This situation leads to superconductivity for thorium, protactinium and uranium, for instance, and to magnetic ordering for curium, berkelium and californium (DABOS-SEIGNON et al. [1993]). The contribution of 5f-electrons to References: p. 45.
34
W Sreurer
Ch. 1, $ 3
C The structure of oCbGa, Crnca, No. 64, 8f 0 0.155 0.081, (a) in a perspective view and projected (010) and (c) (100).showing the distorted hexagonal layers.
bonding leads to low symmetry, small atomic volumes and high density in the case of the light actinides while the heavier actinides crystallize at ambient conditions in the hcp structure type. The position of plutonium at the border of itinerant and localized Sf-states causes its unusually complex phase diagram, with structures typical for both cases. Thus, monoclinic a-Pu can be considered as a distorted hcp-structure with about 20%higher packing density than cF4-Pu owing to covalent bonding contributions from Sf-electrons (fig. 36) (EK et al. [1993]). This ratio is quite similar to the above-mentioned one of a-Ce and y-Ce, which are both ccp. The phase diagram of americium is very similar to
Ch. 1, $ 3
Crystal structure of the metallic elements
35
Fig. 29. The structure of tI2-In, space group Wmmm, No. 139, 2a 0 0 0.
Fig. 30. Relationships between the structures of the two tin allotropes: (a) grey a-Sn, c F 8 4 type, space group Fd3m, No. 227, 8a: 0 0 0, % % %, and (b) white p-Sn, t I w - S n type, space group I4,/amd, No. 141, 4a: 0 0 0. Note the large difference in the minimum distances: d z z = 1.54 A and d z ? =3.02 A.
those of lanthanum, proseodymium and neodymium. Owing to the localization of 5felectrons it is the first lanthanide-like actinide element. Both lanthanides and actinides crystallize in a great variety of polymorphic modifications (fig. 37).
References: p . 45.
36
T[K]
P[GPa]
PS
PT
cla
>25.0
Y
>8 >28
PT
cla
AS 2.617 mP4 p-Sb cI2 w
Te 52 Va,=33.98A3 1s~s'p63szpdd104szp6d105s2p4 a hP3 a-Se 1.330 P >4.0 mP4 P-Te Y > 6.6 OP4 s > 10.6 hR1 &PO E > 27 cI2 w
hR2
ff
PS
hR2 (Y-As 2.805 cP1 ff-Po
Sb 51 Vat=30.21A3 1s'2s$63s2p6d104s$6d'05s2p3
P
P[GPa]
Se
1 ff
T[K]
34 v =27.27A3 ls~s~63s2p~~104s~4 a hP3 a-Se P > 14 mP3 Y > 28 tP4 8 > 41 hR2
AS 33 Vat=21.52A3
P
Ch. 1, 4 3
M! Steurer
Bi 83 V,,=35.39A3 1s22s2p&js2p6d'04s2p6d'af145s2p6d106s~3 ff -hR2 ~ - A S2.609 P > 2.6 mC4 &Si Y > 3.0 mP4 P-Sb s > 4.3 E > 9.0 cI2 w
84 V,=38.14A3 at 311 K 1s22s2p63s$6d104s2p6d10f'45s$6d'06s~4 CY cP1 a-Po P >327 hR1 p-Po
Po
1.135
Ch. 1, $ 3
37
Crystal structure of the metallic elements
Fig. 31. The structure of bR2-As, space group Rjm, No. 166, 6c 0 0 0.277.
Table 13 Intralayer (d,) and interlayer
(4)distances in a-As-type layer structures, y-de-type helix structures and primitive cubic a-Po (PF.~RsoN[1972]).
Element
Distance d,
Distance d,
Wdl
a-As a-Sb a-Bi y-Se y-Te a-PO
2.51 A 2.87 A 3.10 8, 2.32 8, 2.86A 3.37 A
3.15A 3.37 A 3.47 A 3.46A 3.46 A 3.37 A
1.25 1.17 1.12 1.49 1.31 1.oo
References: p . 45.
38
W Steurer
Ch. 1, 5 3
Fig. 32. (a) The structure of h P 3 - k space group P3,21, No. 152, 3a 0.237 0 %, and (b) its projection upon (001) compared with (c) one unit cell of cP1-Po, space group Pmjm, No. 221, l a 0 0 0, and (d) its projection along [ 11 11. The hexagonal outline of the projected cubic unit cell is drawn in.
Ch. 1, 3 3
39
Crystal structuw of the metallic elements
Table 14 Smcture information for the lanthanides and actinides. In the first line of each box the chemical symbol, atomic number Z, and the atomic volume Vat under ambient conditions is listed. In the second line the electronic ground state configuration is given. For each phase there is tabulated: limiting temperature T[K] and pressure P[GPa], Pearson symbol PS, prototype structure PT, and, if applicable, the lattice parameter ratio cla.
T[Kl
P[Gl?al
PS
Ce 58 Vat=34.72A3 1s22s$63s2p6d'04s2p6d'ofzSs*p66s2 a e96 cF4 B hP4 y >326 cF4 s >999 cI2 a' >5.1 oC4 E > 12.2 t12
PT
T[K]
eta
Th 90
P[GPa]
PS
PT
cla
Vat=32.87A3
...3s2p6d104s~6d10f145s$6d106s2p6d~s2 Cu
Ly
a-La 2X1.611 p
>1633
cF4 cI2
Cu
w
Cu
w a-u? In
Pr 59 V =35.08A3 1s22sZp63s~~~104sZp6d10f35~~66s2 a
p
21068
Y
> 3.8
S
> 6.2 > 20
E
hP4 cI2 cF4 hp6 oc4
Pa 91 V =25.21A3 ...3s2p6d104sZ~d10f145s2p6d10f26szp6d17s2 a-La 2x1611 LY t12 @-Pa 0.825 w B >1443 cI2 w Cu
Pr
3x1622
a-U
Nd 60 Ve=34.17A3 U 92 Vat=20.75A3 ls22szp63s~6d104s2p6d10~5s2p66s2 ...3s~6d'04s2p6d10f145s~6d10f36s2p6d17~2 oc4 a-u a hP4 a-La 2~1.612 Ly p >1136 cI2 w p >941 tP30 p-u Y >5.8 cF4 Cu y >lo49 cI2 w 3~1.611 6 > 18 hp6 pr E > 38 mC4 ?
Pm 61
Vat=33.60A3 1s22s~63s2p6d104s2p6d'ofs5s2p66sz a hP4 CY-L p ~1163 cI2 w Y > 10 cF4 Cu s > 18 hp6 Pr E >40 ?
Np 93
Ve=19.21A3
...3s~6d104s2p6d10f145s2p6d10f56s~67~2
2x1.60
LY
B y
>553 >849
oP8 tP4 cI2
a-Np
p-Np 0.694
w
Continued on next page
References: p . 45.
40
Ch. 1, 0 3
W Steurer Table 14-Continued
T[K]
P[GPa]
PS
FT
V =33.17A3 ls~s2p63s2p6~'04s~6d'of65s2p66s2 a hR3 a-Sm p >loo7 hP2 Mg y >1195 cI2 w 6 >4.5 hP4 a-La cF4 Cu e > 14 Pr hp6 t > 19 mC4 ? e >33
Pu 94
Sm 62
Eu 63 V =48.10A3 ls22s~63s$'~'04s$6d4s2p6d10f75SZp66SZ a B > 12.5 Y > 18
T[K]
c/a
w
hP2
Mg
PS
PT
cia
Va,=19.88A3
...3s2p6d'04s%6d'of'5s2p6d10f66s$%'s2
4.5x1.60f a 1.596 p y
2~1.611 6 6' 3X1.611 E
t
>388 >488 >583 >I25 >I56
>40.0
Am 95 cI2
P[GPa]
mP16 a-Pu mC34 p-Pu oF8 y-Pu cF4 Cu tI2 In 1.342 cI2 w hP8 1.65712
Va=29.21A3
...3s2p6d'04s~6d'Df'45~2p6d10f76s~61 s2
a
1.553
?
p y
6 E
>1042or>5 >1350 > 12.5 > 15
hP4 cF4 cI2 mP4 oC4
a-La 2~1.621 Cu
w
6-Am
a-u
Gd 64
v =33.04A3 ls22s2p63s2p'~'04s2p6d10f75s~65d16~2 a hP2 Mg 1.591 p >I508 cI2 w Y >2.0 hR3 a-Sm 4.5~1.61 6 >5 hP4 a-La 2X1.624 E > 25 cF4 Cu t > 36 hp6 Pr
Cm 96 V,=29.98A3 ...3s2p6d'04s2p6d'of145~2p6d10f76s2p6d11s2 a hP4 a-La 2X1.621 p >155001>23 cF4 Cu Y >43 OC4 a-U
Tb 65 V =32.04A3 1s22s2p63s~'~'04s2p6d10~5s~66s2 a e220 0c4 ~ - D a' hP2 Mg p >I562 cI2 w Y >3.0 hR3 a-Sm 6 > 6.0 hP4 a-La E >29 cF4 Cu b >32 hp6 Pc
Bk 91 Va=21.96A3 ...3s2p6d'04s2p6d10f''5s~6d'of86s2p6d'7s2 a hP4 a-La 2~1.620 p >125001 > 8 cF4 Cu Y >25 OC4 a-U
Y 1.580 4.5x1.6C
3~1.616 Continued on nextpage
Ch. 1, 4 3
41
Crystal structure of the metallic elements Table 14-Continued
T[K]
Dy 66
P[GPal
PS
FT
Cf 98
V -31.57A3
1s~s2p63sZp6ad~4s2p6d10f105s2p66sz a' e86 d34 ~ - D Y a hP2 Mg 1.573
p
>1654
Y
s E
>5.0 >9.0 > 38
cI2 hR3 hP4 cF4
w
PT
PS
Vat=27.41A3
a
P
s
>863or >17 > 30 >41
hP4 cF4 aP4 OC4
a-La 2X1.625 Cu
y-Cf
a-U
Cu
V -30.65A3
Es 99
...3sZp6d104s~6d10f'45s~6d10f1'6~Zp~~Z a
P
?
hP4 cF4
a-La
Cu
Fm 100
...3s~6d'04sZp6d10f145~2p6d10f126s~'%'s2
1s~szp63s~6~'04s2~6d6d10f125SZp66SZ -
P
>7.0
hp2 hR3
Y
> 13
hP4
Mg
1.569
a-Sm a-La
Tm 69
Va=30.10A3 1s22sZp63s$6d104s2p6d10f135szp66~2 a hP2 Mg 1.570 fl >1800 cI2 w Y >9 hR3 a-Sm s > 13 hP4 r~-b 4.5X1.5
Md 101 ...3szp6d'04s$6d'of145s~6d10f136szp%'sz
Yb 70
No 102
Vat=41.24A3 ls~szp63s$6d'04s~6dp6d"45s2p66s2 a 34 hP2 Mg B cF4 Cu y >lo47 or >3.5 cI2 w
...3~$~d'~4s~p~d'~f ''5s2p6d10f'46s~67s2 1.646
LU 71
Lr 103 . . . 3 ~ ~ p ~ d ' ~ 4''5s$6d'of'46s~6d'7s2 s~~d'~f
Vat=29.52A3 ls~s$63s$6d'04szp6d10f145szp6d16s2
P Y
> 18 > 35
C h
a-La
V -31.12A3 1s~s~63s$~1~4s~6d'Of"5s2p66s2 a hP2 Mg 1.570 p >1660 cI2 w Y >7.Q hR3 a-Sm 4.5x1.6 s > 13 hP4 a-La
#
P[GPal
...3s$6d'04s2p6d10f145s$6d10f106s2p'%'s2
Y
a-Sm 4.5x1.6C
Ho 67
Er 68
T[K]
cla
hP2
Mg
hR3 hP4
a-Sm 4.5x1.5:
1.583
&-La
Rflerences: p. 45.
42
W Steurer
Fig. 33. The structure of hP&Pr, space group P3,21, No. 152, 6c 0.28 0.28 0.772.
Ch. 1, 9 3
Ch. 1, 03
43
Crystal structure of the metallic elements
5.20
I
I
--
1
I
1
1
1
I
-
.y
z a b c zn
7
5.00
0 10
2
4.80
I= I-
a -I
4.60
480 PRESSURE, GPa
2p
m
6.0
Fig. 34. Pressure dependence of the atomic volume of cerium (from DONOHUE [1974]).
5.80
oa-
I
1
I
I
I
I
I
1
I
I
5.79
Y, c
c
0
c
m c
5.78 1
Q)
.-0 c
0
J
I
100
200
300
Temperature, O K Fig. 35. Variation of the lattice parameters of gadolinium with temperature. There are no structural changes in [1974]). this temperature range (from DONOHUE References: p . 45.
44
*I
Ch. 1, $ 3
W! Steurer
I
24
- 200
0
200
400
600
TEMPERATURE .OC
Fig. 36. The variation of the atomic volume of the various allotropes of plutonium with temperature (from DONOHUE [19741).
Ch. 1, Refs.
45
Crystal structure of the metallic elements
P = 0.1 MPa
Ac
Th
Pa
U
NP
PU
Am
Cm
Fig. 37. Combined binary alloy phase diagrams for the light actinides (from YOUNG[1991]).
Acknowledgements The author would like to express his sincere gratitude to Mss. M. Krichel for the preparation of the structure plots.
References AHUJA,R., J. M. WILLS,B. JOHANSSON and 0. ERIKSSQN,1993, Phys. Rev. B48, 16269-79. BORCHARDT-OTT, W., 1993, Kristallographie (Springer Verlag, Berlin). 1971,Z. Kristallogr. 133, 127-133. BRUNNER,G. O., and D. SCHWARZENBACH, DAAMS,J.L.C., P. VILLARSand J.H. N. VAN VUCHT,1991, Atlas of Crystal Structure Types for Intermetallic Phases (American Society for Metals, USA), Vols. 1 to 4. DAAMS, J. L. C., J. H. N. VAN VUCHTand P. VILLARS,1992, J. Alloys and Compounds 182, 1-33. S., J. P. DANCAUSSE, E. CERING, S. HEATHMAN and U. BENEDICT,1993, J.Alloys and DABOS-SEIGNON, Compounds 190,237-242. DONOHUE,J., 1974, The structures of the elements (John Wiley & Sons, New York). EK, J. VAN, P.A. STERNEand A. CONIS, 1993, Phys. Rev. B48, 16280-9. HAHN, T. (ed.), 1992, International Tables for Crystallography, Vol. A (Kluwer Academic Publishers, Dordrecht). KOCH, E. and W. FISCHER,1992, Sphere packings and packings of ellipsoids. In: International Tables for Crystallography, ed. by A. J. C.Wilson (Kluwer Academic Publishers, Dordrecht). LEIGH,G. J. (ed.), 1990, Nomenclature of Inorganic Chemistry. Recommendations 1990 (Blackwell Scientific Publications, Oxford). MASSALSKI, T. B., 1990, Binary Alloy Phase Diagrams, Vols. 1-3, (ASM International, USA).
46
U! Steurer
Ch. 1, Refs.
PEARSON, W. B., 1972, The Crystal Chemistry and Physics of Metals and Alloys (Wiley-Interscience, NewYork). SCHULTE,O., and W.B. HOLZAPFEL,1993, Phys.Rev. JM8,14009-12. SKRIVER, H. L., 1985, Phys.Rev. B31, 1909-23. B. K., V. M. FRIDKINand V. L. INDENLIOM, 1982, Modem Crystallography IP Structure of VAINSHTEIN, Crystals (Springer-Verlag, Berlin). VILLARS,P. and L. D. CALVERT, 1991, Pearson’s Handbook of Crystallographic Data for Intermetallic Phases (American Society for Metals, USA), Vols. 1 to 4. YomG, D. A., 1991, Phase Diagrams of the Elements (University of California Press, Berkeley).
Further reading BARRETT,C. S., and T. B. MASSALSKI, 1980, Structure of Metals, 3rd edition (F‘ergamon Press, Oxford). BOER,F. R. DE, R. BOOM,W. C. M. MATTENS,A. R. MIEDEMAand A. K. NIESSEN,1988, Cohesion in Metals (North-Holland, Amsterdam). HAFNER,J., F. HULLIGER,W. B. JENSEN,J. A. MAJEWSKI, K. MATHIS,P. VILLARSand P. VOOL,1989, The Structure of Binary Compounds. (North-Holland, Amsterdam). PEmFOR, D. G., 1993, Electron Theory of Crystal Structure, in: Structure of Solids, ed. V. Gerold, Volume 1 of Materials Science and Technology (VCH, Weinheim). VAINSHTEM, B. K., 1994, Modem Crystallography I: Fundamentals of Crystals (Springer-Verlag, Berlin).
CHAPTER 2
ELECTRON THEORY OF METALS D.G. PETTIFOR Department of Materials University of Oxford Ogord, UK
R. U? Cahn and II Haasen?, eds. Physical Metallurgy;fourth, revised and enhanced edition Q Elsevier Science BR 1996
48
D.G.Petrifor
Ch. 2, 0 1
I . Introduction The bulk properties of a metal depend directly on the bonding between the constituent atoms at the microscopic level. Thus, in order to provide a fundamental description of metals and alloys, it is necessary to understand the behaviour of the valence electrons which bind the atoms together. The theory which describes the electrons in metals is couched, however, in a conceptual framework that is very different from our everyday experience, since the microscopic world of electrons is governed by quantum mechanics rather than the more familiar cZussical mechanics of Newton. Rather than solving Newton’s laws of motion the solid state theorist solves the Schrodinger equation, AZ
1
v2-I-v(r)
(-G
+(r)= E#(r),
where V2 = + a2/ay2 + a2/aZ,m is the electronic mass and A is the ubiquitous Planck constant (divided by 277). -(A2/2m) V2 represents the kinetic energy and v(r) the potential felt by the electron which has total energy E. +(r) is the wave function of the electron where [+(r)l2 is the probability density of finding the electron at some point r = (x, y, z). The power of the Schrodinger equation is illustrated by solving eq. (1) for the case of a single hydrogenic atom. It is found that solutions exist only if the wave function is characterized by three distinct quantum numbers n, E and m whose significance has been discussed at the beginning of the preceding chapter. A fourth quantum number, ms, representing the spin of the electron results from a relativistic extension of the Schrodinger equation. Thus, the existence of different orbital shells and hence the chemistry of the Periodic Table follows naturally from quantum mechanics through the Schrodinger equation. WIG= and SEITZ[1933] were the first to apply the Schrodinger equation to the problem of bonding in metals. In their classic paper they studied the formation of the bond in monovalent sodium and obtained the cohesive energy, equilibrium lattice constant, and bulk modulus to within 10% of the experimental values. However, it took nearly another fifty years before the same accuracy was achieved for the polyvalent metals. Whereas WIGNERand SEITZ[1933] could assume that the single valence electron on a sodium atom feels only the potential due to the ion core, in a polyvalent metal a given electron will also feel the strong coulomb repulsion from other valence electrons in its vicinity. Thus the problem becomes much more complex. Firstly, the potential v(r) must be computed self-consistently in that v(r) now depends on the coulomb field of valence electrons whose wave functions and hence average charge distributions themselves depend on v(r) through eq. (1). Secondly, it is necessary in order to obtain bonding to go beyond the average self-consistent field of the Hartree approximation and to include the correlations between the electrons. Pauli’s exclusion principle keeps parallel spin electrons apart, thereby lessening their mutual coulomb repulsion and lowering the energy by an amount called the exchange energy. These statistical correlations are described by the Hartree-Fock approximation. In addition, dynamical correlations also exist between the anti-parallel spin electrons, which lower the energy of the
+
Ch. 2, 0 1
49
Electron theory of metals
system by an amount called the correhtion energy. A major breakthrough in solid-state physics occurred with the realization that these very complicated exchange and correlation effects could be accurately modeled by adding a simple local exchange correlation potential uxc(r)to the usual Hartree coulomb potential
I
0
U S c
I
Ca
Ti
V , M n l C o , Cu Cr
Fe
Ni
Ga
Zn
I
J
/ Sr
b
I
I
1
I
,
Zr
I
I
Tc Mo
, Ru
R 'h
Pd
Cd
Fig. 1. The equilibrium Wiper-Seitz radii, cohesive energies, and bulk moduli of the 3d and 4d transition series. Experimental values are indicated by crosses and the computed LDF values by the connected points. (From MORUZZIet QI. [1978].) References: p. 129.
50
D. G.Pettifor
Ch. 2, $ 2
in eq. (1). The resulting so-called local densityfunctional (LDF) equations (HOHENBERG and KOHN[1964] and KOHNand S w [1965]) have been shown to yield a surprisingly good description of the energetics of atoms, molecules, and solids (GUNNARSSON and LVM>QUIST[1976], HARRIS and JONES [1978], MORUZZIet aZ. [1978], JONES and GUEMARSSON [1989] and Frms [1992]). The success of the LDF scheme is illustrated in fig. 1 by the results of MORUZZIet al. [1978] for the cohesive properties of the elemental metals across the 3d and 4d transition series. We see that for the nonmagnetic 4d series the equilibrium Wigner-Seitz radius (or lattice constant), cohesive energy and bulk modulus are given to better than 10%. The large deviations in lattice constant and bulk modulus observed amongst the 3d series is due to the presence of magnetism and is removed by generalizing the LDF theory to include spin polarization (JANAKand W n ~ u m[1976]). It must be stressed that there are no arbitrary parameters in the theory, the only input being the nuclear charge and crystal structure. This success of the LDF theory in describing the bonding between atoms allows the interpretation of the results within a band framework, since the motion of a given electron is governed by the one-electron Schrodinger equation (1). As is well-known, the energy levels, E, of the free atom broaden out into bands of states as the atoms are brought together to form the solid. In this chapter the nature of these energy bands in simple metals, transition metals and binary alloys is discussed, thereby unraveling the microscopic origin of the attractive and repulsive forces in the metallic bond. In $ 2.1 we begin with a detailed description of the constituent atoms, since we will see that many bulk properties are related to the relative position of the atomic energy levels and to the size of the ionic cores. In 0 2.2 the diatomic molecule is used to illustrate bond formation and in $2.3 the general principle of band formation in solids is outlined. The nature of simple- and transition-metal bands is then discussed in QQ 3 and 4 respectively, the former being treated within the nearly-free-electron approximation, the latter within the tight-binding approximation. In $ 5 the knowledge of the energy band behaviour is used to provide a microscopic picture of metallic bonding which is responsible for the cohesive properties of the elemental metals displayed in fig. 1. In $ 6 structural stability is discussed both in the elemental metals and in binary intermetallic phases. In $ 7 the ideas on metallic bonding are extended to a discussion of the h a t s of fornation, AH, of binary alloys. Finally in $ 8 the band theory of mugnetism is presented which accounts for the antiferromagnetism of Cr and Mn and the ferromagnetism of Fe, Co, and Ni amongst the 3d transition metals.
2. Band formation 2.1. The constituent atoms The hundred basic building blocks of nature, which are enshrined in the Periodic Table, lead to matter having a wide range and variety of physical properties. This diversity reflects the essential uniqueness of each element in the Periodic Table. For example, even though copper, silver and gold lie in the same noble-metal group, nobody except possibly a theoretician would be prepared to regard them as identical. In this
Ch. 2, 92
Electron theory of metals
51
subsection the differences between the elements are quantiJied by discussing the behaviour of the atomic energy levels and the radii throughout the Periodic Table. The structure of the Periodic Table results from the filling-up of different orbital shells with electrons, as outlined in the previous chapter. The chemical behaviour of a given atom is governed by both the number and the angular-momentumcharacter of the electrons in the outer partially filled shells. (We shall refer to these electrons as valence in contrast to the filled shells of core electrons.) The angular-momentum character is determined by the orbital quantum number 1, since the magnitude of the total orbital angular momentum L is given by quantum theory as:
where 1 = 0, 1, 2,... A free-atom electron can, therefore, take only discrete values of angular momentum (i.e. 0, A@, A 6 ,...) unlike a classical particle which would have a continuous spectrum. However, as in the classical case, the angular momentum is conserved because the electron is moving in the central spherically symmetric potential of the free atom. Electrons with I = 0, 1,2 and 3 orbital quantum numbers are referred to as s, p, d and f electrons, respectively (after the old terminology of sharp, principal, diffuse and fine spectroscopic lines). Angular momentum is a vector. Therefore, in addition to the magnitude L of the orbital angular momentum L, the electronic state is also characterized by the components of the angular momentum. Within quantum theory the component in a given direction (say along the z-axis, specified experimentally by the direction of a very weak applied magnetic field) is quantized and given by
L, = mi?,
(3)
where the magnetic quantum number, m, takes the (21+ 1) values 0, f 1,..., f ( I - l), f 1. Because the energy of the electron can not depend on the direction of the angular momentum in a spherically symmetric potential, these (21+ 1) states have the same energy and are said to be degenerate. Allowing for the additional spin quantum number, ms,which can take two values (corresponding to an up, t, or down, &, spin electron), each I-state will be 2(2E+ 1)-fold degenerate. Thus an s-shell can hold 2 electrons, a p-shell 6 electrons, a d-shell 10 electrons and an f-shell 14 electrons as discussed in ch. 2, 0 1. The state of angular momentum of the electron determines the angular dependence of the wave function II/ and hence the angular dependence of the probability-density [+I2. The s-state has zero orbital angular momentum corresponding to a spherically symmetric probability density which is illustrated schematically in fig. 2a. The p-state, corresponding to 1 = 1, m =0, has an angular variation given by cos 8, where is the polar angle. Because the Cartesian coordinates (4y, z) can be related to the spherical polar coordinates (r, 0, +), and in particular z = r cos 8 , it is customary to refer to the E = 1, m = 0 state as the p,: orbital. Its probability-cloud is illustrated by the left-hand diagram in fig. 2b. We see that it has lobes pointing out along the z-axis, in which direction there is a maximum probability of finding the electron (cos 28 = 1 for 8 = 0, T).On the other References:p . 129.
52
Ch. 2, 82
D. G.Petrifor
hand, there is zero probability of finding the electron in the x-y plane (cos *8 = 0 for 8 =.rr/2). Since we often deal with atoms in a cubic environment in which all three Cartesian axes are equivalent (e.g., fcc or bcc crystals), we form the p, and p,, orbitals by taking linear combinations of the two remaining states corresponding to in = + 1. They are illustrated in fig. 2b. The probability clouds of the five d orbitals corresponding to Z=2 are shown in fig. 2c. We might expect from fig. 2 that the nature of the bonding between atoms will be very dependent on the angular momentum character of the atomic valence electrons. This will be discussed in 5 2.2. Historically it was the discrete lines of the atomic spectra and their ordering according to Balmer’s formula that led Bohr to postulate his famous model of the hydrogen atom from which he deduced that the energy levels were given by
E, = -(me4/ 32.rr2t$d2)/ n2,
(4)
S
1.0 P
m=O Z
b
Z
X
C
d
dsz*-rZ
m= 0
dxz
ZY
m =?l Z
Fig. 2. The probability clouds corresponding to s, p and d orbitals are shown in
(a), (b) and (c),
respectively.
Ch. 2, $ 2
53
Electron theory qf metals
where e is the magnitude of the electronic charge, E,, is the permittivity of free space, and n is a positive integer. The corresponding radii of the so-called stationary orbits were given by a,, = (4m,-,~~/ m 2 ) n 2 .
(5)
Substituting into eqs. (4) and (5) the SI values m=9.1096 x lod3'kg, e = 1.6022 x C, 4m0c2=lo7,c=2.9979 x 10' m / s and R = 1.0546 x lo-%Js, we have:
E,, = 2.1799 x 10-"/n2
J
(6)
and a,, = n2 ail.
(7)
The ground stute of the hydrogen atom, which corresponds to n= 1, has an energy, therefore, of 2.18 x lo-" J and an orbital Bohr radius of 0.529 x lo-'' m or 0.529 A. Because of the small magnitude of the energy in SI units, it is customary for solid-state physicists to work in atomic units, where the unit of energy is the Rydberg (Ry) and the unit of length is the atomic unit (au). The former is the ground-state energy of the hydrogen atom, the latter is the first Bohr radius. Thus, in atomic units we have
E,, = -n-2 Ry
(8)
a,, = n2 au.
(9)
and
It follows from eqs. (4),(5), (8) and (9) that A2/2m = 1 in atomic units. Another frequently used unit is the electron-Volt, where 1 Ry = 13.6 eV. In this chapter electronic energy levels, E, will be given in either eV or Ry, whereas total energies will be given in either eV/atom or Ry/aQm. Conversion to other units may be achieved by using 1 mR~/atom=0.314 kcallmole= 1.32 kJ/mole. Length scales will be given either in au or in A, where 1 au=0.529 A. Solution of the Schrodinger equation (1) for the hydrogen atom leads directly to Bohr's expression (4) for the energy levels, E, where n is identified as the principal quantum number. For the particular case of the hydrogen atom where the potential v(r) varies inversely with distance r from the nucleus, the energy levels do not depend on the angular-momentum quantum numbers I and m. Figure 3 shows the energy levels of atomic hydrogen given by eq. (S), where use has been made of the quantum-theory result that for a given n the orbital quantum number I must be such that 0 I I I(n - 1). The total degeneracy of each orbital including spin, namely 2(2l+ l), is given at the bottom of the figure and accounts for the structure of the Periodic Table, discussed in the previous chapter. In practice, the energy-level diagram of elements other than hydrogen is different from fig. 3, because the presence of more than one electron outside the nucleus leads to the potential v(r)no longer showing a simple inverse distance behaviour, so that states with the same principal quantum number n but different orbital quantum References:p . 129.
54
Ch. 2, 52
A G. Pettqor
-k -3s
-4P
-3P
-4d
-4f--k2
3d
~
1
- 1.2
-2P
1-2s
.-%
-> -5(Y I
GLY
W
W z
-10
-
-1s -15
14 6 2(21+1l '0 DEGENERACY OF ORBITAL INCLUDING SPIN
Fig. 3. The energy levels of atomic hydrogen.
numbers I have their degeneracy lifted. This is illustrated in fig. 4, where it is clear, for example, that the 2s level of the second-row elements B to Ne lies well below that of the corresponding 2p level. These atomic energy levels were taken from the tables compiled by HERMAN and SKILLMAN [1963] who solved the Schriidinger equation (1) selfconsistently for all the elements in the Periodic Table. Figure 4 illustrates several important features to which we will be returning throughout this chapter. Firstly, the valence energy levels vary linearly across a given period. As . .
0
-> a
-I
.
-10-
2 J
-
>u
.
[r W
z W
u
. -20 -
E
.
0
.
+ a
Li
ES
\''..4s
-30-
Be-
B
C
Rb Sr--'-Cd
in
Sn Sb
N
0
F Ne
Te
I
Xe
__
Fig. 4. The valence s and p energy levels (after HERMAN ANDSKILLMAN [1963]).
I
Ch. 2, 52
55
Electron theory of metals
the nuclear charge Ze increases, the electrons are bound more tightly to the nucleus. However, rather than varying as Z2, which would be the result for the energy levels of a hydrogenic ion of charge Ze, the presence of the other valence electrons induces the linear behaviour observed. Secondly, the valence s and p energy levels become less strongly bound as one moves down a given group, which is to be expected from the hydrogenic energy levels displayed in fig. 3. But there is an exception to this rule: the 4s level has come down and crosses below the 3s level to the left of group VB. This is a direct consequence of the presence of the occupied 3d shell (cf. table 2, ch. 2) whose electrons do not completely screen the core from the valence 4s electrons, which therefore feel a more attractive potential than their 3s counterparts in the preceding row. We will see in $6.2 that this reversal in the expected ordering of the valence s energy levels is reflected in the structural properties of binary AB compounds containing group IIIB elements. Thirdly, it is clear from fig. 4 that the energy diflerence Ep- E, decreases as one goes from the rare gases to the alkali metals, from right to left across a given period. This will strongly influence the nature of the energy bands and the bonding in the bulk, since if Ihe energy difference is small, s and p electrons will hybridize to form common sp bands. Figure 5 shows the valence s and d energy levels across the 3d and 4d transition metal series, after HERMAN and SKILLMAN [1963]. The energy levels correspond to the atomic configuration where N is the total number of valence electrons, because this is the configuration closest to that of the bulk metal. Again there are several important features. Firstly, we see that the energy variation is linear across the transition metal series as the d shell is progressively filled with electrons. However, once the noble metal group IB is reached the d shell contains its full complement of ten electrons, so that any further increase in atomic number Z adds the additional valence electrons to the
e%,
‘4 d
I6 IIB
IIIAE7AYAEIAZIA Cr Mn Fe Co Ni
Sc
Ti
V
Y
Zr
Nb Mo Tc
Cu Zn
Ru Rh Pd Ag Cd
Fig. 5. The valence s and d energy levels across the 3d and 4d transition series (after HERMAN AND SKILLMAN [19631).
References: p. 129.
56
D.G.Pettifor
Ch. 2, $ 2
sp outer shell and pulls the d energy rapidly down as is evidenced by the change of slope in fig. 5. Secondly, whereas the valence s energy level becomes slightly less strongly bound as one moves down a given group, the valence 4d energy level becomes more strongly bound than the valence 3d away from the beginning of the transition-metal series. This behaviour appears to be related to the mutual coulomb repulsion between the negatively charged valence electrons. The 3d orbitals are much more compact than the 4d orbitals, so that the putting of electrons into the 3d shell leads to a more rapid increase in repulsive energy than in the 4d shell. The Sd and 6s energy levels have not been plotted in fig. 5 because relativistic effects, which are not included in the Schrodinger equation (I), become important for heavy atoms in the Periodic Table. Relativistic corrections are discussed in ch. 2 of HERMAN and SKILLMAN [1963]. Thirdly, since E, - Ed is about 3 eV in copper but 6 eV in silver, it is not surprising that the noble metals display different physical characteristics. A concept that is often used in physical metallurgy to discuss and order properties is that of atomic size. The microscopic description of the atom, which is provided by quantum mechanics, should be able to give some measure of this quantity. We have seen that quantum mechanics replaces the stationary Bohr orbits of radius a, by orbitals which are not located with a fixed radius but are smeared out in probability-cloudsdescribed by ]+I2. The angular dependence of these probability-clouds has been displayed in fig. 2. We now discuss their radial dependence. The solution of the Schrodinger equation for a central spherically symmetric potential can be written in separable form, namely:
where r, 8 and 4 are spherical polar coordinates. As expected, the anguEar distribution depends only on the angular-momentum quantum numbers E and m, the functions Y,"(6 4 ) being the so-called spherical harmonics (see, e.g., SCHIFF[1968]). Y," is a constant and Yp is proportional to cos6 as we have already mentioned. The radiaE function &(r) depends on the principal and orbital quantum numbers, n and 1 respectively, and therefore changes with energy level E& For the hydrogen atom the first few radial functions are (in atomic units)
R,, ( r )= 2e-',
(11)
1 &(r) = -(I
11z - 3r)e-'",
A conceptually useful quantity is the probability of finding the electron at some distance r from the nucleus (in any direction), which is determined by the radialprobability density, Pn,(r)= r2Rzl(r). Figure 6 shows the radial function R,,, and the probability density, P,,, as a function
Ch. 2, 8 2
Electrun theory of metals
57
of r for the Is, 2s and 2p states of hydrogen. We see that there is maximum probability of locating the electron at the first Bohr radius a, for the Is state and at the second Bohr radius a2 for the 2p state. The average or expectation value of the radial distance r is given by:
r;, = n Z [ l + f ( l - l ( ~ + l ) / n 2 ) ] ,
(14)
so that fls=1.5a,, FZs=1.5% and F2p= 1 . 2 5 ~Therefore, ~ the 2s orbital is more extended than the corresponding 2p orbital, as is evident from fig. 6. This is due to the fact that all solutions of the Schrodinger equation must be orthogonal to one another, Le., if &,,, and $,,,,m. are any two solutions and $* is the complex conjugate of @, then
j
$L+nlr’m*
dr = 0.
(15)
If the states have dzperent angular-momentum character then the angular integration over the spherical harmonics [cf. (eq. lo)] guarantees orthogonality. But if the states have the same angular-momentum character then the orthogonality constraint implies that: Bohr radius a,
Fig. 6. The radial function R,,, (dashed lines) and the probability density, P,,, (solid lines) as a function of r for the Is, 2s and 2p states of hydrogen.
References: p . 129.
58
D. G. Penifor
:j
Z&
Ch. 2, $ 2
(I-) Rn,, (I-), dr '= 0.
(16)
For the orbitals drawn in fig. 6, therefore, we must have
:j
R,, ( r )R,, (r)r' dr = 0,
(17)
which can be verified by substituting eqs. (11) and (12) into this equation. This is the origin of the node at r = 2 au in R&), where the radial function changes sign. The 3s radial function must be orthogonal to the 2s and, therefore, has two nodes, the 4s has three nodes, etc. Just as the energetically lowest I s state has no nodes, so the 2p, 3d and 4f states are nodeless since they correspond to the states of lowest energy for a given l (see fig. 3). The position of the outer node of the valence electron's radial function may be used as a measure of an I-dependent core size, since we have seen that the node arises from the constraint that the valence state be orthogonal to the more tightly bound core states. This relationship between node and core size has been demonstrated quantitatively for the [19771 and 4 4.3) and has been case of the sp core of the 4d transition metals (PETTIFOR [1981]. A not unrelated discussed for other elements by BL~CHand SCHATTEMAN measure of size has been adopted by ZUNGER[1980] who defined I-dependent radii R, by the condition (cf. ST. JOHN and BLOCH [1974]) that
v;"(R,)= 0,
(18)
where v:"(r) is some effective angular-momentum dependent atomic potential (which is given by a first-principles screened pseudopotential, cf. 53.3). Figure 7 shows the resultant values of -RS-' and -R i l for the sp bonded elements. We see a linear variation across a given period and a close similarity with the valence energy level behaviour
Fig. 7. The negative of the inverse s and p pseudopotential radii (after ZUNGER[1980]).
Ch. 2, 92
Electron theory of metals
59
illustrated in fig. 4. As expected, the s and p radii contract across a period as the nuclear charge Ze increases, and they expand down a column as additional full orbital shells are pulled into the core region. Figure 7 clearly demonstrates that the sizes of the second-row elements B, C, N and 0 are a lot smaller than those of the other elements in their respective groups, a fact which manifests itself in their different alloying behaviour (cf. fig. 38, below). 2.2.
Bond formation
In this subsection we consider what happens to the atomic energy levels and wave functions as two atoms A and B are brought together from infinity to form the AB diatomic molecule. Suppose the A and B valence electrons are characterized by the free atomic energy levels EA and E, and wave functions @, and @ ,, respectively. Let us assume, following the experience of theoretical quantum chemists, that the rnoZecuZur wave function $M can be written as a linear combination of the atomic orbitals, +AB
= cA+. + cBk9
(19)
where c, and c, are constant coefficients. Then it follows from the Schrodinger eq. (1) that
( A - EXCAICA + c,+B> = 0
(20)
where fi is the Hamiltonian operator for the AB dimer, namely fi= - V2+ V,, where we have used the fact that A2/2m= 1 in atomic units. Multiplying by $, (or +,) and integrating over all space we find the well-known secular equation (taking +* = q!J as q!J is real)
HAB- ESAB
HAA- E HBA- ES,,
HBB- E
where the Hamiltonian and overlap matrix elements are given by
and ~4=
j
+a
IC~dr.
(23)
The Hamiltonian matrix elements can be simplified by assuming that the molecular potential VABis given by the sum of the free atom potentials V, and V,. The diagonal elements HAAand HBBthen take the free atom values EA and EB respectively, provided the energy shift due to the neighbouring potential fields can be neglected. The off-diagonal element Hm can be written
HAB= J.t,9,vt,bB dr
+ ES
(24) References: p . 129.
60
Ch. 2, $ 2
D. G. Peltgor
where E = & ( E A + E B ) , the secular equation
V=*(VA+VB),
and S=S,.
Substituting in equation (21) we obtain
- AE - (E - E ) h - (E - E)s
+AE- (E - E )
=O
(25)
where AE=(E, - EA) is the atomic energy level mismatch and h=h,hAV$Bdris the hopping or bond integral between atoms A and B. For s orbitals h is negative since the average potential V is attractive. Equation (25) may be solved for the eigenvalues and eigenvectors. To first order in the overlap integral S
E:, = E - h S f ( 1 + S 2 ) ' h
c,i = k
(26)
' [1T (6 + S)/(l+ 62 ) x ]x
7T
with 8 =AE/2JhJ. Therefore, as shown in Fig. 8 s valent diatomic molecules are characterized by bonding and anti-bonding states which are separated in energy by the amount w, such that wiB = 4hZ+ (AE)*.
(30)
The formation of the bond is accompanied by a redistribution of the electronic charge. It follows from equation (27) that the electronic density which corresponds to occupying the bonding state with two valence electrons of opposite spin, namely PAB= 2 ( $ i B ) 2 may be written in the form PA,
(1'
= (1 + ai)P,(1 ' + (1 - ai)PB(r)+ a c ~ b n ('d)
(31)
where
(1'1
PA(B)(')
= [@A(B)
Pbond(')
= 2$A(rl$B(')
2
(3 la)
and
- S[PA(')
+ PB(')]'
aiand a, are determined by the normalised energy level mismatch 6 through
(32)
Ch. 2, $ 2
cyi
Electron theory of metals
= S/(1+
61
S2)K
(33)
q.
(34)
and
g = 1/(1+
For the case of homonuclear diatomic molecules S =0, so that the change in the electronic charge distribution on forming the molecule is given solely by the bond charge in equation (31). This is illustrated in fig. 9 for the case of the contribution pbond hydrogen molecule where we see that,as expected, the electronic charge has moved from the outer regions of the molecule into the bond region between the atoms. We should note from equations (32) and (23) that the total charge associated with phndover all space is identically zero. Equation (32) shows explicitly that the formation of the bond is a quantum interference effect, the charge piling up in the bond region because of the interference contribution +A +B. In practice, in order to satisfy the virial theorem, the formation of the bond is accompanied by some modification of the free-atom orbitals $A,B. which has been discussed by RUEDENBERG [1962] and SLATER [1963]. This leads to the energy levels EABnot being directly identifiable as thefree-atom energy levels, a point which will be discussed further in Q 5.2 on transition-metal bonding. For the case of a heteronuclear diatomic molecule 6 # 0, so that the electronic charge distribution in equation (31) contains the ionic contributions aipAand -qpB in addition to
(a)
Fig. 8. The bonding (lower lines) and antibonding (upper lines) states for (a) the homonuclear and (b) the heternnuclear diatomic molecule. References: p. 129.
62
D.G. Penifor
Ch. 2, $ 2
the covalent bond charge contribution aCpbond. cyi and a, are said to measure the degree etal. [1962], PHILIPS of ionicity and covalency of the bond (see, for example, COULSON [1970] and HARRISON [1980]). Note that a ~ + c y ~ = l . The term covalency will be used in this chapter to describe the bonding which arises from the quantum mixing of valence states on neighbouring sites into the final state wave function. It is not necessarily associated with pairs of electrons of opposite spin, as the lone electron in the hydrogen molecular ion Hi,for example, shows all the characteristics of the covalent homonuclear bond discussed above. A diatomic molecule has cylindrical symmetry about the internuclear axis, so that angular momentum is conserved in this direction. Quantum-mechanically this implies that the state of the molecule is characterized by the quantum number m, where mi? gives the component of the angular momentum along the molecular axis. However, unlike the free atom where the (21+ 1) different m values are degenerate, the degeneracy is lifted in the molecule. By analogy with the s, p, d, ... states of a free atom representing the orbital quantum numbers 1=0, 1, 2, ..., it is customary to refer to v, T , 6, ... states of a molecule as those corresponding to m=O, f l , 3 2 , ... respectively. Figure 10 illustrates the different characteristics of the v,T and 6 bonds. We have seen from our previous discussion on the homonuclear molecule that a given atomic energy level will split into bonding and antibonding states separated by 2 Ih I, where h is the matrix element that couples states +A and t,hB together through the atomic potential
INTERNUCLEAR AXIS
Fig. 9. The electron density of the homonuclear molecule (upper panel) can be regarded as the s u m of the noninteracting freeatom electron densities (lower panel) and the quantum-mchunicully induced bond density (middle panel). The dashed curve represents the first-order result, eq. (32), for the bond density.
Ch. 2, $ 2
63
Elecmn theory of metals PP
b
n n
Fig. 10. The formation of u,?r and S bonds from s, p and d orbitals, see text.
are spherically symmetric s orbitals, then a sw bond is formed as shown schematically in fig. loa. If (cIkB are p orbitals whose probability clouds are drawn in fig. 2, then the threefold degenerate free atom level (excluding spin degeneracy) splits into the singly degenerate ppa molecular state (m=O) and the doubly degenerate p p r molecular state (m=+l) shown in fig. lob. If +A,B are d orbitals, whose probability clouds are sketched in fig. 2, then the fivefold degenerate free atom level splits into the singly degenerate dda molecular state (m = 0) and the two doubly degenerate molecular states ddw (m=kl) and dd8 (m=k2) as shown in fig. 1Oc. For the case of a heteronuclear molecule such as NbC where the carbon p orbitals overlap the niobium d orbitals, a pd bond will be formed from the pdu and pdm- states illustrated in fig. 1Od. It is clear from fig. 10 that the u bond is relatively strong since the angular lobes point along the molecular axis and can give rise fo a large overlap in the bonding region. On the other hand, the ppw and dds bonds will be relatively much weaker since their angular lobes extend in the plane perpendicular to the molecular axis. The importance of u,T and 6 bonding in determining the behaviour of the bulk band structure will be demonstrated in 0 4.1. The term covalency will be used in this chapter to describe the bonding which arises from the quantum mixing of valence states on neighbouring sites into the final-state wave function. It is not necessarily associated with pairs of electrons of opposite spin, as the lone electron in the hydrogen molecular ion H i , for example, shows all the characteristics of the covalent homonuclear bond discussed above.
v. If
2.3. Band formation Figure 11 illustrates how the &-atom energy levels E, and Epbroaden into bands as the atoms are brought together from infinity to form the bulk. Just as the single atomic energy level splits into two energy levels on bringing two atoms together (cf. fig. Sa), so the single level on a free atom splits into N levels on bringing N atoms together, thereby conserving the total number of electronic states. These levels lie between the bottom of the band, which: represents the most bonding state, and the top of the band, which represents the most antibonding state. Since N = 10’’ for 1 cm3 of bulk material, these N References:p . 129.
64
Ch. 2, 6 3
D. G. Penifor
I
VOLUME
Fig. 11. Energy band formation.
levels form a quasi-continuous band of states and it is customary to work with the density of states, n(E). where m=n(E) dE gives the number of states in the energy range from E to E + dE.The conservation of states requires that: ca
J- n , ( E ) d E =
I: {: 6 for a = p,
(35)
where n,(E) is the density of states per atom associated with a given atomic s, p or d level. In metals at their equilibrium volume, the bands corresponding to different valence energy levels overlap and mix as shown on the left-hand side of fig. 11. The mixing or hybridization in simple metals is such as to produce nearly-free-electron-like behaviour of the energy bands and density of states, which is discussed in the following section. On the other hand, the density of states in transition metals is dominated by a well defined d band, which is accurately described within the tight-binding approximation by a linear combination of atomic d orbitals and is discussed in 5 4.
3. Simple-metal bands 3.1. The free-electron approximation It had been realized before the advent of quantum mechanics that some metallic properties such as electrical or thermal conductivity could be well understood by regarding the valence electrons as a non-interacting gas of particles which were free to travel throughout the metal without being affected by the parent ions. However, it remained for quantum mechanics to remove a striking failure of the classical model, namely its inability to explain the linear temperature dependence of the electronic heat capacity, since according to classical statistical mechanics a free particle has a constant heat capacity of 5, where kB is the Boltzmann constant. The SchriSdinger equation for a free-electron gas may be written in atomic units as
Ch. 2, $ 3
Electron theory of metuls
-(-$ + -$+ $)$(r)
65
= E$(r)
If the electrons are contained within a box of side L then a normalized solution of eq. (36) is the pZane wave:
which can be seen by writing k w as k s + k,y + k j and substituting eq. (37) into eq. (36). This solution corresponds to an electron with kinetic energy E given by:
E = k,’
+ ky” + k,‘
= kZ.
(38)
Since the kinetic energy equals p2/2m where p is the electronic momentum, it follows from eq. (38) that
p z = 2mE = 2mk2 = A2k2,
(39)
using h2/2m= 1. Thus, we have recovered the de Broglie relation
p = Ak = h/A,
(40)
because k=2rr/A where A is the wavelength of the plane wave. The wavelength, A, of the plane wave is constrained by boundary conditions at the surface of the box. For the case of the Bohr orbits in the hydrogen atom, de Broglie had argued that A must be such that integer multiples of the wavelength fit around the circumference of the orbit. Similarly, imposing periodic boundary conditions on the box, which in one dimension corresponds to joining both ends in a closed ring, we have that
n,A, = nyAy = n,A, = L,
(41)
where nx,ny, n, are integers. Therefore,
so that the allowed values of the wave vector k are discrete and fall on a fine mesh as illustrated in fig. 12. By Pauli’s exclusion principle each state corresponding to a given k can contain IWO electrons of opposite spin. Therefore, at absolute zero all the states k will be occupied within a sphere of radius kF,the so-called Fermi sphere, because these correspond to the states of lowest energy (cf. fig. 13a). The Fermi wave vector kF may be related to the total number of valence electrons,N, by
4 & 2 ~ / ( 2 ~ =) ~N ,
(43)
where V = L3, since it follows from eq. (42) that unit volume of k-space contains V / ( ~ T ) ~ states capable of holding two electrons each. Thus, k, = (37~’N/V)1‘~
Fig. 12. The fine mesh of allowed k values. At absolute zero only the states k within the Fermi sphere are occupied.
and the corresponding Fermi energy, EF is given by
EF = (37r*N/v)2/3.
(45)
The electron concentration, N N , for sodium, magnesium and aluminium at their equilibrium atomic volumes is such that the Fermi energy EFequals 3.2, 7.1 and 11.6 eV respectively. The free-electron densify ofstares n(E) may be obtained from eq. (43) by writing it in the form
N ( E ) = (V/k2)E”2,
(46)
where N(E)is the total number of states of both spins available with energies less than E. Differentiating eq. (46) with respect to the energy gives the density of states:
n(E) = (V/2v2) E’’2,
(47)
which is illustrated in fig. 13b. We can now see why the experimental electronic heat capacity did not obey the classical result of #kB. By Pauli’s exclusion principle the electrons can be excited only into the unoccupied states above the Fermi energy EF.
E,
E
Fig. 13. The free-electron energy dispersion E(k) (a) and density of states n(E) (b).
Ch.2, $3
Electron theory of metals
67
Therefore, only those electrons within about kBTof EF will have enough thermal energy to be excited acmss EF.Since kBT= 0.03 eV at room temperature, these electrons will comprise a very small fraction, f = kBT/EF,of the total number of electrons N. The classical heat capacity is accordingly reduced by this factorJ as is observed experimentally. Using the correct Fermi-Dirac statistics to describe the occupation of the electron states, we find (see,e.g., KITTEL [1971]): v' 2
C, = -k,(k,T/E,)
(48)
in agreement with the previous qualitative argument.
3.2.
Nearly-free-electron approximation
The electrons in a real metal are affected by the crystalline lattice, since the potential which they feel is not uniform but varies periodically as
v(r + R) = u(r)
(49)
where R is any lattice vector. (For simplicity we will be considering only those crystaI structures, such as fcc or bcc, in which there is only one atom per primitive lattice site, in contrast to hcp or the diamond structure, for example, which have a basis of two atoms, cf. KITTEL[1971].) Consider first an infinite one-dimensional periodic lattice of atoms with repeat distance a such that
v(x
+ nu) = 4.).
(50)
Because all the atoms are equivalent, the probability of locating the electron about a site must be the same for all sites, so that:
For n= 1 this implies that *(x
+ a) = eik.+(x),
where k is a number (in units of V u ) which specifies the phasefactor eikalinking the wave functions on neighbouring sites. Repeating eq. (52) n times gives:
+k(x +.a)= eih"+k(x),
(53)
which is the usual statement of Bloch 's theorem in one dimension. Thus the translational symmetry of the lattice leads to the eigenfunctions being characterized by the Bloch vector, k. However, k is only defined modulo ( 2 ~ / a )since , k+m(2?r/a) results in the same phase factor in eq. (53) as k alone. It is, therefore, customary to label the wave function a,bk by restricting k to lie within the first Brillouin zone, defined by - r / a 5 k I+ r i a .
(54)
References: p. 129.
68
D.G. Penifor
Ch.2 , 9 3
We note that in one dimension IM is a direct lattice vector, whereas m(21r/a) is a reciprocaE lattice vector. Their product is an integer multiple of 21r. Extending these ideas to three dimensions, Bloch’s theorem, eq. (53) may be written as:
qk(r+ R ) = eikaRqk(r),
(55)
where R is any direct lattice vector which may be expressed in terms of the fundamental translation vectors a,, a,, a3 as:
R = ?a2 + n2a,+ $a3,
(56)
where n,,n,, nj are integers. The corresponding reciprocaE lattice vectors are defined by:
G = m,b, + m2b2+ %b3,
(57)
where m,, in2, m3 are integers and the fundamental basis vectors are:* b, = (27i-/T)U, X U ,
1
b2 = (21r/T)U3X U I , b3 = (27i-/T)U,X U ,
(58)
with T = lalo(a,x a,) 1 being the volume of the primitive unit cell defined by a,, a, and u3.It is apparent from their definition (58) that
ai e bj = 2d,,
(59)
where 6,= 1 for i =j but zero otherwise. The phase factor in eq. (55) only defines the Bloch vector within a reciprocal lattice vector G since it follows from eqs. (56)-(59) that G.R is an integer multiple of 21r. Just as in the one-dimensional case, it is customary to label the wave function by restricting k to lie within the Jirst Brillouin zone which is the closed volume about the origin in reciprocal space formed by bisecting near-neighbour reciprocal lattice vectors. For example, consider the simple cubic lattice with basis vectors a,, a,, a3along the Cartesian axes x, y, z respectively. Because a,=a,=a,=a it follows from eq. (58) that the reciprocal space basis vectors b,, b,, b, also lie along x, y and z respectively, but with magnitude (2m/a). Thus, the reciprocal lattice is also simple cubic and it is shown in fig. 14 in the x-y plane. It is clear that the bisectors of the first nearest-neighbour (100) reciprocal lattice vectors form a closed volume about the origin which is not cut by the second or any further nearest-neighbourbisectors. Hence, the Brillouin zone is a cube of ~ . eq. (42) it contains as many allowed k points as there are volume ( 2 ~ / a )From primitive unit cells in the crystal. Figure 15 illustrates the corresponding Brillouin zones for the body-centred cubic and face-centred cubic lattices (see, e.g., Krrm. [1971]). The solutions Ek of the Schrodinger equation for k lying within the Brillouin zone
* Note the additional factor of 29r compared to the definitionof reciprocal lattice vectors in the appendix of ch. 11.
Ch. 2, $ 3
69
Electron theory of metals
Zndzone
3rd zone
4th zone
Fig. 14. The first four zones of the simple cubic lattice corresponding to k, = 0. The dotted circle represents the cross-section of a spherical Fermi surface.
determine the band sfrucfure.Figure 16 shows the band structure of aluminium in the IlOO) and 1111) directions, after MORUZZIet al. [1978]. It is very similar to the freeelectron band structure
Ek =@+Cy
(60)
which results from folding the free-electron eigenvalues shown in fig. 13a into the first Brillouin zone. This “folding-in” is illustrated in fig. 14 for the case of the simple cubic lattice. For this two-dimensional cross-section we see that the four contributions to the second zone 2 may be translated through (100) reciprocal lattice vectors into the four
fcc
bcc
r
Fig. 15. The fcc and bcc Brillouin zones. labels the centre of the zone. The intersections of the 1100) and 1111) directions with the Brillouin-zone boundary are labelled X and L in the fcc case and H and P in the bcc case. References: p . 129.
70
D.G.Pettifor
Ch. 2, $ 3
zones 2: which together completely fill the reduced Brillouin zone in the x-y plane. Similarly, the third and fourth zones shown in fig. 14 may each be translated through reciprocal lattice vectors to fill the first Brillouin zone. For the fcc lattice the two lowest eigenvalues given by eq. (60) in the I l00} direction are:
E!) = k 2 ,
E?) = (k+g)2,
(61)
where k = (k, 0,O) and g= (2?r/a)(Z, 0,O).These two eigenvalues are degenerate at the zone boundary X,where k=(2?r/a)(l, 0, 0) because from eq. (61) they both take the value 4$/a2. For aluminium a=7.60 au and 4 d / a 2 = 9 . 3 eV, so that the two freeelectron eigenvalues given by eq. (61) reflect the broad behaviour of the band structure shown along rX in fig. 16. However, in order to recover the energy gap at the zone boundary X, it is necessary to lift the free-electron degeneracy by perturbing the free-electron gas with the periodic potential of the crystalline lattice. Within the nearly-free-electron (NE) approximation this is achieved by writing the wave function Jlk as a linear combination of the plane-wave eigenfunctions corresponding to the two free-electron eigenvalues given by eq. (61); that is:
& = C , g p + e’@,
(62)
where from eq. (37):
+f’ = v-“~ exp(ik r),
(63)
+f’ = v-”~ exp[i(k + g ) r]. Substituting eq. (62) into the Schrtidinger equation (l), pre-multiplying by +)*: or$:)* and integrating over the volume of the crystal, V, yields the NFE secular equation:
k’ - E 4200) u(200) (k + g)’ - E
)[::) =
4200) is the ( 2 ~ / a ) ( 20,, 0) Fourier component of the crystalline potential, where v(g) =
-IV1 v(r)eig”dr.
(66)
The energy, E in eq. (65) is measured with respect to the average potential ~(000). Non-trivial solutions exist if the secular determinant vanishes, i.e. if
k’ - E ~(200) ~(200) (k + g)2 - E
j=O.
This quadratic equation has solutions
Ek = + [ k 2 + (k + g ) 2 ]f +{[(k + 9)’ - k2]2 + [ 2 v ( 2 0 0 ) ~ ~ .
Ch. 2 , 9 3
71
Electron theory of metals
L
r
X
Fig. 16. The band strufture of fcc aluminium (after MORUZZIetal. [1978]).
Therefore, at the zone boundary X where k2 = (k +g)',
the eigenvalues are given by
E, = 47.r2/u24 ~(200)
(69)
and the eigenfunctions are given from eqs. (62) and (65) by:
h =(
2 W
{
cos ( 2 4 4 sin(2m/u) *
Thus the presence of the periodic potential has opened up a gap in the free electron band structure with energy separation
Because the energy gap at X in aluminium is about 1 eV (cf. fig. 16), the magnitude of the Fourier component of the potential within this simple IWE treatment is only 0.5 eV. This is small compared to the free-electron Fermi energy of more than 10 eV in aluminium and, therefore, the band structure Ekand the density of states n(E) are nearlyfree-electron-like to a very good approximation. The NFE behaviour has been observed experimentallyin studies of the Fermi surface, the surface of constant energy Ep in k-space, which separates filled states from empty states at T=O.For a free-electron gas the Fermi surface is spherical as illustrated in fig. 12. However, in simple metals we have seen that the he-electron band structure is perturbed by the periodic lattice potential, and energy gaps open up across zone boundaries. As illustrated in fig. 14 for the simple cubic lattice, a spherical free-electron Fermi surface (whose cross-section is represented by the circle of solid dots) will be folded into the first Brillouin zone by the relevant reciprocal lattice vectors. The states in the second zone 2, for example, are folded back into 2' in the reduced zone, thereby giving rise to the shaded occupied regions of k-space and the corresponding Fermi surface indicated in the lower panel of fig. 14. Similarly, the occupied states in the third and fourth zones are folded back into the reduced Brillouin zone as shown. Therefore, even though the crystalline potential may be very weak, it is sufficient to destroy the spherical free-
References: p . 129.
72
D. G. Pemyor
151
ZONE-FULL
3rd ZWIE-REGIONS OF EL'NS
Ch. 2, $ 3
2nd ZONE-POCKET OF HOLES
4fi ZWsE-POCKETS OF EL'NS
Fig. 17. The free-electron Fermi surface of aluminium (after HARRISON [19591).
electron Fermi surface and to create a new Fermi surface topology, as is illustrated in fig. 14 by the appearance of the electron pockets in the third and fourth zones. A very simple procedure for constructing the Fermi surfaces of free-electron-like materials has been suggested by HARRISON [1959, 19601 and fig. 17 shows the resulting Fermi surface of fcc aluminium. A much more detailed treatment of Fermi surfaces may be found in HARRISON [1966], HEINEand WFAIREE19701 and K I ~ [1971], L where the interested reader is also referred for a discussion of transport properties and concepts such as holes and effective mass. 3.3. Volume dependence Although the energy bands of simple metals appear to be describable by the NFE approximation as discussed in the previous subsection, there is a major difficulty. If the (200) Fourier component of the aluminium lattice potential is estimated from Jirst principles using eq. (66),then
But the magnitude of this is ten times larger than the value we obtained byjtting to the first-principles band structure of MORUZZIet al. [1978], namely lv(200)I =0.5 eV. Moreover, by looking at the symmetry of the eigenfunctions at X, we see from fig. 16 that the bottom of the band gap corresponds to X4, or p-like symmetry whereas the top of the band gap corresponds to X, or s-like symmetry (see, e.g., TINKHAM [1964]). It follows from fig. 2 and eq. (70) that the NFE states at the bottom and top of the band gap correspond to sin ( 2 7 4 ~and ) cos (2TX/a),respectively. Therefore, in the state with lower energy the electron is never located in the planes containing the ion cores, which
Ch. 2, $3
Electron theory of metals
73
correspond to x =: na/2 for the fcc lattice, since sin ( 2 w d a ) vanishes. Instead, the electron has maximum probability of being located midway between these atomic planes. This implies that the relevant Fourier component of the atomic potential is repulsive, thereby driving the electrons away from the ion cores, i.e. ~“(200) = + O S eV.
(73)
The origin of the discrepancy between eqs. (72) and (73) is easily found once it is remembered that the NFE bands in aluminium are formed from the valence 3s and 3p electrons. These states must be orthogonal to the s and p core functions as outlined in 0 2.1 and they, therefore, contain nodes in the core region as illustrated for the case of the 2s wavefunction in fig. 6. In order to reproduce these very-short-wavelength oscillations, plane waves of very high momentum must be included in the plane-wave expansion of I)k, so that a linear combination of only the two lowest energy plane waves in eq. (62) is an extremely bad approximation. In 1940, HERRINGcircumvented this problem by starting at the outset with a basis of plane waves that had already been orthogonalized to the core states, the so-called orthogonalized plane-wave (OPW) basis. The OPW method led to a secular determinant for the eigenvalues that was identical to the hiFE determinant, except that in addition to the Fourier component of the crystal potential v(G) there is also a repulsive contribution coming from the core-orthogonality constraint. This tended to cancel the attractive coulomb potential term in the core region, thereby resulting in much weaker net Fourier components and hence nearly-free-electronlike behaviour of the band structure Ekfor the simple metals. This led to the concept of the pseudopotential in which the true potential u(r) in the Schrodinger equation (1) is replaced by a much weaker potential ups@)which is chosen to preserve the original eigenvalues Ekso that
(see, e.g., HARRISON[1966] and HEINEand WEAIRE[1970]). The pseudo-eigenfunctions,
&, however, differ from the true eigenfunctions I)k because in general they do not contain the nodes in the core region as these have been pseudized-away by the inclusion of the repulsive core component in up. A plane-wave expansion of 4ktherefore, leads to rapidly convergent eigenvalues .& in eq. (74). Thus, the NFE approximation will provide a good description of the band structure of simple metals provided the Fourier components of the pseudopotential rather than the true potential are taken in the NFE secular equation (67). Pseudopotentials are not unique, and certain criteria have been given for their choice (see, e.g., BACH~LET et al. [19821 and VANDERFIILT [19901). However, in this chapter we shall describe only the Ashcroft empty-core pseudopotential because of its simplicity. In 1966, ASHCROFT assumed that the cancellation between the repulsive core-orthogonality contribution and the attractive coulomb contribution is exact within some ion core radius R,, so that:
v?(r) =
{
r c R, O for -2Z/r r > R,
(75)
References: p. 129.
74
D. G.Pettifor
Ch. 2, $ 3
Fig. 18. The Ashcroft emptycore pseudopotential.
where the ionic potential falls off coulombically outside the core (cf. eZ=2 in atomic units). The Ashcroft empty-core pseudopotential is shown in fig. 18. The resulting ionic lattice has Fourier components given by eq. (66), namely:
:v (4)= - (87rZ/fkq2)COS qR,, where SZ is the volume per atom. In the absence of the core R,=O and the Fourier components are negative as expected. However, in the presence of the core the Fourier components oscillate in sign and may, therefore, take positive values. For the case of aluminium the Ashcroft empty-core radius is about 1.2 au (cf. table 16-1 of HARRISON [ 19801) and uE(200) will, therefore, be positive. The corresponding Fourier components u&) are obtained from eq. (76) by allowing the free-electron gas to screen the bare ionic lattice. The resulting Fourier components of the aluminium potential are illustrated in fig. 19 for the more sophisticated HEINEand ABARENKOV [1964] pseudopotential. We see that the values of ups(111) and ~ ~ ( 2 0 are 0 ) in good agreement with the values, 0.17 and 0.53 eV respectively, which are obtained from fitting the first-principles band structure within the NFE approximation (cf. fig. 16, eq. (71) and p. 52 of MORUZZIetal. [1978]). Figure 20 shows the densities of states, n(E> of the sp-bonded simple metals, which have been computed from first principles by MORUZZIet al. [1978]. We see that Na, Mg
Fig. 19. The HEINEAND ABARENKOV [I9641 aluminium pseudopotential u,,(q). The two points give the values of u,(lll) and u,(200) deduced from fig. 16 using eq. (71).
Ch. 2, 0 3
75
Electron theory of metals
wl W
t
a
t lA U
0
>
tVI. z
LLI
0
Fig. 20. The density of states, nQ.
of sp-bonded metals (after MORUZZIet al. [1978]).
and A1 across a period and Al, Ga and In down a group are good NJ3 metals, because their densities of states are only very small perturbations of the free-electron density of states shown in fig. 13b. However, we see that Li and Be display very strong deviations from free electron behaviour. This is a direct consequence of these elements having no p core electrons, so that there is no repulsive core-orthogonality component to cancel the attractive coulomb potential which the valence 2p electrons feel. This leads to sizeable Fourier components of the potential and hence very large band gaps. For example, in fcc Be, Eg!p =5.6 e V compared to the gap of only 0.34 eV in Al, where L is the point ( 2 ~ / a ) ( i , i ,in i ) fig. 15. In fact, the band gaps in different directions at the Brillouin zone boundary (cf. fig. 16) are nearly large enough for a gap to open up in the Be density of states, thereby leading to semiconducting behaviour. We note that the effective potential which the valence electrons feel in Li or Be depends on whether they have s- or p-type character, because there are 1s core states but no p core states. Such an 1-dependent potential is said to be non-local (cf. HARRISON[1966] and HEINEand WEAIRE[1970]), whereas the Ashcroft empty-core pseudopotential of fig. IS is local. The heavier alkalis K and Rb and alkaline earths Ca and Sr have their occupied energy levels affected by the presence of the respective 3d or 46 band which lies just above the Fermi energy (cf. the relative positions of the s and d free-atom energy levels in fig. 5). This leads to a more than free-electron admixture of 1 = 2 component into the occupied energy states, which requires the use of non-local pseudopotential theory for accurate agreement with experimental properties (see e.g., TAYLORand MACDONALD [1980] and MOR.IARTY [1982]). It is clear from fig. 20 that Sr is not a simple NFE metal since the perturbation is very strong and the hybridized bottom of the d band has moved References: p . 129.
76
D.G. Pertifor
Ch. 2, $ 3
below the Fermi energy. Just as in Be, a gap has nearly opened up at EF,and theoretically it requires only 0.3 Gpa of pressure to turn Sr into a semiconductor, which is in reasonable agreement with high-pressure resistivity data (JANand SKRIVER [1981]). The group-IIB elements Zn and Cd, on the other hand, have their valence states strongly distorted by the presence of the$lled d band. In fig. 5 we see that the 5s-4d energy separation in Cd is larger than the 4s 3d separation in Zn, which results in the Cd 4d band lying about 1 eV below the bottom of the valence 5sp band (p. 152 of MORUZZI et al. [1978]). Figure 20, therefore, demonstrates that not all simple metals display good NFE behaviour and particular care needs to be taken with Li, Be and the group-II elements on either side of the transition metal series. The presence of the ion core in simple metals determines the volume dependence of the energy bands. Wigner and Seitz had calculated the behaviour of the bottom of the NFE band in sodium in their classic paper of 1933. They argued that since the bottom of the band corresponded to the most bonding state, it satisfied the bonding boundary condition implicit in eq. (27), namely that the gradient of the wave function vanishes across the boundary of the Wgner-Seitz cell. This cell is formed in real space about a given atom by bisecting the near-neighbour position vectors in the same way that the Brillouin zone is formed in reciprocal space. The Wigner-Seitz cell of the bcc lattice is the fcc Brillouin zone and vice versa (cf. KI~TEL[1971]). Since there are 12 nearest neighbours in the fcc lattice and 14 first and second nearest neighbours in the bcc lattice, it is a very good approximation to replace the Wigner-Seitz cell by a Wigner-Seitz sphere of the same volume (cf. fig. 15).Imposing the bonding boundary condition across the Wigner Seitz sphere of radius S, where
n = 471s3,
(77)
the energy of the bottom of the band rl is fixed by
[dR,(r9 E)/drlr=s,E=r, = 0, where Rs(c E) is the 1 =0 solution of the radial Schrodinger equation within the WignerSeitz sphere. The bonding boundary condition is determined by the 1=0 radial function because the bottom of the NF% band at rl is a pure s state (cf. fig. 16). Figure 21 shows the resulting behaviour of the bottom of the band rl,in sodium as a function of S after WIGNERand SEITZ[1933]. We see that as the free atoms are brought together from infinity, the bonding state becomes more and more bonding until about 3 au when r,.turns upwards and rapidly loses its binding energy. This behaviour is well described ut metallic densities by the Frohlich-Bardeen expression,
ry = -(3z/s)[1-
(RJS,’]
(79)
since the single valence electron of sodium is assumed to feel only the potential of the ion at the Wigner-Seitz sphere centre so that over the boundary
v(s) = -2z/s,
(80)
Ch. 2, $ 4
77
Electron theory of metals
2
4
6
0
1
0
S (auI
Fig. 21. The total energy, V,as a function of Wigner-Sei@ radius, S, for sodium (after WIG= and SEITZ [1933]). The bottom of the conduction band, I?, is given by the lower curve, to which is added the average kinetic energy per electron (the shaded region).
where Z = 1 for the monovalent alkali metals (see, e.g., 9 3.2 of CALLAWAY [1964]). R, may be identified as the radius of an Ashcroft empty-core pseudopotential, because the potential energy of one electron distributed uniformly throughout the WignerSeitz sphere with an Ashcroft ionic potential at its centre is given by eq. (79). It follows from eq. (79) that the maximum binding energy of this state rl,occurs for
S, = *RC.
(81)
Since for sodium R, = 1.7 au (ASHCROFT and LANGRETH [1967] and HARRISON [1980]), eq. (81) predicts that rl,has a minimum at about 2.9 au. This is in good agreement with the curve in fig. 21, which was obtained by solving the radial Schrijdinger equation subject to the boundary condition eq. (78). WIGNERancl Smz [1933] assumed that the valence electrons of sodium have freeelectron-like kinetic energy and density of states, which from fig. 20 is clearly a good approximation. It follows from eqs. (45) and (77) that the Fermi energy EF may be written as:
4
=
rF + (9.rr/4)"/s2.
(82)
In 9 5 we follow up our understanding of the behaviour of the energy bands by discussing the total energy of simple metals and the different factors influencing bulk properties such as equilibrium atomic volume and bulk modulus.
4.
Transition-metalbands
4.1.
Tight-bindingapproximation
Transition metals are characterized by a partially filled d band, which is well described within the tight-binding(TB) approximationby a linear combination of atomic d orbitals, We shall illustrate the TB method (see, e.g., CALLAWAY [1964], PETTIFOR E19921 and SUTTON[1993]) by considering first the simpler case of a lattice of atoms with overlapping s-state atomic wave functions #, and corresponding free atomic energy levels E,. Generalizing eq. (19) for the diatomic molecule to a periodic lattice of N References:p. 129.
78
D.G.Pe?t$or
Ch. 2 , # 4
atoms, we can write the crystal wave function t,bk as a linear combination of the atomic orbitals:
where the phase factor automatically guarantees that satisfies Bloch's theorem, eq. (55). Assuming that the crystal potential is the sum of the atomic potentials v(r - R) and following the method and approximationsoutlined through eqs. (19)-(30), the eigenvalue Ekmay be written as: Ek = E,
+ Ce"" [#:(r)~(r)#~(r - R) dr, R#O
where the non-orthogonality and three-centre contributions have been neglected because they do not contribute to first order. Since the atomic s orbitals are spherically symmetric, the SM hopping matrix elements in eq. (84) do not depend on the direction of R but only on the magnitude R (see fig. lo), so that
Ek = E,
+ xeik'RssuR. R#O
The TB band structure Ek for a simple cubic lattice with s orbitals may now be quickly found. Assuming that the hopping matrix elements couple only to the sixJirst nearest-neighbour atoms with position vectors R equal to (h, 0, 0) (0, & a, 0) and (0, 0, +a) eq. (85) gives
Ek = E,
+ ~ S S ~ , ( C Ok.p S + COS kyu + COS kp),
(86)
where k = (k, ,$, k,) Thus the eigenvalues vary sinusoidallyacross the Brillouin zone. The bottom, E and top, E+ of the s band correspond to the Bloch states at the centre of the Brillouin zone (0, 0, 0) and at the zone boundary ( v / u ) ( l , 1, 1) respectively. It follows from eq. (86) that
E* = E, 1 G/ssu,~
(87)
because SM, is negative as can be deduced from fig. 10 and eq. (84). Comparing F with eq. (26) and fig. 8a for the diatomic molecule, we see that the most bonding state in the simple cubic lattice corresponds to maximum bonding with all six nearest neighbours simultaneously, which from fig. 10 is only possible for the spherically symmetric s orbital case. The structure of the TB p band may be obtained by writing I,%k as a linear combination of the three p Bloch sums corresponding to the atomic p,, py, and p, orbitals, where x, y and z may be chosen along the crystal axes for a cubic lattice. That is,
C
#k(r)= N - ~ caxeik*R+a(r - R), a=x,y.z
R
which leads to the 3 x 3 TB secular determinant for the p band, namely
Ch. 2, 54
EZectron theory of metals
79
where
It is clear from fig. 10 that the hopping matrix elements in eq. (90) do depend on the direction of R btecause the px. p,., and p, orbitals are angular dependent. SLATERand KOSTER[1954] showed that they can be written directly in terms of the two fundamental hopping integrals ppo, and p p Rand the direction cosines (1, m,n) of R. For a simple cubic lattice with only first-nearest-neighbour hopping the matrix elements Taa. may be evaluated to give
T, = 2ppv, cos k,a
+ 2 p p ~ , ( c o ks p + cos k p ) ,
with T, and T, obtained from T, by cyclic permutation. The off-diagonal matrix elements vanish for the simple cubic lattice. Therefore, at the centre of the Brillouin zone, r, the eigenvalues are triply degenerate (if spin is neglected) and given from eqs. (89) and (91) by
This degeneracy is partially lifted along the 1100) symmetry direction, because from eq. (91) the band structure consists of the singly degenerate level
E:) = E,
+ 4pp.rr, + 2ppv, cos k,a
(93)
and the doubly degenerate level
E:) = E,
+ 2(ppv, + ppr,) + 2ppn, cos kxa,
(94)
where the former results from the p, orbitals and the latter from the p,. and p, orbitals. The degeneracy is totally lifted along a general k direction as from eqs. (89) and (91) there will be three distinct non-degenerate energy levels. Finally, the structure of the TB d band may be obtained by writing +kk as a linear combination of thejve d Bloch sums corresponding to the five atomic orbitals illustrated in fig. 2. This results in a 5 x 5 TB secular determinant from which the d band structure may be computed (SLATERand KOSTER[1954]). Starting from first-principles band theory, ANDERSEN [19731 has shown that within the atomic sphere approximation (ASA) canonical d bands may be derived which depend neither on the lattice constant nor on the particular transition metal, but only on the crystal structure. This approximation leads to hopping integrals of the form ddu, =-6
(95) dd8, =-1 References: p . 129.
80
Ch. 2, 94
D.G.Penifor
where W is the width of the d band, which is obtained by imposing the bonding and antibonding boundary conditions over the Wigner-Seitz sphere of radius S . It follows from eq. (95) that the hopping integrals scale uniformly with the band width W and do not depend on the lattice constant as it is the ratio S/R that enters. They fall off quickly with distance as the inverse fifth power. Figure 22 shows the resulting d band structure for the fcc and bcc lattices along the 1111) and 1100) directions in the Brillouin zone (ANDERSENE19731). We see that at the centre of the Brillouin zone, r, there are two energy levels, one of which is triply degenerate, the other doubly degenerate. The former comprises the xy, yz and xz, Tzg orbitals which from fig. 2 are equivalent to one another in a cubic environment. The latter comprises the 2 - y', 3 2 - ?Eg orbitals which by pointing along the cubic axes are not equivalent to the TQ,orbitals. The degeneracy is partially lifted along the 1111) and /loo) symmetry directions as indicated in fig. 22, because eigenfunctions which are equivalent at k =0 may become non-equivalent for k # 0 due to the translational phase factor exp (ik0R) (see fig. 8.8 of TMKHAM[1964]). The band structure of NiO (MATTHEIS[1972]) is shown in fig. 23 because it illustrates s, p and d band behaviour. The three bands arise from the oxygen 2s, 2p and the nickel 3d valence levels, respectively, the ordering being determined by the relative positions of their atomic energy levels in figs. 4 and 5. The Brillouin zone is face-centred cubic since the NaCl crystal structure of NiO consists of two interpenetrating fcc lattices, one containing the sodium atoms, the other the chlorine atoms. In the 1100) direction along rX the s and p band structure is not too dissimilar from that given for the simpZe cubic lattice by eqs. (sa), (93) and (94). The d band structure along r X in NiO is also similar to that of the fcc canonical d band in fig. 22, except that one level, which joins the upper state at I' to the bottom of the canonical d band at X, has been pushed up and runs across the top of the d band in NiO. This is the result of mixing or hybridization between the s, p and d blocks in the TB secular determinant (SLATERand K o m [1954]), whose strength is determined for example by the non-vanishing pda and pdm hopping matrix elements shown in fig. 10. This mixing can [1964]). At the zone only occur between Bloch states with the same symmetry (TINKHAM boundary X there is only one d band state which has the same symmetry XIas the s band state. (There are no d band states with the same symmetry as the p band states at X.) The influence of the hybridization on the band structure is enhanced by orthogonality constraints which can add a further repulsive contribution to the d states because they must be orthogonal ta the valence s and p levels lying beneath them in energy. fcc
bcc
Fig. 22. The fcc and bcc d band structure (after ANDERSEN [1973]).
Ch. 2, $4
Elecfron theory of metals
81
d band
p band
s band
Fig. 23. The band structure of NiO (after MAT~~IEISS [1972]).
The bands in fig. 23 illustrate an apparent failure of one-electron theory. NiO is an insulator. However, adding the ten nickel and six oxygen valence electrons to the bands shown results in the d band containing only eight of its possible ten electrons [cf. eq. (35)]. Thus, the band structure presented in fig. 23 predicts that NiO is a metal. The origin of this dramatic failure of band theory was investigated by MOTT [1949], who considered what happens to a lattice of hydrogen atoms as the lattice constant is decreased from some very large value. Initially each atom has a single Is valence electron associated with it as in the free atom state. The system will, therefore, be insulating, because in order for an electron to hop through the lattice it requires an energy given by the difference between the ionization potential of 13.6 eV (corresponding to the atomic 1s level) and the electron affinity of 0.75 eV. This energy difference of about 13 eV is a measure of the coulomb repulsion U between two 1s antiparallel spin electrons sitting on the same atomic site. However, as the lattice constant decreases the atomic 1s level broadens into a band of states of width W so that the insulating gap will decrease like U - W.Therefore, for some sufficiently small lattice spacing W will be large enough for the system to become metallic and the hydrogen lattice undergoes a Mott metal-insulator transition. The very different conducting behaviour of the 3d valence electrons in metallic nickel and insulating nickel oxide can now be qualitatively understood. The width of the d band in NiO is about 2 eV (MATTHEISS[1972]), whereas in pure Ni it is about 5 eV (MORUZZIet al. [1978]) since the Ni-Ni internuclear separation is smaller than in the oxide. Because the value of the screened intra-atomic coulomb integral U in 3d transition metals is about 4 eV, U/ W is greater than unity for NiO but less than unity for Ni. Thus, we expect the former to be insulating and the latter metallic as observed experimentally. The breakdown of conventional band theory at large lattice spacings can best be illustrated by considering the hydrogen molecule (cf. fig. 8a). In the ground state the two valence electrons 1 and 2 occupy the same bonding molecular orbital +iB with opposite spin, so that the total molecular wave function may be written within the one electron approximation as References: p . 129.
82
Ch. 2, 34
D. G.Petrifor
+(VI= + i B (1)+A (2)-
(96)
Substituting from eq. (27), multiplying through and neglecting the normalization factor [2(1 + a]-'we have
+(u)=
(+A
(1)+B (2) + #B (I)+* (2) + +A
U)+A
(2) + +B (I)+, (2)).
(97)
The first two contributions correspond to the two possible neutral atom states with a single electron associated with each atom, whereas the latter correspond to the two ionic states A-B' and A'B- respectively. Since the hydrogen molecule dissociates into two neutral atoms, we see that $(1, 2) gives the wrong behaviour at large separations (see, e.g., SLATER[1963]). In practice, the Mott transition to the insulating phase is accompanied by the appearance of local magnetic moments (BRANDOW [19771) so that the band model must be generalized to allow for antiferromagnetic solutions of the Schrodinger equation ( S u m [1951a]; cf. 0 8). Within local spin density functional (LSDF) theory (cf. Q 1) this leads to a good curve of total energy versus internuclear separation for the hydrogen molecule because the theory now goes over to the neutral free-atom limit (GUNNARSSON and LUNDQUIST[ 19761). However, although the antiferromagnetic state leads to a band gap opening up at the Fermi level in NiO (SLATER[1951a]), a proper understanding of COO and the temperature-dependent properties of these insulators can only be obtained by using a more sophisticated non-local treatment of exchange and correlation (BRANDOW [1977], JONES and GUNNARSSON [1989]). Fortunately, the bulk properties of simple and transition metals considered in this chapter can be well understood within the local approximation, even though non-locality can play a role in the finer details of the band structure (see, e.g., Ni; COOKE et al. [1980]). 4.2.
Hybrid NFETB bands
Transition metals are characterized by a fairly tightly-bound d band that overlaps and hybridizes with a broader nearly-free-electron sp band as illustrated in fig. 24. This difference in behaviour between the valence sp and d electrons arises from the d shell lying inside the outer valence s shell, thereby leading to small overlap between the d orbitals in the bulk. For example, from eq. (14) the average radial distance of the hydrogenic 3d and 4s wave functions are in the ratio 0.44 :l. Thus, we expect the band structure of transition metals to be represented accurately by a hybrid NFE-TB secular equation of the form (HODGESet al. [1966] and MUELLER[1967}):
C-EI
H
/H'
D-Ej-0
where C and D are sp-NFE and d-TB matrices respectively [cf. eqs. (67) and (89)l. H is the hybridization matrix which couples and mixes together the sp and d Bloch states with the same symmetry, and I is the unit matrix. A secular equation of this H-NFE-TB form may be derived (HEINE[1967], HUBBARD
Ch. 2,94
Electron theory of metatals
83
Fig. 24. A schematic representation of transition metal sp (dashed curve) and d (solid curve) densities of states when sp-d hybridization is neglected.
[1967] and JACOBS [1968]) by an exact transformation (PETTIPOR [1972a]) of the first[1954] principle band structure equations of KORRINGA[1946], KOHN and ROSTOKER (KKR). They have solved the Schrtjdinger equation (1) by regarding the lattice as a periodic array of scattering sites which individually scatter the electrons with a change in phase qr.Transition-metal sp valence electrons are found to be scattered very little by the lattice so that they exhibit NFE behaviour with qo and q, close to zero. Transitionmetal d electrons, on the other hand, are strongly scattered, the 1 = 2 phase shift showing resonant behaviour given by tan 7)2 ( E ) =
4r/(Ed - E),
(99)
where Edand r determine the position and width of the resonance. This allows the KKR equations to be transformed directly into the H-NFE-TB form, in which the two centre TB hopping integrals and hybridization matrix elements are determined explicitly by the two resonant parameters Ed and r. The non-orthogonality contributions to the secular equation (MUELLER[19671) are obtained by linearizing the implicit energy-dependent matrices C, D and H in a Taylor expansion about E,. The nonmagnetic band structure of fcc and bcc iron is shown in fig. 25, being computed from the H-NFE-TB secular equation with resonant parameters Ed=0.540 Ry and r =0.088 Ry (PETTIFOR[197Oa]). The NFE pseudopotential matrix elements were chosen by fitting the first-principle values of WOOD [1962] at the pure p states N,. ~ ( ~ ~ ~ = 0Ry), . 0 4 L,’ 0 (unI=0.039 Ry) and X,. (u2,=0.034 Ry). Comparing the band structure of iron in the 1100) and Ill 1) directions with the canonical d bands in fig. 22, we see there is only the am d level with symmetry A, and A, respectively which hybridizes with the lowest NFE band, the remaining four d levels being unperturbed. Because of the canonical nature of the pure TB d bands (ANDERSEN [1973]), the band structm~of all fcc and bcc transition metals will be very similar to that shown in fig. 25 for iron. The transition-metal density of states, n Q , is not uniform throughout the band as shown schematically in fig. 24 but displays considerable struchlre that is characteristicof the given crystal lattice. This is seen in fig. 26 for the bcc, fcc and hcp densities of states, which were calculated by the H-NFE-TB secular equation neglecting nonorthogonality contributions with Ed= 0.5 Ry and r = 0.06 Ry (PETTIFOR [1970b]). These early histogram densities of states are displayed rather than more accurate recent References:p. 129.
84
Ch. 2, $4
D. G. Perrifor
-P
X
L
K
r
N
H
7
P
Wg. 25. The H-NFE-TB band structure of fcc and bcc iron in the nonmagnetic state. The solid circles represent the first-principle energy levels of W m D [1962]. (From PETTIFOR [1970a].)
calculations (see, e.g., RAm and CALLAWAY [1973], JEPSEN et al. [1975], MORUZZIet al. [1978], PAXTONet ul. [1990]) because they allow a direct comparison between the bcc, fcc and hcp densities of states for the sume model element. This will be important when discussing the relative stability of the three different crystal structures in 5 6.1 and the stability of the ferromagnetic state in the a,y and 8 phases of iron in 5 8. The structure in the calculated densities of states in fig. 26 is reflected in the behaviour of the experimental electronic heat constant, y, across the nonmagnetic 4d and 5d transition metal series. It follows from eqs. (43, (47) and (48)that the electronic heat capacity may be written as
C = yT,
(100)
where
y = 5 T?k&(E,). Therefore, ignoring any renormalization effects such as electron-phonon mass enhancement, the linear dependence of the heat capacity gives a direct experimental measure of the density of states at the Fermi level. Figure 27 shows that the H-NFETB densities of states in fig. 26 reflect the experimental variation in y across the series. 4.3. Volume dependence Figure 28 illustrates the volume dependence of the energy bands of the 4d transition metals Y, Tc and Ag, which were calculated by PETTIFOR [1977] within the atomic-
a.2,94
Electron theory Dfrnetals
85
loot
Fig. 26. The density of states for the three structures (a) bcc, (b) fcc, and (c) hcp for a model transition metal. The dotted curves represent the integrated density of states. (From PETT~FOR[197Ob].)
sphere approxinnation of ANDERSEN [1973, 19751. Similar bands have been obtained by GELATTet al. [19771 for the 3d metals Ti and Cu with the renormalized-atom approximation of WATSONet aZ. [1970]. We see from fig. 28 that the bottom of the NFE sp band rl, which was evaluated within LDF theory, is well fitted by the Frohlich-Bardeen expression (79). The values of R, obtained are found to scale within 1% with the position of the outer node of the 5 s free-atom radial wave function. This demonstrates quantitatReferences: p. 129.
86
D. G. Petrifor
OL
i
5
Ch. 2,94
7
’ 9 ’ li N Fig. 27. A comparison of the theoretical and experimental 4d and 5d heat capacities. The theoretical values were obtained directly from eq. (101) and fig. 26, neglecting any changes in the density of states due to band width changes or mass renonnalization. ’
’
ively that it is the core-orthogonality constraint which is responsible for the rapid turn up in the energy of and that the outer node of the valence s electron is a good measure of the s core size. The free-atom d level broadens into a band of states of width W as the atoms come together from infinity to form the bulk (see figs. 24 and 28). BINE [1967] has shown that the Wigner-Seitz boundary conditions imply that W should vary approximately as S-*, where S is the Wigner-Seitz radius. Assuming a power-law dependence of W on S, we can write
rl
w = K(So/S)”,
(102)
where Woand So are the values of the d-band width and Wigner-Seitz radius respectively at the equilibrium lattice spacing of the transition metal. Table 1 gives the values of So, Woand n for the 4d transition metals (PETTIFOR [1977]). Because of the more extended nature of the d wave functions at the beginning of the transition metal series, n takes a value closer to four than to five which we will see in 9 5.2 is reflected in their bulk properties. Values of the band width W for the 3d, 4d and 5d series may be obtained from the table in ANDERSEN and JEPSEN[1977] and are given explicitly in table 20-4 of HARRISON [1980]. The 3d and 5d band widths are approximately 30% smaller and 20% larger respectively than the corresponding 4d widths. The centre of gravity of the TBd band, Ed, in fig. 28 rises exponentially (PETTIFOR [19771) as the volume decreases because the potential within the Wigner-Seitz sphere renormalizes due to the increase in the electronic charge density (GELATT et al. [1977]). This renormalization in position of the free atomic d level plays an important role in transition-metal energetics and will be discussed further in 5 5.2. The different volume dependences of the NFE-sp and TB-d bands displayed in fig. 28 will lead to changes in the relative occupancy of the two bands with volume. This is illustrated in fig. 4 of PETTIFOR [1977] where Y and Zr show a rapid increase in d-band occupancy under compression as the d band widens and the bottom of the sp band moves
Ch. 2, $ 5
87
Electron theory of metals
Table 1 Equilibrium values of Wiper-Seitz radius So and d band parameters W, n and IUS, for 4d series (from PETTIFOR [1977]). Quantity
So (au) W,(eV)
n dS0
Element
Y
zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
3.76 6.3 3.9 1.03
3.35 7.8 4.0 1.19
3.07 9.3 4.1 1.33
2.93 9.5 4.3 1.47
2.84 9.1 4.5 1.58
2.79 8.5 4.6 1.65
2.81 7.6 4.8 1.71
2.87 6.0 5.1 1.77
3.02 3.9 5.6 1.84
up (cf. fig. 28a). Eventually r, moves up through the Fermi level EF at which point all the NFE-sp states have been emptied into the TB-d states and Nd=N.On the other hand, the transition metals with more-than-half-filled d bands display a marked degree of constancy in N, for volumes about their equilibrium values, because the sp core effects are largely cormter-balanced by the rapid rise in Ed due to the increasing coulomb repulsion between the d electrons (cf. fig. 28c). However, under very high pressures the bottom of the sp band does eventually move up through the Fermi level, and transition metals with ten valence electrons (Ni, Pd and Pt) may become semiconducting (MCMAHANand ALBERS[1982]). We will return to this dependency of the d-band occupancy on volume and core size when discussing crystal structure stability in 5 6.
5. Bulk properties 5.1. Simple metals Within the free-electron approximation the total energy per electron may be written (see, e.g., HEINEand WEAIRE[1970]) as:
Ueg= 2.21/$
-05:
- 0.916/~ - (0.115 - 0.0313 In q),
(103)
I
Fig. 28. The energy bands as a function of WignerSeitz radius S for (a) Y,@) Tc, and (c) Ag. The observed equilibrium Wigner-Seitz radii are marked eq. The dotted curve gives the Frohlich-Bardeen fit (eq. 79) to the bottom of the conduction band r,.Ed,E, and &, mark the centre of gravity, and top and bottom of the d band, respectively. (After PEITlFOR [1977].) References: p. 129.
88
D. G. Pettifor
Ch.2, $ 5
where r, is the radius of the sphere which contains one electron so that
q=
~-1t3s
(104)
for a metal with valence Z and Wigner-Seitz radius S. The first term in eq. (103) is the average kinetic energy of a free electron gas, namely ;EF, where EFis given by eq. (45). The second term is the exchange energy which is attractive, because parallel-spin electrons are kept apart by Pauli’s exclusion principle, thereby leading to weaker mutual coulomb repulsion. The third term is the correZation energy which gives the additional lowering in energy due to the dynamical correlations between the electrons. It follows from eq. (103) that the free electron gas is in equilibrium for r,=4.2 au with a binding energy per electron of 0.16 Ry or 2.2 eV. If the electron gas is perturbed to first order by the presence of the ionic lattice (HEN! and WEAIRE[1970], CIRIFALCO [1976] and HARRISON [1980]), then the total binding energy per atom may be written as:
where
The first and second terms in eq. (106) give the electron-ion [cf. eq. (79)] and the electron-electron potential energies, respectively. The potential energy has been evaluated within the WIGNER-SEITZ[19331 approximation of neglecting the coulomb interaction between different Wigner-Seitz cells as they are electrically neutral. Within the freeelectron approximation the ion cores had been smeared out into a uniform positive background so that there was zero net potential energy and Vi, vanished. The equilibrium Wigner-Seitz radius, So, which is found from eq. (105) by requiring that U is stationary, depends explicitly on the core radius R, through the equation 0.102
+
0.0035S0 -~ 0.491
z
PS,, ’
where the first four terms are coulomb, exchange, correlation and kinetic contributions respectively. GWFALCO[1976] has taken the experimental values of the Wigner-Seitz radius So to determine an effective Ashcroft empty-core radius R, from eq. (107). The resultant values are given in table 2 where, as expected, the core size increases as one goes down a given group in the Periodic Table. It is clear from table 2 that only sodium has an equilibrium value of r, that is close to the free-electron-gas value of 4.2 au. The bulk modulus (or inverse compressibility), which is defined by
B = V(d2U/dV2), may be written from eqs. (105) and (107) in the form B/B,, = 0.200 + 0.815Rz/rs
(108)
Ch. 2, $ 5
89
Electron theory of metals Table 2 Equilibrium bulk properties of the simple and noble metals.
at equilibrium, where the correlation contribution has been neglected since it contributes less than a few percent. Bkeis the bulk modulus of the non-interacting free electron gas, namely
It follows from eq. (109) and table 2 that the presence of the ion core is crucial for obtaining realistic values of the bulk modulus of simple metals, as was first demonstrated by ASHCROFT and LANGRETH [1967]. However, the simple Jirst-order expression eq. (109) is leading to large errors for the polyvalent metals with valence greater than two because the second-order contribution is not negligible and must be included (ASHCROFT and LANGRETH[1967]). Table 2 also demonstrates that the noble metals are not describable by the NFE approximation, the theoretical bulk moduli being a factor of five too small. We will return to this point in 55.2. The cohesive energy of the simple metals is observed in table 2 to be about 1 eV per valence electron. For example, Na, Mg and A1 have cohesive energies of 1.1,O.S and 1.1 eV per electron respectively. These are an order of magnitude smaller than the corresponding binding energies given by eq. (105), the experimental values being 6.3, 12.1, and 18.8 eV per electron respectively. Although NFE perturbation theory can yield good estimates of bulk properties such as the equilibrium atomic volume, structural stability References: p . 129.
90
D. G. Pettifor
Ch. 2, 5 5
and heat of formation, it can not provide reliable cohesive energies which require an accurate comparison with the free atom whose wave functions are not describable by weakly perturbed plane waves. It is necessary, therefore, to perform similar calculations in both the free atom and the bulk as, for example, WIGNERand SEITZ[1933] and MORUZZI et al. E19781 have done in their evaluation of the cohesive energies in figs. 21 and 1 respectively. We should point out, however, that eqs. (103)-(106) do yield a bulk binding energy for sodium that is very similar to Wigner and Seitz’s [cf. eq. (82)], because the additional exchange, correlation and self-energy terms in eqs. (105) and (106) give a net contribution of less than 0.01 eV per sodium atom. CHELIKOWSKY [1981] has linked the cohesive energy of simple metals to a kinetic-energy change which accompanies the transformation of the exponentially damped free-atom wave function to plane-wave bulk states. As expected from table 2 and fig. 20, it is necessary to include an additional non-local bulk bonding contribution in order to obtain the stronger cohesion of Li and Be and the weaker cohesion of Zn, Cd and Hg. The anomalously large cohesion of the noble metals Cu, Ag and Au will be discussed in the next subsection. 5.2. k s i t i o n metals The theoretical points in fig. 1 were computed (~TORUZZI et al. [1978]) by solving the Schrodinger equation (1) with the potential u(r)given by
where u, is the usual Hartree potential and u,, is the exchange-correlation potential evaluated within the local density functional (LDF) approximation of HOHENBERG and KOHN [1964] and KOHN and SHAM[1965], namely
is the exchange and correlation energy per electron of a homogeneous electron gas of density p. It follows from eqs. (103) and (112) that the exchange contribution to the potential may be written as:
e X ( r )= -1.477[~(r)]1’~. Thus the exchange potential varies as the third power of the local density, due to the exclusion of parallel spin electrons from the immediate neighbourhood (SLATER [1951bl). The total energy can not be written simply as the sum over the occupied one-electron energies Eiof the Schrunger equation, because the eigenvalue Eiof the ith electron contains the potential energy of interaction with the jth electron and vice versa. Thus, Ei + Ej double-counts the coulomb interaction energy between electrons i and j . The total LDF energy is, therefore, given by
Ch. 2, 55
91
EIectron theory of metals
Ei - 3JJ 2p(r)p(‘r’) drdr’ - p(r)[vxc- ~ ~ ~ ] d r ,
U=
I t
i
- r’l
where the second and third contributions correct for the “double-counting” of the coulomb and exchange-correlation energies respectively. The potential energy has been written down in eq. (1 15) within the Wigner-Seitz sphere approximation, the coulomb interaction between neighbouring Wigner-Seitz cells, or Madelung contribution, being neglected. (Note that e*= 2 in atomic units, which accounts for the factor of two in the integrand of the coulomb integral.) The presence of the double-counting contribution in eq. (115) does not allow for a direct interpretation of the total energy in terms of the one-electron eigenvalues Ei whose behaviour we have studied in the previous sections. For example, as can be seen from fig. 28b the oneelectron sum alone would lead to no binding in Tc because the delectron eigenvalues at the equilibrium atomic volume are everywhere higher than the free-atom d level. The inclusion of the double-countingterm is crucial for bonding since it counters to a large extent the shift in the centre of gravity of the d bands E,, due to the renonnalizution of the potential under volume change. In copper, for example, GELATT et al. [1977] found that the band-shift energy of 78.6 eV/atom, which accompanies the formation of the bulk metal, is almost totally cancelled by a change in the doublecounting term of 77.7 eV/atom. The remaining net repulsive contribution of about 1eV/atom is typical for the 3d and 4d transition metal series (see fig. 4 of GELATTet al. [1977]). The problems associated with double-counting can be avoided, however, by working not with the total energy, U,but with thefirst-order change in energy, SU,on change in the Wigner-Seitz sphere volume, Sa,for the bulk metal (PETTIFOR[1976]) or change in 11978al). By starting the internuclear separation, SR, for the diatomic molecule (PETTIFOR either from the virial theorem in the form derived by LIBERMAN [1971] or from the totaland HODGES[1976], PETTIFOR [1976, energy expression (115) following NIEMINEN 1978aI showed that the first-order change in total energy, SU,may be written,neglecting the Madelung contribution, as:
SU = CSE,, i
where SEi is the first-order change in the eigenvalue which accompanies the first-order volume or distance change while the potential is kept unrenormalized. The general applicability of this first-order result has been proved by ANDERSEN [1980] for force [19821 for embedding problems involving arbitrary atomic displacements and by NORSKOV problems involving a change in the local atomic environment (cf. 07). SKRIVER[1982], MCMAHAN and MORIARTY[19831and PAXTONand PETTIFOR [19921have demonstrated the applicability of eq. (116) to the evaluation of structural energy differences (cf. 5 6). The first-order expression (116) is important because it allows a direct identification of the different roles played by the valence sp and d electrons in bulk transition metal energetics. The eigenstates can be decomposed within the Wigner-Seitz sphere into their different angular momentum components, 1, so that eq. (116) may be written as: References: p . 129.
92
Ch. 2, $ 5
D. G.Penifor
su = -Pa2 =
-x
lp2,
(117)
1
where P is the pressure, given by P = - dU/dQ. By working within the atomic-sphere [19731 the partial pressures Pl may be expressed (PETTIFOR approximation of ANDERSEN [1976]) directly in terms of parameters describing the energy bands, namely:
3p,a = ~ N , ( T- ,cxc)+ 2u;, 3 4 i - l ~2Nd(Ed- &,)/md + 5Udbond,
(118) (119)
where
Usp" =
1" ( E - r,)nsp(E)dE,
with E ~ ~ = E ~ ~md ( Sis) .the d-band effective mass which is related to the width W through W=25/(mdSz).Additional small contributions to eqs. (118) and (119) have been neglected for simplicity in the present discussion (cf. eqs. (13) and (14) of PETTIPOR [1978bl). The sp partial pressure consists of two terms which give the first-order changes in the bottom of the sp band, r,, and in the kinetic energy, respectively. In the absence of hybridization with the d band, nsp(E)is free-electron-like and eq. (118) is consistent with the pressure which would be obtained from the simple-metal expression (105) if correlation is neglected. This follows from eqs. (ill), (113) and (79) because within LDF theory the bottom of the band is given by
r, = r Y + 2 . 4 z / s + + ,
(122)
since the electron sees the average Hartree field of the valence electrons and the exchange potential v, in addition to the ion core pseudopotential. The d partial pressure also consists of two terms which give the first-order changes in the centre of gravity of the d band, Ed, and the d bond energy, respectively. In the absence of hybridization we may assume that nd(E)is rectangular as illustrated in fig. 24, so that from eq. (121) the d bond energy may be written
upd = -6WNd(l0- Nd).
(123)
Assuming that $-Eltom and W vary inversely as the fifth power of S, Pd may be integrated with respect to volume to give the d contribution to the cohesive energy, namely: u d
= N,j(Ed - Edm)/4md
+ Nd($ E,"""
- EX,)/2fnd
+ upd.
(124)
It follows from fig. 28a that for Tc at its equilibrium volume Ed-Eitom= 6 eV, $Ed"" - eXC = 1 eV and md= 5. Therefore, taking, from table 2, W= 10 eV and Nd= 6, we have
Ch. 2, 5 5
Electron theory of metals
U, = 1.8 .t 0.6 - 12 = -1OeV/atom,
93
(125)
which is in reasonable agreement with the LDF value of -8 eV/atom for the Tc cohesive energy in fig. 1. The dominant contribution to the cohesive energy of transition metals is, therefore, the d bond term in eq. (125) as emphasized by FRIEDEL[1964, 19691 and illustrated by GELATTet al. [I9771 in their fig. 4. From eq. (123) it varies parabolically with band filling and accounts for the observed variation of the cohesive energy across the nonmagnetic 4d and Sd series shown in fig. 1. It attains a maximum value of - 5 W/4 for Nd = 5 when all the bonding and none of the antibonding states are occupied. Equation (124) shows that the shift in centre of gravity of the d band contributionN, (Ed-E,"t"") is reduced by at least an order of magnitude through the factor (4md)-', thereby accounting andytically for the cancellation arising from the double-counting term in eq. (115). Figure 29 shows the sp and d partial pressures for Tc. As expected from eq. (123) there is a large attractive d bond contribution which is pulling the atoms together in order to maximize the strength of the bond. This is opposed for S < 4.0 au by a rapidly increasing repulsive d centre-of-gravity contribution which reflects the renormalization in Ed. The resulting total d partial pressure is attractive at the observed equilibrium volume of Tc (see fig. 29b). As expected from the behaviour of rl in fig. 28b the bottom of the sp band contribution is attractive for large values of S but becomes repulsive in the vicinity of the equilibrium volume as I', moves up in energy. Thus, whereas in simple metals this contribution is attractive because the ion cores occupy only about 10% of the atomic volume (see fig. 21 and table 3), in transition metals it is repulsive because 5-0
.t..
Sloul
Fig. 29. (a) The individual and (b) the total sp and d partial pressures as a function of the Wiper-Seitz radius S for T,.''eq" marks the observed equilibrium Wigner-Seitz radius. (From PETTIFOR [1978b]). References: p . 129.
94
Ch.2, 8 5
D.C. Pettifor Table 3 The values of A for the 3d, 4d, and 5d transition metal series.
Period
I
Element and value of A (in a d ) sc
a
3
1.08
1.23
Y 1 .OS
zr
4
Lu
Hf
1.11
1.25
5
I .23
V 1.37
cr
Mn 1.61
Fe 1.74
co
1.49
Nb 1.37
Mo
Tc 1.60
Ru 1.72
Rh
1.49
Ta 1.38
w
Re
os
Ir
Pt
1.72
1.84
2.01
1.49
1.a
1.88 1.85
Ni 2.07 Pd 2.02
the ion cores occupy a much larger percentage due to their smaller equilibrium atomic volumes (cf. fig. 1). Together with the sp kinetic energy contribution, the bottom of the sp band contribution provides the necessary repulsion to counter the attractive d partial pressure at equilibrium. The size of a transition-metal atom, which is defined by the equilibrium atomic volume of the pure metal, is not necessarily a helpful quantity for discussing alloy energetics. We have seen that it will be very sensitive to the nature of the local atomic environment, since it is the d bond contribution which is responsible in fig. 1 for the skewed parabolic behaviour of the equilibrium Wigner-Seitz radius across the nonmagnetic 4d series. This may be demonstrated by modifying the simple model of DUCASTELLE [1970] and approximating the total energy of a transition metal by
where the Born-Mayer contribution, VP, is: with a being constant across a given series. This form is suggested by the nature of the repulsive d centre-of-gravity contribution in eq. (124) and fig. 29, although we have assumed that II"p is proportional to N2 rather than N,' as a reminder that the sp electrons also contribute to the repulsion. The d bond contribution, eq. (123), is proportional to the band width W which is assumed to vary exponentially as W = bA2e-* with b being constant across a given series. The cohesive energy, equilibrium Wigner-Seitz radius and bulk modulus are given from eqs. (126)-(128) by:
u,,
= 3u
y,
so[ '"(-2aNZ/u:d)]
/2A,
B = -(A2/ 12~s,)U,""~.
Ch. 2, 36
Electron theory of metals
95
a and b for a given period are obtained from the known bulk modulus and band width of 3d Cr, 4d Mo and 5d W, the values of (a, b) being given in atomic units by (24.3,
11.6), (77.2,25.0) and (98.9,31.9) respectively. A is found by fitting to the nonmagnetic Wigner-Seitz radius, assuming that the transition metals have only one sp valence electron. We see from fig. 30 and table 3 that although the equilibrium atomic volume has a minimum in the vicinity of N=8, A varies nearly Linearly across the series as expected for a parameter characterizing the free atom (cf. figs. 4, 5 and 7). Thus, although Mo and Ag have almost the same size factors with their equilibrium WignerSeitz radii of 2.93 and 3.02 au, respectively, they are immiscible because Mo will lose a large part of its attractive d bond contribution in a Ag environment. The logarithmic derivative of the band width, -A, predicted by this model is in good agreement at the equilibrium atomic volume with the first-principles value, -n/S,, as can be seen by comparing tables 1 and 3 for the 4d series. The simple model breaks down at the noble-metal end of the series because the Born-Mayer repulsive term in eq. (126) does not describe correctly the d electron behaviour. This can be seen in fig. 31 where the d partial pressure in Cu is attractive at the equilibrium atomic volume, the d electrons contributing about 25% to the cohesive energy (WILLIAMSet al. [l9SOa]). Thus, as first pointed out by KOLLARand SOLT [1974], the filled d shells in copper interact attractively rather than repulsively as assumed by the Born-Mayer contribution (127). This is due to the second term in eq. (124) which dominates at larger atomic volumes. The sp partial pressure of Cu at its minimum is also more attractive than that of K due to the incomplete screening of the Cu ion core by the 3d valence electrons. The net result is that whereas the simple metal M has a cohesive energy of 0.9 eV/atom and a bulk modulus of 0.3 x 10" N/m2, the noble metal Cu has a cohesive energy of 3.5 eV/atom and a bulk modulus of 13.7 x 10" N/m2, which is reflected by the behaviour of the curves in fig. 31.
6. Structural stability 6.1. Elemental metals The crystal structure of the simple metals can be studied (see, e.g., HARRISON [1966], HEINEand WEAIRE[1970], HAFNER[1974,1989] and MORIARTY[1982,1983 and 19881) by perturbing the free electron gas to second order in the pseudopotential, thereby extending the first-order expression (105) considered in 5 5.1. The resulting binding energy per atom is given in the real-space representation (FINNIS[1974]) by
where K% is the compressibility of the free electron gas. +(R=O; r,) represents the electrostatic interaction between an ion and its own screening cloud of electrons, whereas 4(R # 0; rs)is a. central interatomic pair potential which for a local pseudopotential may be written as:
References:p . 129.
Ch. 2, 56
D. G. Pettgor
96
Fig. 30. The theoretical (open circles) and experimental(crosses) values of the equilibrium Wigner-Seitz radius, cohesive energy, and bulk modulus of the 3 4 4d, and 5d transition metals.
x
X
0.0 I
x = In (o/oo) oo = EQUILIBRIUM LATTICE CONSTANT
0.4 I
0.8 I
x = In (a/o,) oo = EQUILIBRIUM LATTICE CONSTANT I
+5
COPPER
\
POTASSIUM
Fig. 31. (a) The sp and d partial pressures for Cu and (b) the sp pressure for K as a function of the WignerSeitz radius. The independent variable x is the logarithm of the ratio of the lattice constant a (or Wigner-Seitz radius S)to its equilibrium value a, (or S&, so that equilibrium corresponds to the zero value of x on the upper horizontal axis. The cohesive energy associated with a given pressure curve is the area between the curve and et al. [198Oa].) the axis, as illustrated in (b). From WILLIAMS
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Electron theory of metals
97
f$(q) is proportional to the Fourier component of the ionic pseudopotential, taking the value cosqR, for the Ashcroft potential [cf. eq. (76)l. x(q, rs) is the free-electron-gas [1981]). response function which screens the ion cores (see, e.g., JACUCCI and TAYLOR The first term in eq. (133) gives the direct ion-ion coulomb repulsion, the second the attractive ion-electron contribution. [1982]) at The interatomic potential (133) may be expressed analytically (PETTIFOR metallic densities as the sum of damped oscillatory terms, namely
cb(R f 0;4>4 9 4 9 4 9 4,4747 494-43 4,4949 49494, 49494’ 4,494 Coordination formula: 2 [BiLi,,Li,,] (ionic description) For the layer stacking symbols, see under the following description of the NaTl type. For the prototype itself, Li,Bi, a=672.2 pm. This structure (or BiF, structure) could also be described as derived from a cubic closepacked array of atoms (Bi atoms) by filling all the tetrahedral and octahedral holes with Li (or F) atoms. The cFl6-NaTl type structure is face-centered, cubic, space group Fdgm, No. 227. Atomic positions: 8 “1 in a) O,O,O; 0 11. 1 0rZY 1. 1 1 0. 2 1 3 . 3 3 1. 11.1.1 3 2 2929 4,4943 4,494, 4,4349 4,494’ 1 I 1 ‘00.010.1~0. ~ ~ 341r24 r 34 ; 14,4r4. 11 8 Na in b) T , ~ , ~T ,; , ,, ,T, , 2 r 2 r 4,4,4;112. 4r4r4; For the prototype, NaTI, a=747.3 pm. ,2329
25
9
I
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Structure of intermetallic compounds and phases
273
LiZn, LiCd, LiAI, NaIn have this structure. This structure may be regarded as a completely filled-up fcc arrangement in which each component occupies a diamond like array of sites (see sec. 6.3.1. and, in sec. 3.4., “Zintl Phases”.) The structure may thus be presented as NaTl: D + D’ (see the descriptions in terms of combination of invariant lattice complexes reported in sec. 3.1.). The coordination formula is $3 Na[T14,4]4,4. All the three cF16-NaT1, Li,Bi and MnCu,Al types, which may also be considered as composed of four interpenetrating face centered cubic arrays (F + F’ -k F” + F”’), correspond to the same space filling as in 8 b.c. cubic (or in 8 CsCl type) cells (see fig.
24). The layer stacking symbols of the NaTl structure are here reported in comparison with those of the cF16-Li3Bi and cF16-MnCu,Al types. Triangular (T) nets: NaTl type: N 4 4 Na,B, TGlZ =,A, Nag12 N a g mi4 Na,c,,12 Na;: T C l Z Li,Bi type Bit Li;,L Li;6 Li,P4B i g Li& L i i Li&zB i g L4:i Li& Lik,lz MnCu,Al type MnE6 cuG4 A1& cu&’2 Mn:’, cu& A1i3 c u i 4 Mn& cu&Z Square (S) nets: NaTl type: Nai Na: T1: Na:;, Tl:, Tl;], TlP,,Na:, Tli4N a 3 Li,Bi type Si: Bi,“LiiLip/4:Li:/4Li~/2Lip/zBi~,2Li,6,Li~,4 MnCu2Al type: Ali Ali Mni CuE4CuC4Mn:, Mnfn Al:, Cui4Cui4
A1t
6.1.5. Commentson the bcc derivative structures In the family of bcc derivative structures we may include several other structural types. As an important defect superstructure based on the bcc structure we may mention the cPB-Cu,Al, type structure (AgJn,, Au91n4,Pd,Cd4,, Co,Zn,,, Cu,Ga4, Li,,Pb, can be considered reference formulae of selected solid solution phases having this structure). The large cell (a = 870.4 pm in the case of Cu,Al,) can be considered to be obtained by assembling 27 CsCl type pseudocells with two vacant sites. One vacant site occurs on each sublattice PJ,6Cu,,0 and Cu,,O. The y-brass, c152-Cu5Zn,-type structure can be similarly described as a distorted defect superstructure of the W type structure, in which 27 pseudocells are assembled together with two vacant sites (corner and body center of the supercell). In this case, however, the atoms, are considerably displaced from their ideal sites. The structure could also be described as built up of interpenetrating, distorted, icosahedra (each atom being surReferences: p . 363.
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Riccardo Ferro and Adriana Saccone
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rounded by 12 neighbours). This description applies also to the cP52-Cu,A14 type structure. (Ag,Cd,, Li,Ag,, Ag,Zn,, V,Al,, Au,Cd,, Au,Hg,, Fe,Zn,,, NiGa,, V,Ga,, Ni,Zn,,, etc. crystallize in the cI52-Cu5Zn, structural type). Martensite. The iron-carbon martensite structure can be considered a tetragonal distortion of the body-centered cubic cell of Fe (a = 285 pm, c = 298 pm at = 1 mass% C (= 4.5 at%), in comparison to a = 286.65 pm for a-Fe, cI2-W type). Carbon is randomly distributed in the octahedral holes having coordinates O,O,i and i,i,O. Typically an occupancy of these sites of only a few % has to be considered. For a 100% occupancy the structure of the tI4-COO type (low-temperature form) is obtained with 2 Co in a) O,O,O; i,i,i;and 2 0 in b) O,O,i; $,i,O in the space group 14/mmm, No. 139. In the martensitic cell the position parameters of the Fe atoms have a range along the fourfold axis, so there is a displacement from the cell comers and body center and an enlargement of the octahedral holes containing carbon. (Notice, however, that “martensite” is also a general name used by metallurgists to denote all phases formed by diffusionless shear). Al-Cu-Ni continuous sequence of ordered structures. An interesting series of superstructures have been described by Lu and CHANG[1957a, 1957bl. For an assessed description of the system and of the intermediate phases see PRINCE[1991]. They all have hexagonal unit cells (some corresponding to rhombohedral structures) based on ordered sequences of pseudo cubic subcells slightly distorted in rhombohedra having the constant qhom included between 289 and 291 pm and the interaxial angle arhom included between 90.34 and 90.10’. (These data may be compared with the values a = = 288 pm and, of course, a =90’, for the cubic CsCl type unit cell of NiAl at the 50 at% Al composition). The hexagonal cells of the superstructures have a certain number of subcells stacked along c. A1 atoms occupy the corners of the subcells and Ni,Cu (Me) atoms or vacancies (Vac) occupy the centers in ordered array, vacancies occurring along the three triad axes (O,O,O; 3.3,~;?,i,z). All together these phases corresponds to the .r-region lying in the ternary system in a domain included between = 7 and 12 at% Ni and between =27 and 38 at% Cu. The different T~ ordered structures correspond to the stacking of i subcells centred according to a definite sequence of Vac or Me atoms. Following stacking variants have been described: 7, = (Ni,Cu),Al,, hR24, a=411.9 pm, c=2512.5 pm (=5*502.5) stacking sequence VacMeMeMeVac = VacMe,Vac .r6 = (Ni,Cu),Al,, hP30, a=411.3 pm, c=3013.5 pm (=6*502.3) stacking sequence VacMe,Vac .r7 = (Ni,Cu),Al,, hP36, a = 410.6 pm, c = 3493.8 pm ( = 7 *499.1) stacking sequence VacMe,Vac 7 8 = (Ni,Cu),Al,, hR42, a = 410.5 pm, c = 3990 pm ( = 8 *498.8) stacking sequence VacMe,Vac T~~= (Ni,CU),Al,,, hP51, a=411.5 pm, c=5528.9 pm ( = 11*502.6) stacking sequence VacMe,Vac,Me,Vac 713 = (Ni,Cu),Al,,, hR63, a=411.3 pm, c=6517.3 pm ( = 13*501.3) stacking sequence VacMe,Vac,Me,Vac
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hw5, a = 409.6 pm, c = 7464.5 pm ( = 15*497.6) stacking sequence VacMe,Vac,Me,Vac T ~ ?,: (Ni,Cu)&l1,, hP87, a=410.1 pm, c=8449.9 pm ( = 17*497.1) stacking sequence VacMe,Vac,Me,Vac These structures appeared to be determined by the free electron concentration, They represent a so-called “continuous sequence of ordered structures” or, infinitely adaptive and ANDERSON[ 19891). These structures occupy a single-phase field structures (HYDE: in the system: it has been observed that, in such cases, may be ambiguous to define a phase in terms of a unit cell of structure. T~~+5 (Ni,Cu),&E,,,
6.2. Close-packed structures and derivative structures
In this section, a. few important elemental structures are described. Particularly the cubic (cF4-Cu type) and hexagonal close-packed (hP2-Mg) structures are presented. A few other stacking variants of identical monoatomic triangular nets are also reported. A group of structures which can be considered as derivative structures of Cu are also described. Normalized interatomic distances and numbers of equidistant neighbours are shown in figs. 25 and 26. 6.2.1. Structural type: cF4-Cu Face-centered cubic, space group Fmgm, No. 225. Atomic positions: 4 Cu in a) O,O,O; (I, +,O,+; +, +,i,O; ;; Coordination formula: 3 [Cu,,,,] Layer stacking symbols: Triangular (T) nets: C u t C u h Cu& Square (s)nets: CU; c u i CU; For the prototype itself, Cu, a=361.46 pm. The atoms are arranged in close packed layers stacked in the ABC sequence (see sec. 3.5.2.). Several metals, such as AI, Ag, Au, a Ca, a Ce, y Ce, a Co, Cu, y Fe, Ir, /3 La, Pb, Pd, Pt,Rh, Q Sr, Q Th and the noble gases Ne, Ar, Kr, Xe crystallize in this structural type. Several bin;uy (and complex) phases having this structure have also been reported (solid solutions with random distribution of several atomic species in the four equivalent positions). 6.2.2. Cu-derivative, substitutional and interstitial superstructures (tetrahedral and octahedral holes) Derivative structures may be obtained from the Cu type structure by ordered substitution or by ordered addition of atoms. As examples of derivative structures obtained by ordered substitution (andor distortion) in the Cu type we may mention the AuCu,, AuCu, Ti,Cu types, which are described here below. (In the specific case of the AuCu, type structure and the Cu-AuCu, types interrelation, see also sec. 3.5.5.). For a References: p . 363.
276
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Riccardo Ferro and Adriana Saccone
24
N
-
a) --type
16 -
a0-
I
+
b) AuCu,-type
+
*
+
*
24 -
*:
c ) AuCu,-type
N
a-
i*
+
01
1.2
1.4
1.6
1.8
2
d /d-
Fig. 25. Distances and coordinations in the c F 4 - c ~and cP4-AuCu3 types structures. (Compare also with figs. 14 and 15.) a) cF4Cu type structure b) cP&AuCu, type structure: coordination around Au (+) Au-CU; (*) Au-Au. c) AuCu, type structure: coordination around Cu (+) CU-CU;(*) CU-Au.
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Structure of intermetallic compounds and phases
N
24] a] Mg-type(ideal)
1 bf
N
Mg-type
Fig. 26. Distances and coordinations in the hexagonal close-packed (Mg-type) structure. a) Ideal structure, c,la= 1.633 (first coordination shell corresponding to 12 atoms at the same distance). b) Mg-type structurtss with c/a= 1.579. The group of the first 12 neighbours is subdivided into 6 + 6 atoms at slightly different distances.
systematic description of the derivative structures which may be obtained from the Cu type by oldered$lling-up it may be useful to consider that in a closest packing of equal spheres there are, among the spheres themselves, essentially two kinds of interstices (holes). These are shown in fig. 27. The smallest holes surrounded by a polyhedral group of spheres are those marked by T. An atom inserted in this hole will have four neigh-
bours whose centres lie at the vertices of a regular tetrahedron (terruhedrul holes). The larger holes (octahedral holes) are surrounded by octahedral groups of six spheres. In an infinite assembly of close-packed spheres the ratios of the numbers of the tetrahedral and octahedral holes 1.0 the number of spheres are, respectively, 2 and 1. Considering the Cu type structure (in which the 4 close-packed spheres are in O,O,O; O,$,i; i,O,i; &&O) the centers of the tetrahedral and octahedral holes have the coordinates: 4 octahedral holes in: 111. '00., 0729 ' 0 . 001. 2,2929 2' 92, 2 sets of 4 tetrahedral holes in: 3
Fig. 27. Voids in the closest packing of equal spheres; tetrahedral (T) and octahedral holes (0)are evidenced within two superimposed hiangular nets.
Several cubic structures, therefore, in which (besides O,O,O; O,&$; f,O,&; &f,O) one (or more) of the reported coordinate groups are occupied could be considered as filled-up derivatives of the cubic close-packed structures. The NaCl, CaF,, ZnS (sphalerite), AgMgAs and Li,Bi type structures could, therefore, be included in this family of derivative structures. For this purpose, however, it may be useful to note that the radii of small spheres which fit exactly into tetrahedral and octahedral holes are 0.225.. and 0.414..., respectively, if the radius of the close-packed spheres is 1.0. For a given phase pertaining to one of the aforementioned types (NaCl, ZnS, etc.) if the stated dimensional conditions are not fulfilled, alternative descriptions of the structure may be more convenient than the reported derivation schemes. Notice, moreover, that a fc cubic cell of atoms X in which all the interstices are occupied (the octahedral by X and the tetrahedral by Z atoms) is equivalent to a block of 8 X Z , CsCl type, cells (see figs. 3 and 24). This relationships (and other ones with other structures such Li3Bi and MnCu,Al) should be kept in mind when considering, for instance, phase transformations occurring in ordering processes. Similar considerations may be made with reference to the other simple close-packed structure, that is to the hexagonal Mg type structure. In this case two basic derived structures can be considered the NiAs type with occupied octahedral holes and the wurtzite (ZnS) type with one set of occupied tetrahedral holes.
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6.23. Struc:tural type: cP4-AuCu3 Cubic, space group P d m , No. 221; Atomic positions: 1 Au in a) O,O,O 3 in c) 0"; 101.110 7292 29 $29 2127 Coordination formula: 333 [ A ~ 6 1 [ C ~ , l , , Layer stacking symbols: Triangular, kagcimC (T,K) nets: Aut C: A u k Cu;, Au;~ Cup Square ( s ) nets: AU; c u i CU,, For the prototype itself, AuCu,, a = 374.8 pm. (See also sec. 3,5.5. for a detailed description of this structure.) This structure can be considered a derivative structure (ordered substitution) of the cF4-Cu type. A discussion of the characteristics of a number of ordered layer (super)structures involving a XY, stoichiometry has been reported by MASSALSKI [1989]. Sequences of layer structures (among which those corresponding to the cP4-AuCu3, hP16-TiNi3, hP24-VCo3, hR:36-BaPb3 types) as observed in V (or Ti) alloys with Fe, Co, Ni, Cu are described. The relative stabilities of the different stacking sequences have been analyzed in terms of a few parameters which characterize the interactions between various layers. 6.2.4. Structural types: tP2-AuCu (I) and oI4O-AuCu(II) t P 2 d u C u ( l ) is tetragonal, space group P4/mmm, No. 123; Atomic positions: 1 Au in a) O,O,O; 1 Cu in d) +,$,$; For the prototype itself, AuCu(I), a = 280.4 pm, c = 367.3 pm, c/a = 1.310. The unit cell could be considered either as a distorted CsCl type cell greatly elongated in the c direction or, better, as a deformed (and orderly substituted) Cu type cell. This is apparent from fig. 20 where the tp2 unit cell and two tP4 supercells having a' = a@ = 396.61pm, c' = c = 367.3 pm are also shown. The larger cell is similar to a Cu type cell, slightly compressed (c'/a' =0.926) and in which the atoms placed in the center of the sidefaces have been orderly substituted. The coordinates in the tP4 super(pseudo)celll are: Au in O,O,O,and i,$,O; Cu in f,O$ and 0912 $2I . and the correspoinding square nets stacking sequence is AuA Au: Cu f.' The long period superstructure of AuCu(I), discussed in sec. 4.2., resulting in the antiphase-domain structure of AuCu(II) is shown in fig. 20c. 9
6.2.5. Structural type: tP4-Ti3Cu Tetragonal, space group P4/mmm, No. 123; References: p . 363.
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Atomic positions: 1 Cu in a) O,O,O 1 Ti in c) ;,;,o; 2T i in e) 011. '01. 92.21 2, 12, Coordination formula: 323 [ C k J [Ti8/8],u4 Layer stacking symbols: Square (S) nets: Cu: Ti: Ti For the prototype itself, Ti3Cu, a = 415.8 pm, c = 359.4 pm, c/a= 0.864. This structure can be described as a tetragonal distortion of the AuCu, type structure. It may also be considered a variant of the previously described AuCu(1) type (compare with its tP4 pseudocell).
.,:
6.2.6. Structural types: hP2-Mg, hP4-La and hR9-Sm hP2-Mg type. Hexagonal, Space group P6Jmmc, No. 194. Atomic positions: 1 2 1 2 1 3 2 Mg in c) T,T,T; T,T,T; Coordination formula: 2 [Mg,+,,g)] and ideally: 2 [Mg,,,,]. For the prototype itself, Mg, a = 320.89 pm, c = 521.01 pm, c/a= 1.624. Normalized interatomic distances and numbers of equidistant nei hbours are presented in fig. 26a for an "ideal" hexagonal close-packed structure (c/a = 813 = = 1.633), which corresponds to 12 nearest neighbours at the same distance, and, in fig. 26b, for a slightly distorted cells. The atoms are arranged in close-packed layers stacked in the sequence ABAB... (or BCBC... see sec. 3.5.2.). The corresponding layer symbol (triangular nets) is Mg:2, Mg&. Several metals have been reported with this type of structure, such as: aBe, Cd, ECO, aDy, Er, Ho, Lu, Mg, Os, Re, Ru, Tc, aY,Zn, etc. Several binary (and complex) phases have also been described with this type of structure. These are generally solid solution phases with a random distribution of the different atomic species in the two equivalent positions. Other stacking vur-iunts of closepacked structures are the La type and Sm type structures. Characteristic features of these types are presented here below.
$-
hP4-La type. Hexagonal, Space group PGJmmc, No. 194. Atomic positions: 2 La in a) O,O,O; O,O,+; 2 La in c) i2.L213,3349 373*49 For the prototype itself, aLa, a = 377.0 pm, c = 1215.9 pm, c/a = 3.225. Layer stacking symbols: ~ r i a n g ~ l(T) a r nets: La: La,B, La& ~ a & .
hR9-Sm type. Rhombohedral, space group H m , No. 166. Atomic positions: 3 Sm in a) o,o,o. 211. rzz. 333937 393939 3
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Structure of intermetallic compounds and phases
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The La and Sm type structures belong to the same homeotect type set as Mg and Cu (see sec. 4.3.). All these close-packed element structures are stacking variants of identical slab types (monoatomic triangular nets). 6.2.7. Structural type: hPS-Ni,Sn Hexagonal, Space group P6Jmmc, No. 194. Atomic positions: 2 Sn in c): $,+,+; $+,;:, 6 Ni in h): x,2x,i; -2x,-x$; x,-x,$; -x,-2x$; 2x,x,$; -x,x,$. For the prototype itself a=527.5 pm, c=423.4 pm, c/a=0.802 and x=O.833. (A projection of the cell is shown in fig. 28 and compared with that of the hP2-Mg type). The layer stacking symbol (triangular and kagom6 nets) is: B Sn0.2, Nit25 G 7 5 NilL5. (which may be compared with the symbol Mgz2, Mg& of the Mg type). This type is a superstructure of the closed packed (hP2-Mg) hexagonal structure in the same way as the AuCu, type is of the close-packed cubic (cF4-Cu) structure. It can, therefore, be considered a stacked polytype of the AuCu, type. Several phases belong to this type, for instance, Ti,Al, Fe3Ga, Fe,Ge, Fe,Sn, ZrNi,, ThAl,, YAI,, etc:. In the specific case of the rare earth trialuminides REAl,, the Ni,Sn type structure has been observed for LaAl, to GdA1, (and YAl,). For ErA1, to YbAl, and ScA1, the AuCu, type structure is formed. For the intermediate REAl,, intermediate stacking variants of similar layers have been described and their relative stabilities discussed (VAN VUCHT and BUSCHOW 1119651). In fig. 28b, the oP8-p TiCu, type structure is also shown. The close relationship between the two structures may be noticed.
Struc~tudtype: hP6-CaCb As another example of structures in which more complex stacking sequences can be observed we may mention here the hP6-CaCu5 type structure, which is the reference type for a family of structures in which 36 nets (and 63) are alternatively stacked with 3636 (kagomk) nets of atoms. The hP6-CaCu5 structure is hexagonal, space group P6/mmm, No. 191, with: 1 Ca in a) O,O,Ol, 2 Cu in c) +,$,O; +,$,O; 3 Cu in g>$,O,$; O,;,;; + ,;+ ,; For the prototype, a=509.2 pm, c=408.6 pm, c/a=0.802. The layer stacking symbol, triangular (TA,B,C), hexagonal (H: a,b,c) and kagomC ( K a,p,r) nets is: Cat Cu: Cu;, (see fig. 29). 6.2.8.
References: p . 363.
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Riccardo Ferm and Adriana Saccone
0 Sn
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1: 2=1/4
0 Ni
3: 2=3/4
b)
0 Ti 0
cu
o:z=o 2:
z=1/2
Fig. 28. hPL-Ni,Sn type unit cell. a) Projection of the hPS-Ni,Sn type unit cell on the x.y plane (the values of the coordinate z are indicated). A Mg-type subcell is represented by the dotted lines. b) Projection of the oPB+-TiCu, type cell. Compare the similar arrangements of the atoms in the two structures.
A large coordination is obtained in this structure: Ca is surrounded by 6 Cu + 12 Cu + 2 Ca at progressively higher distances and the Cu atoms have 12 neighbours (in a nonicosahedral coordination). Several phases belonging to this structure are known (alkali metal compounds such as KAu,, RbAu,, alkaline earth compounds such as BaAu,, BaPd,, BaPt,, CaPt,, C a n , , etc., rare earth alloys such as LaCo,, LaCu,, LaPd,, LaPt,, LaZn,, etc., The compounds as ThFe,, ThCo,, ThNi,, etc.). Ternary phases have been also described, both correspon-
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Structure of intetmetallic compounds and phases
i@\
I@\ @j 17 \P /
@
\
0 Cu at 2=1/2 @
Cu at z=O
@
Ca at z=o
@
Fig. 29. Projection of the hP6CaCu5 type unit cell on the x,y plane.
ding to the ordered derivative hP6-CeCo3B, type (1 Ce in a), 2 B in c) and 3 Co in g)) and to disordered solid solutions of a third component in a binary CaCu, type phase. According tcr PEARSON [1972] several structures may be described as derived from the CaCm, type (for instance, the t126-ThMn1, type; hR57-ThJn1, type; hP38-Th2Ni,, type;etc.). As for the lbuilding principles of the CaCu, type some analogies with the Laves phases (see sec. 6.6.4) may be noticed. Cobalt-based rare earth alloys such as SmCo, (hP6-CaCus type) are important materials for permanent magnets. A short review on the properties of alloys for permanent magnetic materials has been reported by RAGHAVANand ANTIA[1994]. Complex (especially iron) alloys have been mentioned starting from the Alnico (Fe-Al-Ni-Cs) alloys introduced in the thirties followed by ferrites and Co-based rare earth alloys (such as SmCo,) and then by Sm,(Co,Fe,Cu),, and Nc&Fe,,B (tP68) with a progressive decreasing of volume and weight of magnets per unit energy product.
6.3. Tetrahedral structures This section is mainly dedicated to the presentation of a few typical so-called tetrahedral structures. For the simplest ones, normalized interatomic distances and numbers of equidistant neighbows are shown in fig. 30. The graphite structure will also be described.
6.3.1. cF8-C (diamond) and tI4-/3Sn structural types cF&C (diamond) type. Face-centered cubic, space group F&m, No. 227. Atomic positions:
Referenes: p. 363.
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Riccardo Ferro and Adriana Saccone
N
“1
L4
1
a)
C (diamond)-type
b)
ZnS (sphalerite)-type
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16-
0-
8 C in a) O,O,O; 0,2929 11.1 0 1. 1 1 0- 11 1. 1 3 3. 3 13. 3 31. 2 r 92, 2929 4,4947 4*4?4?4’4949 4,474’ (This group of atomic positions corresponds to the so-called invariant lattice complex D; see sec. 3.1.). The coordination formula is 2 [C,,,] The layer stacking symbols are: Triangular (T) nets: C t C& C:3 C:,, C& C,:,, Square (s)nets: C: C: cP, c ; ~ c;~ For the prototype itself, C diamond, a = 356.69 pm. The diamond structure is a 3-dimensional adamantine network in which every atom is surrounded tetrahedrally by four neighbours. The 8 atoms in the unit cell may be considered as forming two interpenetrating face centered cubic networks. If the two networks are occupied by different atoms we obtain the derivative cF8-ZnS (sphalerite) type structure. As a further derivative structure, we may mention the t116-FeCuS2 type structure (See fig. 31). These are all examples of a family of “tetrahedral” structures which have been described by ParthC and will be briefly presented in sec. 7.2.1. Si, Ge and aSn have the diamond-type structure. The tI4-pSn structure (a = 583.2 7
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Structure of intermetallic compounds and phases
pm,c=318.2 pnn) (4 Sn in a) O,O,O; O&,; i,;,;;$,O,;; space group 14,/amd, No. 141) can be considered a very much distorted diamond type structure. Each Sn has 4 close neighbours, 2 more at a slightly larger (and 4 other at a considerably larger) distance. The PSn unit cell is reported in fig. 32.
6.3.2. Structural types: cF8-ZnS sphalerite and hP4-ZnO (ZnS wurtzite) cFS-ZnS sphalerite Face-centered cubic, space group F43m, No. 216. Atomic positions:
a)
as 0 Zn
ea cu 0 Fe O S Fig. 3 1. a) cF8-ZnS sphalerite and b) t116-FeCuS2 (chalcopyrite) type structure?. References: p . 363.
286
Riccardo Ferro and Adriana Saccone
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Fig. 32. tI4+Sn type structure.
4 Zn in a) O,O,O; O,$,$;
i,o,$;
4 s in c) 11.1.. r3.33.L. 494943 4,474, 1.33. 4,4943 3494343 In terms of a combination of invariant lattice complexes (see sec. 3.1) we may therefore describe the sphalerite structure as ZnS: F+F”. Coordination formulae: 3 [ZnS,] (ionic or covalent description) 333 [Zn,,,J[S,,,,]4,4 (metallic description) For the prototype itself, ZnS sphalerite, a=541.1 pm.
Structural type hP4-ZnO or ZnS wurtzite hexagonal, P6,mc, No. 186. Atomic positions: 2 Zn in b (1) 3,#,z; #,i,i -Iz; (z = zl) 2 0 or 2 S in b (2) +,$,z; $,$,i+z; (z=z;?) Coordination formula: 3 [Zn04,J For the prototypes themselves, ZnO: a = 325.0 pm, c =520.7 pm, c/a = 1.602; ZnS (wurtzite): a = 382.3 pm, c = 626.1 pm, c/a = 1.638. The atomic positions correspond, for both types of atoms, to similar coordinate groups (to the same Wyckoff positions) with different values of the z parameter. For ZnO z,,=O, z,=O.382, and for ZnS zzn=O, zS= 0.371.
63.3. General remarks on “tetrahedralstructures’’ and polytypes. tI16-FeCuS2, hP4-C lonsdaleite, oPlfGBeSiN2 types and polytypes Compounds, isostructural with the cubic cF8-ZnS sphalerite include AgSe, Alp, AlAs, AlSb, AsB, AsGa, AsIn, BeS, BeSe, BeTe, BePo, CdSe, CdTe, CdPo, HgS, HgSe, HgTe, etc. (possibly in one of their modifications). The sphalerite structure can be described as a derivative structure of the diamond type structure. Alternatively we may describe the same structure as a derivative of the cubic close-packed structure (cF4-Cu type) in which a set of tetrahedral holes has been filledin. (This alternative description would be especially convenient, when the atomic diameter ratio of the two species is close to 0.225: see the comments reported in sec. 6.2.2.).
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Stmeturn of intermetallic compounds and phases
In a similar way the closely related hP4-ZnO structure can be considered as a derivative of the: hexagonal close-packed structure (hP2-Mg type) in which, too, a set of tetrahedral holes has been filled-in. Compounds, isostructural with ZnO include some forms of AgI, BeO, CdS, CdSe, CIS (X = H, C1, Br, I), MnX (X = S, Se, Te), MeN (Me = Al, Ga, In, Nb), ZnX (X = 0, S, Se, Te). In order to have around each atom in this hexagonal structure, four exactly equidistant neighbourintg atoms, the axial ratio should have the ideal value that is = 1.633. The experimental values range from 1.59 to 1.66. The ideal value of one of the parameters (being fixed at zero the other one by conventionally shifting the origin of the cell) is z=3/8=0.3750. The C diamond, sphalerite and wurtzite type structures are well-known examples of the “normal tetrahedral structures” (see sec. 7.2.1.). Several superstructures and defect superstructures based on sphalerite and on wurtzite have been described. The tI16-FeCuS2 (chalcopyrite) type structure (tetragonal, a = 525 pm, c = 1032 pm, c/a = 1.966) (see fig. 31b), for instance, is a superstructure of sphalerite in which the two metals adopt ordered positions. The superstructure cell corresponds to two sphalerite cells stacked in the c-direction. The c/2a d o is nearly 1. As another example we may mention the oPl6-BeSiN, type structure which similarly corresponds to the wurtzite type structure. The degenerate structures of sphalerite and wurtzite (when, for instance, both Zn and S are replaced by C) corresponds to the previously described cF8-diamond type structure and, respectively, to the hP4-hexagonal diamond or lonsdaleite which is very rare compared with the cubic, more common, gem diamond. The unit cell dimensions of lonsdaleite (prepared at 13 GPa and lO0OT) are a=252 pm, c=412 pm, c/a= 1.635. (Compare with ZnS wurtzite). While discussing the sphalerite and wurtzite type structures we have also to remember that they belong to a homeotect structure type set. (See sec. 4.3.) The layer stacking sequence s mbols (triangular nets) of the two structures are: Sphalerite: Znt S;4 Zn& S;,, Zn,!I Slf)12
m,
wurtzite: zn; s : ~ zn,C, ~ s,CS7. In the first case we have (along the direction of the diagonal of the cubic cell) a sequence ABC of identical “unit slabs” (“minimal sandwiches”) each composed of two superimposed triangular nets of Zn and S atoms. The “thickness” of the slabs, between the Zn and S atom nets is 0.25 of the lattice period along the superimposition direction (cubic cell diagonal: a 6 ) . It is (0.25 *541) pm = 234 pm. In the WUrtzite structure we have a sequence BC of slabs formed by sandwiches of the same triangular nets of Zn and S atoms (their thickness is = 0.37 * c=(O.37 * 626.1) pm=232 pm). With reference to the aforementioned structural unit slab the Jagodzinski-Wyckoff symbol of the two structures will be: ZnS sphalerite: c; ZnS wurtzite: h. In the same (eqniatomic tetrahedral structure type) homeotect set many more structures occur often with very long stacking periods. Several other polytypes of ZnS itself have
6
References: p. 363.
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Riccardo Ferro and Adriana Saccone
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been identified and characterized. The largest number of polytypic forms and the largest number of layers in regular sequence have, however, been found for silicon monocarbide. A cubic form of Sic is known and many tenths of rhombohedral and hexagonal polytypes. (In commercial S i c a six-layer structure, hcc, is the most abundant). All have = 308 pm, the c,of their hexagonal (or equivalent hexagonal) cells are all the same kx multiples of = 252 pm and range from 505 pm to more than 150000 pm (up to more than 600 Si-C slabs in a regular sequence). 6.3.4. An important non-tetrahedral C structure. The hP4-C graphite In comparison with the previously described tetrahedral structures of C we may mention here the very different structure that carbon adopts in graphite (see fig. 33). hP4-C graphite. Hexagonal, space group P6/mmc, No. 194. Atomic positions: 2 C in b) O,O,f; O,O,i; 2 C in c-121. 212. 3,3347 3.334, Coordination formula: 2 [C3J The lattice parameters are a = 246.4 pm and c = 67 1.1 pm; c/a = 2.724. Different varieties, however, of graphite may be considered: the actual structure, in fact, and unit cell dimensions and layer stacking can vary depending on the preparation conditions, degree of crystallinity, disorientation of layers, etc. In crystalline hP&graphite, sheets of six-membered rings are situated so that the atoms in alternate layers lie over one other, and the second layer is displaced according to the stacking symbol Ck4C:,4 (compare with fig. 9). Whereas in diamond the bond length is 154 pm, in graphite the C-C minimum bond lenght is 142 pm in the sheets and 335 pm between sheets. This may be related to the highly anisotropic properties of this substance. (It may be said, for instance, that properties of graphite in the sheets are similar to those of a metal while perpendicularly are more like those of a semiconductor). In conclusion, notice also that in terms of combinations of invariant lattice complexes the positions of the atoms in the level f may be represented by $,$,f G and those in the level $ by &$,$ G (where G is the symbol of the “graphitic” net complex, here presented in non-standard settings by means of shifting vectors; see sections 3.1. and 3.5.2.). 6.4. cF8-NaC1, cF12-CaF,, and cF12-AgMgAs types In this section the NaCl type, CaF2type (and the related AgMgAs type) structures are described. In fig. 34 the normalized interatomic distances and the equidistant neighbours are shown for the NaCl and CaF2 structures. 6.4.1. cFS-NaC1 type structure and compounds Face-centered cubic, space group Fmgm, No. 225. Atomic positions: 4 Na in a) O,O,O; O,&,&; &O,f; f,$,O; 4 Cl in b) $,+,+; $,O,O; O,i,O; O,O,$;
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Structuure of iniermetallic compounds and phases
289
Fig. 33. Graphite sbucture. a) unit cell with the indication of the atoms at the levels z=a and (part of a second, superimposed cell is also shown). b) the hexagonal net formed at level z=a is shown (four adjacent cells are indicated).
Coordination formula: 333 [Na,u,2][C1,,,&,6 Layer: stacking symbols: Triangular (T) nets: Naf Cl:6 Na:3 Cl& Nai3 Cl& Square ( S ) nets: Nai Nai Cli Cl;2 Cli2Na;, For the prototype itself, NaC1, a = 564.0 pm. (A sketch of the NaCl unit cell is shown in fig. 18.) A large number of compounds belong to this structure type, besides several alkali metal halides, for instance, nearly all the (partially ionic covalent) 1:1 compounds formed by the rare earths and the actinides with N, P, As, Sb, Bi, S, Se, Te, Po, by the alkaline earths with 0, S, Se, Te, Po, etc. Notice that we may also describe this structure as a derivative of the cubic closeReferences:p. 363.
,
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Riccardo Fern and Adriana Saccone
290
N
. a) 16 8-
+
0-
,
24
NaCl-type
I*
+
bf CaF,-type
N
16
*
I
+
8
0
“1
c) CaF,-type
t 1
1.2
1.4
1.6
1.8
+
*
*
,i,,
2
d /d-
2
d/d,,
Fig. 34. Distances and coordinations in the cF8-NaC1 and cFlZCaF, types structures. a) XY compounds of cFGNaC1 type structure: (*) X-X (or Y-Y) ~00rdination. (+) X-Y (or Y-X) coordination. b) cF12-CaF2 type structure. Coordination around Ca: (*) CaCa; (+) Ca-F; c) CaF, type structure. Coordination around F (*) F-F; (+): F-Ca.
packed structure (cF4-Cu type), in which the octahedral holes have been filled in. This description, however, may be specially convenient when the atomic diameter ratio between the two elements is close to the theoretical value 0.414. In this case the small
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Structure of intermetallic compounds and phases
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spheres will fit iexactly into the octahedral holes of the close-packed arrangement of the metal atoms. (!See sec. 6.2.2.). This could be the case of a number of “interstitial compounds”. Compounds of the transition metals having relatively large atomic radii with non metals having small radii (H, B, C, N, possibly 0) may be simple examples of this type. (General properties of these compounds were discussed by HAGG [1931]). Examples of typical phases belonging to this group may be a number of “mono” carbides, nitrides, etc. The NaCl type structure is shown by several monocarbides MeC (or more generally MeC,,) such as Tic,, (homogeneous in the composition range c- 32-49 at% C), ZrC, (= 33-50 at% C), HfC,, (= 33-50 at% C) and ThC, (with a very large homogeneity range at high temperature). All the aforementioned monocarbides are stable from room temperature up 1.0 the melting points (which are among the highest known: = 3500°C for ZrC and = 4000°C for HfC). The carbides VC,, (37-48 at% C), NbC,, (40-50 at% C) are stable only at high temperature: at lower temperature, transformations associated with C-atom ordering have been reported, resulting in the formation of V,C7, V&, Nb& structures. WC,., is a NaCl type high temperature phase homogeneous between 37-39 at% C. At 50 at% C another structure is formed: the hP2-WC type. Among the NaC1 type mononitrides we may mention VN,,. At high temperature (up to the melting point = 2340°C) we have a large homogeneity field (= 33-50 at% N). The composition chalnges result from variation in the number of vacancies on sites in the N sublattice, with x being the fraction of sites randomly vacant. At lower temperature, in the composition range 43-46 at% N, an ordering of the N atoms has been observed, resulting in a tetragonal superstructure containing 32 V atoms and 26 N atoms in the unit cell. In the W-N system, a WN-,, NaCl type phase, has been observed in the composition range = 33-50(?) at% N; hP2-WC type structure, however, has been described at 50 at% N. As a final example, we may mention the NaCl type phases formed in the V-O and Ti-O systems. The (VO,$) phase is homogeneous in the composition range 42 to 57 at% 0.Lattice parameter determination in combination with density measurements evidenced that, in the structure, vacancies occur in both V and 0 sub-lattices through the entire range of composition. At the stoichiometric composition VO there are = 15% of sites vacant in each sublattice. In the T i 4 system, yTi0 (high temperature form, homogeneous in the composition range 35 to 55 at% 0) has the NaCl type structure. (Other forms of the monooxide PTiO, aTiO, pTi,-xC), aTi,-,O have ordered structures based on yTi0.) In the structure there are atoms missing from some of the sites. According to what is summarized by HYDEand ANDIZRSSON [1989], in TiOO.@= 36% of the oxigens are missing, in TiO,,, (which, of course, can be represented also with the stoichiometry Ti,,Q) = 20% of the Ti atoms are missing and in = Ti0 both kinds of atoms are missing (= 15% of each): see fig. 35. 6.4.2. cF12-CaF2 type and antitype structures and compounds Face-centered cubic, space group Fm3m, No. 225. Atomic position:s: References: p. 363.
292
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Riccardo Ferm and Adriana Saccone
42 OA Q6 48
TiO,
1.0
l2
1.4
1.6
1.8
(XI
Fig. 35. Experimental densities of titanium oxides (continuous lines). The upper dotted line gives the values computed for a 100%occupancy of the cation sites in the NaCl structure type (from HYDEand ANDERSON [1989]).
4 Ca in a) O,O,O; (I,; $,, O,; ;;;;,;,(I; 1 3 I 3 1 3 % r l3 3 1 3 1 2 8 F i n c) $11 1 1 2 .!.A3 ,494; 4,494; 4,4$ a,+$ 4.434; 4,494; 4,494; 4,434; Coordination formula: 323 [Ca,2,,2][F6/618/4 Layer stacking symbols: Triangular (T) nets: c a t G I 2 F i 4 Ca,Bn G I 2 F7B112 caE3 F i 4 F,,C,12 Square (S) nets: Cai Ca: Fi4Fi4Ca& Fi4Fi4 For the prototype itself, CaF,, a = 546.3 pm. As pointed out in the description of the cubic close-packed structure (cF4-c~type) this structure may be described (especially for certain values of the atomic diameter ratio) as a derivative of the Cu type structure in which two sets of tetrahedralholes have been filled in. A ternary ordered derivative variant of this structure is the cF12-AgMgAs type. Several (more or less ionic) compounds such as CeO,, UO,, Tho2etc. , belong to this structural type. Several M%X compounds, with Me = Li, Na,K,X =0, S, Se, Te), also belong to this type. In this case, however, the cation and anion positions are exchanged, Me in c) and X in a) and these compounds are sometimes referred to a CuF2-untitype. Typical (more
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Structure of intermetallic compounds and phases
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metallic) phases having this structure are also, for instance, AuAl,, PtAl,, Mg,Pb, Mg,Sn, Mg,Ge, Mg,Si.
6.4.3. Struc:tud type: cF12-AgMgAs Face-centered cubic, space group F43m, No. 216. Atomic positions: 4 As in a) O,O,O;O,i,i; i,O,i; i,),O; 4 Ag in c) GI f 1, .~ .L4,494, 3 3 . 312. 231. 4 9 4 9 4 9 494Y4Y 4 Mg in d) 222. /Ira I L L - ? _ I / 4’4743 49494, 494749 434’43 Layer stacking symbols: Triangular (T) nets: As; MglL A g i k q 3 M$, fg;IzAS& Mg$4Ag42 square nets: Ag114 MgG4 Mg4: Ag314 For the prototype itself, AgMgAs, a=624 pm. In systematic investigations of MeTX ternary alloys (Me = Th, U, rare earth metals, etc., T=transition metal, X element from the V, IV main groups) several tens of phases pertaining to this structure type have been identified. For the same group of alloys, however, other structural types are also frequently found. The hP6-CaIn2 type and its derivative types often represent a stable alternative. The relative stabilities of the two structures (especially as a function of the atomic dimensions of the metals involved) have been discussed, for instance, by DWIGHT[1974], MARAZZA etal. [1980, 19881, WENSKI and MEWS [19&6].
4
6.5. hP4-NiAs, cP3-CdI,, hP6-Ni21n, oPlZ-Co,Si, oP12-TiNiSi types; hP2-WC, hP3-AI&, hP6-CaIn,, hP9-Fe2P types, tI8-NbAs, tI8-AgTlTe, and tI10-BaAl, (ThCr,Si,) types, t112-ThSi2 and tI12-LaPtSi types In this section a number of important interrelated structures are presented. A first group is represented by the cP3-Cd12, hP4-NiAs and hP6-Ni21n types. Some comments on the interrelations between these structures have been reported in sec. 4.1. A further comparison may also be made by considering their characteristic triangular net stacking sequences: hP3-Cd1, Cd; Iz4IZ4 hP4-NiAs Nit AsZ4N i h AsZ4 hP6-Ni2In Nit NiE4 N i i Ni:4 Ini4 YVe see, on passing from CdI, to the NiAs type the insertion of a new layer at level 3 and, from NiAs to NiIn,, the ordered addition of atoms at levels f and i. In this section, moreover, the typical non-metal atom frameworks characteristic of the AB,, and derivative structures (“graphitic” layers) and of LuThSi,, and derivative structures (“hinged”, tridimensional framework) will also be presented, compared and discussedl. The groups of more or less strictly interrelated structures which will be considered in this section are those corresponding to the hP2-WC, hP3-A1B2, hP6-CaIn2 and hP%Fe,P References: p . 363.
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types, and, respectively, to the tI8-NbAs, tI8-AgT1Te2, tIlO-BaAl, (and tIlO-ThCr,Si,) types and to the tI12-cr-ThSi2and tI12-LaPtSi types. 6.5.1. Structural type: hP4-NiAs Hexagonal, space group P6Jmmc, N.194, Atomic positions: 2 Ni in a) O,O,O;O,O,$; 2 As in c) $,$,$; $8,;; Coordination formula: 1,3 [Ni,,]As6,, For the prototype itself, a = 361.9 pm, c = 504 pm, c/a = 1.393. According to HYDEand ANDERSON[1989], the data reported have to be considered as correspondiag to an average slightly idealized structure, corresponding for several compounds to the form which is stable at high temperature. At room temperature, in the real structure, there are very small displacements of both Ni and As from their ideal average positions. The structure should, therefore, be better described by: 2 Ni in a) O,O,z; O,O,h + z; (z = 0) 2 As in b) $,%,z;+,$,$ + z; (z = $) in the space group P6,mc, No. 186. The small (probably correlated) displacements of the atoms produce several sorts of modulated structures (see sec. 4.4.). 6.5.2. Structural type: hP3-Cd12 Hexagonal, space group P3m1, No. 164. Atomic positions: 1 Cd in a) O,O,O 2 I in d) f,$,z;$,$,-z; Coordination formula: 2 [CdI,,] For the prototype itself, CdI,, a = 424.4 pm, c = 685.9 pm, c/a= 1.616 and z= 0.249. Typical phases pertaining to this structural type are COT%,HfS,, PtS,, etc. and also Ti,O (which, owing to the exchange in the unit cell of the metaVnon-metal positions may be considered to be a representative of the CdZ2-antitype). 6.5.3. Structural type: hP6-NiJn Hexagonal, space group P~~/IIIIIIC,No. 194. Atomic positions: 2 Ni in a) O,O,O; O,O,); 2 In in c) 1 2 1 3,3,T; 2 1 3 2 Ni in d) $,23 924 9- 211. 393949 Coordination formula: 2 [InNi6/6Ni5,5] For the prototype itself, a=419 pm, c=512 pm, c/a= 1.222. Typical phases assigned to this structural type are, for instance: Zr2Al, Co,Ge, La21n, Mn,Sn, Ti& and several ternary phases such as: BaAgAs, CaCuAs, CoFeSn, LaCuSi,
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Structure of intermetallic compounds and phases
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VFeSb, KZnSb, etc. A distorted -#aria&of the InNi, type structure is the oP12-orthorhombic structure of the Co,Si, (or PbC1.J type: 2 [SiCo,,Co,,,], that is total coordination 10 of Co around Si with f +$ = $' = 2 Co atoms for each Si atom). A ternary derivative of this type is the oPl2-TiNiSi type (prototype of the so-called E phases).
6.5.4. Structural types: oP12-Co2Si (PbCl;) and oPl.2-TiNiSi Orthorhombic, space group Pnma, N.62. In these structural types the atoms are distributed in three groups of positions corresponding (obviously with different values of the x and z free parameters) to the same type of Wyckoff positions (Wyckoff position c). Atomic positions: in Co2Si in TiNiSi c (1)) x,+,z; +-x.,:,++z; 4 co 4 Ti r Lz. -x 3 -Z' I+x 2 ,492 949
9
9
c (2)) x,;,z; +-x.,$,++z; -x 3 -Z' L+x r LZ. 2 ,492 , 94,
4 co
4 Ni
Y
c (3)) x,f,z; &-x.,$,++z; 4 Si 4 Si L+x r LZ. -x ~34-Z' 9 2 9492 For the prototypes: Co2Si: a=491.8 pm, b=373.8 pm, c=710.9 pm, a/c=O.692; x,(,,=O.O38, z,(,,=O.782; x,(~) = 0.174, z,(~,= 0.438; x,(~,= 0.702, z ~ (=~0.389. , TiNiSi: a = 614.84 pm, b = 366.98 pm, c = 701.73 pm, a/c = 0.876; x,(,) = 0.0212, z,(,,=0.8197; ~,(,,=0.1420,~,(,,=0.4391;~,(,,=0.7651,~,(,,=0.3771. Co,Si is the prototype of a group of phases (also called PbC1, type) which can be subdivided into two sets according to the value of the axial ratio a/c which is in the range from 0.6'7 to 0.73 for one set (for instance, Co,Si, Pd,Al, Rh,Ge, Pd,Sn, Rh,Sn, ek.) and in the range from 0.83 to 0.88 for the other set (for instance PbCl,, BaH,(h), CqSi, Ca2Pb, GdSe,, ThS,, TiNiSi, etc.) (PEARSON[1972]). The ternary variant TiNiSi type is also called E-phase structure. Many ternary compounds belonging to a MeTX formula (Me=rare earth metal, Ti, Hf, V, etc., T=transition metal of the Mn, Fe, Pt groups, X=Si, Ge,Sn, P, etc.) have this structure. 7
9
4 - 5 5 Structural type: hP2-WC Hexagonal, space group P6m2, No. 187. Atomic positions: 1 W in a) O,O,Cl,; 1 C in d) $,$,+; For the prototype itself, a = 290.6 pm, c = 283.7 pm, c/a = 0.976. This structure type with the axial ratio c/a close to 1 is an example of the Hagg interstitial phases formed when the ratio between non-metal and metal radii is less than about 0.59. The structure can be described as a tridimensional array of trigonal prism of Watoms (contiguous on all the faces). Alternate trigonal prisms are centered by C-atoms. References: p . 363.
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Compounds belonging to this structure type, for instance, are: IrB, OsB, RUB,MoC, WC (compare, however, with the NaCl type phase), NbN, WN, MOP, etc. 6.5.6. Structural types: hP3-AIB2 and hP3-BaPtSb; h B w , Cr-Ti phase
Structural type: hP3-A1B2 Hexagonal, space group P6/mmm, No. 191. Atomic positions: 1 A1 in a) O,O,O; 2 B in d) L3,3z923 L. zrl. 3,3729 Coordination formula: 323 [Al&j[B3,3]1U6 For the prototype itself, a = 300.5 pm, c = 325.7 pm, c/a= 1.084. The structure can be considered a filled-up WC structure type. The B-atoms form a hexagonal net and center all the AI trigonal prisms. The arrangement of the boron atoms in their layers is the same as that in graphite (see fig. 9 and sec. 6.3.4). (See sec. 6.5.10. for a comparison between the planar graphitic net and similar threedimensional networks). Several B, Si, Ge, Ga, etc., binary and ternary compounds have been described as pertaining to this structure or, possibly, to its variants (many of them deficient in the second component and corresponding to different stoichiometries in the 1:2 to 1:1.5 range). The axial ratio of phases with this structure varies between very wide limits. The relationships between axial ratio, atomic radii ratio of the elements involved and the role of the different bonds have been discussed by PEARSON [1972]. In the specific case of AlB, (c/a= 1.08) the important role of the graphite-like net of B-atoms in determining the phase stability has been evidenced. A disordered, AlB, type, ternary phase (= Ce,NiSi,) has been presented in table 3. A variant (ordered derivative structure) of the hP3-A1B2 type, previously discussed in sec. 4.1 and presented in fig. 17, is the hP3-BaPtSb type, hexagonal, space group P6m2, No. 187. Another compound with this structure is, for instance, ThAuSi. The atomic positions are the following: 1 Ba (or Th)in a): O,O,O; 1 Pt (or Au) in d) 1 Sb (or Si) in f): $$,.;, The layer-stacking sequence symbols of the three previously mentioned structures are: WC type, tiangular (T) nets; w; co”,; AlB, type, triangular, hexagonal (T, H) nets: Alt B;,; ThAuSi type, triangular, hexzgonal (T,H) nets: Tht Au,”, Si&.
;,$A;
Another ordered derivative structure of the AlB, type is the Er,RhSi3 type, hexagonal, space group P62c, No. 190 with the following atomic positions:
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Structure of intermetallic compounds and phases
297
2 Er in b): O,O,+;O,O$; 4 in 0: 1 2 z . 1.2L-Z. 239391 z*2L.L+z. 3,392 3,392 6 Er in b): x,y,i; -y,x-y,i; y-x,-x$; y,x,:; x-y,-y,$; -x,y-x,$ 12 Si in i): x,y,z; -y,x-y,z; y-x,-x,z; x,y,&z; -y,x-y,hz; y-x,-x,&z; y,x,--z; x-y,-y,-z; -x,y-x,-z; y,x,;+z; x-y,-y,;+z; -x,y-x,;+z. (with z,=O; x,,=0.481; y,=0.019; xsi=0.167; ysi=0.333; zsi =O). The different ordering relationships between these structures have been discussed in sec. 4.1. (see also fig. 17). Finally, while considering the structural characteristics of the AlB, type phases, we may mention that boron-centered triangular metal prisms are the dominating structural building elements in the crystal structures of simple and complex metal borides. Building blocks of centered triangular prisms as base units for classification of these substances have been considered by ROGL [1985, 19911 in a systematic presentation of the crystal chemistry of borides. We may mention here, also as an example of “modular” description, that several Structures may be described in terms of cyclic translations about a 6, axis of blocks of AlB, type columns: see fig. 36. 3939
i
?
9
9
Structural type: hP3-o, Cr-Ti The w-phase, a iabiquitous metastable phase in Ti (or Zr or Hf)-transition metal systems, is approximately isotypic with AlB,. (The axial ratio of the unit cell, however, instead of being close to unity, is very much smaller and has a value of about 0.62.) The components are randomly arranged. One third of the atoms are distributed in a triangular net at z = 0 forming trigonal prisms. Two thirds of the atoms are placed near the centers of the prisms (slightly displaced alternately up and down) forming a rumpled 63 net at 2-3. (The space group is PSml.) 6.5.7. Structural type: hP6-CaIn2 Hexagonal, space group P4/rnmc, No. 194. Atomic positions: 2 Ca in b) O,O$; O,O,$; 4 In in f) 1393, 2 z’ 211 +z. 21. - z. 13 ,23 ,1F z ; $9392 3739 Layer stacking symbol: Triangular (T) nets: ca;4 b ; O In&o Ca44 I n & O For the prototype itself, a=489.5 pm, c = 775.0 pm, c/a = 1.583 and z = 0.455. This structure can be described as a distortion (a derivative form) of the AIB, type structure. Ca-atoms form trigonal prisms alternatively slightly off-centered up and down by In-atoms. In fig. 37 the normalized interatomic distances and the equidistant neighbours are shown for the NiAs and CaIn, structures. 9
7
9
hL
References: p . 363.
298
Riccardo Ferro and Adriana Saccone
Ch.4,$6
Fig. 36a,b,c. AIBz-type derivative structures generated by cyclic translation of blocks of AIBz-type columns. The projections of the unit cells (all having the same c value) on the x,y plane are shown. a) hP22X!e,Ni2Si, structure (a= 1211.2 pm, c=432.3 pm). b) hP40-=Ce,Ni2Si, structure (a= 1612.0 pm, c=430.9 pm). c) hP64-=Pr,,Ni,Si1, structure (a= 1988.1 pm, c=425.5 pm)
Ch. 4,56
299
Structure of intermetallic compounds and phases
0 G3
0
0
0
0
0
O O O
O
0 .
d)
.\,,.' 0
\ \
/
Fig. 36d. AlB,-type derivative structures generated by cyclic translation of blocks of AIB,-type columns. The projections of the unit cells (all having the same c value) on the x,y plane are shown. d) hP22-Ce6Ni,Si, structure (compare with a)): the arrangement of the building blocks around the z-axis (6, symmetry axis) is shown. Black circles represent the rare earth atoms (Ce or Pr), open circles Si (and Si+Ni); small circles are atoms at level 4, large circles at level 8. Double circles (at cell origin) represent Ni atoms at level 0 and at level ).
6.5.8. Structural type: hP9-Fe2P Hexagonal, space group P&m, No. 189. Atomic positions: 1 P iii b) O,O,i . 1 0. 2 p c>$923 , 0. 2333, 3 Fe in f) x,O,O; O,x,O; - x,- x,O; 3 Fe in g) x,O,i; O,X,~;- x,- x,$; For the prototype itself, a = 586.5 pm, c = 345.6 pm, c/a = 0.589 and x (f) = 0.256 and x (g) = 0.594. In the F%P type structure there are 4 different groups of equipoints. The distribution of P and Fe atoms in different groups of positions is reported. A number of isostructural binary compounds are known. To the same structure, however, ternary (or even more complex) phases may be related if different atomic species are distributed in the different sites. This structure can be considered as an example of more complex structures built up by linked triangular prisms of Fe-atoms. Several ordered ternary phases have structures related to the Fe,P type. 9
9
6.5.9. Structural types: tI8-NbAs, t18-AgT1Te2 and tI10-BaAl, (ThCr,Si2)
The three structural types tI8-NbAs, tI8-AgTlTe2 and tI10-BaAl, (with its ordered ternary variants such as the t110-ThCr2Si,) belong to a group of interrelated structures. References: p. 363.
300
Ch. 4, $ 6
Riccardo Ferro and Adriana Saccone
N
2 4 [ a) NiAs-type
I+
I*
1
2
d/dh
N 2 L [ b) NiAs-type
16.
8- +
1
* *
+
* 2
+
I+
d/dh
Fig. 37a,b. Distances and coordinations in the hP4-NiAs and hP6-C!aIn2types structures. a) hP4-NA type structure. Coordination around Ni: (+) Ni-As; (*) Ni-Ni. b) hP4-NiAs type structure. Coordination around As: (+) As-Ni; (*) As-As.
All these structures contain among their building parts layers of (metal atoms) triangular prisms with specific distributions of the (non-metal) atoms centering the prisms (PEARSON [1972]). The prisms are parallel to the basal planes of t.he tetragonal unit cells. Features of the hP2-WC type structure (characterized by an array of trigonal prisms alternatively centered by C-atoms) are, therefore, present in the aforementioned structures. (In the hP2-WC structure, of course, the prism axes are lying in the e-direction of the hexagonal cell.) Another convenient description of these group of structures may be in term of 44 net layer stacking. The corresponding square net symbols for the 8-layers stacks are the following ones: tIS-NbAs:
Ch. 4, $ 6
30 1
Structure of intermetallic compounds and phases
. c)
iv
CaIn,-type
16 .
N
"[ 8
d) Caln,-type
1
*
* 1
1.2
1.4
1.6
1.8
2
I*
d/d,
Fig. 37c,d. Distances and coordinations in the hP4-NiAs and hP6-CaIn2 types structures. c) hP6-CaIn2 type structure. Coordination around Ca: (+) Ca-In; (*) Ca-Ca; d) hP6-CaIn2 type structure. Coordination around In: (+) In*, (*): In-In.
Structure type: tl'8-NbAs Body-centered tetragonal, space group I4,md, No. 109. +z; $ $,;,i ,+ +z; 3,0,$ +z; 4 Nb in a(1): O,O,z; (I, 4 As in a(2): O,O,z; O,&$ + z; &,&,$ + z; f,O,+ + z; For the prototype itself a=345.2 pm, c=1168 pm, c/a=3.384, z(Nb) =0, z(As)=0.416. Structural type: tI8-Ag TlTe, Body-centered tetragonal, space group I4m2, No. 119. 2 TI in a): O,O,O,;;$;,; 2 A g i n c ) : O,2149 ~ ~2 ,- ? - O ~ . 4 Te in e): O,O,z; O,O,- z; i,;,; + z; &&&z; For the prototype itself, a = 392 pm, c = 1522 pm, c/a = 3.883 and z(Te) = 0.369. 949
Structural type: tIl&BaAl, ana' tI10-ThCrJi2 The ThCr,Si, is one of the ordered ternary variant of the BaAl, type, frequently found in several ternary compounds. The two structures may be described by the following occupation of the same atomic positions in the space group I4/mmm (No.139). References: p , 363.
For the prototypes themselves: BaAl,, a =453.9 pm, c = 1116 pm, c/a=2.459 , z=O.38 ThCr2Si2:a = 404.3 pm, c = 1057.7 pm, c/a = 2.616, z = 0.37, The unit cell is presented in fig. 38. Normalized interatomic distances and numbers of equidistant neighbours are reported in fig. 39 for the ternary ThCr2Si2type. Many ternary alloys MeT2X2(Me=Th, U, alkaline-earth, rare earth metal, etc., T=Mn, Cr, Pt family metal, X=element of the fifth, fourth and occasionally third main group) and CHABOT[1984], ROSSI have been systematically prepared and investigated (PARTHE et al. [ 19791). A few hundreds of them resulted in the ThCr2Si2(or other A14Ba derivatives) structure. The peculiar superconductivityand magnetic properties of these materials have been reported. The ThCr2Si2type structure, can be described as formed by T2X2 layers interspersed with Me layers. The bonding between Me and T2X2layers has been considered as largely ionic. In the T2X2layers T-X (covalent) and some T-T bonding have to be considered. A detailed discussion of this structure and of the bonding involved has been reported by HOFFMANN [1987]. In the specific case of the RET2X2phases (RE = rare earth metal) the data concerning
Th
0
Fig. 38. Unit cell of the t110-ThCr2Si2 type structure (a derivative structure of the t110-BaA14 type).
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Stmcture of intermetallic compounds and phases
303
r
16
-
a) ThCr2Si2-type
0
N
+
+o
8-
*
+ I/
.
0.
1
* ,
2 d/d,,
b) ThCr2Si2-type
16 N
+
8+
13-
16
1
*
o
*
o
* *
+
I
C)
ThCr2Si2-type
N i9
+
+
0
*
**
+
*
o *
0
Fig. 39. Distances and coordinations in the tIlO-ThCr,Si, type structure. a) Coordination around Th: (+) Th-Si; (*) Tn-m, (0) Th-Cr; b) Coordination around Cr: (c) Cr-Si; (*) Cr-Cr; (0) Cr-Th. c) Coordination around Si: (+) Si-Cr; (*) Si-Si; ( 0 ) Si-Th.
ten series (T=Mn, Fe, Co, Ni, Cu; X=Si, Ge) have been analysed by PEARSON 11985aI. It has been observed that the cell dimensions are generally controlled by RE-X contacts. In the case of M:n, however, the RE-Mn contact has to be assumed to control the cell dimensions (see sec. 7.2.5.). Magnetic phase transition in RET2X, phases have been described by SZYTULA [1992]. References: p. 363.
304
Riccardo Fern ana' Adriana Saccone
Ch. 4, 96
Structural distortions in some groups of RET,X, phases (REI\Sn,), leading to less et al. [19921. symmetric cells, have been reported by LATROCHE An interesting compound belonging to the RET,X, family is EuCo,P,. In a neutron diffraction investigation of this phase carried out by REEHurs et al. [19921 the positional (nuclear) and the magnetic structures were determined. The ordering of the magnetic moments of the Eu-atoms and the relation (commensurability)between this ordering and that of the atomic positions were studied (see sec. 4.4.).
6.5.10. Structural types: tI12-aThSi2 and tI12-LaPtSi The aThSi, type structure, and its lattice-equivalent ternary LaPtSi type derivative can be considered, filled up t18-NbAs type derivative. These structures can be described in terms of layers of (metal atoms) triangular prisms parallel to the basal planes of the tetragonal cells, the prism axes in one layer being rotated 90" relative to those of the layers above and below. In the NbAs type structure the As atoms only center alternate Nb prisms. In the CwThSi, type structure all the Th-prisms are centered by Si instead of only half of them (PEARSON [1972]). We may also compare the three structures in terms of 44net layer stacking (along the c-direction of the tetragonal cells): See also fig. 40. tI8-NbAs: %?25 Nb& As,:6,NbC?75 As& t112-cuThSi2:
mi
~
%408
%217
n b 3 2 5s3:i3
%!42
nb45
si,k7
n b 2 7 S s&3
tI12-LaPtSi:
Ld-408 sb217 L%?25b 2 3 3 si142 w 4 5 b : S 8 s6:i7 L%275 &8!3 si19Z Structural type: tIl2-aThSi, body-centered, tetragonal, space group 14,/amd, No. 141, 4 Th in a): O,O,O; O,;,:; ;,$; ;,O,$; 8 Si in e): O,O,z; O,;,: + z; ;,O,$-z; ;,$,$z; &,$,& + z; $,O% + z; O$$-z; O,O, - z. For the prototype itself a = 412.6 pm, c = 1434.6 pm, c/a = 3.477 and z(Si) = 0,416,. Structural type: tIl2-LaPtSi body-centered, tetragonal, space group I4,md, N. 109. 4 La in a(1): O,O,z; O,$,+ +z; $$,;, +z; ),O$ +z; 4 Pt in a(2): O,O,z;O,&i + z; + ,; + z; f,o,: + z; 4 Si in a(3): O,O,z; O,+$ +z; &+,+ +z; ),O,$ + z. For the prototype itself a = 424.90 pm, c = 1453.9 pm, c/a = 3.422 and z(La) = 0 (fixed conventionally), z(Pt) = 0.585 and z(Si) = 0.419. The unit cells of the two structures are presented in fig. 40. The ThSi, type structure according to PEARSON [19721 is primarily controlled by the ThSi contacts, with the Si-Si contacts exerting a certain influence. Each Si atom has three close Si neighbours resulting in the three dimensionally connected framework schematically shown in fig. 40d. This framework (and the Si-Si coordination) can be compared
Ch. 4, 56
Structure of intermetallic compounds and phases
a)
b)
Th Si2
LaPtSi
NbAs
La
Nb
@
Th si
Pt
vac
0
Si
Si
AS
0
305
Fig. 4Oa,b. Crystal structures of ThSi, and LaF’tSi (a) and NbAs (b) with the indication of the atoms which, in the three structures, occupy the different sites. (Notice the defective character of the NbAs type. structure in comparison with the ThSi, type.) In c) different sections of the LaF’tSi structure unit cell are presented with the indication of the heights along the z-direction and of the codes used for the different atomic position in a square net (compare with fig. 11). In the NbAs structure, the sections at z=0.08, 0.33, 0.58 and 0.83 are not occupied by any atoms. The dotted lines in a) show a part of the three-connected framework of Si (or Pt,Si) atoms. A larger portion of the framework is presented in d).
with the planar graphite hexagonal nets and therefore with, for instance, the hP3-A1B2 type structure (and its ordered variants). In the case of ThSi,, however, one vertex of each hexagon is always missing and we have parallel sets of planar chains interconnected to similar perpendicular sets. It may be interesting to mention that the characteristic structure of this network described as “hinged’ network should have the peculiar feature that the entire framework could undergo reorganisation by a nearly barrierless twisting type motion. According to References: p . 363.
306
0
Riccardo Ferro nnd Adriana Saccone
Z=0.08
Ch. 4, $6
Z=037
Z =0.25
Z=0.33
Z0.42
2.
3
2
1
Z =083
Z= 0.92
1
Z0.5
4 Z~ 0 . 5 8
2~0.75
0 4
1
3
2
3
El 4
C)
d)
Fig. 40c,d.
BAUGHMAN and GALVAO[1993]and MOORE[1993], unusual mechanical and the& properties may be predicted for substances having all their atoms arranged in such a framework. These special properties, therefore, may be envisaged for- hypothetical compounds such as polyacetylene, polydiacetylene, polyphenylene, (BN), phases, etc. and perhaps for substances containing the hinged network as a part of their structure (“cmwded” hinged network crystals) such as ThSi, compounds. Finally considering the AB, and the aThSi, type structures we may notice that the similarity of their bonding arrangements may be further confirmed by the existence of the AlB, structure also for a different form of ThSi, (p form, high temperature form) and (as a defective structure) for = Th,Si,. Following the description presented by BAUGHMAN and GALVAO [1993]) the AlB, type structure could be called a “crowded” graphitic network structure.
6.6.1. General remarks A number of structures of several important intermetallic phases can be classified as tetrahedrally close-packed structures. As an introduction to this subject we may remember, according to SHOEMAKER and SHOEMAKER [1969] that in packing spheres of equal sizes the best space filling is obtained in the cubic or hexagonal close-packed structures (or in their variants). In those arrangements there are tetrahedral and octahedral holes (see the comments on this point reported in the description of the C F M Utype structure in sec. 6.2.2). The local mean atomic density (the average space filling) is somewhat higher at the tetrahedral holes than in the larger octahedral ones, A more compact arrangement might, therefore, be obtained if it were be possible to have only tetrahedral interstices. It is, however, impossible tofill space with regular tetrahedra throughout. By introducing some variability in the sphere dimensions it is possible to obtain packing containing only tetrahedral holes. The tetrahedra are now no longer regular: the ratio of the longest tetrahedron edge to the shortest, however, needs not exceed about 9 in a given structure. The corresponding crystal structure can be considered to be obtained from the space filling of these tetrahedra (which share faces, edges and vertices). In structures containing atoms of approximately the same size and within the aforementioned limits of edge-length ratio, the sharing of a given tetrahedron edge (i.e. an interatomic link ligand) either among 5 or 6 tetrahedra has to be considered the most favored and situation (according to the systematic analysis of these structures carried out by FRANK KASPER [1958, 19591). On the assumption that only 5 or 6 tetrahedra may share a given edge the number of tetrahedra that share a given vertex is limited to the values 12,14, 15 and 16. The 12 (or 14, 15, 16) tetrahedra sharing a given vertex form, around this point, a coordination polyhedron with triangular faces. The radii of this polyhedron are the edges shared among 5 or 6 component tetrahedra and connect the central atom with the polyhedron vertices, five-fold or six-fold vertices, that is vertices in which 5 or 6 faces meet. The four possible coordination polyhedra are shown in fig. 41 and correspond to the following properties: coordination12 (regular, or approximately regular, icosahedron): 12 vertices (12 five-fold vertices) and 20 faces. coordination 14: 14 vertices (12 five-fold and 2 six-fold ) and 24 faces. coordination 1 5 15 vertices (12 five-fold and 3 six-fold) and 26 faces. coordination 16 16 vertices (12 five-fold and 4 six-fold) and 28 faces. (For symbols used in the coding of the vertex-characteristics see sec. 7.2.7). Several structures {Frank-Kasper stnrctures) can be considered in which all atoms have either 12 (icosahedral), 14, 15 or 16 coordinations. These can be described as resulting from the polyhedra presented in fig. 41. These polyhedra interpenetrate each other so that every vertex atom is again the center of another polyhedron. All structures in this family contain icosahedra and at least one other coordination type. Frank and Kasper demonstrated that structures formed by the interpenetration of the four polyhedra contain planar or approximately planar layers of atoms. (Primaly layers made up by tessellation of triangles with hexagons and/or pentagons were considered, References: p . 363.
308
Ch.4, $ 6
Riccardo Fern, and Adriana Saccone
CN 15
CN 16
a)
b)
Fig. 41. The coordination polyhedra of the Frank-Kasper structures, are shown in two different styles. a) the relative positions of the coordinating atoms are shown (the central atoms are not reported). (For the coordination numbers (CN) 12 and 14, one atom of the coordination shell is not visible). b) the corresponding triangulated polyhedra are shown. Vertices in which 5 or 6 triangles meet are easily recognizable.
Ch. 4, 0 6
Structure of intennetallic compounds and phases
309
and intervening secondary layers of triangles and/or squares). For a classification and [1972] and also SHOEMAKER and coding of the nets and of their stacking see PEARSON SHOEMAKER [1969] or FRANK and KASPER[1958, 19591. A short summary of structural types pertaining to this family is reported in table 6; for a few of them, some details or comments are reported in the following.
6.6.2. cpS-Cr,Si type structure This structure is also called W30 or p-W type (it was previously believed to be a W modification instead of an oxide) or A15 type (see section 3.4). Cubic, space group P m h , No. 223. Atomic positions: 2 Si in a) O,O,O; +,$,&; 921 47 2949 2949 6Crin~)$O.I.rO1-.'1O.I2.0.Olr.012 ,2949 ,294. This structure type is observed for many phases formed in the composition ratio 3:l by several transition metals with elements from the 111, IV, V main groups (or with Pt metals or Au). Phases having this structural type are, for instance, Mo,AI, Nb,Al, V3A1, Ta3Au, Ti3Au, cr3Pt, Cr30s, cr6A1si, V6A1Sn, Nb6GaGe, etc. A number Of compounds with this structure have been found to have significantly high superconducting transition temperature, T, (among the highest known, before the discovery of the families of superconducting complex oxides, such as Ba2YCu30,, or = Bi,(Ca,Sr),Cu,O,, etc.). Examples of superconducting, Cr3Si type, phases are: Nb3Ge (T,=23.1! K, sputtered films), Nb3Ga (Tc=20.7 K, bulk), Nb,Sn (T,= 18.1 K), V3Si (T, = 16.8 IC), V3Ga (T, = 14.1 K), Nb3Au (T, = 11.5 K), Nb3Pt (T, = 9.2 IC), Mo,Ir (T, = 8.8 K), etc. The Cr,Si type structure does not always remain stable in these materials down to 0 K, yet the change in crystal structure, when it occurs (for instance, with a tetragonal structure formed at low temperature as a result of a martensitic transformation) seems not correlated with T,. Solid solutions in general have lower T, values than the stoichiometric compounds. (Other superconductingintermetallic phases belonging to different structural types are, for instance, LuRh,B, (T,= 11.7 K, YPd,B3C, (T,=23 K), quaternary lanthanum nickel. boro-nitrides, etc. See CAVAet al. [1994a, 1994b1). 9
92,
3
9
6.6.3. u phase type structure, (tP3O-aCr-Fe type) In the space group P4Jmnm, No. 136, the two atomic species, Cr and Fe, are distributed in several sites with a nearly random occupation. Different atom distributions have been proposed in the literature (also owing to different preparation methods and heat treatments). The following distribution is one of those reported in DAAMSet aZ. [1991]: two atoms in sites a) (with a 12% probability for Cr and 88% for Fe), 4 atoms (75% Cr, 25% Fe) in sites f), 8 atoms (62% Cr, 38% Fe) in a set i) of sites, 8 atoms (16% Cr, 84% Fe) in another set i) and 8 atoms (66% Cr, 34% Fe) in j). The structure can be considered as made up of primary hexagon-triangle layers containing 3636 + 3'6' and 6, nodes (in a 3:2:1 ratio) at height = 0 and 3 separated (at height = and %)by secondary 3'434 layers (that is layers, in which every node is surrounded, in order, by 2 References: p. 363.
310
Ch.4,96
Riccardo Ferro and Adriana Saccone Table 6 Examples of tetrahedral closepacked structures.
structural types
Unit cell dimensions (rounded values) for the reported prototype [pm]
% of atoms in the center of a polyhedron with CN 12 14 15
16
cp&cr,Si (also called W30 or B-W type or A15 type phase).
a = 456
25
75
tP3OaCr,Fe,* u phases
a = 880 C = 456
33
53
13
hR39-
w&-/
15
15
k
a= 476 c=2562
55
CL P
a = 543 c = 539
43
28
28
oP52=Nbai,&l,,* M phases
a = 930 b= 493 c = 1627
55
15
15
15
oP56=Mo2,Cr~i,* P phases
a=1698 b= 475 c = 907
43
36
14
7
hR159 =Mo,,Cr,,Co,,* R phases
a = 1090
51
23
11
15
cI162Mgl I&,,&*
a=1416
61
7
7
25
Laves Phases: cF24-chMg
a= 704
67
33
hPlZMgZn,
a = 522 c = 856
67
33
hP24-Ni2Mg
a = 482 c=1583
67
33
15
c=1934
* For these phases the reported formulae generally correspond to an average composition within a solid solution field. This also in relation with a (partially)disordered occupation of the different sites. triangles, 1 square, 1 triangle and 1 square). As pointed out by Pearson (by studying the near-neighbours diagram: see sec. 7.2.5.a) the a-phase structure is a good example of a structure which is controlled by the coordination factor: all the known phases are closely grouped around the intersection of lines corresponding to high coordination numbers. (The most favorable radius ratio for the component atoms is included between 1.0 and 1.1.) It is also possible that the electron concentration plays some role in controlling the phase stability. The different phases are grouped in the range 6.2 to 7.5 electrons (s, p and d) per atoms.
Ch. 4, $ 6
Structure of intermetallic compomds and phases
311
6.6.4. Laves phases: cF24-CuzMg(and cF24-Cu4MgSn and cF24-AuBe$, hP12-MgZn, (and hPl%UzOsA13) and hP24-NizMg types General remarks The Laves phases form a homeotect structure type set (a family of polytypic structures). In all of them (described in terms of a hexagonal cell) three closely spaced 36nets of atoms are followed (in the z direction of the same cell) by a 3636 net (see figs. 8 and 10). The 36 nets are stacked on the same site as the kagom6 3636 nets which they surround (for instance: p-BAC-y-CAB in the “two slabs” MgZn, type (h) structure, /3-BAC-y-CBA-a-ACB in the “three slabs” MgCu, type (c) .structure, a-ABC-yCI3A-a-ACB-P-BCA in the “four slabs (hc)” NizMgtype structure, etc.: see sec. 4.3. on homeotect structure type). The Laves phases, as Frank-Kasper structures, (see table 6), can also be described by alternative stacking of pentagon-triangle main layers of atoms and secondary triangular layers (parallel to (110) planes of the hexagonal cell). The importance of the geometrical factor in determining the stability of these phases has been pointed out (PEILRSON[1972]). The role of the electron concentration in controlling the differential stability of the different Laves phase types has been also observed. By studying, for instance, solid solutions of Cu,Mg and MgZn, with Ag, AI, Si (LAVES and W m [1936], KLEE and WITTE [1954]) it was observed that for an average VEC (valence electron concentration) between 1.3 and 1.8 e/a (electrons per atom) the Cu,Mg structure is generally formed, for VEC values in the range from = 1.8 to 2.2 e/a generally the MgZn, type structure is obtained. The Ni,Mg type can be observed for intermediate values of VEC between 1.8 and 2.0. It may be useful, however, to mention that depending, for instance, on the temperature different Lives type structures may be observed in the same chemical system. An example may be: the Ti-Cr system for which 3 different structures have been described a-TiCr, (MgCu;, type, homogeneous in the composition range = 63-65 at% Cr), stable from room temperature up to = 122OOC; p-TiCr, (MgZn, type, homogeneous from 64 to 66 at% Cr), high temperature phase stable from 80OOC up to 127OOC; and y-TiCr, (Ni2Mg type, -65-66 at% Cr), high temperature phase stable from 1270°C up to the melting point (137OOC). Notice that the cu and P forms, which can coexist in the temperature range from 800°C up to 122OOC have slightly different compositions. Many (binary and complex solid solutions) Laves phases are known.Typically Laves compounds XM, are formed in several systems of X metals such as alkaline-earths, rare earths, actinides, Ti, Zr, Hf, etc, with M=Al, Mg, VIII group metals, etc. Before passing to a detailed description of the principal Laves types, a few more remarks can be made concerning the Luvespolytypes. An interesting example may be given by the [1974], MALLIK[1987]). This system is one of the richest in the Li-Mg-Zn alloyis (MELNIK Laves phases among the known ternary systems. It contains, besides MgZn,, eight ternary compounds L, (the index n denotes the number of slabs) in the following sequence: b: MgZn, (hP), a = 521.4 pm; c = 856.3 pm (= 2*428.1) L,: Mg(Lio.o+hl.93)(W), a = 521.3 pm; c = 3422 pm (E8 *427.8) L14: Mg(Lio,,,Sh,.,9)(hP), a = 521.5 pm; c = 5989 pm (= 14*427.8) I+: Mg(Li,,2$5nl.80) (hR), a = 522 pm; c = 3841 pm (=9 *426.8) References: p . 363.
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Mg(Lb.uZnl,n) (hP),a=522.3 pm; c=4278 pm ( E 10*427.8) Mg(Li,2Szn,75)(hP), a = 522.7 pm; c = 1709 pm (E4 *427.3) Mg(Lb,&n,,) (hP), a = 1046 pm; c = 1705 pm (=4*426.3) Mg(Lii.&n,J (hR), a= 1051 pm; c = 1285 pm (=3*428.3) b: Mg(Li,,,7Zn,,) (cF), a = 522.6 pm; c = 1290 pm ( E 3 *430) Notice that the structures with n = 3 and 4 exist not only in an ordinary form L4,Ni,Mg, and &, MgCu, type cubic, (a=744.8 pm, here described in terms of an equivalent set of hexagonal axes) but also with doubled unit cell edge a (Ni2Mg type and MgCu, type superstructures L’4 and L’&.
Llo: L4: L’4: L’3:
Structural type: cF24-Cuflg and derivative structures Face-centered cubic cF24-Cu2Mg, space group Fdjm, No. 227. Atomic positions: r01. 110. 313. 3 3 1 - 111-1 3 3 . 8 Mg in a) O,O,O; 011. 92721 29 2929 4,4949 4,459 494¶4¶ 4r4*49 16 Cu in d) _5898,s; _5 _5 -__. 5 1 1 1 5 1 _1 _1 _5 _ 3_ 71 _ _ 3_ 3_ 5 7 7 5 7 3 1 7 1 3 7 5 7 3 1 1 3 5 3 1 3 7 1 7 1 8,8989 89K8,K;8,898; 89898; 8?898; 8,898; X181K; 8,898; 8,898; 8,898; 898.8; X38,g; 89x98; _ 5 3-3.- _ 5_ 7 -7. 89898- 898*89 Coordination formula: 333 [Mg4,4][Cu,6]1w6 For the prototype itself, Cu2Mg, a=704 pm. Fig. 42 shows the MgCu, packing spheres structure. Normalized interatomic distances and numbers of equidistant neighbours are shown in fig. 43. Ordered variants of this type of structure are the Cu,MgSn type structure and the AuBe, type structure. The packing spheres structure of AuBe, is shown in fig. 44.The atomic positions of the two structures correspond to the following occupation of the same equipoints in the space group F43m (No. 216). in Cu4MgSn in AuBe, 4 Sn 4 Au a) O,O,O; O,$,+; h,o,+; h,+,O 4 Mg 4 Be c) ill. 34-49 133. 4,494, 1_ 3_ . 3- 3_ 1_ . _34’4743 4*4*4, e) x,x,x; -x,-x,x; -x,x,-x; x,-x,-x 16 Cu 16 Be x,;+x,ax; -x,;-x,;+x; -x,;+x,;-x; x321 x921 x; i+x,x,+x; kx,-x,+x; &x,x,&x; $+x,-x,&x; @x,++x,x; &-x,&-x,x; $-x,++x,-x; +x,gx,-x. (x E 0.625 =8) We can see that the 8-atom equipoint of the Cu2Mg type structure has been subdivided into two different, ordered, 4-point subsets in the two derivative structures. Layers stacking symbols, triangular, kagomk (T,K) nets: 929
9
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Structure of intermetallic compounds and phases
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Fig. 42. cF24-MgCu2 type structure (1 unit cell is shown).
Structural type: hP12-MgZn2 Hexagonal, space group P6Jmmc, N. 194, Atomic positions: 2 Zn in a) O,O,O; O,O$; 4 Mg in f) i93, 2 z*2 1 1 +z* 21 - z*1 2 1 - z3,332 3939 3,392 6 Zn in h) x,2x,i; -2x,- xf; x,- ;:,x x,-2x,$; 2x,x,%; - x,x 94’ 3* For the prototype itself, MgZn,, a=522 pm, c=856 pm, c/a= 1.640, zMg=0.O62and X, = 0.830. Coordination formula: 333 [Mg4,4][Zn,,] 12/6 Layer stacking symbols, triangular, kagomC (T,K) nets: MgO% z d 2 5 MgO%O%:z M&C56 z%?75 MgO?W Fig. 45 shows the packing spheres structure for the MgZn, compound. A ternary ordered variant of this structure corresponds to three different atomic species in the three equipoint set. An example may be U20sA13(2 Os in a), 4 U in f) and 6 A1 in h)). Structural type: hP24-Niflg Hexagonal, space group P6,/mmc, No. 194. Atomic positions: 4 Mg in e) O,O,z; O,O,i + z; O,O, - z; O,O,i - z; 4 M g i n f ) f 2 ~ . 3’3’2 2 1 1 + z + z ; ~ , i ,39392 -z.121-~. 4 N i i n f )393, 1 2 ~3,.32921 1 + 393,~ . 2 1z; T,J,T 121 z; 6 Ni in g) $,O,O; O,$,O; i,$,% $,O,$; Oh,;; 4,i.i; 6 Ni in h) x,2x,i; - 2x, - x,$; x, - x,$; - x, - 2x,$; 2x,x,:; - x,x,i; For the prototype itself, Ni,Mg, a =482 pm, c = 1583 pm, c/a = 3.284 and z (eMg)= 0.094, z (fMg)=0.8442,z (fNi)=0.1251,and x (hNi)=0.1643. 9
9
9
-
?
Znt
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9
9
9
-
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16 - a) CupMg-type
N
.
a. 1
2 16
[
d/d,
h
b) Gu2Mg-type
Fig. 43. Distances and coordinations in the cF24-MgCU, type structure. a) Coordination around Mg: (+) Mg-CU; (*) Mg-Mg. b) coordination around Cu: (*) Cu-cU; (+) CU-Mg.
The structure can be described by the following layer stacking sequence triangular, hgom6 (T,K) nets: Coordination formula: 323 l-Mg4/41 [Ni6/6112/6
6.6.5. Structures based on frameworks of fused polyhedra, Samson phases In addition to the Franl-Kasper phases, other structures may be considered in which the same four coordination polyhedra prevail although some regularity is lost. Many of these structures and, in particular the giant cell structures studied by SAMSON[1969] can be described as based on frameworks of fused polyhedra rather than the full interpenetrating polyhedra. Among the most important polyhedra we may mention the truncated tetrahedron: it is shown in fig. 46. It can be related to the CN 16 polyhedron (Friauf polyhedron) of fig. 41. The two polyhedra can be transformed into each other by removing (adding) the 4 six-fold vertices of the CN 16 polyhedron (corresponding to positions out from the center of each of the 4 hexagons of the truncated tetrahedron).
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Structure of intermetallic compounds and phases
Fig. 44.Unit cell of the cF24-AuBe5 type
315
structure.(Compare with the cF24-MgCu2 type structure, fig. 42.)
Several other coordination polyhedra occur in giant cell structures in addition to the Frank-Kasper polyhedra and to the truncated tetrahedron. (The most important are polyhedra corresponding to CN between 11 and 16). The following phases represent a few examples of structures to which the aforementioned considerations specially apply: cI58-a-Mn (a = 891.4 pm) type structure (and its binary variants, c158-Ti,Rez4 or X-phase and cIS8-y-Mg,,Al,,), cFl124-Cu4Cd3 type (a= 2587.1 pm); cF1192-NaCd, type (a=3056 pm); cF1832-MgzA1, (a=2823.9 pm), etc. (In the giant cell structures partial disorder and/or partial occupancy in some atomic positions have been generally reported, for cF1124-Cu4Cd3,for instance, the structure was described as corresponding to the occupation, in several Wyckoff positions, of 388 atomic sites by Cd-atoms, 528 by Cu-atoms and of 208 by Cu- and Cd-atoms in substitutional disorder.)
7. On some regularities in the intermetallic compoundformation and structures 7.1. Preliminary remarks As already mentioned in the previous sections, several thousands of binary, ternary and quaternary intermetallic phases have been identified and their structures determined. In a comprehensive compilation such as that by VILLARSand CALVERT about 2200 (in the first edition, [1985]) or about 2700 (second edition, [1991]) different structural types have been described. The specific data concerning about 17500 different intermetallic phases (pertaining to the aforementioned structural types) have been reported in the lst edition and 26000 in the 2"d one. As an introductory remark, a little statistical information about the phase and structure type distributions may be interesting. References: p. 363.
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Zn
Fig. 45. Unit cell of the hP12-MgZn2 type structure.
For this purpose, we may consider the group of phases described in the compilation by VILLARSand CALVERT [1991]. This, in fact may be considered a fairly representative sample even if the number of new intermetallic phases (and structural types) is constantly increasing. As a first observation we may notice that the number of phases pertaining to each structural type is not at all constant. Table 7 shows that a very high number of phases crystallize in a few more common structure types. About 25% of the known intermetallic phases belong to the first 12 more common structure types and about 50% of the phases belong to 44 types (that is less than 2% of the known structural types). This kind of distribution seems to be significant even if table 7 contains only an approximate list. (Some changes may actually be obtained by a more accurate attribution of different phases to a certain structural type or to its degenerate or derivative variants). The distribution of the phases among the different types is summarized in fig. 47, where (in a double logaritmic scale) the number of phases belonging to each structural type is plotted against the rank order of the type itself. According to a suggestion of the authors of this chapter, in the same figure a curve is presented which has been computed by fitting the reported data by means of eq. (1):
N, = A(r + ro)-B where N, is the number of phases corresponding to the structure type having rank r (A, B and r, are empirical constants whose values have been determined by the fitting (see et al. 119951). also FERRO It may be interesting to point out that the aforementioned equation is that suggested by MANDELBROT [1951] as a generalization of ZIPF’Slaw [1949] (which corresponds to the special case of r,,= 0 and B = 1). This law, in linguistics, relates for a given text the recurrence frequency (NJof a word to its rank (recurrence order). The formula had been deduced to define a cost function for the transmission of linguistic information and minimizing the average cost. (The word “cost” was considered to be related to the complexity of the word
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Structure of intermetallic compounds and phases
Fig. 46. Truncated polyhedron (12vertices) related, by the addition of 4 more coordinate atoms out from the centers of the hexagonal faces to the Friauf polyhedron (CN 16), reported in fig. 41.
itself). (Eq.(€)may be considered a special case of a general “Rank Size Rule”.) We note, moreover, the larger numbers of phases having highly symmetric structures (cubic, hexagonal or tetragonal structures). The most frequent orthorhombic and monoclinic structures are the 6th and the 58th respectively in a general list such as reported in table 7. This may be partially related to a certain greater ease in solving highly symmetric structures but probably also contains an indication of a stability criterion. The Laves’ stability principles (presented in sec. 7.2.3.) and, specially, the “symmetry principle” may be mentioned. Considering ,then the phase composition as a significant parameter, we obtain the histogram shown in fig. 48 for the distribution of the structural types and of the intermetallic phases (as obtained from the 2nd edition of Villars-calvert) according to the stoichiometry of binarypmtotypes (that is, for instance, the binary and ternary Laves phases, the AIB,, CaIn,, etc., type phases are all included in the number reported for the 66 to 67.99 stoichiometry range, even if the real stoichiometry of the specific phase is different). We may note the overall prevalence of phases and (to a certain extent) of structural types, which, at least ideally, may- be related to simple (1:2, 1:1, 1:3, 2:3, etc.) stoichiometric ratios. The restriction of the phases concentration to a limited number of stoichiometric ratios is also valid (and, perhaps, more evident) for the ternary phases. We may notice in fig. 49 (adapted from a paper by RODGERSand VILLARS[19931) that seven stoichiometric ratios (l:l:l, 2:1:1, 3:1:1, 4:1:1, 2:2:1, 3:2:1, 4:2:1) are the most prevalent. According to Rodgers and Villars they represent over 80% of all ternary known compounds. We have, ho’wever, to remark that, considering only selected groups of (binary or ternary) alloys, quite different stoichiometric ratios may be predominant. As an example we may mention the binary alloys formed by an element such as Ca, Sr, Ba, rare earth metals, actinides, etc., with Be, Zn, Cd, Hg and, to a certain extent, Mg. Many compounds are generally formed in these alloys. Among them, phases having very high stoichiometric ratios are frequently observed, such as, for instance: CaBe,,, LaBe,,, BaZn,,, BaCd,,, B a g , , BaHg,,, Bag,,, WZn,,, LaZn,,, h c d , , , LaCd,,, ThZZn,,, Pu,Zn,,, Ce,Mg,,,, La,Mg,,, LaMg,,, etc.
73. On some factors which control the structure of intermetallicphases A systematic:description of bond characterization from thermodynamic properties in intermetallic compounds (and considerations concerning the stability of intermetallic References:p. 363.
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-4.5 t
0
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Structure of intermetallic compounds and phases
C)
I
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5
Log(rank) Fig. 47c. Distribution of the inkmetallic phases among the structural types. In a double logarithmic diagram the phase numbers (expressed as ratios to the total number) are plotted versus the rank order of the structural type. The continuous line corresponds to the Mandelbrot’s equation. a) Number of pha!;es belonging to the overall different structural types. (Compare with Table 6). b) Number of pha!;es belonging to the cubic structural types. c) Number of phases belonging to the hexagonal structural types.
phases) has been reported by ELLNERand PREDEL[1994]. Some information about the computation of the enthalpy of formation of alloys according Miedema’s model will be given in sec. 8.5. On this subject we may mention the peculiar properties of alloys of extraordinary stability formed by elements such as Al, Ti, Zr,Hf with the transition metals Re, Ru, Os, Rh, Ir, Pd, Pt, characterized by very high formation heats and discussed by BREWER E1973, 19901 as example of generalized Lewis acid-base interactions in metallic systems. A general presentation and discussion of the origin of structure of crystalline solids and the structural stability of compounds and solid solutions have been given by Pettifor (see chapter 2 of this book). In this section and in the following one a brief sampling of some semiempirical useful correlations and, respecively, of methods ofpredicting phase (and structure)formation will be summarized. The search for regularities and criteria for the synthesis of new representatives of particular structure types has been carried out by many authors. Several factors References: p . 363.
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25 a)
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b)
50
55
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Atomic % Fig. 48. Distribution of binary intermetallic phases and structural types, according to the stoichiometry. a) Distribution of the structural types. b) Distribution of the intermetallic phases.
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Structure of intermetallic compounds and phases
c x>y>z
B
I
x=y
\
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\
\
x=3y x=4y
Fig. 49. Distribution of the known ternary intermetallic phases according to their stoichiometry. a) In a representative portion of a general composition triangle, the more common stoichiometries are shown.
were recognized to be important in controlling the structural stability and some of them were used as coordinates for the preparation of “classiJicationand prediction maps”, in which various compounds can be plotted and separated into different structure domains. Intermetallic phases, therefore, could be classified following the most important factor which controls their crystal structure (PEARSON [1972], WESTBROOK [1977], GIRGrS [1983], HAFNER [1989]). According to PEARSON [1972], following factors may be evidenced: - Chemical bondfactor, - Electrochemicalfactor, (electronegativitydiference) - Energy band factor, electron concentration - Geometricalfactor - Sizefactor References: p. 363.
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Fig. 49. Distribution of the known ternary intermetallic phases according to their stoichiometry. b) For the same compositions shown in a), an indication is given of the number of phases.
In the following paragraphs a few comments will be reported on this matter. Emphasis, however, will be given only to those aspects which are more directly related to a description of the “geometrical” characteristics of the phases. For the other questions reference should be made to other parts of this volume. For an introduction to the electronic structure of extended systems, see HOFFMANN [1987, 19881.
7.2.1. Chemical bond factor and electrochemical factor A chemical bondfactor can be said to control the structure when interatomic distances (and as a consequence unit cell dimensions) can be said to be determined by a particular set of chemical bonds. Two different situations can be considered: bonds having high ionic characteristics (largely non-directional, the larger anions tend to form symmetrical coordination polyhedra subjected to the limitation related to the aniodcation atomic size ratio) or bonds having covalent character (the directional characteristic of which tend to determine the structural arrangement in the phase). To an increasing weight of the chemical bond factor (ionic and/or covalent bonding) will, of course, correspond, in the limit, the formation of valence compounds. According to PARTHE[1980b] a compound C,A, can be called a normal valence compound if the number of valence electrons of cations (ec) and anions (eA) correspond to the relation (normal valence compound rule): me, = n(8 - e A )
(2)
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Table 7 Intermetallic phases: The most common structural types (from the data reported in VILLARS and CALVERT [1991]). structwal type
If we consider only the s and p block elements, the number of valence electrons of the elements correspond to their traditional group number) In this case (considering that no anions are formed from the elements of groups I, 11 and 111) following formulae can be deduced for the normal valence compounds (formed in binary systems with large electronegativity difference between elements): - 144 - 224 - 3443 - 135 - 2352 - 35 - 4354 - 126 - 26 - 3263 - 462 - 5263 - 17 - 272 - 373 - 474 - 575 - 67, (in these formulae each element is indicated by a number corresponding to its number of valence electrons; for instance: 17 represent NaC1, KC1, etc, 3263A1203,etc.) In the more general case where some electrons are also considered to be used for bonds between cations and anions we have (general valence compound rule):
m(ec - ecc) = n(8 - e,
- eM)
(3)
In this formula, which can only be applied if all bonds are two-electron bonds and additional electrons remain inactive in non-bonding orbitals (or, in other words, if the compound is semiconductor and has not metallic properties) e,, is the average number of valence electrons per cation which remain with the cation either in non-bonding orbitals or (in polycationic valence compounds) in cation-cation bonds; similarly e, can be assumed to be the average number of anion-anion electron pair bonds per anion (in polyanionic valence compounds). In a more limited field than that of the previously considered general octet rule, it may be useful to mention the “tetrahedml structures” which form a subset of the general valence compounds. According to PARTHE[1963, 1964, 19911, if each atom in a structure is surrounded by 4 nearest neighbours at the comer of a tetrahedron, the structure is called “normal tetrahedral structure”. The general formula of this structure, for the compound C,A,,, is (normal tetrahedral structure):
(me,
+ n e A )= 4(m + n)
(4)
(This may be considered a formulation of the so-called GRIMM-SOMMERFELD [19261 Rule). For the same elements previously mentioned the possible combinations are: 4& (all compositions, for instance, C, Ge, Sic) 35 (BP, NSb, etc.), 26 (BeO, MgTe, ZnS), 17 (CuBr, AgI), 326, 3,7, 252 (ZnP,, Z ~ S , )2372, , 15, and 1263. (ternary or more complex combinations may be obtained by a convenient addition of different binary formulae; for instance: 14,5, = (15, + 44): for instance CuGe2P, 136, = (1263+ 3,6)/2: CuAlS,, CuInTe,,, etc. 1,246, = (126,+ 26 +4): for instance C%FeSnS4 (Fen), etc.) The aforementioned rule may be extended to include the “defect tetrahedral structures” where some atoms have less than four neighbours (general tetrahedral structure):
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(me,
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Structure of intermetallic compounds and phases
+ n e A )= 4(m + n) + ~
~+ a ) ~
~
(
m
In this formula NNBois the average number of non-bonding orbitals per atom. By adding the symbol 0 (zero) to the described notation, vacant tetrahedral sites can be represented. Examples of formulae of defect tetrahedral structures are: 40374(Si&, Sn14);406, (GeS,); 360546, (Ga,As,Se,), 1,5,06, (CuSbS,); etc. Notice that the aforementioned compositional scheme is a necessary condition for building the tetrahedral structures, but not every compound that fulfills this condition is a tetrahedral compound. The influence of other parameters, such as the electronegativity difference, has been pointed out. By means of a diagram as shown in fig. 50, the separation of tetrahedral structures from other structures may be evidenced (MOSER and W O N [1959]). As a final comment to this point, we may mention that when one component in a binary alloy is very electropositive relative to the other, there is a strong tendency to form compounds of high stability in which valence rules are satisfied (PEARSON[ 19721). Such alloys are considered to show a strong electrochemicalfactor. 7.2.2. Energy band factor, electron concentration The properties of a solid on principle could be calculated on the basis of the states of the electrons in the crystal. The status of the understanding of the structures of the solids and indications on the technical and computational problems have been presented in other chapters. We may mention here that if the stable crystal structure may be described as controlled by the number of electrons per atoms, the phase is called an “electron compound)). An important class of electron compounds (generally showing rather wide homogeneity ranges) are the Hume-Rothery phases. These include several groups of isostructural phases, each group corresponding to a given value of the so-called valence electron concentration (VEC). Three categories of Hume-Rothery phases are generally considered: those corresponding to VEC values of 3/2 (that is three valence electrons every two atoms), 21/13 and 7/4, respectively. Representatives of the Hume-Rothery phases are the following: VEC = 3/2, body centered cubic, (cI2-W type): CuZn, = Cu,Al, = Cu,Sn, etc. VEC 3/2, complex cubic, (cP20-p Mn type): Cu,Si, Ag,Al, Au,Si, etc. VEC = 21/13, complex cubic, 52 atoms in the unit cell (or superstructures) (cP52: = Cu$J4, = CqGa,, Aga,, = Co,Z%,, etc.; cI52: = CusZn8,y-brass, = Ag,C$, Ag,Zn8, Ru3Belo,etc.; cFM8: Fe,,Znz9,etc.) VEC = 7/4, hexagonal close-packed, (hP2-Mg type or superstructures): = AgZn,, = Au,Ge, = Ag,Al,, etc. The VEC in all the aforementioned cases, for which approximate “ideal” formulae have been indicated, were calculated assuming the following “valence”: transition elements with nonfilled d-shells: 0; Cu,Ag,Au: 1; Mg and Zn,Cd,Hg: 2; Al,Ga,In: 3; Si,Ge,Sn: 4; Sb: 5. The given ratios indicate ranges (which can even overlap). It has to be noted, moreover, that the number of electrons to be considered may be uncertain. The VEC pi:
References: p. 363.
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. 5-
43-
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' . ' * ' . ' * ' . ' - ' ~ ' " .
Fig. 50. Mooser-Pearson diagram separating AB compounds into covalent (0) and ionic (+) types after Hum-ROTHERY [1967]. The representative points of the different components are plotted in the map ii, average quantum number, versus the electronegativity difference multiplied by the radius ratio. (RAand R, radii of the anion and cation elements).
values, therefore, indicate only a composition range where one of the aforementioned structure types may occur. According to GIRGIS[1983] the existence field of the electron phases may be especially related to the combinations of d elements with the elements of the Periodic Table columns from 11 to 14 (from Cu to Si groups).
7.23. Geometricalprinciples and factors, Laves' stability principles LAWS[1956], when considering the factors which control the structures of the metallic elements, presented three principles that are interrelated and mainly geometric in character. a) The principle of efficient (economical) use of space (space-jiillingprinciple). b) The principle of highest symmetry. c) The principle of the greatest number of connections (connection principle). These principles may be considered to be valid to a certain extent for the intermetallic phase structures and not only for the metallic elements. (See also some comments on this point as a result of the atomic-environmentanalysis of the structure types summarized in sec. 7.2.7.)
a) Spacefilling principle The tendency to use the space economically (to form structures with the best space-filling) which is especially exemplified by the closest-packing of spheres is considered to be the
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Structure of intermetallic compounds andphases
327
result of a specific principle which operates in the metal structures (and also in ionic and, to a lesser degree, in van der Waals structures). This principle is less applicable to covalent crystals because the characteristic interbond angles are not necessarily compatible with an efficient use of the space. Among the metallic elements, 58 metals possess a close-packed arrangement (either cubic or hexagonal) which, in the assumption that the metal atoms are indeformable spheres having fixed diameters, corresponds to the best space-filling;23 of the remaining metals crystallize in another highly symmetric structure, the body-centered cubic, which corresponds to a slightly less efficient space-filling. (The space-filling concept has been analysed and discussed by several authors: we may mention LAVES[1956], Pmm [1961], PEARSON[1972]. A short summary of this discussion will be reported in the following, together with some considerations on the atomic dimension concept itself). b) The principle of highest symmetry (symmetry principle) According to Laves a tendency to build conJigurationswith high symmetry is evident and is called the symmetry principle. This tendency is particularly clear in metallic structures, especially in the simple ones. However, according to HYDE and ANDERSSON[1989], for instance, the validity extension of this principle is difficult to evaluate. As time passes, crystallographers are able to solve more and more complex crystal structures and these tend to have low symmetry. The symmetry principle could perhaps be restated by observing that a crystal structure has the highest symmetry compatible with efficient use of space and the specific requirements of chemical bonding between nearest neighbours. For a discussion on the “symmetry principles”, its alternative formulations and the [1977] and by B~NIGHAUSEN [1980] history of its development, papers by BRUNNER may be consulted. In these papers a number of statements have been reported which perhaps may be considered equivalent. When considering close sphere packings, the following statements are especially worthy of mention. a) A tendency to form arrangements of high symmetry is observable. b) Points are disposed around each point in the same way as around every other. c) Atoms of the same type tend to be in equivalent positions.
c) The principle of the greatest number of connections (connection principle) To understand the meaning of this principle it may be at first necessary to define the concept of connection. To this end we may consider a certain crystal structure and imagine connecting each atom with the other atoms present in the structure by straight lines. There will be a shortest segment between any two atoms. We will then delete all links except the shortest ones. After this procedure, the atoms that are still connected constitute a “connection”. The connection is homogeneous if it consists of structurally equivalent atoms, otherwise it is a heterogeneous connection. Such connections may be finite or 1,2, 3 dimensionally infinite and are respectively called islands, chains, nets or lattices. Symbols corresponding to the letters I, C, N, L (homogeneous connections) or i, c, n, 1(heterogeneousconnections) have been proposed. (see also the dimensionality indexes reported in sec. 3.5.1.). As pointed out by Laves (for instance, LAVES[1967]) metallic elements and References:p. 363.
328
Riccardo Fern andddriana Saccone
a.4,87
intermetallic phases show a tendency to form multidimensional (possibly homogeneous) connections (connection principle).
7.2.4. Atomic dimensions and structural characteristicsof the phases a) Atomic radii and volumes A few comments about the atomic dimension concept may be useful also in order to present a few characteristic parameters and diagrams (such as space-filling parameters, reduced strain parameters, near-neighbours diagrams, etc.). Quoting from a comprehensive review on this subject (SIMON[1983]) we may remember that ever since it has been possible to determine atomic distances in molecules and crystals experimentally, efforts have been made to draw conclusions from such distances about the nature of the chemical bonding and to compare interatomic distances (dimensions) in the compounds with those in the chemical elements. Distances between atoms in an element can be measured with high precision. As such, however, they cannot be simply used in predicting interatomic distances in the compounds. In rational procedure, reference values (atomic radii) have to be "extracted" from the individual (interatomic distances) measured values. Various functions have been suggested for this purpose. In the specific case of the metals it has been pointed out that interatomic distances depend primarily on the number of ligands and on the number of valence electrons of the atoms (PEARSON [1972]). Pauling's rule (PAULING [19471):
I?,,
= Rl
- 30 log n (pm)
(6)
relating radii for bond order (bond strength) n (number of valence electron per ligand) to that of strength 1, gives a means of correcting radii for coordination and/or for effective [1972], SIMON[1983]) that, no matter what the valencies. It has been shown (PEARSON limitations may be of any particular set of metallic radii (or valencies) that is adopted, the Pauling's relation appears to be reliable, giving a basis for comparing interatomic [1983] slightly better results could be obtained distances in metals. According to SIMON changing the Pauling's formula to:
R, = Rl(l - A log n)
(7)
where A is not constant but can be represented as a function of the element valency. The subsequent point is to select some system of (a set of) atomic radii which can be used when discussing interatomic distances. The radii given by ~ A T U Met al. [19681 (and reported in table 8, together with the assumed "valencies") are probably the most useful for discussing metallic alloys. These radii have been reported for a coordination number of 12; they were taken from the observed interatomic distances in the fc cubic (cF4-Cu type) structure and in the hexagonal close-packed hP2-Mg type structure (averaging the distances of the first two groups of 6 neighbours, if the axial ratio has not the ideal 1.633.. value) or from the bc c12-W type. Since the coordination is 8 in the cI2-W type structure, for the elements having this structure the observed radii were converted to coordination 12 by using a correction given by the formula:
Ch.4, 87
329
Structure of intermetallic compounds and phases
Table 8 Radii (CN 12) of the Elements (from TEATUM et al. [1968])” Element
a) The elements are arranged according to their atomic number.
Noble gases and halogens are not included. RMerences: p . 363.
330
Riccardo Ferro and Adriana Saccone
RCNI
= 1.0316 RcNs- 0.532 (pm)
ch. 4 , 4 7 (8)
which was empirically obtained from the properties of elements having at least two allotropic modifications, cI2-W type and either cF4-Cu type or hP2-Mg type. The radii in the two structures (calculated at the same temperature by means of the known expansion coefficients) were compared and used to construct the reported equation. For the other metals (that is for the more general problem of the radius conversion from any coordination to coordination number 12) a percentage correction was applied (by using a curve which ranges from about +3% for the conversion from CN 8 to CN 12 to about +20% for the conversion from CN 3 to CN 12) as suggested by LAVES[1956] in a detailed paper dealing with several aspects of crystal structure and atomic sizes. While dealing with atomic dimension concepts, atomic volumes may also be considered. A value of the volume per atom, V , in a structure may be obtained from the room temperature lattice parameter data by calculating the volume of the unit cell and dividing by the number of atoms within the unit cell. See also the table reported by KING[1983]. An equivalent atomic radius could be obtained by computing, on the basis of the space-filling factor of the structure involved, the corresponding volume of a “spherical atom” using the relationship Vsph=(4 7~ R3/3). In the cP2-W type (CN 8) structure we have V,,= 0.68 V, (only a portion of the available space is occupied by the atomic “sphere”, see the following paragraph b). In the cF4-Cu type, and in the “ideal” hP2-Mg type (CN 12) structures we have V, = 0.74 Vat.Considering now the previously reported relationship between qm 12) and , R we may compute for a given element, very little volume (VaJ changes in the allotropic transformation from a form with CN 12 to the form with CN 8. (The radius variation is nearly counterbalanced by the change in the space filling). This generally is in agreement with the experimental observations (PEARSON[1972]). We will see that on the basis of the atomic dimensions of the metals involved (expressed, for instance, as RrRy or R,/Ry) many characteristic structural properties of an X,Y, phase may be conveniently discussed and/or predicted (sizefactor effect). As a further comment to this point we may mention here two “rules”, the VeguniZ and the Biltz-Zen’s ruZes, which have been formulated for solid solutions and to a certain extent for ordered compounds. These rules, mutually incompatible, are very seldom obeyed; they may, however, be useful either as approximations or for defining reference behaviours. The first one, VEGARD’Srule [1921], corresponds to an additivity rule for interatomic distances (or lattice parameters or “average” atomic diameters). For a solid solution &B1-x (x=atomic fraction) between two components of similar structure it takes the form:
dAB= xdA + (1 - x)dB
(9)
The BILTZ[1934], (or ZEN [1956]) rule has been formulated as a volume additivity rule:
v, ‘ X 4
+(l-x)V,
(10)
These rules are only roughly verified in the general case (for the evaluation of interatomic distances weighted according to the composition and for a discussion on the calculation and prediction of the deviations from Vegard’s rule see PEARSON [I9721 and SIMON[1983]).
a.4, $7
Structure of intermetallic compounds and phases
33 1
As contributions to the general question of an accurate prediction of the variation of the average atomic volume in alloying we may mention a few different approaches. MIEDEMA and NIESSEN[ 19821 calculated atomic volumes and volume contractions on the basis of the same model and parameters used for the evaluation of the formation enthalpy of the alloy (see sec. 8.5). In a simple model proposed by HAPNER[1985] no difference of electronegativity and no charge transfer were considered. Volume (and energy) changes in the alloy formation were essentially related to elastic effects. Good results have been obtained for alloys formed between s and p block-elements.An empirical approach has been suggested by MERLO[ 19881. Deviations from Biltz-Zen trend have been discussed and represented as a function of a “charge transfer atomic parameter” which correlates with Pauling’s electronegativity. This approach has been successfully employed for groups of binary alloys formed by the alkaline earths and the bivalent rare earth elements. Negative experimental deviations from Vegard’s rule (and values of the volume contractions) have been sometimes considered as an approximate indication of the formation of strong bonds and related to more or less negative enthalpies of formation (KUBASCHFJWSKII [1967]). This indication is only very poor in the general case. For selected groups of alloys, however, the existence of a correlation between the formation volume and enthalpy (A,,V and AfomH)has been pointed out (even if only as an evaluation of relative trends). This is the case of the rare earth (RE) alloys. As noticed by GSCHNEIDNER [19691 considering the trivalent members of the lanthanide series, we may compare the atomic volume decreasing observed in the metals (RE) (lanthanide contraction) with the decreasing of the average atomic volume measured in a series of lUZMex compounds. If this diminution is more (less) severe in the compounds than in the RE metal series, this is considered an indication that the bonding strength in the REM% compounds increases (decreases) as we proceed along the series from La to Lu; the heats of formation are expected to increase (decrease) in the same order. To make this comparison the unit cell volumes of the compounds are divided by the atomic volumes of the pure metals. The volume ratio for the series of compounds are then divided for that corresponding to a selected rare earth, this giving a relative scale. If the resultant values increase, with the atomic number of the rare earth, then the lanthanide contraction is less severe in the compounds (in comparison to the rare earth element) and a decrease of the heat of formation is expected (conversely if the relative volume ratio decreases, an increase of the heat of formation (more negative enthalpy of formation) is expected). (Examples of this correspondence will be examined in sec. 8.6., see also fig. 59.)
b) Spacefilling parameter (and curves) The spucelfillingpurumerer introduced by LAVES [1956] and by PARTHE[1961] gives a means of studying the relationships between atomic dimensions and structure. For a compound, it is defined by the ratio between the volume of atoms in a unit cell and the volume of unit cell. (471. / 3)(Zi?ziRi3) P=
v,,
(q,Ri number and radius of type i atoms). References: p. 363.
332
Riccardo Fern and Adriana Saccone
Ch. 4, $ 7
To calculate the space filling value for a specific compound, one has to know the radii of the atoms and the lattice constant. Neither of these is needed for the construction of a spacejlling curve of a crystal structure type: it is sufficient to know the point positions of the atoms and the axial ratios. The curve is based on a hard sphere model of the atoms: the cell edges are expressed as functions of the atomic radii (R, and R, for a binary system) for the special cases of X-X, X-Y and Y-Y contacts. The parameter can then be given (and plotted) as a function of the Rx/R, ratio. Considering, for instance, the cF8-ZnS-sphalerite type structure (PAR& [19641) the space filling can be given by:
where a is the cubic cell edge and R, and R, are the radii of the atoms in the a) and c) positions (4 Zn and 4 S, respectively) in the unit cell. (See the description of the structure in sec. 6.3.2.). In the case that the two atoms (or, more accurately, the hard spheres) occupying the Zn and S sites are touching each other, then the sum of the two radii must be equal to one-quarter of the cubic cell diagonal.
R,
+ R~ = a 8 / 4
(13)
By expressing the unit cell volume as a function of the sum of the radii we obtain:
Fig. A25. The shapes of the hyperatoms in ico-AI,Cu,,Fe,,. (a) Fe/Cu/Al triacontahedral hyperatom in the origin n,, (b) mid-edge (n,) Fe/Cu/Al hyperatom, and (c) CdAl hyperatom at bc, (from CORNIERQUIQUANDON, QUIVY, LEFEBVRE, ELKAIM,HEGER,KATZ and GRATIAS[ 19911).
References ANLAGE,S. M., B. FULTZand K.M. KRISHNAN, 1988, J. Mater. Res. 3,421-5. AUDIER,M., M. DURAND-CHARRE and M. DEBOISSIEU,1993, Phil. Mag. B68, 607-18. AUDIER,M., J. PANNETIER,M. LEBLANC, C. JANOT,J.M. LANGand B. DUBOST,1988 Physica B153, 13642. BANCEL,P. A., and P.A. HEINEY,1986, Phys. Rev. B33,7917-22. BANCEL,P. A., P. A. HEZNEY,P. W. STEPHENS,A.I. GOLDMANand P.M. HORN, 1985, Phys. Rev. Lett. 54, 2422-5. BARBIER,J.-N., N. TAMURA and J.-L. VERGER-GAUGRY, 1993, J. Non-Crystall. Solids 153,154, 126-31. BEELI,C., and H.-U. NISSEN,1993, J. Non-Crystall. Solids 153,154, 463-7. BEELZ,C., H.-U. NISSENand J. ROBADEY,1991, Phil. Mag. Lett. 63, 87-95. BOUDARD, M., M. DEBOISSIEU, C. JANOT,J. M. DUBOISand C. DONG,1991, Phil. Mag. Lett. 64, 197-206. BOUDARD,M., M. DEBOISSIEU, M. AUDIER,C. JANOT,G. HEGER,C. BEELI,H.-U. NISSEN,H. VINCENT,R. IBBERSON and J. M. DUBOIS,1992, J. Phys.: Condens. Matt. 4, 10149- 68. BURKOV,S. E., 1991, Phys. Rev. Lett. 67, 614-7. BURKOV,S. E., 1992, Phys. Rev. B47, 12325-8. CAHN,J. W., D. GRATIASand B. MOZER,1988, J. Phys. France 49,1225-33. CAO,W., H. Q. YE and K.H. Kuo, 1988, Phys. Status Solidi (a) 107, 511-9.
App. 42-1.4, Refs.
The structure of quasicrystals
409
CHAR, and R.C. O'HANDLEY, 1989, Phys. Rev. B39,8128-31. CHA-ITOPADHYAY, K., S. LELE,N. THANGARAJ and S . RANGANATRAN, 1987, Acta Metall. 35,727-33. CHEN, H., D.X. LIand K.H. Kuo, 1988, Phys. Rev. Lett. 60, 1645-8. CHEN, H.S., J. C. -LIPS, P. VILLARS,A.R. KORTANand A. INOUE, 1987, Phys. Rev. B35,9326-9. COOPER,M., and K. ROBINSON,1966, Acta Crystallogr. 20, 614-17. CORNIER-QUIQUANDON, M., R. BELLISENT, Y. CALVAYRAC, J. W. CAHN,D. GRATIASand B. MOZER, 1993, J. Non-Crystall. Solids 153,154, 10-4. CORNIER-Q~IQUANDON, M., A. QUIVU,S. LEFEBVRE,E. ELKAIM,G. HEGER,A. KATZand D. GRATIAS,1991, Phys. Rev. B44,2071-84. DAULTON,T.L., and K.F. KELTON,1992, Phil. Mag. B66, 37-61. DAULTON,T.L., and K.F. KELTON,1993, Phil. Mag. B68,697-711. DAULTON,T. L., K.F. KELTON,S. SONGand E. R. RYBA,1992, Phil. Mag. Lett. 65.55-65. DEBOISSIBU,M., C. JANOT, J.M. DUBOIS,M. AUDLER and B. DUBOST,1991, J. Phys.: Condens. Matt. 3, 1-25. DONG, C., J.M. DLJBOIS, M. DEBOISSEUand C. JANOT, 1991, J. Phys.: Cond. Matt. 3, 1665-73. DONG,C., J.M. DUBOIS,S. S. KANG and M. AUDIER,1992, Phil. Mag. B65, 107-26. DONG, J., K. Lu, H. YANGand Q. SHAN,1991, Phil. Mag. B64,599-609. EBALARD, S., E SPAEPEN,1989, J. Mater. Res. 4, 3943. EDAGAWA, K., M. ICHIHARA, K. SUZUKI and S. TAKEUCHI, 1992, Phil. Mag. Lett. 66, 19-25. FREY,E,and W. STEURER,1993, J. Non-Crystall. Solids 153,154, 600-5. FUNG, K. K., C. Y.YANG, Y.Q. ZHOU, J. G.ZHAO, W. S . ZHANand B. G. SHEN,1986, Phys. Rev. Lett. 56, 2060-3. GRATIAS,D., J. W. CAHN, B. MOZER,1988, Phys. Rev. B38, 1643-6. GRUSHKO,B., 1993, Mater. Trans. JIM 34,116-21. HAIBACH,T., B. ZHANGand W. STEURER, 1994, in preparation. HE, L.X.,X. Z. Lr, Z. ZHANGand K. H. KUO, 1988, Phys. Rev. Lett. 61, 1116-8. HE, L. X., Y. K. Wu, K. H. Kuo, 1988, J. Mater. Sci. 7, 1284-6. HE, A.Q., Q.B. YANG and H.Q. YE, 1990,61,69-75. HE, L. X, Z. ZHANG,Y. K. WU and K. H. Kuo, 1988, Inst. Phys. Conf. Ser. No. 93,2,501-2. HEINEY,P.A., P. A. BANCEL,A.I. GOLDMAN and P. W. STEPHENS,1986, Phys. Rev. B34, 6746-51. HENLN, C. L., 1993, J. Non-Crystall. Solids 153,154, 172-6. HENLEY,C. L., and V. ELSER, 1986, Phil. Mag. B53,L59-66. HIRACiA, K., 1991, J. Electron Microsc. 40, 81-91. HIRAUA,K., 1992, Electron Microscopy 2, EUREM 92, Granada, Spain. WOA, K., M. KANEKO, Y. MATSUOand S. HASHIMOTO, 1993, Phil. Mag. B67, 193-205. HIRACIA, K., E J. LINCOLNand W. SUN, 1991, Mater. Trans. 32, 308-14. HIRAGA,K., and W. SUN,1993, Phil. Mag. Lett. 67, 117-23. HIRAGA,K., W. SUNand F.J. LINCOLN,1991, Jap. J. Appl. Phys. 30, L302-5. HIRAOA,IC, W. SUN,E J. LINCOLN,M. KANEKO and Y. MATSUO,1991, Jap. J. Appl. Phys. 30,2028-34. HIRAGA,K., B.P. ZHANG,M. HIWAYASHI, A. INOUEand T. MASUMOTO, 1988, Jpn. J. Appl. Phys. 27, L951-3. Hu, R., T. EGAMI,A.P. TSAI,A. INOUEand T. MASUMOTO,1992, Phys. Rev. B46,6105-14. HUDD,R.C., and W.H. TAYLOR, 1962, Acta Crystallogr. 15,441-2. INOUE, A., H. M. KIMURA, T. MASUMOTO, A. P. TSAIand Y. BIZEN,1987, J. Mater. Sci. Lett. 67,771-4. ISHIMASA,T.,H.-U. NISSEN and Y. FUKANO, 1985, Phy~.Rev. Lett. 85,511-3. JANOT, C., 1992, Quasicrystals. A Primer (Clarendon Press, Oxford). JANOT', C., M. DFBOISSIEU,J. M. Dusors and J. PANNETIER, 1989, J. Phys.: Condens. Matt. 1, 102948. JANSSEN,T., 1986, Acta Crystallogr. A42, 261-71. KANG,S., and J.M. DUBOIS,1992, J. Phys.: Condens. Matt. 4, 10169-98. KANG, S. S., 9.M. DUBOIS,B. MALAMAN and G. VENTURINI, 1992, Acta Crystallogr. B48, 77C-6. KARKUT,M. G., J.M. TRISCONE, D. ARIOSAand 0.FISCHER,1986, Phys. Rev. B34,4390-3. KEK,S., 1991, Thesis, Univ. Stuttgart, FRG. KELTON, K. F., P. C. GIBBONSand P. N. SABES,1988, Phys. Rev. B38,781&3.
.
410
W! Steurer
App. Ch.4, Refs.
KORTAN,A.R., R. S. BECKER, EA. Tma and H. S. C m , 1990, Phys. Rev. Lett. 64,200-3. KORTAN,A.R, H. S. CHEN,J.M. PAR= and L.C. KIMERLING,1989, J. Mater. Sci. 24, 1999-2005. KOSHIKAWA,N., K. EDAGAWA, Y. HONDAand S. TAKBUCHI, 1993, Phil. Mag. Lett. 68, 123-9. KOSHIKAWA, N.,S. SAKAMOTO, K. EDAGAWA and S. TAKEUCHI,1992, Jpn. J. Appl. Phys. 31, 966-9. m o w , W., D. P. DIVINCENZO, P. A. BANCEL, E. COCKAYNE and V. ELSER,1993, J. Mater. Res. 8, 24-7. KRUMEICH,E, M. CONRADand B. HARBRECHT, 1994, 13th International Congress on Electron Microscopy, ICEM, Paris. Kuo, K.H., 1987, Mater. Sci. Forum 22-24, 131-40. Kuo, K.H., 1993, 2'. Non-Crystall. Solids 153,154, 40-4. Kuo, K.H., D.S. Z m u and D.X. LI, 1987, Phil. Mag. Lett. B55, 33-7. LAUNOIS,P., M. AUDIER, E DENOYER,C. DONG,J. M. DUBOISand M. LAMBERT, Europhys. Lett. 13,629-34. LAD.W., and R. A. DUNLAP.1993, J. Non-Crystall. Solids 153,154,611-4. LEMMERZ,U., B. GRUSHKO,C. FREIBURG and M. JANSEN,1994, Phil. Mag. Lett.69,1416. ~ I N ED , .,and P. J. S-m, 1986, Phys. Rev. B34,596-616. LI, X. Z., and K. H. Kuo, 1988, Phil. Mag. Lett. 58, 167-71. LI, X.Z., and K.H. Kuo, 1993, J. Mater. Res. 8,2499-503. LIAO,X.Z., K.H. Kuo, H. ZHANGand K. URBAN,1992, Phil. Mag. B66,549-58. LN, W., U. KOsni~,E MULLERand M. ROSENBERG, 1992, Phys. Stat. Sol. (a) 132, 17-34. Luo, Z., S. ZHANG,Y. TANGand D. ZHAO,1993, Scr. Metall. and Mater. 28, 1513-8. MA, X.L.,and K.H. Kuo, 1994, Met. and Mater. Trans.25A, 47-56. MA, Y., and E. A. STERN,1987, Phys. Rev. B352678-81. MA, Y., and E. A. STERN,1988, Phys. Rev. B38, 3754-65. m, L., R. WANGand K.H. Kuo, 1988, S a Metall. 22, 1791-6. MUKHOPADHYAY, N. K., K. CHATTOPADWAY and S . RANGANATHAN, 1988, Met. Trans. Au),805-12. MUKHOPADHYAY, N. K., N. THANGARAJ,K. CHATWPADHYAY and S. RANGANATHAN, 1987, J. Mater. Res. 2, 299-304. NANAO,S., W. DMOWSKI,T. &AMI, J. W. RICHARDSONand J.D. JORGENSEN,1987, Phys. Rev. B35435-40. N ~ U R AA., , A.P. TSAI,A. INOUE,T. MASUMOTO and A. YAMAMOTO, 1993, Jpn. J. Appl. Phys. 32, L1160-3. NISSEN,H.-U., and C. BEELI,1993, J. Non-Crystall. Solids 153,154, 68-71. OHASHI,W., and E SPAEPEN, 1987, Nature 330,5556. OKABE, T., J. I. FURIHATA, K. MORISHITAand H. FUJIMORI, 1992, Phil. Mag. Lett. 66,259-64. PAVLOVITCH, A., and M KLEMAN, 1987, J. Phys. A Math. Gen. 20,687-702. PLACHKE, D., T. KUPKE, H. D. CARSTANJEN and R. M. EMRICK,1993, J. Non-Crystall. Solids 153,154,724. POON,S . J., A. J. DRBHMANNand K. R. LAWLESS, 1985, Phys. Rev. Lett. 55,2324-7. QIU, S.-Y., and M. V. JARIC,1993, J. Non-Crystall. Solids 153,154, 221-6. REYES-GASGA, J., A. LARA,H. RIV'EROS and M. JOSE-YACAMAN, 1992, Mater. Sci. Eng. A150, 87-99. R o w , D., 1993, Phil. Mag. B67, 77-96. SAINTFORT,P., and B. DUBOST,1986, J. Phys. France 47, C3-321-30. SASTRY, G. V.S., V. V. RAo, P. ~ M A C H A N D R A R A and O T. R. ANANTHARAMAN, 1986, Scr. metall. 20, 191-3. SCHURER,F? J., B. KOOPMANS,E VANDERWOUDE, 1988. Phys. Rev. B37,507-10. S,D., I. BLECH,D. GRATIASand J. W. C m , 1984, Phys. Rev. Lett. 53, 1951-3. SHEN,Y.,G. J. SUIFLET and S. J. POON,1988, Phys. Rev. B38,5332-7. SHOEMAKER, C. B., 1993, Phil. Mag. B67, 869-81. VANSWLEN, S., J.L. DEBOERand Y. SHEN,1991, Phys. Rev. B43,929-37. SOCOLAR,J.E. S., and P.J. STEINHARDT, 1986, Phys. Rev. B34,617-47. SONG,S., and E.R. RYBA,1992, Phil. Mag. Lett. 65, 85-93. SONG,S., L. WANGand E.R. RYBA,1991, Phil. Mag. Lett. 63, 335-44. SRINIVAS,V., R A. DUNLAP,D. BAHADURand E. DUNLAP, 1990, Phil. Mag. B61,177-88. STEURER, W., 1991, J. Phys.: Condens. Matter 3,3397-410. STEURER, W., T. HAIBACH,B. ZHANG,C. BEBLIand H.-U. NISSEN,1994, J. Phys.: Condens. Matter 6,613-32. STcuRnl W., T. HAIBACH, B. ZHANG,S. KEK and R. LUCK, 1993, Acta Crystallogr. B49,661-75. STEURER, W., and K.H. KUO,1990, Acta Crystallogr. B46,703-12.
.
App. Ch. 4, Refs.
The structure of quasicrystals
411
TANG,Y., D. ZHAO,Z. Luo, N. S ~ andGS. ZHANG,1993, Mater. Lett. 18, 148-50. TODD,J., R. MERLIN, R. CLARKE,K.M. M o m m y and J.D. AXE,1986, Phys. Rev. Lett. 57, 1157-60. TSAI,A.P., A. INOUE and T. MASLIMOTO,1987, Jpn. I. Appl. Phys. 26, L1505-7. TSAI,A.P., A. INOUE and T.MASUMOTO, 1988a, Trans. J I M 29,521-4. TSAI, A.P., A. INOUE andT. MASUMOTO, 1988b, J.J. Appl. Phys. 27, L1587-90 TSAI,A.P., A. INOUEand T. MASUMOTO, 1988c, J. J. Appl. Phys. 27, L5-8. TSAI, A. P., A. INOUEand T. MASUMOTO, 1988d. J. Mater. Sci. Lett. 7, 3226. TSAI, A.P., A. INOUEand T.MASUMOTO, 1989, Mater. Trans. JIM 30,463-73. TSAI,A.P., A. INOUEand T. MASUMOTO,1991, Phil. Mag. Lett. 64, 163-7. TSAI,14.P., A. INOUEand T. MASUMOTO, 1993, Mater. Trans. JIM 34, 155-61. TSAI,A.P., A. INOUE, T. MASUMOTO, A. SATOand A. YAMAMOTO, 1992, Jpn. J. Appl. Phys. 31,970-3. and T. MASUMOTO, 1990, Mater. Trans. JIM 31,98-103. TSAI,A.P., A. INOUS, Y. YOKOYAMA TSAI,A.P., T. MASUMOTO and A. YAMAMOTO, 1992, Phil. Mag. Lett. 66,203-8. TSAI,A.-P., Y. YOKOYAMA, A. INOUE and T. MASUMOTO,1990, Jpn.J. Appl. Phys. 29, L1161-4. TSAI,A.-P., Y. YOKOYAMA, A. INOUEand T.MASUMOTO, 1991, J. Mater. Res. 6,2646-52. WANG,N., H. CHENand K.H.Kuo, 1987, Phys. Rev. Lett. 59, 1010-13. WANG,N., K. K. FUNGand K. H. Kuo, 1988, Appl. Phys. Lett. 52, 2120-22. WANG,N., and K. H. Kuo, 1988, Acta Crystallogr. A44, 857-63. WANG,N., and K.H. Kuo, 1990, Phil. Mag. Lett. 61,6343. WELBERRY, T. R., 1989, J. Awl. Cryst. 22,308-14. WIDOM,M., and R. PHILLIPS,1993, J. Non-Crystall. Solids 153,154, 282-7. YAMAMOTO, A,, 1992, Phys. Rev. B45,5217-27. 1992, Mater. Trans. JIM 33, 1012-9. YQKO'YAMA, Y., A. INOUEand T. MASUMOTO, YOKOYAMA, Y., A.P. TSAI, A. INOUE, T. MASUMOTO and H.S.C m , 1991, Mater. Trans. JIM 32,421-8. ~ N GH.,, and K. H. Kuo, 1989, Sa. Metall. 23,355-8. ZHANG. H.,D.H. WANGand K.H. Kuo, 1988, Phys. Rev. B37,6220-5. ZHANG,Z., H. Q. YE and K. H. KUO, 1985, Phil. Mag. A52, L49-52.
Further reading FUJIWARA, T.,and T. OGAWA,(eds.) 1990, Quasicrystals, Springer Series in Solid State Science 93. GOLDIMAN, A. I., and K.F. KELTON, 1993, Rev. Mod. Phys. 65, 213-30. KELTON, K.F., 1995, in: Intermetallic Compounds - Principles and Practice, eds. J. H. Westbrook and R. L. Fleischer (Wiley, Chichester), Vol. 1, pp. 453491. JANOT.C., 1992, Quasicrystals. A Primer (Clarendon Press, Oxford). JANSSEN, T., 1988, Phys. Rep. 168,55-113. STEURER,W., 1990, 2. Kristallogr. 190, 179-234.
CHAPTER 5
METALLURGICAL THERMODYNAMICS D.R. GASKELL School of Materials Engineering Purdue University West Lafayette, IN 47907, USA
R W Cahn and l? Haasen?, eds. Physical Metallurgy; fourth, revised and enhanced edition 0Elsevier Science B'C: 19%
414
D.R. Gaskell
Ch. 5, 8 1
I . Introduction Metallurgical thermodynamics is concerned with the equilibrium states of existence available to systems, and with the effects of external influences on the equilibrium state. The thermodynamic state of a system is defined in terms of state variables (or state functions) and the state variables occur in two categories; intensive variables such as pressure, P, and temperature, T, the values of which are independent of the size of the system, and extensive variables such as internal energy, U,and volume, V, the values of which are dependent on the size of the system. The simplest equation of state is the ideal gas law,
PV = nRT where n is the number of moles of the gas and R is the universal gas constant. In considering a fixed quantity of ideal gas, only two of the state functions in eq. (1) are independent and the other is dependent. Thus, in a three-dimensional diagram employing P, V and T as ordinates, the equilibrium states of existence of the fixed quantity of gas lie on a definite surface. In any reversible change of state of the gas the path of the process lies on this equilibrium surface, such that, in moving from the initial to the final state, the gas passes through a continuum of equilibrium states. Under such conditions the work, w,done on or by the gas during the process is given by:
and thus the magnitude of w is dependent on the actual process path taken over the equilibrium surface between the final and initial states. In an irreversible process the state of the gas momentarily leaves the equilibrium surface while moving between the initial and final states.
1.1. The First and Second Laws of Thermodynamics When a system undergoes a process in which it moves from one state to another, the change in the internal energy of the system, AU,is given by:
where q is the heat entering or leaving the system and w is the work done on or by the system during the change of state. For an increment of the process the change is: d U = dq - dw.
(4)
Equations (3) and (4) are statements of the FirstLaw of Thermodynamics.By convention, heat entering the system and work done by the system are positive quantities. Equation (3) is remarkable in that, although the individual values of q and w are dependent on the path taken by the system between the initial and final states, their algebraic sum (which is the difference between U,and U,)is independent of the process path. Thus integration of eq. (4) to obtain eq. (3) requires that the process path be known and that the process
Ch. 5, J 1
Metallrrrgical rhemodynamics
415
be conducted reversibly. The Second Law of Thennodynamics states that, for a reversible change of state, the integral of dq/T is independent of the process path. As one of the properties of a state function is that the difference between the values of the hnction in any two thermodynamic states is independent of the process path taken by the system in moving between the two states, the term dq/T is the differential of a state function. The state function entropy, S, is thus defined as:
dS = dq, 1 T . If change in volume against an external pressure is the only form of work performed during a reversible change of state of a closed system, the work performed is given by eq. (2), and substitution of eqs. (2) and (5) into eq. (4)gives: (6) du = T ~ -Spav. Equation (6), which is a combination of the First and Second Laws of Thermodynamics, gives the variation of U (as the dependent variable) with S and V (as the independent variables). From consideration of the difference between reversible and irreversible processes and the Second Law, eq. (6) gives the following criteria for thermodynamic equilibrium in a closed system of fixed composition: (i) S is a maximum at constant U and V; (ii) U is a minimum at constant S and V. Equation (6) involves the extensive thermodynamic properties S and U as independent variables. Although it is possible to measure and, with sufficient ingenuity on the part of the experimenter, to control the volume of a system, experimental control of the entropy of a system is virtually impossible, and consequently the criteria for equilibrium obtained from eq. (6) are not of practical use. From the practical point of view it would be desirable to have an equation as simple in form as eq. (6) but in which the independent variables are the intensive properties P and T, both of which are amenable to experimental measurement and control. Such an equation would also provide a criterion for equilibrium in a constant pressureconstant temperature system.
1.2. Auxiliary thermodynamic functions The required auxiliary state functions are generated by Legendre transformations of U. For example, in eq. (6), written as
u=UCS,v), a Legendre transform, H,of U is obtained using:
U-H v-0
(7)
At constant S, the tangent to the variation of U with V passes through the points U = U, V = V and U=H, V =0. Rearrangement of eq. (7) gives: References: p . 469.
D.R. Garkell
416
Ch. 5 , p l
H=U+PV, which, on differentiation, gives:
dH = dU + PdV
+VU.
(8)
Substitution of eq. (6) into eq. (8) gives:
cW = TdS -+ VdP,
(9)
in which the extensive variable V has been replaced by the intensive variable P . The transform H is called the enthalpy. Writing eq. (9) as H = H(S, P), a Legendre transform, G, of H is obtained as:
T = ( S ) =-H - G p
or: G = H
s-0
- TS,
which, on differentiation, gives:
dG = dH - TdS - SdT = -SdT
+VU,
in which the extensive variable S has been replaced by the intensive variable T. This transform, G, is called the Gibbsfree energy. Being dependent on the independent variables T and P , the Gibbs free energy is the most useful of thermodynamic functions and provides the practical criterion that, at constant T and P, thermodynamicequilibrium is established when the Gibbs free energy is minimized. A third Legendre transform yields the Helmholtz free energy, or work function A, defined as A = U-TS. In a multicomponent system containing n1 moles of component 1, n2 moles of component 2, ni moles of component i, etc.: G = G(T, P , nl, n2,...,ni) and thus,
The derivative
is of particular significance and is called the chemical potential, pi, or the partial molar of the component i. Thus, in view of eq. (ll),eq. (12) can be written as free energy,
q,
ch.5, $ 2
Metallurgical thermodynamics
dG = -SdT
+ VdP +
@ni,
417
(13)
and the equilibrium state of any system undergoing any type of reaction at constant temperature and pressure can be determined by application of this equation.
2. Metallurgical thermochemistry 2.1. The measurement of changes in enthalpy In order to distinguish between the value of an extensive property of a system containing n moles and the molar value of the property, the former will be identified by the use of a prime (3, e.g., with respect to enthalpy, H' =nH. From eqs. (5) and (9), for a process occurring reversibly at constant pressure P: dH' = dq, which, on integration, gives: AH' = qp. Thus, in a system undergoing a process in which the only work performed is the work of expansion or contraction against the constant pressure P, the change in enthalpy, A H f , can be measured as the heat qp entering or leaving the system during the constant pressure process. In the case of heat entering the system the process involves an increase in the temperature of the system and the constant pressure molar heat capacity, c,, is defined as:
The constant pressure molar heat capacity of a system can be measured by the methods of calorimetry. In metallurgical applications the measured values are fitted to an equation of the form cp = a
+ b~ + CT-~.
For example, the constant pressure molar heat capacity of solid silver varies with temperature in the range 298-1234 K as: cP.&(.)
- 21.3
-
+ 8.54 x 10-3T + 1.51x 105T-2J/K mole
and hence, from eq. (14), the difference between the molar enthalpy of solid Ag at a temperature T and the molar enthalpy at 298 K is
Refarences: p , 469.
418
D.R. Gaskell
= 21.3(T - 298) -!-4.27 x 10"(T2
Ch. 5, 52
- 2982)
- 1.51 x 105(L T - L298 )J/mole, which is thus the quantity of heat required to raise the temperature of one mole of solid Ag from 298 K to T. Transformation of a low-temperature phase to a high-temperature phase involves the absorption of the latent heat of the phase change, e.g., the transformation of one mole of silver from the solid to the liquid state at the normal melting temperature of 1234 K requires a heat input of 11.09 kJ. Thus at 1234 K the molar enthalpy of melting of Ag, AH,, is AHm,Ag.1234K
- *Ag(1),1234K - *Ag(s),1234K
-
= 1'09kT*
The molar heat capacity of liquid Ag is independent of temperature, ~ ~ , ~ ~ ~ ,J/K ~=30.5 mole, and the difference between the molar enthalpy of liquid Ag at a temperature T and the molar enthalpy of solid Ag at 298 K is
As chemical reactions involve the absorption or evolution of heat, they also necessarily involve changes in enthalpy. For example, when conducted at 298 K, the oxidation reaction 2Ag(,) + 3 0 2 ( g ) = *g20(,) is accompanied by the evolution of 30.5 kJ of heat per mole of Ag20 produced. Thus,
q = AH = -30.5kJ, or the system existing as one mole of Ag20 has an enthalpy of 30.5 kJ less than the system existing as two moles of Ag and half a mole of oxygen gas at 298 K. As the enthalpies of substances are not measurable quantities, i.e., only changes in enthalpy can be measured (as the evolution or absorption of heat), it is conventional to designate a reference state in which the relative enthalpy is zero. This reference state is the elemental substance existing in its stable form at 298 K and P = 1 atm. In practice the designation of P = 1 atm is relatively unimportant as the enthalpies of condensed phases are not significantly dependent on pressure and the enthalpy of an ideal gas is independent of pressure. Thus, in the above example:
AH,% = HAg$3(~),298
- 2HAg(s),298
- - H02(g),298.
Ch. 5, $ 2
419
Metallurgical fhermodynamics
As HAg(s),298 and Ho(g),298are arbitrarily assigned values of zero, the relative molar enthalpy of Ag,O at i98 K is simply equal to the experimentally-measuredmolar heat of formation of Ag20 at 298 K. At any other temperature T: mT
= HAg20,T
- 2HAg.T
- 3H 0 2 , T
where “p
= ‘p.Ag20 - 2cp,Ag
- T1 cp,02‘
The enthalpy-temperature diagram for the oxidation of silver is shown in fig. 1.
2.2. The measurement of entropy From eqs. (5) and (14), we find:
Thus, the variation of entropy with temperature at constant pressure is obtained from measured heat capacities as
ST = So +
T C
XdT. OT
Nernst’s heat theorem, which is also known as the Third Law of Thermodynamics, states that all substances at complete internal equilibrium have zero entropy at 0 K, Le., S,,=O. Thus, in contrast to enthalpies, the entropies of substances have absolute values. According to Gibbs, entropy is a measure of the degree of disorder in a system. Thus the entropy of the gaseous state is greater than that of the liquid state, which, in turn, is greater than that of the solid state. The transformation of a solid to a liquid at the normal melting temperature, T,, involves the absorption of AH, per mole. Thus, at T,, the molar entropy of the liquid exceeds that of the solid by the molar entropy of fusion, AS,, given by eq. ( 5 ) as:
ASm = AH,,, i T,. This corresponds with the fact that the liquid state is more disordered than the solid state, and ASm is a measure of the difference in degree of order. For simple metals, with similar crystal structures and similar liquid structures, AS, lies in the range 8-16 J/K. This correlation is known as Richard’s rule. Similarly, at the normal boiling temperature, Tb, the molar entropy of boiling, AS,, is obtained from the molar heat of boiling as: References: p . 469.
Ch. 5, 82
D.R. Gaskell
420
1
200
1
400
1
1
600
1
1
800
1
1
1000
1
1
1200
1
l
1400
temperature, K Fig. 1. The enthalpy-temperature diagram for the reaction 2Ag+fO,=Ag,O.
For simple metals ASb= 88 J/K, which indicates that the difference in disorder between the gaseous state at 1 atm pressure and the liquid state significantly exceeds the corresponding difference between the liquid and solid states. The correlation = 88Tb is known as Trouton’s rule. Although the degrees of disorder, and hence the entropies of condensed states, are not noticeably dependent on pressure, the entropy of a gas is a significant function of pressure. As the internal energy, V , of an ideal gas is dependent only on T, an isothermal compression of an ideal gas from P, to P2 does not involve a change in V . Thus, from eq. (3), the work of compression, w,equals the heat transferred from the gas to the isothermal surroundings at the temperature T. This transfer of heat from the gas decreases its entropy by the amount
Ch. 5, $ 2
Metallurgical thennodynamics
42 1
which, from eq. (l), gives:
Thus
Si - S; = nRln(P,/P,), which corresponds with the fact that a gas at high pressure is a less disordered state than a gas at low pressure. As changes in entropy are caused by the transfer of heat, chemical reactions involving heat changes necessarily involve changes in entropy. At 298 K and 1 atm pressure, the molar entropies of Ag,,,, OZk,and Ag,O,, are 42.7, 205 and 122 JK, respectively. Thus the entropy change for the oxidation
at 298 is:
AS = 122 - (2 x 42.7) - (0.5 x 205) = -65.9 J/K mole. This can be viewed in two ways: (i) the entropy decrease is due to the loss of the heat of oxidation from the reacting system, or (ii) the degree of disorder in the system existing as one mole of Ag,O is less than that when the system exists as two moles of Ag and half a mole of oxygen gas at 1 atm pressure. The variation, with temperature, of the entropy change for the reaction is determined by the heat capacities of the reactants and products as:
The entropy-temperature diagram corresponding to fig. 1 is shown in fig. 2. From the definition of Gibbs free energy, eq. (lo), the change in Gibbs free energy due to a chemical reaction occurring at a temperature T, AGp is AG, = AH, - TAS.
Thus, the variation of the change in Gibbs free energy with temperature can be determined from measurement of the variation, with temperature, of the constant pressure molar heat capacities of the reactants and products and measurement of the enthalpy change of the reaction at one temperature. For the oxidation of solid silver, such data give
AG, = -34200
+ 87.9T - 1.76TlnT - 10.8 x 10-3T2
+ 3.2 x 105T-'Jlmole Ag20.
(16)
References: p . 469.
422
Ch. 5, 8 3
D. R. Gaskell
350
300
s g 150 9. C
a,
100 50
I , , , , , , , , , , , I
200
400
600
800
1000
1200
1400
temperature, K Fig. 2. The entropy-temperature diagram for the reaction 2Ag+@, = Ag,O.
3. Phase equilibrium in a one-component system At constant T and P the equilibrium state is thit in which the Gibbs free energy has its minimum possible value. In a one-component system the states of existence available are the gaseous and liquid states and the various allotropic or polymorphic forms of the solid state. At any T and P the state with the lowest Gibbs free energy is the stable state. For the transformation solid + liquid:
AG,,, ( P , T ) = G(ll(P,T ) - G(s,(P, T ) = AH, ( P ,T ) - TAS, ( P , T).
(17)
If AGmis negative, the transformation decreases the Gibbs free energy of the system and hence the liquid is stable relative to the solid. Conversely, if AG, is positive the solid is stable relative to the liquid. As absolute values of enthalpy cannot be measured it follows that absolute values of Gibbs free energy cannot be measured. Thus only changes in G can be measured. The solid and liquid phases coexist in equilibrium with one another in that state at which AGm=O, Le., where G(,,=G,,. From eq. (15), at any pressure P this equilibrium occurs at the temperature T, given by
Ch.5, $ 3
Metallurgical thermodynamics
423
T, = AH,/hS,, and hence T, is the equilibrium melting temperature of the solid at the pressure P. From eq. (lo), G is decreased by decreasing H and increasing S and hence nature prefers states of low enthalpy and high entropy. As H,,, >Hco and S,, >S(s,the enthalpy contribution to G favors the solid as the stable state and the entropy contribution favors the liquid as the stable state. In eq. (17) the entropy contribution to AG is temperature-dependent and the enthalpy contribution is not. Thus, at high temperatures the former contribution dominates, at low temperatures the latter contribution dominates, and at a unique temperature T, the two contributions cancel to make AG=O. For the two-phase equilibrium to exist,
G(l) = G(.$,, and maintenance of the two-phase equilibrium with variation in T and P requires that T and R be varied in such a manner that dG(1, =
qs,
or, from eq. (ll), such that -S(,)dT t yl,dP = -S(,dT t ys)dP, i.e.,
(0 1d~)eq= ('(1)
-))3('
/(~11-
YE))
= Urn/Avrn
As equilibrium between the two phases is maintained, AH,,,= TAS,:
(dP/dT)eq = AH,/TAV,.
(18)
Equation (18) is the CZupeyron equation, which, on integration, gives the variation of T and P required for maintenance of the two-phase equilibrium. Strictly, integration requires knowledge of the pressure and temperature dependences of AH, and AV,. However, for relatively small departures from the state P = 1 atm, T, AH, and AV, can be taken as constants, in which case:
Equation (18) can be applied to condensed phase-vapor phase equilibria by making the approximation AV= V(",- V,,,, phase, V,, and assuming ideal behavior of the vapor phase, Le., V,,,=RT/P, Le.,
Equation (19) is the CZuusius-Clupeyron equation. References:p. 469.
424
D.R. Garkell
Ch. 5, $ 4
If AHb (the molar enthalpy of boiling) is not a function of temperature (which integration of eq. (19) gives requires cHv)=cpg)), 1 n P = --mb RT
+ const.,
and if AH, is a linear function of T (which requires that Acp be independent of temperature) given by AHbsT=AH, + AcpT, integration gives
AH, Ac In P = - -+ -In RT R
T + const.
as either (i) the variation of the saturated vapor pressure with temperature or (ii) the variation of the equilibrium boiling temperature with pressure. Experimentally measured vapor pressures are normally fitted by an equation of the type 1 n P = -A/T
+ B l n T + C.
The solid, liquid and vapor states exist on surfaces in G-T-P space. The solid- and liquid-surfaces intersect at a line (along which G,,,= G(sJand projection of this line onto the basal P-T plane of the G-T-P diagram gives the pressure dependence of T,. Similarly the vapor- and liquid-surfaces intersect at a line, projection of which onto the basal P-T plane gives the variation, with temperature, of the saturated vapor pressure of the liquid. Similar projection of the line of intersection of the surfaces for the solid and vapor states gives the variation, with temperature, of the saturated vapor pressure of the solid. The three lines of two-phase equilibrium in G-T-P space intersect at a point, called the triple point, at which all three phases are in equilibrium with one another. Consideration of the geometry of the intersections of the surfaces in G-T-P space shows that, in a one-component system, a maximum of three phases can exist in equilibrium. Alternatively, as the three phases co-exist in equilibrium at fixed values of T and P the equilibrium is invariant, Le., has no degrees of freedom. The phase diagram for H,O is shown in fig. 3 and a schematic representation of the section of G-T-P space at 1 atm pressure is shown in fig. 4.In fig. 4,the slope of any line at any point is -S for that state and hence the “steepness” of the lines increases in the order solid, liquid, vapor. Also the = - (as/agp =-C / T . curvatures of the lines are
(a2~/a~”>,
4. Chemical reaction equilibrium From eq. (13), at constant T and P, the Gibbs free energy varies with composition in a chemically reacting system as
dG‘ =
c.dni.
The reaction proceeds spontaneously in that direction which involves a decrease in Gibbs free energy, and reaction equilibrium is attained when, thereby, the Gibbs free energy is minimized, i.e., when dG’ =O. Consider the water-gas reaction
Ch. 5 , 54
425
Metallurgical thermodynamics
lo
1
r-
--------
/
10-1 -
normal melting temperature
E c td
frn
-
v)
SOLID triple point, P = 0.006 atm
2
T= 0.0075'C
Q
VAPOR
10-4
I 1
10-5
20 40 60 80 100 120 140
-60 -40 -20 0
temperature, "C Fig. 3. The phase diagram for H,O.
The isothermal transfer of a mole of ideal gas i from the pure state at the pressure Pi and temperature T to an ideal gas mixture at the partial pressure pi involves a change in Gibbs free energy:
AG = q.- Gi = RTln(pi/4).
(21)
Rejhmes: p . 469.
Ch. 5, $ 4
D.R Gaskell
426
u
u 0
I I I I I I I
E
SOLID
LIQUID
I I
I
I
I
I
I
I
I
I
I
I
'
I \
temperature, "C Fig. 4. Schematic representation of the variations of G,,, Gm and G(")with temperature at P = 1 atm for H,O.
Again, as only changes in Gibbs free energy can be measured, it is convenient to select a standard state for the gas and consider the Gibbs free energy of the gas in any other state in terms of the difference between the free energy of the gas in this state and the free energy of the gas in the standard state. The standard state for an ideal gas at the temperature Tis the pure gas at 1 atm pressure and in this state the Gibbs free energy is Thus eq. (21) can be written as: the standard free energy, designated
e.
=G :
+ RTlnpi.
(22)
Substitution of eq. (22) into eq. (20) and rearrangement gives: (G& + GL2 - G&,
PH2 PCO2 - G&) = -RT In ___.
PHZOPco
Being the difference between the standard free energies of the products and the standard free energies of the reactants, the left-hand side of eq. (23) is termed the standardfree energy for the reaction at the temperature T, AG,", and, being dependent only on T, it has a definite fixed value at any T. Consequently the quotient of the partial pressures of the reactants and products occurring in the logarithm term on the right-hand side of eq. (23) has a fixed value at any T. This term is called the equilibrium constant, Kp, and hence the equilibrium state in any reacting system is such that
ch.5,04
Metallurgical thernmdynamics
AG; = -RTIn K,.
427
(24)
For the general reaction
Dalton’s law of partial pressures in an ideal gas mixture gives
pi = XiP, where X, being the ratio of the number of moles of i in the gas to the total number of moles of all species, is the moZefpactian of i in the gas and P is the total pressure of the gas. Thus
where K, is the equilibrium constant expressed in terms of the mole fractions of the reactiants and products occurring at reaction equilibrium. From the definition of AG:, Kp is independent of pressure and hence, from eq. (25), K, is only independent of pressure if c+d-a-b=O. From eqs. (24) and (15):
The reaction of a moles of CO with b moles of H,O produces x moles of each CO, and H, and leaves (a - x ) moles of CO and (b - x ) moles of H,. Thus at any point along the reaction coordinate in a reacting mixture at the constant pressure P:
References: p . 469.
428
Ch. 5, $ 4
D.R Gmkell
and at reaction equilibrium:
Pco2Pn2 PCOPH20 (a -
X2
- 1‘
=
K, = exp(-) 36400
8.3144T
(-)
exp -32.0 8.3144
If one or more of the reactants and/or products occurs in a condensed state the attainment of equilibrium involves both phase and reaction equilibrium. For example, at a temperature T the equilibrium (26)
2Ag(s)+ 3 0 2 ( g , = Ag20,,, requires the establishment of the phase equilibria and Ag20, = Ag,o(,),
Ag@)= &(,)
and, in the vapor or gas phase, requires establishment of the reaction equilibrium
2Ag(”)+ 30qg)= Ag,O(,).
(27)
Conditions for the phase equilibria are pAg=pig (the saturated vapor pressure of solid 0 silver at temperature T ) and pAg,.= pAgZO (the saturated vapor pressure of solid Ag20 at temperature T), and thus, as the equilibrium constant K for the vapor phase reaction, given by eq. (27), has a fixed value at temperature T, the equilibrium oxygen pressure, po2, is uniquely fixed by:
Alternatively, reaction equilibrium in the vapor phase requires that: 2GAg(v)
+
3
‘02(g)
--
- GAg20(~)’
(28)
and the two-phase equilibria require that:
and ‘A&O(v)
-- ‘Ag20(s)’
(30)
From eq. (1l), at constant T, dG = VdP, and hence eq. (29) can be written as: Gpc,) + RTlnPZ = GL(.) +
p
v*g(s)de
(31)
where G&) is the standard molar free energy of solid Ag at temperature T. The integral
Ch. 5, $5
Metallurgical thermodynamics
429
on the right-hand side of eq. (31) is negligibly small and hence eq. (31) can be written as: G:g(v) + RT In
Pig = G:g(s,.
Similarly, eq. (30) can be written as:
Substitution of eqs. (32) and (33) into eq. (28) gives:
+
2G:g(s) + G:z(g) + RTln Pt2* = G&20(s).
where AG: is the standard free energy change for the reaction given by eq. (26) and po2(es,nis the value of po, required for equilibrium between Ag,,, Ag,O,, and oxygen gas a t temperature T. The variations of AH:, -TAS; and AG: [given by eq. (16)] are = 1 atm, at which temperature AGo=O. shown in fig. 5. Thus, from eq. (34), pol(en,485K) At T485 K,AG: is a positive quantity and hence polces.n > 1 atm.
5. Ellingham diagrams In 1944 ELLINGHAM published diagrams showing the variation, with temperature, of the standard free energies of formation of a number of oxides and sulfides, and pointed out that these diagrams “would show at a glance the relative stabilities of the various substances within a given class at any temperature, and would thus indicate, in a direct fashion, the range of conditions required for their reduction to the corresponding elements. It would provide, in fact, what might be described as a ground plan of metallurgical possibilities with respect to the reduction of compounds of the specified class”. Such diagrams, which are now available for a wide range of classes of compounds, are known as EZZinghum diagrams, and the Ellingham diagram for oxides is shown in fig. 6. (See also ch. 14, p 2.1). In order to facilitate comparison of the stabilities of the various oxides, the standard free energies are for the reaction (2-dy)M + 0,= (2/y)MXOp Le., for reactions involving the consumption of one mole of 0,. By choosing this basis:
and hence, in addition to being a plot of AG: versus temperature, the Ellingham diagram is a plot of the variation, with temperature, of the oxygen pressure, po2(q,n, required for equilibrium between the metal and its oxide. The free energy change for the change of state 02(T, P = 1 atm) 02(T,P = p,) is: References: p . 469.
Ch. 5, $ 5
D.R. &&ell
430
l-----
50 40
-
30
-
20
-
AH"
-40'
I
300
I
I
I
500
I
700
I
I
900
I
I
1100
temperature, K Fig. 5. The variations of AZ& -TA$ and AQT with temperature for the reaction 2Ag +& =Ag,O.
AG, = RT In po, , and thus, in the Ellingham diagram, lines of constant po, radiate from the origin, A@ = 0, T = 0 K, with slopes of R In pol. Consequently, a nomographic scale of p4 can be placed on the edges of the diagram and pol(es)at any point on an Ellingham line is obtained as the reading on the nomographic scale which is collinear with the given point and the origin of the diagram. The Ellingham diagram is thus a stability diagram, in that any point in the diagram T) lying above the Ellingham line for a given oxide is a state in which pozm >P,,(~,
Ch. 5, 5.5
431
Metallurgical thermodynamics
0 -1
-2
-3 -4 -6 -8 -10
-12
44 -16 -18
-20
-22 -24
-26 -28
ZL. .KELVIN WpO2 ( a m
temperature,%
\.loo
\-8o
\a
yo
-30
p \-38 \-34
Fig. 6. The Ellingham diagram for several oxides.
and hence, in d l states above the line the oxide is stable relative to the metal. Conversely, any point lying below the Ellingham line for the given oxide is a state in whichpOo ~ p , , ( ~and , hence, below the line, the metal is stable relative to the oxide. f i e Ellingham line thus divides the diagram into stability fields and, if it is required that a given oxide be reduced, the thermodynamic state must be moved from a point above the Ellingham line for the oxide to a point below the line, Le., must be moved from a positlion within the oxide stability field to a position within the metal stability field. The magnitude of AG: is a measure of the relative stability of the oxide and hence, with increasing stability, the Ellingham lines occur progressively lower in the diagram. Consequently, in principle, the element A can reduce the oxide B,O,,,if, in the diagram, References: p. 469.
432
D.R. Gaskell
Ch. 5, $ 5
the Ellingham line for A,O,., lies below that for BxOy. Over the ranges of temperature in which no phase transitions occur the Ellingham lines are virtually linear, being given by
AG; = A + BT. In this expression A, the intercept of the line with the T=O K axis, is identified with AH", the standard enthalpy change for the oxidation, and B, the slope of the line, is identified with -A$', the standard entropy change for the reaction. The Ellingham lines for the oxidation of solid and liquid metals are more or less parallel with one another, with slopes corresponding to the disappearance of one mole of oxygen gas in the standard oxidation equation. Consequently, the stabilities of these oxides are determined primarily by the magnitudes of their enthalpies of formation. At the temperature of a phase change the slope of the Ellingham line changes by an amount equal to the entropy change for the phase transition. The slope increases at the transition temperatures of the metal and decreases at the transition temperatures of the oxide. These changes in slope are most noticeable at normal boiling temperatures, e.g., at 1090°C the slope of the Ellingham line for MgO increases by 190.3 J/K, which is the entropy of boiling of 2Mg, and at 1484°C the slope of the Ellingham line for CaO increases by 174.2 J/K, the entropy of boiling of 2Ca. Carbon is unique in that it forms two gaseous oxides, CO and CO,, and the positions of the Ellingham lines for these oxides are of particular significance in extraction metallurgy. The Ellingham line for CO has a negative slope due to the fact that the oxidation 2C+O, = 2 c o
involves the net production of one mole of gas, and, because the oxidation
c t 0,
= co,
does not involve a change in the number of moles of gas, the Ellingham line for CO, is virtually horizontal. The enthalpy change for the oxidation of C to form CO as C+fO, = CO is -1 11 700 J and the enthalpy change for the oxidation of CO to CO, as CO +$O, = CO, is -282 400 J. Thus the standard enthalpy change for the Ellingham line for CO is 2 x (-1 11700)=-223400 J and the standard enthalpy change for the Ellingham line for CO, is (-11 1700)-I-(-282400) =-394100 J. Thus, on the basis that the stability of an oxide is determined primarily by the magnitude of AH", it would appear that CO, should be more stable than CO. However, as the Ellingham line for CO has a negative slope, which means that the stability of CO increases with increasing temperature, the Ellingham lines for the two oxides intersect. Consequently, although CO, is more stable than CO at lower temperature, the reverse is the case at higher temperatures. The gaseous phase in equilibrium with solid carbon is a CO-CO, mixture in which the ratio pco/peo, increases with increasing temperature. For a total pressure of 1 atm, the equilibrium gas contains less than 1% CO at temperatures less than 400"C, contains less than 1%CO, at temperatures greater than 980°C, and is an equimolar mixture at 674°C. The "carbon
Ch. 5, $ 5
Metallurgical t h e d y n a m i c s
433
line" in the diagram, which is the continuum of states in which carbon is in equilibrium with a CO-CO, mixture at 1 atm pressure, follows the CO, Ellingham line up to about 400°C and then curves down gently to tangentially meet and join the Ellingham line for CO at about 1OOO"C. Along the carbon line the ratio p,.,/pm2 is fixed by the equilibrium
c + eo, = 2c0, and, by virtue of the equilibrium
co++o,=co, the oxygen pressure is also fixed. Thus the carbon line divides the other oxides into two classes, those with Ellingham lines which lie above the carbon line, and those with Ellingham lines which lie below the carbon line. With respect to the former class, the carbon line lies in the stability field of the metal and hence carbon is a potential reducing agent for these oxides, whereas, with respect to the latter class, the carbon line lies in the oxide stability field and hence carbon cannot reduce the oxide, Furthermore, if the Ellingham line for a metal oxide intersects the carbon line, the temperature of intersection is the minimum temperature at which the oxide may be reduced by carbon. Thus, for example, FeO cannot be reduced by carbon at temperatures less than 675°C. Whether or not carbon can be used as a reducing agent is determined by the stability of any carbide phase which may form, Le., by the sign of the standard free energy for formation of the carbide from metal and carbon. For example, in the Ellingham diagram the carbon line intersects with the Ellingham line for SiO, at 1676"C, and hence above this temperature liquid Si is stable relative to SiO, in the presence of C and its equilibrium CO-CO, gas mixture at 1 atm pressure. However, for the reaction
Si(,)+ C = Sic, the standard free energy change is AG,"=-122600+37.02' J and hence Sic is stable relative to liquid Si in the presence of carbon at 1676°C and P= 1 atm. The stability fields in the system Si-0-C at 1676°C are shown in fig. 7 as functions of log pco and log pa,. Line A is the variation of pco and pco, required for the equilibrium
+
Si(,) 2C0, = SiO,
+ 2CO.
Line B is the corresponding variation required for the equilibrium
Si(,)+ 2CO = Sic + CO,, and line C is the variation for the equilibrium
+
Sic 3C0, = SiO,
+ 4CO.
These lines divide the diagram into stability fields for Si, Sic and SiO, and meet at the values of pm and pCs2 required for the four-phase equilibrium involving the three condlensed phases Si, Sic and SiO, and the CO-CO, gas phase. Line D is the variation of pco and pco, required for the equilibrium between carbon and the gas phase at 1676°C Rtferences:p . 469.
434
Ch. 5, 8 5
D.R. Gaskell
Si02
-10 -12 -14 -10
-8
-6
-4
-2
0
log PCO Fig. 7. The stability diagram for the system S i - O C at 1949 K.
and, as such, represents the compositions of CO-CO, gas mixtures which are saturated with carbon. The field below line D is designated “unstable” gas, as any gas mixture in this field is supersaturated with carbon and hence will spontaneously undergo the carbon deposition reaction
2‘0
+ ‘(graphite)
+ cop
until, thereby, the composition of the gas lies on line D. The dashed line is the (pco+ pco2)= 1 atm isobar. Consequently, the system containing solid carbon and a gas phase at 1 atm pressure exists at the state a, and as this state is in the field of stability of Sic, SiO, is not reduced to Si by carbon at 1676°C. However, if the standard free energy for formation of S i c had been positive, lines B and C would have occurred below line D in the diagram and, as shown by the dashed-dotted extension of line A, the equilibrium Si-SiO,-C would occur at the state a, which is the state of intersection of the carbon line with the Ellingham line for SiO, in the Ellingham diagram.
Ch. 5, $ 6
435
Metallurgical thermodynamics
6. The thermodynamicproperties of solutions 6.1. Mixing processes The relationship between entropy and the “degree of mixed-up-ness” is quantified by Boltzmann’s expression as:
S’ = k In W, where S’ is the entropy of the system and W is the number of microstates available to the system”. In the simplest of mixing processes, W is the number of distinguishable arrangements of the constituent atoms on the sites available to them. Consider the mixing of NA atoms of solid A and NB atoms of solid B as the process: state 1 + state 2, Le., unmixed A and B + mixed A and B. In state 1, interchange of the positions of A atoms in the crystal of pure A and/or interchange of the positions of B atoms in the crystal of pure B does not produce a distinguishably different arrangement and hence W sub1 = 1. However, the NAatoms of A and NB atoms of B can be placed on the NA+ NB lattice sites of the mixed crystal are distinguishable. Thus (state: 2) in (NA+ NB)!ways, of which (NA+ NB)!/NA!NB! A + NB)! w, = ( NNA!NB!
Thus, for the process: AS’ = ,Ti - S: = kln W,- kln
= kln (NA + NB)!
(35)
NA!NB!
If NA and NBare sufficiently large numbers, Stirling’stheorem can be applied as
NB)!= (NA+ NB)ln(NA+ NB)- N A *nNA- NB NA!NB! = -NAIn XA - NBIn X,,
In (NA
where, respectively, XA and X , are the mole fractions of A and B in the mixed crystal. Thus, the change in entropy, As””, due to mixing, is
A S M = k In ( N AIn X,
+ NBIn XB),
and, if NA+ NB =No(Avogadro’s number) then the molar entropy of mixing is
* The equivalence between this definition of entropy and the definition in terms of heat flow (51.1) is demonstrated in general terms in many texts; a particularly clear treatment is provided in ch. 2 of FAST’Sbook (see bibliography). References: p. 469.
436
D.R. Gaskeil
+ XBIn XB).
ASM = -R( X, In X,
Ch. 5, $ 6
(36)
This increase in entropy is caused by the increase in the number of spatial configurations made available to the system as a result of the mixing process and, hence, is conjigurational in origin. If there is no change in enthalpy on mixing, the Gibbs free energy change due to the mixing process is given by
AGM = -TASM = RT(XAIn X,
+ X, In X,).
(37)
Alternatively, consider the following. Consider that p l and p,” are the saturated vapor pressures of pure A and pure B at temperature T and that pAand pBare the partial pressures of A and B exerted by the mixed crystal (or solid solution) of composition X, at temperature T. Consider that one mole of A is isothermally evaporated from pure solid A to form A vapor at the pressure p i , that the mole of A vapor is isothermally expanded to the pressure p , and is then isothermally condensed into a large quantity of the solid solution. As the evaporation and condensation processes are conducted at equilibrium, they do not involve any change in Gibbs free energy and hence the change in Gibbs free energy for the three-step process is simply that caused by the change in pressure from 0 P A to P A , is.,
AG = ’*(in
the solution)
- ~ i ( p u r e )=
(PA
/pi)*
Similarly, for the corresponding three-step process for B,
Thus, for the mixing of nA moles of A and nB moles of B:
AG’ = G’(so1ution) - G’(unmixed A and B)
+
= (nACA nBCB) - (n,G; = nA(CA - G:)
+ nBGi)
+ nB(cB- G:),
which, from eqs. (37) and (38), can be written for one mole of solution as
A G =~ R$X, ln(p,/p;)
+ X,
ln(pB/pi)].
(39)
Comparison of eqs. (37) and (39) indicates that, if the mixing process does not involve a change in enthalpy,
Equation (40) is an expression of Raoult’s Law and a solution conforming with this behavior is said to exhibit Raoultian ideal behavior. If the energies of the pure states and the solution are considered to be the sums of the pair-wise bond energies between neighboring atoms, Raoultian ideal mixing requires that:
Ch. 5, 56
Metallurgical thermodynamics
431
where Em, EM and E B B are the pair-wise bond energies of A-By A-A and B-B pairs, respectively. If the condition given by eq. (41) is not met, the isothermal mixing process is accompanied by the evolution or absorption of heat, which, for mixing at constant pressure, represents a change in the enthalpy of the system. In such a situation random mixing of A and B atoms does not occur and hence the entropy of mixing is no longer given by eq. (36). Any change in the enthalpy on mixing arises from a redistribution of the atoms among their quantized energy levels and this gives rise to a change in the thermal (as distinct from the configurational) component of the entropy of the system. Boltzmann’s equation can be written as
where Wan, is the number of distinguishable ways in which the atoms can be distributed on the available sites and Wtb, is the number of ways in which the energy of the system can be distributed among the particles. Thus, for the mixing process,
and hence AS’ is only given by eq. (35)if Wt,,-,(l) = WM(2), i.e., if no redistribution of the energy occurs, and hence no change in enthalpy occurs. This condition is required for Raoultian ideal mixing. If
the solution exhibits a tendency towards ordering, i.e., towards maximizing the number of A-B contacts, and if
the solution exhibits a tendency towards clustering or phase separation, i.e., towards minimizing the number of A-B contacts. Configurationalentropy is responsible for the occurrence of vacancies in metals. Consider a perfect single crystal containing N atoms on N lattice sites. If a single atom is removed from a lattice position within the crystal and is placed on the surface of the crystal, random placement of the vacancy on N + 1 sites gives rise to a configurational entropy of
S = kln-.
( N + l)!
N!
This process involves an enthalpy change AHv and, as the vibration frequencies of the nearest-neighbor atoms to the vacancy are altered, a change occurs in the thermal entropy, AS,,,.Thus, for the formation of N, vacancies, References:p . 469.
438
Ch. 5, 56
D.R. Gaskeil
AG' = AH'
- TAS'
= N,AH,
- N,AS,,T + kT In
= Nv(AH, - N,AS,,)
(N+N,)! N!N,!
+
N
+ N , In- N N+vN,
1.
The formation of vacancies in an initially perfect crystal is thus a spontaneous process which proceeds until, thereby, the Gibbs free energy of the crystal is minimized, in which state
From eq. (42), this condition occurs when
--N
+ N,
The fraction of vacant sites in a crystal can be determined from simultaneous measurement of the thermal expansion of a sample, A V l , and the change in the lattice parameter, Audu,,, as measured by X-ray diffraction (see ch. 18, 52.2.2.2). As the former is influenced by both the increase in the average spacing between lattice planes and the creation of vacancies, and the latter is a measure only of the average spacing between planes, the increase in the fraction of vacant lattice sites is proportional to the difference between A V l and Audu,. Measurements of this type on aluminum give:
N
+ N,
from which AH,=73.3 kJ/mole and AS,=20 J/K mole. At the melting temperature of 660°C this gives the fraction of vacant sites as 9 x lo4. The thermodynamic properties of solutions which do not exhibit Raoultian ideal behavior are dealt with by introducing the concept of activity. The activity, a , of the component i in a solution is defined as: ai = pi/Pi
0
(43)
and, from eq. (40),is equal to the mole fraction, Xi, in a Raoultian ideal solution. Thus, the molar free energy of formation of a binary A-B solution, AG", is given by
AGM = RT(XA In uA + X, In a,).
(44)
The free energy of formation of n moles of a solution, AG'', can be written in terms of the partial molar free energies of mixing of the components as:
where AGM = - GP (the difference between the molar free energy of i in the solution and the molar free energy of pure i ) is termed the partial molarfree energy of mixing of i. The partial molar free energy of mixing of i and the molar free energy of formation of the solution are related as:
A G = ~ A G +~ (1- xi) T.P
Comparison of eqs. (39) and (45) shows that in a Raoultian ideal solution
AGM = RT In Xi, and comparison of eqs. (39) and (44) shows that, generally,
AGM = RTlna,.
(47)
A typical ideal variation of AGMwith composition is shown in fig. 8. In this figure the tangent drawn to the free energy curve at any composition intercepts the XA=1 axis at AGf and intercepts the X,,= 1 axis at A ?:. This construction is a geometric representation of eq. (46). Also, as Xi + 0, ai + 0 and hence, from eq. (47), A??: + -, i.e., the vertical axes are tangents to the curve at its extremities. The relationship between the variaFions of the tangential intercepts with composition is given by the Gibbs-Duhern equafion:
X,d In U,
-t XBdhl U B =
0.
(48)
Usually, the activity of only one component of a solution is amenable to experimental measurement, and the activity of the other component, and hence AGM,are obtained from integration of the Gibbs-Duhem equation. The activity coeflcient, yi, is defined as yi = a/Xi and hence eq. (44)can be written as: AGM = RT(X, In X,
+ X, In XB)+ RT(XAIn yA + XBIn yB).
(49)
The first term on the right-hand side of eq. (49) is the molar free energy of formation of a RaouItian ideal solution, AG'jd, and the second term, being the difference between the actual molar free energy of solution and the ideal value, is called the excess molar free energy of mixing, G"'. 6.2. Regular solution behavior
A regular solution is one which has an ideal entropy of mixing and a nonzero enthalpy of mixing. The properties of such a solution are best examined by means of a References: p . 469.
440
Ch. 5, 8 6
D.R. Gaskell
E
k II
A
0.6
0.4
0.2
0.8
B
mole fraction of B Fig. 8. The variation of AGMwith composition in an ideal system at loo0 K.
simple statistical model of the mixing of NA atoms of A and N B atoms of B. If the internal energy, U‘, of the solution can be taken as the sum of the pair-wise bond energies then
’‘
= A‘B
EAB
+ A‘A
EAA
+ B‘B
EBB
(50)
9
where Pi is the number of i-j pairwise bonds and E@is the energy of the bond relative to i and j at infinite separation. If the coordination number of an atom is z, the number of bonds involving A atoms, NAz,is given by 2PAA+PABand, similarly, the number of bonds involving B atoms, NBz,is given by 2PBB +PAB.Thus:
Pfi = 3NAZ- 3 PAB and PBB = 3N B z
-4
PmI
substitution of which into eq. (50) gives:
u’ = 3 N A Z E A A + 3N B z E B B + p A B [ E M - (Efi f E B B ) / 2 ] * The first two terms on the right-hand side represent the internal energies of NAatoms of A and NB atoms of B before mixing and hence, for the mixing process:
Ch. 5, $ 6
AU'
Metallurgical thermodynamics
= P,[E,
- (E, + &,)/2].
441
(5 1)
If the mixing process, conducted at constant pressure, does not involve a change in volume, then, as PAV' =0, AH' = AU' and eq. (51) is the expression for the enthalpy of mixing. As random mixing of the atoms is assumed, the number of A-B bonds is calculated as the product of the probability of occurrence of an A-B pair and the number of pairs of atoms. The former is given by:
2
NA NB NA+NB NA+NB'
and the latter is $(NA+NB)z,and hence:
For the mixing of nAmoles of A (=nANoatoms of A) and nB moles of B (=nBN,atoms of B), eq. (52) becomes:
or, per mole of solution: AHM = 'AxBNO2[
EAB
- (EAA
+
If IE,I>I(E,+EBB/21, AHM is negative, which leads to exothermic mixing, and if 1EI, < I(E,+EBB/21, A H is positive, which leads to endothermic mixing. On the other hand, if EABis the average of EM and EBBI AH is zero and Raoultian ideal mixing occurs. For any given system,
'
= NOz[EAB - (EAA + EBB)/2]
is a constant, and hence, in a regular solution, A P is a parabolic function of composition, given by: AHM = 'XAXB, and
ASM = -R(XA In XA + XBIn XB).
(53) (36)
For any extensive thermodynamic property Q, the relationship between AGY and AQMin a binary system is given by:
and thus, in a regular solution, from eq. (53):
References: p . 469.
442
Ch. 5, 16
D.R. Gmkell
AFy = a(1-xi)2, and from eq. (36):
AqM = -RInXi. The partial molar free energy of mixing of i can be expressed variously as
- TAqM= AqM*id+
AqM=
= RTInXi
+ RTln yi,
and hence, in a regular solution:
9" = bqM = ~ ~ l n=ya(i , - xjr. Consequently, the limiting values of yi as Xi + 1 and Xi+ 0 are unity and exp(CI/RT), respectively; i.e., with increasing dilution, the solvent approaches Raoultian ideal behavior and the activity coefficient of the solute approaches a constant value designated 7:. The tendency of yi towards a constant value as Xi + 0 is expressed as Henry's Law, i.e.: 'yi
+ 7:
asxi
+ 0,
and if yi is constant over some finite range of composition of dilute solution of i, component i is said to exhibit ideal Henrian behavior in this range, its activity being given by:
Application of the Gibbs-Duhem relation, eq. (48), shows that, over the composition range in which the solute B exhibits ideal Henrian behavior, the solvent A exhibits ideal Raoultian behavior. The occurrence of Henrian ideal behavior gives rise to the concept of the Henrian standard state, illustrated in fig. 9 which shows the activity of B as a function of composition in the system A-B. The Raoultian standard state is pure B, located at the point R where a, = 1. If, however, pure B behaved as it does in dilute solution in A, extrapolation of its activity along the Henry's Law line would give an activity of y; in the hypothetical pure state at X, = 1, relative to the Raoultian standard state. This hypothetical pure state is the Henrian standard state, located at the point H in fig. 9, and, relative to this standard state, the activity of B in any solution, h,, is
k
= fflxB,
where fB is the Henrian activity coeficient. In the range of dilute solutions over which B exhibits Henrian ideal behavior, fe = 1 and hence:
hg = x,. If the vapor pressure of B in the Raoultian standard state is p i , then the vapor pressure of B in the Henrian standard state is y i p ; , and hence the change of standard state,
Ch. 5 , 57
443
Metallupgical thennodynamics
1.o
0.8
0.6
aB
1.o
0.4
hB 0.5
0.2
-
n
0
0.4
0.2
A
0.6
0.8
mole fraction of B, XB
0 1.0 B
Fig. 9. Illustration of the Raoultian and Henrian standard states.
i7. i%e thermodynamic origin of phase diagrams In the definition of activity, given by eq. (43), pp is the vapor pressure of pure i at the temperature of interest. However, depending on the convenience of the situation, either pure solid i or pure liquid i can be chosen as the standard state. At temperatures below the triple point, & , l i d ) < P&q"id)y and so the activity of i in a solution, relative to pure solid i as the standard state, is larger than the activity relative to pure liquid i as the standard state. Conversely, at temperatures higher than the triple point temperature the reverse is the case. The activities on the two activity scales are related as 0 'i(reiative to solid standard state) %(relative to liquid standard state)
=-Pi(1iquid) = exp(AG;,JRT). 0 Pi(so1id)
References: p. 469.
444
D. R. Gaskell
Ch. 5, 57
Consider the molar free energies of mixing in the system A-E, the phase diagram for which is shown in fig. loa. For simplicity of discussion it will be assumed that both the solid and liquid solutions exhibit ideal Raoultian behavior. The molar free energies, at temperature T, are shown in fig. lob. Pure liquid A and pure solid B are chosen as the reference states and 0 0 are located at points a and b respectively. G:(d is located at c, where G&, - GAG) = -AGmA at temperature T, and G& is located at d where G&- G&) =AG:, at temperature T. Thus, relative to unmixed pure liquid A and pure solid B as the reference state, the molar free energy of the unmixed pure liquids (given by line ad) is X,AG:,B and the corresponding free energy of the unmixed pure solids (given by line eb) is -X,AG&. Upon mixing to form Raoultian ideal solutions, the molar free energies decrease by LRT(XA lnXA+XBlnX,I and hence, relative to the chosen reference state: AG'(solid
solutions) = -XAAG;,,
+ RT(X, In X, + X , In XB),
and AGM(liquidsolutions) = X,AG:,
+ RT(XAIn X, + X, In XB).
The double tangent drawn to the two free energy curves touches the curve for the solid solutions at g and the curve for the liquid solutions at f, with the intercepts at X, = 1 and X, = 1 being e and h respectively. As the equilibrium state is that of minimum free energy, points f and g divide the composition range into three regions. At compositions between a and f the homogeneous liquid solution has the lowest possible free energy and at compositions between g and b the homogeneous solid solution has the lowest possible free energy. However, at compositions between f and g, a two-phase mixture of liquid solution of composition f and solid solution of composition g, the free energy of which lies on line fg, has a lower free energy than both the homogeneous solid solution and the homogeneous liquid solution. Thus point f is the limit of solution of B in liquid A and g is the limit of solution of A in solid B, and so points f and g are, respectively, the liquidus and solidus compositions at temperature T. Furthermore, for phase equilibrium:
CA(inliquid solution f) = GA(insolid solution g), and CB(inliquid solution f) = CB(insolid solution g) or
AGF(in liquid f) = AG:(in
solid g),
AGf(in liquid f ) = AG:(in
solid g).
and
These requirements state that, for phase equilibrium, the tangent to the molar free energy curve for the liquid solutions at the liquidus composition f is also the tangent to the molar free energy curve for the solid solutions at the solidus composition g. Geometrically, this condition is such that, simultaneously,
Ch. 5, $ 7
Metallurgical themtodynamics
445
liquid solutions
solid solutions
(a)
I
- AG:,A
1. e
A
mole fraction of B
B
Fig. 10. (a) The phase. diagram for the system A-B. @) The ideal free energy of mixing curves for the system A-B at temperature T.
c a + a e = c e and db+bh=dh,
(54)
where: ce = AEf (relative to solid A as the standard state) = RT In XA(at the composition g), ae = AZZ (dative to liquid A as the standard state) = RT In XA (at the composition f), dh = AEF (relative to liquid B as the standard state) = RT In X, (at the composition f), and bh = A??: (relative to solid B.as the standard state) = RT In X, (at the composition g).
References: p . 469.
446
a.5, $7
D.R. Gaskell
Thus eqs. (54)become: AG:,
+ RT In X, (liquidus) = RT In XA(solidus),
and
-AG:,
+ RT In X,(solidus) = RT In XB(liquidus).
As X,(liquidus) +XB(liquidus)= 1 and X,(solidus) +X,(solidus) = 1, the solidus and liquidus compositions (in a Raoultian system) are thus uniquely determined by the values of AG;., and AG;,, as: xA(liquidus)
-
1- exp(-AG;,,
- [exP(-AG:,B/RT)][exP(AG;,A
/ RT)
(55)
/RT)]
and
The phase diagram for the system Si-Ge, calculated from eqs. (55) and (56) and the known variations of A&m,si and AG:,oe with temperature, is compared, in fig. 11, with the liquidus and solidus lines determined experimentally by thermal and X-ray analysis. As is seen, the behavior in the system is very close to Raoultian. Raoultian behavior is very much the exception rather than the rule, and even complete mutual solid solubility between A and B requires that A and B have the same crystal structure, similar atomic sizes, similar electronegativities, and similar valences. The requirement of similar atomic size arises from the introduction of a strain energy into the lattice when the solvent and substitutional solute atoms are of differing size. This strain energy always increases the Gibbs free energy and, hence, can significantly influence the phase relationships in the system. It is found that terminal solid solutions extend only a few atomic percent into a binary system if the atomic diameters differ by more than 14%. Significant differences in electronegativity cause the formation of intermetallic compounds such as Mg,Si, Mg,Sn and Mg,Pb, and differences in valences can cause the formation of electron compounds such as occur in the systems Cu-Zn and CuSn. Although Cu and Ag are chemically similar, the atomic radius of Ag is 13% larger than that of Cu and hence, as shown in fig. 12a, Cu and Ag form a simple eutectic system. In this system it is presumed that Ag exhibits Raoultian ideal behavior in the Agrich a-solid solution and that Cu exhibits Raoultian ideal behavior in the Cu-rich P-solid solution. Consequently Cu in a and Ag in /3 exhibit Henrian ideal behavior and, at 1000 K, the activities of the components, relative to the pure solids as standard states, are as shown in fig. 12b. At 1000 K, saturation of the a-phase with Cu occurs at X+.=O.9, and hence, as Ag obeys Raoult's law in the a-phase, aAg=0.9 at this compos~tion.Phase equilibrium between a saturated with Cu and /3 saturated with Ag requires that the
Ch. 5, 9 8
447
Metallurgical thermodynamics
I
I
I
Fig. 11. Comparison of the phase diagram for the system Si-Ge as determined experimentally by X-ray and thermal analysis, with that calculated assuming Raoultian ideal behavior in both the solid and liquid solutions.
activities of both Ag and Cu be the same in both phases, and hence aAg=0.9in the Agsaturated &phase of composition X,, = 0.04. Similarly, a, = 0.96 in the Ag-saturated fi (at , X = 0.96) and in the Cu-saturated a (at XAg= 0.9). Thus, in the a-phase, Henrian behavior of Cu is given by: a, = 9.6Xc,,
(57)
and in the &phase, Henrian behavior of Ag is given by: aAg= 22.5XA,.
8. Reaction equilibrium involving solutions and the Gibbs phase rule 8.1. The dependence of the equilibrium state on activity
In $4 it was shown that, at constant temperature and pressure, equilibrium is established in the reaction
aA + bB = cC + m>, when
aGA+ bcB = cG,+ dG,.
(58) References: p . 469.
448
Ch. 5, $ 8
D.R. Gash11 1400
I
I
I
I
Y
$ 1000 E 800
600
IC"
As 1.a
0.8
0.6
.3 >
-
v 0.4 0.2
0 mole fraction of copper Fig. 12. (a) The phase diagram for the system Ag-Cu. (b) The activities of Ag and Cu in the system Ag-Cu
atlooOK.
As:
= G:
+ RTlna,,
eq. (58) can be written as: aEag AG; = -RT In -
aiai '
Ch. 5, 8 8
Metallurgical thermodynamics
449
where the quotient in the logarithm term is Krthe equilibrium constant for the reaction. Consider the oxidation, at 1000 K, of Cu from an Ag-Cu alloy of X,=O.OS. From eq. (57),the activity of Cu in this alloy, relative to pure solid Cu as the standard state, is = 9.6 x 0.08 = 0.768.
a, = 9.6X,
For the reaction: 4 q S )+ Oqg)= 2CU20(,)
AG: = -336810
+ 142.5T J.
Thus, A G ~ m=-194300 J=-8.3144x 10o0 lnK,,
and so:
2
K,,,
acu20
= 1.41 x 10" = 7. a," Po,
(59)
Oxidation of the Cu occurs when the oxygen pressure in the system has been increased to the level at which a,,o = 1. From eq. (59) this oxygen pressure is: 1 = 2.04 x lO-''atrn. - (0.768)4 x 1.41 x 10"
From eq. (16), AGP,
+
for the reaction
2Ag(,) + Oqg) = A&O(,) has the value 31 062 J. Thus:
Thus, with uAg= 0.92 (Raoultian behavior in the a-solid solution) and po2= 2.04 x lo-'' atm: 'Ag20
- 2.9
x 10-7,
which shows that the equilibrium oxide is virtually pure Cu,O. As the oxygen pressure in the system is further increased, the Cu content in the alloy decreases in accordance with eq. (59). Thus the alloy in equilibrium with virtually pure CbO and air (oxygen fraction 0.21) at 1000 K is that in which
[
= 1.41 x 10" l x] 0.21 "
"
= 0.0043,
or X, = %/9.6 = 4.5 x lo4. At this oxygen pressure the activity of AgzO in the equilibrium oxide phase, with aAg= 1, is: References: p. 469.
450
D. R, GaskeN
aAg20
Ch. 5, $ 8
- 0.024 x 1x 0.21”*= 0.011,
and so the equilibrium oxide phase is still essentially pure ChO.
8.2. The Gibbs phase rule The complete description of a thermodynamic system containing C components existing in P phases requires specification of the temperatures, pressures and compositions of each of the P phases. As the composition of each phase is defined when the concentrations of C - 1 of its components are known, the total number of variables in the description is P pressures + P temperatures + P(C - 1) concentrations =P(C+ 1). For thermodynamic equilibrium in the system, each of the P phases must be at the same temperature and same pressure and the activity (or partial molar free energy) of each of the individual components must be the same in each of the P phases. Thus, for equilibrium, there are (P - 1) equalities of temperature, (P - 1) equalities of pressure and (P 1)C equalities of activity, and hence the total number of equilibrium conditions, given as the number of equations among the variables of the system, is (P - 1)(C+2). The number of degrees of freedom, F, which the equilibrium system may have, is defined as the maximum number of variables which may be independently altered in value without disturbing the equilibrium in the system. This number is obtained as the difference between the total number of variables available to the system and the minimum number of equations among these variables that is required for maintenance of the equilibrium, 1.e.:
F = P(C + 1) - ( P - 1)(c+ 2)
=C+2-P. Equation (60) is the Gibbs phase rule and is a powerful tool in the determination of possible equilibria which may occur in multicomponent, multiphase systems. In the simplest of applications, i.e., in a one-component system, F = 3 - P. Thus, with reference to the phase diagram for H,O, shown in fig. 3, for the existence of a single phase F = 2 and so the pressure and temperature can be varied independently without disturbing the equilibrium, i.e., with F=2 the state of the system can be moved about within the area of stability of the single phase in the pressure-temperature diagram. However, for a two-phase equilibrium the state of the system must lie on one of the lines in fig. 3 and thus only the pressure or the temperature can be varied independently. From the phase rule, F = 1 and hence the two-phase equilibrium is univariant. The triple point, where the three phases are in equilibrium, occurs at fixed values of temperature and pressure, in accordance with F = 0 from the phase rule. The three-phase equilibrium is thus invariant and three is the maximum number of phases which can be in equilibrium with one another in a one-component system. In a binary system, the inclusion of a second component adds an extra degree of freedom to each equilibrium and hence the maximum number of phases which can be in equilibrium with one another in a two-component system is four. However, phase diagrams for binary systems of metallurgical interest are normally presented for a
Ch. 5 , $8
Metallurgical thermodymamics
451
pressure of 1 atm, i.e., they are the 1 a m isobaric sections of the phase equilibria occumng in pressure-temperature-composition space, and hence one of the degrees of freedom is used in specifying the pressure. Thus, at an arbitrarily selected pressure such as 1 atm, the maximum number of phases which can exist in equilibrium with one another in a binary system is three (unless, by chance, the arbitrarily selected pressure happens to be that at which an invariant four-phase equilibrium occurs). In the binary system A-B, considered at constant pressure, the available variables are T, a, and aB.For the existence of a single phase, such as a,p or liquid in fig. 12a, the phase rule gives F = 2 , and hence any two of T, U, and aB may be varied independently. For any twophase equilibrium, F = 1 and hence the specification of any one of the three variables fixes the state of the system. For example, specification of the temperature at which the two-phase equilibrium exists fixes the compositions of the equilibrated phases on the appropriate liquidus, solidus or solvus lines; and specification of the composition of one of the equilibrated phases fixes the temperature at which the chosen composition lies on the appropriate liquidus, solidus or solves line and fixes the composition of the second phase at the other end of the tie-line between the two equilibrated phases. The threephase equilibrium with F = 0 is invariant, and, in fig. 12a, the eutectic equilibrium occurs at a fixed temperature at which the compositions o f the a,/3 and liquid phases are also fixed. If some, or all, of the components of a system can react chemically with one another to produce new chemical species, a distinction must be drawn between the terms component and species. For example the components silver and oxygen in the binary system Ag-0 are capable of reacting to form the new species Ag,O, and hence an equilibrium among the three species Ag, AgzO and 0, can occur in the two-component system. The equilibrium among Ag, Ag,O and 0, is called an independent reaction equilibrium. In a system containing N species and existing in P phases among which there are R independent reaction equilibria, the number of variables is P(N+ l), Le., P pressures + P temperatures + P(N - 1) concentrations. However, if the species i and j react to form the species k, reaction equilibrium requires that
q + q= 9, and this is an additional equation required among the variables. Thus, if R independent reaction equilibria occur, the number of equations among the P(N+ 1) variables, required for equilibrium is (P - 1) equalities of temperature f (P - 1) equalities of pressure + (P - l)N equalities of activity+R=(P-l)(N+2)+R, and hence the number of degrees of freedom, F, is
F
= P(N
+ 1) - (P - 1)(N + 2) - R
= ( N - R ) + 2 - P.
Comparison with eq. (60) indicates that
C=N-R,
Refewnces: p . 469.
452
D.R. Gaskell
Ch.5, 58
Le., the number of components in a system equals the number of species present minus the number of reaction equilibria. Equation (61) is normally used to calculate the number of independent reaction equilibria from knowledge of the number of components and the number of species. For example, in the two-component system Ag-0, the independent reaction equilibrium among the three species is
2Ag + 30, = Ag,O.
For equilibrium among the phases metal, metal oxide and oxygen gas in the twocomponent system, F = 1 and thus only T or pol can be selected as the single degree of freedom. Selection of T fixes AG: and hence, via eq. (34), fixes poz, and vice versa. Consider the various equilibria which can occur in the ternary system Si-C-0, for which a stability diagram is shown in fig. 7. It can be considered that this system contains the six species Si, SiO,, Sic, C, CO and CO,, and hence R = 6 - 3, i.e., there are three independent reaction equilibria. These are derived as follows. The chemical reaction for formation of each compound from its elements is written: Si + 0, = SiO,, Si + C = Sic, c + 0, = co,,
c++o,
=co.
These equations are then combined in such a way as to eliminate any elements which are not considered as species in the system, and the minimum number of equations so obtained, is the number of independent reaction equilibria, R. In this case oxygen is not considered as species, and elimination of 0, gives:
c + co, = 2c0, Si + C = Sic,
(9 (ii)
and
Si + 2C0, = SiO,
-t
2CO
(iii)
as the independent equilibria. From the phase rule, the maximum number of phases which can coexist in equilibrium is five (the condensed phases Si, SiO,, Sic, C and the gas phase CO-CO,). This equilibrium is invariant and occurs at the temperature T, at which AG& = O and at the pressure P=p,+pm2 at which K,,, = pm/pm2 andK(ci),Tq = (pm/p,)’ are simultaneously satisfied. If the temperature is ar%itrarilyfixed, as is the case in fig. 7, the maximum number of phases which can coexist in equilibrium is four (three condensed phases and a gas phase). One such equilibrium occurs in fig. 7 at point b. For the coexistence of two condensed phases and a gas phase at the arbitrarily selected temperature, F = 1, and such equilibria lie on the univariant lines A, B, C and D in fig. 7, and for equilibrium between a single condensed phase and a gas phase, F = 2 , corresponding to areas of single condensed phase stability in fig. 7. Occasionally situations are found in which it might appear, at first sight, that the
a.5, $ 9
Metallurgical thermodynamics
453
phase rule is not obeyed, and usually, in such situations a degree of freedom is used by a condition of stoichiometry in the system. For example, in the reduction of ZnO by graphite to produce Zn vapor, CO and C02, it might appear that the three-phase equilibrium (ZnO, C and the gas phase) in the three-component system (Zn-042) has F = 5 - 3 = 2 degrees of freedom, and that, with the five species ZnO, C, ZqV),CO and C02, two independent reaction equilibria occur, which can be selected as ZnO(s) +
qgr) = q”) + qg)
(iv)
and (Vii)
However, selecting T, which fixes the values of K(iv)and K,, and any one of pzn,pCoorpmz as the two apparent degrees of freedom does not fix the state of the system, i.e., does not allow simultaneous solution of eqs. (vi) and (vii). This difficulty arises because the stoichiometry requirement has not been taken into consideration, i.e. that, as all the Zn and 0 occurring in the gas phase originates from the stoichiometric ZnO, the condition (viii) must also be satisfied. This stoichiometric requirement decreases F to unity and hence selecting T as the single degree of freedom fixes the partial pressures of Zn, CO and CO, as the values required for simultaneous solution of eqs. (vi), (vii) and (viii).
9, The thermodynamics of suflaces and inteflaces 9.1. The Gibbs adsorption isotherm In passing from one phase to another in a heterogeneous system, some of the properties undergo significant changes as the boundary between the two phases is traversed. The thin region over which these changes occur is called the inteflace, and a complete thermodynamic analysis of the system requires consideration of the thermodynamic properties of the interface. Consider fig. 13 which shows the variation of the concentration, cl, of the component 1 across the interface region in a system comprising equilibrated CY and p phases. Calculation of the total number of moles of component 1 in the system as the sum cf V” + cfV6, where V and b#? are the volumes of the phases, involves the assumption References: p . 469.
454
D.R. Gaskell
Ch. 5 , $ 9
t z
0 .c
!c ! C al 0 K
8
distance Fig. 13. The variation, with distance, of concentration on passing through the interface between two phases.
that the values cy and c f occur up to some plane in the interface region, and evaluation of cPV" + cfVB requires that a mathematical plane be located somewhere in the interface region. In fig. 13 it is seen that the number of moles of component 1 in the system, calculated as C ~ V +" c f V B , is only equal to the actual number of moles of 1 in the system, n, when the boundary plane X-X is located such that the shaded areas in fig. 13 are equal. If the boundary plane is located to the right of X-X, say at X'-X', then: n, < CPV"
+ c,BvB
or, if the boundary plane is located to the left of X-X: n, > CPV"+ c,BvB.
The difference between n, and C ~ V +" cf3Vp defines the surface concentration of component 1 , r,,(moleskm'), as:
T,A, = n, - (cPV" + cfVs), where A, is the area of surface between the two phases. Thus, with the boundary located to the left of X-X, r, is a positive quantity and with the boundary located to the right of X-X, rl is a negative quantity. In a single-component system where the boundary is between a condensed phase and a vapor phase, it is logical to locate the boundary at X-X so that the surface concentration is zero. However, with two or more components in the system it is not generally possible to locate the interface at a position at which more than one of the surface concentrations are zero. In such a case X-X is located such that the surface concentration of the solvent, rl,is zero and the surface concentration of the solute, r2,is not zero. This is illustrated in fig. 14.
Ch. 5, 8 9
455
Metallurgical thermodynamics
distance Fig. 14. The variations, with distance, of the concentrations of solvent and solute on passing through an interface, and illustration of the origin of surface concentration of the solute.
The definition of surfiace free energy per unit area, G,, is analogous to that for the surface concentration, i.e.:
where G’ is the total free energy of the system. The surfiace tension, (+,is defined as:
cT=(g) 9
T.P.ni
and hence, when surfaces are included in the discussion, eq. (13) is written as:
dG‘ = -S’dT
+ V’dP + d, + qdni
(63)
If the surface area is increased by dA, at constant T,P, and n , combination of eqs. (62) and (63) gives:
G,dA, = adg +
(CFn; +CG’n!).
As phase equilibrium is maintained,
dn,u
+ dnf
=
(64)
Gf ; mass balance requires that:
= -q.d4,
in which case eq. (64)can be written as:
Gsdq = ad4 + C G r i d 4
References: p. 469.
456
D. R. Gaskell
a.5 , § 9
or:
G, = u + x q r i , i.e., the surface free energy is the surface tension plus the free energy due to the surface concentrations of the components. Complete differentiation of eq. (65) gives:
dG, = d u +
Gdri +
ridq,
(66)
and the differential of G, for conditions of fixed surface area and fixed P gives: dG, = -S,dT
+ q.d.dT;:.
Combination of eqs. (66) and (67) gives: d u = -S,dT -
TdG,,
which is Gibbs' equationfor surface tension. At constant T, eq. (68) gives, for the binary system A-B in which r,=O:
Equation (69, which is known as the Gibbs adsorption isotheim, indicates that any solute which lowers the surface tension has a positive value of r and hence is concentrated in the surface, and, conversely, any solute which raises the surface tension has a lower concentration in the surface than in the bulk phase. The influence of dissolved oxygen on the surface tension of liquid iron at 1550°C is shown in fig. 15 as the variation of cr with the activity of oxygen relative to the 1 weight percent standard state. The surface concentration of oxygen at any concentration of oxygen in the bulk phase is obtained from the slope of the line and the Gibbs adsorption isotherm. At high oxygen contents the slope of the line approaches the constant value of -240 dyne/cm, which corresponds to saturation coverage of the surface by adsorbed oxygen. From the Gibbs adsorption isotherm this saturation coverage is calculated as
r,, = 6.023 x iou x 8.3144 x240 = 9.5 x 1 0 ' ~atoms/cm2 lo7 x 1823 9.2. The Langmuir adsorption isotherm Consider the equilibrium between the component i in a vapor phase and i adsorbed on the surface of a condensed phase. If is is considered that the atoms of i are adsorbed on specific adsorption sites on the surface of the condensed phase, the limit of adsorption
Ch. 5, $ 9
457
Metallurgical thennodfnamics
0.001
1.8
E 1.6
2
weight percent oxygen 0.005 0.01
0.05
0.1
-
i
.-0fn
s
1-4
4-.
- corresponding limiting slope of -0.24 N/m to saturation surface coverage of 9.5 x I014 atoms/crn*
Q
‘t
2
1.2
-
1.o
-8
-7
-6
-5
-4
-3
-2
In wt% 0 Fig. 15. The variation of the surface tension of Fe-0 melts with activity of oxygen at 1550°C.
occurs when all of the available sites are occupied by adsorbed atoms. This limit corresponds to the surface being covered by a monolayer of adsorbed atoms at the surface concentration r:. At surface concentrations, Ti, less than that corresponding to monolayer coverage, the fraction of surface sites occupied, Bi (or the fractional saturation of the surface) is defined as:
r. r;
8. = I *
(70)
At equilibrium, the rates of adsorption and desorption of i are equal, the former being proportional to the pressure of i in the vapor phase, pi,and the fraction of unoccupied surface sites, (1 -e,), and the latter being proportional to the fraction of surface sites occupied by i, i.e.:
k,pi(l - Oi)= kdei, where k, and k,, are the rate constants for the adsorption and desorption reactions, respectively. Thus: i‘ pi = Ki -
i-ei’
(71)
where
Ki = k,/k, = exp(-AG:/RT),
References:p . 469.
458
D. R. Gaskell
Ch. 5, $ 9
and AGP is the change in molar free energy accompanying the transfer of one mole of i from the vapor state at 1 atm pressure to the adsorbed layer on the surface at the Equation (71), which is Langmuir 's adsorption isothenn, shows surface concentration that 8, is proportional to pi at small Oi and (1 - 8J is inversely proportional to pi at large Oi. Alternatively, eq. (71) can be written as:
c.
a,
= K(-. 4
1 - e,
BELTONhas combined the Gibbs and Langmuir adsorption isotherms by substitution of eqs. (70) and (72) into eq. (69) to give:
da - -RT --
= -RT8,r;" = -RT
d In a,
Ko -, Kla.
1+ K,la,
which, on integration between the composition limits Xi' and Xi" , becomes:
1+ K'a!' 1 + K'a!
d' - a' = -RT yoIn -
(73)
a
If Langmuir's isotherm holds at all compositions, one limit can be taken as the pure solvent, in which case eq. (73) becomes
a'
- (T = -RT :7I
ln(1 + K'a,).
(74)
where c f refers to the surface tension of the pure solvent. Curve-fitting of eq. (74) with the experimental data shown in fig. 16 and d = 1788 dyne/cm, Po=240 dyne/cm, gives K = 220. Thus, if oxygen adsorbed on liquid iron exhibits ideal Langmuir behavior:
e,
=
220 = [wt%O] 1 + 220 * [wt%O] *
(75)
Equation (75) is shown in fig. 16 in comparison with the variation of 8, obtained from the slopes in fig. 15 as Bo = J?dr", A number of applications of the Gibbs and Langmuir absorption isotherms will be found in ch. 13, 592 and 4.
9.3. Curved interfaces The existence of surface tension gives rise to the interesting phenomenon that the equilibrium vapor pressure exerted by a spherical droplet is a function of the radius of curvature of the droplet. This phenomenon, which was first discussed by Kelvin in 1871, is of importance with respect to the dependence of the limit of solid solution of one component in another on the particle size of the second phase. The general equation dG' = -S'dT
+ V'dP + d, +
Gdn,
(63)
Ch. 5, $ 9
459
Metallurgical themdynamics 1.0 1
I
I
I
I
I
e cr)
$? 0.8 U
.-a
-
$ 0.6 M
.= v,
{
0.4
u)
c
0
c
from tangents to the curve in Fig. 15 and Eq. (70)
._
5 0.2
_m c
0
0
0.01
0.02
0.03
0.04
0.05
weight percent dissolved oxygen
Fig. 16. The variation, at 155OoC, of the fractional coverage of the surface of liquid iron by adsorbed oxygen with concentration of oxygen in the melt.
was tacitly applied to systems containing flat interfaces. However, provided that CT is not a function of the radius of curvature of the interface, and that the interface within the system does not influence the exterior pressure, eq. (63) can be applied to the transfer of matter across curved interfaces. The partia2 molurfree energy, E ; , defined from eq. (63) as:
pertains to the addition of i to the system in such a manner that A, remains constant. However, in a process involving the transfer of matter to a small spherical droplet, A,, being dependent on the volume, and hence on the amount of matter in the droplet, is not an independent variable. The incremental increase in volume of a droplet caused by the addition of dn, moles of the various components is:
7
is the partial molar volume of i in the system. From the relationship between where the surface area and the volume of a sphere,
substitution of which into eq. (63) gives:
References: p . 469.
460
D.R. Gaskell
dG’
+ V‘dP +
-S’dT
(
a.5, Q 10
+ ?)Ini.
Comparison with eq. (63) gives the identity -
Gi = G:
2 p +I
(76)
as the variation of partial molar free energy with spherical particle size. From the relationship between partial molar free energy and activity, eq. (76) can be written as
+-.2 F u
lna, = ha,*
RT,
(77)
In a limited terminal solid solution of B in A, in which B obeys Henry’s Law, the activity of €3 at the limit of solubility is: % = YixB(sat)y and hence, from eq. (77),the solubility limit varies with particle size of the second phase as
where XB(sat,rl is the solubility limit when the second phase occurs as a dispersion of spherical particles of radius r and is the solubility limit when the second phase is massive. Equation (78), which is known as the Thomson-Freundlich equation, provides a thermodynamical explanation of the phenomenon of Osnvald ripening (see ch. 9, Q 3.2.2). When the second phase, precipitating from a primary solid solution, occurs in a range of particle sizes, it is observed that the particles of radius greater than some average value grow and that the smaller particles redissolve in the matrix. As the concentration of solute in the matrix at the interface between the matrix and a small precipitate is greater than that at the interface between the matrix and a large precipitate, a concentration, and hence activity, gradient exists between the two interfaces. This, in turn, provides the driving force for the diffusion of dissolved solute from one interface to the other, with the overall result that the larger particle grows and the smaller particle dissolves. Equation (78) is also of interest in that it indicates that no such quantity as “maximum solubility” exists.
10. The measurement of thermodynamic activity Although activities are thermodynamic functions of state, their magnitudes and variations are determined by the interactions among the constituent particles of the system, which, in turn, determine bond energies and influence the spatial configurations assumed by the particles. Thus measurement of activities within a class of similar simple
Ch. 5, 0 10
Metallurgical thermodynamics
461
systems can be expected to provide, at best, some fundamental understanding of the natures of these interactions or, at least, a basis for correlation of the behavior, which can then be used for extrapolation of the behavior of more complex systems. The molar free energy of formation of a solution or compound from its pure components is obtained from the activities via eq. (44) and as the various phase equilibria occurring in a materials system are determined by the variations, with composition, temperature and pressure, of the relative free energies of the various phases, such equilibria can be most precisely determined by accurate measurement of activity. Also, the activity of a component in a solution is a measure of the minimum free energy required to convert the component from its state in solution to the pure state in any proposed extraction or refining process. En the majority of the experimental methods the activity of only one component is measured. In such cases the activities of the other components can be obtained by integration of the Gibbs-Duhem equation. For constant temperature and total pressure this expression is ZXid lna,=O or, in a more convenient form, XX,d lny,=O where yi=nJXj is the activity coefficient of i. Applied to the binary system A-B in which the variation of yA is known across the entire range of composition:
10.1. Determinationof activity by experimentalmeasurement of vapor pressure The experimental technique for the measurement of vapor pressure is determined by the magnitude of the pressure to be measured, and the various techniques which have been developed can be classified as absolute methods (direct and indirect static methods) and indirect methods (effusion and transpiration methods). The earliest activity measurements were made on binary alloys of Hg with Zn, Au, Ag and T1 at temperatures near the boiling point of Hg. The partial pressure of Hg exertsd by an amalgam is so much greater than the partial pressure of the other component that the former can be equated with the total vapor pressure of the amalgam. In the first studies the alloy was used as the sealing liquid in a U-tube null-point manometer. The vapor in equilibrium with the alloy is contained in the closed arm of the manometer, and hydrogen, the pressure of which is measured at a second manometer, is introduced to the other arm until the meniscuses in both arms are at the same level. The vapor pressures of amalgams at lower temperatures have been measured using various devices such as membrane manometers, quartz spiral manometers and ionization gages. Tbe partial pressures of Zn and Cd over a-Ag-Zn-Cd alloys and of Zn over a-brasses have been measured by resonance absorption spectroscopy. In studying the Zn alloys, light produced by a spark between Zn electrodes, is passed through a sample of vapor in equilibrium with the alloy, and the absorption of the 3076 A resonance line is measured. As absorption of the 3035 8, resonance line does not occur, it is used as an interrial standard and the vapor pressure, p , of Zn is obtained from Beer's Law as -ln(Z3m~Ims)= Kpd/T where I is the intensity of the transmitted light, K is the absorpReferences: p . 469.
4.62
D. R. Gmkell
a.5, B 10
tion coefficient, T i s the absolute temperature and d is the distance travelled by the light through the sample of vapor. The dew point method is well-suited to systems containing a distinctly volatile component and has been applied to measurement of the activity of Zn in binary alloys containing Cu, Al, Ag, Au, Zr,Th, U,and Y, and the activity of Cd in Ag-Cd alloys. Experimentally, the alloy is placed at one end of a long initially evacuated tube which is heated to the desired temperature T,. The temperature of the other end of the tube is lowered until condensation of the volatile component is observed at the temperature T,. As the pressure within the tube is uniform, the partial pressure of the volatile component exerted by the alloy at TI equals the saturated vapor pressure of the pure volatile component at T,. The use of fused silica tubes, which permits visual observation of condensation at the cooler end, has limited the temperature to less than 1100°C and, generally, measurements have been made in the range 400-900°C. In a similar isopiestic technique, the pure volatile component is placed in the cool end of an initially evacuated tube maintained in a known temperature gradient, and weighed quantities of the pure second component are placed at intervals along the temperature gradient. The volatile component is transferred from the vapor phase to the specimens of nonvolatile component until the alloys in equilibrium with the prevailing pressure of the volatile component are formed. In this technique, which has been applied to measurement of the activities of A1 in solid A1-Fe and A1-Ni alloys, the compositions of the equilibrated alloys are determined gravimetrically. Application of the dew point and isopiestic techniques to measurement of activity requires knowledge of the temperature dependence of the saturated vapor pressure of the volatile component. In the rrunspirution technique, an inert carrier gas is passed over a sample at a flow rate which permits evaporation of the alloy to occur to the extent necessary to saturate the carrier gas. This technique has been used to measure the activities in liquid F e C u and Fe-Ni alloys. The material evaporated from the sample is condensed downstream and is chemically analyzed. The total amount of evaporation into unit volume of the carrier gas at the total pressure P is determined by measuring the weight loss of the sample or by quantitative analysis of the amount of condensate recovered from a known volume of gas. If nFe.n, and nHeare the numbers of moles of Fe, Cu and He carrier gas in the sampled volume, the partial pressure of Fe is calculated, from the ideal gas law, as p R =PnFe/(nFe+ n, + n&). An advantage of this technique is that the activities of both components are measured and hence internal consistency of the results can be checked using the Gibbs-Duhem equation. However, in order that surface depletion of the more volatile component be avoided, the rates of diffusion in the alloy must be faster than the rates of evaporation. In the efsusion technique the alloy to be studied is placed in a Knudsen cell (a sealed crucible containing a small orifice in its lid) and the crucible is heated in vacuum to the desired temperature. Phase equilibrium is established between the vapor phase and the condensed phase in the cell and, if the dimensions of the orifice are small in comparison with the mean free path of the vapor species, the passage of vapor species through the orifice is not disturbed by collisions. Thus the rates of effusion of the vapor species are
a.5, § 10
Metallurgical thermodynamics
463
proportional to their vapor pressures within the cell. From gas kinetic theory, the number of particles in a vapor phase striking unit area of the containing wall in unit time is Q.25IZC,where n is the density of vapor species and F = (8RT/IW)”2 is the average speed of the particles. Consequently, the weight loss, W,due to effusion through an orifice of area .A in time t is pAt/(211MRT)”2 and hence the pressure, p, of the species in the cell is p = (W/At)(211RT/M)’/2. If a radioactive tracer is added to the alloy, very small amounts of effusing substance can be detected. For example, gamma-ray spectrometry of neutron-irradiated Au-Cu alloys has facilitated estimation of quantities as small as lo-’’ g. The transpiration and effusion techniques require that the molecular weights of the vapor species be known and hence they can only be used to study systems in which no complex vapor molecules are formed. The problems caused by complex molecule formation can be eliminated by massspectrometric analysis of the vapor effusing from the Knudsen cell. In the Knudsepa celltime ofJtight mass-spectrometer combination, the beam of particles effusing from the cell is introduced to the ionization chamber of the mass-spectrometer through a slit, Ionization iis produced by a pulsing electron beam and after each pulse the ionization chamber is cleared of ions by a pulse of small negative potential. The ions are then subjected to a continuously maintained high negative potential which accelerates them into a field-free drift tube, and the time required for a given ion to traverse the drift tube and be detected is proportional to (m/e)’/2. The ion current, Z,+, measured for the species i is related to the vapor pressure of i as:
pi = KI,!T, where the constant K is determined by the ionization cross-section of the ion, the detector sensitivity and the geometry of the Knudsen cell-ion source. The application of the technique to measurement of activities in binary systems was greatly facilitated by a manipulation of the Gibbs-Duhem equation which allows the variations, with composition, of the activity coefficients of the individual components to be obtained from the corresponding measured ratio of the activity coefficients as:
From eq. (80):
substitution of which into eq. (81) gives:
Use lof a mass-spectrometer requires that a pressure of less than lo-* atm be maintained in the areas of the ion source, analyzer and detector. This technique has been applied to References: p. 469.
464
D.R. Gaskell
ch.5, $10
measurement of activities in a large number of binary and ternary systems containing V, Cr, Fe, Co, Ni, Cu, Ag, Au, Al, T1, Pb, Sn, Bi, Sb, and In. 10.2. Determination of activity by establishing heterogeneousequilibrium Heterogeneous equilibrium at constant temperature and pressure requires that the partial molar free energy, and hence activity, of each component of the system be the same in each of the phases present, i.e., ai(in phase I) =ai(in phase 11)=a, (in phase 111) =.... Thus, if the activity of a component can be fixed at a known value in any one of the phases, its value in every other phase is known. One of the more simple heterogeneous equilibria involves a binary liquid, saturated with one of its components. In a simple binary eutectic system exhibiting virtually complete mutual immiscibility in the solid state, the saturated liquids on the liquidus lines are in equilibrium with virtually pure solids. Thus, in the melt of A-liquidus composition at the liquidus temperature T, the activity of A relative to pure liquid A as the standard state equals the activity of pure solid A relative to liquid A as the standard state, both being given by aA= exp( -AG;,/RT) where AG:,, is the molar free energy of melting of A at temperature T. Activities have been calculated in this manner along liquidus lines in such systems as Ag-Si and Ag-Pb. Fe and Ag are virtually immiscible in the liquid state, and when Si is added as a solute to coexisting liquid Fe and Ag it is distributed between the two liquids such that its activity is the same in both phases. The activities of Si in liquid Fe and liquid Fe-C alloys have been determined by chemical analysis of equilibrated Fe and Ag liquids containing Si, and knowledge of the activity of Si in Ag-Si alloys. In a similar manner the activity of Ag in A1-Ag alloys has been determined from measurement of the equilibrium partitioning of Ag between the virtually immiscible liquids AI and Pb, and the activity of A1 in AI-Co alloys has been determined by partitioning Al between the virtually immiscible liquids Ag and Co. The respective equilibrium constants for the reactions C02+ C,,,,,,,, = 2CO and CO +io2 = COz are: 2
K4 = Pco ~
Pco2ac
and
K5 = Pco,t2 ~
Pco Po,
-
(83)
Thus, at a fixed temperature, which determines the values of K4 and K5,a CO-CO, gas mixture of known pco and pco, has an activity of carbon given by eq. (82) and a partial pressure of oxygen given by eq. (83). Similarly, by virtue of the equilibrium H2+$O2= H20, an H2-H20 mixture of known pHaand p%o exerts a unique partial pressure of oxygen at any temperature; by virtue of the equilibrium H2+iS,=H2S, an H2-HzS mixture of known pH,and pHaSexerts a unique partial pressure of sulfur at any temperat-
Ch. 5, 0 10
Metallurgical thermodynamics
465
ure; and, by virtue of the equilibrium CwNire)+2H2=CH4, a CH4-H, mixture of known pa4 and p., has a unique activity of carbon at any temperature. Consequently, CO-CO, and CH,-H, mixtures can be used as gas phases of fixed activity of carbon for use in the establishment of heterogeneous equilibria between a gas phase and a condensed phase. Similarly, CO-CO, and H2-H,O mixtures can be used as gas phases of fixed oxygen pressure and H,-H,S mixtures can be used as gas phases of fixed sulfur pressure. The activities of carbon in liquid and solid iron have been determined by equilibrating iron with CO-CO,and CH4-H, mixtures and measuring the equilibrium carbon content of the metal phase, and the activities of oxygen and sulfur in liquid iron have been determined by equilibrating iron with H,O-H, and H,S-H, mixtures, respectively. In more simple gas-metal equilibria the activities of hydrogen and nitrogen in iron have been determined by measuring the solubilities of the gases as functions of gas pressure. Activities in the system Fe-Fe,O, have been determined by experimental observation of the variation of the composition of small samples of condensed phases with temperature and oxygen pressure imposed by an equilibrating gas phase. The variation, with composition, of the activity of Fe in the system is determined by Gibbs-Duhem integration of the corresponding measured variation of the equilibrium partial pressure of oxygen. The oxygen contmt of liquid iron in equilibrium with pure liquid iron oxide at 1600°C is 0.23 wt%. If the oxide of a less noble metal than iron is dissolved in the liquid iron oxide, the activity of iron oxide, and hence the equilibrium oxygen content of the liquid iron are decreased. If the latter is x wt%, the activity of FeO, relative to pure Fe-saturated iron oxide as the standard state, in the oxide solution is d 0 . 2 3 . This technique has been used to determine the activity of FeO in CaO-FeO and CaO-FeO-SiO, melts saturated with liquid iron. One step more complex is the establishment of equilibrium between a gas phase and two condensed phases. The equilibrium between manganese, manganous oxide and a CO-CZO, mixture, expressed as Mn + CO, = MnO + CO requires:
Thus, at a given temperature, the equilibrium between pure Mn (at unit activity) and Mnsaturated pure MnO (at unit activity) occurs at a unique value of the ratio p,-Jpco, given by eq. (84). If a metal more noble than Mn is embedded in an excess of MnO and subjected to a lower p,-Jpco, ratio, manganese is transferred from the MnO to the metal phase until the activity of Mn required by eq. (84) and the i m p o s e d p d p m zis established. The manganese content of the alloy corresponding to the imposed activity is determined by chemical analysis. The other component of the alloy must be sufficiently more noble than Mn that formation and solution of its oxide in the MnO phase is negligible. The activity of Mn in Mn-F't alloys has been determined in this manner. Having determined this relationship, the activity of MnO in oxide melts containing oxides more stable than MnO can be determined by equilibrating a small sample of Pt with an excess of oxide melt and a CO-CO, gas mixture. Again, as Mn is distributed between the Pt-Mn alloy and the oxide melt in accordance with eq. (84) and the imposed References: p . 469.
466
D.R. Gaskell
Ch. 5 , 5 10
pc,/pco,, chemical analysis of the equilibrated R-Mn alloy yields a,,, and hence, from eq. (84), the value of uMnO in the oxide melt. In this application the other oxide component must be of a metal which is sufficiently less noble than Mn that the extent of its solution in the Pt Mn phase is negligible. This technique has been used to determine the activity of MnO in systems such as MnO-SiO, MnO-TiO,, MnO-Al,O,, MnO-B,O, and MnO-CaO-Si0,. Other examples of determination of activities by establishing equilibrium between a binary alloy, a nonmetallic phase of known composition and a gas phase include: Fe(in Fe-Ni alloys) + H,O = FeO + H,; 2Cr(in Cr-Ni alloys) + 3H,o = Cr,O, + 3H,; 3Mn(in Mn-Cu alloys) + CH, = Mn,C + 2H,; 2C~(inCU-AU alloys) + HZS = C U ~+SH2; 3Si(in Si-Ag alloys) + N, (in N,-H, mixtures) = Si,N,. Again, in this application, the “inert” metal must be sufficiently more noble than the primary component metal that its occurrence in the equilibrium nonmetallic phase is negligible. Corrections are required in systems where the nonmetallic phase is not a line compound. Thus, for example, in eq. (85), the activity of FeO is that in the wustite equilibrated with the imposed partial pressure of oxygen, relative to Fe-saturated wustite as the standard state. Equation (85) has also been used to determine the activity of FeO in FeO-SiO, melts by establishing the equilibrium Fe +H,O = FeO (in FeoSiO, melts)
+H,.
If the difference between the nobilities of the metals is small enough that an oxide solution is produced in equilibrium with the binary alloy phase a different approach is taken. For example, if a small specimen of an Fe-Mn alloy is equilibrated with an excess of an FeO-MnO solid solution, the exchange equilibrium Fe+MnO = Mn+FeO is established, wherein K =aMnuFJaku~m From chemical analysis of the equilibrated in Fe-Mn alloy and knowledge of the activities in the system Fe Mn, the ratio yFedyMnO the equilibrating oxide solution is obtained as: YFeO YMnO
-K
aFeXMnO aMnxFeO
’
and Gibbs-Duhem integration of the variation of this ratio with composition in the oxide solution according to eq. (81) yields the individual activity coefficients, and hence activities, of the components of the oxide solution. This technique has been used to determine activities in the systems Fe,Si04-Co,Si04 and Fe,SiO,-Mn,SiO,. Activities have been determined by establishing equilibrium among three condensed phases and a gas phase. As an example, the activity of SiO, in CaO-MgO-A1,03-SiOz melts has been determined by establishing the equilibrium
SiO,
+ 2C = Si + 2CO
(86)
in systems comprising a silicate melt, solid graphite, liquid iron and CO gas at 1 atm pressure, and by establishing the equilibrium
a.5, 0 10 SiO,
Metallurgical tlzemodynamics
+ 2SiC = 3Si + 2CO
467
(87)
in systems comprising a silicate melt, solid Sic, liquid iron and CO gas at 1 atm pressure. The activity of SiO, is obtained from chemical analysis of the equilibrated liquid phases, knowledge of the equilibrium constants for the reactions given by eqs. (86) and (87) and knowledge of the activity of Si in Fe-Si-C melts. Gibbs-Duhem integration of the results yielded activities in the systems CaO-SO,, CaO-Al,O,, MgO-SiO,, CaOA1,0,-Si02, Mg0-Ca0-Si0, and Mg0-A120,-Si0,.
10.3. Electrochemicalmeasurement of activity The Eh4F of a reversible galvanic cell, 8 , is related to the free energy change, AG, for the cell reaction as AG=-zFE where F is Faraday's constant and z is the number of Faradays required for the cell reaction. Thus, in a concentration cell of the type pure metal A lionic conductor containing metal A ions of valence z,lalloy A-B, the cell reaction is A(pure) + A(in the A-B alloy) for which AG=RT lna, (in the A B alloy). Thus the activity of A in the alloy is obtained as lna,=-(zAFE/RT). The determination of activity by measurement of the EMF of an electrochemical cell requires that the electrolyte be a purely ionic conductor and that the valency Z, be defined. A further requirement is that the extent of the exchange reaction at the cathode-electrolyte interface between B in the alloy and A in the electrolyte be negligible. If this condition is not met, the measured EMF contains a contribution of unknown magnitude arising from the transfer of electrolyte between regions of different composition. In practice the extent of the exchange reaction is rendered negligible by ensuring that B is significantly more noble than A. Molten chlorides are purely ionic conductors and hence these melts are popular as liquid electrolytes. The concentrations of low valent cations in the electrolyte are minimized by dissolving the chlorides in mixtures of alkali chlorides. The activity of A1 in A1-Ag melts in the range 700-8OO0C has been obtained from measurement of the EMFs of cells of the type Ala, I A13'(in KCl-NaCl) I Al-Ag(,). Similarly the activities of Cd in Cd-Pb, Cd-Bi, Cd-Sb and Cd-Sn alloys, and the activities of Cu in Cu-Au melts and Ag in Ag-Au melts have been determined from concentration cells with liquid chloride electrolytes. The cell
is a formation cell in which the cell reaction is Mg +Cl,=MgCl,. With pure liquid Mg, pure liquid MgCI, and C1, at 1 atm pressure, the free energy change is the standard free energy change, AGO, and the EMF is the standard EMF, ao=-A@/2F. Alloying the anode with a more noble metal such as A1 alters the free energy change for the cell reaction to AG= AGo-RT lna,, (in the alloy) and hence the cell EMF to E
= E0
RT In aMg(in the alloy). +2F
(88) References: p. 469.
468
D.R. Gash11
Ch. 5, 0 10
Equation (88) has been used to determine the activities of Mg in Mg-Al melts from EMF measurements in the range 700-880°C. Similarly, the formation cell
has a standard EMF of &'=-AGo/2F. Alloying the PbO electrolyte with the oxide of a less noble metal, such as SO,, changes the cell EMF to: E = &0
--RT In uPbo(in PbO-SiO,), 2F
and this has been used as the basis for electrochemical determination of the activities in the system P b e s i O , in the range 850-1O5O0C. Within wide ranges of temperature and oxygen pressure, Zro, and Tho, in the fluorite structure, stabilized by solid solution with CaO and Y,03, respectively, exhibit unusually high conductivities and transport numbers for 0'- of essentially unity. Consequently CaO-ZrO, and Y,O,-Tho, have been used as solid electrolytes in oxygen concentration cells of the type in which the cell reaction is 02(g,at prersurcp , ) + 02(g,at pressun pL) and the cell EMF is E =-AG/4F = -(RT/4F) InP J P , . The oxygen pressure at the electrodes can be fixed by using equilibrated metal-metal-oxide couples, e.g., with Fe-FeO and Ni-NiO the cell becomes
Fe, FeO I CaO-ZrO, I Ni, NiO, with a cell reaction of NiO + Fe = FeO + Ni. With the electrodes Fe-FeO and (Fe-Ni j FeO, the cell reaction is Fe@,, + Fe,, the FeNi a,,oy) and the cell EMF is E=--
RT In uFe(in the alloy). 2F
(89)
This method is similar to that discussed in connection with eq. (85). In the chemical equilibration technique the oxygen pressure is imposed, and the Fe-Ni alloy in equilibrium with FeO and the imposed oxygen pressure is produced in the experimental apparatus. In the EMF technique the oxygen pressure in equilibrium with a given Fe-Ni alloy and FeO is measured. Equation (89) has been used as the basis for electrochemical determination of the activities in a large number of solid and liquid binary alloy systems, the majority of which contained Fe, Co, Ni or Cu as the less noble metal. The activity of Si in FeSi alloys at 1550°C and 1600°C has been determined with electrodes of Cr, Cr,O, and SiO,, Fe-Si and activities in the systems Ta-W and Ta-Mo have been determined with a Y,03-Th0, electrolyte and Ta, T%O, and Ta-X, T%O, electrodes. The activities of SnO in SnO-SiO, melts and PbO in PbO-SiO, melts have been determined from cells of the type M, MOICa0-ZrO,IM, MO-SiO,. Other solid electrolytes which have been used include /?-alumina and soft soda glass for measurement of the activity of sodium in alloys, and glasses containing K" and Ag+
Ch. 5 , Refs.
Metallurgical thennodynamics
469
for study of K and Ag alloys, respectively. It can be expected that, as new solid state electrolytes are developed for possible use in fuel cells, they will be applied to the determination of activities by EMF measurements.
Bibliography ALCWK, C. B., Principles of Pyrometallurgy (Academic Press, London, 1976). BELTON,G. R., Langmuir Adsorption, the Gibbs Adsorption Isotherm and Interfacial Kinetics in Liquid Metal Systems, Metallurg. Trans. B7 (1976) 35. BELTON,G. R., and R. J. FRUEHAN, The Determination of Activities by Mass-Spectrometry- Some Additional Methods, Metallurg. Trans. 2 (1971) 291. C A L L ~H.B., , Thermodynamics (Wiley, New York, 1960). CHAW-IAH, M. S., 0.M. SR~FDHARAN and E. CHATTOPADHYAY, Thermodynamic:Studies of Alloys and Intermetallic Compounds, in: Solid Electrolytes and Their Applications, ed. E. C. Subbarao (Plenum, New York, 1980). ELLINGHAM, H. J.T., Reducibility of Oxides and Sulfides in Metallurgical Processes, J. SOC. Chem. Ind. 63 (1944) 125. ELLrwm, J. E, Physical Chemistry of Liquid Metal Solutions, in: Metallurgical Treatises, eds. J. E Elliott and J. Tien (The Metallurgical Society of AIME, Warrendale, PA, 1981). ELLIOIT, J.F., M. GLEISERand V. RAMAKRISHNA,Thermochemistry for Steelmaking (Addison Wesley, Reading, MA, 1963). FAST,J. D., Entropy (McGraw Hill, New York, 1962). GASKELL,D.R., Introduction to Metallurgical Thermodynamics, 2nd Ed. (McGraw-Hill, New York, 1981). GOKCEN,N. A,, Thermodynamics (Techscience, Hawthorne, CA, 1977). D. T. HAWKINS,M. GLEISERand K. K.KELLEY, Selected Values of the ThermodyHULTGREN, R., P. D. DESAI, namic Properties of Binary Alloys (American Society for Metals, Metals Park, OH, 1973). KIJBASCHEWSKI, O., C.B. ALCOCKand P.J. SPENCER,Materials Thermochemistry, 6th Ed. (Pergamon Press, New York, 1993). LEWIS,G.N., and M. RANDALL,Thermodynamics, 2nd Ed., revised by K.S. Spitzer and L.Brewer (McGrawHill, New York, 1961). LUPIS,C. H. P., Chemical Thermodynamics of Materials (North-Holland, Amsterdam, 1983). MOORE,J. J., Chemical Metallurgy, 2nd Ed. (Buttenvorths, London, 1990). ROSENQLJIST, T., Principles of Extractive Metallurgy (McGraw-Hill, New York, 1974). STEINER, A., and K.L. KOMAREK, Thermodynamic Activities in Solid Ni-A1 Alloys, Trans. Metallurg. Soc. AIME 230 (1964) 786. SWALIN, R A., Thermodynamics of Solids, 2nd Ed. (Wiley, New York, 1972). TURKDOGAN, E.T., Physical Chemistty of High Temperature Technology (Academic, New York, 1980). WAGNER, C., Thermodynamics of Alloys (Addison-Wesley, Reading, MA, 1952).
CHAPTER 6
PHASE DIAGRAMS ARTHUR D. PELTON Dkpartement de G h i e MLtallurgique Ecole Polytechnique, Case Postale 6079, Station “Centre Ville” Montrbal, QuLbec H3C 3A7, Canada
R. W Cahn and I? Haasen?, e&. Physical Metallurgy;fourth, revised and enhanced edition 0Elsevier Science BY 19%
472
A.D. Pelton
Ch. 6,
01
1. Introduction The study of phase equilibria and phase transformations is central to nearly all branches of metallurgy and materials science. Although departures from equilibrium will occur in any real system, a knowledge of the equilibrium state under a given set of conditions is the starting point for the understanding of most processes. A phase diagram is a graphical representation of the loci of thermodynamic variables when equilibrium among the phases of a system is established under a given set of conditions. The phase diagrams most familiar to the metallurgist are those for which temperature and composition are the axes. These are discussed in $5 2 and 3 for binary (two-component) and ternary (three-component) systems, and in $ 4 for multicomponent systems. However, the effect of other variables such as total pressure and chemical potential of the components (e.g., the partial pressure of oxygen) may often be of interest. In $6, different types of phase diagrams are discussed along with the general rules governing their construction. Throughout the chapter, the thermodynamic origin of phase diagrams is stressed. With the advent of modern computer techniques, the relationship between phase diagrams and the thermodynamic properties of the system has become of increasing practical importance. As discussed in 0 2.10, a quantitative coupling of the two is now possible. Furthermore, as discussed in $ 5 , the computer-assisted thermodynamic approach often permits good estimates of unknown multicomponent phase diagrams to be made, and can often significantly reduce the experimental effort required to measure the phase diagram of a system.
2. Binary phase diagrams The temperature composition (T-X) phase diagram of the Bi-Sb system is shown in fig. 1 (HULTGREN et al. [1963]). The abscissa is the composition, expressed as mole fraction of Sb, XSb.Note that X,, = 1-XBi. Phase diagrams are also often drawn with the composition axis expressed as weight percent. At all compositions and temperatures in the area above the line labelled liquidus, single-phase liquid alloys will be observed, while at all compositions and temperatures
Mole f r a c t i o n X S b Fig. 1. Phase diagram of the Bi-Sb system (after HULTGRENetal. [1963]).
Ch. 6, $ 2
Phase diagrams
473
below the line labelled solidus, alloys exist as single-phase solid solutions. An alloy sample at equilibrium at a temperature and overall composition between these two curves will consist of a mixture of solid and liquid phases, the compositions of which are given by the liquidus and solidus compositions at that temperature. For example, a B i S b sample of overall composition &=0.6@ at T=700 K (at point R in fig. 1) will consist, at equilibrium, of a mixture of liquid alloy of composition X , = 0.37 (point P) and solid alloy of composition xsb=o.82 (point Q). The line PQ is called a tie-line or cunude. As the overall composition is varied at 700 K between points P and Q, the compositions of the liquid and solid phases remain fixed at P and Q, and only the relative proportions of the two phases change. From a simple mass balance, one can derive the Zever rule for binary systems: (moles of liquid)/(moles of solid) = RQRR. Hence, at 700 K a sample of B i S b alloy with overall composition X, = 0.60 consists of liquid and solid phases in the molar ratio (0.82-0.60)/(0.60- 0.37)= 0.96. Were the composition axis expressed as weight percent, then the lever rule would give the weight ratio of the two phases. Suppose that a liquid B i S b alloy with composition XSb = 0.60 is cooled very slowly from an initial temperature of 900 K. When the temperature has decreased to the liquidus temperature of 780 K (point A) the first solid appears, with a composition at point B (xsb =: 0.93). As the temperature is decreased further, solid continues to precipitate with the compositions of the two phases at any temperature being given by the liquidus and solidus compositions at that temperature and with their relative proportions being given by the lever rule. Solidification is complete at 630 K, the last liquid to solidify having composition xsb =0.18 (point C). Tbe process just described is known as equilibrium cooling. At any temperature during equilibrium cooling the solid phase has a uniform (homogeneous)composition. In the preceding example, the composition of the solid phase during cooling varies along the line BQD. Hence, in order for the solid particles to have a uniform composition at any temperature, diffusion of Sb from the center to the surface of the growing particles must occur. Since solid state diffusion is a relatively slow process, equilibrium cooling conditions are only approached if the temperature is decreased very slowly. If a B i S b alloy of composition xSb=O.60 is cooled very rapidly from the liquid, concentration gradients will be observed in the solid grains, with the concentration of Sb decreasing towards the swface from a maximum of xsb = 0.93 (point B) at the center. Furthermore, in this case solidification will not be complete at 630 K since at 630 K the average concentration of Sb in the solid particles will now be greater than X,=0.60. These considerations are discussed more fully in ch. 9. At Xsb= 0 and Xsb= 1in fig. 1 the liquidus and solidus curves meet at the equilibrium melting points, or temperatures offusion, of Bi and Sb, which are: T&) =544.5 K, Z'&,) =903 K. The phase diagram is influenced by the total pressure, P. Unless otherwise stated, T-X diagrams for alloy systems are usually presented for P = const. = 1 atm. However, for equilibria involving only solid and liquid phases, the phase boundaries are typically shifted only by the order of a few hundredths of a degree per bar change in P (see ch. 5, $3). Hence, the effect of pressure upon the phase diagram is generally negligible
References:p . 531.
414
A.D. Pelton
Ch.6 , 9 2
unless the pressure is of the order of hundreds of atmospheres. On the other hand, if gaseous phases are involved then the effect of pressure is very important (5 2.12).
2.1. The thermodynamic origin of phase diagrams In this section we shall consider first of all the thermodynamic origin of simple "lensshaped" phase diagrams in binary systems with complete liquid and solid miscibility. An example of such a diagram was given in fig. 1. Another example is the Ge-Si phase diagram in the lowest panel of fig. 2 (HANSEN[1958]). In the upper three panels of fig. 2 are shown, to scale, the molar Gibbs energies of the solid and liquid phases, gs and g', at three temperatures. As illustrated in the top panel, gsvaries with composition
Pig. 2. GeSi phase diagram (after HANSEN[1958]) and Gibbs energy-compositioncurves at three temperatures, illustrating the common tangent construction.
Ch. 6, 5 2
Phase diagrams
475
between the standard molar Gibbs energies of pure solid Ge and of pure solid Si,gz' and g z ' , while g' varies between the standard molar Gibbs energies of the pure. liquid components go,00) and g;'). The molar Gibbs energies of mixing of the solid and liquid phases, Ags and AB1, are negative and are equal to the difference between the Gibbs energy of the solution and a simple weighted average of the Gibbs energies of the pure unmixed components in each phase. The difference between g;") and g?' is equal to the standard molar Gibbs energy of - gl!)). fusion (melting) of pure Si, Ag& = (g:") - g;")). Similarly for Ge, A&) = (si:,"' The Gibbs energy of fusion of a pure component may be written as:
Agf = Ah:
- TAS;,
(1)
where Ah: and Asp are the standard molar enthalpy and entropy of fusion. Since, to a fist approximation, Ah," and As," are independent of T, Ag! is approximately a linear function of T. If T > T!, then Ag! is negative. If T c T!, then Agfo is positive. Hence, as seen in fig. 2, as T decreases, the gs curve descends relative to g'. At 1500"C,g' c at all compositions. Therefore, by the principle that a system always seeks the state of minimum Gibbs energy at constant T and P,the liquid phase is stable at all compositions at 1500°C.At 13OO0C,the curves of gs and g1cross. The line P,Q,, which is the common tangent to the two curves, divides the composition range into three sections. For compositions between pure Ge and P,, a single-phase liquid is the state of minimum Gibbs energy. For compositions between Q, and pure Si, a single-phase solid solution is the stable state. Between P, and Q1, total Gibbs energies lying on the tangent line P,Q, may be realized if the system adopts a state consisting of two phases with compositions at PI and Q1 and with relative proportions given by the lever rule. Since the tangent line P,Q, lies below both gs and g', this two-phase state is more stable than either phase alone. Furthermore, no other line joining any point on g' to any point on gs lies below the line P,Q,. Hence, this line represents the true equilibrium state of the system, and the compositions PI and Q1 are the liquidus and solidus compositions at 1300°C. IU may be shown that the common tangency condition also results in equal activities of each component in the two phases at equilibrium. That is, equality of activities and miniimization of total Gibbs energy are equivalent criteria for equilibrium between phases. As T is decreased to 11OO"C,the points of common tangency are displaced to higher concentrations of Ge. For Tc 937"C,g" c g' at all compositions. It should be noted that absolute values of Gibbs energies cannot be defined. Hence, the relative positions of gtf) and g z ' in fig. 2 are completely arbitrary. However, this is immaterial for the preceding discussion, since displacing both g:") and g:@) by the same arbitrary amount relative to g,"' and g z ) will not alter the compositions of the points of common tangency. It should also be noted that in the present discussion of equilibrium phase diagrams we are assuming that the physical dimensions of the single-phase regions in the system are sufficiently large that surface (interfacial) energy contributions to the Gibbs energy can be neglected, For very fine grain sizes in the sub-micron range however, surface energy effects can noticeably influence the phase boundaries. References: p. 531.
476
Ch.6, 92
A.D. Peltom
The shape of the two-phase (solid + liquid) “lens” on the phase diagram is determined by the Gibbs energies of fusion, Ag,“ of the components and by the mixing terms, Af and Ag’. In order to observe how the shape is influenced by varying AS:, let us consider a hypothetical system A-B in which Ags and Ag’ are ideal Raoultian (52.2). Let TiA)= 800 K and T&) = 1200 K. Furthermore, assume that the entropies of fusion of A and B are equal and temperature-independent.The enthalpies of fusion are then given from eq. (1) by the expression Ah: = T,“As~since Ag: = O when T = T,“. Calculated phase diagrams for As: =3, 10 and 30 J/mol K are shown in fig. 3. A value ofAs,“ 0 is typical of most metals (Richard’s rule). However, when the components are ionic compounds such as ionic oxides, halides, etc., then As,“ can be significantly larger since there are several atoms per formula unit. Hence, two-phase “lenses” in binary ionic salt or oxide phase diagrams tend to be “fatter” than those encountered in alloy systems. If we are considering vapour-liquid equilibria rather than solid-liquid equilibria, then the shape is determined by the entropy of vaporization, As: (5 2.12). Since As: = lOAs:, two-phase (liquid + vapour) lenses tend to be very wide.
’
~
~
-
IO00
(30)
-
800
I
I2O0Y
I
I
I
4
l
l
l
I
-
(IO)
t = I
c
z
1000
-
-
Q
n
E IQ
800
I
1200-
1000
I
I
1
1
I
I
I
I
I
I
1
I
I
I
(3)
-
BOO 1
A
0.2
0.4
0.6
l
0.8
l
-d B
XB
Fig. 3. Phase diagrams for a system A-B with ideal solid and liquid solutions with I-,= 800 K and ‘I& = 1200 K, calculated for entropies of fusion As&, = As&, = 3, 10 and 30 Jlmol K.
Ch. 6,$ 2
477
Phase diagrams
2.2. Minima and maxima in two-phase regions As discussed in ch. 6, 8 6, the Gibbs energies of mixing, Ag’ and Agl, may each be expressed as the sum of an ideal (Raoultian) term which is purely entropic and which is given by the Boltzmann equation for a random substitutional solution of A and B particles, and an excess term, 8.
Ag = R T ( X , In X ,
+ X, In XB)+ gE,
(2)
where X, and X, are the mole fractions of the components. An ideal or Raoultian solution is defined as one in which ?=O. Both the solid and liquid phases in the Ge-Si system (fig. 2) are approximately ideal. With two ideal solutions, a “lens-shaped” twophase region always results. However, in most alloy systems, even approximately ideal behaviour is the exception rather than the rule. If $>O then the system is said to exhibit positive deviations from ideality. If 8 ~ 0 , then we speak of negative deviations. Curves of gs and g’ for a hypothetical system A-B are shown schematically in fig. 4 at a constant temperature below the melting points of pure A and B such that the solid state is the stable state for both pure components. However, in this system gwl)cg‘s)so that gspresents a flatter curve than does g’ and there exists a central composition region in which g’8(’) to a sufficient extent, then a two-phase region with a maximum will result. In alloy systems, such maxima are nearly always associated with the existence of an intermetallic phase, as will be discussed in 0 2.8.
A
B
X B Fig. 4.. Isothermal Gibbs-energy-compositioncurves for solid and liquid phases in a system A-B in which
pee 2“. A phase diagram of the type in fig. 5 results.
References:p . 531.
478
Ch.6 , 5 2
A.D. Pelton Y
t
1 Liquid
L
3
* 0
1200 Solid 1
1
0.2
I-
1
1
1
0.4
1
0.6
1
1
0.8
1
cn
x cu
Fig. 5. Phase diagram of the Au-CU system (after H U L ~ R E etal. N [1%3]).
2.3. Miscibility gaps If 8 > 0 for a solution, then the solution is thermodynamically less stable than an ideal solution. In an alloy system this can result from a large difference in atomic diameter of the components, which will lead to a (positive) lattice strain energy, or from differences in valence, or from other factors. In the Au-Ni system, g" is positive in the solid phase. In the top panel of fig. 6 is plotted gw'' at 1200 K (HULTGREN et al. [1963]) as well as the ideal Gibbs energy of mixing, Ag"", also at 1200 K. The sum of these two terms is the Gibbs energy of mixing Ag' = Agide*+ $(')', which is plotted at 1200 K as well as at other temperatures in the central panel of fig. 6. Now,
AgiM = RT(XAuIn XAu+ XNiIn XNi)
is always negative and varies directly with T, whereas g" varies less rapidly with temperature. As a result, the sum, Ag'= Agided + 8,becomes less negative as T decreases, curve at X,,= 1 and X,= 1 are both infinite However, the limiting slopes to the Agid""l
( lim d(Agide")/dXAu= lim d(Agide*)/dXNi= XA" +1
XNi+l
whereas the limiting slopes of gE are always finite (Henry's Law). Hence, Ags will always be negative as X,, + 1 and X, + 1 no matter how low the temperature. As a result, below a certain temperature the curves of Ag' will exhibit two negative "humps". Common tangent lines P,Q,, P& P 3 Q to the two humps define the ends of tie-lines of a two-phase solid-solid miscibbiliby gap in the Au-Ni phase diagram which is shown in the lower panel in fig. 6 (HULTGRENet al. [1963]). The peak of the gap occurs at the critical or consolute temperature and composition, T, and X,. When p)is positive for the solid phase in a system it is usually also the case that $")cgEf'), since the unfavourable factors (such as a difference in atomic dimensions) which are causing g(') to be positive will have less of an influence upon gE(') in the liquid phase owing to the greater flexibility of the liquid structure to accommodate different atomic sizes, valencies, etc. Hence, a solid-solid miscibility gap is often associated with a minimum in the two-phase (solid + liquid) region as in the Au-Ni system. Below the critical temperature the curve of Ag' exhibits two inflection points
Ch. 6,$ 2
479
Phase diagram
indicated by the letter “s” in fig. 6. These are known as the spinodalpoints. On the phase diagram their locus traces out the spinodal curve as illustrated in fig. 6. The spinodal 4 2
0 -2
-4 -6
2
I 0 -I
-2 -3
-4
Liquid 1400
Solid
12 00
800
2 Solids 6 00
A- spinoda I
400
AU
0.2
0.4
0.6
0.8
Ni
Ni Fig. 6 . Phase diagram and Gibbs energy curves of solid solutions for the Au-Ni system (after HULTGREN et al. [1963]). Letters “s” indicate spinodal points. References: p . 53I.
480
Ch. 6, $ 2
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curve is not part of the equilibrium phase diagram, but it is important in the kinetics of phase separation as discussed in ch. 5 15. 2.4. Simple eutectic systems The more positive g" in a system is, the higher is T, and the wider is the miscibility gap at any temperature. Suppose that 8'"is sufficiently positive that T,is higher than the minimum in the (solidi-liquid) region. The result will be a phase diagram such as that of the Ag-Cu system shown in fig. 7 (HULTGRENet al. [1963]). In the upper panel of fig. 7 are shown the Gibbs energy curves at 1100 K. The two common tangents define two two-phase regions. As the temperature is decreased below 1100 K, the gs curve descends relative to gl, and the two points of tangency, P, and Pz, approach each other until, at T = 1052 K, P,and Pzbecome coincident at the composition E. That is, at T=1052 K there is just one common tangent line contacting the two portions of the gs curve at compositions A and B and contacting the curve at E. This temperature is known as the eutectic temperature, TE, and the composition E is the eutectic composition. For temperatures below TE,g' lies completely above the common
1100 K
I300
-
Y c
La t
0
Ag
O.'
0.4
0.6
O.*
cu
X C" Fig. 7. Phase diagram and Gibbs energy curves at 1100 K of the Ag-Cu system (after HULTGREN etal. [1963]). Solid Ag and Cu are both fcc.
Ch. 6, $ 2
Phase diagrams
48 1
tangent to the two portions of the g" curve and so, for T< TEa solid-solid miscibility gap is observed. The phase boundaries of this two-phase region are called the solvzrs lines. The word eutectic is from the Greek for "to melt well" since an alloy has the lowest melting point at the eutectic composition E. This description of the thermodynamic origin of simple eutectic phase diagrams is strictly correct only if the pure solid components A and B have the same crystal structure (see 02.6). Suppose a Ag-Cu alloy of compositionX,, = 0.28 (composition PI) is cooled from the liquid state very slowly under equilibrium conditions. At 1100 K the first solid appears with composition QI. As T decreases further, solidification continues with the liquid composition following the liquidus curve from P, to E and the composition of the solid phase following the solidus curve from Q1to A. The relative proportions of the two phases at any T are given by the lever rule. At a temperature T= (TE+ 8) just above TE, two phases are observed: a solid of composition A and a liquid of composition E. At a temperature T=(T,-S) just below T,, two solids with compositions A and B are observed. Therefore, at T,, during cooling, the following binary eutectic reaction QCCUTS:
liquid + solid,
f
solid,.
(3)
Under equilibrium conditions the temperature will remain constant at T = TBuntil all the liquid has solidified, and during the reaction the compositions of the three phases will remain fixed at A, B and E. For this reason the eutectic reaction is called an invariant reaction. The morphologies of two-phase grains resulting from the co-precipitation of two solids during eutectic reactions are discussed in detail in ch. 8.
2.5. Binary phase diagramswith no intermediatephases 2.5.1. Thermodynamicorigin illustrated by simple regular solution theory Many years ago VANLAAR[1908] showed that the thermodynamic origin of a great many of the observed features of binary phase diagrams can be illustrated at least qualitatively by simple regular solution theory. As discussed in ch. 5, 86.2, a regular solution is one in which:
gE = A2XAXB,
(4)
where fi is a parameter independent of temperature and composition. In fig. 8 are shown several phase diagrams calculated for a hypothetical system A-3 containing a solid and a liquid phase with melting points of TiA)= 800 K and TiB,= 1200 K and with entropies of fusion of both A and B set to 10 J/mol K, which is a typical value for metals. The solid and liquid phases are both regular with ~(s)=QsXAXB and gE(I) = fi'XAXB.The parameters IR' and R' have been varied systematically to generate the various panels of fig. 8. In panel (n) both phases are ideal. Panels (1-r) exhibit minima or maxima depending upon the sign and magnitude of (gE(')-gE(')), as has been discussed in $2.2. In panel (h) the liquid is ideal but positive deviations in the solid give rise to a solid-solid miscibility gap as References: p . 531.
482
Ch. 6, $ 2
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- 20
-10
+IO
0
(h)
c)
x
Y
u)
c: Y
D c
fl$r
E
: P
E
F (01
800
I
603 400
( P)
(9)
(rl
Fig. 8. Topological changes in the phase diagram for a system A-B with regular solid and liquid phases, brought about by systematic changes in the regular solution parameters N and a'. Melting points of pure A and B are 800 K and 1200 K. Entropies of fusion of both A and B are 10.0 J/mol K. (PELTONand THOMPSON [19751.)
discussed above in $2.4. On passing from panel (h) to panel (c), an increase in gHs' results in a widening of the miscibility gap so that the solubilities of A in solid B and of B in solid A decrease. Panels (a-c) illustrate that negative deviations in the liquid cause a relative stabilization of the liquid with resultant lowering of the eutectic temperature. Eutectic phase diagrams are often drawn with the maximum solid solubility occurring at the eutectic temperature (as in fig. 7). However fig. 8d, in which the maximum solubility of A in the B-rich solid solution occurs at approximately T=950 K, illustrates that this need not be the case even for simple regular solutions.
Ch.6, 52
483
Phase diagrams
2.5.2. Liquid-liquid immiscibility - monotectics In fig. 8e, positive deviations in the liquid have given rise to a liquid-liquid miscibility gap. An example of a real system with such a phase diagram is the Cu-Pb system et al. [1963]). If a Cu-Pb alloy with X,=O.lO is cooled shown in fig. 9 (HULTGREN slowly from the liquid state, solid Cu begins to appear at 1260 K. Upon further cooling the liquid composition follows the liquidus curve to point A at T= 1227 K. The foIlowing invariant monotectic reaction then occurs:
liquid,
+ liquid, + C U ( ~ ~ , ~ ~ ) ,
(5)
where liquid, and liquid, are liquids with compositions at points A and B. The temperature remains constant at the monotectic temperature and the compositions of all phases remain fixed until liquid, is completely consumed. Cooling then continues with precipitation of copper with the liquid composition following the liquidus line from B to the eutectic E. Returning to fig. 8, we see that in panel (d) the positive deviations in the liquid are not large enough to produce immiscibility but they do result in a flattening of the liquidus which is often described as a “tendency to immiscibility”. An example of such a flattened (or “S-shaped”) liquidus resulting from a positive gE(l)is shown later for the Cd-IPb system in fig. 12. 2.5.3. Peritectics The invariant which appears in fig. 8i is known as a peritectic. The Au-Fe system shown in fig. 10 (HULTGRENet al. [1963]) exhibits a peritectic PQR at 1441 K as well as another at about 1710 K. The Gibbs energy curves, gl and gfcc,of the liquid and solid face-centred cubic phases are shown schematically at the peritectic temperature of Tp=1441 K in the upper panel of fig. 10. One common tangent line PQR to g’ and to the two portions of gfcecan be drawn.
Fig. 1 Phase diagram and Gibbs energy curves at the peritectic temperature of 1441 K .-r the Au-Fe system (after HULTGKHN et al. [1963]).
Suppose that a Au-Fe alloy of composition X,,= 0.65 is cooled very slowly from the liquid state. At a temperature (T’+S) just above 1441 K, a liquid phase of composition P and an fcc phase of composition R are observed at equilibrium. At a temperature (T,-S) just below 1441 K, the two phases at equilibrium are liquid and solid with compositions P and Q respectively. The following invariant binary peritectic reaction thus occurs upon cooling: liquid
+ solid, + solid,.
(6)
This reaction occurs isothermally with all three phases at fixed compositions (at points P, Q and R). In the case of an alloy with overall composition between P and Q, the reaction occurs isothermally until all solid, is consumed. In the case of an alloy with overall composition between Q and R, it is the liquid which will first be completely consumed. A peritectic reaction between a liquid and solid, occurs on the surface of the particles
Ch. 6, $ 2
Phase diagrams
485
of solid, which can rapidly become coated with solid,. By preventing contact between liquid and solid,, this coating may greatly retard further reaction to such an extent that equilibrium conditions can only be achieved by extremely slow cooling. 2.5.4. Syntectics The invariant in fig. 8k in which a solid decomposes upon heating into two liquids is known as a syntectic. It is rarely observed in alloy systems. Examples are found in the K-Pb and K-Zn systems (HANSEN[1958]). A phase diagram similar to fig. 8j, although without the tiny miscibility gap, is exhibited by the Au-R system (HANSEN[1958]).
2.6. Limited mutual solid solubility In 0 2.4 the region of two solids in the Ag-Cu phase diagram of fig. 7 was described as a miscibility gap in the solid phase. That is, only one gs curve was drawn. If, somehow, the appearance of the liquid phase could be suppressed, then the two solvus lines in fig. 7,when projected upwards, would meet at a critical point (as in the Au-Ni system in fig. 6) above which one continuous solid solution would exist at all compositions. Such a description is justifiable only if the pure solid components have the same cryst(a1structure. This is the case for Ag-Cu since solid Ag and Cu are both fcc. The same assumption was made in our treatment of the peritectic Au-Fe system (fig. 10) in which the region of two solids was treated as a miscibility gap. Again in this case this description is permissible since Au and Fe are both fcc in this temperature range. However, consider the simple eutectic system A-B in fig. 11 in which pure solid A and B are hcp (hexagonal close-packed) and fcc respectively. In this case, if the formation of the liquid phase could be suppressed the two solvus lines could not project upward to meet at a critical point, since this would imply that above this critical temperature a continuous series of solid solutions varying smoothly from hcp to fcc coulcl exist. Such a situation is prohibited by symmetry conditions. That is, one continuous curve for gscannot be drawn. Each solid phase must have its own separate Gibbs energy curve, as shown schematically in the upper panels of fig. 11. In this figure,gl(fcc) is the standard molar Gibbs energy of pure fcc A and g:‘h”p’is the standard molar Gibbs energy of pure hcp E. Such quantities may be defined in a number of different and nonequivalent ways as will be discussed below. A real system with a phase diagram similar to fig. 11 is the Cd-Pb system shown in fig. 12 (ASHTAKALA et al. [1981]). Gibbs energy curves at a temperature below the eutectic are shown schematically in the upper panel. Let us derive an expression for g“‘ under the assumption that the Pb-rich fcc solid solution is a Henrian solution. As discussed in ch. 5 , s 6.2, when a solution is sufficiently dilute in one component, Henrian behaviour may be assumed. That is, the activity of the solvent is ideal (uso,vent=Xso~vent; 0 0 ysOlven, = 1), while for the solute, asolute = yso,utdysso,ute, where the Henrian activity coeflcient, ysOlute, is independent of composition. At TE=247.8’C in fig. 12, Cd in the Pb-rich fcc solution at X, = 0.940 exists in equilibrium with virtually pure solid hcp Cd. Thus, in the fcc solution, acd=l.Owith respect to pure solid hcp Cd as standard state. Hence, 0 ycd= u c d X , = 1.O/O.O6O = 16.67 at 2473°C. We can now express gfccas: References:p. 531.
A.D. Pelton
486
Ch. 6 , 6 2
Fig. 11. Phase diagram and Gibbs energy curves at two temperatures for a simple eutectic system A-B in which pure solid A and B have different crystal structures.
+ RT(X, In a, + x,, In a,) = (x,g,4""'+ xpbggw)) + RT(X, h(&xa)+ X , In x,)
gfcc= (xagp)+ X,g:p))
(7)
0 However, since ycd is independent of composition we can combine terms as follows:
+ X,gzw'] In xPb).
gfcc= [ X c d ( g F )+ RT In y&)
+ RT(X,,
In x,,
+ X,
Let us now define:
ggw)= (g:fq) + RT In y&).
Ch. 6 , 9 2
487
Phase diagmms
Liquid
w
2 00 c C d (hcp)
I50
0.2
Cd
0.4
0.6
Pb
Pb(fcc)
0.8
Pb
Fig. 12. Phase diagram of the Cd-Pb system (after ASHTAKALA et al. [1981]) and Gibbs energy curves (schematic) at a temperature below the eutectic. Dashed lines indicate limiting liquidus slopes calculated for zero solid solubility.
From eq. (8) it can be seen that relative to g i p ’ defined in this way and to g r ’ the fcc solution is ideal. This is illustrated in fig. 12. At X7.8”C in Cd-Pb, ( g z ’ - g?’) =RTln -y& =R(247.8 + 273.15) In 16.67 = 12.19 kJ/mol. As a first approximation we could take this value to be independent of T, or as a second approximation we could evaluate &, at other temperatures along the solidus and express (gza’-g$p’) as, say, a linear function of T. Although the above treatment has the advantage of numerical simplicity, it suffers from the difficulty that the numerical value of (g,qlf“e’-giFq’) is solvent-dependent and will be different for, say, solutions of Cd in fcc Cu and Cd in fcc Pb. For purposes of predicting binary phase diagrams from first principles or for estimating ternary phase could be diagrams from binary phase diagrams ($5.5) it would be desirable if defined to be system-independent so as to be truly the “standard molar Gibbs energy of metastable fcc Cd”. A great deal of effort has been expended by the international CALPHAD group under the impetus of Kaufman (KAUFMANand BERNSTEIN [1970]) and References: p . 531.
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co-workers to compile tables of lattice Stabilities for metals in the fcc, hcp, bcc, and liquid states (that is, to obtain a set of relative values of gqfm),gwq),g-’ and gw) for every metal). In some cases, these can be calculated by extrapolating thermodynamic data from regions of T and P where the phases are stable. In other cases, lattice stabilities can be estimated partly from theoretical calculations and partly from the analysis of a large number of binary phase diagrams followed by a judicious choice of the “best” values which most closely fit the greatest number of systems. Tabulations of [1991]). lattice stabilities are now available for many metals (DINSDALE
2.7. Calculation of limiting slopes of phase boundaries In fig. 12 we see that the solubility of Pb in solid Cd is very small. The actual solubility at TE is about 0.14 mol% (HANSEN [1958]). In thermodynamic terms this means that 9““ increases very rapidly as Pb is added to solid Cd (see fig. 12), or that the Henrian activity coefficient 7: is very large. The fact that the solubility of Cd in solid Pb is much greater than that of Pb in solid Cd can be understood in terms of the Hume-Rothery rule (ch. 4) that solubilitiesare greater when the solute atoms are smaller than the solvent atoms, since the lattice strain energy will be less and hence g will rise less rapidly upon addition of solute. As discussed later in $7, it is usually more difficult experimentally to determine a solidus than it is to measure liquidus temperatures. However, if the liquidus has been 1, then the limiting slope of the solidus can be m e h r e d in the limit as XsOlvent+ calculated. Let component B be the solvent in a system A-B. The partial Gibbs energies of B along the liquidus and solidus are equal (gi - 8,” =O). Hence:
(gi
But: - g,””) = RT In ad and ( g i - g?)) = RT In a i , where a i and ai are activities of B on the liquidus and solidus with respect to the pure liquid and pure solid standard states respectively. Hence, eq. (9) may be written as:
RT In a;
- RT In a;
0 = -Agf(B).
In the limit X, + 1, Raoult’s Law holds for both phases. That is, a; Xi. Hence, in the limit, eq. (IO) may be written as:
RT In X i / X i = -A&,,.
(10)
+X;
and
ai + (11)
Furthermore, in the limit, T+ T&) and from eq. (1) Ag& + Ah&,( 1- T/T&)). Finally, liqn+l(ln X,) = (X,- 1). Substituting these limiting values into eq. (11) we obtain: x,lim(dXi/dT 4
- dXi/dT)
= Ah&)/R(T&y.
If the limiting slope of the liquidus, lim, ,,(dX,$dT), is known, then the limiting slope of the solidus can be calculated via eq. (12) if the enthalpy of fusion is known. For the Cd-Pb system, limiting liquidus slopes were calculated for both components
Ch. 6, $ 2
489
Phase diagmms
from eq. (12) under the assumption that there is no solid solubility (that is, that dXi/dT =O). These are shown as the dashed lines on fig. 12. In Cd-rich solutions, agreement with the measured limiting liquidus slope is very good, but in Pb-rich solutions the poor agreement indicates the existence of appreciable solid solubility as has been confirmed by direct measurement.
2.8. Intermediate phases The phase diagram of the Ag-Mg system (HULTGREN etal. [1963])is shown in fig. 13. An intermetallic phase, p', is seen centered approximately about the composition T = 1050K'
T=744K
Liquid
0.2
0.4AiMg0.6
*
0.0
-
M~
MII
Fig. 13. Ag-Mg phase diagram (after HULTGREN el al. [1963])and Gibbs energy curves (schematic) at 744 K and 1050 K. References: p . 531.
490
Ch. 6 , 9 2
A.D. Pelton
XMg= 0.5. The Gibbs energy curve for such an intermetallic phase has the form shown schematically in the upper panel of fig. 13. rises quite rapidly on either side of its minimum which occurs near XMg=0.5.As a result, the p' phase appears on the phase diagram only over a limited composition range. This form of the curve results from a particularly stable crystal structure exists in which Ag and the fact that when XAg=XMg Mg atoms preferentially occupy different sites. The two common tangents P,Q, and P,Q, give rise to a maximum in the two-phase @' +liquid) region in the phase diagram. (Although the maximum is observed very near XMg=0.5,there is no thermodynamic reason for the maximum to occur exactly at this composition.) The Na-Bi phase diagram is shown in fig. 14 (HANSEN[1958]). Gibbs energy curves at 700°C are shown schematically in the upper panel. g(Na3/4Bi1/4)rises extremely rapidly on either side of its minimum which occurs at X,, = 3/4, XBi= 1/4. (We write g(Na3/4Bi1/4)rather than g(N%Bi) in order to normalize to a basis of one mole of metal atoms.) As a result, the points of tangency Q1 and Qz of the common tangents P,Q, and P2Q2are nearly (but not exactly) coincident. Hence, the composition range over which single-phase Na,Bi exists (sometimes called the range ofstoichiometry or homogeneity
8'
8'
0 0 I
E 3 c W L 0)
Q.
E 0)
I-
Bi Fig. 14. Na-Bi phase diagram (after HANSEN[1958]) and schematic Gibbs energy curves at 700°C.
Ch.6,$ 2
Phase diagrams
49 1
range of Na,Bi) is very narrow (but never zero). The two regions labelled (Na,Bi+ liquid) in fig. 14 are the two sides of a two-phase region which passes through a maximum just like the (p’ +liquid) regions in fig. 13. Because the Na3Bi single-phase region is so narrow we refer to Na3Bi as an intermetallic compound. In the case of Na3Bi, any slight deviation from the stoichiometric composition causes a very large increase in Gibbs energy. Owing to the large difference in electronegativities of Na and Bi, Na,Bi could be considered to be a semi-ionic compound. Deviations from stoichiometry would require the substitution of Na on Bi sites or vice versa which would be energetically very unfavourable. If stoichiometric Na3Bi is heated, it will melt isothermally at 775°C to form a liquid of the same composition. That is, the melting behaviour of Na3Bi is similar to that of a pure element. Such intermetallic compounds are called congruently melting or simply congment compounds. The p‘ phase in fig. 13 might also be called a congruent intermetallic compound AgMg (or AgMg,,). It is debatable, however, whether a phase with such ;a wide range of composition should really be called a “compound”. It should be noted with regard to the congruent melting of Na3Bi in fig. 14 that the limiting slopes dT/dX of the two liquidus curves at the congruent melting point (775OC) are both zero, since we are really dealing with a maximum in a two-phase region and not with the melting of an element. Another intermetallic phase, the E phase, is also observed in the Ag-Mg system, fig 13. This phase has a narrow range of stoichiometry around the composition AgMg,. This phase is associated with a peritectic invariant ABC at 744 K. The Gibbs energy curves are shown schematically at the peritectic temperature in the central panel of fig. 13. One common tangent line can be drawn to g’, g’,and 8“. Suppose that a liquid alloy of composition XMg= 0.7 is cooled very slowly from the liquid state. At a temperature just above 744 K a liquid phase of composition C and a p’ phase of composition A are observed at equilibrium. At a temperaturejust below 744 K, the two phases at equilibrium are p’ of composition A and E of composition B. The following invariant peritectic reaction thus occurs upon cooling (cf. 0 2.5.3):
liquid + P’(so1id) + c(so1id).
(13)
This reaction occurs isothermally at 744 K with all three phases at fixed compositions (at points A, B and C). For an alloy with overall composition between points A and B the reaction proceeds until all the liquid has been consumed. In the case of an alloy with overall composition between B and C, the p’ phase will be the first to be completely consumed. The AgMg,(&) compound is said to melt incongruently. If solid AgMg, is heated, it will melt isothermally at 544 K, by the reverse of the above peritectic reaction (141, to form a liquid of composition C and another solid phase, p’, of composition A. Another example of an incongruent compound is the compound NaBi in fig. 14. This compound has a very narrow range of stoichiometry. When heated, it melts incongruently (or peritectically) at the peritectic temperature of 446°C to form another solid, Na,Bi, and a liquid of composition XBi= 0.53. An incongruent compound is always associated with a peritectic. (The word peritectic comes from the Greek for (loosely) “to melt in an indirect way”.) However, the converse References: p. 531.
492
A.D. Pelton
Ch. 6, 42
is not necessarily true. A peritectic is not always associated with an intermediate phase. See, for example, fig. 10. For purposes of phase diagram computations involving very stoichiometric compounds such as Na,Bi, we may, to a good approximation, consider the Gibbs energy curve, g(Na,,Bi,,), to have zero width. Then all we need is the numerical value of g(Na,,,Bi,,,) at the minimum. This value is usually expressed in terms of the Gibbs energy of fusion of the compound, Ag&%,4Bi,,),or in terms of the “Gibbs energy of formation”, Ag&,,(N%14Bi,,4) ,of the compound from solid Na and Bi according to the reaction :Nu, +$Bi, =Na,,,Bi,,,,,. Both these quantities are interpreted graphically in fig. 14. 2.9. Topology of binary phase diagrams
In ch. 5, 3 8.2 the Gibbsphase rule was derived.:
F = C - P+2, where C is the number of components, P the number of phases in equilibrium, and F the number of degrees of freedom or variance. That is, F is the number of parameters which can and must be specified in order to completely specify the state of the system. In the present context, the thermodynamic parameters are temperature, total pressure, and the compositions of the phases at equilibrium. Since binary temperature-composition phase diagrams are plotted at constant pressure, usually 1 bar, one degree of freedom is already used up. In a binary system, C=2. Hence, for binary isobaric T-X diagrams the phase rule reduces to:
F=3-P.
(15)
Binary T-X diagrams contain single-phase areas and two-phase areas. In the singlephase areas, F = 3 - 1= 2. That is, temperature and composition can be varied independently. These regions are thus called bivuriunt. In two-phase regions, F = 3 - 2 = 1. If, say, T i s chosen, then the compositions of both phases are fixed by the ends of the tie-lines. Two-phase regions are thus termed univariant. Note that the overall composition can be varied within a two-phase region at constant T, but the overall composition is not a variable in the sense of the phase rule. Rather, it is the compositions of the individual phases at equilibrium that are the variables to be considered in counting the number of degrees of freedom. When three phases are at equilibrium in a binary system at constant pressure, F = 3 - 3 = 0. Hence, the compositions of all three phases as well as T are fixed. There are two general types of three-phase invariants in binary phase diagrams. These are the eutectictype andperitectic-type invariants as illustrated in fig. 15. Let the three phases concerned be called CY, /3 and y, with p as the central phase as shown in fig. 15. CY, /3 and y can be solid, liquid or gaseous phases. At the eutectic-type invariant, the following invariant reaction occurs isothermally as the system is cooled: P+CY+Y,
whereas, at the peritectic-type invariant the invariant reaction upon cooling is:
(16)
Ch.6, $ 2
493
Phase diagrams
(17)
ff+Y+B
Some examples of eutectic-type invariants are: (i) eutectics (fig. 7) in which a = solid,, p =liquid, y = soli&. The eutectic reaction is 1+ s, + s,; (ii) monotectics (fig. 9) in which a =liquid,, p = liquid,, y = solid. The monotectic reaction is 1, + 1, + s; (iii) eutectoids (fig. 10) in which CY =solid,, p =solid,, y =solid,. The eutectoid reaction is s2+ s1 + s,; (iv) catutectics in which CY =liquid, /3 = solid,, y = solid,. The catatectic reaction is s1 + 1+ s2. Some examples of peritectic type invariants are: (i) peritectics (fig. 10) in which a =liquid, j? = solid,, y =soli&. The peritectic reaction is 1+ s, + s,; (ii) syntectics (fig. 8k) in which a =liquid,, p =solid, y =liqui&. The syntectic reaction is 1,+12+ s; (iii) peritectoids in which a = solid,, p = soli&, y = solid,. The peritectoid reaction is s, + s3+s,. An important rule of construction which applies to invariants in binary phase diagrams is illustrated in fig. 15. This entension rule states that at an invariant the extension of a boundary of a two-phase region must pass into the adjacent two-phase region and not into the single-phase region. Examples of both correct and incorrect constructions are given in fig. 15. To understand why the “incorrect extension” shown is not correct, consider that the (a+ y ) phase boundary line indicates the composition o f the y-phase in equilibrium with the a-phase as determined by the common tangent to the E u t e c t i c - type invariant
incorrect extensions
Peritectlc-type invariant
correcf
correct extensions
P+Y
----Maximum
a+P
a+P
Fig. 15. Some topological units of construction of binary phase diagrams illustrating rules of construction.
References:p . 531.
494
Ch. 6,$ 2
A. D.Pelton
Gibbs energy curves. Since there is no reason for the Gibbs energy curves or their derivatives to change discontinuously at the invariant temperature, the extension of the (a +y) phase boundary also represents the composition of the y-phase in equilibrium with the a-phase. Hence, for this line to extend into a region labelled as single-phase y is incorrect. Two-phase regions in binary phase diagrams can terminate: (i) on the pure component axes (at X,= 1 or X,= 1) at a transformation point of pure A or B; (ii) at a critical point of a miscibility gap; (iii) at an invariant. Two-phase regions can also exhibit maxima or minima. In this case, both phase boundaries must pass through their maximum or minimum at the same point as shown in fig. 15. All the topological units of construction of binary phase diagrams have now been discussed. The phase diagram of a binary alloy system will usually exhibit several of [1982]) is shown these units. As an example, the Fe-Mo phase diagram (KUBASCHEWSKI in fig. 16. The invariants in this system are: peritectics at 1540, 1488, and 1450°C; eutectoids at 1235 and 1200°C; peritectoids at 1370 and 950°C. The two-phase (liquid + y ) region passes through a minimum at X,,=O.Z. Between 910°C and 1390°C is a two-phase (a+ y ) y-loop. Pure Fe adopts the fcc y structure between these two temperatures but exists as the bcc a phase at higher and lower temperatures. Mo however, is more soluble in the bcc than in the fcc structure. That is, g, O@cc-Fe) < g;y’ as discussed in 02.6. Therefore, small additions of Mo stabilize the bcc structure. 2.9.1. Orderdisorder transformations In fig. 13 for the Ag-Mg system, a transformation from an a‘ to an a phase is shown occurring at approximately 390 K at the composition Ag,Mg. This is an orderdisorder
0.0
0.2
0.6
0.4
Mo
Fig. 16. Fe-Mo phase diagram (KuB-
[1982]).
0.8
1.0
Ch. 6, 0 2
Phase diagrams
495
transformation. Below the transformation temperature, Zong-range ordering (superlattice formation) is observed. An orderparameter may be defined which decreases to zero at the transformation temperature. This type of phase transformation is not necessarily a first-order transformation like those considered so far in this chapter. Unlike transformations which involve atomic displacements over distances large compared with atomic dimensions, order-disorder transformations, at least at the stoichiometric composition (Ag,Mg is this example), occur by atomic rearrangement over distances of the order of atomic dimensions. The slope of the curve of Gibbs energy versus T is not necessarily discontinuous at the transformation temperature. For a detailed discussion see ch. 4, 54.1.1, INDEN [1982], and Prrsc~and INDEN[1991]. A type of order-disorder transformation of importance in ferrous metallurgy is the magnetic transformation. Below its Curie temperature of 769"C, Fe is ferromagnetic. Above this temperature it is not. The transformation involves a change in ordering of the magnetic domains and is not first-order. Additions of alloying elements will change the temperature of transformation. Magnetic transformations are treated in ch. 29. See also MIODOWNIK [1982], INDEN [1982] and HILLERTand JARL[1978].
2.10. Application of thermodynamics to phase diagram analysis In recent years, the development of solution models, numerical methods and computer software has permitted a quantitative application of thermodynamics to phase diagram analysis. Computer programs are available which permit phase diagrams to be generated from equations for the Gibbs energies of the phases. Other programs have been written to perform simultaneous critical evaluations of available phase diagram measurements and of available thermodynamic data (calorimetric data, measurements of activities, etc.) with a view to obtaining optimized equations for the Gibbs energies of each phase which best represent all the data. These equations are consistent with thermodynamic principles and with theories of solution behaviour. Vie phase diagram can be calculated from these optimized thermodynamic equations, and so one set of self-consistent equations describes all the thermodynamic properties and the phase diagram. This technique of analysis greatly reduces the amount of experimental data needed to characterize a system fully. All data can be tested for internal consistency. The data can be interpolated and extrapolated more accurately, and metastable phase boundaries can be calculated. All the thermodynamic properties and the phase diagram can be represented and stored by means of a small set of coefficients. Finally and most importantly, it is often possible to estimate the thermodynamic properties and phase diagrams of ternary and higher-order systems from the assessed parameters for their binary sub-systems as will be discussed in $ 5 . The analysis of binary systems is thus the first and most important step in the development of databases for multicomponent systems. The computer coupling of thermodynamics and phase diagrams is a growing field of much current research interest. The international Calphad Journal, published by Pergamon Press, and an annual international meeting, the Calphad Conference, are now devoted to this subject. References: p. 531.
496
A.D. Pelton
Ch. 6, $ 2
2.10.1. Polynomial representation of excess properties Empirical equations are required to express the excess thermodynamic properties of the solution phases as functions of composition and temperature. For many simple binary substitutional solutions, a good representation is obtained by expanding the molar excess enthalpy and entropy as polynomials in the mole fractions X, and XBof the components:
hE = X,XB(hO+ h,(XB- XA)+ 4 ( X B -X*)' + 4 ( X B - x A ) 3 +...)
(18)
where the hj and si are empirical coefficients. As many coefficients are used as are required to represent the data in a given system. For most systems it is a good approximation to assume that the coefficients hi and si are independent of temperature. If the series are truncated after the first term, then:
gE = hE - TsE = XAXB(hO - Ts,)
(20)
This is the same as eq. (4) for a regular solution. Hence, the polynomial representation can be considered to be an extension of regular solution theory. When the expansions are written in terms of the composition variable (XB-XJ as in eqs. (18) and (19)they are said to be in Redlich-Kisfer form. Other equivalent polynomial expansions such as and BALE[1986]. orthogonal Legendre series have been discussed by PELTON Differentiation of eqs. (18)and (19)yields the following expansions for the partial excess properties:
h," = x:chi[(xB
- XA)i + 2iXB(XB- x,)'-']
i=O
2.10.2. Least-squares optimization Eqs. (18), (19)and (21)to (24) are linear in terms of the coefficients. Through the use of these equations, all integral and partial excess properties @, hE,3, gr, hE, s:) can be expressed by linear equations in terms of the one set of coefficients {hi,si}. It is thus possible to include all available experimental data for a binary phase in one [19831, simultaneous linear least-squares optimization as discussed by BALEand PELTON
Ch. 6, $ 2
497
Phase diagrams
LUKASet aZ. [1977] and D~RNER et al. [1980], and specialized software for such optimizations is available. The technique of coupled thermodynamic/phasediagram analysis is best illustrated by an example. The phase diagram of the Cd-Na system with points measured by several authors is shown in fig. 17. From electromotive force measurements on alloy concentration cells, several authors have measured the activity coefficient of Na in liquid alloys. These data are shown in fig. 18 at 400°C. From the temperature dependence of E ,g =RT In y,,, the partial enthalpy of Na in the liquid was obtained via the GibbsHelmholtz equation. The results are shown in fig. 19. Also, hE of the liquid has been measured (KLEINSTUBER [19611) by direct calorimetry. Along the Cd-liquidus in fig. 17 the partial Gibbs energy of Cd in the liquid is equal to that of essentially pure solid Cd with which it is in equilibrium:
Weight P e r c e n t Sodium 0 450
........,.........,
Cd
10
. . . , . . ! . . . ,....
,
20
30
Atornic P e r c e n t Sodium
4,O
50
60 70 BO I
'
1 d
Ka
Fig. 17. Cd-Na phase diagram calculated from optimized thermodynamic parameters (Reprinted from PELTON [1988]:). 0 KIJRNAKOW and KUSNETZOV [1907] A MATHEWSON [1906] x WEEKS and DAVIES [1964] References: p . 531.
498
Ch. 6, 32
A.D. Pelton
Cd
Atomic
Percent Na
Nu
Fig. 18. Sodium activity coefficient in liquid Cd-Na alloys at 400°C. Line is calculated from optimized thermodynamic parameters (Reprinted from PELTON[1988]). 0 HAUFFE 119401 0 LANTRATOV and MIKHAILOVA [I9711 A MAIOROVA et al. [ 19761 V ALABYSHEV and MORACHEVSKII [1957] 0 BARTLETT etal. [I9701
RT In XLd + g:!)
= -AgiCd)
(27)
Therefore, from the experimental liquidus composition x & , and from a knowledge of the Gibbs energy of fusion, g z ) at the measured liquidus points can be calculated from eq. (27). Similar equations relating the liquidus compositions along the Cd,,N%- and CGNaliquidus lines to the partial excess Gibbs energies of the liquid and to the Gibbs energies of fusion of Cd,,Na, and Cd,Na can be written based upon the graphical construction shown in fig. 14. The thermodynamic data for g:# hiaand hEas well as the measured liquidus points and the Gibbs energies of fusion of the compounds were optimized simultaneously by a least-squares technique to obtain the following optimized expressions (PELTON[19881):
Fig. 19. Partial excess exthalpy of sodium in liquid Cd-Na alloys. Line is calculated from optimized thermodynamic parameters (Reprinted from PELTON [ 19881). 0 LANTRATOV and MIKHAILOVA [1971] A MAIOROVA et al. [1976] 0 BARTLEITet al. [1970]
AG;(:cd*Na) = 8368 - 12.737 T Jlg-atom AG&, = 6201 - 10.4344 T Jig - atom AG,",,,
= 2598 - 7.0035 T J/g-atom
(32) (33)
The Gibbs energies of fusion of Cd and Na were taken from CHASE[1983] and were not changed in the optimization. The optimized enthalpies of fusion of 6816 and 8368 J/g-atom for the two compounds in eqs. (30, 3 1) were modified from the values of 6987 and 7878 J/g-atom measured by Roos [1916]. These changes are within the experimental error limits. Eq. (28) reproduces the calorimetric data within 200 J/mol-'. Eqs. (22, 24) can be used to calculate and y"N,. The calculated curves are compared to the measured points in figs. 18 and 19, The phase diagram shown in fig. 17 was calculated from eqs. (28) to (33). Complete details of the analysis of the Cd-Na system are given by PELTON [19881. It can thus be seen how one simple set of equations can simultaneously and selfconsistently describe all the thermodynamic properties and the phase diagram of a binary system. The exact optimization procedure will vary from system to system depending on the type and accuracy of the available data, the number of phases present, the extent of Rt.ferences:p. 531.
500
A.D. Pelton
Ch.6, 32
solid solubility, etc. A large number of optimizations have been published in the Calphad Journal since 1977.
2.10.3. Calculation of metastablephase boundaries In the Cd-Na system just discussed, the liquid exhibits positive deviations from ideal mixing. That is, gE‘”>0.This fact is reflected in the very flat liquidus in fig. 17 as was discussed in 0 2.5.2. By simply not including any solid phases in the calculation, the metastable liquid miscibility gap as well as the spinodal curve (0 2.3) can be calculated as shown in fig. 17. These curves are of importance in the formation of metallic glasses by rapid quenching (see ch. 19, 0 2.1). Other metastable phase boundaries, such as the extension of a liquidus curve below a eutectic, can also be calculated thermodynamically simply by excluding one or more phases during the computations.
2.11. Solution models Polynomial expansions, as in eqs (18, 19), give an adequate representation of the excess mixing properties for simple substitutional solutions in which deviations from regular solution behaviour are not too large. In other cases, more sophisticated models are required. The Gibbs energy of a regular solution is given by combining eqs. (2) and (4). The ideal mixing term in eq. (2) is a consequence of the assumption that A and B atoms form a random substitutional solution. The parameter fl in eq (4) can be interpreted as resulting from the fact that the energy of A-B bonds in the solution is different from that of A-A and B-B bonds. Suppose that A-B bonds are energetically favourable. The solution is thereby stabilized, 0 and 9 i ni When B atoms are isotopes A* of the element A, eq. (13) becomes:
JA* = -DA.VnAt
+ n kTD
ZreE -
n D
QT
It can be shown, (HOWARD and LIDIARD[1964]), that:
-1= I + - LA*A f,
-%*A*
The apparent valency and heat of transport are therefore given by:
where f, is the correlation factor for self-difision; its presence stems from the nonD is the ramdom character of the tracer atom displacements by a vacancy mechanism. , self-difision coeficient, given by:
13.4. The various diffusion coefficients Diffusion coefficients have the dimension Length2Time-'. In the international system of units they are expressed in m2s-'. The CGS system (em's-') is still widely used. We will show now which experimental situations correspond to these various coefficients.
References: p . 651.
544
Ch.7,8 1
J. L Bocquet, G.Brebec, !I Limoge
The chemical dz@sion coeficient 0 describe the interdiffusion of A and B (fig. 4a); it can be measured from the curve CA (or C,) versus x; in general it depends on the concentration. The intrinsic coej'3icients DA and D, correspond to a similar experiment; but to obtain them it is necessary to determine fi and v [see eqs. (9) and (ll)]. v is obtained from the displacement of inert markers (see Kirkendall effect, 5 5.3.1.1). These coefficientsdepend also on the concentration. The solute diffusion coeficient at inJinite dilution D p corresponds to the experimental situation shown in fig. 4b. A thin layer of B* atoms has been deposited on the A surface so that CB* 0 and B* diffuses in pure A. The self-difision coeficient D, corresponds to a similar situation when B* is replaced by A*. Two other diffusion coefficients are defined as shown in fig. 4c; they are the selfdifision coeficients in an homogeneous alloy AB which are denoted by D E . The B* (or A*) concentration is always negligible so that the alloy composition is not modified by the diffusing species. These coefficients depend on the concentration. An alternative notation often used for dilute alloys is:
-
where C, is the concentration of B. The macroscopic description presented above cannot account for the A* and B* diffusion into AB alloys; it would be necessary to derive the flux equations for four species A, A*, B, B* (plus vacancies); this is beyond the scope of this review and we and LIDIARD [1964] for more details. Thus it is possible to refer the reader to HOWARD show that the self-diffusion coefficients in the alloy and the intrinsic diffusion coefficients are related by:
where p is the thermodynamic factor and rA and r, are terms which will be made [1948] in a explicit in 3 5.1.2. These relations, (17), were first established by DARKEN A* or B *
a
C
Fig. 4. Different types of diffusion experiments: (a) chemical diffusion + b ;(b) self or solute diffusion in pure metals -+ D,. or DB.; (c) self-diffusionin homogeneous alloys + @or
e
Ch. 7, 0 1
D i e i o n in metals and alloys
545
simplified form for the case when rA=rB= 1. 1.2.5. Fick’s second Law We have seen that the fluxes in a binary alloy have the form:
Ji = -DiVni or Ji = -DiVni
+ ni c v
>i
By using the conservation equation:
dn.
1- -divJi
at
we obtain Fick’s second Law; this partial differential equation can be solved for given initial and boundary conditions. Di and can then be obtained from a comparison between the experimental and the calculated concentration curve C(x). When Di and are constant and the diffision is along the x direction Fick’s second Law has the form:
OE
The geometry which is most commonly used for measuring Di is a thin layer deposited onto a “semi-infinite sample” (see fig. 4b and 4)in ;this case the solution of eq. (18) has the well-known form:
ni(x,t)= Q exp(-L) 4Dit where Q is the quantity of the diffusing species deposited per unit surface, so that Di is obtained from the slope of the straight line: log ni versus x2. In the presence of an electric field, the equation to be solved is eq. (19); very often the geometry used is a thin layer sandwiched between two semi-infinite samples. The solution is then:
Q ni(x,t) = ___ 2J.rrD,texp
[-
(x- < v
>i
44t
t) 2]
is obtained from the displacement of the maximum of the curve ni(x) with respect to the origin (defined by the welding interface). For chemical diffusion (see fig. 4a), fi is not constant, we have then to solve:
References: p. 651.
546
J. L Bocqwt, G. Brebec, E: Limoge
MATANO[1933] has shown that, when
Ch. 7, 5 1
Bi depends on x through ni:
ni
J x d ni 1 o i ( n i )= -o 2t (dn,ldx)
-
the x origin must be chosen so that:
ni'F
d ni = 0
0
This origin defines the Matanoplane. In fig. 5 the different terms of eqs. (23) and (24) are illustrated. Numerous solutions of the diffusion equation can be found in CRANK[1956] and CARSLAW and JAEGER [1959]. We will see that in some cases Fick's first law is not valid; the first restriction is related to the discontinuous nature of crystals (lattice effect) and will be discussed in $1.3.5. The second restriction is met in chemical diffusion (spinodal decomposition: CAHN[1967]; ch. 15, $ 3.1). In both cases the discrepancy with Fick's law becomes noticeable only for harmonics of concentration with short wavelengths.
1.3. The random walk theory of diffusion The aim of the random walk theory is to describe the observed macroscopic effects from the atomic jumps which are the elementary processes in diffusion.
1.3.1. Einstein relation and flux expression For a random walk motion, EINSTEIN [ 19051 has shown that the diffusion coefficient of species i along the x direction is given by:
where 9 is the mean square displacement along the x direction for the duration T. If X, is the displacement of the kthatom along the x direction during 7,we have:
k=l
where N is the number of diffusing atoms of species i. In many cases the motion is not random but the expression (25) still holds provided that T + 0. According to LE CLAIRE [1958] and MANNING[1968], the flux Ji measured with respect to the lattice reference frame is equal to:
hi dDi Ji =< v >, n, - 4.-- n, dX
dX
Ch. 7, 0 1
547
Difision in metals and alloys
I
I
I
, 4
MaLano plane
0
F
X
Fig. 5. Matano method for the calculation of b . The Matano plane is defined by the equality of the two areas POW and FOM (hatched surfaces). k x hi is equal to the area HPFO (doubly hatched surface), dn,/dx is the slope of the tangent to the concentxation curve at P.
where:
X c v >i = limT+o
(28)
7
Di is given by eq. (25) when T -+ 0, 9 is the mean displacement during T for species i. These relations, (25), (27) and (28), are valid for anisotropic media but to save space we have omitted the more precise notation Dk, < v > ~etc... ~
x and 9 in terms of jump frequencies
1.3.2. Calculation of It is easy to show that:
i=l
References: p , 651.
548
J. L Bocquet, G. Brebec, L Limoge
-
x2 =
Ch. 7, $ 1
cxi" c 2% n -
n-1
n
-I-2
i=l
1=1 & i t 1
where xi is the i th displacement along x and n is the mean number of atomic jumps during T. The overbar denoteLan average over a large number of atoms. 1.3.2.1. Expression for X2. For a truly random walk motion the last term in eq. (30), P = 2 XX vanishes. When X differs from zero (chemical diffusion, electro and thermal diffusion, etc.) this term P is also different from zerohut is has been shown that the X contribution to P is of the order to ? whereas the C xi2 term, eq. (30), is of the order of T ; as a consequence the contribution to P is negligible when T + 0. But even if = 0, the P ,term is not necessarily equal to zero, owing to the mechanism of diffusion. We will see later that for most diffusion mechanisms the successive atomic jumps are not independent of each other, and that the motion is not a truly random walk. This can be easily understood for the vacancy mechanism: the vacancy concentration is so low (- lo4 to the melting point) that two consecutive atomic jumps are likely due to the same vacancy and it is obvious that after one jump an atom has a greater than random probability of making a reverse jump; there is correlation. This correlation between the directions of two successivejumps initiated by the same vacancy reduces the efficiency of the walk with respect to a truly random walk. Correlation occurs for all defect-assisted diffusion mechanisms except for the purely interstitial and exchange mechanisms; it is related to the low concentration of point defects (vacancies, divacancies, interstitials, etc.) and decreases when this concentration increases (WOLF [19801). How to take this effect into account will be reported in Q 1.3.4. To summarize, we can always calculate X 2 by assuming X = 0, because when r + 0, X 2 does not depend on X. For a truly random walk motion, P = 0 and we have:
xi,
x
x
where z is the number of jump directions, rkthe mean atomic jump frequency for the k direction and xk the displacement along x for a k-jump. Hence:
De=+
2rkx;. k=l
For cubic lattices all the frequencies rkare equal, and:
ri2
Ddm = -
(33)
6
where I? =Xkrkis the total jump frequency and 1 is the jump distance (;a@ $ a 6 for bcc). 1.3.2.2. Expression for With the same notation as for ??we have:
x.
for fcc,
Ch. 7 , 5 1
549
DifSusion in metals and alloys
w
For the case where is not zero, the potential energy of the atoms versus their position is schematized in fig. 6 (for simplicity we have shown regular energy barriers which correspond to a mean displacement independent of x). The shape of this energy diagram is due to a force Fi acting on the atoms such that (see fig. 6):
x
The atom jumps are easier towards the right than towards the left (in fig. 6) and if AW
3000
4000
O
1000
2000
3000
4000
Tm
b)
Fig. 8. Van Liempt relation for metals. Fig Sa) fcc metals, fig Sb) bcc metals. The straight line represents the Van Liempt relation: Q=0.14TMin kJ/mole. bcc structures are widely dispersed around the line, contrarily to fcc ones. References: p . 651.
574
. I L. .Bocquet, G. Brebec, Y: Linwge
a.7 , 4 3
Tm /T Fig. 9. Diffusion coefficients in an h h e n i u s plot normalized to the melting point of the element considered. a) in fcc metals; the range of diffusion values is quite narrow and the curvatures, if any, are small. b) same curves in bcc metals; the diffusion coefficient varies widely from one element to the other according to a systematics controlled by the row number in the Mendeleev classification, the curvatures also are frequently strong.
3.1. Self-diffusion in fcc and hcp metals The vacancy assumption for the diffusion in these metals is now well accepted by everyone (but see also 0 4.2.2). Nevertheless there are still some controversies as regard to the origin of the possible curvature of the Arrhenius plots. In order to explain this curvature, three hypotheses can be retained among all the possibilities which have been discussed in 5 1.4.3.1 .: - A vacancy mechanism occurs over the whole temperature range but because of a strong thermal expansion coefficient for the vacancy, (GILDER and LAZARUS[ 1975]), or due to the variation of the elastic constants with the temperature, (VAROTSOSand A ~ x o m [1986]), ~ s Do and Q increase with temperature. - Both vacancies and &vacancies contribute to the diffusion, with an increasing participation of the latter at high temperatures (SEEGERand MEHRER [1970]).
Table 2 Self-diffusionparameters for pure elements. Element (see corn- C.S. ments at end)
Ag
fcc
T,,, (K)
1234
D,, (m's-') x 104 Q (kJlmole)
Dol=0.046 D, = 3.3
Q,= 169.8
2.25
144.4
(K)
Q =.1422Tm D(T,,,)(m*s-') D (ph. tr.) (m*s-') Van Liempt
Comments on table These self diffusion data have been extracted from the compilation by MEHRERet al. (Ref. B). Column 1: Symbol of the metal. Column 2 Crystal stucture. bcc = body centered cubic, fcc = face centered cubic, hex = hexagonal, m = monoclinic, ort = orthorhombic, tetr = tetragonal, trig = trigonal. Column 3: Melting temperature. For the phases which do not melt (for instance Ce y , Fe LY etc.) we have given the temperature of the phase transition. column 4 Experimental D, The value in m*s-' is multiplied by lo4 (so that it is in cm*s-'). For some of the metals the Arrhenius plot is curved and D has the form: D= D,,exp (- Q,lRr)+ D,,exp (- Q21R~), in these cases Doland D, are given (they are also multiplied by lo4). For Ti and Zr which have strongly curved Arrhenius plots special expressions are given for D (in m*s-'without any multiplying factor). column 5: Experimental Q in kJ mole-'. Same remadcs as for column 4. column 6 Temperarange of the experimental determination of D. Column 7: Empirical value of Q according to the Van Liempt relation. For the phases which do not melt this value is followed by an *. Column 8: Value of D at the melting point. Column 9: For metals which display several phases the values of D are given at the temperature boundaries of the phase. For instance Up is stable between 941 and 1048 K, D values at these temperatures are given in column 9. Column 10 References.
- A vacancy mechanism occurs and the curvature is due to the dynamical correlation between successive jumps (vacancy double jumps) (DA FANOand JACUCCI [1977]). Experimentally the following data are available: frequency factor Do,activation energy Q, isotope effect E and activation volume AV. When the Arrhenius plot is curved, we notice that Doand Q increase with T whereas E decreases; for example, for silver self-diffusion, E decreases from 0.72 to 0.58 when T increases from 673 to 954°C. Any of the three assumptions can explain these experimental data: the decrease with temperature of the isotope effect is obvious for the mixed vacancy-divacancy mechanism since the correlation factor for the divacancy mechanism is smaller than for the vacancy mechanism. As a result, since the contribution of the divacancies to the diffusion incre:ases with T, the apparent correlation factor and then the isotope effect will decrease. But this variation of E with T can also be explained with the two other assumptions. Likewise the variation of Doand Q with T is compatible with all three hypothesis. The variations of AV with P and T have not been frequently studied; in the case of silver AV increases with T, but remains constant for gold and aluminium. The increase with T has been interpreted as resulting from an increase of the divacancy contribution at high temperatures (REIN and MEHRER[1982]). However, measurements of defect properties after quenching can only be understood [1978]); in addition, the analysis of if vacancies and divacancies are present (PITERSON tracer and NMR data on self-diffusion in sodium seems also to favour the mixed vacarncy-divacancy mechanism ( B R ~ G EefRal. [19801). Although these two statements are riot very general a consensus does exist in favour of the mixed vacancy-divacancy mechanism. Thus, in general when the Arrhenius plots are curved the data are fitted by assuming a two-defect mechanism; in addition a possible dependence of enthalpies and entropies on wmperature is sometimes taken into account (see for instance SEEGERand References: p . 651.
580
J. L Bocquet, G.Brebec, L Limoge
Ch. 7,1 3
MEHRER[1970] or PETERSON [1978]). Nevertheless the discussion is still open, since divacancies might also be formed during the quench, and the role of divacancies is among the “Unsolved Problems” for some experts (MUNDY[1992]). In hcp metals the limited number of available data is compatible with a slight decrease of the ratio of the activation energies of the diffusion parallel to perpendicular to the c axis with increasing c/a ratio, the activation energies being the same in the ideal lattice (HOOD[ 19931).
3.2. Diffusion in bcc metals Self-diffusion in bcc metals presents three characteristics which do not comply with the previous picture. At first there is a much larger scatter of the diffusivity in bcc metals
than in the compact phases, and some of them display an unusually large absolute value of D (fig. 9b); second, they frequently exhibit much larger curvatures than the fcc or hcp systems, much to large to be accounted for by a divacancy contribution; last they show a systematic variation of D with the position in the classification which has to be explained, e.g. metals of the same column, like Ti, Zr, Hf in the group 4, have for all of them a very small activation energy and a large curvature (fig 9b). Many explanations have been proposed in order to account for these anomalies: strong contribution of shortcircuits, presence of extrinsic vacancies due to impurities, interstitial mechanisms, etc. All these assumptions have been ruled out by experiments. The very origin of this behaviour is now recognized to be linked to the electronic structure of the metal and to the structural properties of the bcc lattice. At first the diffusion mechanism is now proved by quasi elastic neutron scattering experiments, to be the vacancy one with nearest-neighbour jumps, either in sodium (AIT SALEM etal. [1979]) or in p-Ti (PETRYetal. [1991]). A small fraction of N.N.N. jumps could also contribute, the fraction being independent of temperature. The same mechanism very likely is also at work in other bcc metals. The key point now is the recognition that the bcc structure is intrinsically soft with respect to some specific shear deformations; moreover this intrinsic softness can be enhanced (as in @Ti) or lowered (as in Cr) according to specific features of the electronic structure controlled by the number of d electrons (Ho etal. [1983, 19841). This softness is the very origin of the numerous martensitic phase transformations observed between bcc and hcp or GI phases, under ambient or high pressure in several of the metals displaying a range of stability in the bcc structure. It is also manifested by the presence in the phonon dispersion curves of a whole branch of soft phonons at large wave vector, from the longitudinal q = 2/3[11I] to the q = 1/2[110] phonons. These phonons are precisely the ones which control most efficiently both the jump of the vacancy and the martensitic bcc to hcp phase transformation (1/2 [llO]) or to w phase ( U 3 [lll]). Being of low frequency, they contribute to large fluctuations of the reaction coordinate and therefore give rise to a small migration enthalpy as well as to high diffusion coefficients (see P 1.4.2.2 and eq. (52-53)) (HERZIGand K ~ H L E[1987], R PETRY et al. [1991]). Using experimental dispersion curves, in the framework of the dynamical theory, it is possible to calculate migration enthalpies in good agreement with the
Ch. 7 , $ 3
Diffusion in metals and alloys
581
experimental values (SCHOBER etal. [1992]). In this respect the 1/2[1 IO] phonon is twice as efficient as the 2/3[111] one to promote the jump (WILLAIME [1991]). Moreover, using inelastic neutron diffraction methods, the 1/2[110] phonon has been shown to be strongly anharmonic and to soften as temperature decreases in the “most curved” metals (Ti, Zr and Hf)(PETRYand col. [1991]). In this approach the curvature of the Arrhenius plots also can be qualitatively explained, as well as the decrease of the isotopic effect with decreasing temperature (from 0.285 at 916°C to 0.411 at 1727°C in Zr), in contrast with the data of isotope effects in self-diffusion in other structures. In this picture the whole of the effect appears to be due to the migration term, being small and T-dependent. However we can also expect that these soft phonons will be linked with large relaxations around the vacancy, corresponding to specific features also for the formation contribution in bcc metals. Indeed it is recognized (SCHULTZ[1991], SCHOBERet al. [1992]) that in this respect Cr displays an anomalously large formation enthalpy and Ti an anomalously small one. In Cr the 1/2[110] phonon softens with increasing temperature. Since the diffusion activation enthalpy appears to be a constant in the whole temperature range, the formation enthalpy should then increase with T according to the preceding analysis (SCHOBERet al. [1992]). The analysis of the electronic structure of bcc metals indeed allows for a systematic variation of the vacancy properties with the number of d electrons: due to the presence of a quasi-band gap in the band structure for a number of electrons of 4, and a maximum around 2, the above mentioned variations of formation terms can be understood (WILLAIMEand NMTAR ri9941). hTegative activation volumes have been found for 6-Ce and E-Pu, pointing possibly to an interstitial diffusion mechanism resulting from specific electronic structure effects (CORNET[19711). In alkali metals the migration enthalpy is very low, of the order of one tenth of the formation part (SCHULTZ[ 19911). The calculated vacancy formation enthalpy also forms a very important part of the experimental activation enthalpy, or is even greater than it. An interpretation in term of a Zener ring mechanism (see 0 1.1.1), has been recently proposed (SEEGER[1993]).
3.3. Prediction of the self-diffusion coefficients There are three possible ways to predict the diffusion coefficients: - by theoretical calculations; - by simulation (see 0 1.5.) - by empirical laws. 3.3.1. Theoretical calculations of D Using one of the theories given in paragraph 1.4 and 1.5, the calculation of the enthalpies and entropies of formation and migration of the defect involved in the diffusion mechanism allows the determinationof the diffusion coefficient.The techniques used in this type of calculation are beyond the scope of this review and we refer the
References:p. 651.
582
J. L Bocquet, G.Brebec, I:Limoge
Ch. 7,$ 4
reader to the general references at the end of this chapter and to specialized treatises, for instance GERLand LANNOO[1978] (see also ch. I8 by WOLLENBERGER).
3.3.2. Empirical relations Empirical relations are numerous, and we only present the most important; - The Zener formula (ZENER [1951]). This has been established for interstitial solutions and therefore deals only with migration. The idea is that the migration free enthalpy is due to the elastic work required to strain the lattice so that the interstitial can jump. The relation has been empirically extended to self-diffusion. This expression relates the entropy of diffusion AS to the activation energy Q via Young’s modulus (or shear modulus):
where h is a constant which depends on the lattice (A =OS5 for fcc and 1 for bcc);
P =-d (,u/,uo)/d(T/TM), where p is Young’s modulus (or shear modulus) and ,uo the value of p at 0 K, TM is the melting temperature. The review by LAZARUS 119601 shows that there is a pretty good agreement between experimental and calculated values of AS, - The Varotsosformula (VAROTSOS [1978], VAROTSOS and ALEXOPOLJLOS [1986]). This is based on the idea that the free enthalpy of diffusion has the form AG=CBil, where C is a constant which depends on the lattice, B is the bulk modulus (the inverse of the compressibility x ) and il the atomic volume. Thus for cubic materials:
The agreement with experimental data seems fairly good. - Other empirical relations. These include the Van Liempt relation: Q=32 TM (at present one prefers Q = 34 TM); the Nachtrieb relation: Q= 16.5 LM (at present one prefers Q= 15.2 LM), LM is the latent heat of melting; finally the Keyes relation: AV = 4xQ, where AV is the activation volume.
4. Sev- and solute-difision in dilute alloys This section recalls the expressions of the tracer diffusion coefficients, correlation factors, and phenomenological coefficients L,’s as functions of the atomic jump frequencies in the frame of standard models which are today widely accepted as good descriptions of impurity effect in diffusion studies. The two methods which have been currently used in the past to establish the expression of the Lij’s are also briefly reviewed. Finally, it is recalled how to determine the atomic jump frequencies starting from the experimental determination of various macroscopic quantities, together with the difficulties usually encountered. The first part of this section deals with the substitutional alloys for which the vacancy mechanism is expected to be dominant. A short second part deals with the interstitial
Ch. 7, 94
Difision in metals and alloys
583
dumbbell mechanism in substitutional alloys, since this case is encountered in irradiation experiments. The third part deals with those alloys which do not meet the requirements of a “normal” diffusion behaviour and in which the solute diffusivity is often much larger than the solvent diffusivity. 4.1. Vacancy diffusion in dilute A-B alloys 4.1.1. Standard models for bcc and fcc alloys In the fcc lattice, the difference between the first and second neighbour distances is large enough to allow us to ignore the interaction between a solute atom and a vacancy beyond the nearest-neighbour distance. The same dissociative jump frequency w, is therefore attributed to the three possible dissociative jumps (fig. 10) which separate a vacancy from a neighbowing solute atom; w, is the frequency of the reverse jump. w2 stands for the solute-vacancy exchange and w1 for the vacancy jump around the solute atom which does not break the solute-vacancy complex. wo is a jump not affected by the solute atom. Detailed balancing implies that: w4/w3 = exp(-E&’)
where EB is the binding energy of the vacancy-solute pair (E,is negative for an attractive binding). This is the so-called “five-frequency model”. All the physical quantities which will be compared to experimental diffusion data in dilute alloys are functions of only three independent ratios of these five jump frequencies, namely w.Jwl, w,lw, and w41w,. In the bcc lattice, conversely, the second-neighbour distance is close to the firstneighbour distance and the solute.-vacancy interaction energy is not negligible at the second-neighbour distance. Four distinct dissociative frequencies are defined for a vacancy escaping from the first-neighbour shell (w,, w; and w r ) and from the secondneighbour shell (w5). The frequencies of the reverse jumps are w,, wi, w: and w6, respectively (fig. 11). The solute-vacancy exchange frequency is w,. If we denote the interaction energies at the first- and second-neighbour distances by E,, and EB2, respectively, detailed balancing requires that: wi/wi = wz/w: = exp(-E,,/kT) w6/ws = exp(-EB2/kT) W6W4/W5W3
= w:/w;
The calculation of tracer diffusion coefficients has never been performed with the whole set of frequencies. Simplifying assumptions have always been made to reduce the large number of unknown parameters. -MODEL Z assumes that wb = w: = w6= w,,. These equalities imply in turns w; = w: and w3w5= wJ’w,. All the physical quantities which will be compared to experimental data can be expressed as function of w3/w; and w21wi only. - MODEL ZZ restricts the interaction to first neighbour distances and assumes that w, = wi = w: and w, = w, = w,. These equalities imply w, = wi = w:. The physical References: p. 651.
584
. I L.Bocquet, G. Brebec, I:Limoge
Ch. 7, $4
fig. 10. Standard fivefrequency model for solute diffusion in fcc lattices by a vacancy mechanism. The encircled figures denote more and more distant shells of neighbours around the solute atom (solid circle).
quantities which are to be compared with the experimental data are function of w2/w3and w41woonly.
4.13. Kinetic expressions of the phenomenologicalcoefficientsLM, L,, LBA and L B B The purpose of the calculation is to express these coefficients as functions of the jump frequencies, the solute and vacancy concentrations, and the various interaction energies between the species. Two methods have been used so far. 4.13.1. Kinetic theory. In this theory, also-called pair association method, the stationary fluxes JA, JB and J, are calculated in the presence of a constant electric field E, which biases the jump frequencies of the vacancy. The bias can take two distinct values, and E ~ according , to the chemical nature of the atom which exchanges with the vacancy. Hence:
w: = w,(l f E ~ ) , 'w = W,(I_+ s A ) fori # 2 where the superscript k stands for a jump fequency in the direction of the electric field (+) or in the reverse direction (-). It can be shown that eAand E~ are proportional to the thermodynamic forces ZLeE and &*eE, respectively, which act upon the species A and B. The final kinetic expressions of the fluxes are then compared with the phenomenological expressions in order to deduce the Lij's. For an fcc lattice, the calculation has been carried out at first order in C, and to an increasing degree of accuracy by including more and more distant shells from the solute (HOWARD and LIDLARD [19631, MANNING[1968], BOCQUET [1974]). For a bcc lattice the calculation has been published in the frame of the two approximations quoted above (SERRUYSand BREBEC[1982bl). For both structures, the common form of the results is the following:
Ch. 7, $ 4
Di#+swn in metals and alloys
585
Fig. 11. Standard model for solute diffusion in bcc lattices by a vacancy mecharism.
where n is the number of lattice sites per unit volume; DA*(0) and DB*(0) are the solvent and solute tracer diffusion coefficients in pure A (CB= 0); G is the vacancy wind term L m b B which accounts for the coupling between J, and .IB through the vacancy flux J,{. Tables 3 and 4 summarize the expressions of D,, (0), DB, (0), b, and G for both structures. A comprehensive series of papers by FRANKLIN and LIDIARD[1983, 19841, and LIDIARD[1985, 19861 gives a full account of a synthetic reformulation for this method. The function F,always smaller than unity, is a correction to the escape frequency w3 or w,’ which accounts for that fraction of the vacancies which finally returns in the neighbourhood of a tracer atom: the same function appears in the expression of the correlation factor and in the phenomenological coefficients for all the models where the solute vacancy interaction is restricted to a first neighbour distance (fcc model and bcc model II). More functions appear in the other case (bcc model I). The accuracy in the calculation of these functions increases with the size of the matrix used for the random walk calculation of the defect. The first evaluations (MANNING[1964]) have been recently revised by integral methods (KOIWAand ISHIOKA[1983]). In the same way, the
References: p . 651.
586
J. L. Bocquet, G. Brebec, E Limoge
Ch. 7, $4
Table 3 Theoretical expressions of various quantities entering the phenomenological coefficients in an fcc lattice (f, = 0.78 145)
DB*(0)= 2S'C 3 w2fe:
" w3
u = w2/w,,
v = WJW,,
w = w,/w, ;
D A * ( c ~=)D~*(o)(1b1CB);
f, = (2 + 7Fv)/(2 + 2~ + ~ F v;) b, = -34
+ 16% + w(4$ + M,); fo
fo
( 3 -~2) + (7 - ~ F ) v ( ~-/1) w 1 + 3.5FV 14(1- F)(I- w)[3v - 2 + (1 + u + 3.5v)(l/w - l)] + (w/v)(3v - 2)' bA = -19 + w ( ~ /+v 14) 1 + u + 3.5Fv low4 + 180.3122~~924.3303~~ + 1338.0577~ 7F=72w4 + 40.1478~~253.3~' 595.9725~+ 435.2839
G=
3
+
+
3
+
C, is the vacancy concentration in pure A, s is the jump distance.
expression of the linear enhancement factor for self-diffusion b, contains coefficients X,, X, and X, for the fcc lattice and X,, X,,X, and X, for the bcc one: these coefficients are functions of the partial correlation factors for the different solvent jump types in the vicinity of an impurity; they reduce identically to fo when all the jumps frequencies are equal, that is, for the case of self-diffusion. They have been numerically tabulated for the and MANNING[1967]) as well as for the bcc lattice (LE CLAIRE fcc lattice (HOWARD [197Ob], JONES and LE CLAIRE[1972]). Defining a larger number of solvent jump types, and KOIWA revised and more accurate values have been obtained recently (ISHIOKA [1984]). 4.1.2.2. Linear response method. In the linear response method, a timedependent (but spatially uniform) external field E(t) is applied to the alloy and instantaneous values of the fluxes JA, JB and J, are calculated. It is shown that the calculation of the Lij's reduces to the solution, by a Green's function method, of closely related random-walk problems in the unperturbed (E= 0) state of the system. This general formalism has been adapted for the first time to mass transport in solids (ALLNATT [19651): all the possible trajectories of the vacancy around the tracer atom are automatically taken into account and not only those contained in a few coordination shells, as was done in the pair association method. The formalism has been illustrated by an application to various cubic structures (ALLNATT[1981], OKAMURAand ALLNATT[1983a]) and has confirmed (and generalised) the results previously obtained by the kinetic method, namely the general
a.7,04
587
Di@sion in metals and alloys
Table 4 Theoretical expressions of various quantities entering the phenomenological coefficients in a bcc lattice (f0=0.72714) 4 DA*(0)= - s ~ c , w , ~ , 3
%
4 DB.(0)= - szC,,w, f B w3 3
7Fw; + 7Fw;
= 2w,
Expression
Quantity
Model I
Model II
U
WJW;
WdWO
b,
-38+
+-
6x,
+
ax,
fo 6X3+ 18X4 A3
F2
fB
(U
-38+
+-
6X, + 8X, ~
fo
u
6X3+ 18X4
fo
- 2)2+ 2F4(U - 2 ) ( -~ 1)
+3F3(3u+3.096)
F4
u +0.1713 u + 0.8082
G
-2
bA
F, + 2F3v 7+6~-22v + 7F
(U -2) + F4(u - 1)
7F A,
u -7(1 -F)(u - 1) 7Fu
- u z -7 (1 -F)(u - 1)’ (2v + 7F9 u 7F
2u2+5.175u+2.466 u + 0.8082
3u3+33.43u2+97.38u +66.06 u3+8.68u2+18.35u+9.433
forms for the phenomenological coefficients, and the number of distinct functions F to be used (ALLNATTand OKAMURA [1984]). Finally, the equivalence between the kinetic and linear response methods has been demonstrated by LIDIARD[1987], and ALLNATT and LJDIARD[1987a]: the former theory focusses on the jumps of a given chemical References: p. 651.
588
J. L Bocquet, G.Brebec, Z Limoge
Ch.7, $ 4
species when paired to the defect which causes its migration and is well suited to dilute alloys where such pairs can be easily defined; the latter follows the path of a given species by separating it into a direct part when in contact with a defect, and a correlated part where the immobile species waits for the return of the defect. It is more general and can be applied to concentrated alloys (see below 8 5).
4.1.3. Experimentallyaccessible quantities We restrict ourselves to the experiments which are commonly used to deduce the vacancy jump frequencies at the root of the models for bcc and fcc lattices. The measurements performed on pure solvent A consist in determining: - the solvent and solute tracer diffusivities D, (0) and DB, (0); - the isotope effect for solute diffusion, f, AK,. The AK, factor must be evaluated in some way to extract fB. Several theories have tried to determine AKB as a function of the ratio m B / qwhere mBand m, are the masses of the solute and of the solvent respectively (ACHAR[1970], FEIT[1972]): but they apparently do not fit with the experiments and MCFALL[1973]). performed in lithium (MUNDY The measurements of alloying effects are performed on dilute A-B alloys and comparison is made with the same quantities determined in pure A, in order to extract the slope of the linear resulting variation. These measurements usually determine: - The linear enhancement factor b, for solvent tracer diffusion D,, (C,), defined by
Tables 3 and 4 give the expressions for the enhancement factor b, which contain the coefficients X,, X, and X, for the fcc lattice and X,, X,, X,and X, already defined above. The solute diffusion coefficient D,, also varies linearly with the solute concentration, according to:
The expression of B, has been calculated only in the frame of simplified models which do not take into account the solvent partial correlation factors in the presence of solute pairs. But it introduces additional frequencies of the vacancy in the vicinity of two solute atoms (which were not necessary for b,) as well as the binding energy between solute atoms. A thorough overview has been presented recently on this point (LE CLAIRE [1993]). It is experimentally observed that b, and B, often have the same sign and are roughly of equal magnitude whenever the diffusion mechanism is the same for A' and B' in the alloy (it is not true in Pb-based alloys, $4.2.2). This means physically that the preponderant effect of the solute is to increase (or decrease if b,, is negative) the total vacancy concentration, which affects solvent and solute diffusivity roughly to the same extent. - The linear enhancement factors b, and b, for the shift of inert markers and solvent tracer markers in an electric field. If we denote the rates of these shifts by V, and V,, & and are defined according to:
Ch. 7, 34 vM(cB)
Diffusion in metals and alloys
= vM(o)(l
+ bMCB),
'A*('B)
=
589
+ bTCB)
b, and b, have been calculated as functions of the vacancy jump frequencies (DOAN [1972]; BOCQUET [1973]; DOANand BOCQUET [1975]; LIMOGE[1976a]) and are given by:
- The vacancy wind term G=L,/LBB can be measured from the solute enrichment or depletion in the neighbourhood of a sink (ANTHONY[1971, 19751) or by combining tracer diffusion experiments with Kirkendall shift measurements in differential couples and HEUMANN A+A- B (HEUMANN [1979]; HOSHINOet al. [1981a]; HAGENSCHULTE [19891). 4.1.4. Determination of vacancy jump frequencies
Jump frequencies depend on the interatomic potential which should, in principle, be deduced from ab-initio calculations. Unfortunately an accurate knowledge of these potentials is far from being currently acquired, except for particular systems, and one usually proceeds differently. Jump frequencies are instead fitted to the experimental results. As already mentioned, diffusion data yield only three jump frequency ratios for an fcc lattice and only two for a bcc one; thus only three independent measurements are required in the former case and two in the latter. Any additional result is highly desirable and is used to check the consistency of the experiments. If this consistency cannot be maintained in view of a new result, this may mean that one (or more) experimental results are not worthy of confidence or that the model does not correctly represent the experimental system. All the dilute alloys of fcc structure, for which we know the jump frequency ratios, are displayed in table 5. Whenever the number of experiments is equal to three, one reference only is quoted. When the experimental data are redundant, several references are given. The error bars on the final values of these ratios are large: at least 50% for the best cases, up to an order of magnitude for the worst. We have to keep in mind that any ratio which departs too much from unity (say less than or larger than lo2) may be an indication that the weak perturbation assumption at the root of the model is violated in the alloy under consideration. A similar table of jump frequency ratios has been published elsewhere (HERZIGel al. [1982]). For bcc alloys, similar tables can be found in fairly recent reviews &E CLAIRE [1978], AGARWALA [19841). The search for the frequency ratios is not always straightforward, as can be seen from the following examples: Al-Cu: the value of the self-diffusion coefficient is still today highly controversial. At 585K it is measured or evaluated to be 1.66 m2/s (FRADIN and ROWLAND[1967]), m2/s (SEEGERetal. [1971]), 3.66 m2/s (BEYELER and ADDA[19681), 3.73 3.03 lo-'' m2/s (LUNDYand MURDOCK[1962]) and 4.51 m2/s (PETERSON and ROTHMAN
-
References: p . 651.
590
Ch.7, 54
J. L. Bocquet, G. Brebec, 1! Limoge Table 5 Jump frequency ratios for dilute fcc alloys.
4 9.2 6.5 13.7
1060 1133 1153 1197
3.8 3.28 3.18 2.96
0.41 0.71
1064 1043 1010 1153 1153
5.7 5.8 4.1 3.9 3.9
0.35
1075 1175
8.6 7.5
0.26 0.26
71 49
e
1059
0.16 0.16
130 73 73
e
1129
16.4 16.4 12.93
1058 1117 1133 1076 1W6
6.2 5.7 1.15 10.2 10.2
0.15 0.15 0.9 0.22
24 23 8.1 35 35
1133 1133
0.81 0.81
0.85 0.88
0 0
1293 1005 1089 1089
1.1 13.3 11.4 12
0.8
-5 42 43 43
Cu-Mn CU-Ni
1199 1273 1273
4.2 0.36 0.36
CuSb CuSn
1005 1014 1014 1089 1089
AgCd
Ag-In Ag-Sn Ag-Zn
Au-~
Au-Sn
AU-ZII
CU-AU Cu-Cd cu-co CU-Fe &-In
Cu-Zn
1168 1168 1220
0.62 0.46
0.52 0.57
0.07 0.36
24.1 15.5 17 13.6 14.1
0.15 0.15
3.56 3.3 3.4
0.47 0.47
17.5 15.6 12.6 12.7 12.7
5 -5 -5.3 79 40 40 48 48 7.3 8 8.8
C
b
a b
-12
C C
d d
6
a
e
-0.5
4.7 1.8 1.53 1.54 1.20
0.7 0.2 0.27 0.39 0.26
212 40
45 7.3
0.85 0.52
0.46 1.7 1.9 1.1
1.15 1.30 1.12 5.5 4.2
NO SOLUTION
1.5 31.2
1.2 4.2
6.3 7.1
e
942 973 0.2 0.1 7.6
85 85 0.1 1 0.6
2.9 2.6 0.6 3 2.8
2 0.3
4.2 0.4
1.2 0.76
0.4 18 11 33
0.09 0.5 1 0.8
0.3 3 4 3.2
0.35 1 0.42
0.95 1 0.53
g
h h g g
-0.71 -0.57
0.3 0.07 0.07 0.8
f n
e
-0.7
2.6 0.49 0.5 1.7
i j j e C
3.4 0.2 0.27
0.07 0.12
k 1
-1.2 -1.06
j j e j e
15 13 7.5 7 11
0.40 0.2 0.14 0.33 0.5
5 2 1.7 3 3.3
m
2.5 3 3.6
0.5 0.5 0.9
1.2 1 1.5
-0.84
-0.22
k m
DOANand BOCQUET 119751; BHARATIand SINHA[1977]; HERZIGet al. [1982]; ROTHMAN and PETERSON 119671; e HILGEDIECK 119811; REINHOLD el d. [1980]; ECKSELER and HERZIG [1978]; HOSHINOel al. [1981bl; BOCQUET[19721; HOSHINOet al. [19821; HIRANO119811; DAMKOHLER and HEUMANN [1982]; PETERSON and ROTHMAN [1971]; " HAGENSCHULTX and HEUMANN [1989].
a
'
'
Ch. 7.$ 4
Dtfision in metals and alloys
59 1
[19701). Using ANTHONY'Sresult, which establishes that no detectable solute redistribution occurs in the neighbowhood of a vacancy sink, very different values of G = LAB/LBB are deduced according to the value which is retained for the self-diffusion coefficient. It is easy to check that one obtains G = -0.4; - 0.01; + 0.203; + 0.226, and + 0.43, respectively. The jump frequency ratios which stem from such scattered values of G are highly different of course; in addition they do not fit with the measurement of inert marker shifts in dilute alloys (LIMOGE[1976a]). Finally, according to SEEGERet al. [1971], 40% of the total diffusivity at 858 K is due 'to divacancies. This fact cannot be ignored any longer, and a revised version of the atomic model should be presented to take properly into account the contribution of the divacancies to diffusion and electromigration. - Au-Sn: the extracted value forb, is sometimes very sensitive to the way chosen for the fitting whenever DA*(CB)exhibits a pronounced curvature. A rough fitting extracts a value which is not compatible with the other data and does not allow to deduce the jump and HEUMANN[19721); a more careful fitting gives reasonable frequency ratios (€IERZIG et al. [1980]). It must be noted however that the direct measurement values (REINHOLD of th,e vacancy flow factor G at a slightly different temperature on dilute couples yields noticeably different values (HAGENSCHULTE and HEUMANN[ 19891): the departure from the previous ones cannot be accounted for by the small temperature difference, or would imply unusually high activation energies for these frequency ratios. Although the partial correlation factors are not analytically known, it is possible to check the internal consistency of the experimentally determined quantities in the frame of a given diffusion mechanism. For instance, once the ratio D B J D p is known, a constraint on the possible values for u,v and w is imposed, which in turn, restrains the possible range for other quantities like b, or G. For instance, b, is kept to a minimum if the vacancy spends most of its time in exchanging with the solute (u= wz/wl + ce) and keeping the exchanges with the solvent to the lowest possible value which is compatible with the solute diffusion (v = w = 0). Assuming that X, =f, and using the tabulated value X,(u + 00, v =w = 0) =0.4682 yields (MILLER[19691):
-
E the experimental value for this term is noticeably smaller, it means that the vacancy mechanism alone cannot account for the diffusional behaviour of the system and that, probably, other diffusion mechanisms must be looked for. A similar limitation has been established for the bcc structure, although no simple analytical formula is available (LE CLAIRE [1983]). In the same spirit, it has been shown that the vacancy flow term G in bcc alloys ranges from -2 to a maximum value which depends on the same ratio DBJD, and on the model (I or rr) to be chosen (IIJIMAet al. [1985]). - Pb-Cd: self-diffusion in lead meets the usual requierements of normal diffusion. On the other hand, the solute diffusivity is roughly 20 times larger than the solvent diffusivity: this fact alone is not an indisputable proof that another mechanism is operating. MLLER
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[1969] pointed out that the linear enhancement factor b, exhibited a value which was noticeably smaller than b?. This is the reason why he proposed a new mechanism with interstitial-vacancy pairs (04.2.2). Like the fcc alloys, there are several bcc systems in which the b, factor is too small to be compatible with the high value of the solute-to-solvent diffusivity ratio, namely based alloys (Co, Cr, Fe), Ti-&, E F e and &Co. The isotope effect measurements, and VIGNES[1978]), are when available in these systems (ABLITZER [1977]; AFSLITZER not compatible with the frequency ratios in the frame of a pure vacancy mechanism: another mechanism resting on a dissociative model similar to MILLER’S one for Cd in Pb is commonly thought to come into play.
a-
4.1.5. Determinationof the solute-vacancy binding energy The only relevant quantity for determining the binding energy E&, of the solute-vacancy complex is the ratio w4/w3,which cannot be deduced from the knowledge of wdw,, wJwl and w4/wV DIRKESand HEUMANN[1982] worked out a simple procedure for simulating the vacancy trajectory around the solute and proposed to extract from this trajectory the desired quantity. It is true that the only knowledge of the ratios w2/wl,w,/wl and w4/wo is sufficient to determine, at each step of a Monte Carlo simulation, the direction of the most probable next jump. But these authors used an incorrect definition of the vacancy concentration on a first neighbour site of the solute. This concentration is not related to the number of times that the vacancy was located on a first-neighbour site of the solute, but rather to the time the vacancy actually spent on this site. This definition needs the knowledge of the mean residence time of the vacancy on each site (that is, the inverse of the total escape frequency from this site). It is easily checked that the fraction of the total time which has been spent on a first-neighbour site involves one more independent frequency ratio w,/w, (BOCQUET [1983a1). Moreover, the assumption w, + 4w1+ 7w3= 12w0 which is invoked here and there in the diffusion literature for the fcc alloys has no physical justification and is totally arbitrary. Diffusion experiments by themselves are not sufficient to determine this binding energy. Experiments of another kind must be added: for instance a direct determination of the total vacancy concentration in a dilute alloy, by comparing the macroscopic thermal expansion and the increase in lattice parameter as already done for Al-Ag and A1-Mg (BEAMANet al. [1964]; BEAMAN and BALLUFFI[1965]).
-
43. Dumbbell interstitial diffusionin dilute A-B alloys The self-interstitial atom in a compact structure is too large to content itself with an octahedral or tetrahedral position as smaller solute atoms do; it minimizes the distortion of the surrounding lattice by sharing a lattice site with a neighbouring atom and making up a dumb-bell-shaped defect denoted by IAAaligned along () direction in a fcc (bcc) structure. The migration mechanism involves a translation to a first neighbour site combined to a rotation of its dissociation axis (see chap. 18). The diffusion coeffi-
Ch. 7, $ 4
Dzfision in metals and alloys
593
cient of a substitutional solute atom has been calculated with this mechanism at work for bcc and fcc lattices, under the assumption that it can be incorporated into the defect under the form of a mixed dumb-bell IABwhich does not possess necessarily the same symmetry (BOCQUET[1983b, 19911); the phenomenological coefficients have been calculated for the fcc lattice by the kinetic method (ALLNATT et al. [19831) as well as by the linear response one (OKAMURA and ALLNATT[1986], CHATURVEDI and ALLNATT [1992], SINGHand CHATURVEDI [1993]). But these models cannot be checked experimentally as thoroughly as in the vacancy case, since the frequencies cannot be determined by a clever combination of diffusion experiments; the interstitial defects are necessarily produced by irradiating the solid, and their contribution to diffusion is intricately linked with that of thermal and irradiation-produced vacancies.
4.3. A-B alloys with a high solute diffusivity 4.3.1. Purely interstitial solutes Light elements like H, C, N, 0 are known to dissolve interstitially in many bcc and fcc metals. No theoretical criterion has yet been found to predict with confidence the localization of the interstitial atom in the host lattice. In many bcc metals C, 0 and N are believed to be located on octahedral sites: but dual-occupancy models (octahedral + tetrahedral position) have been invoked to account for the upward curvature of their Arrhenius plot at high temperatures (FARRARO and MCLELLAN [1979]). For the case of hydrogen, a simple empirical rule has been proposed (SOMENKOV and SHIL’STEIN 119791): H dissolves in the tetrahedral position in all the host metals which have an atomic radius larger than 0.137 nm (Sc, Ti, Y, Zr,Nb, La, Hf, Ta, W) and in the octahedral position for the others (Cr, Mn, Ni, Pd). Vanadium is the link between the two groups and is believed to have a dual occupancy. In Fe, H is expected to be located in octahedral sites although no clear experimental proof has ever been given. The insertion into the host lattice is accompanied by a (generally) large distortion of the surroundings, which can give rise to Snoek-type or Gorsky-type relaxations (5 2.2). Although in an interstitial location, the solute atom is believed to interact with vacancies of the host; the diffusivity and the phenomenological coefficients have been calculated with the linear response method (OKAMURA and ALLNATT [1983b]). The diffusivity of such interstitials in metals has been measured over orders of magnitude by complementary techniques (relaxation methods, tracers, out-gassing, etc.. .). The Arrhenius plot is straight or exhibits a small curvature at high temperatures. This curvature has been tentatively explained by different models (FARRAROand MCLELLAN [1979]), either a single mechanism with a temperature-dependent activation energy or several mechanisms (or defects) acting in parallel. For very light interstitials like hydrogen and its isotopes, or the positive muon /A+, quantum effects play a significant role at low temperatures. Several regimes are expected to be observed in the following order with increasing temperature (STONJBAM [1979]; KEHR [1978]): (i) coherent tunneling, the interstitial propagates through the lattice like a free electron;
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(ii) incoherent (orphonon-assisted)tunneling, the ground state levels of an occupied and an unoccupied interstitial site have different energies; the tunneling process requires the assistance of phonons which help to equalize the levels of neighbouring sites; (iii) classical regime, the jumping atom receives from the lattice the amount of energy which is required to overcome the potential barrier of the saddlepoint configuration; (iv) high-temperatureregime, the residence time on a site is comparable to the time of flight between two neighbouring sites. The second and third regimes have been observed in many systems. Whether coherent tunneling can actually be observed in real systems or not is still controversial (STONEHAM [1979]; GRAFetal. [1980]). Let us mention the reversed isotope effect which is observed in fcc metals at low temperatures: tritium is found to diffuse faster than deuterium, which diffuses faster than hydrogen. Several models have been proposed to account for this anomaly (TEICHLER [1979]; KAURand PRAKASH [1982]). See also ch. 18, 8 3.3.2.7 for the interaction of selfinterstitials with solute atoms.
4.3.2. Complex diffusion mechanisms The most widely studied case is that of dilute Pb-based alloys. In lead, several solute atoms (Cu, Ag, Au, Pd, Ni, Zr) diffuse from lo3 to lo6 times faster than the solvent tracer. Other elements (Na, Bi, Sn, T1) diffuse roughly at the same rate. A third group (Cd, Hg) diffuses at rates between the two extremes. It is well established that these properties are in no way related to any short-circuit diffusion path and that they reflect a bulk property. We already mentioned in 84.1.4 why a pure vacancy mechanism should be rejected for cadmium diffusion in lead. The high value of the diffusivities led many investigators in the past to think in terms of an interstitial-like diffusion mechanism: it can be shown however, by particular examples, that a purely interstitial mechanism would not yield a value of the linear enhancement factor b, consistent with experiment. This is why many authors proposed more complex mechanisms involving interstitial-vacancycomplexes, interstitial clusters, and today the consensus is roughly as follows: - very fast diffusers dissolve partly as substitutionals and partly as interstitials in lead. The total diffusivity is therefore the sum of both contributions; pairs made up of an interstitial solute and a host vacancy are expected to play a dominant role; the phenomenological coefficients Lij have been calculated for this mechanism (HUNTLEY [1974], OKAMURA and ALLNATC [1984]); - multidefects (interstitial solute atoms sharing one substitutional lattice site) are necessary to account for the diversity of experimental results, especially for the signs and the orders of magnitude of the linear enhancement coefficients b, and B, (WARBURTON [1975], KUSUNOKIet al. [19Sl]), as well as for the low value of the isotope effect measurements; - solute atoms which diffuse roughly as fast as the solvent dissolve presumably as substitutionals (except Sn: DECKER et al. [19771); A general and detailed atomic model including all these defects is still lacking, apart from an attempt by VANFLEET [1980]. The reader is referred to an extensive review by
Ch. 7, 3 5
595
Diffusion in metals and alloys
WARBURTON and TURNBULL [1975]. Lead is not a unique case however, since similar problems arise in other polyvalent metals like Sn, In or Tl (WARBURTON and TURNFWLL[19751; LE CLAIRE[1978]), in the a-phase of Zr, Ti and Hf (HOOD [1993], NAKAJIMA and KOIWA[1993], K~PPERS et al. [1993]), in bcc metals like Nb (ABLITZER [1977]; SERRUYS and BREBEC[1982a]), and for rare-gas diffusion (He) in fcc metals like Au, Ni, A1 (WILSONand BISON [1973]; MELIIUS and WILSON[1980]; SCHILLING [1981]). The interaction energy between the smaller solutes and the intrinsic point defects of the host, namely the vacancy one, is believed to be high (above 1 eV); this feature, when combined with a very low solubility in the host, can lead to behaviours, which have puzzled the experimentalists for long. The general interpretation (K~PPERS et al. [19931) distinguishes three different temperature ranges: in the first (high-temperature) one, the native intrinsic vacancies are more numerous than those trapped by the impurity atoms, and the self-diffusion is normal; at intermediate temperatures (second range), the extrinsic vacancies trapped by the impurity atoms become dominant, and the apparent activation energy for self-diffusion is markedly decreased; at the lower temperatures (third range) where the impurity atoms precipitate into clusters, the number of trapping sites is reduced to such an extent that the intrinsic defects play again the dominant role. It ensues an unusual downward curvature of the Arrhenius plot over the low and intermediate temperature ranges. Depending on the ratio of the melting temperature to the a-p transformation temperature, the interaction energies between impurity and vacancies and between impurities themselves, not all the three regimes are automatically observed. In a-Zr, which has been for long the archetype, the (practically unavoidable) Fe impurity has been found to give rise to the regimes 2 and 3 with the downward curvature observed for self- as well as solute-diffusion; the determining experiments have been carried out only recently since ultra-high purity Zr was not available before (HOOD [1993]). For a-Hf, only regimes 1 and 2 are observed, but the impurity which is responsible of the upward curvature is not yet identified (K~PPERS etal. [1993]). At last for a-Ti, the impurity is believed to be oxygen which is easily incorporated into this highly reactive metal (NAKAJIMA and KOIWA[1993]).
5. Difision in concentrated alloys We shall restrict ourselves to binary alloys. The first two sections are devoted to the diffusion of A* and B* tracer atoms in homogeneous disordered and ordered alloys. The third section will deal with chemical diffusion, that is, diffusion in the presence of chemical gradients.
5.1. Diffusion of A* and B* tracers in homogeneous disordered alloys 5.1.1. Experimental results Diffusion measurements in concentrated binary alloys are legion, but only few alloys have been investigated throughout the whole composition range: Ag-Au (MALLARD et al. [1963]), Au-Ni (KURTZet al. [1955]; REYNOLDS et al. [1957]), Co-Ni (MILLIONand KUCERA[1969, 19711, HIRANOet al. [1962]); Cu-Ni (MONMAet al. [1964]), Fe-Ni References: p . 651.
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(CAPLAINand CHAMBRON [1977], MILLIONet al. [1981]), Fe-Pd (FILLON and CALAIS [1977]), Ge-Si (MCVAYand DUCHARME [1974]), Nb-Ti (GIBBSetal. [1963]; PONTAU [1961]). For Fe-Ni, the diffusion and LAZARUS [1979]), Pb-Tl (RESING and NACHTRIEB has been studied both through a magnetic relaxation method which yields apparent values for the formation and migration energies of the vacancy and by tracers. Two general trends can be outlined: - The same kind of empirical correlation as for self-diffusion in pure metals are observed between the preexponential factors Do and the activation energy Q, or between Q and the melting temperature T,,, of the alloy. - The diffusion coefficients and D E for a given temperature and composition do not differ by more than one order of magnitude. When they do, it might be an indication that the diffusion mechanism for the two tracers is not the same (Ge-Si or Pb-TI). Some cases still offer matter for controversy, like Ge-Si alloys (PIKEetal. [1974]). For brevity, DZ and @: will be denoted by DA*and DB*in what follows.
DE
5.1.2. Manning's random alloy model In this model, the simplest which can be thought of, the alloy is assumed to be random and the vacancy exchanges at rate W, with A atoms, and W, with B atoms, whatever the detailed atomic configuration of the local surroundings (fig. 12). The most important finding lies in the fact that the vacancy no longer follows a random w&, its successive jumps are correlated and a vacancy correlation factor f, smaller than unity shows up in the final expressions (MANNING 119681, [1971]): DAYB*)
=~s2cvf~(B)~,(B)
= k2fvw
where f, = (CAwAfA +cBwBfB)/fo and fo= w(M+o 2) is the correlation factor for selfdiffusion, W =CAW, + CBWB,and finally, fA@)= MJV W/(&fv W + 2wA@))for A(B). Consistent expressions of the phenomenologicalcoefficients Lijhave been established in this frame: LAA=n-
'ADA*
kT
(1 '22) ;
where n is the average number of sites per unit volume, and D* is the average CAD,* + CBDp. At last it can be easily shown that the vacancy wind corrections showing up in the expressions of the intrinsic diffusivities [see eqs. (17)] are given by:
ch. 7 , 3 5
597
Diffusion in metab and alloys
Fig. 12 Manning’s random alloy model.
The ]Lij’sare not independent since they are functions of DA*and D,, only; they obey the relationship:
The same expressions have been later recovered following two different routes: i) in a formal derivation resting on two macroscopic assumptions related to the invariance of the functional relationship between mobility and flux between the pure substance and the average alloy (LIDIARD[1986]); ii) in a mean-field treatment of the diffusion problem, resting on the adoption of a preliminary consistency equation over the diffusivities, namely: C A DA*+ CBDp = foCvDv
the right-hand side of the above equality being nothing but the tracer self-diffusivity in the average alloy (BOCQV~T 119871). Manning’s approximation appeared fascinating and very appealing since the only independent quantities are the easily accessible tracer diffusivities. It has been the object of very numerous Monte Carlo simulations, which can take into account the detailed occupancy of the sites surrounding the vacancy and can check the accuracy of the approximation. These simulations essentially show that the approximation is indeed quantitatively excellent over the whole concentration range, as long as the disparity c W,/W, e (BOCQUET 119731, between the jump frequencies is not too large, say DE BRUINet al. [1975], 119771, ALLNATT and ALLNATT119841). An analytical more sophisticated method for the self-consistent decoupling of the kinetic equations has been worked out and yields the same conclusion (HOLDSWORTH and ELLIOTT[1986], ALLNATT[1991]). Even in dilute alloys, the approximation turns out to be satisfactory
la
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for all quantities but the linear enhancement factor b, (ALLNATT and LIDIARD[1987b]). In the same spirit, the same kind of approximation has been worked out for the dumbbell interstitial mechanism in random two-frequency alloys on fcc and bcc lattices. Although no simple analytical expressions can be established for the tracer diffusivities (BOCQUET[1986]), a similar functional dependence of the Lij’s versus the Di*’sas above can be proposed after replacing f, by the product f,+, (BOCQUET [1987]), where p, is the ratio of the tracer average squared jump length to that of the defect. po=1/2 for fcc structures and 7/15 for bcc ones (BOCQUET [1983b, 19911). The numerical simulations show a good agreement only in special cases for the fcc lattice, and a disagreement for all the cases investigated in the bcc lattice (Bocqm [199Ob]): the reason for these discrepancies has not been elucidated so far, in spite of a recent treatment involving the more sophisticated linear response method (CHATURVEDI and ALLNATT[1994]).
5.1.3. Atomic models for diffusion in non-random disordered alloy The attempts to improve the alloy model beyond the random approximation and to include the effect of short-range ordering on diffusion have historically followed two different routes. The first one consisted in extending the dilute alloy models by including more and more solute clusters of increasing size together with the corresponding modifications of the solute and solvent jump frequencies in their neighbourhood. This route turned out to be not well fitted to this purpose, due to the rapidly increasing number of unknown parameters which yielded intractable results, together with the intrinsic impossibility to deal with cluster overlap (BOCQUET [1973]); only rough approximationscan be proposed by selecting a few solute clusters which are believed to have a dominant influence (FAUPEL and HEFIENKAMP [1987]). But this choice is totally arbitrary and physically unjustified; as a consequence, this route has now been abandoned. The second route, at the expense of some loss of accuracy, approximates the effect of the local surroundings on the height of the potential barrier by using a small number of pair interaction energies for the stable (Eij) and the saddle-point (E;) configurations. The merit of such a description lies in the fact that it connects simply and consistently the. thermodynamics (reflected in the Qj’s) and the kinetic behaviour of the alloy (reflected in the Ei’s). The model was used first to account for the kinetics of shortrange ordering in Ag-Au alloys (RADELAAR [1968, 19701) and Fe-Ni alloys (CAPLAIN and CHAMBRON [1977]). Later it was improved to take into account correlation effects in short-range ordered alloys (STOLWUK [1981], ALLNATT and ALLNATT[1992]): the analytical formula obtained for the tracer diffusivities and the associated correlation factors are in fair agreement with Monte-Carlo simulations over a reasonably large range of the thermodynamic parameter mastering the order, namely [2 EAB- (EAA+.b,)]/kT. The agreement deteriorates to some extent for the lower temperatures where systematic departures show up. Independently from the search for better expressions of the D,’s, a systematic investigation of the phenomenologicalcoefficients Lij’s has been carried out numerically by simulating non-random alloys using such pair energies; and the most intriguing result of the last ten years is that the functional dependence of the Lij’s upon the Dj*’s
a.7, g5
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599
established by Manning for random alloys is still preserved in non-random ones with a fairly good quantitative agreement (MURCH [1982a, 1982b, 1982~1,ZHANG et aZ. [1989a], ALLNATTand ALLNATT[1991]), except at the lower temperatures or dilute concentrations; the latter restrictions are not a surprise, since they correspond to those actual situations where the departure from randomness is expected to be the largest. The basic reason for such an agreement is still not understood. Following previous tentative papers (HEUMANN [1979]; DAYANANDA[1981]), an active research effort has recently been undertaken to establish more general relationships between the 4 ' s and the various diffusivities (tracer, chemical) (ALLNA-ITand LIDIARD[1987c], LIDIARDet al. [1990], QIN and MURCH[1993a]). Before closing this section, the main limitations of such models in the present state of the art must be recalled - they do not calculate D,, and DB* but only the activation energies for diffusion QA* and QB*,with the correlation effects included in the best case. The preexponential factors D, and DOB are not known and are arbitrary assumed to remain constant, since no model is available which would account for their variations throughout the whole composition range. - they use pair energies and assume implicitly that the energy of the alloy can be summed in this way, which is not always true (namely, transition metals). Even if effective pair energies can be defined, the electronic theory of alloys must be used to predict the variations of these pair energies as function of the composition on physically grounded arguments for each specific alloy (DUCASTELLE [1978]). - finally, they cannot have any predictive power: while pair energies in the stable position can be deduced from thermodynamic measurements, saddle-point pair energies conversely can only be deduced from experiments involving diffusion jumps, that is, from the diffusion experiments themselves. 5 2 Diffusion of A' and B*tracers in ordered binary alloys
In the last ten years, intermetallics have been the object of intensive study for their attractive practical properties: some of them are indeed characterized by a high melting temperature, high elastic limit (see ch. 24), high resistance against corrosion and (or) creep (LIUet a2. [19921). Before reviewing in more detail the different ordered structures, some preliminary and general ideas should be recalled here. The progress in the understanding of phase stability from ab initio calculations based on the local density functional approximation (LDF) has allowed research people to address very basic points, namely, the physical reasons leading a given alloy to adopt a well-defined structure or symmetry. Such calculations are able to explain the reason why Ti Al, is tetragonal (DO,), while Ti, Al is hexagonal (DO,,)and Ti A1 is cubic &lo); or why Ni, Al (Ll,) exists whereas there is no corresponding closepacked phase for Ni A13 (PETTIPOR 119921). However, the problem of the point defects has not yet been addressed. The existence of the so-called constitutional (or structural) defects is probably connected to the preceding point but has not yet received an unambiguous experimental References: p . 651.
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confirmation as well as a firmly grounded theoretical explanation, Starting with the simple case of the B2 stnucture as an illustration, the stoichiometric A-B alloy is perfectly ordered at 0 K and the N, (NB)atoms occupy the a @)-sites. When the temperature is raised, several kinds of defect are believed to appear: - antistructure or substitutional defects: A atoms can occupy &sites and are denoted 44, their number is NW,B atoms can occupy a-sites (B,, Nk); - vacancies denoted by V, on a-sites and Vs on &sites. Depending on the atomic interactions, the alloy will choose preferentially one type of defect or the other (or some combination of the two). Up to this point, these defects have been introduced as a pure manifestation of thermal excitation in a stoichiometric alloy. However, for a non-stoichiometric alloy, one must think of the way to accommodate the departure from stoichiometry. The same defects (antistructure atoms or vacancies) have been also invoked; but in this case they are expected to be much more numerous than in the thermal case, since their concentrations will be of the order of the stoichiometry offset (up to several percent) and to survive even at 0 K unlike thermal ones. The difficult point in looking at actual systems at finite temperatures is to decide which part of the observed vacancies or antisite defects has a thermal origin and which part has a structural one. The undisputable fingerprint of structural defects (their non-null concentration at 0 K) is unfortunately very difficult to use practically: in many systems indeed, high Concentrations of vacancies (at room temperature and above) have undoubtedly been evidenced by a careful comparison between density and lattice parameter measurements: but their apparent migration energy has often be found to be large, and one cannot safely state that equilibrium properties rather than quenched-in defects ones are measured. Phenomenological models like bond-breakingpictures (NJXJMANN [19801, KIM 119911) or the Miedema “macroscopic atom” model (DE BOERet al. [1988]) using rough expressions for configurational entropies cannot claim to be anything but guiding approximations to decide which type of defect is most likely to appear. A rapidly growing number of model defect calculations using semi-empiricalpotentials is presently et al. [1993]) but the approximations observed (CLERIand ROSATO[1993], REY-LOSADA involved are probably still too crude to solve this question. The problem requires undoubtedly accurate ab initio calculations of ground-state energies, together with a minimisation procedure which would allow charge transfers (KOCHand KOENIG[1986]), local relaxations as well as the settlement of an arbitrary vacancy concentration: such calculations would tell us whether the ternary (A,B,V) is most stable in the investigated lattice structure at low temperatures or whether a phase separation between ordered phases of other symmetries (and, or) concentrations occurs. If the existence of these structural vacancies can be theoretically proven, one must remember however that their properties are in no way different from those of the so-called thermal ones although they have received a different name. Indeed, the total vacancy concentration at finite temperatures minimizes the free-energy of the alloy: but in the present case, the existence of two sublattices and of suitable atomic interactions implies that the result of the calculation is more sensitive to a small variation of the composition, than to a temperature change, unlike the case of the disordered alloy at the same concentration. Tracer diffusion measurements are still performed and are still highly desirable, as a
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first insight into the transport mechanisms. Correlation factors have been calculated in various structures and for various jump mechanisms (BAKKER[1979], ARITAet al. [1989], WEILERetaZ. [1984], SZABOet al. [1991]). While isotope effects were measured with the hope of determining the jump mechanism, it was shown later that, for B2 structures, such a measurement does not yield the correlation factor (ZHANGet al. [1989b]); this is rather unfortunate since these (very difficult) experiments have been performed only on alloys with this symmetry (PETERSONand ROTHMAN[1971], HILGEDIECK and HERZIG[19831). Spectroscopic methods like nuclear magnetic resonance (NMR), quasi-elastic Mossbauer line broadening (QEMLB) and quasi-elastic neutron scattering (QENS) appear today as the best candidates to clarify the atomic diffusion mechanisms in ordered alloys; NMR techniques measure the frequency (ies) of the diffusing species (TARCZONet ai. [1988]), while QEMLB and QENS give besides access to the individual jump vectors (see 0 2.2.2 and VOGLet al. [1992]). Although some intermediate modelling is still necessary for the final interpretation, they yield the most confident information gained so far. The most important result obtained up to now is that diffusion in ordered structures seems to proceed simply via nearest-neighbour jumps of a vacancy defect. Finally, the ability of the positron annihilation (PA) technique to measure vacancy concentration with confidence is also being currently improved (BALOGHet al. [1992]). The preceding point helps to solve old ill-formulated problems about the migration mechanisms in ordered alloys. The &jump cycle was initially designed for transporting and ELWK [1958], ELCOCK atoms without altering the long-range order (MCCOMBIE [1959], HUNTINGTON et ak. [19611): this condition is unnecessarily stringent since local and thermally activated fluctuations of the long-range order (LRO) must necessarily occur in a real system, the only requirement being the conservation of the average LRO through detailed balancing: this remark has been the starting point of a new formalism (path Probability Method or PPM) for the evolution of cooperative systems (KIKUCHI [1966]; SATO[1984]). The &jump cycle is thus not necessary. Moreover, it is also very improbable: many computer simulations show that such cycles never go to completion and are destroyed while underway by strongly correlated backward jumps (ARNHOLD [198l]). In the same way, the triple defect has been introduced only for thermostatistical reasons (large difference between vacancy formation energies on both lattices): but it was implicitly thought that it should migrate as a whole, that is, without dissociating. This unnecessary constraint has led previous investigators to imagine a mechanism of highly concerted vacancy jumps (STOLWUKet al. [1980], VAN OMNENand DE MIRANDA [198l]), which has never been clearly evidenced neither experimentally nor theoretically. At last, a growing body of practical knowledge has been gained through the use of [1992]), kinetics of macroscopic measurements like chemical diffusivity (DAYANANDA long range order recovery after irradiation or plastic deformation (CAHN[1992]), internal friction (GHILARDUCCI and AHLERS[ 1983]), degradation of superconductingtemperature in A,, compounds (BAKKER[19931): these experiments yield effective quantities which are of importance for mastering the practical properties of these materials. But a detailed atomistic model is still lacking which would link these effective energies to the usual parameters deduced from tracer diffusion experiments. References:p . 651.
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For sake of space, the reader will be referred to a recent compilation of experimental reSUltS (WEVER [1992]). 5.2.1. Ordered alloys with B2 structure The B2 structure has been more extensively studied than the others: it is made of two interpenetrating simple cubic lattices, a and p. Each a-site is surrounded by eight firstneighbour p-sites and conversely. The existing alloys belong to two distinct groups: (i) In the first group (AgCd, AgMg, AgZn, AuCd, AuZn, BeCu, BeNi, CuZn, NiZn) the defects are mainly antistructure defects on both sublattices (A,and B,) the departure from stoichiometry is compensated by ABdefects for A-rich alloys and B, defects for Brich alloys. The apparent formation energy E,” of thermal vacancies can be different on the two sublattices. (ii) In the second group (CoAl, FeA1, NiA1, PdAl, CoGa, NiGa, PdIn), for which a will denote the sublattice of the transition metal A, maintenance of equal a and /3 site numbers allows formation of paired defects only (A,+ B, or V, +V,). If Vp costs more energy than A, + V,, then V, + V, converts into the triple defect A, + 2V,. Symmetrically, if B, costs more than 2V, (mainly due to size effects), then B, +A, converts also into the same triple defect. The departure from stoichiometry is therefore compensated in two different ways: for an A-rich alloy the major defect is 4;for a B-rich alloy, the major defect is V,. In the latter case, very high structural vacancy concentrations on one sublattice are expected and (indeed) experimentally observed (up to 10% in CoGa on the gallium-rich side). All the theoretical calculations performed so far (e.g., EDELIN[19791) are based on a zeroth-order treatment (BRAGGand WILLIAMS[1934]); not withstanding their crudeness, they account qualitatively well for all the presently known experimental situations, provided reasonable values of the adjustable pair energies qjare chosen. A first and simple explanation has been proposed to account for the fact that a particular alloy belongs to the first or to the second group (NEUMANN [1980]). Using a crude bond-breaking picture, this author shows that the number of substitutional defects is dominant whenever the mixing enthalpy AHf is (algebrdically) higher than -0.3 eV/atom; the number of triple defects is dominant otherwise. It is very gratifying to ascertain that this correlation is very well obeyed. The existence of structural defects (namely in CoGa) has been however questioned recently on the basis of a similar model (KIM [1991]): but the controversy rests entirely on the relative values of the bond energies, which are nothing but phenomenological parameters and which cannot be extracted from experimental quantities by undisputable procedures. In a growing number of experimental systems, a combination of lattice parameter and sample length measurements (SIMMONSand BALLUFFI’S technique; ch. 18, Q 2.2.2.2) yields the total vacancy concentration increase between a reference state at room temperature and the high temperature state: (CoGa: VAN OMNENand DE MIRANDA {198l]; AlFe: Ho and DODD[1978], PARISand LESBATS[1978]; GaNk Ho etal. [1977], CoSc and InPd WAEGEMAEKERS [1990]). The concentration of vacancies for the reference state is determined by a density measurement at room temperature. PA techniques have also been used, which confirm the previous determinations. But the concentration of anti-site defects is usually not directly reachable through spectroscopic
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methods; only an indirect determination of their number is possible if such defects can be associated with some macroscopically measurable quantity. As an example, from the measurement of the quenched-in magnetisation of a Co-Ga alloy, the number of antisite atoms Co,, which are the only Co atoms to be surrounded by like neighbours and, as such, are assumed to be the only ones to bear a magnetic moment, is indeed found equal to half of the number of vacancies. This beautiful result points strongly in favor of the very existence of the triple defect in this alloy 0.0 CASCIO[1992]). Other experimental techniques are necessary in order to gain a sharper insight into the defect populations on each sublattice. First results have been obtained through positron annihilation in CuZn (CHABIKand ROZENFELD[1981]) or direct observation in a field ion microscope in AlFe (PARISand LESBATS[1975]); but extracting meaningful values from the raw data requires a delicate analysis of positron trapping at vacancies for the first technique, and a careful analysis of image contrast for the second. 5.2.1.1. Experimental results. Most of the experiments measured the tracer , and D,, as a function of temperature and composition. The diffusion coefficients D reader is referred to a recent compilation for the detailed results and references (WEVEX [1992]]. Without entering into details, the following trends can be outlined: - At constant composition, the activation energy for diffusion is higher in the ordered than in the disordered phase (when it exists). There is a break of the Arrhenius plot at the critical temperature T, of ordering, and a large fraction of the increase in activation energy is due to correlation effects. In the ordered phase, the Arrhenius plot is often more or less curved (KUPERet al. [1956]). Simple models show that the migration and formation energies (EM,E&) of the vacancy and, therefore, the total activation energy Q, exhibit a quadratic dependence upon the long-range order parameter S (GIRIFALCO [19641): EM= EL(1f aMS2)EF = EF(1+ a$')
and Q = Qo(lf cr,S2).
The experiments are not entirely conclusive however: and Zn* tracers (KUPER et aE. [1956]) have been plotted logarithmically as function of (1+a,S2)/T (GIRIFALCO [1964]). The Arrhenius plot is a straight line only if the theoretical values SBWof the long-range order parameter (BW stands for Bragg and Williams) are arbitrarily replaced by the experimental values S , which have been determined by X-ray measurements. It has been checked however that S is not well accounted for by a Bragg-Williams approximation and that f"P a more sophishcated treatment including short-range order (SRO)must be used instead (COWLEY[1950]). An interesting observation is that Sim,, is equal to S,, at the same temperature: therefore the quadratic dependence of Q upon S i p can be interpreted as a linear dependence of Q on S,, as well. The last difficulty lies in the fact that, as already mentioned above, most of the change in the activation energy comes from the temperature dependence of the correlation factor, which is not included in Girifalco's analysis. - In AlFe alloys, the migration energy of the vacancies which have been retained by quenching varies roughly as S2 (RIVIEREand GWHE [1974]). But it is clear from the data that the results, within the error bars, can as well be accounted for by a linear law. - In CoFe alloys, the observation of a Portevin-Le Chatelier effect is related to vacancy
- In CuZn, the diffusion coefsiccients of Cu'
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migration and the effective migration energy varies quadratically over a large range of S extending from 0.1 to 0.9 (DINHUTeta2. [1976]). - At constant temperature, the diffusion coefficients vary with composition and exhibit a minimum at stoichiometry (or in the close neighbourhood of stoichiometry). This minimum is more or less pronounced (V-shaped curve for AgMg or AlNi) and corresponds to a maximum of the activation energy. The existence of this maximum is understandable, since the formation and migration energies of the vacancy are both increasing functions of the long-range order parameter which goes through a maximum at stoichiometry. D, and DB*differ by no more than a factor of two or three for the alloys in which the defects are predominantly of substitutional type (AgMg, AuCd, AuZn). - For alloys belonging to group ii), a marked asymmetry between hypo-and-hyper stoichiometric compositions is exhibited: very high vacancy contents show up which correspond to an excess of B component (Ga in NiGa and CoGa; Al in FeAl or CoA1). The difference between D,, and D,, is more pronounced than above for the alloys (between one and two orders of magnitude).The apparent vacancy formation energy is usually low (typically 0.4 eV per vacancy), and a minimum shows up at stoichiometry. An effective migration energy can also be determined by following the kinetics of thermal equilibration through the macroscopic length of the sample: the previous analysis of NiGa and CoGa in terms of two diffusion mechanisms (nearest-neighbourplus nextnearest-neighbour jumps: VAN OMNENand DE MIRANDA[1981]) has been recently revisited: with the only assumption that the departure from the equilibrium value of the vacancy concentration follows a first order kinetics, it turns out that a simple vacancy mechanism with NN jumps only can account fairly well for the observed kinetics (WAEGEMAEKERS [1990]). A puzzling result however is that the sum of the effective formation and migration enthalpies is approximately equal to the activation energies for tracer diffusion in NiGa, but significantly lower in the case of CoGa. 53.1.2. Atomic mechanisms for diffusion in ordered B2alloys. Several atomic mechanisms have been proposed nearest-neighbour (NN) or next-nearest-neighbour W)jumps. The triple-defect (TD) has been unnecessarily assumed to migrate as a whole and the migration of the divacancy ZV, was supposed to occur through a correlated sequence of NN vacancy jumps with species A and NNN vacancy jumps with B. The direct determination of jump vectors has been performed only very recently on FeAl alloys. The most probable path for Fe diffusion consists of sequences of two consecutive NN jumps, implying a transitory residence on a &site and resulting in the net displacements along ello>, and depicted on figure 13a (SEPIOLand VOGL[1993b]).
5.2.2. Ordered alloys with L1, structure The L1, structure of the A,B compound is such that the B component occupies one of the four sc lattices which make up the host fcc lattice: each B atom has twelve nearest-neighbour A atoms, whereas each A atom has four unlike neighbours and eight like ones. Due to this last property, it is commonly believed that A should diffuse
Ch. 7, $5
DiMion in metals and alloys d
111
Fig. 13. Observed jump mechanisms for ordered alloys by QEMLB: (a) B2 structure with a + b + a jump sequences; (b) DO, structure with a c)p, a t)y. p t)/3 and y t)S jumps; (c) B8 structure with (oi) H (dti) jumps.
markedly faster than B species: it has been experimentally checked only recently on Ni,Ge where Ni (as well as Fe or Co tracer) diffusivity is indeed found to be one order of magnitude larger than Ge diffusivity (YASUDA etal. [1993]). Direct measurements of vacancy concentrations in Ni,Al suggest that mainly antistructureatoms accommodate the departure from stoichiometry (AOKI and Izuhlr [1975]): model calculations with the embedded atom method (EAM) agree with this picture and predict low vacancy concentrations on both sublattices, with a marked preference for the sublattice of the major component (FOILES and DAW [1987]; X I E and FARKAS[1994]); the same results are suggested by EAM calculations for Cu,Au (JOHNSON and BROWN[1992]). It is worth noticing that Ni,Al is the only alloy in which the vibrational entropy has been measured in the ordered and disordered phase (ANTHONY et al. [1993]): the reduction in entropy when passing to the ordered phase is equal to 0.3 kB per atom. In Co3Ti alloys, on the References: p . 651.
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Ti-poor side, the diffusivity of Co tracer increases with approaching stoichiometry and the isotope effect, which can be shown to equal f AK in L1, structures (ITO et al. [ 1990]), is small (ITOHet al. [1989]).
5.2.3. Ordered alloys with L1, structure The L1, compound AB is such that the (001) planes of the host fcc lattice are alternately filled with A and B atoms. In TiAl alloys, the diffusion of Ti tracer complies the empirical rules of normal diffusion in compact structures (KROLLetal. [1993]) and no structural vacancies are expected from PA measurements on either side of the stoichiometric composition (SHIRAIet al. [19921).
5.2.4. Ordered alloys with DO, structure The DO, structure for this A,B compound can be viewed as the occupancy by B atoms (Al, Si, Sn, Sb) of a fcc lattice (named p), the parameter of which is twice that of the host bcc lattice; the others sites belong to three other fcc lattices with the same lattice parameter (a,y, 8 ) which are occupied by A atoms (Fe, Ni, Cu, Ag). Aa and A8 have 4 A,, +4BB as first neighbours, while A,, and B, have 4%, +4A, (fig. 13b). As a consequence, B atoms have only unlike nearest neighbours. The major component has generally the larger diffusivity which increases with increasing the concentration of the minor component: Cu,Sn (PRINZ and WEVER [1980], AMTA et al. [1991]); Cu,Sb (HEUMANNet al. [1970]), Ni,Sb (HEUMANNand ST&R [1966]). QEMLB in stoichiometric Fe,Si (SEPIOLand VOGL[1993a]) and QENS in Ni,Sb,, (SEPIOLet al. [1994]) indicate that the transition metal (Fe, Ni) atoms diffuse by NN jumps between a,y and S sublattices; the departure from stoichiometry for F%,Si,, is accommodated by antisite Fe, which are shown to participate also strongly to diffusion. PA measurements in Fe,,Al,, cannot separate V, from V,, and gives an apparent vacancy formation energy of 1.2 eV (SCHAEFER et al. [1990]): structural vacancies are not expected from the data.
53.5. Ordered alloys with B8 structure The B8 structure for this AB compound is made of a compressed hcp lattice for the B component (In, Sn, Sb, As, Ge) with a c/a ratio of the order of 1.3; the A component (Ni) occupies either the octahedral interstices (oi) or the doubly tetrahedral ones (dti). Antistructure atoms Ni, are however believed in NiIn. The large number of (oi) + (dti) sites allows the compounds to accommodate a significant positive departure of Ni atoms from stoichiometry, while still maintaining high concentrations of vacancies on the (oi) sites as large as several percent, even for Ni-rich alloys. The Ni* diffusivity is roughly 10' times that of Sb* (HAHNELetal. [1986]) or Sn* (SCHMIDT etal. [1992a, 1992bl). The determination of Ni jump vectors in NiSb compounds shows that Ni atoms jumps essentially from (oi) to (dti) sites, the vacancies on (oi) sites being crucial for allowing easy (dti) to (oi) backward jumps (fig 13c). Direct (oi-oi) or (dti-dti) jumps are excluded (VOGLet al. [1993]).
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5.2.6. Ordered alloys with B3, structure The B3, structure for this AB compound is made of two interpenetrating diamond lattices. Only two ordered alloys have been investigated so far, namely p-Li-A1 and p-LiIn. Structural vacancies VLi and antisite Lid (or LiI,) are believed to be the dominant defects, both defects coexisting at stoichiometry with noticeable concentrations. Li diffusion studied by NMR relaxation exhibits an activation energy of the order of one tenth of an eV and a diffusion coefficient in the range 104-10-’ em's-' at room temperature. A significant interaction is found between the immobile Li,, or Li, antisite atom and the vacancy on the Li sublattice (attraction for the first, repulsion for the second). (TARCZON et al. [1988], TOKLMRO et al. [1989]).
5.2.7. Ordered alloys with A15 structure The A15 structure for this A$ compound is made of a bcc lattice for B atoms (Gay Sn, Au, Si), together with a split interstitial A-A (A=%, V, Cr) dissociated along , < 0 1 b and d o l > directions in the faces (OOl), (100) and (010) of the elementary cubic cell, respectively. When bringing together the cubic cells, the split interstitials make up linear chains along the corresponding directions. The only alloy in which both diffusivitieshave been measured is V,Ga: the activation energy of the transition element is high (4.3 eV), and Ga, which is found to diffuse in grain-boundaries with an unexpectedly high activation energy, is probably the slowest component in bulk diffusion (BAKKER [1984]). Superconductivityoccurs along the chains of the transition metal; the thermal disorder, which is believed to be mainly antistructural by analogy with Nb$n (WELCHet al. [1984]), can be retained by quenching from higher temperatures: it degrades the superconducting transition temperature T, in a reversible way, since a subsequent annealing restores the original value. A simple model relates the drop of T, to the amount of antistructural defects (the vacancies, which are necessary for atomic transport, are neglected) (FAHNLB[1982]]. An apparent formation energy of 0.65 eV for antisite defects is deduced from the variation of T, with the quenching temperature (VAN WINKELet al. [1984]). The healing kinetics of T,, attributed to vacancy bulk migration, is dominated by the slowest bulk diffusivity of Ga atoms: it has been measured at different temperatures with an apparent migration energy of 2.2 eV, however one is left with the contradiction that Ga is the slowest component with the lower activation energy (VANWINKELand BAKKER[1985]). Further studies on these compounds are currently in progress (Lo CASCIOet al. [1992]).
5.3. Chemical diffusion When diffusion takes place in a region of the sample where the chemical gradients cannot be ignored, the diffusion coefficients of the various components are no longer constant, as in homogeneous alloys, but depend on space and time through the composition. In what follows, we examine the case of chemical diffusion and the Kirkendall effect in binary alloys. The reader is referred to more extensive reviews for the case of multiphase and multi-component systems (ADDA and PHILIBERT[1966], KIRKALDYand Refemnces: p. 551.
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YOUNG [1987]). The interdiffusion of two elements having different partial molar volumes implies a volume change of the sample which must be taken into account for an accurate measurement of chemical diffusivities (BALLUPPI[1960]). The change of the average atomic volume in neighbouring parts of the sample induces however the birth and the development of stresses, which are usually partially released by some amount of plastic deformation. The inclusion of such effects in the analysis of Kirkendall effects started only recently and is currently under progress (STEPHENSON [1988]; SZABO et al. [1993]); they will be ignored in what follows.
5.3.1. Chemical diffusion in binary systems and Kirkendall effect 5.3.1.1. Description and interpretation of a typical experiment. The simplest diffusion experiment to carry out consists in clamping together two pieces of pure metals A and B, to anneal this couple long enough and to determine, at the end of the run,the concentration profile all along the sample. What is observed is a spreading of the initially step-like profile together with a shift of the initial welding interface (defined by inert markers such as oxide particles or tungsten wires) with respect to the ends of the couple which have not been affected by the diffusion (fig. 14). This shift results from the Kirkendall efsect and finds its origin in the fact that the diffusivities DA and D, are not equal. Indeed, if D, is larger than D,, species A penetrates into B at a faster rate than B into A as a consequence, the B-rich part of the sample must increase its volume to accommodatethe net positive inward flux of matter. This increase will be achieved at the expense of the A-rich part by shifting the interface towards A. This observation was reported for the first time by SMIGELKAS and KIRKENDALL[19471 on copper-zinc alloys: the zinc is the faster diffuser and the welding interface (called Kirkendull plane) shifts towards the zinc-rich side of the couple. This experiment was a milestone in the history of solid-state diffusion: it definitely ruled out the assumption of a direct exchange A c)B mechanism which was formerly proposed and which would have implied equal diffusivities for both species. It must be noted that a Kirkendall effect has also been observed in fluids: it is expected indeed to be very general, since the first convincing interpretation of the phenomenon is not based on any detailed mechanism for matter transport (DARKEN [1948]). The simultaneous measurements of the displacement rate v of the Kirkendall plane and of the chemical diffusivity fi in that plane yield the intrinsic diffusion coefficients D, and D, for the composition of the Kirkendall plane. In order to know D, and DB at several concentrations, one should prepare the corresponding number of differential couples, which are made of two alloys with different compositions. In fact it can be shown that a single experiment is needed, provided that a complete set of inert markers has been inserted on both sides of the welding interface (CORNETand CALAIS[1972]). In what follows we suppose that the observed effect is unidirectional, and that only one space coordinate x is needed, in conjunction with the time variable t, to describe the evolution of the system. The transformation x / 6 - A in Fick's second Law shows that the solution C(x,t) can be expressed as a one-variable function C(A). We know from
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Fig. 14. Kirkendall effect experiment with a diffusion couple made of two pure metals A and B.
experiment that the Kirkendall plane has a constant concentration during the diffusion anneal, and accordingly that it is characterized by a constant value of A. As a conseno exception to this simple law has ever quence, the Kirkendall shift Ax varies as been reported. A similar behaviour has also been observed for any inert marker which is not located in the Kirkendall plane at t = 0; after a time lag, the duration of which depends on the distance from the Kirkendall plane, the inert marker starts moving with the same time and PHILIBERT [1967]; MONTY[1972]). law (LEVASSEUR Up to now no atomic mechanism for matter transport has been mentioned; but if we know it, something more can be said about the Kirkendall plane. We suppose in the following that the vacancy mechanism is operating. In all the experiments performed so far, the inert markers are invariably made of materials which have a high melting temperature. The formation and migration energies of the vacancy in such materials are significantly larger than in the surrounding matrix. As a consequence, the markers are impermeable to the vacancy flux. Under this condition, it can be [1969]), shown that such a marker shifts along with the lattice planes (KRIVOGLAZ whatever the type of its interface with the matrix (coherent or incoherent). Thus the measurement of the Kirkendall shift is nothing but the measurement of the lattice plane
6;
shift.
The above formalism can be easily enlarged to account for the case in which the [1960]). average atomic volume varies with the concentration of the alloy (BALLUPPI 5.3.1.2. Vacancy wind effect Manning’s approximation. In the original formulation of the Kirkendall effect, the flux J, of species A stems only from the chemical potential gradient V p Aof species A (DARKEN[1948]). At infinite dilution, the solid solution becomes ideal (p = 1) and the intrinsic diffusion coefficient D, must tend towards the tracer diffusion coefficient DA*. Hence:
-
DA = DA*- 9,
D, = DB** 9.
These relationships are known as Darken’s equations; we know however, from the thermodynamics of irreversible processes, that the offdiagonal term cannot be neglected. More general expressions can be established [see eqs. (6)]: References: p . 651.
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There is no simple way to relate theoretically the Lij’s to experimentally accessible quantities such as tracer (or intrinsic) diffusion coefficients. This has been done only in the particular case of a simplified random-alloy model (MANNING[1968]) for which cp = 1. These expressions are arbitrary assumed to hold even for a non-random alloy where the thermodynamic factor cp is no longer unity. Hence the final expressions for the intrinsic diffusivities are still given by eqs. (17) with the random values of the vacancy wind corrections r, and r, recalled in 0 5.1.2. Hence:
The last term in the brackets is called a vacancy wind term since it reflects the coupling between the transport of species A and B through the vacancy flux. We note that Manning’s equations predict a chemical diffusion coefficient b always larger than that given by Darken’s equations. The match of both sets of equations with experimental results will be reviewed in the following section. Before closing this section, a last remark should be made concerning the structure. of Darken’s or Manning’s expressions: in both sets of equations the thermodynamic factor cp enters in a multiplicative way. In some cases the variations of cp with the respect to concentration or temperature may outweigh the variations of other factors. This situation can be met accidentally as in Au-80 at 8 Ni (REYNOLDS et aZ. [1975]) but is also expected to happen in well-defined situations: for any alloy which tends to unmix at low temperatures, cp goes through zero at the top of the coexistence curve at some critical temperature T,. It is easy to show that the maximum of the coexistence curve is such that the second derivative of the molar free energy, d2f/dCi, vanishes. A short derivation yields:
where yA is the activity coefficient of species A. A convincing illustration of a vanishing 6 has been reported for Nb-34 at % H (WLKL and ALEFELD [ 19781). At critical temperature T,,the Arrhenius plot of b bends downwards and fi falls several orders of magnitude, whereas the Arrhenius plot of the hydrogen tracer diffusion exhibits a normal behaviour. This phenomenon is called critical slowing down; the top of the coexistence curve is the very point where the alloy hesitates between two conflicting forms of behaviour: - high-temperature behaviour where all the concentration fluctuations flatten out (B >0); - low-temperature behaviour where the concentration fluctuations of large wave-lengths are amplified (fi f The reason for the discrepancy is not yet clearly understood. As pointed out by CARLSON [1978], Manning’s correction to Darken’s expressions holds only for a random alloy, a condition which is never fulfilled in real systems. But, as outlined above, Manning’s approximation is quantitatively reasonable even in the non-random case; the problem of the experimental accuracy should be clarified first. 5.32. Ternary alloys The expressions of the three matter fluxes J,, J,, J3 in the lattice reference frame introduce nine independent phenomenological coefficients (or intrinsic diffusion coefficients if the chemical potentials gradients are expressed as concentration gradients). Neglecting the vacancy concentration C, against the matter concentrations C1, C2and C, and eliminating one of the concentrations (say C,) yields flux expressions with only six independent new coefficients. Expressing at last, the three fluxes in the laboratory reference frame, together with the condition Jp +J,” +Jp =O we are finally left with only four independent chemical diffusion coefficients fi fi;, fi;, 82,,the superscript ‘3’ recalling that C, is the dependent concentration and is evaluated through C3= 1- C, - C, and the tilde (-) recalling that interdiffusion coefficients are determined ( B O C Q ~ T [199Oa]). A beautiful analytical approach has been worked out on simplified systems, where the above diffusion coefficients are assumed to be concentration independent, a condition which holds in practice whenever the terminal concentrations of the diffusion couples are close to one another (differential couples). This analysis enlightens all the
:,
References: p . 651.
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characteristic features to be encountered in the practical studies of ternaries, namely, the existence of maxima in the concentration-penetration curves, the possible occurrence of and zero-flux planes, together with the general properties of diffusion paths (THOMPSON MORRAL[1986]). The extension to concentration dependent diffusivities can be made straightforwardly with the help of numerical methods.
6. Electro- and thermomigration At temperatures where diffusion is noticeable, atoms of a pure metal, or of an alloy, are caused to drift by a gradient of electric potential or temperature. We saw (04.1.2) that this phenomenon, also called the Sore? eflect in the case of thermal gradients, has been used to study phenomenological coefficients. It has also been used practically to purify some refractory metals. Last, but not least, it is a way to study the electronic structure of point defects (vacancies, impurity atoms) at high temperatures and its variation during a jump. Careful reviews of all aspects of electromigration can be found [1988] and Ho and KWOK[1989]. in VERBRUGGEN
6.1. Thermodynamic aspects Starting with the equations (1)-(5) in 8 1.2.2., if J, and J, are the electron and the heat flux, respectively, we define (DOAN[1971]) the valency and the heat of transport by:
The effective valence Z i and the reduced heat of transport QL introduced in 0 1.2.2. are then given by:
and the equivalent relations for the €3 component. The form of %* is due to the effect of the temperature gradient on the vacancies. It is derived under the hypothesis of a local equilibrium concentration of vacancies. It has therefore no counterpart in the electric field case. Any deviation from this equilibrium (see 3 8) invalidates the comparison between microscopic evaluations of q* and experimental Q*. In self-difhsion, B stands for an isotope of A, so eqs. (1)-(5) give, in the case of electromigration:
where fo is the self-diffusion correlation factor. The thermomigrationcase is given by an analogous equation, QL and -VT/T replacing Z,* and E.
Ch. 7,$ 6
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Dimion in metals and alloys
Such self-diffusion experiments then give access to the true values ZL and For solute diffusion, one calculates easily (dropping the Vn,, term):
QL.
Measurements can then give access only to the apparent effective valence Z : (or heat of transport Q;). This value differs from the true one, Z i , by the vacancy wind term Z i LAB/LBB(MANNING [1968]). The ratio L m b B varies approximately from +2 to -2 and can then give a very large correction to Z i , especially in polyvalent solvents. Equations (1)-(5) are written in the lattice frame, and so are defined the Z y andQ*,* values. But if the fluxes are,for some reason, measured in another reference frame, they give access to other values of coupling coefficients. For example in the laboratory frame, one obtains:
where the bracketed term defines the apparent effective valence in the fixed frame.
6.2. Microscopicanalysis Atoms in a metal under a gradient of potential or temperature are submitted to a force which has a double origin. On one hand, one finds a static part called direct in the electric case, or intrinsic in the thermal one. The direct force is due to the unscreened action of the electric field on the true ionic charge [eq. (72),term zA] and the intrinsic contribution corresponds to the enthalpy transfer due to an atomic jump (WIRTZ[1943], [1954]). In this approximation the heat of transportqi BRINKMAN [1954], LE CLAIRE [eq.(72)] is nothing else than a part of the migration enthalpy (HUNTINGTON 119681). On the other hand electrons and phonons in metals are highly mobile carriers, either thermal or electrical. Therefore their scattering at atoms which are neighbours of a vacancy gives rise to a second contribution: the electron or phonon breeze. In the case of electromigration FIKS[1959] and HUNTINGTON and GRONE[1961] have given a model of this scattering part, treating electrons as semiclassical particles. BOSVJEUX and FRIEDEL [1962] have used the free-electron model in the Born approximation to give a quantum-mechanicalexpression of the z* term. More rigorous treatments of this term have been developed later, either in the framework of the linear response [1975], TURBAN etaZ. theory, or of the muffin-tin approach (KUMARand SORBELLO [1976], SCHAICK [1976], RIMBEYand SORBELLO [1980], GUPTA[1982], VANEK and LODDER[1991]). Controversies are still running on the existence either of a screening effect in z*, which could partially or exactly cancel the direct force (TURBAN et aZ. [1976], LODDER[1991]), or other contributions behind the carrier scattering (GUPTA [1986]). However all these treatments give essentially the same basic results, their main interest being to define more precisely the range of validity for the preceding models. The results are the following: References: p. 451.
614
J. L Bocquet, G. Brebec, I:Limoge
Ch. 7 , # 6
(i) For a free electron gas the scattering part of the effective valence is given by:
p y and Apy*”” are the residual resistivities (expressed in y h m per at%) of where A + Ap~E”b’e, atoms of species i (i = B or A) in saddle or stable position (their sum, Ap? is denoted by Ap, in table 8, below); po is the matrix resistivity and fo is a correction term due to the neighbouring vacancy (zeroed for an interstitial solute). We find that in normal metals, owing to the order of magnitude of AI and po, the (possible) direct term is completely negligible. (ii) In polyvalent metals, or transition metals, with a hole conductivity, one has to take into account the details of the F e d surface and of the scattering atom, electron velocities, wave function character, anisotropic scattering. Schematically two opposite contributions like eq. (73) are found, one for electrons and one for holes, which yields a partial compensation between them. The link with the residual resistivities is lost. In that case, the effective valence is much lower, and the calculations are quite involved (RKS [1973], HUNTINGTONand Ho [1963], LIMOGE [1976b], GUPTA[1982], VAN EK and LODDER[1991]). The situation is more troublesome in thermomigration. FIKS[1961], GERL[1967] and SORBELLO [19721have calculated the phonon scattering contribution.The result, as given by Gerl, is a positive term, of the order of 100 kJ/mole (or lower after CROLET[1971]) and linear in temperature, contrary to SCHOTTKY’Scalculation [19651. The electron term is more firmly established and according to GERL[1967]:
dl
OC
z*
and so gives a negative contribution in normal metals. The final Q*is then the result of the compensation between four terms, and theoretical calculations are very questionable (DOANet ul. [19761). Some years ago, it was proposed to use directly the thermodynamic definition of q* eq. (71) to calculate it (GILLAN[1977]); but this way has not been much followed till now to give quantitative results.
63. Experimental methods In electro- or thermotransport, three techniques have been used. In the first, one measures the total atomic flux J, + Jb=- J,. This is done by measuring the displacement of inert markers with respect to the ends of the sample. This method can be used only for self-diffusion but is able to yield a good accuracy if vacancy elimination conditions are well controlled (GERL[1968]; LIMOGE[1976a]). In the second method one establishes a steady state between the external force, either E or VT, and the induced concentration gradient. Measurement of the contration profile gives access to effective valence, or heat of transport, in the Zuborutoryfrarne (fig. 15a). The accuracy is generally not very high and the method is restricted to solute diffusion. Moreover the assumptions concerning the equilibrium vacancy concentration must be
Ch. 7,56
615
Dimion in metals and alloys
carefully checked. In the third method one uses a thin deposit of tracer between two bulk samples of solvent. This deposit will spread (0 1.2.5.), as a Gaussian in electromigration, and simultaneously displace (fig. 1%) due to the external force. This displacement with The accuracy is very respect to the welding interface gives the coefficient Z**, or Q**. high and the method is as suitable for self-diffusion as for solute diffusion (GILDERand LAZARUS [1966], DOAN[1971]), although its use in thennomigration needs some care ( ~ O L E T[19713). 6.4. Experimental results and discussion
The reader can find an exhaustive review of experimental results on electromigration in h n and SELLORS’monograph [1973]. For thennomigration he is referred to ORIANI’Sarticle [1969], see also WE= [1983]. Let us first discuss thennomigration results.
Thermomigration In table 6, the heat of transport q for interstitial solutes are displayed: this case does not raise of course the delicate problem of the vacancy local equilibrium! It can be noticed first that q i has generally the same sign for all solutes in a given solvent. There is also some correlation between Z i and q i , but opposite to the one predicted by Gerl’s et al. [1987] there is a good correlation between theq; model. According to NAKAJIMA and the migration enthalpies of the three isotopes of Hydrogen in V, Nb and Ta. In table 7, we display the heat of transport in self-diffusion in common metals. The strong scattering of the experimental values can be seen at once, either for a given 6.4.1.
Lefore
-.
anode
I
pure A
1
j
steady s t a t e c o n c c n f r a tion
solvent 3!
Icalhodc
A
thin l a / e r of t r a c e r
onode i n i t i a l salute c o n c c n t r a t i o n
z; O a O’
-
N
-
>Ob
0
O = 0.5 oto2g -
-
-
12’ 54 - 67
28.5 j -2Ot0-80g - 10 to - 40
PRATTand SELLORS [1973]; ORIAM[19691; ~MAREcHEe r d . [1979]; ERCKMANN and WIPF [1976]; e MATHUNIer al. [1976]; PETERSON and SMITH.[19821; MAel al. [1979]; CARLSONand SCHMIDT [19811; UZ and CARLSON [1986]; NAKAJIMA er aL [1987].
a
element or for similar elements. This underlines the experimental difficulties and also a possible departure from equilibrium of the vacancy distribution ($5 1.2.2, 6.1 and 8.1). Transition metals display large QL values. This has been explained by HUNTINGTON [19661 as the result of additive contributions of electrons and holes, contributions which are of opposite sign in electromigration, leading to small Z*. 6.4.2. Electromigration In table 6 are also given the Zi values for interstitial solutes. As in thermomigration, most intentitial solutes migrate in the same direction in a given solvent. The hole
Ch. 7,96
Difision in metals and alloys
617
Table 7 Thennomigration - effective heats of transport in self-diffusion, after ORIANI[1969]. Metal
e l (Id/mole)
Na
- 6.3
Al cu
- 22.6; 0; + 16.7
Ag Au Pb
Zn
a
-6.3to-8.4;+46
0 -27; 0 + 8.8 - 0.8; 0; + 9.6 to 14.6
> l), the solution is equivalent to Suzuoka’s solution (see 0 7.1.1). The expression for the parameter /3 is:
p=--D’
ka
- r’ Ti
1
D (Dt)Y2 -
Since the bulk diffusion coefficient is r b 2 (b is the lattice parameter), the comparison of the two solutions yields D’ = r‘b2; the segregation factor k is equal to riKoand the grainboundary thickness is b. In the case where the grain boundary is modelled as p parallel planes, it is found that its thickness is pb. It must be noticed that this thickness is not altered even if the bonds between the sites in the bulk and the sites in the boundary are stretched perpendicularly to the boundary plane. The “thickness” of the grain-boundary is not related to the actual atomic relaxations at the grain-boundary but only to the number of high-diffusivity paths which are available for the tracer. - In the limit of a small penetration depth into the bulk, the identification with the continuous solution is impossible. At very short times ( r t c 0.1) the exact solution tends towards a Gaussian with I”b2 as diffusion coefficient. This model has been modified to account for more realistic grain-boundary structures, but still disregard the correlation effects. We refer the reader to the original papers (COSTEet al. [1976]). For long, the sophistication of the modelling has been several steps forward with respect to the available experimental information. Only recently, an impressive series of grain-boundary Ag diffusivity measurements, using a clever accumulation method of improved accuracy, has been undertaken in Au bicrystals of well-controlled tilt angle (MA and BALLUPFI[1993a]); the diffusion coefficients (and the activation energies as well) do not exhibit any cusp at those particular orientations which correspond to coincidence site lattice boundaries (CSLB) of short-period and low-2. This was taken as an indication that the core of the boundary is made up of several structural units derived from relatively short-period delimiting boundaries which are nearby in the series; in this picture, the change in tilt angle is reflected in a continuous change in the mixture of these structural units. An atomistic modelling resting on the embedded atom method (EAM) suggests that vacancy, direct interstitial and intersticialcy mechanisms are References: p . 651.
626
J. L Bocquet, G.Brebec, E Limoge
Ch. 7, 57
probable candidates for matter transport along the boundary. The change in activation energy experimentally observed is accounted for by additional jumps of higher energies and BALLUFFI[1993b]). At last, the magnitude of the correlation factor for the intersticialcy mechanism is found to be roughly equal to that of the vacancy one. Thus, a relatively large correlation factor is no longer the indisputable fingerprint of a vacancy mechanism at work, contrarily to previous findings (ROBINSONand PETERSON [1972]). This last result, together with that concerning the diffusion in a dissociated dislocation, suggests that the vacancy mechanism is not necessarily the dominant mass transport mechanism, as thought before from preliminary simulation work (BALLUFFI et al. [198 11; KWOKet a2. [1981]; CICCOT~ et al. [1983]). It is worth mentioning however that the activation volume for self-diffusion in a tilt boundary of Ag bicrystals is consistent with the vacancy mechanism (MARTINet al. [1967]).
7.4. Surface diffusion Although free surfaces can actually play the role of short-circuits for bulk diffusion (inner surfaces of cavities, surfaces along a crack), they have been mostly studied for their own sakes. We shall not repeat hereafter the continuous approach which has been already used for interface or grain-boundary diffusion; grain 2 in fig. 16 has only to be replaced by vacuum and the exchanges between the surface and the vacuum suppressed. As in the case of a grain boundary, the characteristic quantity which appears in equations is 6D,, where 6 is the “thickness” of the surface layer and D, the surface diffusion coefficient. We will focus in the following on the atomistic point of view.
7.4.1. Atomic structure and point defects A surface is essentially made up of terraces which are portions of low-index surfaces; these terraces are separated by ledges of atomic height, along which kinks are present (TLK model: fig. 18). Ledges and kinks have a double origin: - A geometrical one, to provide the misorientation of the actual surface with respect to the dense planes of the terraces (0and a angles in fig. 18). - A thermally activated one for entropy purposes. Such a description is thought to hold in a range of low temperatures where the formation free energy of ledges is large enough to keep their thermal density at a low level and where reconstruction or faceting are not observed (in practice between 0 K and OST,,,). As predicted by BURTONetal. [1951] a dramatic change in the surface topology occurs at some transition temperature TR,at which the formation free energy of the ledges vanishes (or becomes very small): as a consequence the surface becomes delocalized (ch. 8, Q 5.1). This transition (called roughness transition) is due to a large number of steps of increasing height which make the edges of the terraces indistinguishable. This has been clearly illustrated by Monte Carlo simulations on (100) surfaces of a simple cubic lattice (LEAMYand GILMER[1974]; VAN DER EERDENet al. [1978]) Figure 17 of ch. 8 shows examples of LEAMYand GILMER’S computations. TRis roughly
Ch. 7, 5 7
Difision in metals and alloys
627
Fig. 18. Terrace-Ledge-Kink (TLK) model for low-index surfaces. The formation of adatoms (the extra atoms bulging out from the plane of a low-index surface) and advacancies (the anti-defects to adatoms) is represented.
given by
TR=0.5~/k, where E is the strength of the first-neighbour bond. This transition has indeed been observed on several metals using He scattering spectroscopy (for a recent overview, see LAPUJOULADE [19941). In what follows, we restrict ourselves to surfaces maintained below TR. Point defects are also present, namely adatoms and advacancies (see fig. 18); they can be created pairwise at a site of a terrace or separately at a ledge or a kink. The latter case is energetically favoured with respect to the others and is thought to be dominant. Multi-defects can also form by clustering adatoms or advacancies. Theoretical calculations of pointdefect properties on low-index surfaces have so far been performed first with very crude potentials (WYNBLATTand GJOSTEIN[1968]; PERRAILLON etal. [1972]; FLAHIVE and GRAHAM [198Oa]), and later, refined with atomic potentials derived from the embedded atom method ( E M (THOMPSON and HUTTINGTON [1982], DESJONQUERES and SPANJAARD[1982], Lnr et al. [1991], LIU and ADAMS [1992], SANDERSand DE PRISTO [1992]). It is worth noticing that their results do not differ very much, even quantitatively: this is undoubtedly an indication that the formation energies of point defects depend only on very fundamental and simple properties of the surfaces (likethe number of lateral neighbours or the packing): - The formation and migration energies for adatoms and advacancies are found to be highly sensitive to surface orientation. - The formation energies for both kinds of defect are comparable, except for the (100) surface of an fcc lattice, where the formation energy of the advacancy is significantly smaller than the corresponding energy for the adatom. Therefore, both defects are expected to contribute significantly to matter transport. They will be created in roughly equal amounts, either separately at k i n k s or pairwise at terrace sites. - The migration energies have been mainly calculated for adatoms on fcc and bcc References: p. 651.
628
J. L.Bocquet, G.Brebec. Z Limoge
Ch. 7, $ 7
surfaces, and for the vacancy on Cu (FERRAILLON et al. 119721) and Ni (LILTand ADAMS [1992]) surfaces; the advacancy is in most cases the slower-diffusing defect. For fcc lattices, the migration energies of adatoms increase roughly with increasing surface roughness: E,, (lll)cE,,, (113)- E,,, (331)<E,,, (OOl)<E, (210). For bcc lattices, the migration energies are roughly in the following order: E,,, (110)- E,,, (211)- E,,, (321)<E,,, (310)<Em (001)cE, (111). It is worth noticing however that, due to the presence of the defect concentration term, the su$ace self-dz@sivities are not necessarily in the same order: a compensation effect occurs, which pairs a low migration to a large defect formation energy (on the (111) surface of fcc lattice, namely). As a result, the surface diffusivity of Ni is expected to be noticeably larger on (113) and (133) surfaces than on any other (LIUand ADAMSt19921). A further difficulty stems from the fact that diffusion is expected to be highly anisotropic on non-perfect surfaces: the migration energy along dense rows or in deep channels is usually smaller than the migration energy across these rows or channels. This may obscure the ordering of low-index surfaces with respect to their migrational properties. Diffusion is thought to take place through individual jumps only at very low temperature and to exhibit a marked anisotropy due to atomic roughness (T3 d s ) and slows as the interface moves across the droplet. This decrease in velocity is due to the evolution of the latent heat at the S-L interface and the resultant reduction in the interface supercooling. The effect of changing the heat transfer coefficient by two orders of magnitude is primarily to alter the velocity after the fraction solid exceeds the dimensionlessinitial supercooling (0.5 in this case). Growth at small fraction solid is controlled primarily by heat flow inside the powder particle, while growth at large fraction solid is controlled by external heat flow. If no initial supercooling were present, the growth velocities across the entire particle would be near those seen at large fraction solid (fig. 4), which are typically less than 10 MS. 2.5. Softwarepackages
Different numerical methods have been used to treat solidification. RAPPAZ[1989] discusses five main computational techniques: i) The finite difference method (FDM) with or without the alternative implicit direction (ADD time stepping scheme, ii) The finite element method (FEM),iii) The boundary element method (BEM), iv) The direct finite difference method (DFDM),v) The control volume element method (VEM). He discussed the basic advantages and/or inconveniencesof these methods using schematic 2-dimensional enmeshments that are associated with the five main computational techniques. OHNAKA[19921 analyzed solidification for a thermal conduction model and [1988] give a reviewed critically these computational techniques. BERRYand PEI-IL~ comprehensive view of the steps to be taken when solidification modeling is used, stressing the fact that the thermophysical properties and details about mould material are often poorly known. Indeed, conditions such as moisture, property dependence on temperature, etc., would require an almost limitless data base. Software packages permit, through suitable interactions, the generation of maps displaying the variation of specific criteria functions that affect casting soundness, such as local temperature gradient, freezing time, front speed or cooling rate. DANTZIGand WIESE[1986] and WJESE and DANTZIG[1988] have focused on a technique to reduce the computation time involved in FEM methods for sand castings. Considering that the number of nodes located within the mold is far greater than those within the casting, the authors replace the sand mold by a set of boundary conditions applied at each element on the surface of the casting. This method was initially proposed by NIYAMA[1977] and later by WEI and BERRY [1980] and is known as the Q-Dot
Ch. 8, $ 2
SolidiJcation
0.1
0.2
0.3 0.4 FRACTION SOLID
0.5
0.6
0.7
Fig. 4. Calculated interface temperatwe WW and MEHRABIAN [1982]) for the solidification of a powder particle initially undercooled by 1/2 UC;.The temperature rises and the velocity falls as growth proceeds from the point of nucleation on the powder surface across the powder particle. The effect of various values of the heat transfer coefficient 4 is also shown. The velocity scale on the right was added by BOBTTINGER and PERIPEZKO[1985].
method. More recently DANTZIGand Lu [1985] and DANTZIGand WIESE[1985] developed the Boundary Curvature Method (BCM). In both techniques the heat flow from the surface of the casting is treated using a library of special functions, related to the local geometry, which contain the information about heat flow from that shape into a mould material. The BCM method seems to adapt better to arbitrary shapes in three dimensions than the Q-Dot method. DANTZIGand WIESE[ 19861 present an example of a complex casting (a cylinder block section) stressing the fact that this problem would have required much greater computation time without the BCM.
2.6. Experimentalmethods involving controlled solidification The difficulties associated with predicting external heat transfer in casting has lead to the development of two research techniques of controlled solidification: unidirectional solidification and solidification with prescribed bulk supercooling. Much of the understanding of solidificationlaws comes from unidirectional solidificationexperiments based on the simple principle that the extraction of latent heat must be achieved without allowing the melt to supercool sufficiently to permit the nucleation of crystals ahead of the solidification front. In practice this requires a heat sink that removes heat from the References: p . 830.
682
H.Biloni and KJ. Boerringer
Ch. 8, 53
solid and a heat source that supplies heat to the melt. The extensive use of such and his school (CHALMERS [1964], 119711) produced the basis techniques by CHALMERS of modern understanding in solidification. The basic heat flow objectives are to obtain a unidirectional thermal gradient across the interface and to move it so that the interface moves at a controlled rate. For a planar S-L interface the gradients are related by eq. (3). Based on this idea different techniques have also been developed to obtain single crystals employed for research purposes. FLEMINGS[1974] gives details of the different techniques used. Heat flow in controlled directional solidification (DS) is treated in Q 2.4.1. Another directional controlled process corresponds to the extraction of heat via a botton chill. Growth occurs in a direction parallel and opposite to the heat flux direction. In this situation a better control of microstructure and properties is obtained in comparison with conventional casting. However the microstructure is not uniform as in the Bridgman method, because V and G decrease with the distance from the chill. It is possible to improve the microstructure uniformly by programmed furnace temperature and withdrawal rates. The above examples of controlled solidification do not involve bulk liquid supercooling. However, when a crystal is nucleated at a specified temperature and grows freely into the liquid, the bulk supercooling AT plays a major role in determining the structure observed during the solidification process. This type of study has been performed with both low and high supercooling and important structural information has been obtained in organic material analogues (GLICKSMAN[1981]) and metallic alloys (FLEMINGS and SHIOHARA [1984], WILLNECKER et al. [1989], [1990]). More details will be given in Q 11.
3. Thermodynamics of solidijcation 3.1. Hierarchy of equilibrium The process of solidification cannot occur at equilibrium. However it is clear that different degrees of departure from full equilibrium occur and constitute a hierarchy which is followed with increasing solidification rate. This hierarchy is shown in Table 2 (BOJZTINGER and P E ~ P E Z K[1985]). O The conditions required for global equilibrium, (i), are usually obtained only after long term annealing. Chemical potentials and temperature are uniform throughout the system. Under such conditions no changes occur with time. Global equilibrium is invoked for descriptions of solidification that apply the lever rule at each temperature during cooling to give the fraction of the system that are liquid and solid as well as the compositions of the (uniform) liquid and solid phases. This situation is only realized during solidification taking place over geological times. During most solidification processes, gradients of temperature and composition must exist within the phases. However one can often accurately describe the overall kinetics using diffusion equations to describe the changes in temperature and composition within
Ch. 8 , 9 3
Solidfiation
683
Table 2 Hierarchy of Equilibrium.
Full Diffusional (Global) JZquilibrium a. No chemical potential gradients (composition of phases are uniform) b. No temperature gradients c. Lever nile applicable 3. Local Interfacial Equilibrium a. Chemical potential for each component continuous across the interface b. Rase diagram gives compositions and temperatures only at Solid-Liquid interface e. Comction made for interface curvature (Gibbs-Thomson Effect) iii. Metastable Local Interfacial Elquilibnum a. Important when stable phase cannot nucleate or grow fast enough b. Metastable phase diagram (a true thermodynamic phase diagram missing the stable phase or phases) gives the inter€ace conditions iv. Interfacial Non-EqUilibcium a. Phase diagram fails to give tempentture and compositions at interface b. chemical potentials are not equal at interface c. Free energy functions of phases still lead to criteria for the “impossible” (Baker and Cahn [1971]) i.
each phase and using the equilibrium phase diagram to give the possible temperatures and compositions for boundaries between the phases, e.g., at the solidification interface. The Gibbs-Thomsm effect is included to determine shifts in equilibrium due to the curvature of the liquid-solid interface. This condition is called the local equilibrium condition, (ii) in table 2. For a dilute alloy the liquidus and solidus of the phase diagram can often be represented as sbxight lines (fig. 5). The local equilibrium condition for a curved interface is given in this case by = T,
+ mLe*, - T,FK,
and
Ci = k,,Cfl, where Ti and T, are the interface temperature an^ pure solvent melting temperature, m, is the liquidus slope, C; and C,* are the compositions at the interface of the liquid and solid, r is the ratio of the liquid-solid surface energy to the latent heat per unit volume, K, is the mean curvature of the interface (defined as positive when the center of curvature is in the solid), and k, is the equilibrium partition coefficient. Local equilibrium is never strictly valid, but it is based on the notion that interfaces willequilibrate much more quickly than will bulk phases. The conditions described in (ii) of Table 2 are widely used to model the majority of solidificationprocesses that occur in castings. For example, under the assumptions of fast diffusion in the liquid phase, negligible diffusion in the solid phase, and local equilibrium at the interface, the Scheil muatReferences:p. 830.
684
H.Biloni and RJ. Boettinger
Ch. 8, $ 3
Fig. 5. Solidus-liquidus relationships for dilute binary alloys. For a liquid composition of C, the equilibrium solid composition is hC,.For a solid of composition, C,,the equilibrium liquid composition is C,&,.
ion (see 0 7.3) gives a reasonable first approximation to the “nonequilibrium” dendritic coring or microsegregationin conventional castings. Clearly phase diagrams constitute an essential part of the data base for the modeling and analysis of solidification problems. Metastable equilibrium, (iii) in Table 2, can also be used locally at interfaces and is important in ordinary metallurgical practice. For example, one can understand the microstructural change of cast iron from the stable gray form (austenite and graphite) to the metastable white form (austenite and cementite) with increasing solidification rate (and interface supercooling) using information from the stable and the metastable phase diagrams combined with a kinetic analysis (JONES and KURZ [1980]). The eutectic temperature and composition for white cast iron are well defined thermodynamic quantities just as they are for gray cast iron. Metastable equilibrium is represented by a common tangent construction to the molar free energy vs. composition curves for the liquid, austenite, and cementite phases and thus minimizes the free energy as long as graphite is absent. When solidification is complete, a two phase mixture of austenite and cementite can exist in a global metastable equilibrium. The concept of local metastable equilibrium i s , especially important during rapid solidification (PEREPEZKO and BOETTINGER [1983], PEREPEZKO [19881) because some equilibrium phases, especially those with complex crystal structures, have sluggish nucleation and/or growth kinetics and are absent in rapidly solidified microstructures. An example of a metastable phase diagram superimposed on a stable phase diagram is given in fig. 6 for the Al-Fe system. If A1,Fe is absent, phase boundaries involving a metastable phase, A16Fe, which is isomorphous with A16Mn, are obtained. In particular a metastable eutectic, L -+ A1 + A&Fe occurs. Transitions of microstructures involving
Fig. 6. Calculated Al-rich portion of the stable Al-Fe phase diagram is shown by solid lies. If AI,Fe is absent, a metastable phase diagram (dot-dashed lines) involves a eutectic L+Al+A16Fe. The To for the solidification of A1 solid solution is shown dashed. The solvus curves are omitted for clarity. MURRAY [1983a].
A13Fe to those involving A&Fe have been observed with increasing solidification speed by ADAM and HOGAN[1972] and by HUGHESand JONES [1976]. The competitive growth kinetics of the two can also be analyzed using the stable and metastable phase diagrams. For local equilibrium, whether stable or metastable, the chemical potentials of the components for the liquid and solid are equal across the interface. In Table 2, however, another situation is described in (iv), and relates to a situation where chemical potentials can not be approximated as being equal across an interface growing at a high rate and large supercooling. These rapid growth rates can trap the solute into the freezing solid at levels exceeding the equilibrium value for the corresponding liquid composition present at the interface. Thus the chemical potential of the solute increases upon being incorporated in the freezing solid in a process called solute trapping.This increase in chemical potential of the solute across the interface must be balanced by a decrease in chemical potential of the solvent in order for crystallization to occur; Le., to yield a net decrease References:p . 830.
686
€ Biloni I .and WJ.Boerringer
Ch.8, $ 3
in free energy (BAKERand CAHN[19711). The free energy change during solidification, AG, is given by
AG = [(d - &)(1-
G)- (d - d)c;],
(20)
where p$ and p! are the chemical potentials for species A and B in the solid, andpt and p: are the chemical potentials in the liquid. These potentials are functions of the temperature and solid or liquid composition (C,’ or C,‘) at the interface during solidification. Despite the loss of interface equilibrium during rapid solidification, the free energy functions of the solid and liquid phases and their associated chemical potentials can be used to define the possible range of compositions that can exist at the interface at various temperatures. This restriction is obtained by the requirement that AG be negative. Figure 7 shows the region of allowable solid compositions at the interface for a fixed liquid composition, C,*, at the interface as a function of interface temperature (BOETTJNGER [1982]). Such allowable regions can be calculated from a thermodynamic model of the system of interest.
3.2. Tocurves For any selected pair of liquid and solid compositions, a thermodynamictemperature can be described that is the highest temperature where crystallization can occur as shown in fig. 7. However one often considers a limiting case, called partitionless solidijication, which is favored at very high solidification rate, where the composition of the solid formed at the interface, Cs*,equals the composition of the liquid at the interface, CL. The To temperature is the highest temperature where this can occur (APTEKARand KAMENETSKAYA[1962]), (BILONIand CHALMERS [19651). This is the temperature where the molar free energies of the liquid and solid phases are equal for the composition of interest; i.e., the temperature where AG = 0 for C,’ = C,* in Equation (20). As illustrated in fig. 7, a To curve represents only part of the thermodynamic information available, when solidification occurs without local equilibrium. To curves exist for the liquid with stable or metastable phases, and lie between the liquidus and solidus for those phases. In fact for dilute alloys the slope of the To curve is mL [(ln&)/(k,,- l)]. Figure 8 shows schematically,possible Tocurves for three eutectic phase diagrams (BOETTINGER [1982]). An important use of these curves is to determine whether a bound exists for the extension of solubility by rapid melt quenching. If the To curves plunge to very low temperatures as in fig. 8a, single phase a or crystals with composition beyond their respective To curves cannot be formed from the melt. In fact, for phases with a retrograde solidus, the To curve plunges to absolute zero at a composition no greater than the liquidus composition at the retrograde temperature, thus placing a bound on solubility extension (CAHNet al. [1980]). Experiments on lasermelted doped Si alloys seem to confirm this bound (WHITE et al. [1983]). Eutectic systems with plunging To curves are good candidates for easy metallic glass formation. An alloy in the center of such a phase diagram can only crystallize into a mixture of solid phases with different compositions regardless of the departure from equilibrium. The diffusional kinetics of this separation from the liquid phase frequently depresses the
Ch. 8, $4
687
Solidificaiion
F3
2 c
L
E
E
Q)
2
2
F
F
Composition
Composition
(a)
(b)
Fig. 7. The shaded regions indicate thermodynamically allowed solid compositions that may be formed from liquid of composition at various temperatures. The Tocurve gives the highest temperature at which partitionless 6& can occur. In (b) the To temperature plunges and partitionless solidification is impossible solidification (6,= for liquid of composition 6,BOET~NCER [1982].
solidification temperature to near the glass transition, TB,where an increased liquid viscosity effectively halts crystallization. In contrast, alloys with Tocurves which are only slightly depressed below the stable liquidus curves, as in fig. 8b, c, make good candidates for solubility extension and unlikely ones for glass formation. In fig. 8b the crystal structures of a and p are different and the Tocurve cross, whereas in fig. 8c the crystal structures are the same and the To curve is continuous across the diagram. At temperatures and liquid compositions below the To curves, partitionless solidification is thermodynamically possible. Ni-Cr and Ag-Cu are examples of the behavior in fig. 8b and c.
4. Nucleation 4.1. Nucleation in pure liquids Nucleation during solidification can be defined as the formation of a small crystal from the melt that is capable of continued growth. From a thermodynamic point of view the establishment of a S-L interface is not very easy. Although the solid phase has a lower free energy than the liquid phase below T,, a small solid particle is not necessarily stable because of the free energy associated with the S-L interface. The change in free energy corresponding to the liquid-solid transition must therefore include not only the change in free energy between the two phases but also the free energy of the S-L interface. From a kinetic point of view it is possible to arrive at the same result on the basis that the atoms at the surface of a very small crystal have a higher energy than the surface atoms of a larger crystal (CHALMERS [ 19641). Therefore, the equilibrium temperature at which atoms arrive and leave at the same rate is lower for a very small
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Fig. 8. Schematic representation of To curves for liquid to crystal transformations in three types of eutectic systems. BOET~WGER [1982].
crystal than for a larger one. Consequently for each temperature below T,, a solid particle can be in equilibrium with the liquid when its radius of curvature has a particular value, known as the critical radius. Because at higher supercooling there is more bulk free energy to compensate for the surface free energy, the critical radius decreases with increasing supercooling. On the other hand, at any supercooling, there exists within the melt a statistical distribution of atom clusters or embryos of different sizes having the character of the solid phase. The probability of finding an embryo of a given size increases as the temperature decreases. Nucleation occurs when the supercooling is such that there are sufficient embryos with a radius larger than the critical radius (HOLLOMON and TURNBULL
[1953]).
4.1.1. Calculation of the critical radius and energy barrier The change in the free energy per unit volume, AG, to form a solid embryo of spherical shape of radius, r, from liquid of a pure material involves the variation of the volume free energy and the surface free energy associated with the S-L interface and is given by
AG = AG,
LAT 4- 4.rry,,P 2 , + AGi = --4 m3-
3
Tm
where AG, is the change in free energy on solidification associated with the volume and AG, is the free energy associated with the interface, ysL is the S-L interfacial free energy, L is the latent heat per unit volume and AT is the supercooling. The critical radius, r*, occurs when AG has a maximum given by the condition, d(AG)/dr=0, as
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Figure 9, due to KURZ and FISHER [1989], gives a comprehensive picture of the variation of the free energy of an embryo as a function of its radius and AT: (a) At temperatures T greater than T, both AG, and AGi increase with r. Therefore the sum AG, increases monotonically with P. (b) At the melting point, AGv= 0 but AG, still increases monotonically. (c) Below the equilibrium temperature the sign of AG, is negative because the liquid is metastable while the behavior of AG, is the same as in (a) and (b). At large values of r, the cubic dependence of AGv dominates over AGi and AG passes through a maximum at the critical radius, r*. When a thermal fluctuation causes an embryo to become larger than P*, growth will occur as a result of the decrease in the total free energy. Nucleation in a homogeneous melt is called homogeneous nucleation and from eq. (21) the critical energy ofactivation for an embryo of radius r* is given by AG
16 3
=-T-
&T: L~AT~
(23)
The unlikelihood that statistical fluctuations in the melt can create crystals with a large radius is the reason why nucleation is so difficult at small values of the supercooling. Thus, homogeneous nucleation is only possible for high supercooling (on the and TURNSULL [1953] theory. However order of 0.25 T,) according to the HOLLOMON small contamination particles in the melt, oxides on the melt surface or contact with the walls of a mould may catalyze nucleation at a much smaller supercooling and with fewer atoms required to form the critical nucleus. This is known as heterogeneousnucleation. In fig. 10, homogeneous and heterogeneous nucleation are compared for a Bat catalytic surface and isotropic surface energies. For this simple case, the embryo is a spherical cap that makes an angle 8 with the substrate given by YCL
- Ycs = YSL cos
$9
(24)
where ‘yeL is the catalyst-liquid interfacial free energy and yCsthe catalyst-solid interfacial free energy. At a supercooling, AT, the critical radius of the spherical cap is again given by eq. (22), but the number of atoms in the critical nucleus is smaller than that for homogeneous nucleation as a consequence of the catalytic substrate. Indeed the thermodynamic barrier to nucleation AG* is reduced by a factor f(f3) to * 16 y3 T’ AG = - r m f ( e ) 3 L~AT~
where f(e) =
(2 + COS e)(i - COS e)’
4
If nucleation occurs in a scratch or a cavity of the catalytic substrate, the number of atoms in a critical nucleus and the value of AG* can be reduced even more. For a planar catalytic surface, the reduction in the free energy barrier compared to that for homoReferences: p . 830.
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T8
8
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a)
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Fig. 9. Volume, surface and total values of the free energy of a crystal cluster as a function of radius, r, at three temperatures: (a) T > T,, (b) T = T,, and (c) Te T,. KURZ and FISHER[1989].
geneous nucleation depends on the contact angle. Any value of 8 between 0" and 180" corresponds to a stable angle. When 8 = M O O , the solid does not interact with the substrate, A8) = 1 and the homogeneous nucleation result is obtained. When 8 = O", the solid "wets" the substrate, A8) = 0, and AG*= 0. As a result, solidification can begin immediately when the liquid cools to the freezing point. From the classical heterogeneous nucleation point of view, a good nucleant corresponds to a small contact angle between the nucleating particle and the growing solid. According to eq. (24) this implies that ycs must be much lower than ycL.However, in general, the values of yCsand ycLare not known and, therefore it is rather difficult to predict the potential catalytic effectiveness of a nucleant. TILLER [1970] pointed out that there is no clear insight into what deter-
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Fig. 10. Schematic comparison of (a) homogeneous and (b) heterogeneous nucleation of a crystal in a supercooled liquid. The interface energies are assumed to be isotropic and in @) the catalytic surface is assumed to be flat.
mines 6 and how it varies with (i) lattice disregistry between substrate and the stable phase, (ii) topography of the catalytic substrate surface, (iii) chemical nature of the catalytic surface and, (iv) absorbed films on the catalytic substrate surface.
4.13. Nucleation rate The rate of homogeneous nucleation, I, is the number of embryos formed with a size that just exceeds the critical value per unit time per unit volume of liquid. Similarly, the heterogeneous nucleation rate is considered per unit area of active catalytic site. To determine the rate of nucleation, it is necessary to find expressions for the number of embryos that have critical size and the rate at which atoms or molecules attach to the critical nucleus. By considering the entropy of mixing between a small number, N,, of crystalline clusters, each of which contains n atoms, and NL atoms of the liquid, an expression for the equilibrium number of clusters with n atoms can be obtained as,
References: p . 830.
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5 = exp( NL
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2),
where kB is Boltzmann’s constant, AGn is the value of AG, obtained from eq. (21) for a cluster of radius, r, containing n atoms. In particular, the number, Nn*,of clusters of critical radius, r*, is given by
where AGn* corresponds to the critical size cluster. If one can assume that an equilibrium number of critical nuclei can be maintained in the melt during the nucleation process, the homogeneous nucleation rate (#nuclei/time/ volume) is then given by
where K,is a constant involving the product of the number of atoms per unit volume and the number of atoms on a nucleus surface, and where v is the rate at which atoms can attach to the critical embryos. This is called the steady state nucleation rate. The value for v is usually thought to scale with DL/u:, where DLis the diffusion coefficient in the liquid and a, is the atomic jump distance. For metals, this attachment rate is fairly independent of temperature and so
K, typically has a value of lO4,/rn3s. For nonmetallic melts, where the diffusion coefficient in the liquid can depend strongly on temperature,
where K4 includes the pre-exponential factor for diffusion, and AG, is the activation energy for diffusion. For heterogeneousnucleation, similar expressions can be developed but is described per unit area of catalytic surface. An evaluation of eq. (30) shows that as the supercoolingis increased, I increases very rapidly at a critical supercooling in the range of 0.2 T,,, to 0.4 T,. Changes in the preexponential term in eq. (30) by orders of magnitude do not appreciably affect the calculated supercooling for sensible nucleation rates. This rapid rise in nucleation rate with temperature effectively defines a nucleation temperature. During rapid cooling of the melt especially to large supercoolings in glass forming alloys, atomic transport may be too slow to maintain an equilibrium number of clusters. This requires the examination of transient nucleation theory and effectively introduces a
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delay time into nucleation kinetics that can be important during glass formation or during devitrification as described by THOMPSON et al. [1983]. One of the major assumptions of the classical nucleation theory is that the free energy per unit volume and free energy per unit surface area are independent of the size of the embryo. Since the interface between solid and liquid is usually considered to be diffuse on the level of a few atomic dimensions (see § 5 ) , embryo that are a few atomic dimensions in radius cannot be described classically. This leads to a radius (or temperature) dependence of the surface energy as shown by LARSONand GARSIDE[1986] and SPAEPEN [1994]. PEREPEZKO [1988] has pointed out that if 8 approaches zero for a heterogeneous nucleation process, the thickness of the spherical cap can approach atomic dimensions, even when the cap radius is much larger, a fact that would also necessitate a nonclassical approach to heterogeneous nucleation.
4.2. Effect of melt subdivision When a volume of liquid metal is converted to an array of liquid droplets, large supercoolings prior to solidification are often obtained in many of the droplets. This fact leads to a method for the study of nucleation and to understand the supercooling often obtained in metal droplets created by atomization. For the study of nucleation this and CECH[1950] and continued most notably by method was pioneered by TURNBULL PEREPEZKO and ANDERSON [1980]. An example of the effect of supercooling on microstructure development is given for atomized AI-8ZFe alloys by BOETTJNGER et al. [1986]. If the nucleating sites contained within a given liquid volume are distributed randomly, the arrangement of nucleants among the droplets may be described by a Poisson distribution. For this case the nucleant free droplet fraction, X,is represented by X = exp(-mv) where m is the average number of nucleants per volume in the melt and v is the droplet volume. Based on experience with droplet emulsion samples (PEREPEZKO and ANDERSON [1980]), supercooling effects become measurable for size refinement below about 100 pm diameter and can become appreciable for powder sizes less than about 10 pm. This suggests that typical values for nucleant densities within the volume of a melt must be in the range from about 106-10g ~ m - The ~ . relatively sharp selection of a given X value (e.g., X = 0.9) with the droplet volume indicates the important role of size refinement in achieving large supercooling. Similar relationships can be developed for surface nucleant distributions. 4.3. Experiments on nucleation in pure metals
PEREPEZKO and ANDERSON[1980] have summarized the principal techniques for nucleation experiments conducted at slow cooling rates as shown in fig. 11. The most common corresponds to the dispersion of a pure metal into droplets within a suitable medium. For metals that melt below 50O0C, organic fluids with added surfactants are used to form the droplet dispersion. In addition to the isolation of nucleants discussed above, the surfactant probably plays a role in rendering some nucleates inactive. For systems with high melting points, molten salts and glasses have been employed. In both References:p . 830.
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LIQUID
a
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METHODS b
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Fig. 11. sample configuration for different supercooling methods. (after PERE.PFZKO and ANDERSON [1980]).
cases, independence and separation of droplets are maintained by a thin inert coating which must be non-catalytic to nucleation. Such dispersions of droplets can be thermally cycled in a DTA or DSC to determine the supercooling of the majority of droplets prior to nucleation. ~EREPEZKO [19841 summarizes the maximum supercoolings obtained by his coworkers and by previous work. Maximum supercoolings in the range of 0.3-0.4T,,, are typically obtained. Often maximum supercoolings are used to compute the liquidsolid surface energy using the homogeneous nucleation temperature. Such procedures can provide only a lower bound on the value of the surface energy, unless it is known that the nucleation is indeed homogeneous. It may be that heterogeneities still limit the observed maximum supercoolings in most metals. In practice the predictive capability of nucleation theory is limited by the unavailability of data for liquid-solid surface energy, which appears in the nucleation rate expression as y X 3 ,and by a lack of knowledge about the catalytic sites present in liquid metals and alloys. Improvements in the independent experimental determination of ysL for metals would be invaluable in allowing a clearer evaluation of theoretical interface models and more reliable nucleation and growth rate calculations. While the measurements of surface energy using techniques such as grain boundary grove experiments (for example GUNDUZ and HUNT [1985] and HARDY et al. [1991]) is preferable, the difficulty of these measurements has led to the development of theoretical models. SPAEPEN[1975] and SPAEPEN and MEYER [1976] estimate the surface energy based on
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the configurational entropy of a structural transition that enforces polyhedral atomic packing in the interfacial region. Their result is equivalent to the expression (32)
where N, is Avogadro’s number, V, is the molar volume and LY is a numerical factor related to the atomic packing (0.866 for fcc and hcp and 0.71 for bcc metals). KIMet al. [1988a], [1988b] have used this model to calculate homogeneous nucleation rates for the fcc and bcc phases for all compositions in Fe-Ni alloys. Due primarily to the higher a values for fcc as described above, nucleation of fcc phase is predicted to dominate in these alloys even for compositions where bcc has a higher liquidus. Indeed this is confirmed in their own experiments in 3-30 p m size powders as well as in the earlier work of m C H [19561. 4.4. AHOYnucleation
For a binary alloy, AG, in eq. (21) depends not only on the temperature but also on the composition of the liquid and of the solid nuclei. Thus for a given liquid composition, critical values of nucleus composition as well as size are required to determine AG*. If the surface energy andfie) are constants independent of cluster composition, the smallest value of r* (hence easiest nucleation) is obtained if the composition of the critical cluster maximizes AG,. For alloy nucleation the appropriate expression for AGv is obtained by dividing the expression given in eq. (20) by the molar volume of the solid. It is apparent from eq. (20) that AG, would be maximized for a composition of the solid where pt - pt = p: - p: ; Le., by a parallel tangent construction as shown in fig. 12. This maximum driving force condition has been proposed (HILLERT [1953], THOMPSON and SPAEPEN[1983]) to find the favoured nucleus composition for a given telliperature and liquid composition. In order to use this condition, one must have a thermodynamic model for the alloy of interest; Le., the free energy functions for the liquid and solid phases must be known. For simple analysis, regular solution models are often employed for the liquid and solid phases. More precise models that fit the measured phase diagram and other thermodynamic data are often available in the literature. In contrast, by including a simple model of the composition dependence of the surface energy, ISHIHARA et aZ. [1986] have shown that the critical nucleus composition can approach the bulk liquid composition at large supercoolings. Experience with alloy supercooling indicates that the composition dependence of the nucleation temperature, TN,reflects the composition dependence of liquidus temperature TL.For example in the Pb-Sb system, the supercooling results shown in fig. 13 reveal that TNfollows a similar trend to TLeven for different TN levels resulting from catalytic sites of different potency, Le., different surface coatings (RICHMONDet al. [1983]). The maximum AG, condition to determine nucleus composition has been used to successfully predict the composition dependence of measured values of TNin various alloy systems (THOMPSON and SPAEPEN [1983]). References: p. 830.
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? G
? G
? G
Fig. 12. Schematic free energy versus composition curves for liquid and solid at three tempecahues: (a) above the liquidus for composition xL, (b) at the liquidus, and (c) below the liquidus at an arbitrary nucleation temperature. The composition of a nucleus, x,, that maximizes the free energy change at the temperature given in (c) is given by the parallel tangent construction. THOMPSON and SPAEPEN[1983]).
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L 30
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0 10
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Atomic Percent Antimony Fig. 13. Summary of nucleation temperatures for Pb-Sb alloys which generally follow the liquidus slope (RICHMONDet al. [1983]). Supercooling trends at different levels are. produced by different droplet surface coating treatments.
4.5. Experiments on heterogeneous nucleation While it is clear that most nucleation processes are heterogenous, and the formalism of embryo/substrate interaction is useful, rarely have the heterogeneities been identified References: p. 830.
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with any certainty. The most common way to control the catalytic substrate is to use binary alloys, typically of the simple eutectic type, and to determine the nucleation temperature of each solid phase in the presence of the other. These experiments can be performed using entrained droplets (SOUTHIN and CHADWICK[1978], with analysis by CANTOR and D~HERTY [1979], MOORE etal. [1990], KIM and CANTOR[1991]) or the droplet emulsion technique used extensively by PEREPEZKO and coworkers. A typical thermal cycle for a collection of droplets employed for this purpose is shown in fig. 14. A composition on one side of the eutectic is first melted to determine the characteristic double peaked endotherm corresponding to eutectic ( a + p ) and primary phase Q3) melting. The sample is then cooled to form solid at TNand then reheated. After eutectic melting, heating is halted and the sample is equilibrated in the (liquid+p) two phase field. During subsequent cooling, the nucleation peak observed at Thet is then the nucleation temperature of a on j3. In Perepezko’s work great care was taken to confirm that p was indeed the catalytic surface and that a formed rather than a metastable phase. A summary of such measurements is given in Table 3 where AT,, is taken as the difference between the eutectic temperature and Th,. One simple idea put forward originally by SUNDQVIST and MONDOLFO [1961] was that of a nonreciprocal potency; Le., that if a is an effective nucleant for p, then p was an ineffective nucleant for a.Writing two expressions as in eq. (24),one for the contact angle, Oup, of a on a p substrate and one for the contact angle, e,, of p on an a substrate and eliminating yup,it is easy to show that
Thus if yBL>yuL, 0, c e, and p is a more effective substrate for the nucleation of a than a is for p. For example, values of liquid solid surface energy for both Pb and Sn phases in contact with eutectic liquid have been measured by GUNDUZand HUNT [1985] (132 and 56 using the grain boundary groove technique. They find that ysnL>ypoL erg/cm2, respectively). Thus the contact angle for nucleation of Pb on a Sn substrate is smaller than the contact angle for Sn on a Pb substrate. Thus both factors in the product, y3slfTO)in eq. (25), contribute to a lower activation energy for nucleation for Pb on Sn. This is consistent with the results in Table 3 for Pb-Sn. Recently HOFFMEYERand ~ P E Z K O [19881 have intentionally added heterogeneous sites to pure Sn. Using the droplet emulsion technique and by carefully changing the Table 3 Heterogeneous Nucleation in the Presence of the Primary Phase (F%REPEW(O [1994])
System Substrate AT,(oc) Substrate ATd’c)
Bi-Cd
Bi-Ag
Pb
Bi
27
61
Bi 28
Pb-Sn
Pb-cd
W-Sb
Pb-Ag
Pb 80 Sn 22
Pb
Pb 43 Sb
Ag
Cd
Ag
23
40
94
160
69 Cd 39
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NUCLEATION CATALYSIS THERMAL SCHEDULE
ENDOTHERM
I
A 90
8+
Fig. 14. Example of the intempted thermal cycle using the droplet emulsion technique that can be used to measure the heterogeneous nucleation temperature, T ,of CY phase on a p substrate. PEREPEZKO [1992].
droplet coatings to prove that spurious nucleation by the coating was unimportant, they measured the nucleation temperature of Sn in the presence of various stable oxides, sulphides, and tellurides. The Supercooling response of each class was identical even though the lattice disregistry within each class varied. This result indicates that a common surface reaction product may form in each class and catalyze the nucleation process. More details about heterogeneous nucleation and grain refining will be given in 5 11.
4.6. Formation of metastable phases by supercooling One of the most dramatic effects of large supercoolings prior to solidification is the possibility of forming metastable phases. An elegant yet simple example occurs for pure Bi (YOONet al. [1986]). A dispersion of Bi droplets was cooled from above the Bi melting temperature of 27 1°C to approximately 50°C where nucleation took place. Upon reheating the dispersion, melting of a metastable phase occurred at 174°C. If one examines the pressure-temperature diagram for Bi and extrapolates the melting curve for the high pressure Bi(II) phase to atmospheric pressure, one obtains a metastable melting point for Bi(1I) very close to 174°C. In addition performing the supercooling experiment with an increase in ambient pressure modified the melting point of the metastable phase in a manner consistent with it being Bi(II). Thus the formation of metastable Bi(II) occured rather than the stable Bi(1) phase at large supercooling. The bulk free energy change for solidification, AGv (eq.21) is always largest for the References: p . 830.
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stable phase. However in the context of the heterogeneous nucleation theory described above, a metastable phase may make a smaller contact angle with a particular catalytic site than does the stable phase. Thus the barrier for nucleation of a metastable phase may be smaller than the barrier for the stable phase. Of course one must always supercool below the melting point of the metastable phase in order for AG, for the metastable phase to be negative. Similarly metastable phases have been formed in alloy systems. In fact in the Pb-Sn system, by avoiding the nucleation of the stable Sn phase, the metastable Pb liquidus and solidus curves have been measured more than 80K below the P b S n eutectic temperature as shown in fig. 15 (FECHT and PEREPEZKO [1989]). When nucleation did occur in this supercooled state, a metastable phase was formed.
4.7. Grain size predictions in castings Perhaps one of the most elusive problems in the prediction of cast microstructure involves the grain size. THEVOS et al. [1989] and STEFANESCU et aZ. [1990] have found it necessary to postulate respectively the existence of a distribution of catalytic sites or a cooling rate dependence for the number of sites to predict accurately the grain size of castings. This subject will be discussed in more detail in 0 9 and 11.
5. Inteij4ace kinetics As mentioned in 03, local equilibrium is often a good approximation for interface conditions during growth of metals and alloys under casting conditions. Here we quantify the degree of nonequilibrium (interface supercooling) required to move an interface between a crystal and a melt at a given velocity. First we describe pure materials and then describe alloy effects focussing on the nonequilibrium incorporation of solute into a growing crystal at high solidification velocity.
5.1. Purematerials The nature of the S-L interface and the rate at which atoms attempt to join the crystal can have a decisive influence on the kinetics and morphology of crystal growth. For solidification of a pure material, the parameter which governs the atomic or molecular attachment kinetics is the interface supercooling, ATk, which is the difference between the thermodynamicmelting point and the interface temperature. The dependence of ATk on growth velocity is the subject of this section. The discussion of bulk supercooling and supersaturation and their effect on transport of heat and solute will be addressed in later sections on dendritic and polyphase growth. An interface can advance by two basic processes depending on the nature of the S-L interface. (i) Non-uniform (or lateral growth) advances the interface by lateral motion of steps that are typically interplanar distances in height. An atom or molecule can attach itself to the solid only at the edge of a step and as a result the crystal only grows by the passage of steps. The relationship between the lateral spreading rate and the effective growth rate normal to the interface is very sensitive to the number and formation
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300 250
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5,
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100
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d w Q
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P 0 20
40 XSn
-
60
80
100
Fig. 15. The stable PbSn phase diagram (solid line), including measured and calculated metastable extensions (dashed line) of liquidus and solidus curves. RCHT and PEREPEZKO[1989].
mechanism of new steps. (ii) Uniform or continuous growth advances the interface without needing steps, that is, growth can equally well proceed from any point. For a given material and supercooling, it is important to determine which type of growth occurs. The supercooling required for lateral growth at a given interface velocity, is typically much larger than that for continuous growth. Moreover, an interface that advances by continuous growth can propagate with a smoothly curved interface on a microscopic scale while lateral growth leads to facets. Whereas growth from the vapor or growth from supersaturated aqueous solutions is easily observed and usually occurs by lateral growth, such atomic scale observations are not usually possible for melt growth. Thus the nature of the S-L interface for metals is the subject of various models. In fact there is strong evidence that most metals freeze by continuous growth.
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5.1.1. Interface structure There are two approaches involved in the description of the transition in order from a liquid to a crystal across an interface. In the first, atoms are considered to belong to either the crystal or the liquid and the interface is considered to be sharp. The geometry of the surface that separates the two types of atoms may be smooth or meandering on an atomic scale. The former is called a facetted interface while the later is called a rough interface. In either case the atomic position of atoms in the crystal at the interface are considered to be in perfect crystallographic positions. The second approach includes the additional possibility of a gradual transition in atomic position from the randomness associated with a liquid to the perfect registry of the crystal. This later is called a diffuse intelface. Indeed if the thickness of the transition layer is of the same order as the roughness, the distinction between rough and diffuse is lost. A facetted interface provides the most difficult situation for growth while a diffudrough interface moves most easily. First we describe sharp interface structure models and later describe models based on diffuse interfaces. JACKSON [1958] has considered a sharp interface model and estimates the conditions when a facetted or a rough interface will occur between liquid and solid. Using a nearneighbor bond model and assuming that a random arrangement of atoms are added to an atomically planar crystal surface, he obtained an expression for the change in free energy as a function of the fraction, x, of the N possible sites occupied by “solid atoms” as
where a* = {+--}E,
(35)
R is the gas constant, L, the molar latent heat and ,f is a factor depending on the crystallography of the interface. This factor is always less than 1 and is usually greater than 0.5 and is largest for close-packed planes. This theory has been successfully used to classify and categorize growth morphologies (JACKSON [1971]). Figure 16 shows plots of eq. (34) for different values of a*.When a*e 2, the minimum value of AG occurs at x= 1/2; i.e., when half the sites are full. This represents a rough interface. Solidification will then occur by continuous growth because there are so many sites for easy attachment. In these circumstances, from a macroscopic point of view the S-L interface is, in general, non-faceted and may exhibit curvature on a scale much larger than atomic dimensions. When cy* >2, minima in x occur at small and large values of x indicating a interfacial layer with only a few filled (or empty) sites. This represents a smooth (facetted) interface. Solidification must then occur by layer or lateral growth. Planes that are not close-packed have smaller values of 6 and thus for some materials, can exhibit roughness while close-packed planes may be facetted. Planes that are rough will grow faster than the close-packed planes leading to a crystal growth shape composed only of slowly growing close-packed interfaces.
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OCCUPIED FRACTION OF SURFACE
sms x Fig. 16. Change in free energy of a monolayer at the liquid-solid interface as a function of the fraction, x, of sites that are occupied. The value of a*depends most strongly on the entropy of fusion and to a lesser extent on the crystallographic orientation of the face. JACKSON [1958].
Another approach to interface roughness comes from the consideration of how thermal vibrations affect the surface energy of a step on an otherwise facetted interface (CHERNOV [1984]). It is found that the step energy vanishes when LJRT,,, falls below a critical value of order unity. When the step energy goes to zero there is no barrier to surface roughening. This analysis gives the same qualitative result as Jackson's approach. Various statistical multilevel models of interface structure and Monte Carlo simulations (TEMKIN References: p , 830.
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[1964,1969], LEAMY and JACKSON [1971], JACKSONf19741)also indicate the importance of the ratio LJRT,,, presented in eq. (35). A common feature of all these models is that the roughness of the interface increases with decreasing LJRT,,,. Figure 17 shows and GILMER[1974]). simulations of an interface at several values of L,,,/RT, (LEAMY Molecular dynamics simulations have also been used to model interface structure. Because the position of each atom is computed as a function of time, the approach allows an interface to be diffuse. Indeed simulations show that the transition between liquid and solid for a material with a Lennard-Jones interatomic potential takes place over several atomic layers (BROUGHTONet al. [1981]). Such potentials are thought to approximate nondirectional metallic-like bonding. Figure 18 shows the calculated structure of successive (111) layers between the liquid and crystal. In another technique, density functional theory, superposition of ordering waves are employed to represent the local atomic density (OXTOBYand HAYMET[1982]). This method also shows the interface to be several atom layers thick. The expansion relating the free energy to the local density uses order parameters that describe the amplitude of the ordering waves through the interfacial region. This is in fact a generalization of the gradient energy approach of CAHN [1960] except that liquid structure factor data are used to determine the interface thickness and gradient energy coefficient.
5.13. Continuousgrowth The growth of a rough, sharp interface is called continuous or normal growth because the interface can propagate normal to itself in a continuous manner due to the large number of sites for easy atom attachment. To obtain the velocity-supercooling function for continuous growth of single-component melts, the growth velocity is typically expressed as a product of a factor involving the thermodynamic driving force for solidification and a kinetic prefactor involving the interface mobility:
V(q) = V,(T# - exp(AG/RI;)],
(36)
where is the interface temperature and AG is the Gibbs free energy change per mole of material solidified (defined to be negative for solidification).The bracketed term in eq. (36) represents a difference between the “forward flux” (liquid + solid) and the “backward flux”. The kinetic prefactor, V,(T),is the rate of the forward flux alone, and corresponds to the hypothetical maximum growth velocity at infinite driving force. Near equilibrium, the exponential can be expanded, resulting in a linear relation between velocity and supercooling:
that is often written as
v = PAT,, (38) where p is called the linear interface kinetic coefficient. [1900], F~ENJLEL [1932], In conventional modeling of interface kinetics (WILSON
Ch.8, $ 5
Solidification
705
Fig. 17. Surface configuration showing increasing roughness with decreasing values of LJRT,. After LEAMY and GILMER[1974].
TURNBULL [1962], JACKSON [1975]) it is assumed that the rate of the forward reaction, i.e., the rate at which atoms can jump across the interface to join the solid, is similar to the rate at which atoms can diffuse in the melt. Consequently, the kinetic prefactor is assumed to scale with the diffusivity in the liquid:
v, = fi D,/a,,
(39)
where a, is an interatomic spacing and f, is a geometrical factor of order unity. Since the shear viscosity, h,, of the melt is usually more readily measured, this scaling is often rewritten in terms of h, using the Stokes-Einstein relation, resulting in:
V, = f2 k,T/h,
(40)
where fi is another geometrical factor of order unity. Because of the temperature dependence of D, (or hs), V will first increase linearly, then go through a maximum and finally decrease as the supercooling increases. This relation has extensive experimental References: p . 830.
a.8, g5
H.Biloni and W:J. Boeninger
706
(111) face
Layer 7
Layer 8
Fig. 18. Trajectoriesof the molecules in layers parallel to a (111) interface going from solid (layer 5) into the liquid (layer 8) that were obtained by molecular dynamics simulations by BROUGIITON et al. [1981].
support for the crystallization of oxide glasses and other covalent materials (JACKSON et al. [1967]). However a prefactor that scales with viscosity has never been verified for monatomic melts such as liquid metals (BROUGHTONet al. [1982]). TURNBULL and BAGL~Y [19751 pointed out that for simple molecular melts in which the intermolecular potential is largely directionally independent, crystallization events may be limited only by the impingement rate of atoms with the crystal surface and therefore can be much more rapid than diffusive events. According to their collisionlimited growth model,
v, = f35,
(41)
where V, is the velocity of sound and f3 is another numerical factor of order unity. The important consequence is that V, is about three orders of magnitude greater than D,/a, for typical metallic melts, resulting in a correspondingly more mobile crystavmelt
Ch. 8, 3 5
Solidfiation
7M
interface. In addition no maximum is expected in the velocity-supercooling curve. The collision-limited growth model has been confirmed by the analysis of the velocities of rapidly growing dendrites growing into pure Ni melts (CORIELL and TURNBULL [1982]), by molecular dynamics calculations on Lennard-Jones systems (BROUGHTON et al. [1982]) and by pulsed laser melting experiments on Cu and Au (MCDONALDet al, [1989]). In the molecular dynamics simulation of growth, BROUGHTON et al. [1982] approximate their results by eq. (36) with V, given by (3kBTJqJ1’* where rn, is the atomic weight. This speed is the average thermal velocity and corresponds to the velocity at which atoms can strike the lattice sites. For Ni at its melting point, this velocity is 8.6 x 104cm/s which is only slightly less than the speed of sound for liquid Ni estimated by CORIELLand TURNBULL [1982]. For Ni at small supercoolings,this value of V, yields a kinetic coefficient, p, of 200cm/sK correspondingto negligible interface supercoolings under ordinary solidification conditions. In addition, RODWAYand HUNT[1991] using the Seebeck effect to measure the velocity-interface supercooling relation for Pb have obtained a value of 28 cm/sK for p that agrees well with that predicted by BROUGHTON et al. [1982] using the average thermal speed for this lower melting point material. Both values are orders of magnitude greater than would be predicted for interface kinetics governed by diffusive jumps.
5.1.3. Growth of a diffuse interface Another approach to modelling continuous growth comes from the consideration of a diffuse interface. MIKHEEV and CHERNOV [19911have estimated the kinetic coefficient for Pb using this approach. For a gradual transition in the density of atomic planes, the model predicts that the kinetic coefficient is proportional to the thickness of the transition width. Thus a diffuse interface propagates faster at a given level of supercooling than a sharp interface. Spatial and temporal fresuency data obtained from bulk liquid structure factor measurements, obtained with neutron scattering experiments on simple monatomic liquids, are employed to estimate the transition width and rate at which atoms can readjust into correct atomic positions. The estimate of the linear kinetic coefficient of 28 cm/sK for Pb is in excellent agreement with the data of RODWAYand HUNT[1991]. An earlier model for a diffuse interface (CAHN[1960], CAHNet al. [1964]) also shows the interface mobility to increase with interface thickness. This concept was also incorporated into models for lateral growth (see item 5.1A.) involving diffuseness of the step edge. In all cases, the diffuseness increases the mobility of the interface over what would be expected for a sharp interface. 51.4. Tho dimensional nucleationcontrolled growth If the interface is atomically smooth and free of any defects, the growth rate is limited by the nucleation of surface clusters. These clusters must form on the interface in order to create the necessary surface steps for lateral growth. The lateral spreading rate is assumed to occur quite rapidly at a speed determined by the continuous growth law described above. The classical theory of two-dimensional nucleation was developed by VOLMER and M A R D F ~ [1931]. The growth law (for the formation of cylindrical surface clusters) has the form:
References: p. 830.
708
H. Biloni and WJ.Boeninger
Ch.8, 85
where ye is the ledge energy per unit area and h is the step height. In the classical theory the constant of proportionality scales with the diffusion coefficient but it could scale with the speed of sound as does the prefactor for continuous growth. According to eq. (42) the growth rate is effectively zero at small supercooling and increases sharply at some critical supercooling. HILLIG[19661has found that the “constant” of proportionally may in fact depend on ATp When the number of nuclei is extremely high, the growth is better described by the continuous mechanism.
5.1.5. Growth by screw dislocations If one or more screw dislocations emerge at the S-L interface it is not necessary to nucleate new layers to provide the sites for lateral attachment. The step generated by each dislocation moves one plane each time it sweeps around the dislocation (FRANK [1949]). It was shown by HILLIG and TURNEXJLL [1956] that the distance between neighboring turns of the spiral is inversely proportional to ATk, and therefore the total length of step is directly proportional to AT,. For small supercooling, the rate of growth, therefore, will be V
- (AT,)’,
(43)
because the rate of growth per unit length of step should also be proportional to ATk (CHALMERS [1964]). Figure 19 schematically shows the growth laws for continuous, 2D-nucleation, and screw dislocation-assisted growth laws. Another source of steps at the S-L interface is the reentrant angle resulting from the emergence of twin planes at the S-L interface. This mechanism is found to be important for Si and Ge. See FLEMINGS[1974].
5.1.6. hnsition between continuous growth and facetted growth The model of CAHN119601, CAHNet al. [1964] predicts a smooth transition from lateral growth at low supercooling to continuous growth at high supercooling. The transition begins at a critical supercooling, AT;, and ends at TAT;. The critical value depends on the surface energy and the degree of diffuseness. For a material with a very diffuse interface, AT: may be so small that lateral growth is impossible to observe. An experiment showing the transition predicted by the theory has been performed by PETEVESand ABBASCHIAN [1986] on Ga. The transition to continuous growth occurs at a supercooling of about 4K and a growth rate of about 5 m d s for the (1 11) face. With increasing supercooling other models can also predict a transition to continuous growth. CHERNOV [1984] considers the nucleation of a disk on an otherwise facetted interface. If LJRT,,, > 2 the edge energy is nonzero and the surface is facetted and the growth will be lateral. At some value of increased supercooling, the driving force is comparable to the work of creating a new disk and the moving interface becomes rough. This kinetic roughening then permits growth to proceed in a continuous manner.
709
Solidification
SUPERCOOLING, AT,
__c
Fig. 19. Growth rate versus interface supercooling according to the three classical laws of interface kinetics.
5.2. Binaryalloys
BAKERand C m [1971] described the general formalism for the interface conditions for solidification of a binary alloy in terms of two response functions. One choice for these functions describes the interface temperature, q,and the composition of the solid at the interface, C,*. Neglecting orientation effects, these response functions can be written as follows: = T(V,Ci)- T,rK, and
c; = c;k(V, CJ,
(45)
where V is the local interface velocity. The functions T(V, C,*)and k(V, C,") must be determined by a detailed kinetic model for the interface. At zero velocity they are very simply related to the phase diagram: T(0, C,*) is the equation for the phase diagram liquidus and k(0, C,*) is the equation for the equilibrium partition coefficient b,which can depend on composition. The dependence of k on curvature is thought to be negligible (FLEMINGS[1974]). The possible forms for the functions T and k are constrained by the condition that AG < 0 as described in fig. 7 above. The kinetic partition coefficient can also depend on the crystallographic orientation of the growing interface. If a crystal References: p . 830.
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Ch. 8, $ 5
grows with a S-L interface having regions that are curved and facetted, the incorporation of solute into the crystal behind the facet can be quite different from the rest of the crystal. BRICE[1973] calls this the facet efect. This effect will be neglected in the remainder of this chapter. Several models for the dependence of the partition coefficient on velocity, eq. (43, have been formulated for continuous growth. The model formulated by BAKER119701 is quite general. Other theories predict that the partition coefficient changes monotonically from its equilibrium value to unity as the growth velocity increases. In these models the interface partition coefficient is significantlychanged from the equilibrium value, k,,,only when a dimensionless growth velocity, V/V, approaches one. Here V, is the diffusive speed for atom exchange between the crystal and the liquid. V, is a ratio of a diffusion coefficient, Di, for that exchange to the interatomic distance, 6.The value of this diffusion coefficient should be bounded by those of the liquid and the solid, but experimentally appears to be closer to that of the liquid. Using a value for a liquid diffusion coefficient typical of metals (2.5 x lo-’ cm2/s) and a length scale of 0.5 nm, V, should be less than 5 m / s . Experimental evidence (SMITHand Awz [1994]) suggests that V, is in the range between 6 and 38 m / s . Thus in ordinary solidification where V < 1 m / s the local equilibrium assumption is valid. The functional form of the models of AZIZ[I9821 and of JACKSON et al. [1980] for non-facetted growth is given by
k=
+ v/vD 1 + V/VD
&lJ
-
This expression is valid only for dilute alloys where the composition dependence of k, can be neglected. At a velocity of V,, the partition coefficient is the mean of the equilibrium partition coefficient and unity (see fig. 28a). Because eq. (46)has no dependence on composition, it cannot treat the situation shown in fig. 7b), in which partitionless solidificationis impossible for some compositions. The Aziz model has been generalized to treat non-dilute alloys by AZIZand KAPLAN[1988] and to include the possibility that solute drag dissipates some of the available free energy. In the continuous growth model of Aziz,the process that accomplishes the formation of the crystal structure from the liquid and the process that establishes the compositions at the interface are considered to be distinct. Indeed for metals, the former may only be limited by the rate at which atoms hit the interface (collision limited growth) whereas the latter process requires diffusive interchanges between liquid and solid to reach equilibrium partitioning. It is this separation of time scales that permits solute trapping at high velocity where there is insufficienttime for the diffusive rearrangementsbefore the solute is buried under additional crystal. For this reason the k(V) expression given above involves a diffusion coefficient, whereas the model assumes that the interface temperature equation can be obtained in an identical manner to a pure material, eq. (36), employing eq. (41). It is however necessary to use the value for AG obtained for alloys, eq. (20) rather than the simple expression based on ATk For dilute solutions, an analytical expression for the interface temperature can be obtained (BOETTINGER and CORIELL [1986]. In this case AG from eq. (20) can be given by
Ch. 8, $ 5
Solidification
711
Expanding eq. (36) for small AG gives V=-Vc(AG/RTi) where V, is taken here as the speed of sound. One can then summarize the two response functions for a flat interface as follows:
and
c; = kc;,
(49)
where mL(V) is given by
with k (i.e. NV)) given by eq. (46). For a curved interface, eq. (48) must include an additional term for the Gibbs-Thomson effect. Note that for V= 0, eq. (48) and (49) revert to the local equilibrium conditions. As k goes to unity, the term in the large brackets in eq. (50) changes the effective liquidus slope from the equilibrium value of mL to the slope of the To curve, which is mJ(lnkJ/(k,,- l)]. The last term in eq. (48) can be identified as a interface kinetic supercooling necessary to drive the formation of the lattice. Figure 20 shows a plot of the response functions obtained using eqs. (46), (48)-(50) superimposed on a phase diagram including the liquidus, solidus, and Tocurves. The plot gives the liquid composition at the interface and the interface temperature as a function of interface velocity for a fixed solid composition. This would be the case for directional solidification at different but constant velocities. The figure is based on a phase diagram for Ag-Cu with T, = 960°C, ki,= 0.44and mL=-5.6 K/at.%, C,*= 5 at%Cu, V, = 5 m / s , and V,=2 x lo3m / s . At zero velocity, the composition of the liquid lies on the liquidus curve, a situation that corresponds to local equilibrium. At intermediate velocities (about 1 4 s ) the composition of the liquid moves towards the composition of the solid but with an increased interface temperature. At higher velocities the liquid composition continues to approach the solid composition but the interface temperature drops below the To curve. At still higher velocities, where the partition coefficient is essentially unity, the temperature drops sharply with increasing velocity. This analysis provides a pair of thermodynamicallyconsistent response functions for dilute alloys. For concentrated alloys no simple expressions can be written because k, depends on composition. However, given a thermodynamicdescription of the liquid and solid phases and values for the two kinetic parameters V, and V,,the response functions can be calculated numerically (AZIZand UPLAN [1988]). Figure 21 shows the result for all compositions of an ideal binary alloy. Effective liquidus and solidus curves are shown for several interface velocities. These curves give the combinations of C i , C;, and Ti References: p . 830.
712
H. Biloni and WJ.Boetiinger
Ch. 8, $ 5
1
Fig. 20. Plot of interface temperature vs. liquid composition at the interface for a fixed solid composition, ( = 5 at 46). forming at the indicated velocities (in d s ) superimposed on the phase diagram.
possible as interface conditions at the indicated velocities. A related model, which includes solute drag (Azrz and KAPLAN [1988]), can also be reduced to a simple analytical expression for dilute alloys (AUZ and BOEITINGER [1994]). Extensive experimental research has been focused on testing these models (Kim et al. [1994]). These solute trapping ideas have been extended to ordered intermetallic phases by BOETTINGER and AZIZ [1989]. Rapid solidification experiments indicate that some compounds, which are normally ordered at the solidus, can be forced to solidify into the chemically disordered form of the crystal structure @Nom,et al. [19841, HUANG, et al. [19861, BOETTINGER et al. [1988a)], HUANG and HALL [1989]). The theory treats the trapping of disorder by a consideration of solute trapping on each sublattice of the ordered phase. At high rates, there is insufficient time to proportion the solute onto each
Ch. 8, $5
713
Solidification
MOLE FRACTION
A
0
Fig. 21. Kinetic interface condition diagram for ideal liquid and solid with pure component melting points as shown, pure component entropies of fusion equal to R, and VJV,= 100. Dashed lines: equilibrium liquidus and solidus labeled zero, dash-dot lines: kinetic liquidus and solidus at different interface velocity given as values of V/V, dotted line: To curve. Auz and KAPLAN[1988].
sublattice and a chemically disordered crystal can be formed. Often however, this disordered phase reverts to the equilibrium ordered phase during solid state cooling with a resultant microstructure consisting of a high density of antiphase domains. An approximate expression giving the solidification velocity at which the long range order parameter, q, at the liquid solid interface goes to zero is given by \-to
= v,[T,/T, - 11,
(5 1)
where T, is the melting point of the ordered phase and Tc is the temperature where the solid phase would disorder during heating if melting could be prevented; i.e. the metastable critical temperature for the order-disorder reaction. Clearly the closer T,is to the melting point, the lower is the velocity to obtain disorder trapping. Strongly ordered compounds cannot usually be disordered by rapid solidification techniques.
References: p . 830.
714
H. Bibni and U?J. Boettinger
Ch.8, 56
6. Solidijication of alloys with planar and nearly planar S-L intetfaces The analysis of the shape of the S-L interface on a scale larger than atomic dimensions begins with a consideration of plane front growth. Plane front p w t h is used extensively to control the solute distribution that remains in a solid after freezing, especially in the area of crystal growth of materials for electronic applications. It is also important as a starting point to understand the more complex interface shapes involving cellular and dendrite growth. To achieve planar growth, it is necessary to obtain a S-L interface that is both macroscopically and microscopicallyplanar. The former is achieved by controlled directional solidification with good furnace design and avoiding convection in the melt. The latter is achieved by avoiding interface instabilities due to constitutional supercooling (C.S). This section will summarize important work in these two areas.
6.1. General formulation of dmsion controlled growth In general, the transport of solute during directional solidification in the absence of convection with a planar S-L interface growing in the z-direction is described by the one dimensional diffusion equation
that must be solved for the composition C in the liquid and solid subject to conditions at the interface given by C, = k,C,
(53)
and
G
(54) D, -- 9. = V(C, - C,). dz dz The use of k, in eq. (53) corresponds to the local equilibrium assumption and can be modified for rapid solidification using eq. (46).
63. Solute redistribution during one dimensional solidiiieation Four limiting cases of the diffusional transport and the resultant solute distribution can occur during unidirectional plane front solidification of a rod (fig. 22). 6.2.1. Equilibrium freezing Here the S-L interface advances so slowly that diffusion in both phases maintains global equilibrium at all time (see 0 3). If the phase diagram predicts the solid to be a single phase for the bulk alloy composition, any difference in solid and liquid composition occurring during the solidification process will disappear after solidification is complete (fig. 22a). In more quantatative terms, a Peclet number (€‘e), defined by
Ch. 8, $6
Solidification
715
DIRECTION OF INTERFACE MOTION
Fig. 22. Solute distribution remaining in the solid after one-dimensional solidilication: (a) complete diffusion in solid and liquid; (b) complete mixing in liquid-no solid diffusion; (c) diffusion in liquid only; (d) partial mixing in liquid, including convection.
p = - 4lv e D '
(55)
for each phase, especially the solid, must be much smaller than unity, where 4 is the length of the sample. Only under unusual circumstances can the lever rule be expected to apply during solidification. One such case is Fe-C growing at very low velocity because of the high solid state diffusivity of C in Fe. In substitutional alloys, equilibrium freezing does not occur in any practical casting or crystal growth situations. 6.2.2. Complete liquid mixing, with no solid diffusion The conditions considered in this limiting case are: i) complete mixing of the liquid; ii) no solid state diffusion; iii) constant and equal solid and liquid densities; iv) local equilibrium at the S-L interface. The assumption of complete liquid mixing and no solid diffusion is often made because diffusion in the liquid is typically orders of magnitude faster than in the solid. In this case eq. (52) is replaced by a statement of solute conservation. When a fraction of the rod, fs, is solidified, the concentration of the remaining liquid is found to obey the following differential equation
which is easily integrated for constant k, to
c,
= co(l- fs)ko-',
(574
where C, is the nominal alloy composition. After freezing is complete, this process leaves a solute distribution in the solid (fig. 22b) given by References:p . 830.
716
H.Bihni ana‘ WJ.Boettinger
c,
= koC0(l-
fSy,
Ch. 8, $ 6 (5%)
where now fsis taken as the fractional distance down the rod. Indeed in this model, the rod geometry with a planar interface is not a necessary assumption if the GibbsThomson effect can be neglected, and the equations are valid for any interface shape as long a s h is taken as volume fraction. The temperature of the interface can be simply obtained by substituing eq. (57a) into eq. (18). GULLIVER [1922], HAYES and CHIPMAN [1938], SCHEIL [1942] and PFANN[1952] all developed the same type of equation, also called the “nonequilibrium lever rule”. Equations (57) can only be considered a limiting case since most real systems have at least some solid state diffusion and some level of incomplete liquid mixing. For alloys containing a eutectic (see 0 8) the liquid composition may reach the eutectic composition. In this case, the remaining liquid will freeze as a eutectic mixture. For cases when depends on composition, eq. (56) can be numerically integrated.
6.2.3. Solid diffusion during solidification BRODYand FLEMINGS [1966] were the first to account for diffusion in the solid. They retained the assumption of complete liquid mixing, approximatedthe solute concentration gradient in the solid at the S-L interface and derived the following expressions:
and
Equation (58) considers the case where V is constant and eq. (59) considers the case where the growth is parabolic, i.e. V- fl’*.The parameter L Y ~is a measure of the extent of diffusion of solute in the solid and is defined as:
where t,=local solidification time, that in the simplest case, where V=constanC is given by V/&. BRODY and FLEMMGS[1966] also obtained numerical solutions of the solid diffusion equation. Equations (58) and (59) are good approximationsonly when l), the VZS is optimum for all of them.
6.3. Lateral segregation In the previous discussion we have assumed the interface to be planar. In the event that the thermal distribution of the crystal growth apparatus is not perfect, macroscopic curvature of the interface can develop. If convective mixing can be neglected, CORIELL and SEKERKA[1979], and CORIELL et al. [1981] have modeled the lateral segregation that will be present for a given shape. Their numerical and analytical results treat the segregation in terms of the distance that the interface deviates from planarity (ap),the sample width, the characteristic diffusion distance (DL/V) and the partition coefficient, k,,. The radial segregation is greatest when k, is small and when Sd(DL/V) >> 1. Detailed calculations of lateral segregation due to convection driven by longitudinal and radial gradients, typical of Bridgman growth upward and downward, are described by CHANGand BROWN [1983]. Experiments performed by SCHAEFER and CORIELL[1984] in the transparent succinonitrile-acetone system show the effect of radial gradients at a S-L interface. 6.4. Morphological stability of a planar interface In $6.2 it was assumed that the S-L interface was microscopically planar during solidification. Thus the composition profile induced in the solid varied only in the direction of growth. However even if the heat flow is controlled to be unidirectional and the isotherms are planar, a planar interface may be unstable to small changes in shape. Lateral composition variations can then be induced in the solid on a scale much smaller than the sample width. The morphological stability theory defines the conditions under
Ch. 8, 0 6
Solidification
721
which this instability can occur. Instability of the interface utimately leads to the development of cellular and dendritic structures. An excellent review of the stability theory has recently been presented by CORIELL and MCFADDEN[19!33]. 6.4.1. Theory The temperature and solute profiles for &< 1, as a function of the distance, z, from a planar S-L interface for growth at velocity V (constrained growth) are shown in fig. 25. A linear stability analysis was first applied to a growing sphere and later to the plane by MULLINSand SEKERKA [1963], E19641 and SEKERKA [1967] under the conditions of local interface equilibrium and isotropic surface energy. One of the first extensions of this approach came when CAHN[1967] treated anisotropic kinetics and surface energy. Many other assumptions of the original theory have been relaxed over the years as summarized by SEICERKA [1986]. The analysis begins by perturbing the shape of a planar S-L interface growing in the z direction by
z = S exp(a t + 27rh/A),
(66)
where 6 is the perturbation amplitude, A is the wavelength, and u is the growth (or decay) rate of the perturbation. The value for u is determined by solving the steady-state heat flow and diffusion equations with appropriate boundary conditions for small values of S (linear theory). The planar interface shape is stable if the real part of u is negative a
T CONSTITUTIONAL
TEMPERATURE
Fa.25. Temperature and solute profile for R, e 1 as a function of distance z measured from the S-L interface when constrained gmwth at a velocity V occurs for a binary alloy. The concept of constitutional supercooling is included. References: p. 830.
722
H. Biloni and WJ.Boettinger
Ch. 8, 96
for all values of A. The conditions giving the stability-instability demarcation (u=0) reduce to an equation with three terms corresponding to the three factors contributing to the overall stability of the interface, the thermal field, the solute field and the capillarity forces. The stability equation is
G - mLG,&
+ 4.n2T,r n2 ~
= 0.
The parameter G is the conductivity weighted temperature gradient given by
where K = (KL+ Ks)/2 is the mean of the liquid and solid thermal conductivities. The parameter G, is the composition gradient in the liquid, which for a planar interface moving at constant velocity, is obtained from eq. (63) and is given by
The parameter 6, can usually be set equal to unity. However 6, may deviate significantly from unity under rapid solidification conditions, where vh /2DL > 1. In general its value is given by
Technically eq. (67) is correct only when VA/2a is small, where a is the liquid or solid thermal diffusivity. This condition is almost never violated even during rapid solidification. See KURZ and RSHER [1989] for a complete description of this detail. If the left hand side of eq. (67) is positive, the interface is stable. The first term is stabilizing for positive temperature gradients; if the temperature gradient is negative (growth into an supercooled melt), this term is destabilizing. If a pure material is considered, this is the only possible destabilizing term. Thus a planar interface in a pure material is only unstable for growth into an supercooled melt. The second term represents the effect of solute diffusion in the liquid and, being negative, is always destabilizing. The third term, involving capillarity, has a stabilizing influence for all wavelengths, though its effect is largest at short wavelengths. This is the sort of stabilizing effect to be expected from surface energy which tends to promote an interface shape with the least area, namely a plane. Figure 26 shows a plot summarizing the stability of a planar interface for dilute Al-Cu alloys. Figure 26, shows the value of u vs. A for selected values of GL,V, and C,, (200 Wcm, 0.1 cm/s and 0.1 wt%Cu). Under these conditions a range of wavelengths have positive u,and are therefore unstable. The smallest unstable wavelength is usually referred to as the marginal wavelength and the wavelength with the largest value of u is
724
H.Biloni and WJ.Boettinger
Ch. 8, 36
called the fastest growing wavelength. For some velocities, u e 0 for all A, and the interface is said to be stable. Figure 26c shows a closed curve of A and V values for fixed GLand Co (200aWcm, 0.1 wt%Cu) for which u = 0. At values of V and A inside the closed curve the interface is unstable. Note that instability only exists over a range between two critical velocities. Outside this range, u e0 for all A and the interface is always stable. Also shown dashed is the wavelength that corresponds to the fastest growing wavelength. Figure 26d shows the two critical velocities as a function of C, for fixed GL (200 Wcm). The curves for the two velocities merge into a smooth curve for low C,. Instability occurs to the right of the curve. Figure 26b shows the critical velocities as a function of GL for fixed Co (0.1 wt%Cu). Note the insensitivity of the upper velocity stability limit to the value of GL. The upper and lower critical velocities may be approximatedby two different limiting cases: (i) If the interface grows at slow velocities, the capillarity forces are small and can be neglected; the stability criterion becomes:
G 2 mLGc.
(71)
E, in addition, eq. (68) is coupled with eq. (3), and VL e
4
3 2 1
0
I
i
0
0.1
0.2
I
0.3
0.4
0.5
0.6
0.7
0.8
CO (mol%) Fig. 33. Comparison of theory and measured dendrite growth velocities as a function of alloy composition for a fixed bulk supercooling below the liquidus of 0.5 K for succinonitrile-acetonealloys. CHOPRA et al. [1988].
kinetics for pure Ni and for Ni-B alloys by ECKLERet al. [1992]. The abrupt increase in velocity at high supercoolings is associated with the transition from solute controlled dendritic growth to thermally controlled growth brought about by the solute trapping (velocity dependent partition coefficient). The predictions of the theory for constrained growth have been compared by KURZ et al. [1988] to measurements of cell compositions for Ag-15 wt% Cu alloy samples prepared by moving electron beam melting performed by BOETTINGER, et al. [1987]. Figure 35 shows bars corresponding to the observed range of composition across cells of the Ag-rich phase as a function of growth velocity. The solid curve shows the results of the present theory. The curve marked IV/MS is the same theory without the inclusion of the velocity dependence of the partition coefficient. The curve BH is the older dendritic growth theory of BURDENand HUNT[1974a], [1974b] which uses the maximum growth rate hypothesis rather than the marginal stability analysis. It can be seen that the present formulation is closer to the observed experimental data.
Ch. 8,87
741
Solidij5catim
>
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supercooling AT (K) Fig. 34. Comparison of theory and m u r e d dendrite growth velocities as a function of bulk supercooling below the liquidus for pure Ni and two Ni-B alloys. Ecmw et al. 119921.
7.2. Cell and dendrite spacings
7.2.1. Numerical calculations of arrayed cell and dendrite primary spacings During constrained growth cells and dendrites form arrays with a characteristic spacing transverse to the growth direction called the primary spacing. To model the primary spacing, interactions with neighboring cells/dendrites must be considered. In a series of papers, HUNT[1990], Lu and HUNT[1992] and Lu et al. [1994] have performed finite difference calculations to study celVdendrite shapes and primary spacings during constrained growth. The diffusion equation for solute in the liquid is solved employing local equilibrium. More recently Lu et a2. [1994] use the non-equilibrium interface conditions, eq. (46) and (48), for similar calculations. Computations are performed for single cells and dendrites with radial symmetry in a cylindrical computational domain of various diameters. This geometry is used to approimate a hexagonal array of cells or dendrites with a primary spacing equal to the cylinder diameter. Two classes of solutions were obtained; those with rounded nearly hemispherical interface shapes and those with parabolic interface shapes. The authors associate these two shapes with cells and dendrites respectively. Side branching was not simulated in the dendrite calculations. Multicell calculations where also performed to examine the stability of arrays of cells and dendrites with various primary spacings. For a very wide primary spacing, only dendritic solutions were obtained and a unique References: p. 830.
742
Ch. 8, $7
H.Biloni and U?J. Boeninger
16 h
3 $
r
14
v
d
E 12
0
6
4
0.01
0.1
1 V (cmls)
10
100
fig. 35. Measured average composition (circles) of cells of the Ag-rich phase in a Ag-15 wt% Cu alloy as a function of solidification velocity (BOFXIWGERet uL [1987l). The bars at each velocity represent the range of composition observed for each profile measured by STEM.The three curves are: i) the predictions of dendritic growth theory with velocity dependent partition coefficient, (solid curve), ii) without the velocity dependent partition coefficient, curve labled IV/MS, iii) the older Burden-Hunt model of dendritic growth (labeled BH). KURZet ul. [1988].
value of tip radius was determined for a given set of growth conditions. This computational regime is thought to approximate isolated dendrites. Although the original work (HUNT[1990]) indicated that anisotropic surface energy was not necessary to obtain this unique solution, it was later determined that slight anisotropy was induced by the calculation mesh. Later work (L,u and HUNT[19921) confirmed the important role of anisotropy on the selection of the radius of the tip. The values of radius determined by these calculations can be approximated by the IV/MS model. For smaller primary spacings, both cell and dendrite solutions could be obtained for the same growth conditions. For each, solutions could be obtained only over a range of primary spacings. The physical processes that define the allowable range of primary spacings are shown in fig. 36. The minimum stable cell spacing is determined by overgrowth of one cell by the adjacent cells. The maximum cell spacing occurs when the cell tips become very flat and on the verge of splitting. For dendrites, the minimum
Ch. 8, $7
743
Solidification
OVERGROWTH
STABLE
TIP SPLllTlNG
OVERGROWTH
STABLE
GROUTH OF TERTIARY ARM
Fig. 36. Schematic illustration of primary spacing adjustment mechanism for cells and dendrites. Lu and HUNT [19921.
spacing is also determined by overgrowth. The maximum spacing is assumed to be twice the minimum spacing based on an argument regarding the transition of tertiary arms into new primary arms. An example of the range of primary spacing that could be calculated is shown in fig. 37 and 38 along with experimental data for slow and rapid solidification respectively. The existence of a range of spacings for cells for given growth conditions agrees with observations of ESHELMAN and TRIVEDI[1988]. The predictions of the dendrite tip radius from the N / M S theory presented above are also shown in fig. 38. It is seen that the primary spacing of the cell solutions is approximately twice the dendrite tip radius from the IV/MS value. The computations described above do not describe the conditions at the root of cells. The walls of adjacent cells never join and a thin layer of liquid persists far behind the cell tips. UNGARand BROWN[ 19851 have performed finite element calculations of deep cells using a composite coordinate technique that allows cell walls to fold over and join at the cell roots. In fact small drop-like structures are found to form at the cell roots that may be indicative of the pinch-off and periodic shedding of liquid droplets. Such shedding of liquid droplets may be the cause of spherical second phase particles that occur in some rapidly solidified alloys, BOETTINGER et al. [1988c].
7.2.2. Analytical expressions for primary spacings Analytical theories of primary spacings often yield a power law expression applicable to a range of solidification conditions not too close to the constitutional supercooling or absolute stability velocities that is given by
References: p . 830.
744
H. Biloni and WJ.Boettinger
Ch. 8, $ 7
h
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Velocity (pm/sec) Fig. 37. Comparison of the predicted ranges of celVdendrite primary spacings with experimental results: (a) succinonitrile-0.35 wt% acetone; (b) A1-0.34 wt% Si-0.14 wt% Mg. E, specifies the anisotropy of the surface energy. Lu and HUNT[1992].
Ch. 8, $ 7
745
Solidification
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References: p. 830.
746
H.Biloni and WJ.Boettinger
A, = AIG?Vn.
Ch. 8, $ 7 (93)
The result of the model by HUNT 119791 coincides closely with the expression derived by KURZ and F~SHER[1981] using quite different assumptions about the dendrite geometry. The values m = 0.5 and n = 0.25 were obtained by both with the constant, A,, being proportional to the fourth root of the alloy composition for dilute alloys. Thus for a given V and GL,the primary spacing is larger for concentrated alloys than for dilute alloys. Traditionally many authors have considered that experimental values of A, are [1974]; better correlated to the local cooling rate, G,V, such that m = n = 0.5 (FLEMINGS OKAMOTOet ul. [1975]; OKAMOTOand KISHITAKE [1975]; YOUNG and KU~KWOOD [1975]). On the other hand, in steels a broad discrepancy exists for the values of m and n, although many of the experimental studies have not been performed under controlled solidification conditions. JACOBI and SCHWERDTEFEGER l19761, controlling GLand V separately, obtained values of m = 0.25 and n =0.72. The complexity of modeling primary spacings is indicated by the fact that values of m and n fit to the numerical computations of Lu and HUNT[1992] depend on the anisotropy of the surface energy. Equation (93) is often used in conjunction with microstructural measurements to estimate the freezing conditions for various rapid solidification processing techniques. In view of the complexity of modelling A, and the disparity in experimental values even at slow cooling rates, such an approach should be used with great caution. 7.2.3. Secondary dendrite arm spacing During dendritic growth of cubic materials, the paraboloid-shape dendrite tip bulges laterally in the four (100) longitudinal planes containing the [lo01 growth direction. Each bulge then develops perturbations down the length of the dendrite shaft that become secondary arms as shown in fig. 39. The initial spacing of the secondary arms near the tip has been observed in transparent pure materials and alloys (HUANGand GLICKSMAN [198l], TRIVEDI and SOMBOONSUK [1984]). For both constrained and free growth, the spacing near the tip is approximately 2.5 times the tip radius over a broad range of growth conditions. Theoretical work by LANGERand MUELLER-KRUMBHAAR [19813 has also confirmed this relationship, which is related to the question of the tip stability discussed previously. However as a result of coarsening effects during solidification, the final secondary dendrite spacing in a fully solidified casting is usually much coarser than the one formed near the tip. The observed mechanism of coarsening is the melting or dissolution of smaller arms at the expense of larger arms. Through the Gibbs-Thompson effect, local differences in curvature give rise to slight temperature and/or composition variations along the liquid solid interface. Diffusion of heat and/or solute in response to these differences cause the dissolution of small arms and the growth of others effectively increasing the average spacing of the secondary arms. For dendritic alloys, the process is practically important because it sets the length scale associated with the microsegregation. Indeed, the interplay between microsegregation and the coarsening process has only
Ch. 8, $7
Solid$cation
747
Fig. 39. Superposition of time-lapse photographs of the growth of a succinonitrile dendrite. Sidebranch [1981]. evolution on the [ 100) branching sheets is evident. HUANCand GLICKSMAN
recently been examined (VOLLERand SUNDARRAJ [1993]). Coarsening in some alloys can be so extreme that the dendritic structure can be lost. For all but the most dilute alloys, the rate of coarsening is determined by the diffusion of solute in the liquid (VOORHEES [19901). Models were obtained for isothermal coarsening of the secondary arms held in the mushy zone by KATTAMISet al. [ 19671 and References: p. 830.
748
Ch.8, 87
H.Biloni and WJ.Boettinger
F~URERand WUNDERLIN[1977]. The latter obtained an expression for the secondary ann spacing, A,, given by
h, = 5.5(M*tf)1/3,
(94)
where
where C, is the final composition of the liquid at the base of the dendrite. When solidification is completed by a eutectic reaction, C, is equal to the eutectic composition. The numerical factor in eq. (95) depends on the details of the geometry of the coarsening process and the dendritic structure and should be viewed as approximate. For directional solidification the local solidification time can be approximated by
AT, tf = -
GLV’
where ATf is the temperature difference between the dendrite tip and the base of the dendrite. Thus for a given growth velocity and temperature gradient A, decreases with increasing composition. KIRKWOOD[1985] obtained the above result only when the cooling rate is small compared to mLCdtpAt larger cooling rates, the strict dependence,A, t:”, was not obtained and he ascribed the scatter in A, vs. tf data to this effect. In real solidification processes, coarsening takes place simultaneouslywith the subsequent increase in the fraction solid during cooling. MORTENSEN [1991a] presents a model that includes this effect.
-
7.2.4. Cell to dendrite transition
As the solidificationvelocity is increasedfor a given alloy compositionand temperature gradient, one observes a transition in structure: planar, cellular, dendritic, cellular, planar. The lower and upper transitions between planar and cellular structures axe given by the modified constitutional supemling and the absolute stability criteria, respectively (06.4). The transitions between cellular structures and dendritic structures at intermediate velocities are not well understood. Experimentaldata for the low velocity transition were obtained and RUTTER[1956] and CHALMERS [1964] and were fit by the expression by TILLER
- c&,,
(97) with a constant of proportionality that depends on crystallographic growth direction. [1981] proposed the condition Later KURZ and FISHER GLV”
Ch. 8, $ 7
749
Solidi$ccztion
which is k, times the critical value of GJV for the plane to cell transition given by eq. (77).
In experiments by SOMBOONSUK etal. [1984], it became clear that cells and dendrites could coexist at the same velocity and that the conditions required for the transition were I Indeed in Hunt's calcumore difficult to calculate. (See BILLIAand T R ~ D[1993]). lations described above, cell and dendrite structures were fqund to exist at the same velocity. Hunt postulated that the cell to dendrite transition might occur when the tip temperature for the cell becomes lower than that for the dendrite (fig. 40).This postulate also predicts an upper transition velocity from dendrites back to cells with increasing solidificationvelocity in the rapid solidificationregime. This upper transition is observed experimentally but has received no theoretical treatment. 7.3. Microsegregation
Microsegregation is the pattern of composition variation that remains in a solidified alloy. It includes the composition variation across cells or dendrites as well as the formation of other phases in the intercellular or interdendritic regions. One of the major goals of microsegregation analysis is the prediction of the volume fraction of eutectic or other secondary phases that may form between the cells and/or dendrites of the primary phase. The simplest approximation for the prediction of microsegregation during dendritic solidification uses the Scheil equation described in 56.2.2 where the volume under consideration is shown in fig. 41. With this approach no assumptions are necessary
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References: p. 830.
Ch. 8, 57
H.Biloni and E!J. Boettinger
750
I
I
zs DISTANCE t
COMPOSITION
zS
zt
DISTANCE z 1 I I
I
/
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Fig. 41. Conceptual model of cellulddendritic freezing showing: (a) portion of the phase diagram; (b) schematic temperatwe distribution in the melt; (c) distribution of solute within the liquid in the mushy zone; (d) representation of plate-like, unbranched dendrites showing the position of a characteristic elemental volume. (After BRODYand FLEMINGS 119661).
Ch. 8, 57
Solidifiation
751
concerning the geometry of the freezing solid until one desires to convert from fraction solid to distance in a cast microstructure. For example, the dependence of fraction solid on distance, rl from the center of a cell or dendrite with radial symmetry with a primary spacing, A,, can be obtained the expression, /
\z
(99) Other geometries require other relationships between fraction solid and distance. The predictions of the Scheil Equation establish an upper bound on the severity of microsegregation, for example, as determined by the predicted fraction of eutectic. Effects that alter the predicted microsegregation are: (a) the variation in cell or dendrite tip composition with growth conditions; (b) the degree of liquid mixing between the cells or dendrites; and (c) diffusion in the solid. All of these effects decrease the severity of microsegregation from that predicted by the Scheil Equation. One of the first treatments of (a) and (b) was performed by BRODYand FLEMINGS [1966] and BOWERet al. [1966]. They assumed complete lateral mixing behind the cell tips and matched the composition gradient in the growth direction with the temperature gradient using the local equilibrium assumption (without curvature). This yields a prediction of the cell tip composition, C[ . A mass balance behind the tips was then used to obtain their final result which can be rewritten as
with C," = C,( 1-a') where a' = DLGL/mLVC,,. other approaches can be used to obtain the celvdendrite tip composition along with the same mass balance. SOLARIand B ~ N [1980] I used the BURDEN-HUNT [1974a], [1974b] model based on the maximum growth rate hypothesis. In this work, C[ = C,( 1a ' + b') with b' = [Tm13'(k,- l)/mLDL Coli". Substitution of this value of C," into eq. (100) gives the S o w and BILONIprediction. Use of the N/MS dendrite model to obtain Cc would update this approach. The appeal of these approximate treatments is found in two limiting cases. If C l = C,, eq. (100) becomes the Scheil Equation. If CL = Cdk,, eq. (100) becomes C, = C, independent off,. This limit corresponds to planar growth where no microsegregation occurs. GIOVANOLA and KURZ119901 employ a method that uses the IV/MS prediction for the tip composition without assuming complete mixing between the dendrites near the tip region. They employ a polynomial to connect the tip composition to a composition well behind the tips where the complete mixing assumption is valid. The solid composition vs. fraction solid relation is determined by numerically solving algebraic equations. The most important feature of microsegregation models that employ dendrite tip kinetics is the alteration of the composition of solid that forms at f s = O . The Scheil equation implicitly assumes that no solute is built up in front of the dendrite tip. Thus the first solid to form has composition, %C,. Increasing levels of solute in the liquid at the References: p . 830.
752
H. Biloni and IJ.Boettinger
Ch. 8, 97
tip correspondingly increase the amount of solute in the solid along the central dendrite trunk leaving less solute to accumulate in the interdendritic liquid regions at high fraction solid. Thus, for example, the amount of eutectic that forms will be reduced compared to the Scheil prediction. Solid diffusionduring solidification also reduces the amount of microsegregation. One approach is to apply the one-dimensional solid diffusion analysis presented in 5 6.2.3 to a small volume in the mushy zone shown in fig. 41. The distance L, is then equated with A, or A, depending on whether the microstructure has fully developed secondary arms. When solid diffusion is added to the Scheil approach, one must know how the fraction solid changes with time. For a linear dependence of fraction solid on time, the term (1-fs) in eq. (100) was changed to (1 -f/( 1+a,&,))(see eq. 58) as was done by SOLARI and B I L ~ M[1980]. Another approach is to numerically solve the diffusion equation in the solid and liquid (e.g. BATTLEand PEHLKE[1990]). While this approach is more accurate for the treatment of solid diffusion, it ignores the dendritic nature of cast alloys and the details of the solute profile in front of a growing dendrite. Corrections due to solid diffusion are very important for extremely slow solidification or when solid diffusion is very rapid as in the case of interstitial solutes, as for C in steels. That the Scheil equation gives a good first approximation to dendritic microsegregation in the columnar zone of ingots has been proven, among other authors, by KATTAMIS and FLEMINGS[19651 for low alloy steels and WEINBERG and TEGHTSOONIAN [19721 for Cu-base alloys. CALVOand BILONI[1971], combining anodic oxidation techniques with electron microprobe measurements, obtained a clear map of the solute segregation together with quantitative measurementsof the Cu concentration in Al-1% Cu alloys. The corrections to the Scheil approach due to solid diffusion have been tested by BOWERet al. [1966] and FLEMINGSet al. [1970]. The fraction of eutectic as well as the composition at the center of the dendrite agrees well with predictions using this approach, In the rapid solidification regime, microsegregation profile must be measured in the TEM due to the small primary spacings. MASURand FLEMINGS[1982] measured the composition profiles across very fine dendritic cells of Al-4.5% Cu solidified at a very high cooling rate (2x lo5 Ws)and a solidification velocity estimated to be 0.6 m / s . PALACIOet al. [19851 and BILQNI[19831 analyzed their data using eq. (100) with values of C,*from the theory of BURDENand HUNT[1974a]. Figure 42 shows the measured and predicted variation of solid composition as a function of f s . The segregation profile described by this equation is in close agreement with the experimental results. Other features of microsegregationprofiles however have been noted especially in the rapid solidification regime. Flat solute profiles have been associated with cellular growth et al. [1987]. Solute rich regions at the at high solute Peclet numbers by BOETTINGER et al. center of cells have been observed by KATTAMIS [1970] and by BOETTINGER 119871. The recent computation method of Lu et al. [1994] may yield a better approach to understanding the details of microsegregation over a broad range of growth conditions. 7.4. Solidificationof ternary alloys
Practical alloys typically contain many components and a brief discussion of
Ch. 8, 97
753
Solidijrcarion
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solidification in ternary alloys is appropriate. A general view of solidification microstructures in a ternary eutectic system has been presented by MCCARTNEY et al. [198Oa], [198Ob] as shown in fig. 43. Depending on the alloy composition, various mixtures of primary phase, monovariant binary eutectic and invariant ternary eutectic can be expected. For the dendritic growth of a primary phase, the Scheil approach can be References: p . 830.
754
H. Biloni and U?J. Boetiinger
Ch. 8, 57
a d
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utilized, if appropriate, or a dendrite tip model constructed. The ternary (or higher order) phase diagram is important for either of these approaches. One requires information concerning the relationship between CG and CG for each of the i alloying additions. Often values are taken from the binaries, but in general the composition of one species in the solid may depend on the composition of other species as well. This information is given by the tie-lines of the phase diagram that are usually only available through a thermodynamic model of the multicomponent alloy of interest, rather than through experimental studies. Use of the Scheil approach to determine the solidification path, which is the composition of liquid (and thus solid) as a function of fraction solid, uses a set of equations
that are uncoupled if the hi (=Csi/C,) do not depend on the composition of the other species. In fact, if the kOiare constants, then for an alloy of initial composition C,,
c, = COi(l- fs)4i-'*
Ch. 8,§8
SoJidijication
755
for each component during primary solidification. Using the phase diagram, one then determines the fraction solid at which the liquid crosses a monovariant eutectic line. Solidification then follows this line and finally ends with the ternary eutectic. If the solidification path for primary solidification crosses a monovariant peritectic line, solidification switches to single phase growth of the new phase. Calculations of solidification path using full integration of phase diagram tie-lines with the Scheil approach (including solid diffusion) have been conducted by CHENand CHANG[1992]. For a treatment of the kinetics of the dendrite tip in a multicomponent alloy, Ivantsov solutions are obtained for each solute to determine the composition of each solute at the dendrite tip. The marginal stability criterion for the multicomponent alloy is applied to determine the tip radius as shown by BOBADILLA et al. E19881 and FLWPAZetal. [1990].
7.5. New approaches to modeling dendritic growth Another approach to modeling dendritic growth is the phase-field method (CAGINALP [1986], KOBAYASHI [1991] [1992], WHEELER et aZ. [1992], [1993b]). With this method the interface is treated as being diffuse and the transition from liquid to solid is described by an order parameter called the phase field. For a pure material, equations that govern the phase field and the temperature are solved over a domain of interest without applying boundary conditions along the liquid solid interface. The position of the interface is then determined from the solution as a surface with a constant value of the order parameter intermediate between a homogeneous liquid and solid. This method, which requires time consuming computations on a supercomputer, has succeeded in obtaining realistic dendritic growth forms for pure materials and alloys with no extra conditions required to specify the operating state of the dendrite tip, such as the marginal stability condition, eq. (85). The velocity and tip radius are naturally selected by the solution to the differential equations (WHEBLERet al. [1993a]). An example of a computed two dimensional alloy and BOETTINGER [19951). Presently, dendrite using this method is shown in fig. 44 (WARREN computations have been performed using finite difference methods and, due to high computation times, are limited to high supercooling of the order of LE:.
8. Polyphuse solid$cation The discussion of solidification is now extended to situations where several phases
are formed from the melt. Eutectic, peritectic, and monotectic solidification involve the freezing of an alloy at or near a special liquid composition. In a binary system, this special composition is defined thermodynamicallyas the liquid composition that can be in equilibrium with two other phases at the same temperature. This equilibrium can only occur at a single temperature in a binary system at fixed pressure. For the eutectic and peritectic cases, the liquid is in equilibrium with two solid phases and for the monotectic case, the liquid is in equilibrium with a solid and another liquid phase. The monotectic case involves alloys that exhibit a miscibility gap in the liquid phase.
References: p . 830.
Ch. 8, 8 8
H.Biloni and WJ.Boettinger
756
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I I
d 0.385 0.408 0.431 0.455 0.478 Fie. Simulation of an allov dendrite mwine into a suuercmled liouid usine the ohase-field method. The e 44.shadings show variation of composition (atomic fraction Cu) in the liquid and solid for parameters approximatand BOETTINGER [1995].) ing a N i C u alloy with 0.41 atomic fraction Cu. (WARREN W
P
-
1
8.1. Eutectic solidification
In a binary alloy, eutectic solidification converts a liquid simultaneously into two solid phases. Alloys near eutectic compositions are very important in the casting industry due to several characteristics: (i) low liquidus temperatures compared to the pure component melting points that simplify melting and casting operations: (ii) zero or small freezing ranges that effectively eliminate the dendritic mushy zone thereby reducing segregation and shrinkage porosity while promoting excellent mold filling; (iii) possibil-
Ch. 8, 8 8
Solidification
757
ities of forming “in-situ’’ composites. Among the most common eutectic or near-eutectic alloys of industrial importance are cast irons, Al-Si alloys, wear-resistant alloys and solders. In many other practical alloys that freeze dendritically, secondary phases are formed near the end of freezing by eutectic solidification. Eutectic solidificationinvolves the following stages: eutectic liquid is supercooled and one of the solid phases nucleates, causing solute enrichment in the surroundingliquid and sympatheticnucleation of the second solid phase. Repeated nucleation and/or overgrowth of one solid phase by the other produces a growth center that defines an individual eutectic grain. For many eutectics, solidificationproceeds by simultaneous growth of two interspersed solid-phases at a common liquid-solid interface. The solute rejected into the liquid by each phase is taken up by the adjacent phase particles. As solidification proceeds, spatial and crystallographic rotation of the solid phases together with competitive overgrowth of adjacent eutectic grains lead to a stable solidification front, with the surviving eutectic grains having, as much as possible, maximized the solidification rate, minimized the a-p interfacial energy and oriented the cy+ interfaces in the heat flow direction. The ability of one solid phase in the eutectic to stimulate nucleation of the other varies widely for different a+ phases and, in general, knowledge about the nuc[19651). Thus, leation process in eutectic solidification remains rather poor (MONDOLFO this section will treat mainly theory and experiments concerning steady state directional solidification, a method which has furnished, as in the case of dilute alloys, a large number of reliable results. Two excellent summaries of the earlier literature on eutectic solidification can be found in reviews by CHADWICK[19631 and HOGANet al. [19711. 8.1.1. Eutectic classification When a eutectic liquid solidifies, the resulting material generally consists of a dispersed two-phase microstructure that is approximately ten times finer than cells or dendrites formed under the same conditions. The exact arrangement of the two phases in the eutectic microstructure can vary widely, depending on the solidification conditions and the particular eutectic alloy being solidified. HUNTand JACKSON[1966] provided a simple classification scheme according to the interface kinetics of the component phases. A correlation was found between eutectic morphology and the entropies of fusion of the two solid phases, concepts discussed in 5 5.1. The classification refers to: non-facettednon-facetted eutectics (nf-nf); non-facetted-facetted eutectics (nf-f); and facettedfacetted eutectics (f-f). Very little is known about the solidification of f-f eutectics but extensive research has been performed on nf-nf and nf-f eutectics. KERRand W ~ A R D [1967] argued that the facetted or nonfacetted nature of the liquid-solid interface of the phases was better described by the entropies of solution of the individual phases because the (eutectic) liquid composition is generally quite different from the solid phase et al. [19731 examined a large number of eutectic microstructures compositions. CROKER and in addition found that the volume fraction of the solid phases in the eutectic structure and the growth velocity must also be considered to obtain a good classification scheme. 8.1.2. Non-facetted - Non-facetted eutectics Eutectic mixtures of two non-facetted phases tend to form regular microstructures References:p . 830.
758
H.Biloni and WJ.Bwttinger
Ch. 8, $ 8
consisting of either alternate lamellae of a and p or rods of a embedded in /3 that grow in a continuous edgewise manner into the melt. Experimentally, it is found that eutectics in which one phase has a very low volume fraction tend to grow in a rod-like manner. This can be explained on the basis that a rod-like structure has lower total a+ surface energy than a lamellar structure when the volume fraction of one of the phases is less than about 0.3. This assumes that the surface energies are isotropic. A 25% anisotropy of surface energy can stabilize lamellar structures at practically any value of volume fraction. In marginal cases, lamellar-to-rod transitions can occur for a given alloy by changing the growth conditions. Three-dimensional analysis of the eutectic structures shows that both phases are continuous in the growth direction over large distances that are called eutectic grains. A eutectic grain can be described approximately as two interpenetrating single crystals of the two component phases having almost constant crystallographic orientation. HOGANet al. [197 11 give a thorough discussion of eutectic grains and a summary of the orientation relationships between phases and the habit plane of the a+ interfaces. The orientation relationship and the orientation of the interfaces usually allow for a high degree of fit between the two crystal structures that minimizes the a+ surface energy. In lamellar or rod eutectic solidification, the two phases, a and /3. solidify side by side with an approximately planar and isothermal S-L interface, supercooled AT below the equilibrium eutectic temperature. During solidification of an A-B alloy, the A-rich phase rejects B atoms into the liquid and the B-rich phase rejects A atoms. The interaction of the diffusion fields in the liquid in front of the two phases gives rise to the term coupled growth, which is commonly used to describe eutectic solidification. At a given solidification rate V, the spacing of the lamellae or rods, A,, and the interface supercooling below the eutectic temperature, AT,are controlled by a balance between: (i) the necessity for lateral diffusion of excess A and B in the liquid just ahead of the S-L interface, which favors a small interlamellar or interrod spacing, and (ii) the necessity to create a+ interfacial area, which tends to favor large A, (fewer interfaces). Over the years the theory of lamellar or rod growth for the slow solidification velocities found in castings or in directional solidification when AEV/2D,a 1 has been developed by ZENER[1946], HILLERT[1957], TILLER [1958], JACKSON and HUNT[1966], MAGNIN and TRIVEDI [1991]. The situation for rapid solidification is treated below. The basis of the analysis is illustrated in fig. 45. The interface temperature at each interface point is controlled by the velocity at each point, the composition of the liquid at the interface at each point and the curvature at each point, The supercooling below the eutectic temperature, TE,for each point on the S-L interface (a-L and p-L) can be described by AT = ATk + ATD -I-AT.
(103)
The first term is the interface attachment kinetic supercooling and is usually neglected compared to the other terms. Thus it is assumed that each point on the a-L and p-L interfaces is at local equilibrium. AT,,is the supercooling below the eutectic temperature due to the local variation in composition from the eutectic composition. This local variation is approximated by solving the steady-state diffusion equation in the liquid for
Ch. 8 , 9 8
759
Solidijcation
(a)
B
.T t Q (0
2 ‘E
7 A
4
LAMELLAR WIDTH
- 10-LCM
Fig. 45. (a) Liquid composition (96 B) across an CY+ interface. (b) Contributions to the total supercooling (AT) existing at the S-L interface, AT,, A T , and AT, are the solute, mature and kinetic supercooliigs respectively. (c) Shape of the lamellar S-L liquid interface. HUNTand JACKSON [1966].
References:p. 830.
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H.Bi60ni and WJ.Boettinger
Ch. 8, 5 8
a planar a+ interface growing at velocity V and spacing A,. AT, is the supercooling due to curvature. It is then assumed that the total supercooling, AT, given by eq. (103) is constant across the interface. Thus across the interface, any variation in ATDmust be balanced by a variation in AT,, giving a constant value for AT= AT,+ AT, at each point of the interface. Due to the solution of the diffusion equation, AT, has a minimum value near the a-&L groove, and thus AT, has a maximum value there. The radius of curvature is, therefore, smallest near the triple junctions and leads to an interface shape similar to that shown in fig. 45c. JACKSON and HUNT[1966] have shown that the predicted interface shape agrees very well with the interface shape observed in the transparent hexachloroethane-carbon bromide model system. By averaging the composition deviation from the eutectic composition in the liquid in front of each phase and the interface curvature of each phase (which depends on the width of the phase and the angle the interface makes with the a-P interface), the total supercooling is found to be
AT = K,&V
+ &/AE,
(104)
where K, and K, are constants. For lamellar growth they are given by
and
with
The parameters m, and m are the liquidus slopes for the alpha and beta phases, C i is Y. the difference in composition between the solid phases, DL is the liquid diffusion coefficient, f , a n d 6 are the volume fraction of the solid phases in the eutectic, PEis a series function of the phase fractions and can be approximated (TRIVEDI and KURZ [19883) by
PE = 0.3383
(fafar'661.
r, and I', are the Gibbs-Thomson coefficients (surface energy/entropy of fusion per unit volume), and 6, and Os are the angles that the alpha and beta interfaces make with a plane perpendicular to the a+ interface. Because a balance of tensions must exist at the adE-,!,triple point, the solid-solid energy exerts its influence through its effect on these angles. A low value of yapleads to small values for 0, and ea making ATc small.
Ch. 8, $8
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Solidijication
Very similar expressions are obtained for rod eutectic growth. Equation (104) is shown schematically for different growth rates in fig. 46. It can be seen that AT is biggest for large lamellar spacings because diffusion is difficult, and also for small spacings where curvature effects are dominant. Clearly, the values of A, and AT are not fixed uniquely by V, yet in experiments it is well established that the value of A, generally decreases with increasing V. Hence an additional condition is required to specify the operation point on each A, vs. AT curve. The simplest additional condition is obtained by assuming that growth occurs at the minimum AT for a given V or, equivalently, a maximum V for a given AT. This condition is called the extremum condition. Using this condition, &V = K,/K,,
(109)
AT2/V = 4K5K6.
(110)
and
120
I00
80
Y me
-
o 60 X
!-
Q 40
20
1 0 0
10
20
30
40
50
60
70
80
90
100 110
120 130
140
A€ *)Lm Fig. 46.Interface supercooling,AT, as a function of lamellar spacing, A , for different growth rates, Vusing eq. (104) for CBr&CI, organic eutectic. The vertical arrows show the theoretical minimum (extremum) and maximum spacings for stable lamellar growth. The experimentally observed range of spacings (hatch marks) and the mean spacing (filled circles) at the different velocities are shown.-S( and T R ~ [1988]). I References: p . 830.
762
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Many investigators (for example JORDAN and HUNT [1971, 19721, TASSAand HUNT [1976]) have found the average spacing and interface supercooling to be close to the values given by these equations using the extremum condition. However the use of the extremum condition is rather ad hoc and a more satisfying selection criterion has been sought for years. Indeed the discovery that the dendrite tip radii could not be described by a maximum growth rate hypothesis (see 07), led to further consideration of this criterion for eutectic growth. Experiments on eutectics (JORDANand HUNT[1972], SEETHARAMAN and TRIVEDI [1988]) have shown experimentally that a small range of spacings are observed for a given growth rate. They find that the minimum spacing is close to that given by the extremum condition but that the average spacing is somewhat larger than that given by eq. (109). Much research has been focused on defining the allowable range. The basic concepts were first proposed by JACKSONand HUNT[19661, who indicated that onIy spacings within a certain range were stable to fluctuations in the shape of the S-L interface as shown in fig. 47. They quoted unpublished work by J.W. CAHNthat argued that spacings smaller than that given by the extremum condition are inherently unstable and thus the extremum spacing is expected to be the minimum observed. This instability is due to the pinching off of an individual lamella or rod that locally increases the spacing (fig. 47a). That the extreemum value corresponds to the minimum spacing [1980], DAYTEand LANGER has been confirmed by extensive theoretical work GANGER [1981] and DAYTEet aZ. [1982].) Less sudden spacing adjustments can occur by the motion of faults (lamellar edges) perpendicular to the growth direction. A value for the maximum possible spacing at any velocity was first proposed by JACKSON and HUNT[1966]. This maximum spacing also follows a AiV = constant law. If the spacing exceeds the extremum value by a critical factor, the larger volume fraction phase develops a pocket that drops progressively back from the interface until growth of the other phase ultimately occurs in it (fig. 47b). They took this condition to occur when the slope of the pocket became infinite. For example, this condition yields maximum
(a>
(b)
Fig. 47. (a) Schematic illustration of the instability of lamellae with A, less than the extremum value. The lamella in the center will be pinched off with time. (b) The shape instability of the interface of one phase that occurs when the spacing becomes too large. A new lamella may be created in the depressed pocket. JACKSON and HUNT[1966].Figure taken from TRIVEDI and Kmz [1988].
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Solidijication
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values for the AiV constant of ten and two times the extremum value for volume fractions of 0.5 and 0.1 respectively. As a consequence of the formation of the new lamella, the local spacing is abruptly reduced by a factor of two. However, the careful experiments of SEETHARA~~IAN and -DI [1988] show that the maximum observed spacing is much smaller than this estimate (see fig. 46),giving an average spacing that is only -20% larger than the extremum value (or minimum stable spacing). Thus the maximum value of spacing occurs before the pocket depression attains infinite slope. While further research is required on this topic, the extremum value is often taken as a good approximation for nf-nf growth. Other stability issues of eutectic growth involving compositions different from the eutectic composition in 5 8.1.4 and 8 8.1.5. Convection in the liquid near the interface of a growing eutectic has been found to increase the value of A, (JUNZE et al. [1984] and BASKARAN and WILCOX[1984]). Flow parallel to the interface distorts the liquid concentration profile in front of the lamellae slightly and alters the diffusion controlled growth. The effect is greatest when the dimensionless parameter, GuA2DL,is large, where G, is the gradient normal to the interface of the fluid flow velocity parallel to the interface. Vigorous stirring is required to alter the spacing significantly.
8.1.3. Non-facetted-facetted eutectics The modeling of nf-f eutectics is quite important given the fact that eutectics of technological importance such as Al-Si and FS-C belong to this class. FISHERand KURZ [1979] and KURZ and RSHER [1979] summarize the main features of (nf-f) eutectic growth. When (nf-f) eutectics are compared with (nf-nf) eutectics, several characteristics can be noted: i) the degree of structural regularity is much lower and a wide dispersion of local spacing is observed. ii) for a given growth rate and fraction of phases, the average spacing and the supercooling for growth of a nf-f eutectic are much larger than for a nf-nf eutectic. iii) for a given growth rate, the supercooling and the spacing decrease as the temperature gradient is increased. No such effect is seen for nf-nf eutectics. Early investigations introduced interface attachment supercooling for the facetted phase h order to explain the increased supercooling. However S ~ andN HJ~LLAWELL [1975] and TOLOUIand HELMWELL [1976] showed that the kinetic supercooling of Si in Al-Si eutectic is too small to explain the increased eutectic supercooling. Indeed Si in Al-Si and graphite in Fe-C both contain defect planes parallel to the plate growth direction that enable easy growth (twins in Si and rotation boundaries in graphite). TOLOUIand HELLAWELL[1976] suggested that the large supercoolings were due to the difficulties of adjusting the spacing to minimize the diffusion distance. These difficulties are related to the anisotropy of growth of the facetted phase. Measurement of spacing and supercoolingon the model system camphor-naphthalene by FISHERand KURZ [1979] permitted impOrtant results to be obtained. The system exhibits two distinct eutectic growth forms: one regular and the other irregular. By assuming that the measured spacing and supercooling for the regular growth were given using the nf-nf theory with the extremum condition, the various materials parameters for References: p. 830.
764
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this system were obtained. When AT vs. A, was plotted for the relevant growth rate, the spacing and supercooling values for irregular growth fell on the same derived theoretical curve for regular eutectic growth but, with spacings much larger than the extremum value. Thus the coarseness of the structure is the cause of the large supercooling of growing nf-f eutectics. Indeed theoretical analysis of the A, vs. AT curves by MAGNIN and KURZ [1987] that relax the assumption of an isothermal interface made in the nf-nf theory, show a deviation from the nf-nf theory only at very slow growth rates where the constants in eq. (103) become functions of G,. Thus for growth at more normal speeds, the theory turns to an analysis of why the spacing is so big for nf-f eutectics. The general argument employed to understand why the average spacing of f-nf eutectics is large focusses on determining the stable range for eutectic spacings at a given velocity. Important contributions have been made by FISHER and KURZ [ 19801, MAGNIN and KURZ [ 19871 and MAGNINetal. [ 19911. For irregular eutectics the growth directions of different lamellae are not parallel. Thus as growth proceeds, the local spacing decreases between two converging lamellae and increases between diverging lamellae (fig. 48). For converging lamellae, when their separation decreases below the extremum value, one of the lamellae is pinched off, just as for nf-nf growth. For diverging lamellae, when the local spacing increases beyond a critical value, FISHERand KURZ [ 19801 have suggested that the facetted phases branches into two diverging lamella. The formation of the new lamella decreases the local spacing. The anisotropic growth kinetics
A br
Fig. 48. Proposed growth behavior of irregular eutectics, showing branching at A, and termination at A,. (a) Fe-graphite eutectic growth at V = 1.7 x lO-*pm/s. (b) schematic representationof solid-liquid interface during and KURZ [1987]. growth. MAGNIN
Ch. 8 , 5 8
Solz@kzfion
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of the facetted phase leads to what is termed branching-limited growth. Several criterion have been proposed to determine the maximum value of spacing where the branching takes place (FISHERand KURZ [1980] and MAGNINand KURZ [1987].) MAGNINand KURZ [1987] suggest that this branching instability occurs when the facetted phase interface develops a depression of some characteristic depth; e. g., when it drops below a line joining the two triple points for the lamella. The average spacing lies between the minimum spacing and the spacing that cause the branching instability. MAGNINetal. [1991] argue that the mechanism that establishes the minimum and maximum spacings remains undetermined and that the inherently nonsteady solidification of nf-f eutectics plays a fundamental role. Many issues remain to be studied in this area especially those regarding orientation relationships and their relationship to the branching mechanism.
8.1.4. Eutectic sells and dendrites In addition to consideration of the stability of the eutectic spacing, two other instabilities can influence the microstnuctureof alloys at or near the eutectic composition. These involve the addition of a ternary impurity or the deviation of the average composition from the thermodynamic eutectic composition in a binary. These will now be discussed in turn. A ternary impurity added to a binary eutectic can lead to a cellular structure (CHADWICK[1963]). The mechanism is similar to the cellular breakdown in single-phase solidificationtreated in 0 7; for a critical value of GL/V the average planar S-L interface of the eutectic structure can become unstable and the solidification front becomes corrugated. The cells (often called eutectic colonies) are quite large containing many (10-100) eutectic spacings with the lamellae curving to remain approximately normal to the liquid solid interface. Thus cells are most noticeable for nf-nf eutectics. BERTORBLLO and BILONI[1969] propose that the inception of the instability occurs at depressions in the interface due to eutectic grain boundaries or at fault terminations at the S-L interface. It might be noted that near the edge of eutectic colonies there is often a transition to a rod structure. If an excessive amount of a ternary element is added, the eutectic colony can actually evolve into a two-phase dendrite with secondary arms. In this case, ternary eutectic is usually found between the two-phase dendrite (SHARPand FLEMINGS [1974] (see also fig. 43).
-
8.1.5. Competitive growth soupled zone As a binary alloy with a composition different from the thermodynamic eutectic composition cools from the liquidus to the eutectic temperature, dendritic growth of the primary phase followed by eutectic growth of the remaining interdendritic liquid is expected. However, there is a range of alloy composition and growth rate (or interface supercooling) where it is possible to freeze these liquids as eutectic microstructures without dendrites. This range of conditions is known as the coupled zone. Pioneering investigations in this field were those of TAMMANN and BOTSCHWAR [19261 and KOPLER [1950] in organic systems, which established that at low or zero temperature gradients, the range of alloy compositions for coupled growth widened with increasing growth Rejerences: p . 830.
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velocity. Later, MOLLARD and FLEMINGS E19671 showed that the widening of the coupled zone was not restricted to high growth rates but, with a positive temperature gradient, could also be obtained at low growth rates with a high GL/V ratio. Solidification with a high value of GL/V suppresses the dendritic growth of the primary phase. Other milestones in the development of the knowledge of the coupled zone were the investigations and HUNT[1974c], TASSAand by HUNTand JACKSON [1967], JACKSON [1968], BURDEN HUNT[19761 and KURZ and FISHER[1979]. Taking into account the fact that the description of the microstructural transition from eutectic to eutectic plus dendrites has not yet been successfully modeled using perturbation analysis (JORDAN and HUNT[1971]; HURLEand JAKEMAN [1968]), the coupled zone width can only be obtained using a simpler approach that employs three concepts [1979]: For each overall liquid composition, (i) consider reviewed by KURZand RSHER all the growth forms possible, i.e., LY dendrites, dendrites, and eutectic. (ii) Analyze the growth kinetics of these forms and determine the interface (or tip) temperatures of the growth forms as a function of V, and possibly of GL. (iii) Apply the competitive growth criterion, e.g., that the morphology having the highest interface temperature for a given growth rate, or the highest growth rate for a given temperature will dominate. The range of temperatures and compositions, within which eutectic growth is fastest, is called the coupled zone and can be plotted on the phase diagram. The composition range of the coupled zone can also be plotted versus velocity because each value of interface temperature corresponds to a known value of the growth velocity for the dominant growth structure. For growth conditions where the eutectic is not dominant, the microstructure consists of a mixture of dendrites and interdendritic eutectic. Figure 49 shows an example of a competitive growth analysis for a system involving a f-nf eutectic (the phase is facetted). The kinetic curves are shown for LY dendrites, p dendrites and (planar) eutectic. The curves for the dendrites depend on the value of the
T!
4
TE
~
I-
Fig. 49. The origin of the coupled zone (hatched) is understood by considering the variation in eutectic interface temperature and dendrite tip temperatures for an off-eutectic alloy. The dominant microstructure for any composition at a given velocity (or supercwliig) is that which grows with the highest temperature (or fastest growth rate given by the solid curves in b). TRrVEDr and KURZ [1988].
Ch. 8 , § 8
Solidification
767
temperature gradient and this dependence leads to the decreased interface (tip) temperatures at low velocity and the widening of the coupled zone at high GL/V ratio. The skewed nature of the coupled zone about the eutectic composition for this f-nf system is due to two factors (KURZ and FISHER[1979]): the nf-f eutectic and the facetted phase dendrite require higher supercooling for a given growth rate than a nf-nf eutectic and a nonfacetted dendrite respectively. The former is due the branching difficulties already discussed and the latter is due to the fact that facetted phase dendrites usually grow as a plate or 2D dendrite rather than a paraboloid or 3-D dendrite. Diffusion of solute away from the tip region of a plate dendrite is more difficult and leads to increased supercooling. Thus for the alloy shown, one expects with increasing growth rate (or supercooling): eutectic, /3 dendrites (with eutectic), eutectic, and (Y dendrites (with eutectic). This kind of behavior leads to much confusion if a simplistic, purely thermodynamic view of solidification is employed and leads to difficulty in determining eutectic compositions by purely metallographic methods. For different alloy compositions, the various kinetic curves are raised or lowered leading to a description of the full coupled zone. For a system with a nf-nf eutectic the coupled zone is symmetric about the eutectic composition and the formation of dendrites phase is not observed on the “wrong side” of the eutectic. It is useful to note that near the growth rate where a microstructural transition from dendritic to fully eutectic structure takes place, the interdendritic eutectic will not have an average composition equal to the thermodynamic eutectic composition (SHARPand FLEMINGS [1973]). The methodology of competitive growth outlined above provides an adequate framework to understand the major features of the transition from eutectic to dendritic growth However more subtle variations in eutectic microstructure occur under conditions close to the transitions that require a more complete analysis of interface stability. JACKSON and HUNT[1966] observed a tilting of lamellae when the growth rate was suddenly increased. ZIMMERMANN et al. [1990] have observed oscillations where the widths of the Al lamellae vary while the A1,Cu lamellae widths remain fixed in the growth direction in Al-rich Al-Al,Cu off-eutectic alloys. GILL and KURZ [1993] observed another instability where the widths of both lamellae vary in the growth direction. KARMA[1987] succeeded in simulating these instabilities using Monte Carlo methods. Me related the appearance of the instability with increasing velocity to critical values of the concentration gradient in the liquid ahead of the interface.
.
8.1.6. Divorced eutectics When the liquid remaining between a primary dendritic phase reaches the eutectic composition, eutectic solidification usually occurs. Typically one observes the same eutectic microstructures already described between the dendrite arms especially if the fraction of liquid remaining between the dendrites when the liquid reaches the eutectic composition is large. If however the fraction of liquid remaining is so small that the width is comparable to the eutectic spacing, the characteristic two-phase structure may not be observed. It is easier for the second phase to form as single particle or layer between the dendrites. This occurs more often for a facetted second phase because coupled nf-f eutectics grow with larger spacings and hence require more space to References:p . 830.
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develop their characteristicmorphology. Thus the final solidified microstructure consists of dendrites or cells with interdendritic single phase. This microstructure is sometimes misinterpreted as resulting from a peritectic reaction in complex alloys where the phase diagram is unknown. 8.1.7. Rapid solidificationof eutectic alloys Rapid solidification produces a very rich variety of microstructures for alloys near eutectic compositions. At extremely high rates of solidification and depending on the thermodynamic structure of the To curves, glass formation or extended crystalline solubility is expected as described in 4 3.2, fig. 8. How microstructures and phase distributions evolve from the classical microstructures described above as the solidification velocity is increased has been the subject of intense investigation over the past 15 years. Eutectic theory - In 0 7, the general theory presented for dendritic growth included the modifications necessary to treat high growth rates; viz., modifications of the tip stability condition for high Peclet numbers (of order unity) and the inclusion of nonequilibrium interface conditions (solute trapping). For eutectic growth the Peclet number becomes large at relatively low velocity (-10 c d s ) where the effects of solute trapping are not too important. Thus TRIVEDIet al. [1987] recomputed the solute field in the liquid in front of a growing eutectic when the Peclet number is large while maintaining the local equilibrium assumption. The theoretical results are similar to those for nf-nf growth at slow velocity except that the function PEin eq. (105) depends not only on the volume fractions of the solid phases but also on the Peclet number, the shape of the metastable extensions of the liquidus and solidus curves below the eutectic temperature, and the partition coefficients. Also at high velocity the supercooling can become sufficiently large that the temperature dependence of the liquid diffusion coefficient must et al. [19841). be considered (BOETTINGER These considerations alter the hiV “constant” at high speed and the spacing vs. velocity relation (fig. 50) in a way that depends strongly on the equilibrium partition coefficients of the two phases, taken to be equal in the etal. [1987] analysis. Two cases are distinguished depending on whether (a) k, is close to unity or (b) k, is close to zero. In case (a), the eutectic interface temperature is found to approach the solidus temperature of one of the constituent solid phases as the velocity is increased. During this approach, the eutectic spacing actually increases with increasing velocity. Indeed eutectic solidification is replaced by single phase planar growth at high velocities. In case (b) the interface temperature can not reach a metastable solidus curve of either phase. The supercooling becomes so large that the temperature dependence of the diffusion coefficient has a major influence and the spacing decreases with velocity faster than predicted by a constant AgV value. In both cases there exists a maximum velocity for coupled eutectic growth. In case (a) the eutectic is replaced by single phase growth of one of the phases whereas in case (b) glass formation is possible if the interface temperature reaches the glass transition temperature where the melt viscosity (diffusion coefficient) plummets. In fact, the cases where the $‘s are close to zero are those that would exhibit plunging To curves and lead to glass formation as described in 0 3. The symmetry of the coupled zone has an impact on these considerations. Glass formation
Ch. 8, 5 8
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.08
.04
.02
0 1
10
100
V (cmls) Fig. 50. At high solidification velocities the AiV "mnstant" depends on the Peclet number and the relationship between A, and V is altered. The changes at high velocity depend strongly on the partition coefficients of the two phases (here taken to be equal). Curves are shown when DLis assumed constant and when DLdepends on temperature. TRWI et aJ. [1987].
often occurs in systems involving f-nf eutectics. Thus the composition with the smallest maximum growth rate for the eutectic (and hence easiest glass formation) may be shifted towards the direction of the facetted phase. Case (b) may also lead to the formation of a metastable crystalline phase if the eutectic interface temperature drops below the liquidus for such a phase. Experiments - It is clear from the above that solidificationvelocity plays a dominant role in controlling microstructure. To control solidification velocity at high rates, surface melting and resolidification employing a moving heat source have been used to create small trails of material that are solidified at speeds close to the scan speed (BOETTINGER et ul. [1984]). This technique is useful for speeds up to u / d where a is the thermal diffusivity and 6' is the diameter of the focussed electron or laser beam. For higher speeds surface melting and resolidification employing a pulsed laser or electron beam must be used. Maps giving the predominant microstructure as a function of speed and alloy composition are then produced. Figure 51 shows such a map for Ag-Cu alloys. Similar maps have been constructed for Al-A12Cu (GILL and KURZ 119941). In fig. 51 four microstructural domains are obtained cells/dendrites and eutectic microstructure at slow speed, bands at intermediate speed and microsegregation-free References: p. 830.
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MICROSEGREGATION FREE
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9 3
CELLS OR DENDRITES
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EUTECTIC 0
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EQUILIBRIUM SOLuBlLln LIMIT \
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COMPOSITION (wt % Cu)
Fig. 51. Experimental results for the variation in microstructure observed for AgCu alloys depending on solidification velocity. BOG~WGER et al. 119841.
single phase FCC at high speed. The boundary on the left is due to absolute stability and was described in $7. Eutectic growth ceases at -2.5 c d s generally following the description above except that an intermediate structure of bands is observed before single phase growth dominates at high velocity. This banded microstructure consist of thin (1 pm)regions paralZeZ to the growth front that alternate between cellular solidification and cell-free solidification. The general
Ch.8, $ 8
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character of this structure is due to details of solute trapping (MERCHANTand DAVIS [1990], BRAUN and DAVIS [1991], GREMAUDet al. [1991], CARRADet aZ. [1992]). Ordinarily, interface kinetics requires that the temperature of a planar single phase growth front decrease with increasing velocity. However over the range of velocity where the partition coefficient is approaching unity the interface temperature actually increases with increasing velocity (fig. 20). This reversed behavior is the basic cause of an instability that leads to the banded microstructure.At high speeds in the nondilute Ag-Cu alloys, the microsegregation-freestructures are caused by the fact the partition coefficient has gone to unity. In the Pd-Cu-Si system, maximum growth rates for eutectic and dendritic growth are also observed as shown by BOETTINGER [1981]). However in this case, partitionless growth is not thermodynamically possible and the liquid cools into the glassy state. Metushble crystallinephaseformation - An analysis of growth competition has also been highly successful at explaining observed transitions from microstructures involving stable phases to those involving metastable phases. In the Fe-C system the transition from gray cast iron (Fe-graphite) to white cast iron (Fe-Fe,C) has been extensively studied (JONES and KURZ 119801). This transition occurs at relatively slow speeds not normally considered to be rapid. However the same principles can be employed at higher rates for other alloy systems using appropriately modified kinetic laws. The competitive growth analysis must include the dendritic and eutectic growth involving the stable and metastable phases. Figure 52 shows the coupled zones for A1-A13Fe (the stable eutectic) and Al-Awe (the metastable eutectic) summarizing experimental and theoretical work et of several groups (ADAMand HOGAN[1972], HUGHESand JONES [1976], GREMAUD al. [1987]). The metastable A&Fe phase forms at an increased solidification velocity when the interface temperature drops below about 920K One of the most striking results of this diagram is the fact that alloys with compositions far on the Fe-rich side of either eutectic can form a microstructure consisting of primary A1 at large supercooling and increased velocity; Le., the A1 phase is the first to freeze. Indeed a determination of the powder size dependence of microstructural transitions in Fe-rich Al-Fe alloys from primary Al,Fe, to eutectic Al+Al,Fe, to primary A1 as the powder size decreases is consistent with increasing velocities (supercoolings) calculated for the different size powders (BOETTINGERet ai. [1986]). Similar microstructural transitions have been observed for other AI based-transition metal alloys. The ability to form a matrix phase of a nf (usually ductile) phase for alloys with a large excess of alloying additions has been a major motivation for alloy development through rapid solidification processing.
8.2. Monoteetic solidification In some metallic systems, the liquid separates into two distinct liquid phases of different composition during cooling. On the phase diagram, the range of temperature and average composition where this separation occurs, as well as the compositions of the two liquid phases are given by a dome-shaped curve that defines the miscibility gap. The maximum temperature of the miscibility gap is called the critical temperature. An example of a miscibility gap is shown in fig. 53 for the Al-In system. References: p . 830.
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CONCENTRATION [ wt. % ] Fig. 52. Theoretical coupled zones for the stable AI-A13Fe and metastable Al-AI& eutectics (hatched). The phases forming outside the coupled zones are also designated. The microstructure present at any value of interface supercooling and average alloy composition has been determine by a competitive growth analysis. TRIVEDI and KURZ [1988]. Subscripts P and N refer to plate (2D) and needle (3-D) dendrites respectively.
Even for alloys outside the miscibility gap, such as for Al-rich alloys in fig. 53, a consideration of the miscibility gap is important in developing an understandingof solidification microstructure. Under ordinary conditions, solidification of these alloys begins with the formation of dendrites of the A1 s.olid phase and enrichment of the liquid remaining between the dendrites with In until the composition reaches the edge of the miscibility gap. This composition (17.3 wt% In, fig. 53) defines the monotectic composition and temperature where the “reaction”, liquid L, +solid S, +liquid &. Formally this reaction is the same as a eutectic reaction except that on cooling, one of the product phases is a liquid, the liquid defined by the other side of the miscibility gap. At much lower temperatures this liquid usually solidifies in a terminal eutectic reaction &+S, +Sz. Some sulphide and silicate inclusions in commercial Fe-based alloys are thought to [1974]). Free-machining Cu alloys form by monotectic solidification (F~MINGS containing Pb also involve this reaction. Considerable research has been focused on directional solidification for fundamental reasons but also because of the potential for producing aligned growth of composites, or (with selective removal of one phase) thin and HELLAWELL [1981]). For this latter purpose it is most fibers or microfilters (GRUGEL
important to describe the possibilities of coupled growth of the SIand & phases from the
L,phase of monotectic composition. 8.2.1. Directional solidificationof monotecticalloys As in the case of eutectic solidification, a wide variety of microstructures can be produced by directional solidification of monotectic alloys. Lamellar microstructures are not observed in monotectic systems because the volume fraction of the L,phase is usually small. Three types of structures are observed. The first and most interesting and useful, typified by AI-In (GRUGEL and WLLAWELL [1981]), is a regular fibrous or composite structure that consists of closely packed liquid tubes of a uniform diameter embedded in a matrix of the solid phase. These liquid tubes solidify at much lower temperatures to solid rods of In by a divorced eutectic reaction. If the growth rate is increased, the distance between rods decreases, and the structure gives way to a second type of microstructure that consists of discrete droplets of L, embedded in a solid matrix. The third type,typified by Cu-Pb (LIVINGSTON and CLINE[1969]), is more irregular consisting of interconnected globules that take on some degree of alignment as the growth rate is increased. Although the L, tubes formed at a monotectic reaction are susceptible to References: p . 830.
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ripening and spheroidization during subsequent cooling, the droplet and irregular structures are not thought to form by coarsening (GRUGELand HELLAWELL [1981]). One of the most important considerations for understanding the different microstructures comes from a consideration of whether a stable triple junction can exist between L,, L,,and S, (CHADWICK[1965]). This condition can only occur if Ys,Ll + ‘YL~L~> Y s , b -
(111)
If the inequality is satisfied, regular fibrous structures can be obtained. The eutectic theory for rod growth can then be applied although some modifications are required to treat the increased diffusion in the L, phase (GRUGELand HELLAWELL [1981]). When this inequality is not satisfied, L, does not “wet” S,, and L, will tend to coat the interface between & and SI; Le., L, preferentially wets S, to the exclusion of L,. CAHN119791 calls this the pe$ecf wetfing case and during monotectic growth, the L, phase will form droplets in the L, phase just ahead of the growing S, interface. As growth proceeds, the droplets are pushed by the interface and the size of the droplets increase until they reach a critical size where they are engulfed into the growing SIsolid. The critical size for engulfment is determined by microscopic fluid flow around the droplet. As the solidification velocity is increased, irregular semicontinuousliquid rods can be partially engulfed in the solid as shown schematically in fig. 54. This irregular engulfment is believed to be the origin of the irregular globular microstructure typified by Cu-Pb. CAHN[1979] showed that in monotectic systems, the perfect wetting case should be expected if the temperature difference between the monotectic temperature and the critical temperature of the miscibility gap is small. Thus irregular composite structures are formed in these systems. When the temperature difference is large, perfect wetting does not occur, a stable triple junction can exist, and regular composite growth is expected. This idea was confirmed by the addition of a ternary element to a binary monotectic alloy. This addition altered the height of the miscibility gap and hence the wetting behavior (GRUGELand HELLAWELL [1981]). GRUGELet al. [1984] found experimentally that the border between systems with regular and irregular composite
Fig. 54. Schematic sequence to show liquid particle pushing, growth, and engulfment during irregular monotectic growth. GRUGELet al. [19841.
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structures occurred when the ratio of the monotectic temperature to the critical temperature is approximately 0.9. For regular fibrous growth, the spacing varies inversely with the square root of the et al. [1984] showed that the velocity” constant was about velocity. GRUGEL an order of magnitude larger for irregular growth than for regular growth. DERBY and FAVIER[1983] have presented a different model for the occurrence of regular and irregular structures similar to those used for irregular eutectics. KAMIOet al. [1991] have shown that the value of the temperature gradient has a large effect on the transitions between aligned growth and the formation of droplets. GRUGEL and HELLAWELL [198l] also examined whether composites could be grown for compositions different than the exact monotectic composition. They found that the dendritic growth of the SIphase could be suppressed by sufficiently large values of GL/V to permit planar composite growth just as for eutectic alloys. Attempts to grow composites with compositions on the other side of the monotectic (within the miscibility gap) failed due to convective instabilities. Reduced gravity experiments have been employed et al. [ 19921 to avoid these difficulties. by ANDREWS
83.2. Rapid solidification of monotectic alloys Some alloys whose phase diagrams do not contain a miscibility gap or a monotectic reaction form microstructures consisting of droplets embedded in a matrix of a primary phase after rapid solidification. If the liquidus curve has a portion where the slope is close to zero, a metastable miscibility gap lies just beneath the liquidus curve. In fact the temperature difference between the liquidus and the metastable critical point is proportional to the liquidus slope (PEREPEZKO and BOETTINGER[1983b]). Thus with the supercooling inherent in many rapid solidificationprocesses, alloy microstructure can be influenced by the presence of the metastable miscibility gap and its associated metastable monotectic reaction. The microstructureof rapidly solidified A1-Be alloys, which consists of fine Be particles in an A1 matrix, have been interpreted in this manner (ELMERet al. [1994]). In fact even some slowly cooled alloys can exhibit microstructurescharacteristic of monotectic solidificationeven though there is no apparent miscibility gap. VERHOEVEN and GIBSON[1978] showed that oxygen impurities raise the metastable miscibility gap in the Cu-I% system so that it becomes stable and produces droplet microstructures. 83. Peritectk solidification The phase diagram for the Pb-Bi system is shown in fig. 55a. If a liquid with 33% Bi is cooled, and global equilibrium could be maintained (see 5 3), the alloy would be composed of L + a at a temperature just above the peritectic temperature of 184°C denoted T,,and would be composed of single phase p just below T,. This gives rise to the notion of a peritectic “reaction” that occurs on cooling that is written as L + a +/3. However the diffusion required to accomplish this “reaction” during any realistic solidification process greatly reduces the amount of the p phase formed.
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PERCENT BISMUTH
S
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Fig. 55. (a) Pb-Bi peritectic phase diagram. (b) Concentration (% Bi) in the solid according to a Scheil model of solidification. FLEMINGS[1974].
83.1. Peritectic solidificationduring dendritic growth Under conditions where the a phase grows dendritically, the p phase will usually begin to form along the surface of the a phase. Although the /3 phase can be formed by three mechanisms, the most important during continuous cooling is the formation of p directly from the melt. The simplest way to treat this situation is to employ the Scheil approach in a small volume of the interdendritic liquid with the usual assumptions: local equilibrium at the solid-liquid interface, uniform liquid composition at each instant (temperature) and no solid diffusion. At small fraction solid, solidification of a phase occurs in the normal way with build up of solute in the liquid between the dendrites following the Scheil equation. When the liquid composition reaches 36%Bi, denoted C,, solidification switches from the a phase to the p phase. A new value of the partition coefficient given by the p liquidus and solidus must then be employed in the Scheil equation to follow the continued enrichment of the liquid composition in the component Bi. Often one must employ a concentration dependent partition coefficient for the p phase in peritectic systems that requires numerical solution of the differential form of the Scheil equation. The solid composition and the fraction of a and phases formed by this mechanism using the Scheil model are shown in fig. 55b. For Pb-Bi alloys the final solidification product is eutectic. One notes that using the Scheil approach, any alloy composition to the left of 36% will contain a phase in the solidified microstructure. Many systems involve a cascade of peritectic reactions, with solidification switching from phase to phase forming separate layers around the initial a dendrite. The second and third mechanisms for the formation of /3 are less important and more difficult to model. The geometry and connectivity of the L, a and p phases determine
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their relative importance. Both decrease the fraction of a phase from that predicted above. They have been referred to as the peritectic reaction and the peritectic tram$.ormation by KERR et al. [1974]. The peritectic reaction requires that all three phases be in contact with each other. This occurs in the vicinity of the liquid-cu+ triple junction and involves partial dissolution of the a phase and solidification of the fl by diffusion of solute through the liquid from the Gp boundary to the Gcr boundary (fig. 56a). HILLBRT [1979] gives an approximate analysis of this process. The third way that p phase can form, the peritectic transformation, involves solid state diffusion and the motion of the a+ interface during cooling as shown in fig. 56b. This mechanism is very important when the solid diffusion coefficient is large; e.g., for interstitial solutes such as carbon in Fe. Indeed the peritectic reaction in low carbon GROWTH RATE V DIFF.
L
Fig.56. Rritectic reaction and periwtic transformation on the side of an a dendrite. (a) in the pentectic reaction a second solid phase, B, grows along the surface of the primary phase a by diffusion through the liquid. (b) In the peritectic transformation, diffusion of B atoms through the already formed solid phase /3 occurs. (After HELERT [1979]). References: p . 830.
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steels (L+ 6-Fe + y-Fe) seems to go to completion; i.e., no 6-Fe is observed in these alloys. The diffusion problem that governs the motion of the a+ interface involves long range transport of solute from the liquid across the p phase to the a-/3 interface. At this interface, this flow of solute causes the ar phase to dissolve at the expense of the growing p phase. The analysis must use compositions for the interfaces that are given by the L + p and a +p two phase fields on the phase diagram. HILLERT [1979] has given an approximation for the growth of the fi phase by solid state diffusion. Due to the long range diffusion, the thickness of the p phase increases with the square root of time if the interface compositions and diffusion coefficients can be assumed to be independent of temperature. FREDRIKSSON and NYLEN[19821 have measured the fraction of L, a, and p phases as a function of distance behind the peritectic isotherm by quenching various alloys during directional solidification. The relative contributions of the three mechanisms are analyzed and compared to the measurements. In one alloy (AI-Mn) the p phase did not grow along the L-rr interface but grew independently from the melt.
8.32. Aligned peritectic growth Several attempts have been made to grow aligned composites of peritectic alloys by directional solidification. By increasing the temperature gradient or slowing the solidification velocity, it is possible to suppress the dendritic growth of the a phase during directional solidification in peritectic alloys with narrow freezing ranges. When cellular growth of the a phase occurs for alloys whose overall composition falls within the (a +p) field, a coarse aligned (a+p) two-phase structure can be formed (BOETTINGER [1974]; BRODYand DAVID [1979]). More interesting were attempts to achieve coupled growth of the a and j?phases to produce a fine two-phase structure. It was thought that if the GL/V ratio were large enough to suppress cellular solidification of the CY phase and force a planar solidification front that coupled growth of the two solid phases might be possible. This would also require that the composition of the liquid near the interface be maintained near Cp, a liquid composition from which both a and p could form. However coupled growth has never been observed in peritectic alloys. Instead, coarse alternating bands of ar and p form from the melt. (BOETTINGER [1974]; OSTROWSKI and LANCER[1979]; TITCHENER and SPITTLE([1975]). BOETTINGER [19741 analyzed the supercooling-velocity-spacingrelation for hypothetical coupled growth in a peritectic alloy and showed that it was intrinsically unstable. The formation of bands can be understood through an examination of the stable and [1979]). In fig. 57, metastable liquidus and solidus curves for the ar and p phases (HILLERT an alloy of the indicated composition can solidify to single phase a at a planar interface at temperature T,if the fi phase does not nucleate. Alternately the same alloy composition can solidify to single phase at a planar interface temperature at T,if a does not nucleate. Thus two steady state solidification situationsare possible. However each situation is precarious in that nucleation of the other phase can occur in front of the growing phase. The system is extremely sensitiveto minor growth rate fluctuations that leads to solidification that alternates between ar and p. This situation has been recently analyzed by TRIVEDI [1995].
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Fig. 57. Peritectic phase diagram with metastable extensions of a and p liquidus and solidus curves below and above the peritectic temperature respectively. For an alloy of the composition of the vertical line, planar steady[1979]. state growth of either a or p is possible at the temperatures T,and T2 respectively. After HILLERT
8.3.3. Rapid solidificationof peritectic systems As shown in fig. 57, the metastable liquidus curve of the a! phase below the peritectic temperature must lie below the stable /3 liquidus. However when compared to a eutectic system, the metastable liquidus is relatively close to the stable p liquidus. Thus there exists considerable opportunity for the formation of the CY phase directly from the melt at modest levels of supercooling at compositions where it is not expected. Whether or not this happens depends on the competition of nucleation and growth kinetics for the a! and p phases. An example of this kind of behavior is found in the classic experiments of CECH[19561 on solidificationof small droplets (3-30 pm) of Fe-3O%Ni alloys in a drop tube. This system contains a peritectic reaction, L + bcc + fcc. Experiments showed that References: p. 830.
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bcc was formed from the melt (and not by solid state transformation) at compositions with high Ni content where only fcc should have formed. In contradiction to the experimental facts, analysis of the dendritic growth kinetics of the competing fcc and bcc structures showed that fcc would be the favored product phase at all supercoolings for Fe30%Ni (BOETMNGER[1988]). This result is due primarily to the fact that the partition coefficient for bcc is much larger than for fcc. On the other hand, analysis of the nucleation behavior indicates that bcc is favored over fcc if the nucleation is homogeneous or if the contact angle on heterogeneities is greater than about 45" (KELLYand VANDERSANDE[1987]). Thus only nucleation can explain the observed bcc structure. In alloys containing a cascade of peritectic reactions, the close proximity of stable and metastable liquidus curves can also explain why one or more phases may be skipped over in the layer structure that coats the initial dendritic phase in rapidly solidified alloys.
9. FluidJEowand casting structure The flow of molten metal or alloy is an important consideration during casting. Although the requirements for mold filling especially in narrow cross sections have been considered for centuries, the importance of flow during the freezing process has been recognized much more recently. In this section we consider: i) the general origins of fluid flow in castings; ii) the development of casting macrostructure (grain structure); iii) the macrosegregationobserved in ingots and castings due to flow in the mushy zone; and iv) the formation of porosity and inclusions. Finally, the foundry concept of fluidity associated with the mould filling capability of alloys will be described.
9.1. Transport processes and fluid flow in casting There are many sources of material transport that can occur during solidification: i) residual flow due to mold filling, ii) thermal and solute driven bouyancy convection, iii) convection due to expansion or contraction upon solidification, iv) floating or settling of free crystals, v) dendritic breakage and transport., vi) convection driven by thermocapillary forces, vii) pushing of equiaxed crystals by the columnar solidification front, viii) external forces (pressure, rotation, magnetic fields). COLE[1971] and WEINBERG [1975] extensively studied the flow in the fully liquid part of castings. Weinberg's use of radioactive tracers techniques proved to be very sensitive and overcame the handicap of other experiments performed with transparent model liquids having much lower thermal conductivity than metals. Convection has the largest effect on thermal transport and macrostructure when the S-L interface (position of dendrite tips) is parallel to the gravity vector. Thus horizontal solidification is dramatically affected by convection due to the horizontal temperature gradients. Flows in metals and the heat transfer due to the flow can be reduced by the application of a magnetic field due to the induced eddy current that exerts a body force on the fluid. Rotation of ingots gives a effect similar to the application of a magnetic field; here the body force is a Coriolis force that deflects particles of fluid in a direction normal to the axis of rotation, and normal to the direction of fluid motion. On the other hand, an
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increase in heat transfer can be accomplished by vigorous fluid motion near the S-L interface. Rotation or oscillations of the crucible, a rotating magnetic field, or electromagnetic field interactions can be used for this purpose (COLE[1971]). On a more microscopic scale, fluid flow due to a variety of causes has also been associated with dendrite fragmentationand subsequentcrystal multiplicationduring solidand TILLER[1967]). In alloys the interification (JACKSON etal. [1966] and O’HARA action of natural or forced convection with the dendritic substructure can be rationalized as follows. When secondary branches form, they must grow through the solute-rich layer that exists around the primary stalk. The initial growth through this layer is slow. After the branch passes through the layer, it enters the bulk liquid of lower solute concentration and grows more rapidly. The result is a thin neck of the secondary branch near the primary stalk. Any slight increase in the local temperature or shear force due to local ~ portions of the dendrites. These crystals may be able to survive in fluid f l can~ detach other portions of the liquid and, if the thermal and constitutional conditions are appropiate, subsequently grow. This leads us into a discussion of ingot macrostructure. 9.2. Ingot structure
The classical representation of ingot macrostructure shows three distinct zones: the chill zone, which i s a peripheral region near the mould surface composed of small equiaxed grains, the columnar zone and a central equiaxed zone. Inside each grain a substructure of cells, dendrites, and/or eutectic exists. Fluid flow during solidification affects the origin and development of the three zones. Extensive research has been performed because of the important influence of macro- and microstructure of ingots and castings upon mechanical properties.
9.2.1. Chill zone The formation of the chill zone structure involves complex interactions of liquid metal flow, metal-mould heat transfer, nucleation catalysis and dendritic growth. CHALMERS [19641 suggested that chill zone grains could form by independent nucleation [19671, B ~ Nand I events or by a copious nucleation mechanism. BOWERand FLEMINGS MORANDO[1968] and PRATES and BILONI[1972] experimentally simulated the thermal conditions existing in the chill zone using thin samples filled quickly by a vacuum [1967] found a dendritic technique. By controlling the fluid flow, BOWERand FLEMINGS substructure in the chill grains and established that a grain muZtipZication (fragmentation) mechanism induced by melt turbulence during pouring was quite important. They used moulds coated with lampblack which drastically reduces the value of hi compared to an [1965] had earlier found chill uncoated Cu mould. In contrast BILONIand CHALMERS grains with a different substructure with uncoated moulds. F’redendritic nuclei with solute-rich cores were formed by partitionless solidification. BILONIand MORANDO [1968] used an identical device as BOWERand FLEMINGS [1967], but coated only part of the chill surface with lamp black (fig. 58). The region with lampblack had the same substructure as was previously observed by BOWERand FLEMINGS [1967], but the chill grains in the region without the lampblack were smaller and contained a predendritic References: p . 830.
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Fig. 59. Longitudinal section of an ingot poured from the bottom (shown to the left), after macroetch. The difference in grain size is due to mould walls with different microgeometries. The small columnar grains start at the asperities of an alumina mould coating presenting a controlled microgeometry; the very large grain started from a wall coated with a very smooth film of lamp black. MORALES et al. [1979].
favourably oriented and a texture arises. In fcc and bcc alloys, a preferred orientation is characteristic of the structure. Figure 60 (RAPPAZ and GANDIN [ 19931) schematically shows the competetive columnar growth of three grains. Some equiaxed grains that nucleate and grow in front of the columnar zone are also depicted which will be discussed later. The columnar grains on the left and on the right contain dendrites whose crystallographic orientations are nearly perpendicular to the liquidus isotherm. These dendrites grow with the same velocity V, as that of the isotherms. The grain in the middle of the figure, having a deviation of its crystalographic direction from the heat flow direction, grows with a velocity V, = V,/cos 8 that is larger than V,. According to the growth kinetic model of the dendrite tip for constrained growth (Q7), the faster growing (misoriented) dendrites are characterized by a larger tip supercooling. Thus the tips lie behind those of the better oriented grains. RAPPAZand GANDIN[1993] give the details of the elimination of the misoriented grains according to the convergence or divergence of the neighbouring grains and the ability or inability to form tertiary arms as shown in fig. 60. After elimination of misoriented grains, the growth of the columnar front in a casting can be modelled using a macroscopic heat code that solves eq. (7). This requires a fraction solid versus temperature (and other variables) relationship obtained from one of the various dendritic microsegregation models presented in section 7.3. FLOOD and HUNT [1987a] give one approach and the various methods are summarized by RAPPAZ and STEFANESCU [1988] and RAPPAZ[1989]. Several researchers have attempted to provide a more detailed numerical procedure to model the development of the columnar zone including the variation of the transverse References: p . 830.
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t"
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Fig. 60. Competing processes during directional dendritic growth: development of preferred orientation in the columnar region, formation of equiaxed grains ahead of the columnar front. RAPPAZand GANDIN[1993].
size of columnar grains described by CHALMERS [1964] and to deal with other issues of crystalline anisotropy, and texture formation. BROWNand SPITTLE[1989] and ZHU and SMITH[1992a], [1992b] used Monte-Carlo methods to model the effect of anisotropy. RAPPAZand GANDIN[1993] developed what they call a probabilistic model that includes an orientation variable for each grain and keeps track of the developing grain shape using a cellular automaton technique. These numerical methods have been able to be compute realistic grain macrostructures as well as crystallographic texture. As predicted by the model, the orientation distribution of the columnar grains narrows as the distance from the mould surface increases and the comparison with the WALTONand CHALMERS [1959] experiments is acceptable. Additionally, the selection of the columnar grains at the chill-columnar transition as well as the columnar extension is also succesfully predicted. This approach is also useful for modeling the grain competition in the grain selector during directional solidification (DS) for the production of modern turbine blades (RAPPAz and GANDIN [1994]). It must be remembered however, that the success of this approach depends strongly on the input of an accurate nucleation law that generally must be determined from experiments. Fluid flow can also affect the columnar zone structure. In conventional ingots,
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columnar growth may not be perpendicular to the mould wall if convection sweeps past the S-L interface due to horizontal temperature gradients. If convection is diminished through magnetic fields or mould rotation, perpendicular columnar growth can be restored (COLE[1971]). When columnar growth occurs in concentrated alloys with a low temperature gradient, a substructure other than cellular-dendritic can sometimes appear. Dendrite groups rather than individual dendrites comprise the solidificationfront. These groups have been called superdendrites, where apparently the normal coupling that establishes the primary spacing between dendrites becomes unstable, with certain dendrites grow ahead of their [1968]; FAINSTEIN-PEDRAZA and BOLLING [19751). neighbors. (COLEand BOLLING In aluminum-base alloys and under some conditions, another unusual structure can appear. Laminar grains grow from a single origin and form “colonies”. These structures have different names in the literature but most commonly are called feather crystals. Generally, they appear in continuous or semicontinuous castings and in welding [1980], [1983] processes. Occasionally they are observed in conventionalcasting. BIL~NI describes the current knowledge about this structure, which is open to further research.
9.2.3. Equiaxed zone Equiaxed grains grow ahead of the columnar dendrites and the columnar to equiaxed transition (CET) occurs when these equiaxed grains are sufficient in size and number to impede the advance of the columnar front. Evidence for the collision of the columnar front with equiaxed grains can be found in the work of BIL~NIand CHALMERS[1965] and BILONI[19681. When this collision occurs, the heat flow direction in the equiaxed grains changes from radial to unidirectional and a modification of the dendritic substructure can be observed by careful metallography in the solidified samples. The situation to be modelled therefore comsponds to that shown in fig. 60. The major challenges to predict the columnar to equiaxed transition and the size of the equiaxed zone in castings involve an accurate description of the source of the nuclei and an accurate description of the growth rates of the columnar and equiaxed crystals under the prevailing conditions. 93.3.1. Origin of the equiaxed nuclei. Three principal sources of nuclei can be considered. i) ConstitutionalSupercooling (CS) driving heterogeneousnucleation (WINEGARD and CIIALMERS [1954]). Because the tips of the dendrites in the columnar grains are at a temperature below the bulk alloy liquidus temperature a region of liquid exists where hetrogenous nuclei may become active. ii) Big-Bang mechanism: Equiaxed grains grow from predendritic shaped crystals formed during pouring at or near the mould walls. These crystals are carried into the bulk by fluid flow with some surviving until the superheat has been removed (CHALMERS [1963]). As outlined in 0 9.2.1., the origin of the grains could be either by nucleation events or crystal multiplication mechanisms. OHNO et al. 119711 has proposed a separation theory (OHNO [1970]) for the origin of the equiaxed grains by the big bang mechanism. iii) Dendriticfragmentation occurring from and TILLER[1967]) or from the columnar grains (JACKSON et al. [1966]), (0’dendritic crystals nucleated at the top of the ingot as a result of the radiation cooling occurring in that region (SOUTHIN [1967]). References: p. 830.
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Many experiments have been performed in metallic alloys and transparent analogues to decide the nucleation mechanism responsible for the equiaxed zone. The available information suggests that in conventional castings there is strong evidence to support the big bang and fragmentation mechanisms, probably a combination of both, in most of the et al. [1970], FLOODand HUNT[1988]). cases where convection is present (MORANDO In these circumstances, the C.S. mechanism would seem not to have a large enough contribution except in the presence of very efficient heterogeneous nuclei. When upward directional solidification experiments are performed and a near-perfect adiabatic lateral mould walls exist, convection is minimized, and the only possible mechanism for equiaxed grain formation is C.S. However, the complete elimination of convection is quite difficult even in this type of growth (CHANGand BROWN[1983], ADORNATOand BROWN[1987]). 93.3.2. Columnar to equiaxed transition (CET). If a reliable nucleation model were available, and convection could be ignored, any one of several models could be employed to predict the columnar to equiaxed transition. The model of HUNT[19841uses selected columnar growth and nucleation models to determine whether the structure will be fully equiaxed or fully columnar. The results depend on whether the temperature gradient is smaller or larger respectively than a critical value given by
where No= density of nucleating sites, AT,, = supercooling required for heterogeneous nucleation and AT,=supercooling of the dendrite tips in the columnar grain. This analysis was expanded by FLOODand HUNT[1987b]. The experimental results by ZIV and WEINBERG [1989] are in close agreement with eq. (112) for Al-3%Cu. Factors which promote a columnar to equiaxed transition by this mechanism are: large solute content (increases the value of AT, for fixed growth conditions), low temperature gradient, which increases the size of the supercooled region in front of the dendritic tips, a small value for ATn (potent nucleation sites), and a large number of nuclei. Hunt’s model ignores many complexities of the dendritic growth of equiaxed grains and nucleation was assumed to take place at a single temperature rather than over a range of temperature. It therefore cannot predict the effect of solidification conditions on equiaxed grain size (KFXR and VILLAFUERTE [1992]). More detailed models of equiaxed growth employing empirical nucleation laws have been combined with numerical solutions of the heat flow to predict the grain size of fully equiaxed structures (THEvoz et al. [1989], RAPPAZ [1989], STEPHANESCU et al. [1990]). These more detailed analyses could be used to predict the columnar to equiaxed transtition. FLOOD and HUNT[1988] have critically reviewed the models and experiments of several researchers. RAPPAZand GANDIN[19931 have used the probabalistic model described in $j9.2.2 to simulate the columnar to equiaxed transition. Figure 61a) corresponds to the simulation of the final grain structure of an AI-5%Si casting with no temperature gradients when cooled at 2.3 Ws. Figure 61b) corresponds to an A1-7%Si casting cooled at 2.3 K/s and fig. 61c) to an A1-7%Si casting cooled at 7.OWs. Comparisons among the
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Fig. 61. Simulation of columnar and equiaxed structures by RAPPAZand GANDIN [1993].
References: p . 830.
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three figures show the effect of the alloy composition and cooling rate upon the grain structure. Very recently RAPPAZ and GANDIN[1994] presented a very comprehensive review of the modeling of grain structure formation in solidification processes. [19681 which showed A set of experiment were performed by BILONIand CHALMERS that the CET could be stimulated by mechanically disturbing the columnar growth front during upward directional solidification (fig. 62). Experiments were conducted in which the value of G/V112decreased with distance down the length of a small ingot. This leads to a columnar to equiaxed transition at some position as previously shown by PLASKETT and WINEGARD119591 and ELLIOT[1964]. The critical values of G / V 1 / 2obtained by BILONIand CHALMERS [1968] for various A1-Cu alloys are shown in fig. 62 as circles. When the experiment was repeated, but with a periodic disturbance (1 min) of the interface, small equiaxed grains were formed at the position of the disturbance at significantly higher values of G/V”2 as shown by the squares in fig. 62. However this band of equiaxed grains reverted to columnar growth upon further solidification. Presumably the disturbance increased the number of potential growth sites by dendrite fragmentation and acccording to eq. (112) momentarily increased the possibilities for equiaxed growth. However the available growth centers where quickly consumed and the growth reverted to columnar. Finally at a position in the ingot with a low value of G/V*” 451
I
I
I
I
I
I
I
I
I
I
4I
35/
309 ’
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COLUMNAR CRYSTALS
I
I
z
/’
/’
/’
/’
A”
Fig. 62. Critical values of WV2 for columnar to equiaxed transition for various A I 4 alloys. BILONIand CHALMERS [1968].
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(triangles, fig. 62) close to that established by the undisturbed experiments, it was found that the equiaxed grains did not revert to columnar grains, but persisted for the remainder of the ingot. These experiments indicate the importance of dendrite fragmentation on the columnar to equiaxed transition.
9.3. Macrosegregation Macrosegregation is defined as variations in composition that exist over large dimensions, typically from millimeters to the size of an entire ingot or casting. We have already considered one form of macrosegregation in the initial and final transients during planar growth of a rod sample (Q 6.2). However in order to define or measure macrosegregation for dendritically solidified samples, it is necessary to determine an average composition over a volume element that contains several dendrite arms. As we will see, changes in the dendritic microsegregation profile in such a volume element due to settling of free-floating solid or flow of solute-rich liquid in or out of the volume element during solidification will change the average composition of the volume element away from the nominal composition of the alloy. Thus macrosegregation is produced. Figure 63 shows a drawing of a large steel ingot showing some of the major types of macrosegregation commonly found. Positive and negative macrosegregation refer to solute content greater or less than the average.
93.1. Gravity segregation Negative cone segregation has been explained by the settling of equiaxed grains or melted off dendrites into the bottom of a casting if they are of higher density than the liquid. If lighter solids such as nonmetallic inclusions and kish or spheroidal graphite are formed they can float to the upper part of a casting forming positive segregation areas. Centrifugal casting clearly can alter the pattern of macrosegation formed by this mechanism (OHNAKA[l988]). It is important to note that gravity produces negligible macrosegration of a single phase liquid, discrete particles with densities different from the average are necessary. 9.32. Interdendritic fluid flow and macrosegregation The first attempt to create models for macrosegregation due to flow of solute rich material was by KIRKALDY and YOUDELIS[1958]. Later the subject was treated extensively at MIT by FLEMINGS and NEREO[1967]) and MEHRABIAN et al. [1970] and has been summarized by FLEMINGS [1974], [1976]. Using a volume element similar to that chosen in fig. 41, a mass balance is performed under the additional possibility that flow of liquid in or out of the volume elements can occur and that the liquid and solid can have different densities. Thus the necessity for flow to feed solidification shrinkage is treated. The result is a modified form of the solute redistribution equation used to describe microsegregatian in Q 7,
References: p . 830.
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"A" SEGREGATES
CONCENTRATES UNDER HOTTOP SEGREGATION
*
"BANDS"
"\r SEGREGATES
+
' + +U+ + + ++ + + v + + + ++ $ + + + -. +
/
CONE O f NEGATIVE SEGREGATION
Fig. 63. Different types of macrosegregation in an industrial steel ingot. REMINCS [1974].
where: fL is the fraction of liquid; p = (ps-pJ/ps = solidification shrinkage; fi = velocity vector of interdendritic liquid; V T = local temperature gradient vector; E =local rate of temperature change. This expression assumes: i) local equilibrium without curvature correction, ii) uniform liquid composition in the small volume of interest, iii) no solid diffusion, iv) constant solid density, v) no solid motion, and vi) absence of voids. In this approach, the appropriate values for fi and E at each location must be determined from a separate calculation involving thermal analysis and flow in the mushy zone, which will be outlined below. However given these values for each small volume element, C, and hence C, as a function of fs can be determined along with the fraction of eutectic. The average value of C, from fs = 0 to 1 gives the average composition at each location in the
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casting. The average composition will not in general be equal to the nominal alloy composition. Several cases can be distinguished. If the interdendritic flow velocity just equals the flow required to feed local shrinkage,
Then eq. (113) reverts to the Scheil equation and the average composition is equal to the nominal. Here E is the unit normal to the local isotherms and V is the isotherm velocity. If on the other hand E i j is greater than or less than this value, negative or positive macrosegregation occurs, respectively. A particularly simple case ocurs at the chill face of a casting. Here E i j must be zero because there can be no flow into to the chill face. This clearly produces positive macrosegregation (for normal alloys where > O and &< 1). This is commonly observed in ingots and is termed inverse segregation (KIRKALDY and YOUDELIS [1958]) because it is reversed from what one would expect based on the initial transient of plane front growth. In order to compute the fluid velocity of the liquid, the mushy zone is treated as a porous media and D’Arcy’s Law is used. The pressure gradient and the body force due to gravity control the fluid velocity according to
where:
l$,= specific permeability; q =viscosity of the interdendritic liquid; VP = pressure gradient; g =acceleration vector due to gravity. Often, heat and fluid flow in the interdendritic region have been computed by ignoring the fact that the fraction of liquid at each point in the casting depends on the flow itself through eq. (115). This decoupling is thought to cause little error if the macrosegregationis not too severe. FLEMINGS[19741 and RIDDFX et al. [19811 solved the coupled problem using an iterative numerical scheme for an axisymmetric ingot. In their work the flow in the bulk ingot was also coupled to that in the interdendritic region. Experiments on a model system showed good agreement. Determining an accurate expression for the permeability of a mushy zone is a difficult problem since the value of K p depends on interdendritic channel size and geometry. In the case of the mushy zone it has been proposed (PIWONKA and FLEMINGS [19661) that
where A, is a constant depending on dendritic ann spacing. Recently, POIRER[1987] analysed permeability data available for the flow of interdendritic liquid in Pb-Sn and borneol-paraffin. The data were used in a regression analysis of simple flow models to arrive at relationships between permeability and the morphology of the solid dendrites. When flow is parallel to the primary dendritic arms the permeability depends upon References: p . 830.
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A I (primary arm spacing) but not A, (secondary arm spacing). When flow is normal to the primary arms the permeability depends upon both A, and A,. These correlations are only valid over an intermediate range of fL, roughly between 0.2 and 0.5. With this model of macrosegregation, FLEMINGS [1974], [1976] was able to explain different types of macrosegregation present in industrial ingots (fig. 63): a). Gradual variations in compositionfrom suvace to center and from bottom to top are due to the interdendritic fluid flow with respect to isotherm movement. b) Inverse segregation as described above. If a gap is formed between the mould and the solidifying casting surface, a severe surface segregation or exudation can arise. c) Banding or abrupt variations in composition that result from either unsteady bulk liquid or interdendritic flow, or from sudden changes in heat transfer rate. d) “A” segregates or “frecket’. These are abrupt and large variations in composition consisting of chains of solute-rich grains. They result from movement of interdendritic liquid that opens channels in the liquidsolid region. Recent work by HELLAWELL [1990] seems to prove that, at least in some cases, the initiation of the channels is at the growth front itself. e) “V” segregates. As the fraction solid in the central zone increases in the range of 0.2 to 0.4, the solid network that has formed is not yet sufficiently strong to resist the metallostatic head and fissures sometimes occur. These internal hot tears open up and are filled with solute rich-liquid. f) Positive segregation under the hot top: Probably occurs during the final stages of solidification when the ingot feeding takes place only by interdendritic flow. More recently KATO and CAHOON[1985] concluded that void formation can affect inverse segregation. They studied inverse segregation of directionally solidified Al-Cu-Ti alloys with equiaxed grains. MINAKAWA et al. [19851 employed a finite difference model of inverse segregation. This model allowed for volume changes due to microsegregation and thermal contractions as well as the phase change. 9.3.3. Further theoretical developments for flow in the mushy zone Many simplifying assumptions regarding the flow in the mushy zone and its interaction with the bulk flow have been required in the past work. More rigorous approaches have recently been performed. In order to formulate a set of governing equations that determines the flow, temperature and composition fields in the mushy zone as well as in the bulk liquid, two approaches have been employed: continuum mixture [1990] and PRESCOTT theory and a volume averaging technique as reviewed by VISKANTA et al. [19911. Both approaches have successfully computed macrosegregation patterns in ingots, including freckles. The continuum mixture approach has been used by BENNON and INCROPERA [1987a)], [1987b)], GANESAN and POWER [1990], and FELICELLI et al. [1991]. The volume averaging technique (BECKERMANN and VISKANTA [1988], NI and BECKERMANN [1991]) may be better suited to flow models involving the formation of equiaxed (free floating) dendritic structures where solid movement must be treated.
9.4. Porosity and inclusions Porosity and inclusions have a strong effect on the soundness and mechanical properties of castings.
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9.4.1. Porosity For most metals the density of the solid is higher than the liquid. Thus liquid metal must flow toward the solidifying region in order to prevent the formation of voids. Much of foundry practice is involved with the placement of chills and risers that maintain proper temperature gradients to retain an open path of liquid metal from the riser to the solidification front. Indeed the major use of macroscopic heat flow modelling of castings is to identi@ potential locations in the casting where the solidifying regions are cut off from the risers. Even if a path of liquid metal remains open to the riser, porosity on the scale of the dendritic structure can still form. When liquid metal flows through the mushy zone to feed solidification shrinkage, the liquid metal pressure in the mushy zone drops below the external atmospheric pressure. The pressure gradient required for flow is given by eq. (115). Microporosity forms when the local pressure in the mushy zone drops below a critical value. Thus detailed prediction of microporosity requires a rather complete description of fluid flow in the mushy zone as does the prediction of macrosegregation. Clearly the larger the freezing range of an alloy and the smaller the temperature gradient, the more. tortuous are the liquid channels in the mushy zone. This leads to greatly increased difficulty of feeding the shrinkage and a greater reduction of the liquid metal pressure deep in the mushy zone (far from the dendrite tips). The critical reduction of the liquid pressure required for the formation of micropores depends on many factors. Indeed the initial formation of a pore is a heterogeneous nucleation problem (CAMPBELL [1991al) that requires the same consideration as described in 8 4, but with pressure substituted for temperature. In principle one needs to know the contact angle of the pore on the solid-liquid interface. It is common to postulate that the critical radius (in the sense of nucleation) of a pore, above which the pore can grow, is related to the scale of the dendrite structure. KUBOand PEHLKE[ 19851 let the critical radius be equal to the primary dendrite spacing, h whereas POWERet al. [19871 relate the critical radius to the space remaining between dendrites. They obtain an expression for the liquid metal pressure where a void can form as
PIP,-
-
A,,
fL
where PG is the pressure of gas in the pore if any, and yLGis the surface energy of the liquid gas interface. If no dissolved gas is present and if the surface energy were zero, porosity would form at locations in the casting where the liquid metal pressure drops to zero. Dissolved gas increases the likelihood of porosity formation whereas the inclusion of the surface energy effect makes it more difficult, possible requiring negative pressure to form a void. Dissolved gas in a liquid alloy causes porosity because the solubility of gas in liquid metal usually exceeds the solubility in the solid. One can define a partition coefficient for the gas, k;, as the ratio of the equilibrium solubilities of gas in the solid and the liquid just like any solute. The value of the coefficient as a function of temperature must be known to make predictions. As an alloy solidifies, the dissolved gas is rejected into the References: p . 830.
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remaining liquid where its level increases. Because the diffusion rate of gas in solid metal is usually quite high, the gas content of the solid phase is usually assumed to be uniform and at the equilibrium concentration. Thus the lever rule can be applied to compute the concentration of gas in the liquid as a function of fraction solid. This increase in the concentration of gas leads to an expression for the equilibrium pressure of the gas in the pores given by
Pd
PG = (f,(l
- k;) + k;)* ’
where Pd is the partial pressure of the gas above the melt (given by the initial concentration of gas in the melt (BRODY[1974]). Thus it can be seen how the presence of dissolved gas in the melt (through its effect on Po) as well as solidification shrinkage both contribute to the formation of dendritic microporosity. This combined effect is particularly important for aluminium castings where the solubility of H in the solid is only tenth of that in the liquid. Equation (118) for the pressure inside a pore is only valid when the volume fraction of porosity is small. To actually calculate the size and fraction of porosity after solidification is complete, a more complex analysis is required. Kmo and PMLKE[1985] have calculated the amount and size of the porosity formed in A1-4.5 wt% Cu plate castings containing hydrogen that match experimental measurements. POIRIERet al. [1987] perform such calculations for A1-Cu in a directional solidification geometry. ZHUand OHNAKA[19911 have also simulated interdendritic porosity considering both H redistribution in the melt and solidification contraction. With this method, the effect of the initial H content, cooling rate and ambient pressure were simulated.
Inclusions At present it is very clear that inclusions exert an important influence on fracture behaviour of commercial materials. As a result, this portion of the field of solidification is receiving much greater attention. One type of inclusions, called primary inclusions, corresponds to: i) exogeneous inclusions (slag, dross, entrapped mould material, refractories); ii) fluxes and salts suspended in the melt as a result of a prior melttreatment process; and iii) oxides of the melt which are suspended on top of the melt and are entrapped within by turbulence. These are called primary inclusions because they are solid in the melt above the liquidus temperature of the alloy. In the steel industry a significant reduction of inclusions is obtained by the their floating upward and adhering to or dissolving in the slag at the melt surface. In the aluminum industry filtering has become a common practice and the development of better filters is an important area of [ 19801, APELIAN [1982]). research (Ross and MONDOLFO Secondmy inclusions are those which form after solidification of the major metallic phase. Although in industrial practice commercial alloys involve multicomponent systems, a first approach to the understanding of the formation of secondary inclusions has been achieved through the considerations of ternary diagrams involving the most [19741). Then, the solidifiimportant impurity elements under consideration (FLEMINGS 9.4.2.
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cation reactions occurring during the process, together with the values of the various partition coefficients of the impurity elements in the metallic phase play an important role in the type, size and distribution of inclusions in the final structure. Important ternary systems to be considered are Fe-O-Si, Fe-OS, and Fe-Mn-S from which the formation of silicates, oxides and sulphides results. As an example of research in this field FREDRICKSSON and HILLERT [1972], through carefully controlled solidification, were able to determine the formation of four types of MnS inclusions formed by different reactions. CAMPBELL [1991a] and TRAJAN[I9881 treat extensively both primary and secondary inclusions and their effect on mechanical properties in ferrous and nonferrous alloys. An important effect to consider when a moving solidification front intercepts an insoluble particle is whether the inclusion is pushed or engulfed. If the solidificationfront breaks down into cells, dendrites or equiaxed grains, two or more solidification fronts can converge on the particle. In this case, if the particle is not engulfed by one of the fronts, it will be pushed in between two or more solidification fronts and will be entrapped in the solid at the end of local solidification. STEFANESCU and DHINDAW [1988] reviewed the variables of the process as well the available theoretical and experimental work for both directional and multidirectional solidification. More recently, SHANGGUAN et al. [1992] present an analytical model for the interaction between an insoluble particle and an advancing S-L interface. There exists a critical velocity for the pushing-engulfment transition of particles by the interface. The critical velocity is a function of a number of materials parameters and processing variables, including the melt viscosity, the wettability between the particle and the matrix, the density difference as well as the thermal conductivity difference between the particle and the matrix, and the particle size. Qualitatively the theoretical predictions compare favorably with experimental observations. As an example of the interaction between the formation of inclusions and porosity, et al. [19931 present a novel theoretical approach to the nucleation of pores in MOHANTY metallic systems. The proposed mechanism is based on the behavior of foreign particles at the advancing S-L interface. Mathematical analysis has been employed to predict gas segregation and pressure drop in the gap between the particle and the S-L interface. The authors discussed the effect of particle properties and solidification parameters, such as wettability, density, thermal conductivity, solidification rate and S-L interface morphology. They recognize, however, that at present quantitative measurements of materials properties are necessary, in particular for interfacial energies. 9.5. Fluidity
Over the years the foundryman has found it useful to employ a quantity called fluidity.The concept arises from practical concerns regarding the degree to which small section sizes can be filled with metal during castings with various alloys. This property is measured through one of several types of fluidity tests. Hot metal is caused to flow into a long channel of small cross section and the maximum length that the metal flows before it is stopped by solidification is a measure of fluidity. The solidification process References: p . 830.
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in the channel is under the influence of many variables: metallostatic pressure, heat transfer coefficient, superheat, latent heat of fusion, density of the alloy liquid, viscosity, liquid surface tension, alloy freezing range, and whether the alloy freezes with plane front or with columnar or equiaxed dendrites. FLEMINGS [1974] reviewed the field and his contributions to the study of fluidity. More recently CAMPBELL [1991a], [1991b] gives a general view of this property and stresses the importance of the factors that influence the fluidity test as they relate to present limitations and future difficulties of numerical modeling of casting. It is worthwhile to follow the approach of CAMPBELL [1991a] who considers three cases: i) Maximum Fluidity Length, &, determined by an experiment where the cross sectional area of the channel is large enough that the effect of surface tension is negligible; ii) Lf, when surface tension is important, and, iii) Continuous Fluidity Length, L,.
95.1. Maximum fluidity Instead of only determining the total distance travelled, which is the maximum fluidity length, MORALES et al. [19771 and AGUILARR r v ~ and s BILONI[1980a1, [198Obl performed tests on Al-Cu alloys that measured the distance flowed vs. time for different metallostatic pressures, superheats and metal-mould heat transfer coefficients, hi.In addition, the use of careful metallographic analysis of the fluidity samples gave information about the vein closing mechanism. The measured data have two stages with different slopes. For a given fluidity test, the first stage represents a high percentage of the total distance flowed but depends on variables independent of the true capacity of flow of the metal or alloy, namely, the liquid superheat and the heat transfer coefficient at the metal-channel surface. The second stage, in general, represents a small percentage of the total distance flowed but reflects the intrinsic ability of the metal or alloy to flow. In a complementary study, MORALES et al. [1979] determined how the channel microgeometry, as influenced by machining, polishing and coating, affects microstructure and L, through variations in the local and average heat transfer rates. In summary, the molten metal entering the channel flows until all the superheat is eliminated in the first stage. The liquid can continue to flow primarily because of the delayed cooling due to the latent heat evolution. This second stage is strongly affected by the solidification mechanisms, the fs(T)relationships, and the nature of the columnar/equiaxed structures. Until recently all fluidity tests performed in fundamental investigations used binary alloys to establish relationships between Lf and alloy composition (FLEMINGS[ 19741, CAMPBELL [1991a]). GARBELLINI et al. [1990] carried out an extensive study of the fluidity of the A1-Cu rich corner of the Al-Cu-Si ternary system, which serves as a basis for many commercial alloys. This paper developed a correlation between Lf and alloy microstructure in the binary (AI-Cu and A1-Si) and ternary (Al-CuSi) systems. CAMPBELL [ 1991~1 discussed this paper expanding on the results. L, for any composition is a balance among three factors primarily related to the phase diagram: a) the latent heat of the proeutectic phase, Le., Al, Si or A1,Cu; b) the amount of interdendritic liquid remaining at the end of the proeutectic solidification; and c) The value of the fluidity for the specific eutectics (binary and/or ternary) that complete solidification. These considerations led to the conclusion that minor changes of compositions can be quite
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important, for example for metal matrix composites (MMC) obtained by infiltration. Also due to the latent heat effect, the high fluidity of alloys with proeutectic Si phase in hyper eutectic alloys was confirmed. The important industrial A1-Si alloys do not display a peak fluidity at the eutectic composition. Typically alloys with the smallest freezing ranges show the best fluidity (FLEMINGS [1974], CAMPBELL [1991a]). Small amounts of Si in hypereutectic alloys dramatically increased Lfbecause of the extraordinarily high latent heat of Si that maintains the fluid state of the alloys for longer times. The relationships that exist between fluidity length and solidification microstructure are open to further research. As examples, there is a large difference in fluidity between binary Al-Al,Cu, a regular eutectic and Al-Si, an irregular eutectic (G-ELLINI et al. [19901). In eutectic cast irons the fluidity is determined by the morphological changes of the graphite phase as documented by fluidity tests on laminar, vermicular and nodular cast irons (STEFANESCU et al. [1988]). 9.5.2. Combined effects of surface tension and fluidity When the channel section becomes thinner than a critical value, considered to be -0.30 cm for most metals and alloys by REMINGS [1974], the resistance to liquid flow increase because of surface tension. This is particularly critical in technologies such as aerofoils, propellers, and turbine blades (CAMPBELL[1991aI). CAMPBELL and OLLIP [1971] distinguish two aspects of filling thin sections:j?owability, essentially, following the rules discussed above andjllability limited by surface tension.
9.5.3. Continuousfluidity length Figure 64a is a schematic representation of the solidification into a channel when a nondendritic S-L interface is considered (MORALES et al. [1977]). In region I, no solidification occurs; in region II the solidificationoccurs in the presence of a decreasing amount of superheat. Region III corresponds to the liquid and solid at the melting temperature, i.e. with no superheat. The length, L,,defines a critical value known as the continuous fluidity length. The physical meaning of L, has been defined by FELIUet al. [1962], who introduce the concept of the flow capacity of a channel. In the case of a very long channel the flow capacity is just the volume of the cast fluidity length Lp In the case of a channel of intermediate length, the flow capacity is the total of the amount which has flowed through, plus the amount which has solidified in the channel. For a channel shorter than I,, the flow capacity becomes infinity (fig. 64b). CAMPBELL [1991a] gives technological applications of these concepts for different types of alloys and moulds.
10. Solid@icationprocesses This section will treat two conventional solidification processes that have an important impact in current technology: continuous casting and welding.
10.1. Continuous casting Continuous casting has emerged as one of the great technological developments of References: p. 830.
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2
3 4
Fig. 64. The concepts of: (a) maximum fluidity length showing the stages of freezing leading to the arrest of flow in a long mould; and @) the continuous flow which can occur if the length of the mould does not exceed a critical length, defined as the continuous-fluiditylength. CAMPBELL [1991a]. In (a) has been included the schematic representation of the solidification considered by MORALES et al. [1977].
this century, replacing ingot casting and slabbing/blooming operations for the production of semi-finished shapes: slabs, blooms and billets. More recently even continuous production of single crystals for research and sophisticated technology has been
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developed. Excellent reviews of the research and technology involved in this field have been presented by TAYLOR[1975], WEINBERG[1979a] [1979b] and BRIMACOMBE and SAMARASEKERA [1990] for steels and EMLEY[1976] and BAKERand SUBRAMIAN [19801 for aluminum and its alloys. Additionally PEHLKE[1988] and Mmom [1986] present comprehensive details of the process both in ferrous and nonferrous alloys together with technical details of the industrial installations currently used.
10.1.1. Continuous casting of steels Figure 65 shows the main components of a continuous casting machine. Molten steel is delivered from a ladle to a reservoir above the continuous casting machine called a tundish. The flow of steel from the tundish into the water cooled mould is controlled by a stopper rod nozzle or a slide gate valve arrangement. To initiate a cast, a starter, or dummy bur is inserted into the mould and sealed so that the initial flow of steel is contained in the mould and a solid skin is formed. After the mould has been filled, the dummy bar is gradually withdrawn at the same rate that molten steel is added to the mould. Solidification of a shell begins immediately at the surface of the water cooled copper-mould. The length of the mould and the casting speed are such that the shell thickness is capable of withstanding the pressures of the molten metal after it leaves the mould. Usually a reciprocating motion is superimposed on the downward travel to prevent sticking. Figure 66 shows schematically the role of fundamental knowledge in analyzing the process for the achievement of quality products (BRIMACOMBE and SAMARASEKERA 119901). Determination of the heutjow is important because it allows the prediction of the shell profile, the pool depth and temperature distribution as a function of casting variables. In the mould heat transfer, the gas gap separating the mould and the strand casting and its relationship to mould heat flux is very important. MAHAPATRA et aZ. [1991a] [1991bl employed measured mould temperatures and mathematical modeling in order to predict formation of oscillations marks, longitudinal or corner depressions and subsurface cracks. Additionally, laboratory experiments and in-plant studies have been undertaken to determine the relationship between spray heat transfer coefficient and spray water flux. As, a result, mathematical models of spray systems have been developed. The main aspects of solidification that must be understood in continuous casting are: i) the cast structure; ii) growth of the solid shell encasing the liquid pool and, iii) segregation. For the first concern, as with static casting, the structure consists of both columnar and equiaxed grains. The lower the pour temperature, the higher the fraction of equiaxed grains. The equiaxed structure is favoured in the medium carbon range, from about 0.17% to 0.38%C. Also induced fluid flow, for example by electromagnetic stirrers, enhance the growth of equiaxed grains. Shell growth is affected by all the variables that influence the mould heat flux distribution. When microsegregation is considered the models discussed in 9 7 can be applied but the macrosegregation associated with the fluid flow during the solidification process is still not completely understood in continuous casting. However it is recognized that the same factors favoring an equiaxed structure diminish the macrosegregation. Very recently BRIMACOMBE [19931 stressed the challenges of transfemng knowledge References: p. 830.
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Ladle
Tundish
Fig. 65. Schematic diagram of continuous-casting machine. BRIMACOMBE and SAMARASEKERA 119901.
from R & D to the steel continuous casting industry. He suggests the development of expert systems in the form of an intelligent billet casting mould. This system effectively transfers knowledge on line to the shop floor through the combination of thermocouple and load cell information, signal data recognition based on years of research, mathematical models of heat flow in the solidifying shell and mould, understanding of the mechanism of quality problems, and the formulation of a response to a given set of casting conditions.
Ch. 8, 3 10
801
Solidification
I
Continous Casting process (Design and Operation Liquid S t e l l Q u a l i t y Ladle Tundish Mould Sprays Sub-muid support
Cleanness Cracks Porosity Segregation Shape
Mechanism ~
-
1
Cast Product Q u a l i t y
,
Mathematical
Physica 1 (Fluid flou)
.
Heat and f l u i d f l o u i n ladle, tundish, mould Electromagnetic s t i r r i n g Shell growth, Steel temperature field, pool b o t t m Mould temperature field. distortion Osci l a t i o n marks Stress generation Shrinkage Bulging (segregation) Chemical interaction
Ladle Tcsldish Mould Shrouds Steams
I
Laboratory
chemistry
Methods (FMD, FEM)
Properties duct i1i t y
Measurements
Sol i d i f icat i o n structure
Fig. 66. Knowledge-based approach to analysis of continuous casting. After BRIMACOMBE and SAMARASEKERA
[iwoi. 10.12. Continuous casting of light alloys The principal casting process for light metals, such as AI,is the direct chill process. Figure 67 shows a schematic representation of the D.C. casting components for conventional open mould casting, the most common method used for blocks and billets References: p . 830.
802
Ch. 8, 3 10
H.Biloni and RJ.Boettinger
CONSTANT METAL LEVEL IN LAUNDER
/
FROM FURNACE
-*.
LEVEL CONTROL VALVE ON FLOATING METAL DISTRIBUTOR
I
TO NEXT MOULD
LIQUIDMETAL --
Fig. 67. Schematic diagram of vertical DC casting as commonly practiced. EMLEY[1976].
(BAKERand SUBRAMANIAN [1980]). There are three factors that influence the separation of the ingot shell from the mould: (i) shrinkage at the ingot shell itself; (ii) thermal strain within the ingot shell; (iii) shrinkage in the block section below the mould and the associated mechanical strains in the shell. All are influenced by the primary and secondary water cooling system and can affect the ingot structure, principally at the surface of the ingot. The air gap developed when the shell separates from the mould can give rise to defects of various types. When gap formation occurs there is increased resistance to heat transfer and, consequently, reheating of the skin. Reheating results in macrosegregation, exudation, runouts, retardation of the solidification in the subsurface zone, and variations in the CelVdendrite size of the outer surface of ingots. Zones of coarse dendritic substructure may extend 2-3 cm below the surface. Associated with the coarse cells are large particles of intermetallic constituents, formed by eutectic reactions, which may be exposed by surface machining that is usually performed before fabrication (BOWERe t d . [1971]). Several methods have been proposed to reduce surface defects. The most succesful are those that reduce the heat extraction at the mould through the control of the microgeometry of the mould surface, for example by machining fine
Ch.8, 5 10
Solidification
803
grooves in the face of the mould. The molten aluminum does not fill the grooves due to surface tension. More recently, mouldless electromagnetic casting has been developed to improve the metal surface quality. Electromagnetic casting was invented by GETSELEV [1971] and the principle behind this process is simple. Molten metal is supported slightly away from a mould by radial electromagnetic body forces concentrated within the upper part of the ingot. These forces are generated by a one-loop induction coil supplied with 5000 A at 2000 Hz (VIVES and Rrcou [1985]). SATOet al. [1989], [1991], [1992] studied extensively the production of Al, Al-Cu, and AI-Si rods with different cross-sectional geometries by a mouldless vertical continuous casting process. The main advantages consist of: (i) near net shape material having small and complex cross sectional configuration; (ii) full automation preventing break-out of molten metal and permitting easy start-up and easy shut down; (iii) cast material having unidirectional solidification structures; iv) geometries having changeable cross sectional configuration along their axes. Elimination of the nucleation source for new crystals during solidification permits long single crystals to be continuously cast. Figure 68 shows the principle of the ingenious Qhno Continuous Casting process (OCC) (QHNO [1986]). The mould is heated to a temperature above the melting point of the metal to prevent the formation of new crystals on the surface of the mould. Cooling is arranged so is that the ingot maintains a small region of molten metal when leaving the mould, which solidifies immediately after leaving the outlet of the mould. If many crystals are nucleated at the end of the dummy, where the ingot begins to solidify, the combination of dendritic growth competition and macroscopic interface curvature eliminates all grains except one having a direction of preferential growth in close proximity to the casting direction. As a consequence a single crystal is formed. The process can also be used with a seed crystal of desired crystal orientation at the end of the dummy. Excellent results have been reported with Al, Pb, Sn, Cu and their alloys (OHNO [1986]). More recently KIM and KOU [1988] and WANG et a2. [1988] studied the experimental variables of the OCC process and performed numerical modeling of heat and fluid flow. TADAand OHNO [19921 extended the OCC principles to the production of aluminum strips using an open horizontal, heated mould. The method was patented under the name, Ohno Strip Casting process (OSC). As in most solidification processes, mathematical modelling of continuous casting of nonferrous alloys has also been undertaken. In the particular case of aluminum alloys, SHERCLIFT et al. [19941 present a comprehensive review.
10.2. Fusion welding structures In most metallurgical processes the scientific approach to process improvement is to obtain relations between operational variables, metallurgical structures and properties. However, for many years in fusion welding technology, only relations between operational variables and mechanical properties were considered. In the last thirty years, the scientific approach has begun to be applied in cases where quality assurance is manReferences: p. 830.
H. Biloni and WJ.Boeitinger
804
a.8, 0 10
MOLTEN METAL
HE4TED MOULD
'MOLTEN METAL FlLM
0::. .. .. UNlOlRECTlONALLY SOLIDIFIED INGOT
C Fig. 68. The principle of the OCC process. After OHNO[1986].
datory for sophisticated technologies. The fusion welding structure is a result of complex transformations and interactions starting with metal-gas and metal-flux reactions in the liquid state followed by the formation of the primary structure by solidification. Pioneering work by SAVAGE and coworkers at R.P.I. initiated the correlation between operational variables and primary structures (SAVAGEet al. [1965]). More recently, DAVIESand GARLAND[1975], EASTERLING [1984] and DAVIDand VITEK[1989] have presented comprehensive reviews of the correlation between solidification parameters and weld microstructures. Metallurgically, a fusion weld can be considered to consist of three major zones, namely the fusion zone (E), the unmelted hear afected zone (HAZ) and the unaffected
Ch.8, 0 10
Solid@cation
805
Fig. 69. Schematic diagram showing the three zones within a weldment. After DAVIDand VITEK[1989].
base mruE (BM), fig. 69. However careful metallographic analysis reveals the zones [ 19671). The (FZ) can be divided schematically shown in fig. 70 (SAVAGEand SZEKERES into subzones: the composite zone (CZ)and the unmixed zone (UZ). In addition between
COMPOSITE REGION
E L D INTERFAC
HEAT-AFFECTED UNAFFECTED BASE METAL
Fig. 70. Schematic representation of the different zones of a weld. After SAVAGEand SZEKERES [ 19671. References:p. 830.
806
H. Biloni and K J. Boeninger
Ch. 8, 8 10
the (FZ) and the (HAZ),a partially melted zone (PMZ) exists. The (UZ) appears in welds made with filler metal additions. It is a thin zone composed of base metal that is melted and resolidified without mixing with the filler metal during the passage of the weld puddle. This zone can be the location of initiation sites of microcracking as well as corrosion susceptibility in stainless steel. The PMZ is a region where the peak temperatures experienced by the weldment falls between the liquid and the solidus. As a consequence, only inclusions of low melting point as well as segregated zones can be melted. These areas resolidify and the contraction during the subsequent cooling can be a source of microcracks (BJLONI[1983]). In solidification processes, cooling rates range from lO-*-lb Ks-' in casting technology to 104-107 Ks-' for rapid solidification technology. Cooling rates in welds may vary from 10-103 Ks-' for conventional processes, but when modern high energy beam processes such as electron beam (EB) and laser welding (LW) are considered cooling rates may vary from IO3 to lo6 Ks-'. Furthermore the local conditions and cooling rates vary significantly within the weld pool. Therefore weld pool solidificationincorporates aspects of both extremes of solidification, i.e. traditional casting as well as rapid solidification. Thus, most of the solidification concepts discussed previously may be applied to the understanding of the different weld microstructures. The most important subjects to be considered are: weld pool geometry and macro- and microstructures of the welds. 10.2.1. Weld pool geometry Figure 69 schematically describes an autogeneous welding process in which a moving heat source interacts with the metal parts to be joined. The weld pool geometry is a function of the weld speed and the balance between the heat input and the cooling conditions. For arc welding processes the puddle shape changes from elliptical to tear drop shaped as the welding speed increases. For high energy processes such as EB or LW, the thermal gradients are steeper and as a result the puddles are circular at lower speeds becoming more elongated and elliptical in shape as the welding speeds increases. Eventually at high speeds they become tear drop shaped. The heat transfer equation for a moving heat source was developed by ROSENTHAL [1941]. MINKOW[1986] and Kou et al. [1981] reviewed this type of analytical model of heat transfer for welding. They consider only heat transfer by conduction neglecting the important convective heat transfer existing in the weld pool. In recent years much effort has been focussed on the dynamics of the heat and fluid flow in the weld puddle through erperimental work and mathematical modeling. The goal is to reproduce actual weld pool shapes and eventually to develop the capability of predicting weld geometries (DAVID and V m K [1989], [1992]), @AVID etal. [1994]). Convection is produced by: a) buoyancy effects; b) electromagnetic forces and, c) surface tension forces. The interactions among these three driving forces have been modeled by WANGand KOU [1987] showing the effects on the shape and weld peneetal. [1992] in tration in G.T.A. (Gas Tungsten Arc) aluminum welds and by ~ACHARIA G.T.A. welding of type 304 stainless steel. It is important to keep in mind that impurities in weld metal are often surface active and alter the surface tension of the liquid metal and its temperature dependence. For pure metals and high purity alloys the surface
Ch. 8 , 5 10
Solidijcation
807
tension decreases with increasing temperatures and the resultant flow is outward, away from the center of the weld pool. The result is a wide and shallow weld pool. When surface active elements exist, a positive temperature coefficient of the surface tension can occur in some cases and the resultant inward flow promotes a deeper and narrower weld pool. An example is the presence of small amounts of 0 and S (less than 150ppm) in stainless steel (HEIPLEet al. [ 19841). As already mentioned, computational models have been developed in order to predict the weld pool shape. Figure 71 shows calculated profiles superimposed on a macrograph of a cross section of an aluminum G.T.A. weld (ZACHARIA etal. [1988]). Although the correlation is reasonable, improvement of the model is necessary considering that the calculations underestimate the depth of penetration.
10.2.2. Macro- and microstructures of welds A weld pool solidifies epitaxially from the parent grains in the (PMZ) surrounding it. As a consequence of competitive growth controlled by the orientation of the temperature gradients and the easy growth direction, favourably oriented grains survive. As in castings, a columnar region develops, favoured by the presence of a continuous heat source, which keeps thermal gradients high at the S-L interface. Figure 72 corresponds to an elliptical weld pool. The local thermal gradient, GLand the local solidijication rate, R, changes from the fusion line to the weld center line. If crystal growth is considered isotropic (fig. 72a) R, = v,
case,.
(119)
P
Fig. 71. Calculated temperature profiles superimposed on a macrograph of a cross-section of an aluminium alloy. DAVIDand VITEK[ 19891. References: p . 830.
H.Biloni and TJ. Boeninger
808
,WOOL€
W *(
Ch. 8, 5 10
ISOTHERM-
WELDING 0I RECTI0 N
INTERFACE AT TIME t + dt
INTERFACE AT TIME t
Fig. 72. (a) Solidification rates at different positions of the weld pool if isotropic growth is assumed. (b) Relationship between welding speed and actual growth rate if anisotropic growth is assumed. NAKAWAKA et al. [1970].
where V, is the velocity in the welding direction. If crystal growth is anisotropic, R, must be corrected according to the expression developed by NAKAGAWA et al. [ 19701,
where R, is the growth rate in a direction normal to the isotherm and 6; is the angle between the welding direction and the direction of favoured growth (fig. 72b). Thus in welding the solidification rate is greatest on the weld center line where 8, = 0". At this
Ch.8, $11
Solid$cation
809
point the temperature gradients are shallow because of the large distance from the welding heat source. The liquid pool shape determines the columnar growth direction as well as the solidification rate and thermal gradient into the liquid. Columnar-grain substructures are determined by the S-L interface morphology. Thus, most of the concepts discussed in 5 6 and $ 7 for unidirectional solidification of alloys and in $ 9 for columnar growth in conventional castings can be applied to columnar growth in welding: micro- and macrosegregation, banding, inclusions, porosity, etc. In addition, when situations involving rapid solidification arise, significant departures from local equilibrium at the S-L interface may occur (see 5 5 ) and as a result metastable structures may be obtained. V m x and DAVID[1992] reported recent research showing that in LW or EB welding, metastable microstructures can be produced in stainless steel welds. Laser surface melting and alloying will be treated in 0 11. Regarding the classification of welding macrostructures, PEREZetal. [1981], BILONI[1983] shown schematically nine types of macrostructures obtained with different fusion welding processes. All of them show a columnar zone which may occupy the entire weld, or be accompanied by grains growing along the welding direction, by feathery crystals (see $9), by equiaxed grains or by mixed coarse and fine grains, the last being characteristic of electroslag welding (PATON [1959]). Recently RAPPAZ et al. [1989] and DAVIDet al. [1990] examined the effect of growth crystallography and dendrite selection on the development of the Ez microstructures. Using single crystals and geometrical analysis that provides a three dimensional relationship between travel speed, solidification velocity and dendrite growth velocity, they were able to reconstruct a three dimensional diagram of a weld pool. The presence of equiaxed grains near the center of welds is believed to be beneficial in preventing solidification cracking and in maintaining good ductility in alloys subjected to brittle fracture. KERRand VILLAFUERTE [19921 reviewed the mechanisms and conditions which give equiaxed grains in castings (see 0 9) and compared them to the situation in welds.
11. Structure manipulation and new processes The concepts discussed in previous sections are the basis used by metallurgists to manipulate microstructure and thus obtain better physical and mechanical properties. However, the manipulation of structure is closely related to the continued development of processing methods. In this section we describe a few examples: single crystal growth from the melt, grain refinement, eutectic modification, rapid solidification, microgravity processing, metal matrix composite fabrication, and semi-solid metal forming. 11.1. Single crystal growth from the melt Single crystal growth is useful to study the laws governing solidification of metals and alloys, to prepare samples for scientific studies, especially mechanical properties, and for technological devices such as those used in the electronics indusbries or the production of gas turbine blades. Thus, we consider single crystal production an important example of structure manipulation. References:p . 830.
810
H.Biloni and N.I. Boettinger
Ch. 8, Q 11
CHALMERS [1964] and THORTON [1968] present reviews of the different methods available for single crystal preparation. In general, the same methods can be used for the growth of bicrystuls if grain boundaries are to be studied. Four main methods and their variations are employed. (i) In the Bridgman method the mould containing the melt is lowered through the furnace and the solidificationbegins either at the lowest point of the mould or on a seed located in the bottom of the mould. (ii) The Chalmers method is a variation of the above using a horizontal boat. An advantage of these methods is the fact that the remaining liquid can be decanted by electromagnetic devices. However with horizontal growth, problems arise due to excessive convection during growth (see 5 6). (iii) The Czochralski orpulling method consists of a melt contained in a crucible and a seed crystal that is lowered into it from above and then slowly rotated and withdrawn. The purpose of the rotation is to maintain axial symmetry of the crystal and decrease the solute enriched layer at the S-L interface. HURLE[1987] discussed the evolution of the method, mathematical modeling, and the advantages and future potential of this technique. (iv) Thefiuting zone method employs zone melting principles ANN [1966]) and, uses no mould. Essentially, the molten zone is held in place by surface tension forces, sometimes aided by magnetic fields. In 5 6 the importance of maintaining a flat interface has been discussed in detail as well as the relationship between different types of convection and stable interfaces. PIMPUTKAR and OSTRACH[I9811present a comprehensive review of convective effects in crystals grown from the melt, especially when Czochralski and Bridgman methods are considered. The directional freezing technique has been applied extensively to turbine blade manufacture. The techniques are employed in preparing aligned eutectic structures, directional columnar structures, and single crystal turbine blades. Unlike single crystals grown for electronic applications, single crystal turbine blades solidify with a dendritic structure. Therefore even though the crystallographic orientation of all the dendrites is the same, the turbine blades contain microsegregation and occasionally second phase particles formed by eutectic reactions. PIWONKA [ 19881 reviewed the different methods used. In the case of single crystal turbine blades, the alloys have no grain boundaries and thus need no grain boundary strengtheners. Elimination of these elements permits solution heat treatment at higher temperatures. Consequently, new high temperature Ni base alloys have been developed having better high temperature properties because they contain a higher percentage of the 7’ strengthening phase. In addition, because no grain boundaries exist, monocrystal turbine blades have better corrosion resistance. 11.2. Grain refinement Various techniques may be used to produce fine grained structures during solidification: thermal methods, innoculation and energy induced methods (BOLLING [19711). Each of these operates principally through one mechanism: nucleation or multiplication.
113.1. Thermal methods From a thermal point of view we are concerned principally with two possibilities, rapid chilling and bulk supercooling.
Ch. 8, 0 11
Solidification
811
(i) Chill eflect: when molten metals contact the cold walls of a mould, the melt superheat is removed from the liquid and it becomes locally supercooled. The number of nucleation centres increase and nucleation takes place catastrophically in the liquid. Techniques such as splat cooling (see below) and die casting, as well as applications using chills employ this approach with varying efficiencies according to the sample size. The reader is referred to Q 9 regarding the influence of wall microgeometry on the grain size in the chill zone of ingots and castings as well as the columnar zone grain size. (ii) Supercooling methods: WALKER[19591 published the first observations of grain refinement phenomena due to bulk supercooling. Samples of about 500 g were cooled under a glass slag and/or inert gas atmosphere and were contained in a fused silica crucible to reduce the probability of heterogeneous nucleation on the crucible walls, and, thus, aid the achievement of large supercooling of the melt. With this method Ni samples doped with Ag (to preserve the grain structure) were supercooled in the range from 30 to 285K. From 30 to about 145K of supercooling the grain size decreased monotonically but at this critical supercooling (AT*), a sharp decrease in grain diameter from about 20mm to 2mm arose. WALKER[1964] found a similar grain size transition in Co but at AT*= 180K. WALKERmeasured the pressure pulse associated with the solidification of Ni and found that it showed a maximum in the vicinity of AT*. He suggested that the grain refinement was brought about by homogeneous nucleation in the liquid ahead of the S-L interface due to a transient local elevation in melting temperature caused by the pressure pulse. Subsequently, many researchers tried to understand the grain refinement mechanism associated with high supercooling. Other containerless processes, such as electromagnetic levitation and drop tubes have been used (HERLACH er al. [19931). These authors have analyzed most of the previous results and the proposed mechanisms as well as the effect of minor amount of impurities on the grain refined structure, which can be columnar and/or equiaxed. The large number of proposed mechanisms fall into three broad categories having much overlap: (i) multiple nucleation in the melt arising from a mechanical disturbance of some kind, generally associated with a pressure pulse generated by the solidification itself. (WALKER [1959]; COLLIGAN at al. [1961]; GLICKSMAN [1965]; HORVAY [1965], POWELL[1965]). (ii) Dendrite fragmentation arising in a number of possible ways including remelting and mechanical fracture (UTTAMIS and FLEMINGS [1966]; JONESand WESTON[197Oa]; MCLEOD[1971]; TARSHIS eta2. [19712; SOUTHIN and WESTON[1973], [1974]; KATTAMIS[1976], MCLEODand HOGAN [1978]; KOBAYASHI and HoGAN[1978]; KOBAYASHI and SHINGU[1988]; DEVAUand TURNEXJLL [1987]). (iii) Recrystallization of the solidified structure (POWELLand HOGAN[1968], [19691; JONESand WESTON[ 197Oa1, [1970bl). The study of the segregation substructure of the grains obtained at different supercooling seems to support the fragmentation mechanism. KATTAMISand FLEMINGS [1966], [1967] and KAITAMIS [I9761 observed a gradual change in dendrite morphology from a normal branched form to a more cylindrical form with decreasing side branching as the supercooling increased. At AT', the transition from coarse columnar grains to fine equiaxed grains was accompanied by a change in the solute segregation pattern from dendritic to spherical. Each of these spherical patterns corresponded to a refined grain. The evolution of the spherical morphology is assumed to occur in two steps: detachment References: p . 830.
812
H.Biloni and WJ.Boettinger
Ch. 8, 8 11
of the secondary dendrite arms and subsequent coarsening into spherical elements during recalescence. Regarding the short time available for the remelting/coarsening mechanism during recalescence, COCHRANE et al. [1991] have shown that in drop tube processed Cu-3O%Ni and Fe-35%Ni alloys, local solidification times as short as 200 ps cannot retain the dendrite structure formed during the rapid growth phase. In addition, WILLNECKER etal. [1989], [1990] measured dendrite growth velocities and grain refinement in Cu-3O%Ni and Cu-30%Ni-l%B as a function of supercooling. The addition of lat % B shifted AT* from 193 to 255 K. The grain refinement transition occurred in both cases at the same supercooling that a sharp change in V-AT behavior occurred (fig. 73). These results open the possibility of a link between grain refinement with the onset of complete solute trapping during the rapid growth phase (see $ 5). As HERLACHet aZ. E19931 pointed out, there are many issues to be resolved in determining the mechanism responsible for grain refinement in highly supercooled melts and containerlessprocessing. SCHWARZ et al. [1994] have presented a simple model of dendrite fragmentation by a Rayleigh instability that appears to predict the correct value for A‘I? for Cu-Ni alloys. A variant of the supercooling method uses so-called denucbation, a concept introduced [1974], working with aluminum alloys. If the nucleants by MARCANTONIO and MONDOLFO that act at low supercooling can be removed (by centrifuging during freezing, among other methods) a denucleated melt can be supercooled well below its freezing point where it can be made to freeze rapidly on a chosen nucleant. These authors were able to reduce considerably the grain size of commercial purity Aluminum and A1-Mn alloys by this method. Also in A1-Mn alloys the method permits the retention of a higher percentage of Mn in solid solution.
11.23. Innoculationmethods One of the most important examples of structure modification in industry is the grain refinement of A1 and its alloys using innoculants that increase heterogeneous nucleation (See $4). A fine grain size in shaped castings ensures the following: (i) mechanical properties that are uniform throughout the material, (ii) distribution of second phases and microporosity on a fine scale, (iii) improved machinability because of (ii), (iv) improved ability to achieve a uniformly anodizable surface, (v) better strength, toughness, and fatigue life, and (vi) better corrosion resistance. The grain refining inoculants used in the aluminum industry employ so-called “master alloys” containing Al with Ti, B and C. Several mechanisms have been proposed for grain refinement and critical reviews exist in the literature (GLASSONand EMLEY[19681, Ross and MONDOLFO[1980], PFREPEZKO and LEBEAU1119821, PEREPEZKO [1988], MCCARTNEY [1989]). It is agreed that when master alloys are added to aluminum alloy melts, the aluminum mattrix dissolves and releases intermetallics into the melt, probably A1,Ti and various borides and carbides. Some compounds appear to act as effective nucleants with disregistry values below 10% and nucleation supercooling of less than about 5°C. In the past, the identities of the active nucleants have typically been studied from thermal and structural results obtained from standard bulk refinement tests. However more recently HOFFMEYER and PEREPEZKO [1989a], [1989b], [1991] utilized the Droplet Emulsion Technique (DET) (see $4) to produce a fine dispersion of master alloys powders containing innoculant particles. In this way the response of A1 to specific
a.8, Q 11
813
Solidi$cation
effective and ineffective nucleants can be separated and identified through highly sensitive DTA and metallographic analysis. On the other hand, very recently JOHNSSON
200
100
200
300
SUPERCOOLING AT,K
IC
IC
AV-225K
t IO*
Cum Ni,
Cu,
-'7 1
5. n
lo3 7
n
W
k! v)
$ 102
-
(3
-
(b) 1
N b 6,
I
I I I
-
lip iI
v
+
A F = I 93K
I I
I
P e
1 1 1
1
I
AF
II I I
, +
2)
Fig. 73. Dendrite growth velocity and grain diameter as function of supercooling for Cu,,,Ni, and Cu&&,. WILLNECKER etal. [1990]. See HERLAcH etaL [1993].
References: p . 830.
814
H. Biloni and a!J. Boeitinger
Ch. 8, 0 11
etaZ. [1993] used highly sensitive thermal analysis in bulk experiments to study the mechanisms of grain refinement in high purity A1 by master alloys. The use of high purity material and a computer automated thermal analysis technique, permitted them to isolate the effect of the Ti and B containing particles. Both types of work are complementary and promise to elucidate the quite complicated refinement mechanisms. Despite the uncertainty about the details of nucleation mechanisms, improvements have been made in theory in order to predict the final grain size as a function of the dispersion and density of nucleant particles. MAXWELLand HELLAWELL [1975] and HELLAWELL [1979] treat the ability of a substrate to act as a surface for heterogeneous nucleation simply using the wetting angle, 8, without entering into the details of the nucleation mechanism. A comparison is made between the final grain density and the initial nucleant density for various freezing conditions. As the melt cools below the liquidus, two processes take place concurrently. Solid nucleates on the available substrate surfaces at a rate which rises exponentially with the supercooling and, when the temperature has fallen below a certain limit, nucleated particles start to grow and evolve latent heat. The local cooling rate decreases as nucleation and growth accelerate until the temperature reaches a minimum and recalescence begins. The nucleation rate rises rapidly to a maximum at a temperature just before the minimum in the cooling curve. Afterwards the number of nucleation events decreases quickly as the available particles are exhausted and particularly because recalescence begins. Consequently, nucleation is almost complete just beyond the minimum temperature in the cooling curve. Subsequently, there is only growth. The number of grains varies according to the nucleation rate and is determined by the time recalescence begins. The minimum temperature reached and the recalescence rate are strongly influenced by the growth rate from each nucleation center. MAXWELL and HELLAWELL [1975], and HELLAWELL [19791 employ a purely solute diffusion model for growth of spheres. More recently the deterministic model of RAPPAZ[19891 employs a more realistic dendritic growth model to predict final grain size for different inputs of nucleation and cooling conditions. An important effect is how grain size is decreased, for fixed nucleation and cooling conditions, by increasing the level of solute in the alloy. This occurs because of the reduction of the dendritic growth rate with increasing composition as described in 0 7. The important effects of convection are not included in these models.
11.2.3. Energy-induced methods A large number of methods have been employed to refine the grain structure through energy-induced methods that increase nucleation through cavitation or that promote crystal multiplication, principally through mechanical vibration, bubbling agitation, rotating magnetic fields, magnetic-electric interactions and mould oscillation. COLEand BOLLING[1969] and CAMPBELL [1981] present comprehensive reviews in the field. Although cavitation may be responsible for grain refinement under some experimental conditions, today it is generally accepted that the best, and cheapest method of grain refinement using energy-induced methods is to promote crystal multiplication during the solidijicution process. The multiplication occurs primarily by fragmentation of the
Ch. 8, 0 11
Solid$cation
815
incipient dendritic structure by forced convection. FLEMINGS[1991] gives a summary of the possible dendrite fragmentation mechanism existing in the literature.
11.3. Eutectic modification Among the most important foundry alloys are those based on the Fe-C and A l S i systems. The mechanical properties of these metal-nonmetal (nonfacetted-facetted) eutectics are dominated by the morphologies in which the non-metals solidify. As nonfacetted-facetted eutectics, the asymmetry of the couple zone must always be considered in the interpretation of microstructure regarding the presence or absence of primary phases as described in 8 8.1.5. In both systems the structure can be modiJied either by rapid cooling or by controlled addition of specific elements. The use of elemental additions has an advantage because their effect is essentially independent of the casting section thickness. Thus we shall only discuss modification by additives. The modification of the structure of these alloys and the resultant effect on the mechanical properties is a clear example of the manipulation of the structure based on the application of fundamental principles.
11.3.1. Aluminum-silicon alloys Many studies have been focussed on the mechanism of modification in A l S i alloy and several reviews are available (CHADWICK[1963]; S m [1968]; HELLAWELL [1970]; GRANGER and ELLIOTT(119881). Most important is the change of the morphology of the silicon phase in the eutectic mixture. In unmodified alloys, the Si phase in the eutectic appears as coarse flakes that grow more or less independently of the AI phase. With small additions of alkaline or alkaline earth metals (especially Na and Sr), the Si phase takes on a somewhat finer branched fibrous form that grows at a common liquid-solid interface with the A1 phase to form a composite-like structure with improved properties. The easy branching of the modified Si leads to a more regular and finer structure as described in 5 8.1.3. Modifiers also change the morphology of primary Si in hypereutectic alloys. Although nucleation studies have been performed (CROSSLEY and MONDOLFO [1966]; Ross and MDNDOLFO [1980]), the modifying effects of Na and Sr are now thought to be growth related (HANNAet al. [1984]). Nucleation remains important however, through the addition of AI-Ti-I3 master alloys to control the A1 grain size in hypoeutectic alloys (8 11.2.2) and through the addition of P (usually Cu-P) to promote heterogeneous nucleation (on Alp) and refinement of primary Si in hypereutectic alloys. During the growth of unmodified AlSi eutectic, the Si flakes contains widely spaced ( 111] twins that provide for easy growth in the [lll] direction and difficult growth normal to [111]. This is called twin plane reentrant edge (TF'RE) growth. In modified alloys the Si fibres contain a much higher density of twins that exhibit an internal zigzag pattern (Lu and HELLAWELL [1987], [1988]). Since both Na and Sr are concentrated in the Si phase, these authors proposed a mechanism whereby the modifying elements are adsorbed on the growth ledges spewing out from the re-entrant corner. These adsorbed atoms cause the formation of new twins due to stacking errors on the growing interface due to size mismatch of the Si and the modifier. A hard sphere model for atomic packing References: p . 830.
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Ch. 8, 5 11
was used to define a critical ratio (1.65) of modifier to Si atomic size that promotes twinning. The growth of the Si is then thought to occur by repeated twin formation in a more isotropic manner than by WRE governed growth. This more isotropic growth permits the Si fibers to branch and adjust eutectic spacing. The formation of the internal zig-zag twin structure is also consistent with the observation of microfaceting on the Al-Si interface (Lu and HELLAWELL [1987]). MAJORand RUTTER[1989] proposed that a certain concentration of Sr is required at the interface to achieve modification (CLAPHAM and SMITH[19SS]). Below a critical concentration, growth of the Si is by the twin plane re-entrant edge ( T P E ) mechanism typical of unmodified eutectic. If growth occurs for a sufficient distance to accumulate Sr concentration at the solid-liquid interface above a critical level, the re-entrant edges are poisoned. Then, new twins form as described above. A continuous cycle of twin formation, TPRE growth, poisoning, new twin formation and so on can occur. More recently QIYANGet al. [1991] confirmed the adsorption of Na on { lll},i in agreement with the poisoning of the twin re-entrant edges. The phenomenon of overmodification can be explained as complete suppression of the TPRE mechanism resulting from elevated quantities of the modifying addition. In this way, formation of A1 bands in overmodified structures (F~EDRIKSSON et al. [1973]) may be explained.
11.3.2. Cast iron It is known that cast iron, belonging to the family of high carbon Fe alloys, can solidify according to either the stable iron-graphite system (grey iron) or to the metastable Fe-Fe,C system (white iron). As a consequence, the eutectic may be austenitegraphite or austenite-cementite (ledeburite). Furthermore, the complex chemical composition of the material has important and powerful effects on the structure of cast iron. Commercial alloys usually contain Si, minor addition of S, Mn and P and usually trace elements such as Al, Sn, Sb and Bi as well as the gaseous elements H, N and 0. Both forms of cast irons (white and grey) have technological importance. Several comprehensive reviews and books have been published in the last decades both from fundamental and technological points of view (MORROG[1968a], [1968b]; MINKOFF [1983]; ELLIOT[1988]; STEFANESCU [1988]; CRAIGet al. [1988]; HUGHES[1988]; STEFANESCU et al. [1988]). Grey iron is the most interesting because of the different morphologies that the graphite can achieve and the resulting differences in mechanical and physical properties. Although semantic problems have confused scientists and foundrymen in the past, a general understanding of the mechanisms of nucleation, growth and modification of the graphite phase has occurred in the last 15 years. In this section, the present status of knowledge in the area will be discussed briefly. It is known that the growth of the stable Fe-G eutectic is favoured over the metastable Fe,C eutectic by slow freezing (See Q 8.1.7.) or by addition of elements such as Si and Al. These elements increase the temperature difference between the stable Fe-G and metastable Fe-Fe,C eutectic temperatures. In addition a wide variety of compounds have been claimed to serve as nuclei for graphite, including oxides, silicates, sulphides, nitrides, carbides and intermetallic compounds. Most of the nucleation mechanisms are connected with impurities existing in the melt or with inoculants that promote heterogen-
Ch. 8, $ 1 1
Solidification
817
eous nucleation of the graphite. Although other inoculants are used, Fe-Si alloys are the most powerful and popular (ELLIOT [1988], STEFANESCU [1988]), SKALAND etal. [1993]).
11.3.3. Cast iron eutectic morphology The morphology and characteristic of the eutectic, whether stable or metastable, with or without modification, are very important in determining the physical and thermal properties. Thus, it is worthwhile to consider the most important eutectic structures observed. The microstructure of these major forms are shown for example by STEFANESCU [1988]. White Iroas: The metastable unalloyed Fe-Fe,C ledeburite eutectic is classified as quasi-regular. HILLERTand SUBBARAO [1968] described the mode of growth of the eutectic as well as the orientations arising between Fe,C and austenite(y). POWELL [1980] has shown that the eutectic structure can be modified by quenching. By adding Cr or Mg, a plate-like Fe,C structure associated with equiaxed grains can be achieved (STEPANESCU [1988]). Grey Irons: For high purity Fe-C-Si alloys, the structure of the Fe-G eutectic is spheroidal (SADOCHAand GRUZLESKY [1975]). However in practice the presence of impurities in the melt cause the graphite to take a flake morphology and grey flake iron is considered to be the characteristic form from a practical point of view. Modification of this structure gives different graphite morphologies: nodular, compact or vermicular, and coral. We shall be concerned only with the growth of eutectic structures without a primary phase and we will refer mainly to the three structures widely used in industry: flake, compact or vermicular, and nodular or spheroidal cast iron. At present there exists a theory for the mechanisms of the evolution from flake to compact and nodular graphite (MINKOFF[1990]): (i) fig. 74 corresponds to cooling curves for the three types of structure (BACKERUD et al. [ 19751). The major characteristics of these cooling curves is the increase of supercooling on going through flake, compact and spheroidal graphite forms. The rate of recalescence after nucleation is determined by the nucleation rate and the growth rate. (ii) Greyflake irons: the growth of the flake structure is well understood. Once graphite has nucleated, the eutectic cell or colony grows in an approximately radial manner and each flake is in contact with austenite up to the growing edge. The crystals of graphite grow in the close packed strong bonding “a” direction using steps created by rotation boundaries. These rotation boundaries are defects in the crystals in the form of rotations of the lattice around the axis. According to MINKOFF[1990], the screw dislocations on { IOIO} planes, which have been proposed as an alternative growth mechanism, are inactive (fig. 75a). ELLIOTT [19881, STEFANEXCU [19881 and SKALAND et al. [19931 discuss the effect of S and O as promoters of the flake graphite morphology on the basis of their adsorption on the high-energy (lOT0) plane. Thus, growth becomes predominant in the “a” direction. The result is a plate like or jhke graphite. (iii) Nodular or Spheroidal Irons. This is considered a divorced eutectic. Until recently has been widely accepted that the growth of this eutectic begins with nucleation and growth of graphite in the liquid, followed by early encapsulation of these graphite speroids in austenite shells. The result is eutectic grains (often named “eutectic cells”) presenting a single nodule (WETTERFALL et al. References: p. 830.
Ch. 8, 0 11
H. Biloni and WJ.Boettinger
818
\
CMh4ATI-I I
w m n ii u v
VERMICULAR GRAPHITE
N
UT
AUSTENITE
/
i
1154°C 11509c
RAKEY \ / GRAPHITE
1137OC
SPHEROIDAL GRAPHITE
//
,/
Fig.
Cooling curves for laminar, vermicular and no_.lar cast irons. BACKERLID et al. [1975].
[1972]). Thus, it is common practice in the foundry industry to associate the number of nodules to the number of eutectic grains. However recent research by SIKORAet al. [1990]) and BANERJEE and STEFANESCU [1991] indicated the existence of simultaneous nucleation of both the dendritic austenite and the spheroidal graphite. The interaction between both phases during solidification gives rise to the formation of eutectic grains presenting several nodules. This is a fact to be taken into account when micromodelling of the structure is attempted. Regarding the spheroidaI growth of the graphite, several theories exist in the literature and they have been reviewed by MINKOPF[1983], ELLIOT [1988] and STEPANESCU [1988]. MINKOFF[1990] considers that the relationship among supercooling, melt chemistry and crystalline defects determine the spheroidal growth of the graphite. In this case the screw dislocation mechanism is considered dominant in causing repeated instability of the pyramidal surfaces, so that a radial array of pyramids is formed (fig. 75b). (iv) Compact or Vermicular Graphite Irons: This intermediate graphite morphology has been studied extensively due to its technological importance (RIPOSANet al. [1985]). The graphite is interconnected within the eutectic cell but its growth differs from flake graphite. As in the case of spherulitic growth, several theories exist (STEPANESCU [1988], ELLIOT[1988]). The influence of the melt chemistry is very important. The occurence of compact graphite form requires a balance between flakepromoting elements, such as S and 0, spheroidizing elements such as Mg, Ce and La, and antispheroidizing elements such as Ti and Al (SUBRAMANIAN et al. 119851). MINKOFF[1990], in his general approach to the interdependence of supercooling,
Ch. 8, 9 11
Solidi@cation
819
b)
Fig. 75. (a) Mechanism of growth of flake graphite from a rotation boundary which provides steps for the nucleation of (1010) faces. (b) Mechanim of growth of spheroidal graphite by repeated instability of pyramidal surfaces forming a radial array. (c) Mechanism of growth of compacted graphite by development of pyramidal forms on the crystal surface at steps due to screw dislocations. MINKOFF[1990].
chemical composition and crystalline defects, considered that compact or vermicular graphite forms are intermediate between flake and spherulitic formation, and that the rounded morphology of the structure is a result of the thickening of graphite crystals at small values of supercooling by growth from the steps of screw dislocations, which have Burgers vector in the cOOOl> direction (fig. 7%). We consider that the mechanisms of graphite growth as well as modification by proper chemical agents is an open research field. Recent unidirectional solidification of flake and compact graphite irons with and without “in situ” modification in front of the S-L interface open questions about the forms of crystalline growth of the graphite phase, as well as the influence of the impurities and chemical agents on the surface tension of the Fe-G eutectic phases (ROVIGLIONE and HERMIDA[19941, ROVIGLIONE and BILQNI[1994]). Refererrces: p . 830.
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11.4. Influence of rapid solidification processes (RSP) Most of the fundamentals of RSP have been included in 5 2-8 and grain refinement due to high supercooling was described in 0 11.1.2. The present section considers how RSP can be used to manipulate microstructure.
11.4.1. Experimental and production methods The evolution of RSP methods goes back to the last century and its potential for the modification of microstructure were initiated by D u w ~ zand his collaborators at the beginning of the 60’s. They designed the Duwez Gun, a device in which a gaseous shock wave smashes a drop of about lOmg of molten alloy into contact with a copper substrate or chill block to produce small foils or splats. This method usually is called splat cooling. These authors were successful in making a continous series of Cu-Ag alloys without any two-phase structure. In the same year the first metallic glasses were also discovered ( D w z et al. [1960]) (see chapter 19, $8 2 and 3). Much earlier work in the field focused on equipment invention and evaluation, for example, the invention of melt spinning by POND[1958]. Subsequently many other methods have been developed and they have been reviewed by JONES [1982], CAHN [1983], LAVERNIA et al. [1992], and SURYANARAYANA et al. [1991]. In all the methods a cast sample is produced where at least one physical dimension is small assuring a rapid removal of the latent heat of fusion by an appropriate heat sink. The methods developed to achieve rapid solidification have been categorized into three main categories, (i) atomization, (ii) chill methods, and (iii) selfsubstrate quenching. (See also detailed discussion in ch. 19, 8 3). In atomization a fine dispersion of droplets is formed when molten metal is impacted by a high energy fluid; as a result of the transfer of kinetic energy from the atomising fluid (gas or liquid) to molten metal, atomization occurs. For example LAVERNIA et al. E19921 summarize the phenomena associated with the method considering that the size distribution of atomized droplets will depend on: (a) the properties of the material, such as liquidus temperature, density, thermal conductivity, surface tension, heat capacity and heat of fusion; (b) the properties of the gas, such as density, heat capacity, viscosity and thermal conductivity and (c) the processing parameters such as atomization gas pressure, superheat temperature and metaVgas flow ratio. One of the most important applications of the atomization method is powder fabrication. Powder is convenient for subsequent consolidation into near final shapes. Variants of the method include, among others, gas atomization, centrifugal atomization and spark erosion (KOCH[ 19881). A recent review by GRANT[1992] indicates that in recent years, atomization processes have been improved, new alloy compositions have been developed, hot isotatic pressing (HIP) has become an important processing tool, and new commercial product areas have emerged. One extension of the atomization method is spray atomization and deposition. Pionneering work in this area was performed by SINGER [1970], [1972]. The general principle of the method is to atomize a stream of molten metal and to direct the resulting spray onto a shaped collector, or mandril. On impact with the collector, the particles, often partially liquid, flatten and weld together to form a high density preform which can be readily forged to form a product. LAVE= et al. [1992] and GRANT[1992] reviewed
Ch. 8, 8 11
Solidi$cation
821
the different variables involved in the method. Additionally, GRANT[1992] reviewed further application of the method to the formation of continuous sheet. It seems clear that high potential exists for continuous forming of steel and other metals as strips, sheet, plate and other forms. Recently ANNAVARAPU and DOHERTY [19931 have developed an understanding of the microstructural development in spray forming. ChillMethods: Melt spinning is the technique most widely used in rapid solidification because it is easy to execute and the quenching rates compare favorably with other available processes. Melt spinning makes possible the production of long narrow ribbons up to a maximum width of 3 mm. This limitation has led to the development of a patented method, planar$ow casting (fig. 76a), and to renewed interest in the melt overfIow process originally patented in 1911 by STRANGE (fig. 76b) and now not protected by patents. Both methods can yield large quantities of wide ribbons cooled at rates approaching lo6 Ks-'. Sev-Substrate Quenching Methods: In this method the main objective is the modification of surface layers by rapid solidification. The heat source is typically a scanned or pulsed laser or electron beam focussed onto the specimen surface to cause rapid melting and resolidification. Spark discharge has also been used as a localized heat source. This field, especially when laser beams are used, have been reviewed among others by DRAPERand POATE[1985] and SLJRYANARAYANA et a2. [1991]. Because of the intimate thermal contact within the different regions of the sample, the flow of heat during solidification can be modeled simply by conduction into the underlying cold material. In addition resolidification does not require nucleation: the unmelted portion of a crystalline sample provides a growth source. Then the supercooling that exists in the process is only associated with the S-L interface and is due to kinetics of the growth.
11.4.2. Relationshipsbetween RSP and solidificationstructures The principal changes which can be brought about in crystalline alloys by RSP include, (i) Extension of solid-state solubility; (ii) refinement of grain size, with possible modification of grain shapes and textures; (iii) reduction or elimination of microsegregation;(iv) formation of metastable phases; (v) achievement of high density of point defects; (vi) surface alloying. Many aspects of RSP have been discussed earlier from thermodynamic and /or kinetic points of view. Several papers and reviews exist for particular metals and alloys as a consequence of the increasing importance of RSP. Among these contributions the reader is referred to LAVERNIA et al. [1992] for aluminum alloys; KOCH[1988] for intermetallic compounds; SURYAMARAYANA et al. [1991] for titanium alloys; GRANT[1992] for powder production by RSP methods; PAWLOSKIand FUCHAIS[1992] for thermally sprayed materials; DRAPER and POATE[1985] for laser surface alloying; CHEN et al. [1988a] [1988b] for laser surface modification of ductile iron. KLJRZand TRIVEDI [1989] have modeled the selection of the microstructure under given laser processing conditions. 11.5. Low gravity effects during solidification
The common features of research in this area are the drastic reduction of sedimentation and buoyancy driven convection. Space flight provides solidification research with References: p . 830.
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c r
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3
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b Fig. 76. (a) Planar flow casting in which the nozzle is positioned near the moving substrate to control the thickness of the cast tape. NARASHIMAM [1980]. See LAVERMAef al. [1992]. (b) Melt overflow process as described by Strange in 1911 patent. STRANGE [1911]. See LAVERNIA er al. [1992].
Ch. 8, 0 11
823
Solidijication
long duration access to microgravity. However there also exist several short duration free fall facilities. Figure 77 shows the available low-g experimental systems (CURRERI and STEFANESCU [1988]). Both drop tubes and drop towers are used extensively. A drop tube is an enclosure in which a molten droplet can solidified while falling freely either in vacuum or in a protective atmosphere. Microgravity conditions are obtained up to a maximum of about 2 s. By contrast in a drop tower an entire experimental package, which may include furnace and instrumentation as well as the specimen is dropped within an enclosure. At present the best facilities are at NASA-LRC and at ZARM (Bremen) with drop towers of 145 m and 110 m height where lod and g during free fall times of 5.2 and 4.7 s respectively are obtained. However, the 500 m drop tower planned at Sunagawa (Japan) and the Eurotube-Saar project of 1200 m high drop tower would give microgravity periods of 10 s and 12 s respectively (HERLACH etal. [1993]). On the other hand parabolic flight and suborbital sounding rockets (fig. 77) provide microgravity levels between 30 s and 5 min. The wide variety of experiments and disciplines involved in microgravity research opens a new field where solidification and mechanisms as modified by the near null value of gravity may be studied. On the basis of different reviews (JANSEN and SAHM
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[1984], MINKOFT[1986], CURRERIand STEFANESCU [1988], HERLACHet ul. [1993]) among others, and the numerous proceedings of microgravity symposia quoted by the preceding authors, the mechanisms under study can be summarized (i) distribution of the constituents in the initial fluid phase; (ii) phenomena of mass and heat transfer in the absence of bouyancy-driven convection. The drastic reduction of this source of convection leads to the increased influence of other mechanisms, which in general are masked on earth, namely: a) Murangoni convection; b) fluid movement as a result of expansion associated with phase changes or in the presence of electric, magnetic or thermoelectric fields; c) mechanisms of solute redistribution through diffusion; d) damping of temperature fluctuations in the fluid phase; e) effect of the absence of gravity on phase interfaces or meniscuses. iii) Techniques for forming or preparing materials without crucibles, direct formation of whiskers, thin films without substrates, production of hollow spheres with extremely thin walls, and float zone melting with large diameter; iv) in the more specific case of alloys in connection with casting structures: a) convection and soIute distribution, b) solidification of off-monotectic and eutectic alloys, c) morphological stability of the S-L interface, d) micro- and macrosegregation, e) grain multiplication and casting structures. 11.6. Solidificationprocessing of metal matrix composites
Thirty years ago KELLYand DAVIES [19651 and CRATCHLEY [19651 summarized their own and other pioneering efforts on metal matrix composites. Recently there has been a surge of interest in using reinforced metal matrix composites as structural materials. ASHBY[1993] indicates that any two materials can, in principle, be combined to make a composite in many geometries. A review of the processes for fabrication of MMCs is presented by GOSH [1990] covering a wide range of materials such as light alloy matrices, high temperature matrices and other special cases such as high thermal conductivity matrices and a variety of particulates, whiskers and fibers as reinforcements. Process methods may be divided as follows (i) liquid metal processes, (ii) solid state processes, (iii) deposition processes and (iv) deformation processes. Of these, solidification processing of MMC is gaining more and more importance because liquid metal is relatively inexpensive, and can flow easily to surround the reinforcing phases and create composites having a wide variety of shapes. ROHATGI[1988] has presented the state of the art related to the technical aspects of solidification processing for metal matrix and JIN [19921 have composite fabrication. MORTENSEN[1991bI and MORTENSEN [1991b] has classified the methods for producreviewed the fundamentals. MORTENSEN tion of MMCs into four classes on the basis of the mechanism by which the reinforcement and metal are combined (fig. 78 a-d). l.Znfi&&n. (fig. 78A) The reinforcement phase is stationary and essentially constitutes a very fine and intricate mould into which the liquid metal flows to fill all open porosity. 2. Dispersionprocesses (fig. 78B) The reinforcement phase is discrete and is added to the metal. Entrainment of the reinforcing phase into the melt is affected by agitation, which acts on the reinforcement via viscous shear stresses. 3. Sprayprocesses (fig. 78C) The metal is divided into molten droplets and sprayed with,
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Solidijcation
A
‘ I C
‘Di
Fig. 78. Schematic representation of the principal classes of metal matrix composite solidification processes: A: infiltration, B: dispersion, C: spray-casting and D: in-situ processes. The reinforcing phase is black, the [1991b]. metal gray. MORTENSEN
or onto, the reinforcement. Incorporation of the reinforcement into the matrix is affected by the added surface energy of the metal and the kinetic energy of moving droplets and particles. 4. In situ-processes (fig. 78D) Essentially consists of obtaining composites via directional solidification. In contrast with the above methods, “in-situ” processes use conventional alloying and not an artificial combination of two phases. Thus the material can have exceptional high temperature capabilities. MORTENSEN and JIN [ 19921 present a comprehensive review of the physical phenomReferences: p . 830.
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Ch.8, 5 11
ena that govern infiltration and dispersion processes, which describes the governing phenomena of the process as being connected with the three main steps of metal matrix composite solidification. The j r s t step involves the interaction between the liquid matrix and the reinforcement material. This step is governing by the wetting between the metal and the reinforcing phase that are being combined. In most cases wetting is generally unfavorable and positive pressure and hence external work must be provided to create the composite (MORTENSEN [1991bl).From an elementary thermodynamic analysis a minimum amount of energy W per unit volume of composite is equal to
w = 4(Ym - YAR),
(121)
where Ai is the surface area of the reinforcemenVmatrix interface per unit volume of = reinforcement/molten metal interfacial energy; ym = reinforcement/ composite; yLR atmosphere interfacial energy. MORTENSENand JIN [1992] discussed the methods to measure wettability and to improve it namely by: (a) reinforcement pretreatment, (b) alloying modification of the matrix and (c) reinforcement coating. From a chemical point of view, knowledge about the surface of the metal matrix, the reinforcement surface and the interface chemistry as well as the influence of alloy additions and reactive wetting are described. The secund step corresponds to fluid flow, heat transfer and solidification phenomena that takes place in the composite material during infiltration and before it is fully solidified. The mechanics of infiltration and thermal and solidification effects, as well as processing of metal matrix composite slumes, (rheology and particle migration) must be considered. The third step corresponds to completion of the solidification process. The solidification of the metal matrix is strongly influenced by the presence of the reinforcing phase that can affect the nucleation, coarsening, microsegregation and grain size. In the case of particulate composites, particle pushing theories and experiments discussed in 5 9 are very important. In conclusion it appears that the solidification processing of MMCs is an open and fascinating research area. It now stands at a point where the most essential phenomena are beginning to be clarified. On the other hand, the engineering potential of metal matrix composites is continuously increasing. Reinforced metals have been introduced in a growing number of applications, ranging from the sports industry, electronics and aerospace to the automotive sector. Current estimates indicate that the number of applications and their commercial significance will grow significantly over the next and OLSON[1990]). decade (WHITE
11.7. Semisolid metal forming processes When metals or alloys solidify in castings and ingots, a dendritic structure forms that develops cohesion when the alloy is as little as 20% solid. Thereafter strength develops rapidly. When the casting is deformed during solidification, deformation takes place preferentially along grain boundaries; grains slide and roll over one another with small welds forming and breaking. Occasional bending of dendrite arms in the neighbourhood
Ch. 8 , Q 11
827
Solidif cation
of the region of deformation also occurs. As deformation proceeds, open fisures may form (hot tears) and be fed by liquid. In this case segregated regions, sometimes called jilled hot tears result, yielding V segregates as mentioned in $ 9 (FLEMINGS and MEHRABIAN [1971]). In the course of this type of macrosegregation research, METZand FLEMINGS [1969a], [1969b] performed experiments where small blocks of AI alloys were isothermally sheared. They found negligible strength below about 0.2 fraction solid. SPENCER er al. [1972] carried out similar tests of Sn-15pct Pb alloy. Their test apparatus consisted of two grooved, counter-rotating cylinders. Figure 79a) shows the result obtained. Cleverly SPENCER [1971] decided to use the same apparatus to conduct a quite different type of test. Instead of partially solidifying the alloy before beginning the shear, they began the shear above the liquidus and then slowly cooled the alloy into the solidification range (fig. 79b). Two major differences are noted: (i) there is a change from ROTATING OUTER CYLINDER
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828
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FRACTION SOLID, fs Fig. 79. (a) Maximum shear strength of semisolid dendritic Sn-15wt pet Pb alloy vs. fraction solid obtained in isothermal experiments. (at a shear rate of 0.16 s-'). Schematic upper right shows test arrangement. (b) Viscosity and shear stress vs. fraction solid for Sn-15 wt pet Pb alloy cooled at 0.006 Ks? and sheared continuously at 200 s-'. Schematic at upper right is an illustration of the test specimen. FLEMINGS[1991].
dendritic to non-dendritic structure; (ii) the shear stress is diminished for the dendritic structure compared to the nondendritic structure. For example, at f, = 0.4 the shear stresses are 200 KPa and 0.2 KPa respectively. The material behaves as a liquid-like
Ch. 8, 0 11
Solidificntion
829
slurry to which an apparent viscosity can be assigned, as has been done in fig. 79b). The fundamentals of the semi-solid metal (SSM) forming processes lie in these results. [1991] presented an extensive review of the development of this Recently FLEMINGS processes during the last 20 years as well as the current industrial applications. Also KENNEYet al. [1988] reviewed this area, especially from an applied and technological point of view. Different names to characterize SSM forming processes exist in the literature. In rheocasting (from the Creek “rheo” to flow) strong shear forces break off dendrite fragments. If the alloy is poured when the viscosity is still low, it can be made to fill the mould. Each dendrite fragment becomes a separate crystal and a very fine grain size can be achieved without the disadvantages inherent in the use of grain-refining additions. The process also affects other casting features because the alloy is already partialy frozen when cast. Thus, shrinkage is reduced and economy in risers and gating may be substantial. Pouring temperature can also be much lower and as a consequence the amount of heat to be removed is lower and thermal stresses are reduced. Through this method all kinds of castings can be made including continuous rheocasting with or without electromagnetic stirring. The method can be used to prepare material to be utilised in subsequent casting processes. The prepared slugs are heatly rapidly to the partially molten state, dropped in the die-casting machine and forged under pressure into the mould. This method is named thixocasting. One important advantage is the dramatic improvement in die life due primarily to reduced metal temperature. Compocasting corresponds to the development and production of metal-matrix composites containing nonmetallic particles, taking advantage of the rheological behaviour and structure of the partially solidified and agitated matrix. The particulate or fibrous nonmetals are added to the partially solid alloy slurry. The problems of wettability and their solutions are similar to those discussed in 0 11.6. and the reader is and JIN [I9921 review of MMC processes. The high viscosity referred to the MORTENSEN of the slurry and the presence of a high volume fraction of primary solid in the alloy slurry prevents the nonmetallic panicles from floating, setting or aglomerating. With increasing mixing times, after addition, interaction between particles and the alloy matrix promotes bonding. The composites are subsequently reheated to the partially molten state in a second induction furnace and forged into shape with hydraulic presses. Very promising wearresistant alloys have been obtained based on work by SATOand MEHRABIAN [1976] in Al alloys containing particulate additions of Al,O, and Sic. MATSUMIYA and FLEMINGS E19811 extended the application of SMM to stripcasting and the basic technology of SMM provides a potential means of metal purification (MEHRABIAN et al. [1974]).
Acknowledgements The authors wish to acknowledge the contributions of their various co-workers over the years and the financial support of many agencies. Special thanks are in order for S.R. Coriell and E W. Gayle for criticisms of this manuscript, and to Miss Graciela Martinez for her effort and patience in the elaboration of the manuscript. References: p. 830.
830
H. Biloni and WJ. Boettinger
Ch. 8,Refs.
References* ADAM,C. M. and L. M. HOGAN,1972,J. Australian Inst. of Metals, 17,81. ADORNATO, M. and R. A. BROWN,1987,J. Cryst. Growth 80,155. AGUILAR RIVAS,R.A. and H. BILONI,1980 a), Z.Metallk. 71,264. AGWLARRIVAS,R. A. and H. BILONI,1980 b), Z.Metallk. 71,309. ANDREWS, J.B., A.L. SCHMALE, and A.C. SANDLIN,1992,J. Cryst. Growth 119,152. ANNAVAIWPU,S. and R.D. D O ~ T Y1993, , Int. J. of Powder Metall. 29, 331. APELLW, D., 1982,in: Aluminum Transformation Technology and Applications 1981,eds. C. A. Pampillo, H. Biloni, L. Mondolfo and F. Sacchi (ASM, Metals Park, OH) p. 423. APTEKAR, J. L. and D. S. KAMENETSKAYA, 1962,Fiz Metal, Metalloved, 14, 123. ASHBY, M.F., 1993,Acta Metall Mater. 41, 1313. AUDERO,M. A. and H.BILONI,1972,J. Cryst. Growth 12,297. AUDERO,M.A. and H.BILONI,1973,J. Cryst. Growth 18,257. AZIZ, M.J., 1982,J. Appl Phys. 53, 1158. AZIZ,M. J. and T. KAPLAN,1988,Acta Metall. 36,2335. AZIZ, M. J. and W. J. BOETCINGER, 1994, Acta Metall. Mater. 42,527. BACKERUD, L.,K.NILSSON and H. STEEN,1975,in: Metallurgy of Cast Iron (Georgy Pub. Co., St. Saphorin, Switzerland), p. 625. *BAKER,J. C., 1970,Interfacial Partitioning During Solidification, Ph. D. Thesis, Massachusetts Institute of Technology, Chapter V, (see also CAHNetal. [1980]). BAKER,J.C. and J. W. CAHN,1971,in: Solidification (ASM, Metals Park, OH), p. 23. BAKER,C. and V. SUBRAMANIAN, 1980,in: Aluminum TransformationTechnology and Applications 1979,eds. C. A. Pampillo, H. Biloni and D. E. Embury (ASM, Metals Park, OH.) p. 335. BANFXJEE, D. K.and D.M. STEFANESCU, 1991,Trans. AFS, 99,747. BASKARAN, V., W. R. WILCOX,1984,J. Cryst. Growth 67,343. BATTLE, T. P., 1992,Int. Mat. Rev. 37,249. BATTLE,T. P., and R.D. PEHLKE,1990, Metall. Trans. 21B, 357. BECKERMANN, C. and R. VISKANTA, 1988,Physicochem. Hydrodyn. 10, 195. BENNON,W. D. and F.P. INCROPERA, 1987 a), Int. J. Heat Mass Transfer 30, 2161. BENNON,W. D. and F. P. INCROPERA, 1987 b), Int. J. Heat Mass Transfer 30,2171. BERRY,J. T. and R. D. RHLKE, 1988,in: Metals Handbook, ninth ed., 15: Casting (ASM, Metals Park, OH), 858. BERTORELLO, H. R., and H. BILONI,1969,Trans. Met. SOC.AIME 245, 1375. BILLIA,B. and R. TRIVEDI,1993,in: Handbook of Crystal Growth 1B: Fundamentals, Transport and Stability, D. T. J. Hurle, ed., (North-Holland, Amsterdam) p. 899. *BILONI,H., 1968,in: The Solidification and Casting of Metals (Iron and Steel Inst., London) Pub. 110,p. 74. BILONI,H., 1970,Metallurgia ABM (Aso. Brasileira de Metais) 26, 803. BILONI,H., 1977,Ciencia Interamencana 18 (M),3. BILONI,H., 1980,in: Aluminum Transformation Technology and Applications 1979,eds. C.A. Pampillo, H. Biloni and D. E. Embury (ASM, Metals Park, OH) p. 1. BILONI,H., 1983, Solidification, in: Physical Metallurgy, 3rd. ed, Ed. R.W. Cahn and P. Haasen (North Holland, Amstdam), p. 478. BILONI,H. and B. CHALMERS, 1965,Trans. Met. SOC.AIME, 233, 373. BILONI, H.,G.F. BOLLINGand H. A. DOMIAN,1965 a), Trans. Met. Soc. AIME 233, 1926. BILONI, H.,G.F. BOLLINGand G. S. COLE,1965 b), Trans. Met. SOC.AIME 233, 251. BILONI,H.,G. F. BOLLING,and G. S. COLE,1966,Trans. Met. SOC.AIME 236,930. BILONI,H., R. DI BELLAand G.F. BOLLING,1967,Trans. Met. SOC.AIME239.2012. BILONI,H. and B. CHALMERS,1968,J. Mat. Sci. 3, 139.
* References marked with an asterisk are suitable for Further Reading.
Ch. 8, Refs.
Solidificatwn
83 1
BILONI,H. andR. MORANDO,1968, Trans. Met. SOC. AIME242, 1121. BLODGETT, J. A., R. J. SCHAEFER, and M. E. GLICKSMAN, 1974, M. E., Metallography 7, 453. BOBADILLA, M., J. LACAZE,G. J. LESOULT,1988, J. Cryst. Growth 89, 531. W. J., 1974, Metall. Trans. 5, 2023. BOETTINGER, BOETTINGER, W. J., 1981, in: Proc. Fourth Intl. Conf. on Rapidly Quenched Metals, edited by T. Masumoto and K. Suzuki, (The Japan Institute of Metals, Sendai, Japan), p. 99. BOETTINGER, W. J., 1982, in: Rapidly Solidified Crystalline and Amorphous Alloys, B. H. Kear and B. C. Giessen, eds., (Elsevier North Holland, NY), p. 15. BOETIINGER, 1988, unpublished research, NIST, Gaithersburg, MD, U.S.A. BOETTINGER, W. J., D. SHECHTMAN, R. J. SCHAEFER,and E S. BIANCANIELLO, 1984, Metall. Trans. 15A, 55. *BOETTINGER, W. J. and J. H. PEREPEZKO, 1985, in: Rapidly Solidified Crystalline Alloys, S. K. Das, B. H. Kear, C. M. Adam, eds., (TMS-AIME), p. 21. BOETTINGER, W. J., L. A. BENDERSKY and J.G. EARLY,1986, Metall. Trans. 17A, 781. BOEITMGER, W. J., and S. R. CORIELL, 1986, in: Science and Technology of the Supercooled Melt, P. R. Sahm, H. Jones and C.M. Adam, eds., (NATO AS1 Series ENo. 114, Martinus Nijhoff, Dordrecht), p. 81. B O E ~ G E W. R , J., L. A. BENDERSKY, S. R. CORIELL,R. J. SCHAEFER, and F.S. BIANCANIELLO, 1987, J. CIyst. Growth 80, 17. BOETTINGER, W. J., L. A. BENDERSKY, E S. BIANCANIELLO and J. W. CAHN,1988a), Mater. Sci. Eng. 98,273. BOETTINGER, W, J., S. R. CORIELL, and R. TRIVEDI, 1988b), in: Rapid Solidification Processing: Principles and Technologies R. Mehrabian and P.A. Parrish, eds., (Claitor's Publishing, Baton Rouge), p. 13. R. J. SCHAEFERand E S. BIANCANIELLO, 1988~.Metall. Trans. 19A, BOETTINGER,W. J., L. A. BENDERSKY, 1101. BOETTMGER,W. J., and M. J. AZIZ, 1989, Acta Metall. 37, 3379. BOLLING,G.F., 1971, in: Solidification (ASM, Metals Park, Oh.), p. 341. BOWER,T. F., H. D. BRODY,and M. C. FLEMINGS, 1966, Trans. Met. SOC.AIME 236,624. BOWER,T.F. and M.C. FLEMINGS, 1967, Trans. Met. SOC. AIME 239, 1620. BOWER,T. F.,D. A. GRANGER and J. KEVERIAN, 1971, in: Solidification (ASM Metals Park, OH), p. 385. BRAUN,R. J., and S. H. DAVIS,1991, J. Cryst. Growth 112, 670. BRICE,J. C., 1973, in: The Growth of Crystals from Liquids (North Holland, Amsterdam), p. 120 BRIMACOMBE, J. K., 1993, Metall. Trans., 24B,917. BRIMACOMBE, J. K., and I. V. SAMARASEKERA, 1990, in: Principles of Solidification and Materials Processing, T. 1, (Trans, Tech. Publication), p. 179. BRODY,H. D., 1974, in: Solidification Technology, eds. J. J. Burke, M. C. Flemings and A. E. Quorum (Brook Hill Pub., Chesnut Hills, MA) p. 53. BRODY,H.D. and FLEMINGS, M.C., 1966, Trans. Met. SOC. AIME 236, 615. BRODY,H. D., and S. A. DAVID,1979, in: Solidification and Casting of Metals (The Metals Society, London) p. 144. 1981, J. Chem Phys. 74,4029. BROUGHTON, J.Q., A. Bo~ssnur,and F.F. ABRAHAM, BROUGHTON, I. Q., G. H. GILMER,and K. A. JACKSON,1982, Phys. Rev. Lett.49, 1496. BROWN,R. A., 1988, J. AiChE J., 34,881. BROWN,S.G.R. and J. A. SPITTLE,1989, Mat. Sc. Tech. 5, 362. BURDEN,M.H., D. J. HEBDITCH and J.D. HUNT,1973, J. Cryst. Growth 20, 121. BURDEN,M. H. and J.D. HUNT, 1974a). J. Cryst. Growth 22,99. BURDEN,M. N., and J.D. HUNT: 1974b), J. Cryst. Growth 22, 109. BURDEN,M. H., and J.D. HUNT, 1974c), J. Cryst. Growth 22 328. BURTON,J.A., R.C. PRIMand W. P. SCHLICHTER,1953, J. Chem. Phys. 21, 1987. G., 1986, Arch. Rational Mech. Anal. 92,205. CAGINALP, CAHN,J. W., 1960, Acta Metall. 8, 554. CAHN,J. W., 1967, in: Crystal Growth, ed. H. S. Peiser (Pergamon Press, Oxford) p. 681. CAHN,J. W., 1979, Metall. Trans. 10A, 119. CAHN,J. W., W, B. HILLIG,and G. W. SEARS,1964, Acta Metall. 12, 1421. CAHN,J. W., S. R. CORIELL and W. J. BOEXTINGER, 1980, in: Laser and Electron Beam Processing of Materials, C. W. White and P. S. Peercy eds. (Academic Press, NY),p. 89.
832
H.Biloni and WJ,Boeitinger
Ch. 8, Refs.
CAHN, R W., 1983, in: Physical Metallurgy, Ch. 28,3er. ed.,eds. R. W. Cahn and P. Haasen (North Holland), p. 1709. CALVO,C., and H. BILONI, 1971.2. Metallk. 62, 664. CAMEL,D. and J. J. FAVIER,1984 a), J. Cryst. Growth 67, 42. CAMEL,D. and J. J. FAVIER, 1984 b), J. Cryst. Growth 67, 57. CAMPBELL,1981, Int. Met. Rev. 26, 71. CAMPBELL,I., 199la), Castings (Butterworth-Heinemann, London). I., 1991 b), Mat. Sc. and Tech. 7, 885. CAMPBELL, CAMPBELL, J., 1991 c), Cast Metals 4, 101 CAMPBELL, J. and I.D.OLLIFF, 1971, AFS Cast Metals Research J. (ASM Metals Park, OH) May, 55. CANTOR,B.and R. DOHERTY, 1979, Acta Metall. 27, 33. CAROLI,B., C. CAROLIand B. ROULET, 1982, I. Physique 43, 1767. CARRAD,M., M. GREMAUD,M. ZIMMERMANN, and W. KURZ, 1992, Acta Metall. Mater. 40,983. CARRUTHERS,J.R., 1976, Thermal Convection Instabilities Relevant to Crystal Growth from Liquids in: Preparation and Properties of Solid State Materials, vol. 2 (Marcel Decker, N. Y.) CECH,RE., 1956, Trans. Met. SOC. AIME 206, 585. CHADWICK,G.A., 1963, in Progress in Materials Science, 12, (Pergamon Press, Oxford) edited by B. Chalmers, p. 97. CHADWICK, G. A., 1965, Brit. J. Appl. Phys. 16, 1095. CHALMERS,B., 1963, J. Aust. Inst. Metals 8,255. *CHALMERS, B., 1964, Principles of Solidification (Wiley, N.Y.). CHALMERS, B., 1971, in: Solidification (American Society for Metals, ASM, Metals Park, OH) p. 295. CHANG,Ch. J. and R. A. BROWN,1983, J. Cryst. Growth 63, 343. CHEN,C. H.,C.P. JU and J.M. RIGSBEE,1988a), Mat. Sci. and Tech. 4, 161. C ~ NC.,H., C. P. Ju and J. M. RIGSBEE, 1988b), Mat.Sci. and Tech. 4, 167. CHEN,S. W., and Y. A. CHANG, 1992, Metall. Trans. 23A, 1038. *CHERNOV,A. A, 1984, Modern CrystallographyIll Crystal Growth, (Springer-Verlag, Berlin). CHEVEIGNE de, S., C. GUTHMANN, P. KUROSKI,E. VICENTE and H. B ~ N I , 1988, J. Cryst. Growth 92,616. CHOPRA,M. A, M.E. GLICKSMAN, and N.B. SINGH, N.B., 1988, Metall. Trans 19A, 3087. CLAP^^^, L. and R. W. SMITH,1988, J. Cryst. Growth 92,263. CLYNE,T. W. and A. GARC~A,1980, Int. J. Heat and Mass Transfer 23,773. CLYNE, T. W., 1980 a), J. Cryst. Growth 50, 684. CLYNE,T. W., 1980 b), J. Cryst. Growth 50, 691. CLYNE,T.W., 1984, Metall. Trans. 15B, 369. CLYNE,T. W. and A. GARCIA,1981, J. Mat. Sci. 16, 1643. COCHRANE,R.F., D.M. HERLACH and B. FEUERBA~HER, 1991, Mat. Sci. Eng. A133,706. *COLE, G. S., 1971, in: Solidification (ASM Metals Park, OH), p. 201. Corn, G.S. and G.F. BOLLING,1968, Trans. Met. SOC.AIME 242, 153. COLE,G. S. and G.F. BOLLING,1969, in: Proc.16th. Sagamore Army Materials Res. Conf. Conference quoted by BOLLING[1971]. COLLIGAN,G. A., V. A. SUPRENANT and F.D. LEMKEY, 1961, J. Metals 13,691. CORIELL,S. R., D.T. J. HURLEand R.F. SEKERKA, 1976, J. Cryst. Growth 32, 1. CORIELL,S.R. and R.F. SEKWKA,1979, J. Cryst. Growth 46,479. CORIELL,S.R, R.F. BOISVERT,R. 0. REHMand R.F. S-, 1981, J. Cryst. Growth 54, 167. CORIELL,S.R. and D. TURNBULL, 1982, Acta Metall. 30,2135. CORIELL,S. R. and R. F. SEKERKA, 1983, J. Cryst. Growth 61, 499. CORIELL,S. R., G. B. MCFADDEN and R. F. SEKERKA,1985, Ann Rev. Mat. Sci 15, 119. CORIEU,S. R., G.B. MCFADDEN, P. W. VOORHEES and R.F. SEKERKA,1987, J. Cryst. Growth 82, 295. CORIELL,S.R., and G.B. MCFADDEN,1989, J. Cryst. Growth 94,513. CORIELL,S. R, and G. B. MCFADDEN,1990, in: Low Gravity Fluid Dynamics and Transport Phenomena, 4s. J. N. Koster and R. L. Sani, vol. 130 of Progress in Astronautics and Aeronautics AIAA, Washington D. C., p. 369. CORIELL, S.R., G. B. MCFADDEN,and R.F. SEKERKA,1990, J. Cryst. Growth 100,459.
Ch. 8, Refs.
Solidijcatwn
833
*CORIELL,S.R. and G.B. MCFADDEN,1993, in: Handbook of Crystal Growth ed. D.T.J. Hurle (Elsevier Science Publishers, Amsterdam) vol. lb, p. 785. CORIELL,S. R. and W. J. BOETTINGER,1994, NIST, unpublished research. and T.K. CLUHAN,1988, Metals Handbook, 9th.d.. (ASM, Metals Park, OH) CRAIG,D. B., M. J. NORNUNG 15 “Casting”, 629. CRATCHLEY, D., 1965, Metall. Rev. 10, (37), 79. CROKER,M.N., R.S. FIDLER, and R. W. SMITH,1973, Proc. Roy. SOC.London A335, 15. CROSSLEY, P. A. and L. F. MONDOLFO, 1966, Modern Casting 49, 89. CURRERI,P.A. and D.M. STEFANESCU, 1988, Metals Handbook, 9th.ed., (ASM, Metals Park, OH) 15 “Casting”, 147. DANTZIG,J. A. and S. C. Lu, 1985, Metall. Trans. la,195. DANTZIG,J. A. and 5. W. WE%, 1985, Metall. Trans. 16B, 203. DANTZIG,J. A. and J. W. WIESE, 1986, Advanced Manufacturing Processes 1, 437. DAS, S. and A. J. PAUL,1993, Metall. Trans. 24B,1073. DAVID,S. A. and J.M. VITEK, 1989, Int. Met. Rev. 34, 213. DAVID,S. A., J. M. V ~ KM., RAPPAZand L. A. BOATNER,1990, Metall. Trans. 2lA, 1753. DAVID, S. A. and J. M. VITEK,1992, in: The Metal Science of Joining, eds. A. J. Cieslak, J. H.Perepezko, S. Kang and M.E. Glicksman, (TMS Pub., Warrendale, PA) p. 101. DAVID, S. A., T. DEBROYand J.M. VITEK,1994, MRS Bull. xix, No 1,29. DAVIES, G. J. and J. 6. GARLAND, 1975, Int. Met. Rev. 20, 83. DAYTE,V., and J.S. LANGER,1981, Phys. Rev. B 24,4155. DAYTE,V., R. MATHURand J.S. LANGER,1982, J. Stat. Phys. 29, 1. DERBY,B., and J. J. FAVIER, 1983, Acta Metall. 31, 1123. DEVAU,G. and D. TURNBULL, 1987, Acta Metall. 35,765. DRAPER,C. W. and J.M. POATE, 1985, Int. Met. Rev. 30,no 2, 85. Duwez, P., R.H.WILLENSand W. KLEMENT,1960, I. Apply Phys. 31, 1136. EASTERLING,ICE., 1984, Mat. Sci. and Eng. 65, 191. ECKLER,K., RF. COCHRAN, D. M. HERLACH,B. FELIERBACHER, and M. JURISCH, 1992, Phys. Rev. B 45, 5019. ELMER,I. W., M. J. AZIZ, L.E. TANNER,P.M. SMITH,and M. A. WALL, 1994, Acta Metall. Mater. 42, 1065. ELLIOT,R., 1964, Br. Foundryman 9,389. ELLIOT,R., 1988, in: Cast Iron Technology (Butterworths, London) EMLEY,E. E, 1976, Int. Met. Rev. 21, 175. ESHELMAN, M.A., V. SEETHARAMAN and R. TRIVEDI,1988, Acta Metall. 36, 1165. ESHELMAN, M. A,, and R. TRIVEDI, 1988, Scripta Metall. 22, 893. FAINSTEIN--PEDRAZA, D. and G.F. BOLLING,1975, J. Cryst. Growth 28,311. FAVW,J. J., 1981a), Acta Metall. 29, 197. FAVIER, J. J., 1981b), Acta Metall. 29, 205. FAVIER, I. J., 1990, 5. Cryst. Growth 99, 18. FAVIER,J. J. and A. ROUZAUD,1983, J. Cryst. Growth 64,367. FAVIER, J. J. and D. CAMEL,1986, J. Cryst. Growth 79, 50. FAVIER, J. J., and L.0. WiLsoN, 1982, J. Cryst. Growth 58, 103. FF.CHT,H. J., and I. H. PEREPEZKO, 1989, Metall. Trans. ZOA, 785. FELICELLI, S.D.,J. C. HEINRICHand D. R. POIRIER,1991, Metall. Trans. 22B, 847. FELLIU, S., L. LUIS,D. SIGUINand J. ALVAREZ,1962, Trans. AFS 70, 145. FEURER,U., and R. WUNDERLIN, 1977, DGM Fachber. (Oberusel, FRG). See also KURZand FISHER [1989]., p. 257. FISHER,D. J. and W. KURZ,1979, in: Solidification and Casting of Metals (The Metals SOC.,London), p. 57. FISHER,D. J., and W. KURZ, 1980, Acta Metall. 28,777. *FLEMINGS, M. C., 1974, SolidificationProcessing (McGraw Hill, New York). FLEMINGS, M. C., 1976, Scand. J. Metall. 5, 1. FLEMINGS, M. C., 1991, Metall. Trans. 22A, 957. FLEMINGS, M. C. and G. E. NEREO,1967, Trans. Met. SOC.AIME 239, 1449.
834
H. Biloni and WJ.Boettinger
Ch. 8, Refs.
FLEMINGS,M. C., D. R. POIRIER,R V. BARONE,and H. D. BRODY,1970, J. Iron and Steel Inst. 208,371. *FLEMINGS, M. C. and R. MEHRABIAN, 1971, in: Solidification (ASM, Metals Park, OH) p. 311. FLEMINGS, M.C. and Y. SHIOHARA, 1984, Mat. Sci. and Eng. 65, 157. FLOOD,S. C. and J.D. HUNT,1987a), J. Cryst. Growth 82, 543. FLOOD,S. C. and J.D. HUNT,1987b), J. Cryst. Growth 82, 552. FLOOD,S.C. and J. D. HUNT,1988, Metals Handbook 9th.ed., (ASM, Metals Park, OH) 15 “Casting“, p. 130. FRANK,F. C., 1949, Disc. Farad. SOC.5 4 8 . FREDRIKSSON. H., and M. HILLERT,1972, Scand. J. Metall. 2, 125. FREDRIKSSON, H., M. HILLERTand N. LANGE,1973, I. Inst. Metals 101,285. FREDRIKSSON, H., and T.NYLEN,1982, Metall. Sci. 16, 283. FRENKEL,J., 1932, Phys. Z. Sowjetunion 1, 498. GANESAN,S. and D.R. P o r n , 1989, J. Cryst. Growth 97, 851. GANESAN, S. and D.R. POIRIER,1990, Metall. Trans. 21B, 173. GARBELLINI,O., H. PALACIO and H. B m m 1990, Cast Metals 3,82. GARCIA, A. and M. PRATES,1978, Metall. Trans. 9B, 449. GARCIA,A., T. W. CLYNEand M. PRATES,1979, Metall. Trans. lOB, 85. GARCIA,A. and T.W. CLYNE,1983, in: Solidification Technology in the Foundry and Casthouse, ed. J.A. Charles (The Metals Society, London), p. 33. GETSELEV, Z. N., 1971, J. Metals 23, 38. GILL, S. C. and W. KURZ, 1993, Acta Metall. Mater. 41, 3563. GIOVANOLA, B., and W. KURZ., 1990, Metall. Trans.21A, 260. GLASSON, E. L. and E.F. EMLEY,1968, in: The Solidification of Metals (Iron and Steel Inst., London) Public. 110, p. 1. GLICKSMAN, M. E., 1965, Acta Metall. 13, 1231. GLICKSMAN, M. E., 1981, in: Aluminum Transformation Technology and Applications 1981, eds. C. A. Pampillo, H. Biloni, L. Mondolfo and E Sacchi (ASM, Metals Park, OH), p. 347. and J. D. AYERS, 1976, Metall. Trans. 7A, 1747. GLICKSMAN, M. E., R. J. SCHAEPER, GLICKSMAN,M. E., S. R. CQRIELLand G. S. MCFADDEN,1986, Ann. Rev. Fluid Mech. 18,307. *GLICKSMAN, M. E., and S. P. MARSH,1993, in: Handbook of Crystal Growth 1B: Fundamentals, Transport and Stability, D. T. J. Hurle, ed.,(North-Holland, Amsterdam) p. 1075. GOSH,A. K., 1990, in: Principles of Solidification and Materials Processing (Oxford and IBH Publication Co.) p. 585. GRANGER, D.A. and R. ELLIOT,1988, Metals Handbook, 9th. ed., (ASM, Metals Park, OH) 15 “Casting”, p. 159. GRANT,N. J., 1992, Metall. Trans. 23A, 1083. GREMAUD, M., W. KURZ, and R. TRIVEDI,1987, unpublished work. See TRIWDIand KURZ [1988]. GREMAUD, M., M. CARRAD and W. KURZ, 1991, Acta Metall. Mater. 39, 1431. GRUGEL,R.N., A. HELLAWELL, 1981, Metall. Trans. 12A, 669. GRUGFX, R.N., T.A. LOGRASSO,A. HELLAWELL, 1984, Metall. Trans. 15A, 1003. GULLIVER,G. It, 1922, Metallic Alloys, ed. Charles Griffin (London) p. 397. GUNDUZ, M., and J. D. HUNT, 1985, Acta Metall. 33, 1651. HANNA,M.D.,SHU-Zu Lu and A. HELLAWELL, 1984, Metall. Trans. lSA, 459. HAO,S. W., Z.Q.ZHANG,J.Y. CHEN and P.C. LN, 1987, AFS. Trans. 95,601. HARDY, S. C., G.B. MCFADDEN,S. R. CORIELL,P.W. VOORHEES, and R. E SEKERKA, 1991, J. Cryst. Growth 114, 467. HAYES, A. and J. CHIPMAN,1938, Trans. Met. SOC. AIME 135, 85. HEIPLE,C. R., P. BURGARDT and J. R. ROPER,1984, in: Modeling of Casting and Welding Processes, m.The Met. Soc. of AIME, Wamendale. Ed. J. A. Dantzing and J.T.Berry, p. 193. HELLAWELL, A., 1970, h g . Mater. Sci. 15, 3. HELLAWELL,A., 1979, in: Solidification and Casting of Metals (The Metals So., London) p. 161. HELLAWELL, A., 1990, in: F. Weinberg International Symp. on SolidificationProcessing, ed. V. E.Lait and I. V. Samarasekera (Pergamon Press), p. 395. HENZEL, J. G. Jr. and J. KEVERIAN, 1965, J. Metals 17, 561.
Ch. 8, Refs.
Solidification
835
*HERLACH, D. M., R.F. COcHRAm, I. WRY,H. J. FECHTand A.L. G m ,1993, Internat. Mat. Reviews 38, no 6, 273. HILLERT,M., 1953, Acta Metall. 1, 764. HILLERT, M., 1957, Jemlumtorets Ann. 141, 757. HILLERT, M., 1979, in: Solidification and Casting of Metals, The Metals Society, London, p. 81. HILLERT, M. and V. V. SUBBARAO,1968, in: Solidificationof Metals (Iron and Steel Inst., London), Publication no 110, p. 204. HILLIG,W.D. and D. Tmmuu, 1956, J. Chem. phys. 24,219. HILLIG,W. D., 1966, Acta Met. 14, 1868. HILLS,A. W. D., S. L. MALHOTRA and M. R. MOORE, 1975, Metall. Trans. 6B, 131. Ho, K. and R.D. Emm, 1984, AFS Trans. 92,587. Ho, K. and R. D. PEHLKE, 1985, Metall. Trans. 16B, 585. HOAGLUND, D. E., M. E. Azrz, S. R. STIFFLER, M. 0. THOMSON, J. Y. TSAOand P. S. PEERCY, 1991, J. Cryst. Growth 109, 107. HOFFMEYER, M. K. and J. H. PEREPEZKO, 1988, Scripta Metall. 22, 1143. HOFFEAEYER, M. K. and J. H. ~EREPEZKO, 1989a) in:Light Metals 1989, Ed. P. G. Campbell, ( T M S , Warrendale, PA), p. 913. H o m m , M.K. and J.H. PEREPEZKO, 1989b) Script. Metall. 23,315. H o ~ R M., K. and J. H. PEXEPEZKO, 1991, in: Light Metals 1991, Ed. E. L. Roy, (TMS, Warrendale, PA), p. 1105. *HOGAN,L. M., R. W. KRAFT, and F.D. LEMKEY,1971, in: Advances in Materials Research, Vol. 5, (Wiley, New York), edited by H. Herman, p. 83. HOLLOMON, J.H., and D. TURNBULL, 1953, Progress in Met. Phys., vol. 4, (Interscience, New York), p. 333. HORVAY, G., 1965, Int. J. Heat Mass Transfer 8, 195. HUANG,S. C. and M. E. GLICKSMAN, 1981, Acta Metall. 29, 717. HUANG,S. C., E. L. HALL,K M. CHANG,and R. P. LAFORCE,1986, Metall. Trans. 17A, 1685. HUANG, S. C. and E. L. HALL,1989, Mater. Res. Soc. Symp. Proc. 133, 373. HUGHES, I. R and H. JONES, 1976, J. Mater. Sci. 11, 1781. HUGHES,I. C. H., 1988, Metals Handbook, 9th.ed., (ASM, Metals Park,OH) 15 “Casting”, 647. HUNT, J.D., 1979, in: Solidification and Casting of Metals (Metals Society, London), p. 3. HUNT, J.D., 1984, Mat. Sci. and Eng. 65, 75. HUNT,J.D., 1990, Acta Metall, Mater. 38, 411. HUNT,J. D., and K. A. JACKSON, 1966, Trans. Met. SOC.AIME 236, 843. HUNT,J. D., and K.A. JACKSON, 1967, Trans. Met. SOC.AIME 239, 864. HURLE,D.T. J., 1969, J. Cryst. Growth 5, 162. HURLE,D. T. J., 1972, J. Cryst. Growth 13 /14, 39. HURLE,D.T. J., 1987, J. Cryst. Growth 85, 1. HURLE, D.T.J., E. JAKEMAN, 1968, J. Cryst. Growth 34,574. INOUE,A., T. W o r n , H. TOMIOKA and N. Ymo, 1984, Int. J. Rapid Solidification 1, 115. ,-sI K.N., M. MAEDAand P. H. SHINGU, 1985, Acta Metall. 33,2113. IVANTSOV, G.P., 1947, Dokl. Akad. Nauk S.S.S.R. 58,567. JACKSON, K. A., 1958, in: Liquid Metals and Solidification(ASM, Metals Park, OH)p. 174. JACKSON, KA., 1968, Trans. Met. Soc. AIME 242, 1275. JACKSON,K.A., 1971, in: Solidification (ASM, Metals Park, OH) p. 121. JACKSON, KA., 1974, J. Cryst. Growth 24/25, 130. *JACKSON,K. A. and J. D. HUNT,1966, Tran. Met. SOC. AIME 236, 1129. JACKSON, IC A., J.D. HUNT,D, R. UHLMANN and T. P. SWARD, 1966, Trans. Met. Soc. AIME 236, 149. JACKSON, K.A., D.R. UHLMANN,and J.D. HUNT,1967, J. Cryst. Growth 1, 1. JACKSON, K. A., 1975, in: Treatise on Solid State Chemistry, Vol. 5, edited by N. B. Hannay, (Plenum, NY), p. 233. JACKSON, K. A., G. H. GILMERand H. J. LEAMY, 1980, in: Laser and Electron Beam Processing of Materials, edited by C. W. White and P. S. Peemy, (Academic Press, NY), p. 104. JACOB],H., and K. SCHWERDTEFEGER, 1976, Metall. Trans. 7A, 811.
836
H.Biloni and WJ.Boettinger
Ch. 8, Refs.
JANEN,R. and R.R. SAHM,1984, Mat. Sci. and Eng. 65, 199. JOHNSON,M., L. BACKERTJDand G. K. SIGWORTH, 1993, Metall. Trans. UA, 481. JONES, B.L. and G.M. WESTON,1970 a) J. Aust. Inst. Met. 15, 189. JONES,B.L. and M. WESTON,1970 b), J. Mat. Sci. 5, 843. JONES, H. and W. KURZ,1980, Metall. Trans. 114 1265. JONES, H., 1982, Rapid Solidification of Metals and Alloys. Monograph, no 8, (The Institute of Metals, London). JORDAN, R.M. and J.D. HUNT,1971, J. Cryst. Growth 11, 141. JORDAN, R. M. and J. D. HUNT,1972, Metall. Trans. 3, 1385. JUNZE, J., K.F. KOBAYASHI and P. H. SHINGU,1984, Metall. Trans. EA, 307. KAMIO,A., S. KUMAI,and H. TEZUKA, 1991, Mat. Sci. and Eng. A146, 105. KARMA,A., 1987, Phys. Rev. Let. 59,71. KATO,H. and J.R. CAHOON,1985, Metall. Trans. 16A, 579. KATTAMIS, T. Z., 1970, Z. Metallk. 61, 856. KATTAMIS, T.Z., 1976, J. Cryst. Growth 34,215. KATTAMIS, T.Z., and M.C. FLEMINGS,1965, Trans. Met. SOC.AIME 233,992. KATTAMIS,T . 2 and M. C. FLEMINGS, 1966, Trans. Met. Soc. AIME 236, 1523. KATTAMIS, T. Z. and M. C. FLEMINGS, 1967, Mod. Cast. 52,97. KATTAMIS, T.Z., J. C. COUGLIN,and M. C. FLEMINGS, 1967, M. C., Trans. Met. SOC.AIME 239, 1504. KELLY, A. and G. J. DAVIES,1965, Metall. Rev. 10 (37), 1. KELLY, T.F., and J.B.VANDERSANDE, 1987, Intl. J. of Rapid Solidification 3, 51. KENNEY, M.P., J.A. COURTOIS, R.D. EVANS, G.M.FARRIOR, C.P. KYONKAand A.A. KOCH,1988, Metals Handbook, 9th.ed., (ASM, Metals Park, OH) 15, “Casting” 327. KERR, H. W., and W. C. WINEGARD,1967, in: Crystal Growth (suppl. to the Physics and Chemistry of Solids), H. S. Peiser, ed., (Pergamon, Oxford), p. 179. KERR, H. W., J. CISSE,and G. E BOLLING,1974, Acta Metall. 22,677. KERR, H.W. and J.C. VILLAFUERTE, 1992, in: The Metal Science of Joining, eds. H.J. Cieslak, J.H. Perepezko, S. Kang and M. E. Glicksman, (TMS Pub., Warrendale, PA), p. 11. KIM, Y. I. and S. Kou, 1988, Metall. Trans. 19A, 1849. KIM, Y.-W., €I.-M. LIN, and T.F. KELLY, 1988 a), Acta Metall. 36,2525. KIM, Y.-W., H.-M. LIN,and T. E KELLY, 1988 b), Acta Metall. 36,2537. KIM, W.T. and B. CANTOR, 1991, J. Mat. Sci. 26, 2868. KIRKALDY,J. S. and W.V. YOUDELIS,1958, Trans. Met. SOC.AIME 212, 833. KIRKWOOD,D. H., 1985, Mat. Sci. and Eng. 73, L1. KIT^, J. A., M. J. AZIZ,D.P. BRUNCO,and M. 0. THOMPSON, 1994, Apply. Phys. Lett. 64,2359. KOBAYASHI, S., 1988, J. Cryst. Growth 88, 87. KOBAYASHI, R., 1991, J. Jpn. Assoc. Cryst. Growth 18(2), 209 in Japanese. KOBAYASHI, R., 1992, in: Pattern Formation in Complex Dissipative Systems, S. Kai, ed., (World Science, Singapore) p. 121. KOBAYASHI, K. and L.M. HOGAN,1978, Met. Fonun 1, 165. KOBAYASHI, K. and P.H. SHINGU, 1988, J. Mat. Science 23,2157. KOCH,C. C., 1988, Int. Mat. Rev. 33, 201. KOFLER,A., 1950, Z. Metallk 41, 221. KOU, S., S. C. HSU and R. MEHRABIAN, 1981, Metall. Trans. 12B, 33. KUBO,K. and R. D. PEHLKE, 1985, Metall. Trans. 16B, 359. *KURZ,W. and D. J. FISHER,1979, Int. Met. Rev. 24, 177. KURZ,W. and T. W. CLYNE,1981, Metall. Trans. 12A, 965. KURZ,W. and D. J. FISHER, 1981, Acta Metall. 29, 11. KURZ,W., B. GIOVANOLA, and R. TRlVeDI, 1986, Acta Metall. 34, 823. KURZ,W., B. GIOVANOLA, and R. TRIVEDI, 1988, J. Cryst. Growth. 91, 123. * K w , W. and D.J. FISHER,1989, Fundamentals of Solidification, 3rd.d. (Trans. Tech. Publication, Switzerland).
Ch. 8, Refs.
Solidification
837
KURZ, W. and R. TRIVEDI, 1989, in: Series M.D., vol. 14, Microstructural Development and Control in Materials Processing, ed. D. R. Durham and A. Saigai (ASM, Metals Park, OH), p. 47. LANGER, J.S., 1980, Phys. Rev. Lett. 44, 1023. LANGER, J.S., and H. MUELLER-KRUMBHAAR,1978, Acta Metall. 26, 1681. LANCER, J.S., and H. MUELLER-KRUMBHAAR, 1981, Acta Metall. 29, 145. LARSON, M. A., and J. GARSIDE, 1986, J. Cryst. Growth 76, 88. LAVERNIA, E. J., J. D. AYERSand T. S . SRIVATSAN, 1992, Int. Mat. Rev. 37, no 1, 1. LEAMY,H. I., and K. A. JACKSON,1971, J. Appl. Phys. 42,2121. LEAMY, H. J., and G. H. GILMER,1974, J. Cryst. Growth 24125,499. LEE, J.T. C. and R. A. BROWN,1993, Phys. Rev. B 47,4937. LEVI,C.G. and R. MEHRABIAN,1982, Metall. Trans. WA, 221. LIPTON,J., A. GARCIAand W. HE.INEMANN, 1982, Archiv. f i r des Einsenhlittenwessen53,469. LIPTON, J., M.E. GLICKSMAN,and W. KURZ,W., 1984, Mat. Sci. and Eng. 65, 57. LIPTON, J., W. KURZand R. TRIVEDI, 1987, Acta Metall. 35, 957. LIVINGSTON, J. and H. CLINE, 1969, Trans. Met. Soc. AIME 245,351. Lu, S.Z. and A. HELLAWELL, 1985, J. Cryst. Growth 73,316. Lu, S. Z. and A. HELLAWELL, 1987, Metall. Trans. 18A, 1721. Lu, S.Z. and A. HELLAWLL,1988, in: Proc. Solidification Processing 1987. The Institute of Metals (London), p. 44. Lu, S.Z., and J.D.HUNT, 1992, J. Cryst. Growth 123, 17. Lu, S. Z., J. D. HUNT,P.GILGIEN,and W. KURZ,1994, Acta Metall. Mater. 42, 1653. M C C A R ~ YD., G., J. D. HUNT,and R.M. JORDAN, 1980a, Metall. Trans. 11A, 1243. M~CARTNEY, D.G., J.D. HUNT, and R.M. JORDAN,1980b, Metall. Trans. 11A, 1251. M C C A R ~ YD. , G., 1989, Int. Mat. Rev. 34, 247. MCLEOD,A. J., 1971, J. Aust. Int. Metals 16, 124. MCLEOD,A. J. and L.M. HOGAN,1978, Metall. Trans. SA, 987. MCDONALD,C. A., A.M. MALWZZI,F. SPAEPEN,1989, J. Appl. Phys. 65, 129. MAGNINP., and W. KURZ,1987, Acta Metall. 35, 1119. MAGNIN,P., J.T. MASON and R. TRIVEDI,1991, Acta Metall. Mater. 39,469. MAGNIN,P., and R TRIVEDI, 1991, Acta Metall. Mater. 39,453. MAHAPATRA,R.B., J. K. BRIMACOMBE, I.V. SAMARASEKERA, N. WALKER,E. A. PATERSON and J.D. YOUNG, 1991 a), Metall. Trans. UB, 861. and LV. SAMARASEKERA, 1991 b), Metall. Trans. 22B, 875. MAHAPATRA,R. B., J. K. BRIMACOMBE MAJOR,J. F. and J. W. RUTTER,1989, Mat. Sci. and Tech. 5, 645. MARCAMTONIO, J. A. and L. E MONDOLFO,1974, Metall. Trans. 5, 1325. MASUR,L. J. and M. C. FLEMINGS, 1982, Proc.4th. Int. Conf. on Rapidly Quenched Metals, Sendai, Japan, edited by T. Masumoto and K. Suzuki (Jap. Inst. of Metals, Sendai), 1557. MATSUMIYA, T. and M. C. FLEMINGS, 1981, Metall. Trans. 12B, 17. MAXWELL,I. and A. HELLAWELL, 1975, Acta Metall. 23,229. *-IAN, R., 1982, Int. Met. Rev. 27, 185. MEELRABUN,R., N. KEANEand M. C. FLEMINGS, 1970, Metall. Trans. 1, 1209. MEHRABIAN, R., D.R. GEIGERand M. C. FLEMINGS, 1974, Metall. Trans. 5,785. MERCHANT,G. J., and S. H. DAVIS,1990, Acta Metall. Mater. 38, 2683. Mmz, S. A. and M.C. FLEMINGS, 1969a), Trans. AFS 77,329. METZ, S. A. and M. C. FLEMINGS, 1969b), 'hns. AFS 77,453. MEYER,6.H., 1981, Int. J. Heat Mass Transfer 24, 778. MIKEEV,L.V., and A.A. CH'ERNOV,1991, J. Cryst. Growth 112, 591. MINAKAWA, S., V. SAMARASEKBRA and E WGINBERG, 1985, Metall. Trans. 16B, 595. *MINKOFF,I., 1983, The Physical Metallurgy of Cast Iron (Wiley, N.Y.). MINKOFF,I., 1986, Solidification and Cast Structure (John Wiley & Sons, N. Y.). MINKOFF,I., 1990, in: E Weinberg Int. Symposium on Solidification Processing, ed. J.E. Lait and L.V. Samarasekera (Pergamon Press, N.Y.) p. 255. MOHANTY, P. S., F.H. SAMUEL and J.E.GRUZLESKI,1993, Metall. Trans. 24A, 1845.
838
Ch. 8, Refs.
H. Biloni and INJ. Boettinger
MOLLARDE, and M.C. FLEMINGS, 1967, Trans. Met. Soc. A M 239, 1534. MONDOLFO,L., 1965, J. Austr. Inst. Metals 10, 169. MOORE, K.I., D.L. ZHANG,and B. CANTOR,1990, Acta Metall. Mater. 38, 1327. MORALES, A., J. J. Rssom and H. BILONI,1977, Z. Metallk. 68, 180. MORALES, A., M. E. GLICKSMAN and H. BILONI,1979, in: Solidification and Casting of Metals (The Metals Society, London) p. 184. MORANDO,R., H. BILONI,G. S. COLEand G.F. BOLLING,1970, Metall. Trans. 1, 1407. 1969, J. Cryst. Growth 5, 361. MORRIS,L. R and W. C. WINEGARD, MORROG,H., 1968a), J. Iron Steel Inst. 206, 1. MORROG, H., 1968b), in: The Solidification of Metals (Iron and Steel Institute, London) Publication no 110, p. 238. A., 1991a), Metall. Trans.22A, 569. MORTENSEN, MORTENSEN, A., 1991b), in. Proceedings of the 12th. RIS0 InternationalSymposiumon MaterialsScience: Metal matrix compositesprocessing.Microstructures ami Properties, Ed. H. Hansen, D. J. Jensen, T. hffers, H. Liholt, T. Lorentzen, A. S. Pedersen, 0. Pedersen and B. Ralph. lUS0 Nat. Lab. Rolskilde, Denmark, p. 101. MORTENSEN, A. and I. JIN, 1992, Int. Mat. Rev. 37, no 3, 101. MULLMS,W. W. and R. F. SEKERKA, 1963, J. Appl. Phys. 34, 323. MULLINS,W. W. and R.F. SEKERKA,1964, J. Appl. Phys. 35,444. MURRAY, J. L., 1983%Mat. Res. SOC.Symp. hoc. 19,249. MURRAY, J. L., 1983b, Bull. Alloy Phase Diagrams 4, 271. MURRAY, B. T., S . R. CORIELL and G. B. MCFADDEN, 1991, J. Cryst. Growth 110, 713. NAKAGAWA, H., H. KATO,F. MATSUDA and T. SENDA,1970, J. Japan Weld. Soc. 39,94. NANDAPURKAR, P. and D. R. POIRIER, 1988, J. Cryst. Growth 92, 88. NARASHIMAN, M.C., 1980, US.Patent 4221257. NARAYAN, J. J., 1982, J. Cryst. Growth 59, 583. NI, J. and C. BECKERMANN, 1991, Metall. Trans. 22B,349. NIYAMA, E., 1977, J. Japan Foundrymen Soc. 49, 26. O’HARA,and W. A. TILLER,1967, Trans. Met. SOC. AIME 239,497. OHNAKA, I., 1986, Trans. ISIJ 26, 1045. OHNAKA,I., 1988, Metals Handbook 9th. ed., (ASM, Metals Park, OH) 15 “Casting”, p. 136. OHNAKA,I., 1991, in: Freezing and Melting Heat Transfer in Engineering, eds. K.C. Ching and H. Seki, Chapter 21: Solidification Analysis of Casting (Hemispher Pub.), p. 1. OHNO,H., 1986, J. of Metals 38, 14. OHNO,H., 1970, J. Japan Inst. Metals 34, 244. OHNO,H., T. MOTEGIand H. SODA,1971, The Iron and Steel 1nst.of Japan 11, 18. OKAMOTO, T. K., K.KISHITAKE,and I. BBSSHO,1975, J. Cryst. Growth 29, 131. OKAMOTO, T. K., and K. KISHITAKE, 1975, J. Cryst. Growth 29, 137. OLDFIELD,W., 1973, Mat. Sci. and Eng. 11,211. O~TROWSKI, A. and E.W. LANGER,1979, in: Solidification and Casting of Metals, (The Metals Society, London), p. 139. OXTOBY, D. W., and A. D. J. HAYMET, 1982, J. Chem. Phys. 76, 6262. PALACIO,H., M. SOLAR1 and H. BILONI,1985, J. Cryst. Growth 73,369. See also H. BILONI[1983]. PATON,B., 1959, Electroslag Welding (Foreing Language Publishing House, Moscow). PAWLOWSKI, L. and P. FAUCKAIS,1992, Int. Mat. Rev. 37, 271. F’EHLKE, R.D., 1988, Metals Handbook, 9th. Ed., (ASM, Metals Park, OH) 15 “Casting”, p. 308. PEREPEZKO, J.H., 1984, Mat. Sci. and Eng. 65, 125. PERHPEZKO,J. H., 1988, in: Metals Handbook, (ASM, Metals Park, OH) 15 - “Casting”, p. 101. PEREPEZKO, J. H., 1992, in: Thermal Analysis in Metallurgy, ed. by R. D. Shull and A. Joshi, (TMS, Warrendale, PA) p. 121. *PBREPEZKO, J. H., 1994, Mat. Sci. and Eng. A178, 105. See also I. E. ANDERSON, Ph. D. Thesis, University of Wisconsin, Madison WI, 1983.
-
Ch. 8, Refs.
Solidification
839
PEREPEzKo, J. H. and I. E. ANDERSON, 1980, in: Synthesis and Properties of Metastable Phases, edited by E. S. Machlin and T. S . Rowland, (TMS-AIME, Warrendale, PA), p. 31. PEREPEZICO, J. H.and S. E. LEBEAU, 1982, in: Aluminum Transformation Technology and Applications 1981, eds. C. A. Pampillo, H. BiloN, L. Mondolfo and E Sacchi (ASM, Metals Park, OH)p. 309. PEREPEZKO,J.H. and W. J. BOFITINGER;1983, Mat. Res. Soc. Symp. Proc. 19,223. PEREZ,T., M. SOLARIand H. BILONI, 1981, Int. Inst. Weld. DOC11-541-81. PETEYES, S. D. and G.J. ABBASCHIAN, 1986, J. -st. Growth 79, 775. PFANN,W.G., 1952, Trans. Met. Soc. AIME 194,747. DANN, W. G., 1966, Zone Melting, 2nd.ed. (Wiley, New York). PIMPUTKAR, M. and S. OSTRACH,1981, J. Cryst. Growth 55,614. mREs, 0.S., M. PRATESand H. BILONI,1974, Z. Metallk. 65, 143. PIWONKA, T. S. and M. C. FLEMINGS, 1966, Trans. TMS-AIME 236, 1157. FWJONKA, T. S., 1988, in: Metals Handbook, 9th. ed., (ASM, Metals Park, OH) 15 “Casting”, p. 319. PLASKEIT,T.S. and W.C. WINEGARD, 1959, Trans. ASM 51,222. POIRIER,D.R.,1987, Metall. Trans. lSB, 245. POIRIER, D.R., K. YEUMand A.L. MAPPLES, 1987, Metall. Trans. lSA, 1979. POND,R. B., 1958, US.Patent 2, 825, 198, Metallic Filaments and Method of Making Same. POWELL,G.L. E, 1965, J. Aust. Inst. Met. 10, 223. POWELL, G. L. E, 1980, Metals Forum 3, 37. POWELL, 6.L.E and L.M. HOQAN,1968, Trans. Met. SOC.AIME 242, 2133. POWELL, G. L.E and L. M. HOGAN,1969, Trans. Met. SOC.AIME 245, 407. PRATES, M. and H. BILONI, 1972, Metall. Trans. 3, 1501. PRESCOTT, P, J., E P. INCROPERA and W. D. BENNON,1991, Int. J. Heat Mass Transf. 34, 2351. QIYANG, L., L. QINGCHUNand L. QIFU,1991, Act. Metall. Mater. 39, 2497. *RAPPAZ,M., 1989, Int. Mat. Rev. 34, 93. RAPPAZ,M., and D. M. STEFANESCU, 1988, Metals Handbook, 9th.ed.. (ASM, Metals Park, OH) 15 “Casting”, p. 883. RAPPAZ,M., S.A. DAVID,J.M. VITER and L. A. BOATNER, 1989, Metall. Trans. 20A, 1125. RAPPAZ, M.. S.A. DAVID.,J. M. VITEK,L. A. BOATNER, 1990, Metall. Trans. 21A, 1767. RAPPAZ,M. and Ch.A. GANDIN,1993, Acta Metall. Mater. 41, 345. RAPPAZ,M. and Ch. A. GANDIN,1994, Mat. Research Bull. xix, nol, 20. RICHMOND, J. J., J.H. PEREPEZKO, S. E. LEBEAU and K. P. COOPER,1983, in: Rapid Solidification Processing: Principles and Technologies 111, edited by R. Mehrabian, (NBS, Washington, DC), p. 90. RIDDER,S. D., S. Kou and R. MEHRABIAN, 1981, Metall. Trans. 12B, 435. RIPOSAN, T., M. CHISAMERA, L. SOFRONIand V. BRABIE,1985, in: Physical Metallurgy of Cast Iron, ed. H. Fredricksson and M. Hillert (North Holland, N.Y.) p. 131. RODWAY,G.H. and J.D. JUNT,1989, J. Cryst. Growth 97,680. RODWAY, G. H. and J.D. HUNT,1991, J. Cryst. Growth 112, 554. ROHATGI,€?, 1988, Metals Handbook, 9th. ed., (ASM, Metals Park, OH) 15 “Casting”, p. 840. ROSENTHAL, D., 1941, Welding J. 20, 220-s. Ross, A. B. De and L. E MONDOLFO,1980, in: Aluminum Transformation Technology and Applications, eds. C. A. Pampillo, H. Biloni and D.E. Embury (ASM, Metals Park, OH) p. 81. ROVIGLIONE,A. and H.D. HERWA, 1994, Materials Characterization 32, 127. ROVIGLIONE,A. and H. BILQNI, 1994, Fifth International Symposium on the Physical Metallurgy of Cast Iron (SCI-S). To be published in Key Engineering Materials (Trans. Tech. Publications, London). SADOCHA, J. P. and GRUZLESKI,1975, in: Metallurgy of Cast Iron (Georgi Pub. Co, St. Saphorin, Switzerland) p. 443. SATO,A. and R. MEHRABIAN, 1976, Metall. Trans. 7B, 443. SATO,T. and G. OHIRA,1977, 3. cryst. Growth 40.78. SATO,A., Y.OHSAWAand G. ARAGANE,1989, Mat. Trans. JIM 30,55. SATO,A, Y.OHSAWAand G. ARAGANE, 1991, Mat. Trans. JIM 32,77. SATO,A., Y. OHSAWAand G. ARAGANE, 1992, Mat. Trans. JIM 33, 66. SAVAGE,W.F., C.D.LUNDIN and A. H.ARONSON,1965, Welding J. 40, 175s.
840
Ch.-8, Refs.
H.Biloni and WJ.Boettinger
SAVAGE,W. F. and E. S. SZEKERES, 1967, Welding J. 46,94-s. SCHAEFER,R. J. and M. E. GLICKSMAN, 1970, Metall. Trans. 1, 1973. SCHA&R,R J. and S.R. CORIELL,1984, Metall. Trans. A E , 2109. SCHEIL,E., 1942, 2.Metallk. 34, 70. SCHWARZ, M., A. KARMA, K. ECKLERand D.M. HERLACH, 1994, Phys. Rev. Lett. 73,1380. SEETHARAMAN, V., and R. TRIVEDI, 1988, Metall. Trans. 19A, 2955. SEKERKA, R. E, 1967, in: Crystal. Growth, ed. H. S. Peiser (Pergamon Press, Oxford) p. 691. SEKERKA, R. F., 1986, Am. Assoc. Cryst. Growth Newslett. 16, 2. SHANGGUAN, D., S. AHUJAand D.M. STEFANESCU, 1992, Metall. Trans. 23A, 669. SHARMA, D. G. R. and KRISHMAN, 1991, AFS Trans. 99,429. SHARP, R.M., and M. C. FLEMINGS, 1973, Metall. Trans. 4,997. SHARP,R.M., and M. C. FLEMINGS, 1974, Metall. Trans. 4, 823. SHERCLIFF,H. R, 0.R. NYHRand S.T. J. TOTTA, 1994, Mat. Research Bull xix, nol, 25 SIKORA, J.A., G.L. RIVERAand H. BILQNI, 1990, in: Proceedings of the F. Weinberg Symposium on SolidificationProcessing, eds. J. E. Lait and L. V. Samarasekera (Pergamon Press, N.Y.) p. 255. SINGER,A.R.E., 1970, Met. Mater. 4,246. SINGER,A.RE., 1972, J. Inst. Metals 100, 185. SKALAND, T., 0. GRONGand T. GRONG,1993, Metall. Trans. 24A, 2321. SMITH,P.M. and M. J. AZIZ, 1994, Acta Metall. Mater. 42, 3515. SMITH,R. W., 1968, in: The Solidification of Metals (Iron and Steel Inst., London) Pub. 110, p. 224. SOLARI, M., and H. BILONI,1980, J. Cryst. Growth 49, 451. SOMBOONSUK, K., J. T. MASON,and R. TRIVEDI, 1984, Metall. Trans. 15A, 967. SOUTHIN, R.T., 1967, Trans. Met. SOC.AIME 236,220. SOUTHIN,R T . and G.M. WESTON,1973, J. Aust. Inst. Met. 18, 74. SOUTHIN,R.T. and G.M. WESTON,1974, J. Aust. Inst. Met. 19, 93. SOUTHIN, R T. and G. A. CHADWICK,1978, Acta Metall. 26,223. SPAEPEN,F., 1975, Acta Metall. 23,729. SPAEPEN, F., 1994, in: Solid State Physics, edited by H. Ehenreich and D. Turnbull 47, (Academic Press, San Diego) p. 1. SPAEPEN,F. andR.B. MEYER,1976, ScriptaMetall. 10,257. SPENCER,D. B. 1971, Ph. D. Thesis. Massachusetts Institute of Technology, Cambridge, MA. Quoted by M. C. Flemings 1991. and M. C. FLEMINGS, 1972, Metall. Trans. 3, 1925. SPENCER D. B., R. MEHRABIAN STEEN,H. A. H., and A. HELLAWELL, 1975, Acta Metall. 23, 529. *STEFANESCU, D. M., 1988, Metals Handbook, 9th.ed., (ASM, Metals Park, OH) 15 “Casting”, p. 168. STEFANESCU, D. M., R. HUMMER and E. NECHTELBERGER, 1988, Metals Handbook, 9th. ed., (ASM, Metals Park, OH) 15 “Casting”, p. 667. STEFANESCU, D.M. and B.K. DHINDAW,1988, Metals Handbook 9th.ed., (ASM, Metals Park, OH) 15 “Casting”, p. 142. STEFANESCU, D.M., G. UPADWA and D. BANDYOPADHYAY, 1990, Metall. Trans. 2 1 4 997. STRANGE, E.H., 1911, U.S.Patent 993904. SUBRAMANIAN, S.V., D. A.R. KAY and G.R. PURDY, 1985, in: Physical Metallurgy of Cast Irons, eds. H. Fredriksson and M. Hillert (North Holland, N.Y.),p. 47. S U N D Q ~ , B.E. and L.F. MONDOLFO, 1961, ’ h n s . Met. SOC. AIME 221, 157. SIJRYANARAYANA, C., F.H. ROSSand R.G. ROW, 1991, Int. Mat. Rev. 36, no 3, 85. TADA,K. and H. OHNO, 1992, Keinzoku 42,321. (In Japanese). TAMMANN, G., and A. A. BOTSCHWAR,1926, Z. Anorg. Chem. 157,27. TARSHIS,C.A., J.L. WALKERand J. W. RUTTER,1971, Met. Trans. 2, 2589. TASSA,M. and J.D. HUNT, 1976, J. Cryst. Growth 34, 38. TAYLOR, C. R., 1975, Metall. ”kans. 6B, 359. TEMKIN, D. E., 1964, Crystallization Processes (Tranl. by Consultants Bureau, New York, 1966) p. 15. TEMKIN, D. E., 1969, Sov. Phys. Crystallogr. 14, 344.
.
Ch. 8,Refs.
Solidificntion
841
THEVOS,Ph., J. L. DFSBIOLLES, and M. RAPPAZ,1989,Metall. Tram. 20A, 311. See also M. RAPPAZ, 1989,Int. Mat. Rev. 34,93. THOMPSON, C.V., A.L. GREER and E SPAEPEN, 1983,Acta Metall. 31, 1883. THOMPSON, C. V. and E SPAEPEN,1983,Acta Metall. 31,2021. THORTON, P.H.,1968,in: Techniques of Metals Research, Ed. R. E Bunshah (Interscience, New York), vol.1, part 2, p. 1069. TILLER,W. A., 1958,in: Liquid Metals and Solidification (ASM, Metals Park, OH) p. 276. TILLER,W., 1970, Solidification, in: Physical Metallurgy, 2nd edition, ed. R. W. Cahn, (North-Holland, Amsterdam), p. 403. TILLER,W. A., K. A. JACKSON,J. W. RUTTERand B. CHALMERS, 1953,Acta Metall. 1,453. TILLER,W. A., and J. W. RL~TER, 1956,Can. J. Phys. 34,96. TIT~NER A.P. , and J. A. SP~TLE,1975,Acta Metall. 23,497. TOLOUI, B. and A. HELLAWELL,1976, Acta Metall. 24 565. TRAJAN, P.K., 1988,in: Metals Handbook, 9th. ed., (ASM, Metals Park, OH) 15 "Casting", p. 88. TRNEDI,R. and K. SOMBOONSUK, 1984, Mat. Sci. and Eng. 65,65. TRNEDI,R, P. MAGNIN,and W. KURZ, 1987,Acta Metall. 35,971. TRMZDI,R., and W. KURZ, 1988,in SolidificationProcessing of Eutectic Alloys, edited by D.M. Stefanescu, G. J. Abbaschii and R. J. Bayuzick, (The Metallurgical Society, Warrendale, PA), p. 3. TRIVEQIR., and W.KURZ, 1994,Int. Mat. Rev. 39,49. TRIVEDI,R.,1995,Met. Trans., 26A, 1583. TURNBULL, D.,1962,J. Phys. Chem. 66,609. TURNBULL, D.and R. E. CECH,1950,J. Appl. Phys. 21, 804. TURNBULL, D. and B. G. BAGLEY,1975,in: Treatise on Solid State Chemistry, vol. 5, edited by N. B. Hannay, (Plenum, NY), p. 513. UNGAR, L.H. and R. A. BROWN,1985,Phys. Rev. B 31,5931. VERHOEVEN, J. D.,and E. D. GIBSON,1978,J. Mat. Sci 13, 1576. VERHOEVEN,J. D., W. N. GILL,J. A. PUSZYNSKI and R.M. GINDE, 1988,J. Cryst. Growth 89, 189. VERHOEWN, J.D., W.N. GILL, J. A. PUSZYNSKI and R M . GINDE,1989,J. Cryst. Growth 97,254. VISKANTA,R., 1990,ISME Int. J. (Series 11) 33,409. VITEK,J. M. and S. k DAVID,1992,in: The Metal Science of Joining, eds. H. I. Cieslak, J. H. Perepezko, S. Kang and M. E. Glicksman (TMS Pub., Warrendale, PA) p. 115. VIVES,Ch. and R. RICOU,1985,Metall. Trans. 16B, 377. VOLLER,V.R., and S. SUNDARRAJ, 1993, in: Modeling of Casting, Welding and Advanced Solidification Processes V, T.S.Piwonka, V, Voller, and L. Katgerman eds., (TMS, Warrendale, PA), p. 251. VOLMER, M.I., and M. MARDER,1931,Z.Phys. Chem. A154, 97. VOORHEES, P. W., 1990,Metall. Trans. 21A, 27. WALKER,J.L., 1959, in: The Physical Chemistry of Process Metallurgy, part 2, ed. G.S.St. Pierre (Interscience, New York), p. 845. WALKER, J.L., 1964, cited in: Principles of Solidification, ed. B. Chalmers, Chap. 4, (wiley, New York) p. 122. WALTON,D. and B. CHALWS, 1959,Trans. Met. SOC.AIMJ3 188, 136. WANG, C.Y. and C. BECKERMANN, 1993. Mat. Sci. and Eng. A171, 199. WANG,Y H. and S. Kov, 1987, in: Advances in Welding Science and Technology, ed. S. A. David, (ASM, Metals Park, OH.), p. 65. WANG,Y. H., Y.I. KIMand S. Kou, 1988,J. Cryst. Growth 91,50. WARREN,J. A., and W. J. BOETTINGER, 1995,Acta Metall. et Mater. 43, 689. WEI, C. and J.T. BERRY,1980, Int. J. Heat and Mass Transfer 25,590. WEINBERG, E, 1975,Metall. Trans. 6A, 1971. WEINBERG, F., 1979 a), in: Solidification and Casting of Metals (The Metals Society, London) p. 235. WEINBERG, F.,1979 b), Metals Technology February, 48. WEINBERG, E and E. TEGHTSOONIAN, 1972,Metall. Trans. 3,93. WETTERFALL,S. E.,H. FREDRICKSSON, H. and M. HILLERT,1972,J. Iron Steel Inst. 210,323. WHEELER,A. A., W. J. BOETTINGER, and G. B. MCFADDEN,1992,Phys. Rev A 45,7424.
842
H.Biloni and WJ. Boettinger
Ch.8, Refs.
WHEELER,A. A., B.T. MURRAY,and R J. SCHAEFER, 1993a, Physica D 66,243. W ~ L E RA.A., , W. J. B O ~ G E Rand , G.B. MCFADDEN,1993b, Phys. Rev E 47, 1893. WHITE,C. W., D. M. ZEHNER, S. U. CAMPISANO and A. G. CULLIS,1983, in: Surface Modification and Alloying by Lasers, Ion, and Electron Beams, edited by J. M. Poate, G. Foti and D. C. Jacobson (Plenum Press, NY), p. 81. WHITE,G. and D. W. OLSON,1990, in: The New Materials Society. Challenges and Opportunities, (Bureau of Mines, U.S.Dept. of the Interior.) Quoted by MORTENSEN and JIM[1992]. WIESE,J. W. and J. A. DANTZIG,1988, Applied Mathematical Modelling 12, 213. WILLNECKER, R., D. M. HERLACH and B. FEUERBACHER, 1989, in: Proc.7th.Europ. Symp. on “Materials and Fluid Sciences under Microgravity”, Oxford, ESA SP-295, p. 193. R., D.M. HERLACH,and R. FEUERBACHER, 1990, Apply Phys. Lett. 56,324. WILLNECKER, WILSON,H.A., 1900, Phil Mag. 50, 238. WILSON,L.O., 1978, J. Cryst. Growth 44, 371. WILSON,L. O., 1980, J. Cryst.1 Growth 48, 363. WINEGARD,W. and B. CHALMERS, 1954, Trans. Quart. ASM 46, 1214. WOLLKIND,D. and L. SFGAL,1970, Phil. Trans. Roy. SOC.London 268,351. Wu, Y., T. J. F’ICONNF,, Y. SHIOHARA, and M. C. FLEMINGS,1987, Metall. Trans. 18A, 915. YOON,W., PAIK,J. S., LACOURT, D., and PEREPEZKO, 1986, J. H., J. Appl. Phys. 60, 3489. YOUNG, K. P., and D. H. KIRKWOOD, 1975, Metall. Trans. 6A, 197. ZACHARIA, T., A. H. ERASLA,and D. K. AIDIPN,1988, Weld. J. 67, 18-s. ZACHARIA, T., S. A. DAVID, J.M. V m , 1992, in: The Metal Science of Joining, eds. H.J. Cieslak, J. H. Perepezko, S. Kang and M.E. Glicksman, (TMS Pub., Warrendale, PA), p. 257. ZENER,C., 1946, Trans Met. SOC.AIME 167,550. ZHU, J.D. and I. OHNAKA,1991, Modelling of Casting, Welding and Advanced Solidification Processes, eds. M. Rappaz, M.R. Ozgti and K. W. Mahin, (TMS Pub., Warrendale, PA), p. 435. ZHU, P. and R. W. SMITH,1992 a), Acta Metall. Mater. 40, 683. ZHU, P. and R W. SMITH,1992 b) Acta Metall. Mater. 40,3369. ZIMMRRMANN, M., A. KARMA,and M. CARRARD,1990, Phys. Rev. B 42,833. ZIV, I. and E WEINBERG,1989, Metall. Trans. 20B,731.
Further reading The publications marked with an asterisk in the adove list of references may he consulted.
R. W Cahn and I? Haasen?, eds. Physical Metallurgy;fourth, revised and enhanced edition 8 Elsevier Science Bt! 19%
844
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Ch. 9, 52
I . Dejinition and outline The microstructure of crystalline materials is defined by €he type, structure, number, shape and topological arrangement of phases andlor lattice defects which are in most cases not part of the thermodynamic equilibrium structure. In the first part (paragraph two) of this chapter, the different types of lattice defects involved in the formation of microstructure (elements of microstructure) will be discussed. As far as the arrangement, shape and crystal structure of phases are concerned, we refer to chs. 1, 4, 8, 15-17, and 28. The third and fourth paragraph of this chapter will be devoted to the characterization and to the present understanding of the development of microstructures.
2. Elements of microstructure 2.1. Point defects, dislocations and stacking faults Point defects, point-defect clusters, dislocations and stacking faults are important elements of the microstructure of most materials. The atomistic structure and properties of these defects are discussed in chs. 18 and 20.
2.2. Grain boundaries Control of the grain size is one of the most widely used means of influencing the properties of materials. Consequently, intense efforts have been directed in recent years towards a better understanding of grain boundaries. The progress achieved is documented in several comprehensive reviews (BALLUPFI[1980], AUST [1981], GLEITER[1982], SUTTONand BALLUFFI[1987], SUTTON[1990], FINNISand R ~ [1991], E WOLFand YIP [1992], SUTTONand BALLUFFI[1995]). The complex nature of interatomic forces and relaxations at interfaces has motivated the development of simple, mostly crystallographic, criteria for predicting the structure, the energy and other physical properties of interfaces. Some of these criteria and the underlying physical concepts of the atomic structure of grain boundaries will be briefly discussed in the following sections.
22.1. Crystallography 2.2.1.1. Coincidence site lattice. The coincidence site lattice (CSL)has proved a useful concept in the crystallography of interfaces and in the description of dislocations in interfaces. It is defined as follows (fig. 1). Two crystal lattices A and B which meet at an interface are imagined to be extended through each other in both directions. Crystal A or B is then translated so that a lattice point in A coincides with one in B. This point which is labelled 0 is designated as the origin of coordinates. Now it is possible that no other lattice points of A and B coincide, in which case 0 is the only common lattice point. However, for many orientations of the two crystals there will be other coincidences. These coinciding lattice points form a regular lattice which is known as the
Ch. 9,$ 2
845
Microstructure
Fig. 1. Coincidence site lattice (25) formed by interpenetration of two crystals. The unit cell of the CSL is outlined to the left, and the linear transformation relating the two lattices is shown on the right.
Coincidence Site Lattice, CSL. Boundaries are classified according to the ratio of the volume of a unit cell of the GSL to the volume of a unit cell of A or B. This value is denoted by Thus a low value of 2 implies a high frequency of coincidences of the interpenetrating lattices, and at the other extreme, 2 = 00 implies a completely incommensurate orientation. Boundaries with a relatively low (i.e., unambiguously measurable) value of 2 are referred to as coincidence boundaries and are sometimes associated with special properties. A boundary plane (e.g., CD in fig. 1) which is also a plane of the CSL has a certain planar density of coincidence sites r per unit area. The area l/r is geometrically important. For CSL boundary planes, periodically repeating unit cells of the bicrystal can be defined (unit cells of the CSL) whose faces pave the boundary. An early idea was that boundaries with a high r would have a low free energy because the atoms occupying coincidence sites are in their bulk equilibrium positions relative to either crystal lattice. 2.2.1.2. 0-lattice. A generalization of the CSL can be made and is called the 0lattice. Let us start with two ideal interpenetrating lattices A and B coinciding at one point which we take as the origin (fig. 2). Consider a linear homogeneous mapping which brings A into complete coincidence with B at every site. In the simplest case, the mapping is a pure rotation or a shear. As all coincidence sites are completely equivalent, any one of them could be regarded as the origin of the transformation. In general, there are even more points which could be taken
c.
References: p. 935.
846
Ch. 9,82
H.Gleiter
as the origin of the mapping and all such points together form a set called the set of 0points (fig. 2, 0 stands for Origin) (BOLLMANN [1970]). A conceptual advantage of the 0-lattice over the CSL is that the set of 0-points moves continuously as crystal A is rotated or deformed with respect to crystal B, whereas CSL points disappear and appear abruptly. There are 0-points even when there are no coincidences. Lines bisecting the array of Opoints in the boundary can be geometrically regarded as the cores of dislocations which accommodate the misfit. There is thus a close relationship between the [1974]) (fig. 0-lattice and the geometrical theory of interfacial dislocations (BOLLMANN 2b). Such geometrical dislocations are a mathematical device, and do not necessarily correspond to physical dislocations, which are observable atomic structures. The principal utility of an 0-lattice construction is that it enables the geometrical location and Burgers vectors of interfacial dislocations to be discussed. A weakness in the concept of an
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Fig. 2. An (001) projection of the 0-lattice between two simple cubic lattices, called lattice A and B in the text, with different lattice parameters. (a) The atom of each crystal are represented by dots and crosses and the 0points are circled. (b) Showing the lines midway between the 0-points which can be geometrically regarded as the cores of dislocations. (After SMITH and FOND[1976].)
Ch. 9, $ 2
Microstructure
847
0-lattice stems from the fact that for a given orientation of A and B, the transformation which brings A into coincidence with B is not representable by a unique matrix. This non-uniqueness is why one must be cautious about attaching any physical significance in terms of observable dislocations to the mathematical dislocations defined by means of the 0-lattice. 2.2.1.3. DSC lattice. Another lattice which is of importance for the discussion of isolated dislocations and steps at boundaries is called the DSC lattice, sometimes referred to as Displacement Shi# Complete (BOLLMANN [1970]) (fig. 3). This is defined as the coarsest lattice which includes the lattices of A and B (in an orientation for which there is some coincidence) as sublattices. Any vector joining a lattice point of crystal A to a lattice point of crystals A or B, and vice versa, is also a vector of the DSC lattice. Thus if either lattice is translated by a DSC vector, the complete pattern of the interpenetrating lattices and the CSL is either invariant or is simply displaced by the same vector. The DSC lattice is unique to the given orientations of A and B and is useful for predicting or explaining the observable (physical) dislocations in boundaries.
2.2.2. Coincidence models The first attempt to correlate predictively the crystallographic parameters of a boundary (e.g., the lattice structure of the crystals forming the interface, the orientation relationship between the two crystals, the boundary inclination, etc.) with the actual atomic arrangement in the interface was made by KRONBERG and WILSON[1949] who
Fig. 3. The DSC lattice corresponding to the interpenetrating lattices illustrated in fig. 1. The DSC lattice points are the points of the fine grid. References: p . 935.
848
H. Gleiter
Ch. 9, 32
applied the concept of lattice coincidence - which was developed independently by several crystallographers-to grain boundaries. An example of a coincidence lattice was shown in fig. 1. It was also at this time that boundaries with “low” values of 2 were called “special” and all other boundaries were called “random” or “general”. No limit on for “special” boundaries was given. Hard-sphere models suggested that every atom at the boundary was at a site of one of the crystals or at a site common to both crystals. This led to the suppositionthat low-I: boundaries had low energies because they contained high densities of atoms at shared sites. Such reasoning embodied the “coincidence model” of grain et aZ. [1964] subsequently pointed out that not all boundaries in boundaries. BRAND~N a given coincidence system had the same density of coincidence sites. In fact, the density depends on the plane of the boundary. Indeed, one can always find boundaries in any coincidence system with extremely low densities of coincidence sites. They therefore proposed that the planar density, r, of coincidence sites in the plane of the boundary should be a more reliable indicator of low energy than 2.This became known as the planar coincidence site density, or I‘ criterion. Two arguments were given in support of this criterion: in a boundary with a high value of there are more atoms at shared sites and hence the boundary “core” energy is lower. Secondly, by St. Venant’s principle, the strain field of the boundary extends into the grains roughly as far as the period of the boundary structure: the higher r the smaller the period, and, hence, the lower the strain field energy. BRANDON et al. [1964] also extended the planar coincidence to noncoincidence boundaries by combining it with the dislocation model (0 2.2.1.1). Boundaries between crystals deviating from an ideal coincidence-orientationrelationship were proposed to consist of areas with an array of coincidence atoms in the plane of the boundary (boundary coincidence) separated by (misfit) dislocations. The significance of boundary inclination was incorporated in the model by suggesting that the boundary follows the planes containing a high density of coincidence sites in order to minimize the misfit., as the boundary consists in these regions of atomic groups with little strain. Boundaries constrained to lie at an angle to the most densely packed coincidenceplane were visualized as taking a step structure.
2.23. Structural unit models The concept of atoms occupying coincidence sites (in terms of boundary or lattice coincidences) had to be abandoned after it was discovered by means of computer simulations of the atomic structure of grain boundaries (WEINSet aZ. [1971]) that two crystals forming a coincidence boundary relax by a shear-type displacement (rigid-body relaxation) from the position required for the existence of coincidence-site atoms at the boundary (figs. 4a and b). This conclusion was confirmed in subsequent years by numerous more sophisticated computer simulations as well as by experimental observations. The physical reason for the rigid-body relaxation may be seen from figs. 4 and 5. Figure 5 shows the boundary structure predicted by the latticecoincidence model. Both closely and widely spaced pairs of atoms exist, resulting in a high-energy structure. The energy of the boundary may be lowered by translating the two crystals (without rotation) so that the “hills” on the “surface” of one crystal coincide with the “valleys” on the
Ch. 9, $ 2
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(b)
Fig. 4. Structure of the boundary shown in fig. 5 after a “rigid-body relaxation” of the two hexagonal arrays of atoms in order to remove large interatomic repulsion or attraction forces (a) and subsequent relaxation of individual atoms into positions of minimum energy (b). The interaction potential assumed between the atoms corresponds to gold. (From WEINSef at. [1971])
“surface” of the other crystal (fig, 4a) followed by the relaxation of individual atoms into minimum-energy positions (fig. 4b). The existence of rigid-body relaxations led CHALMERSand GLEITER[1971] to propose that the boundary periodicity rather than the existence of boundary coincidence per se is the physically meaningful parameter. In fact, the existence of small structural units with an atomic packing density comparable to that in a perfect lattice was hypothesized to result in low-energy boundaries (fig. 4b). By extending the structural unit concept originally proposed by BISHOPand CHALMERS [19681 for unrelaxed coincidence boundaries to boundaries with atomic relaxations, the following (relaxed) structural unit model of grain boundaries was put forward (CHALMERS and G ~ E [1971]): R boundaries of low energy consist of only one type of (relaxed) structural units, whereas the structure of high-energy boundaries may be derived from a simple rule of mixing of the low-energy structural units of the nearest low-energy boundaries (GLEITER[1971]). This boundary model has been worked out in detail for different types of boundaries (PONDand VITEK[1977]). One of the limitations of the model is its applicability to interpolate between two structures of short-periodic boundaries. When the misorientation between two crystals forming a boundary consisting of a mixture of different structural units is described in terms of an axis and an angle of rotation of one crystal relative to the other, and when the rotation axis is of high indices, the Burgers vector of secondary dislocations associated with the minority units may become unrealistically large. “Unrealistically large” means that the dislocation is unlikely to be localized within the minority unit unless the minority unit itself is large. If the dislocation is not localized then the assumption of the model, that local misorientation References: p . 935.
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Fig. 5. Latticecoincidencemodel of a 18’ tilt-type grain boundary between two hexaqonal arrays of atoms. The atoms at coincidence sites are indicated by cross-hatching. The “surfaces” of the two crystals are marked by dashed lines.
changes occur so as to introduce structural units from other boundaries, breaks down. On the other hand, if large minority structural units are required then the misorientation range between the majority and minority structural unit boundaries is small and the predictive capacity of the model is limited. In practice this means that the model may usefully be applied pure tilt and pure twist boundaries with rotation axes of relatively low indices, i.e., , , and possibly . A few years after the discovery of quasiperiodicity in crystals, the same concept was utilized to model grain and interphase boundaries (RIVER [1986], GRATIASand THALAL [1988], S~TTON [1989]) see ch. 4,Appendix). The simplest way to visualize quasiperiodicity at an irrational grain boundary is to apply the structural unit model to an irrational tilt boundary. Along the tilt axis (which is assumed to be rational) the structure of the boundary is periodic. However, perpendicular to the tilt axis in the boundary plane the boundary structure will be an aperiodic sequence of majority and minority structural units.
2.2.4. Broken bond model While for free surfaces a broken-bond description of the structure-energy correlation has been commonly used for over half a century (HERRING[1953]), such an approach has only recently been adopted for grain boundaries, by WOLFand YIP [1992]. In an otherwise perfect crystal, thermal disorder is responsible for a broadening and shift of the interatomic spacings towards larger distances. Owing to the presence of planar defects, polycrystals are structurally disordered even at zero temperature, and their zero-temperature radial distribution function shows the same two effects. However, because of its localization near the interface, this type of disorder is inhomogeneous, by contrast with thermal disorder.
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Microstructure
To illustrate this inhomogeneity in the direction of the interface normal, let us consider the radial distribution function (or the planar structure factor) for each of the atom planes near the interface. As seen from fig. 6 for the case of a symmetrical (100) twist boundary in the fcc lattice, the amount of structural disorder decreases rapidly from one (100) plane to another, indicating the existence of large gradients in structural disorder. Starting from the description of structural disorder in terms of the radial distribution function, G(r), a broken-bond model seems to present a useful step towards simplification. By characterizing the atomic structure in terms of the number of broken nearest-neighbor(nn) bonds per unit area, essentially in a broken-bond model the detailed peak shapes in G(r) are simply replaced by the areas under these peaks. Because all the
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Fig. 6. Planeby-plane zero-temperature radial distribution functions, ?G(r), on the three lattice planes closest to the (001) 0 =43.60° @29) symmetrical twist boundary simulated by means of an embedded atom potential for Cu. The full arrows indicate the corresponding perfect-crystal-8-function peak positions; open arrows mark the average neighbor distance in each shell. The widths of these shells are indicated by dashed lines. (From Wow and YIP[1992].) References: p. 935.
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Fig. 7. Grain-boundary energy (mJ/m2) versus number of nearest neighbor bonds per unit area, C(1). The unit area is normalizedby the lattice constant, a. The computations were performed for a Lennard-Johnes potential fitted to Cu (fig. 7a) and an embedded atom potential fitted to Au (fig. 7b). a is the lattice parameter. For and YIP[1992, p. 1391.) comparison, the related free surface energies, y, are also shown. (From WOLF
information contained in the detailed shapes of these peaks is thus lost, small long-range elastic strain-field effects associated, for example, with interface dislocations or surface steps are therefore not 'seen' in the coordination coefficient. This is an inherent limitation of a broken-bond model. Figure 7 indicates the correlation between the average number of broken nn bonds per unit grain boundary area, C(1), and the grain boundary energy for the two fcc potentials (WOLF [1990]). While, in principle, more distant neighbors should also play a role, in fcc metals their contribution was found to be rather small (typically about 10-20% of the nn contribution). By contrast, in bcc metals the second-nearest neighbors were found to be practically as important as the nearest neighbors.
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2.2.5. Dislocation models The idea of modelling a high-angle grain boundary in terms of an array of closely spaced dislocations (fig. 8b, READ and SHOCKLEY [1950]) is an extension of the well established structure of small-angle boundaries (fig. 8a). If the dislocations are uniformly spaced in the plane of the boundary, a low-energy interface is formed because the strain fields of the dislocations extend into the lattices of both crystals over a distance comparable with the spacing of the dislocation array (St. Venant’s principle), which is a relative minimum for periodic arrays. A uniform dislocation spacing can only result if the dislocation spacing is equal to an integral number of lattice planes terminating at the boundary. For all other tilt angles the boundary may be described as a boundary with a uniform dislocation spacing (e.g., a 53” tilt boundary, fig. 8b) and a superimposed smallangle tilt boundary that accounts for the deviation from the tilt angle required for uniformly spaced dislocations (fig. 8c). The idea of describing a boundary which deviates from a low-energy structure in terms of the superposition of a low-energy boundary and a small-angle boundary has been increasingly used in association with dislocation models as well as with boundary models that are not based on the dislocation concept. The work of READ and SHOCKLEY,which is generally considered as a major achievement of the theory of interfacial structures, suffers from two inherent limitations. First, the singular behavior of the elastic strain fields near the dislocation centers was dealt with by removing the singularity mathematicallywith an inner “cut-off” radius. The second deficiency is the linear superposition of the strain fields of the individual dislocations, which results in complete neglect of the interactions among the dislocations in the array. In order to ameliorate these deficiencies, the cores of dislocations in grain boundaries were modelled either by a hollow-core dislocation description (LI[1961]), by picturing the material in the core region as a second phase (MASAMURA and GLICKSMAN [1974]), or by assuming the dislocation cores to merge into a slab of core material. A special case of the latter group of boundary models is the “random grain boundary model” (WOLF [1991]). It applies to high-angle twist boundaries. From an investigation of the role of the plane of the boundary relative to the crystallographic orientation of both crystals it was concluded that two types of boundaries should be distinguished in metals with a bcc structure. If the boundary plane is parallel to a set of widely spaced lattice planes, the boundary behavior (energy, cleavage fracture energy, etc.) is governed by the interplanar lattice spacing only. In this case, the interaction of the atoms across the interface is independent of the relative orientation of the two crystals. This suggestion is based on the physical reasoning (WOLFand YIP[1992]) that atom are shoved more closely together when creating a twist boundary on a plane with small rather than large interplanat spacings. As a consequence of the short-range repulsion between the atoms, the bicrystal then expands locally at the grain boundary, and this expansion is largest for the grain boundaries with the smaller interplanar spacings, resulting in boundaries with higher grain-boundary energy. More precisely, two criteria for low interfacial energy were suggested (WOLF[19851). The first criterion applies to twist boundaries: On a given lattice plane local minima in the energy vs twist angle relationship are expected at twist angles corresponding to the “locally smallest” CSL unit-cell area. The second criterion References: p . 935.
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(C)
Fig. 8. Dislocation models of symmetricaltilt boundaries in a simple cubic structure: (a) small-angle boundary; (b) 53O (high-angle) boundary; (c) 60" boundary.
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applies to tilt boundaries. This criterion states that local minima in the energy vs tilt angle relationship correspond to planes with “locally large” interplanar spacing. Thus, for symmetrical tilt boundaries high values of d(hk1) are suggested to correspond to local minima in the energy vs tilt angle relationship. In order to see that Wolf’s criterion of the interplanar spacing is distinct from the criterion of a high density of coincidence sites (5 2.1.1) in the boundary plane (r),let us consider non-periadic grain boundaries. There are no exactly periodic { 111) { loo} grain boundaries, for example. Thus, for all { 111) { 100) boundaries I’=0, whereas the interplanar spacing would attain a relatively high value. The arguments discussed so far do not apply to grain boundaries on vicinal planes. Vicinal planes are planes that deviate little from the widely spaced ones. In this case, the “surfaces” of both crystals forming the boundary contain a pattern of ledges. In the case of boundaries of “vicinal” planes, the energy of the boundary will depend on the degree of matching between the two ledge patterns. The spacing of the lattice planes is of little relevance (see also ch. 20, 5 6).
2.2.6. Polyhedral unit models The concept of describing the atomic arrangements in grain boundaries in terms of densely packed atomic groups (e.g., the groups existing in amorphous structures) led to the development of the polyhedral unit models. Apparently, the idea of comparing the atomic arrangements in a grain boundary with the atomic arrangements existing in amorphous structures was first proposed by POTAPOVet al. [1971]. They analyzed the three-dimensional atomic structure of grain boundaries in tungsten by means of field-ion microscopy. Boundaries were found to consist of periodically arranged rings formed by five atoms with a central atom between the rings (fig. 9). On the basis of these observations it was concluded that a grain boundary may be represented in terms of the atomic configurations existing in amorphous metals. Some years later, a similar structural concept was worked out in detail by several other authors (ASHBYet al. [19781and POND et al. [1978]), For example, fig. 10 shows the interpretation of the structure of a 36.9” [loo] tilt boundary between fcc crystals in terms of the polyhedral unit model. The comparison between grain-boundary structures and structural elements of amorphous materials is not without problems as the atoms in a boundary cannot relax to the same extent as in a glass. In an interface, the boundary conditions are given by the periodic structure of the two crystals on both sides, whereas an atomic group in a glass has no such periodic boundary conditions for its relaxation. This difference is borne out by several experiments. For example, positron annihilation measurements (CmN and CHANG[1974]) and Mossbauer studies (OZAWAand ISHIDA [1977]) suggest that the atomic packing in grain boundaries is more “open” than in a glassy structure. This conclusion is supported by recent studies of the atomic arrangements in the interfaces of nanostructured materials (5 5 of this chapter). Investigations of the atomic arrangements formed in the grain boundaries of nanostructured metals (e.g., by means of density measurements, X-ray diffraction, thermal expansion and various types of spectroscopies) suggest that the atomic structures of glasses differ from those of grain boundaries.
References: p . 935.
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Fig. 9. Schematic diagram of the arrangement of the atoms in a 40’ c 110> tilt boundary in tungsten, derived from a sequence of field-ion microscopy images. The position of the < 110> common tilt axis in the two grains is indicated. Numbers 1-5 indicate subsequent layers of the boundary. Letters A,-& label one of the polyhedral nngs proposed by POTAPOV etal. [1971].
2.2.7. Limitations of existing models A detailed comparison of the experimental observations (SUTTON and BALLUPPI [1987]) indicates that none of the above models seems to predict the boundary energy reliably in all cases, i.e., for metal/metal, ionichonic and metalhonic interfaces. Apparently, no geometric criterion can enshrine the universal answer to the question about the atomic structure of intercrystalline interfaces. This is not surprising since a severe shortcoming of all existing boundary models is the neglect of electronic effects. In fact, studies (HERRMANNet al. [1976] and MAURERet al. [ 19851) of low-energy boundaries in metals with the same lattice structure but different electronic structure suggest a division of all low-energy boundaries into two groups: “electron-sensitive”and “electroninsensitive” boundaries. Physically, this different behavior was interpreted in terms of the different atomic arrangements in the boundaries. If the atomic arrangement is similar to the lattice structure (e.g., in a twin boundary), the boundary energy is low irrespective of the contribution of the conduction electrons to the boundary energy. However, for boundaries with complex atomic structures, the electronic contribution to the boundary energy is crucial so that any difference in the electronic structure (e.g., two materials with different Fermi energies) leads to different boundary behavior. In fact, free-electron calculations showed that the positive charge deficit associated with a grain boundary may
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Fig. IO. Structure of a 36.9" loo> tilt boundary between fcc crystals, in t m s of polyhedral units. The boundary is composed of stacks of capped trigonal prisms.
be the dominant part of the boundary energy (SEEGERand SCHOTTKY [19S9]). A first ab initio solution of the electronic and atomic structures of a 2 =5(001) twist grain boundary in Ge was performed recently by PAYNEet al. [1985] by means of the Car-Parrinello (CARand PARINELLO [19851) method. In this method one solves for both the electronic wave functions and the ion positions. The only approximations made in these computations were the use of a local pseudopotential for Ge, the local density approximation for the exchange-correlation potential, and a large but finite number of plane wave basis functions. By modem standards this is a first principles calculation. One of the interesting features reported was that a contraction could occur at the boundary and was associated with a relatively low energy. This is contrary to the common experience with grain boundaries in metals. As the boundary region becomes denser, five-fold coordinated atoms may arise, and this was indeed reported. With only 4 valence electrons per atom, the system cannot form the same type of bond to a fifth neighbor as it has already established to four neighbors. Instead, rehybridization must take place. We might expect a five-fold coordinated atom to cost less energy than a three-fold coordinated atom since the former represents a smaller perturbation to the tetrahedral bond order. These ideas have been developed further by SUTTON [1988], who proposed a bond-angle distribution function to characterize the boundary structure. A further shortcoming of existing boundary models is their limited ability to account for temperature effects. The significance of temperature for the structure and properties of interfaces was demonstrated experimentally several years ago (e.g., ERBet al. [19821). The influence oftemperature on the stability and structures of interfaces has been modelled by a number of groups and was recently reviewed critically by PONTIKIS [1988], Molecular dynamics and Monte Carlo techniques have been employed and attention has focussed on disordering of the boundary region through roughening or premelting. Such disordering processes are nucleated by thermal fluctuations. However, many thermal properties of a grain boundary like those of a perfect crystal do not depend on thermal fluctuations but on thermal vibrations averaged over a long period of time, References:p . 935.
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e.g., such as the thermal expansion coefficient, specific heat and elastic constants. As little is known about these boundary properties, it is difficult at present to evaluate the relevance of the above computations. A great deal of information about the atomic structure of grain boundaries has been deduced by means of computer simulations. All simulations ultimately rest on the assumptions they make about interatomicforces. The majority of simulations assume some form of potential to describe atomic interactions. For simple s-p bonded metals, e.g., Li, NayK, Mg, Al, such interatomic potentials can be derived rigorously from the electrostatics of interacting spherical screening clouds of electrons called pseudopotentials. The presence of d-electrons at or near the Fermi level spoils this simple linear screening picture. For substances of this kind pairwise potentials have been generated by fitting more or less arbitrary functions to bulk properties such as cohesive energy, lattice parameter, elastic constants and vacancy formation energy. A more realistic class of potentials for transition and noble metals goes beyond the purely pairwise description of the interaction. They are referred to as isotropic N-body potentials. They are derived from a simplified tight-binding description of electronic densities of states. The next step up the ladder of realism in metallic solids takes us to 3- and 4-body potentials and to tight-bindingmudeZs beyond the second-moment approximation. For insulators, simulations have used the Born model in which the cohesive part of the total energy is the pairwise summation of the Coulomb interaction of the ions. This is counterbalanced by a short-ranged repulsive energy which is either fitted to the lattice parameter and bulk modulus of the perfect crystal or determined by the electron gas method. In semiconductors, a number of empirical angular-dependent interatomic potentials have been developed, motivated by the stability of the fourfold coordination which results from sp3hybrid bonds. None can be regarded as completely reliable for structural predictions since they are fitted to bulk properties, and the energy associated with dangling bonds or rehybridization remains poorly represented. In semiconductors, ab initio pseudopotential methods have made rapid progress since 1985, when CAR and PARIUNELLO[1985] showed how the Schrijdinger equation could be solved for the wave functions with a simultaneous adjustment of the positions of the ions - either to solve their equation of motion or to minimize the total energy. (For further details see ch. 2). On the experimental side, the most important sources of struchual information about interfaces are high-resolutionelectron microscopy (HREM) and X-ray diflaction studies. A major experimental problem with X-ray diffraction studies is the fact that boundary scattering is relatively weak. For example, for a 25[001] twist boundary in Au the intensity of the weaker diffracted beams of the boundary region corresponds to about 4% of a monolayer of Au. In addition to the resulting poor signal-to-noise ratio, the presence of forbidden lattice reflections, double diffraction from the crystals adjoining the boundary, and scattering from free surfaces enhances the difficulties of obtaining reliable [1980], FITZSIMMONS and SASS structural information (SASS [1980], SASSand BFUSTOWE [1988]. A promising new technique for studying interfaces by X-rays recently became important. This scattering of X-rays at grazing angles is called grazing incidence X-ray scattering, GIXS ( ~ ~ A R RetA al. [1979] and EISENBERGER and MARRA[1981]). In this
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technique, a grazing-incidence X-ray beam of high intensity is reflected not only by the bulk material, but also by the reflectivity of interfaces buried slightly underneath the surfaces. By fitting the computed and measured X-ray reflectivity curves, information about the density profile across an interface may be obtained. GMS studies require relatively large sample areas (typically 0.2 cm x 1 cm) under which an interface is buried. The GIXS study averages over this area. In other words, films of extreme homogeneity are required. Those films can only be obtained so far for epitaxial layers of semiconductors or metals grown on rigid substrates such as semiconductors or ceramics. Modern high-resolutionelectron microscopeshave point-to-point resolutions of about 0.16 nm In order to apply HREM to the problem of imagining the atomic structure of interfaces, only thin-film specimens (typical thicknesses 5 to 10 nm) and crystal orientations that result in close packed rows of atoms being parallel to the electron beam are suitable. In addition to these limitations, one has to keep in mind that the image of the boundary structure is a product of the wave field of the boundary in the objective plane and the contrast transfer function (CTF) of the microscope which depends on the lens errors and the focus conditions. In fact, the image of any object imagined in the electron microscope is severely modified by the CTF if electron scattering to large angle occurs since the influence of the spherical increases rapidly with the scattering angle. This effect is crucial for imagining the atomic structure of boundaries. Good imagining conditions are Mfilled for lattices with large lattice parameters. If, however, deviations in the periodicity exist, components of the diffraction pattern of such an object appear at large diffraction vectors (vJ. This applies, for example, most likely to the atomic arrangement in grain boundaries. Hence, the structure of grain boundaries can only be imagined reliably if the oscillating part of the CTF is outside of v,. Naturally, all HREM studies imply that no structural changes occur during specimen preparation.
2.3. Interphase boundaries
2.3.1. Bonding at interphase boundaries
In comparison to the extensive body of work on grain boundaries, relatively few studies have been directed at the understanding of the structure and properties of interphase boundaries. The bonding between chemically different materials across an interface involves interactions with and without charge exchange. The interaction between induced dipoles (London), between neutral atoms polarized by a dipole (Debye) and dipolddipole interactions (Keesom) constitute the first group and are summarized as Van der Waals amactions. Charge exchange results in the formation of ionic, covalent and metallic bonds. Among the numerous conceivable combinations of materials at interphase boundaries, interfaces between ceramics and transition and noble metals are of particular practical importance. Hence, it is of interest to understand the bonding between them. Most ceramics of technological use today are insulating metaoxides. They bind to afree-electron-likemetal essentially owing to the Coulomb attraction between the ions of the ceramic and their screening charge density in the metal (STONEHAM and TASKER[1988], FINNIS [1992], PINNIs et al. [1990]). This image attraction is balanced mainly by the hard-core repulsion between the ceramic and metal ions and it seems References: p . 935.
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primarily (DUFN et al. [1992]) the hard-core repulsion which determines the relative positions projected onto the plane of the interface of the ceramic and metal ions, i.e., the parallel rigid-body translation. The model of an electrostatic image interaction stems from the classical concept in the continuum electrostatics of point charges near a conducting surface, and makes no reference to discrete atoms. Hence, it usually has no place in simple models of adhesion based on chemical bonding. Nevertheless, its equivalent attractive force close to a real metal surface can be calculated quantum-mechanicallyfor simple geometries. Fortunately, the results of these quantum-mechanicalcalculationscan be simulated by a classical interatomic force model, suggesting a way to incorporate this effect in computations of the atomic structure of interphase boundaries (FINNIS [1992]). The binding of a ceramic to a transition metal is less understood, since here, strong covalent pd-bonds may be formed across the interface between oxygen and the transition atoms (JOHNSON and PEPPER [1982]). The formation of such bonds is supported by the observation that the adhesion correlates with the free energy of oxide formation for the transition metal; both increase in the order Ag + Cu +Ni Fe ... In this situation, it is remarkable that atomically sharp transition-metalkeramicinterfaces exist without interdiffusion and formation of a transition-metal oxide layer. Whereas the hard-core repulsion favors translation states with the metal cores above the holes in the top layer of the ceramic, pd-bonding pulls the transition atoms on top of the oxygens. Noble metals have filled, but polarizable d-shells so that their binding to a ceramic is presumably of intermediate nature. In this case, the question of the translation state is difficult and there could be several metastable states. Experimental results are lacking and, for the system (001) fcc Ag on (001) MgO, which has the rocksalt structure consisting of two interpenetrating fcc sublattices, the semi-empiricalimage-chargemodel predicts (DUPFYet al. [19921) silver to be above the hole between two magnesiums and two oxygens, whereas ab initio electronic density-functional calculations (BLtjCm et al. [1990], F~EEMANet al. [19901) favor silver on top of oxygen. However, no general prediction seems to be possible at this moment. For example, ab initio LDA calculations (SCH~NBERGER et al. [1992]) indicate that the exact atomic positions and the type of bonds formed in interphase boundaries depend on the elements involved. Both Ti and Ag were found to bind on top of the 0 in T i i g O and Ag/MgO interfaces. However, the binding between Ti and 0 is predominantly covalent and weaker than in bulk TiO; it corresponds to a Ti oxidation state less than +l. The bonding between Ag and 0 is weak and predominantlyionic. The bond length and force constants resemble those in AgzO where the oxidation state of Ag is +l. Whether it is the polarization of the Ag d-shell which pulls Ag on top of 0, or if this could also happen for an sp-metal is not yet known. The understanding of the fundamental physics involved in the bonding between a metal and a ceramic requires quantum-mechanicalmodels to be developed. The simplest approach involves cluster calculations (JOHNSON and PEPPER[19821). Such calculations have established that the primary interactions at metavoxide interfaces involve the metal (d) and oxygen @) orbitals, to create both bonding and antibonding orbitals. For copper and silver in contact with A1203, both states are about equally occupied, resulting in zero net bonding. However, for nickel and iron, fewer antibonding states are occupied and net
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bonding occurs. The calculations also reveal that a transfer of valence charge occurs, resulting in a contribution to the net ionic bonding which increases in strength as the metal becomes more noble. Consequently, in the case of metal-to-alumina bonding, strengths are predicted to increase in the order: Ag-Cu-Ni-Fe. This order is generally consistent with the measured trends in sliding resistance as well as with the energies of adhesion. However, it is emphasized that the calculations approximate the interface by an (AlO$ cluster and one metal atom. The selection of the charge to be assigned to this cluster is non-trivial and the choice influences the predicted magnitudes of the energies (ANDBRSON et al. [1985]). To further examine this issue, ANDERSON et al. [1987] performed calculations for the Al,O,/Pt couple that included more atoms: 3 1close-packed platinum atoms and the corresponding numbers of aluminum and oxygen ions. Then, by applying a quantum-chemical superposition technique, including an electron delocalization molecular orbital method, bonding energies were calculated for different atomic configurations of the WA120, interface. These calculations confirmed that the bond was strongest when oxidized platinum atoms opposed close-packed oxygen ion planes. Ab initio calculations seem to be essential for a full understanding of the bonding. LOUIEand coworkers (LOUIEand COHEN[1976], LOUIEet al. [1977]) have performed such calculations on metal-semiconductor interfaces. In these calculations, the metal was described by a jellium, so that insight emerged regarding the bonding mechanisms, but not on the atomistic sbxcture. More recently, supercell calculations have been carried out that include an interface area and adjacent regions large enough to incorporate the distorted (relaxed) volumes of both crystals. With this approach, the electron distribution around all atoms has been calculated and the atomic potentials evaluated. In a next step, interatomic forces may be calculated and strains determined. Such calculations have been performed rather successfully for the interface between Ge-GaAs (KUNCand MARTIN [198l]) and Si-Ge (VANDE WALLEand MARTIN [1985]). The crystals adjacent to those interfaces are isomorphous and very nearly commensurate, such that the misfit between lattice planes is very small. However, misfits between metals and ceramics are typically rather large so that extremely large supercells are required. First attempts in this direction have been made. SCH~NBERGEX et al. [1992] performed ab initio local density-functional calculations of the equilibrium geometries, force constants, interface energies and works of adhesion for lattice-matched interfaces between rocksalt-structured MgQ and a f.c.c. transition or noble metal. The interfaces had (~1)M/l(oo1)~&, and [lOO],ll[lOO]M, (M= Ti or Ag). The full-potential LMTO method was used. Both Ti and Ag are found to bind on top of oxygen. The interface force constants are 3-4 times larger for TilMgO than for AglMgO. These, as well as the M-0 distances, indicate that the T i 4 bonding is predominantly covalent and that the Agbonding is predominantly ionic. The calculated interface energies are both 0.8 eV/MIMgQ and the interface adhesions are, respectively, 1.2 eV/TilMgO and 0.9 eVIAglMg0. With the advent of a new calculational scheme (CAR and PARRINELLO [1985]), involving a combination of molecular dynamics (see, e.g., RAHMAN [1977]) and densityfunctional theory (KOHNand SHAM[1965]), it should be possible to conduct computaReferences: p . 935.
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tions of relaxed interfaces much more efficiently. The scheme should also allow equilibrium computations of metayceramic interfaces at finite temperatures. The conduct of such analysis on model interfaces should greatly facilitate the basic understanding of the bonding phenomenon and allow judicious usage of both cluster calculations and continuum thermodynamic formulations. For recent reviews of this area we refer to the articles by R ~ andEEVANS[1989], m s and RUHLE[I9911 and the Proceedings of an International Symposium on this subject edited by Ret aZ. [1992].
2.3.2. Chemistry of interphase boundaries In multicomponent two-phase systems, non-planar interfaces or two-phase product regions can evolve from initially planar interfaces (BACKHAUSand SCHMALZRIED [1985]). The formalism previously developed for ternary systems can be adapted to metavceramic couples, with the three independent components being the two cations and the anion. In general, the problem is complicated by having several phase fields present, such that intermediate phases form: usually intermetallics with noble metals and spinels (or other oxides) with less noble metals. The actual phases depend on the geometry of the tie lines, as well as on the diffusion paths in the tenary phase field, and cannot be predicted a priori. In general, the following two cases can be distinguished 2.3.2.1. Interfaces without reaction layers. Detailed scanning electron microscopy ~ et al. E [1986], and TEM studies performed for Nb/AlzO, (BURGERet al. [1987], R [1987]) have shown that no reaction layer forms. Concentration profiles revealed that, close to the interface, the concentration of aluminum is below the limit of detectability. However, with increasing distance from the interface, the concentration of aluminum, cM, increases to a saturation value. The corresponding oxygen content is below the limit of detectability. These measurements suggest that at the bonding temperature, c& the local concentration of A1 at the interface possesses a value governed by the solubility limit. Bonding between platinum and AlZO3subject to an inert atmosphere also occurs without chemical reaction. However, for bonds formed subject to a hydrogen atmosphere containing about 100 ppm H,O, aluminum is detected in the platinum, indicative of A1,0, being dissolved by platinum ( R ~ and E EVANS[1989]). 23.2.2. Interfaces with reaction layers. For systems that form interphases, it is important to be able to predict those product phases created. However, even if all the thermodynamic data are known, so that the different phase fields and the connecting tie lines can be calculated, the preferred product phase still cannot be unambiguously determined. Sometimes, small changes in the initial conditions can influence the reaction path dramatically, as exemplified by the Ni-Al-0 systems (WASYNCZUK and R ~ E [1987]). Under high vacuum conditions (activity of oxygen < lo-'') the diffusion path in the extended nickel phase field follows that side of the miscibility gap rich in aluminum and low in oxygen, (path I in fig. 111, caused by the more rapid diffusion of oxygen than aluminum in nickel. This interface composition is directly connected by a tie line to the Al,O, phase field, such that no product phase forms. However, whenever nickel contains sufficient oxygen (about 500 ppm solubility), the Ni(O)/AI,O, diffusion couple yields a spinel product layer. It is noted that the interface between spinel and nickel seems to be
Ch. 9, 52
863
Microst~re
0
Ni
Ni3AI
NiAl
AI
Fig. 11. Ni-AI4 phase diagrams (schematically for T= 1600 K). Two reaction paths are possible when nickel is Lmnded to AZO,:(I) Low oxygen activity: no reaction product forms, @) high oxygen activity: spinel forms. (From RUHLEand EVANS[1989].)
unstable, morphologicalinstabilities becoming more apparent with increasing spinel layer thickness. Bonding of copper to Al,O, seems to require a thin layer of oxygen on the surface of copper prior to bonding and CuA120, or CuAl,O, form (WITTMER[1985]). The spinel thickness can be reduced by annealing under extremely low oxygen activities, leading first to a “non-wetting” layer of Cu,O and then to a direct Cu/Al,O, bond. Bonding of titanium to A120,results in the formation of the intermetallic phases TiAf or Ti& which probably also include oxygen. The thickness of the reactive layer increases with increasing bonding time, and morphologically unstable interfaces develop. A recent study (JANGer al. [1993]) has demonstrated the capability of atom-probe field-ion microscopy to reveal the chemical composition profile across interphase boundaries on an atomic level. By applying this technique to Cu/MgO { 111}-type heterophase boundaries along a common e111> direction it was demonstrated that these boundaries are formed by CulOiMg bonds. Experimental evidence for the formation of reaction layers at interphase boundaries has been obtained by studying nanostmctured alloys (cf. Q 5 of this chapter). These alloys were synthesized by consolidating nanometer-sized crystals with different chemical compositions, e.g., Ag-Fe, Cu-Bi, Fe-Cu, Cu-Ir, Sn-Ge. Although all systems exhibit little mutual solubility at ambient temperature in the crystalline state, it was found (by X-ray diffraction, EXAFS and spectroscopic methods) that solid solutions are formed at the interphase boundaries. In the case of AgFe this result is particularly remarkable because Ag and Fe are immiscible even in the molten state. Most structural ceramics are polyphase materials. They are either composites or References: p. 935.
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different crystalline phases or contain an intergranular vitreous phase in addition to a single crystalline phase. The vitreous intergranular phases can be caused (i) as a result of a liquid-phase sintering process (e.g., in Sialon ceramics, alumina, etc.), (ii) by an incomplete crystallization of a glass (glass ceramic), and (iii) by a condensation of impurities present in the single-phase component at the grain boundaries (e.g., silicates in zirconia). In the materials noted above, most grain boundaries are covered with a glassy film and, in addition, glass is present at grain junctions. This observation may be explained by considering the energy of the grain boundary as a function of misorientation for crystalline and wetted boundaries. The curve of the interfacial energy vs. misorientation of a crystalline boundary exhibits cusps at the special misorientations. In contrast, the energy curve of the wetted grain boundary should be independent of orientation due to the isotropic nature of the glass. On the basis of such descriptions, low-angle grain boundaries will be free of glass, whereas all large-angle grain boundaries will contain an intergranular glass phase with the exception of “cusp” orientations (the “special” boundaries), as was observed.
2.3.3. Crystallographicstructure: “lock-in” model FECHTand GLEITER [19851determinedrelatively low-energyinterfaces between noble metal spheres and low-index ionic crystal substrates by the rotating crystallite method. They observed that in the resulting low-energy interfaces, some close-packed directions in the two phases are parallel and some sets of low-index planes are parallel, as well. On the basis of these observations it was concluded that no CSL orientation of low X exists in the vicinity of the observed relationships and, therefore, the applicability of the “CSL model” was ruled out for the systems studied. Instead, they proposed the following “lock-in” model: a) relatively low interfacial energy is achieved when a close-packed direction (ello>, , , ) in the crystal lattice of one phase is parallel to the interface and also parallel to a close-packed direction in the crystal lattice of the adjoining phase and if two sets of relatively low-index lattice planes are parallel to the interface. In this configuration, close-packed rows of atoms in the “surface” of the metal crystal can fit into the “valleys” between close-packed rows of atoms at the “surface” of the ionic crystal in “locked-in” configurations. Generally, the two phases adjoining the boundary are incommensurate, and unless a small strain is allowed parallel to the interface, locked-in rows of atoms soon begin to ride up the valleys in the interface and cease to lock-in. The lock-in model is not entirely crystallographic in character. Certainly, the conditions for the existence of parallel closepacked directions in the interface and of the interface being parallel to low-index planes are geometrical. But the assumption that, when these conditions are fulfilled, the interface relaxes in such a way that rows of atoms along the close-packed directions form a “locked-in configuration” is equivalent to assuming a particular translation state of the interface is energetically favorable. On the other hand, the planar density of coincidence sites is unaffected by the translation state of the interface. Hence, there is an important distinction between the lock-in criterion and the r criterion (cf. paragraph 2.2.2). Discrepancies from the orientation relationships predicted by the “lock-in” model have
Ch. 9, $ 3
Microstructure
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been found for interphase boundaries between Nb/Al,O, and CU/Al,O,. Relaxation effects are presumably responsible for these deviations.
3. Characterization of microstructure The basic elements of microstructure are lattice defects and second-phase components. A complete description of the topology and crystallography of the microstructure of a crystalline material requires the following information: (a) the spacial distribution of the orientation of all crystals; (b) the characterization of the shape, type, size and spacial distribution of all the elements of the microstructure; (c) the characterization of the orientation distribution of these elements in space (microstructural anisotropy). Naturally, all of these parameters can be represented in terms of orientational and spacial correlation functions. Any special feature of a particular microstructure such as a special type of symmetry, self-similarity, order, etc., of all or of some of these elements are enshrined in the correlation functions. In reality, however, the correlation functions are frequently not known because they are difficult to measure. Hence, in most cases, the discussion of microstructural features had to be limited to a few, relatively simple microstructures (HORNBOGEN [19891). However, in recent years modem on-line data evaluation procedures became available involving, for example, a combination of conventional microscopic methods with an on-line image processing attachment. By means of such systems the spacial orientation distributions of all crystals of a polycrystal can be measured (ADAMS[1993]). This procedure has been termed “orientation imaging microscopy”. Similar systems have been developed to characterize the size distribution, shape distribution, etc., of crystallites and other elements of microstructure. ]Inthe straightforward situation of randomly distributed defects and/or second phase particles, the microstructure may be characterized to a first approximation by an average density, p. of defects and/or second-phase particles. Depending on the type of microstructural element involved, p describes the number of 0-dimensional defects (e.g., vacancies) per volume, the total length of one-dimensional defects (e.g., dislocations), the total area of two-dimensional defects (e.g., grain boundaries) per volume, or the total volume fraction of three-dimensional objects (e.g., pores, precipitates). However, even in the simple situation of uniformly distributed microstructuralelements, the average defect spacing may not be the appropriate parameter to describe the microstructure. For example, in the case of a polycrystal with equiaxed grains of a narrow size distribution, the use of the average grain size as a length scale implies that all boundaries are the same, As this is mostly not so, a new length scale has to be introduced which accounts for the boundary to boundary variability. This new length scale is given by the clusters of grains linked by grain boundaries sharing misorientations in the same category. In fact, it has been shown that it is this length scale of the microstructure that controls crack propagation and failure of polycrystals (PALUMBO etal. [1991a], WATANABE [1984]). The References: p. 935.
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H. Gleiter
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control of properties of materials by means of controlling the structure of interfaces seems to develop into a separate subdiscipline of materials science termed “interfacial [19931). This engineering” or “grain boundary design” ( W A T A N ~[1993], E HONDROS new discipline is based on the structure-property relationship for individual boundaries and relates the boundary-induced heterogeneity of deformation and fracture of polycrystals to the topological arrangement and the spacial distribution of the character of grain boundaries in polycrystalline materials. It is these parameters that seem to be crucial in controlling the high-temperatureplasticity, superplasticity and brittleness. (Ch. 28. 5 3.7). Frequently, the spacial arrangement of the microstructural elements is non-uniform. For example, a non-uniformity in the dislocation density may exist on different length scales; e.g., in heavily cold-worked materials, dislocation cell walls are formed on a nanometer scale whereas in irradiated specimens the formation of helical dislocations introduces roughness on a submicron scale. In some cases an isotropic or anisotropic length scale is required to characterize such microstructures. In other cases this is not so. For example, in some materials fractal microstructures have been reported (e.g., in polycrystalline alloys with localized slip). The significance of fractals in various types of microstructures in metals has been discussed recently by HORNBOGEN [1989]. In multiphase alloys different types of microstructuresresult depending on the volume [1986], [1989]). For the fraction, shape and the distribution of the phases (HORNBOGEN sake of simplicity let us limit the discussion first to two-phase systems (called a and 6 phases) with equiaxed grains or particles. In alloys of this type a variety of microstructures can be formed. A convenient parameter to characterize the various microstructures is the density @) of grain and interphase boundaries. In a microstructure called an “ideal dispersion” of a particles in a p matrix no alp interphase boundaries exist (fig. 12a). The other extreme results if the p particles cover all ala grain boundaries completely. In other words, the ala boundaries disappear and we are left with alp interfaces and PIP boundaries the relative density of which depends on the p crystal size (fig. 12c). In both cases, two-phase microstructures (fig. 12b) exist that are characterized by different degrees of percolation of the a and p phases. One way to characterize this type of microstructure (called duplex structure) is to determine the relative densities of the alp. PIP and alp interfaces. In the ideal duplex structure the ratio between the total number of grain and interphase boundaries is 0.5. In the cases considered so far, the two phases were assumed to have no shape anisotropy. In reality, this is often not so. In order to exemplify the role of anisotropy, a few microstructures with different degrees of anisotropy are displayed in fig. 13 (HORNBOGEN [1984]). Depending on the aspect ratios of the second-phase crystals, a fibrous or 1amelIar structure may result. Obviously, other anisotropic microstructures are formed if the relative aspect ratios of a duplex or a skeleton-type microstructure are modified. Microstructuraltransformations.The well-established scheme of phase transformations in condensed matter systems may be extended in a modified form to the transformation of one type of microstructure to another one. The following three cases have been discussed so far:
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Fig. 12. Principal types of qui-axed two-phase microstructures. (a) Dispersion: Ni+ 18.2Cr+5.7AI, y+y’. (b) Duplex: Fe+9wt%Ni, a + [y+cu,] (austenite transformed to martensite during cooling). (c) Net: Fe+O.6wt%C, a +Fe,C. (From HORNBOGEN[1984].)
a) Transfornation of microstructural elements. Well known examples for this type of transformation of microstructural elements are as follows. The condensation of single vacancies into dislocation loops, the transformation of small-angle boundaries into high-angle grain boundaries by incorporation of additional dislocations, the incorporation of dislocations into high-angle grain boundaries which transforms them, for example, from a special (low-energy) boundary into a random (high-energy) one (cf. References: p . 935.
868
Ch. 9, $ 3
H. Gleiter
Coating
Lamellae
Fibres
Dispersion
Duplex
Sceleton
Type of microstructure
Fig. 13. Different two-phase microstructures depending on the topology of the two phases. (From HORNBOGEN [ 19841.)
0 2 of this chapter), the change from coherent to incoherent interphase boundaries during the growth of precipitates are examples of this kind. b) Transformationof a crystalline material with a high defect density to a new phase Such a transformation may occur if the spacing between microstructural elements (e.g., dislocations, grain boundaries) in a crystal approaches atomic dimensions. If this is so, it may transform into an amorphous phase (OEHRING et al. [1992]). The same applies to the shock wave-induced transformation of a crystal into a glass or the dislocation model of melting. In the dislocation model of melting, melting is proposed to occur by the entropy-driven proliferation of dislocations in crystals (KUHLMANNand WARNER [1979], HOLZ[1979]). WILSDORF [1965], COTTERILL [1979], EDWARDS The phase transition from (Y to /3 quartz has been shown to occur by the incorporation of a high density of regularly spaced Dauphin6 twins. c) Transformationsbetween diferent types of microstructures Transformations of this type are defined by the appearance of new microstructural elements. Two-phase microstructures shall serve as examples for such transformations. A transformation of one type of microstructure to another is associated with
Ch.9, $ 3
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Microstructure
the appearance or disappearance of a particular type of boundary which represents a characteristic topological feature. Transformations can take place as a function of volume fraction of the phases, and of their shape and orientation. For constant shapeand size-distribution,the transformation must take place at a critical volume fraction, f,. This in turn leads to relations with alloy composition x and annealing temperature T and with the equilibrium phase diagram. Figure 14 shows two-dimensional representationsof the followingtransformations: duplex dispersion and net L- dispersion. Microstructural transformationsare found not only under conditions of phase equilibrium, but also if the volume fractions change after nucleation and finally when an equilibrium value is approached. The first stage in formation of a duplex structure is usually the formation of randomly dispersed nuclei. In this case, the transformation, dispersion + duplex, takes place during isothermal annealing. Other examples for the occurrence of microstructural transformations in alloys are bulk concentration gradients as for example for decarburization or carburization of steels. Concentration gradients may cause microstructural gradients.
f = 0.5
f < f, = 116
Fig. 14. Transformations of types of microstructure in planar sections (schematic). (a) Dispersion ==net. (b) Duplex * dispersion. (From HORNBOGEN[1984].) References: p . 935.
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4. Development of microstructure 4.1. Basic aspects The microstructure of a crystalline material may result from structural phase transformations and/or interaction processes between structural defects. In fact, a general and reciprocal relationship exists between microstructures and defect interactions. Any microstructure may be interpreted as the result of the interaction between structural defects and/or phase transformations. On the other hand, it should be possible to synthesize new microstructures (and, hence, materials with new properties) by utilizing this relationship, e.g., by means of more complex defect interactions (possibly combined with phase transformations). The thermomechanicaltreatment of materials represents an example of this kind. Microstructures generated by structural phase transformations are discussed in chs. 8, 15, 16, and 17. Microstructures resulting from the interaction between some specific types of lattice defects are treated ifl chs. 21-25 and 27. In this chapter, attention will be focused on those microstructures that are not the result of phase transformations. Despite the complexity of all conceivable processes for the development of microstructures, a guideline for a systematic understanding of the genesis of microstructures may be provided by considering some relatively simple and well studied types of interaction processes involving only one or two types of structural defects. This "model approach" will be used here.
43. Microstructuralchanges stimulated by interfacial-energyreduction Three classes of microstructural changes driven by interfacial energy may be distinguished
1. Microstructural changes in single-phase materials (without applied potential fields). 2. Microstructural changes in polyphase materials (without applied potential field) when the materials have: (a) a dispersion-type structure; @) a duplex (or a network) structure. 3. Microstructural changes due to the effects of applied potential fields (e.g., temperature or electric field gradients). An excellent review of these classes is the one by MARTINand DOHERTY [1976].
4.2.1. Microstructuralchangesin single-phasematerials,stimulatedby interfacial energy: domain and grain growth Internal interfaces in solids - such as grain boundaries or domain boundaries in ordered systems (atomic order, spin order, polarization order, etc.) -are associated with a positive excess energy resulting in grain or domain growth. Local atomic arrangements at or near moving interfaces can differ significantly from arrangements at or near stationary interfaces, giving rise, for example, to drag effects (solute or defect drag), structural changes of the interfaces due to defect production, etc. A complete theory of interfacial motion would have to account not only for these effects, but also for the topological changes of the array of interconnected interfaces during domain and grain
Ch. 9, 34
Microstructure
871
growth. For far no such theory is at hand. However, models treating various aspects of the problem separately have been put forward. The problems of the growth of grains or magnetic domains are treated in chs. 28 and 29, respectively, As many growth theories make no distinction between grain and domain growth, we refer to ch. 28 for all theories that apply to both processes. Two approaches to describe the motion of a domain wall have been proposed. The earlier, widely used phenomenological approach states that the wall mobility is proportional to the thermodynamic driving force, the proportionality constant being a nositive quantity called the mobility. The driving force in this approach is the product of the mean of the local principal curvatures of the interface and the excess free energy per unit area (cr). This approach leads to a growth law of the type 4 b 2 aut,where is the average diameter of the domains in a polydomain skucture and t is the time. This result has been experimentally confirmed for several ordered alloys (e.g., ARDJXLet uZ. [19791 and ROGEXSet aZ. 19751) and can account for the broad distribution of domain sizes frequently observed experimentally. The approach of ALLEN and CAHN [1979] models the motion of an interface by solving a diffusion equation that has been modified to account for the thermodynamics of non-uniform systems (LANGEXand SEKERKA [1975]). Domain walls have a width in which there are compositional and order-parameter variations, i.e., a crystal containing a domain wall is a non-uniform system. The order-parameter variation is the basis of a diffusion potential whose gradient results in an atomic flux. This description leads to a time-independent Ginsburg-Landau equation for changes in the order parameter and hence for the wall motion. In the limit of large radii (r) of wall curvature, the propagation velocity (V) of the wall is found to be proportional to r, but independent of the excess energy (a)of the wall. This does not imply that the energy dissipation during domain growth is independent of cr. In fact, the energy dissipation may be shown to be proportional to cr. Experimental tests were carried out in which cr was varied by two orders of magnitude. Domain-coarseningkinetics and a were found to scale differently with temperature as had been predicted theoretically. On the basis of the results discussed so far, we are led to conclude that cases exist where V is proportional to cr, and there are clearly other cases where V is independent or even nonlinearly dependent on v (e.g., TURNJWLL [1951] and LI [1969]). No general criterion seems apparent at present for relating Y to a and boundary curvature for different experimental conditions. The development of our understanding of continuous (“normal”) grain growth started when it was recognized that the driving force for grain growth is the decrease of grain boundary energy (SMJTH[1952]). On the basis of this approach several simple and mostly qualitative interpretations of grain growth were given. The first quantitative model (BECK 119541) predicted an increase of the average grain diameter with time as t”’ which, however, was rarely confirmed experimentally. Mostly, time exponents less than 0.5 were observed and attributed to the drag forces caused by inclusions or solute atoms. More insight into the processes involved in grain growth was obtained when more sophisticated theoretical approaches were used. HILLERT[19651 applied the statistical treatment of Ostwald ripening of precipitates to grain growth. Some of the foundations References: p . 935.
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of Hillert’s analysis were reexamined by LOUAT[1974] who pointed out that in grain growth, as opposed to Ostwald ripening, “grain collisions” OCCUT in which faces are gained andor lost. In order to allow for these events, grain growth was considered as a particular case of random walk. HUNDERIand RYUM [1979], [1981] applied a deterministic model considering individual boundaries and described the change of size of the individual grains by an extremely large set of differential equations (one for each grain), which they solved numericany. Very recently, ANDERSONet al. [1984] used the Monte Carlo method to simulate grain growth and included in this treatment also the case of Zener drag. Finally, ABBRUZZESE [1985] further developed the Hillert model by calculating the critical radius. He used discrete grain size classes which led to a reasonably small set of differential equations (only one for each class) and thus to the possibility to calculate numerically the evolution of the grain size distribution. This approach was extended further by including textural effects (ABBRUZZESE and LUcm [1986]) in the form of orientation-dependent grain-boundary energies and mobilities. The most important result of this extension was that, instead of the single critical radius found in the textureless case, now for each orientation a different critical radius is obtained but with a value depending on all orientations. It is shown that (very generally) grain growth leads to pronounced texture changes which are accompanied by characteristic changes of the scattering of the grain size distribution and by peculiarities of the grain growth kinetics, which then does not even approximately follow a 1’’’-law. On the experimental side, the effect of particles has repeatedly been taken into account. The amount of work on the texture effects, however, is still small. For a long time, it was limited to measurements of the orientation of secondary grains. It was first and LUcm [19831) that also the apparently continuous shown for a-brass (BRICKENKAMP grain growth leads to drastic texture changes and that these are connected with peculiarities in grain growth kinetics and grain size distribution. The authors were able to put forward some qualitative interpretations for these effects. “Anomalous” grain growth (also termed secondary recrystallization) is characterized by an increase in size of a small fraction of the crystallites whereas the rest does not grow at all. Anomalous grain growth has been suggested to originate either from the fact that a few of the crystallites have higher grain-boundary energies and/or boundaries of higher mobilities than the majority of the grains. In the experimental literature, anomalous grain growth was demonstrated to exist in pure polycrystals as well as multiphase alloys. PVlAy and RTRNSULL [19581demonstrated the significance of second phase particles for anomalous grain growth for the first time. In Fe-Si alloys, the occurance of anomalous grain growth was shown to depend on the presence of MnS particles which inhibited normal grain growth. In several high-purity materials (SIMPSONetal. [1971], ANTONIONE et al. [1980]) strain and texture inhibitation of normal grain growth was demonstrated. Texture inhibition is based on the following idea. If the grain structure is strongly textured, boundaries of low mobility result. The anomalous grains are assumed to have high mobility boundaries because their crystallographic orientations relative to their neighbouring grains deviate strongly from the texture. A firm link between texture and abnormal grain growth has been made by
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IIARASE et al. [1988]. In a Fe,&,
alloy they found that the largest grains after secondary recrystallization had a high frequency of coincidenceboundaries such as Z 7 or Z 19b with the primary matrix grains.
4.2.2. Microstructural changes in polyphasematerials with a dispersion structure, stimulated by interfacial energy: Ostwald ripening K interfacial energy is the only driving force for an instability and if the rate of development of the instability is governed only by mass transport processes, the linear dimension, d, of any microstructural feature can be shown (HERRING[1950]) to scale with time by the expression:
d”=&+aGt (1) where do is the value of d at time t = O , G is the parameter of the appropriate mass transport process and a is a dimensionless parameter which depends on the geometry. The scaling exponent, n, takes the values: n= 1 for viscous flow, n = 2 for interfacial control, n = 3 for volume diffusion in all phases, n = 4 for interfacial diffusion and n = 5 for pipe diffusion. The growth laws discussed in the following paragraphs for specific processes extend eq. (1) by giving explicit expressions for a and G. Normally, microstructural changes in multiphase alloys involve c h g e s of shape, size andor position simultaneously. For convenience, these three aspects are discussed separately. An array of inclusions or dislocation loops or pores of equilibrium shape, but different sizes, interact because the concentration of solute atoms (or the concentration of vacancies, or the vapor pressure in a gaseous system) in the vicinity of small (large) particles is higher (lower) than the average supersaturation. The solute, therefore, flows from the smaller to the larger particles. Hence, smaller particles shrink and larger particles grow by “devouring” the smaller ones, a process known as Ostwald ripening. For reviews of various aspects of the problem, we refer to the articles by JAIN and HUGHES[1978], KAHLWEIT [1975] and HENDERSON et al. [1978]. The phenomenon of Ostwald dpening was analyzed first for the solid state by G R ~ E N ~[1956], ~ ~ Dand then independently and simultaneously by WAGNER[1961] and by LIFSHITZand SLYOZOV [1961], assuming the common case of spherical precipitates growing by volume diffusion. Analogous expressions have been developed for other types of coarsening. The corresponding constants (a,G, n) of eq. (1) are summarized in table 1. In the case of coarsening of spheres by volume diffusion, which is the most commonly observed case (&e),a steady-state distribution of sizes is predicted to be approached irrespective of the initial size distribution, with a maximum particle size of SF, where r is the mean particle radius (figs. 15a, b). (See also ch. 15, 82.4.2) A critical assumption in the Lifshitz-Slyozov-Wagner (LSW) theory is that the diffusion fields around each particle are spherically symmetrical. This is strictly valid for zero volume fraction of precipitates when the concentration gradients around adjacent particles do not interfere. If the LSW theory is extended (ARDELL[1972]) to account for finite volume fractions, the basic form of eq. (1) is retained but the proportionality constant 01 is increased and the particle size distribution is broadened. The LSW analysis cannot persist to very large times, because ultimately the system should ripen into one References: p . 935.
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large particle. In fact, it was shown (KAHLWEZT[1975]) that the coarsening rate of the largest particles initially increases rather rapidly, then passes through a maximum and slowly approaches zero for long periods of time. The value of the coarsening rate predicted by the LSW theory is reached shortly before reaching the maximum rate. The significance of coherency strains for Ostwald ripening effects will be discussed in the next sub-section on stability against coarsening. In the LSW treatment, solute transport is assumed to be the rate-controlling process. Hence, modifications are required if other processes play a role, for example, ternary additions, solvent transport effects or dissociation of solute andor solvent molecules. The latter case has been discussed by WAGNER[1961]. The effect of solvent transport may be accounted for (ORIANI [1964]) by modifying the diffusion parameter D in the LSW equations, leaving the main result (growth law, size distribution) unaltered. The modification of D becomes important, however, if we use growth-law observations to derive a,D, etc. The same applies to the effect of ternary additions on the ripening rate. Ternary additions alter the rate constants of coarsening by a factor 1/3(1 -K)-2c,,-1, leaving the scaling law unchanged (BJORKLUND et al. [1972]). c, is the ternary alloy content and K is the distribution coefficient. Ostwald ripening of semi-coherent plate-shaped precipitates (Widmansttitten plates) represents yet another case for which the LSW treatment cannot be applied without modifications. Ostwald ripening of W i d m a n s t n plates results in large aspect ratios, as the lengthening of these plates is diffusion-limited whereas the thickening is mobilitycontrolled by the good-fit (semi-coherent) interface (AARONSONet al. [1970] and -ANTE and DOHERTY [19791). The experimental confinnations of the theoretical predictions on Ostwald ripening are [1978]). Although numerous still fragmentary (for a review see JAIN and HUGHES experiments confirming the scaling laws given in table 1 have been reported (for a and DOHERTY [1976]), the relatively small review of the earlier data we refer to MARTIN range of particle sizes that can be obtained experimentally is not sufficiently precise to allow unambiguous identification of the scaling exponent. In most cases, the observed r
I
I
I
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Micmstnrcture
875
NORMALIZED PARTICLE RADIUS ( VP)
F?
I
I
I
I
E
I
1
vi'
y
3c
8
20
0 Icc LL
0
>
t 10 m Z
W 0
PARTICLE DIAMETER, pm Fig. 15. (a) Plot of the particle radius against (time)'" for a nickel-aluminium alloy annealed at three different temperatures (from -ELL et al. [1966]). (b, c) Size distribution developed during Ostwald ripening: (b) theoretical prediction according to the Lifshitz-Slyozov-Wagner theory; (c) experimental observation on silica particles in copper annealed at 1173 K for 27 h. (From BHA~TACHARYA and RUSSELL[1976].)
size distributions are wider than those predicted by the theory (figs. 15b and 1%). Frequently, also a few large particles are found which are not a part of the main system of phcles. Furthermore, a tail on the large-size side of the size distribution is observed, in contradiction to the LSW theory (see also ch. 15, 0 2.4.1). The available experimental evidence on the effect of volume fraction on coarsening is conflicting. Studies on Cu-Co alloys showed clear dependence on volume fraction, whereas other work (Ni,AI in Ni, NbC in Fe, Cu in a-Fe) failed to detect any effect. Recent experimental work has evidenced the possibility of contact between growing References: p . 935.
876
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H.Gleiter Table 1 Ostwald ripening rates.
Rate-controlling process
Shape of Expressions for Constants' particles n a G
Reference
WAGNER [1961] LIFSHITZ and SLYOZOV [1961] DOHERTY [1982]
3 819
DubC,VJRT
Plates
3 3A'(1 +A'/A',)
DfibV,,/2pRT
Grain-boundary diffusion
Spheres
4 9/32
Dbd,V,G/ABRT
Dislocation-pipe diffusion
Spheres
5 (l.03)5(3/4)45/~ DgbC,VmqNq/RT
ARDELL[I9721
2 64/81
DOHERTY
Volumediffusion Spheres
Interfacelimited
u,SC,VJRT
ARDELL [I9721 KIRCHNER [1971]
[1982]
growth a
The symbols used are given below with their meaning or an expression.
-
parameter, A = 2/3 udr + (rdu)*/24. parameter, B=41n l/$ average and equilibrium aspect ratios of precipitate, respectively. precipitate and matrix concentrations, respectively. C,, Cs D , Db, Dd general, boundary, and dislocation-pipe diffusion constants, respectively. function given by f= C(1- CJ(C, - C,)*. f number of dislocations intersecting one particle. N parameter tending to T for large precipitates. P diffusional cross-section of a dislocation. 4 R gas constant. T temperature. vm molar volume. proportionality constant including the interface mobility. B 6 boundary thickness. 7) geometrical parameter. energy of grain boundary or interphase boundary. u(uJ A B A', A',
precipitates. The theoretical discussion of this effect is based on independent growth of adjacent precipitates, the center-to-center distance between the particles remaining fixed (DAVIS et aZ. [198Oa]). It is, however, difficult to see how the solute atoms are led into the narrowing gap between adjacent particles. An alternative hypothesis is that the particles will actually attract each other and move together in order to reduce the elastic strain energy. As similar effects have been seen in alloys with small lattice misfit (Ni-Cr-Al), t h i s hypothesis seems not to provide a convincing explanation. In a number of alloys (e.g., carbides in steels, UAl, in a-U,0 in Al-Cu alloys) evidence has been presented suggesting substructure-enhanceddiffusion, i.e., power laws during coarsening (r"-t) where n is greater than 3. In none of these studies were attempts made to fit the results to one of the theoretically predicted relationships r"-t and no work seems to have been published which attempted to check if the details of the substructure-enhanced diffusion theories are correct. The general conclusion, though, is that the theories described seem successful in accounting qualitatively for the effect of lattice defects on coarsening.
Ch. 9, 54
Micmstmcture
877
43.2.1. Stability against coarsening. The general condition for Ostwald ripening to proceed is a decrease of the free energy. In the case of precipitates surrounded by an elastic strain field, the total energy (E) of an array of precipitates consists of the volume energy, the interfacial energy of the precipitates and the elastic energy of the strain fields. If the precipitate volume is constant, E depends only on the elastic energy and the interfacial energy. In the special case of two precipitates in a solid (volumes VI and V2), the interfacial energy (5‘) and elastic energy (2‘) scale as (VIw3+ VZ”’) and (V, + V,) + V,V2fr$, respectively, where a is the separation of the two precipitates andf is a function that is unity for distant precipitates. If T B S , the total energy (for VI + V,=const.) has a minimum if VI = V,. In other words, the strain energy stabilizes the two particles of the same size against coarsening into one large particle. Basically, the same arguments hold for infinite arrays of particles as was apparently first recognized by KHATCHATURYAN and SHATALOV [1969]. The general conditions for stability of precipitate arrays against coarsening were recently worked out by PERKOVIC et al. [1979], LARCHEand CAHN [1973], [1978] and JOHNSON and ALEXANDER [1986]. Stability was found to be promoted by low interfacial energies, large elastic misfits and large volume fractions of precipitates. The phenomenon of elastic stabilization may be significant for the design of hightemperature alloys. In fact, the growth rates observed experimentally for 8’ precipitates in A1-Cu alloys (Born and NICHOLSON [1971]) seem to support this view. AUBAUER [1972] has attempted to account for certain fine dispersions being stable against coarsening in terms of the diffuseness of the interface between a precipitate and the surrounding matrix, as described by CAHNand HILLIARD [1958]. If one assumes that the diffuse rim surrounding a precipitate is independent of precipitate size, it can readily be seen that the fraction of material that is in the precipitate and not in the rim will increase as the size increases. The bulk of the precipitate has a different structure and therefore a different atomic volume from the matrix while in the interface rim zone it is assumed that the structure changes steadily towards that of the matrix. Consequently, the precipitate and the rim zone are associated with an elastic distortion. The energy associated with this distortion increases as the particle grows, whereas the energy associated with the diffuse interfaces decreases as the total surface area is reduced during coarsening. If the surface energy is sufficiently small and if there is a finite rim thickness and appreciable strain energy, a minimum exists in the total energy, stabilizing the corresponding particle size. This conclusion was questioned because of the treatment of the strain energy used (DE F o m m [1973]) and because the precipitates, even if stabilized against growth, should be unstable against a change in shape, for example towards a disk (MORRALL and LOUAT[1974]). On the other hand, the Aubauer model seems to account successfully for several reported cases (GAUDIGand WARLIMONT f19691, WARLIMONT and THOMAS[1970]) where very fine dispersions of coherent ordered particles were seen to resist coarsening. The stability against coarsening ot two misfitting particles subjected to an applied tensile stress was examined by JOHNSON [I9841 using a bifurcation approach. The stability of precipitates subjected to an applied stress may be enhanced or diminished depending on the relative orientation of the precipitates and the magnitude of the applied stress. The results obtained for cube-shaped particles are qualitatively different from those for spheres at short distances of separation. References: p . 935.
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The effects of the applied stress are manifested only in elastically inhomogeneous systems and are sensitive to the precipitate morphology. The theory predicts that precipitates may be stable up to a level of applied stress (threshold stress) sufficient to induce morphology changes. 42.2.2. Technological applications of coarsening theory. In all types of coarsening, the rate of the process is proportional to the interfacial energy (a)driving the process and the solubility C, of the solute atoms. Furthermore for all situations, except the relatively rare interface-controlled one, the coarsening rate also scales with the diffusion coefficient (0).Hence, alloys for high-temperature application, where low coarsening rates are desirable, may be obtained if (T, C, or D are small. This expectation is borne out by various classes of high-temperature materials. Nickel-based superalloys containin coherent ordered y’ precipitates (Ni,Al structure) in a disordered y-matrix (Ni-A1 solid solution) have exceptionally low inte$uciuZ energies of the order of lo-’ J/m’. In other words, the driving force for the coarsening of the y’ precipitates during creep is minimized. Alloys of this type retain their mircrostructure during creep. If the microstructure of such an alloy is tuned to a maximum creep lifetime, the microstructure changes little during creep loading. This is not so if the lattice misfit between y and y’ is significant.The lattice misfit enhances the driving force for y’ coarsening, which changes the microstructure of the alloy in the coarse of a creep experiment. Hence, the microstructure tuned to maximimum creep life disappears during creep, and reduces the lifetime relative to an alloy with zero misfit. This difference is borne out by the observations. In fact, owing to the different solubilities of most elements in the y and y‘ phases, the y/y’ misfit can be tailored to zero by the addition of solute elements to equalize the lattice constants of the two phases (e.g.. Cr). Low solubility can easily be achieved by using precipitate phases with high energies of formation and with a type of chemical bonding that differs from the surrounding matrix. The best-known examples are low-solubility oxide-dispersed phases, e.g., Al’O, in Al. Except for very special cases, such phases cannot be precipitated from a supersaturated solid solution. Therefore, other techniques, such as powder metallurgy, internal oxidation or implantation are commonly applied. If the atomic radius of the solute atoms differs strongly from the atomic size of the matrix material, the two metals normally show negligible solubility in the solid state. Alloys of this type (e.g., W-Na, Al-Fe) have also been used to obtain coarsening-resisting materials. Low diJJirsion coeficients have been applied in several ferrous alloys to resist coarsening. For example, the addition of a third component which segregates preferentially to the carbide phase (e.g., Cr, Mo, W) can slow down the coarsening of carbides considerably as it requires diffusion of both carbon and the third element and the latter, being substitutional, diffuses much more slowly than the carbon.
42.3. Microstructuralchanges in polyphase materials with a duplex structure, stimulated by interfacial energy A dupZex structure (SMITH[1954]) is an oriented crystallographic unit consisting of two phases with a definite orientation relationship to each other. Technologically and scientifically, the most important group of duplex structures are rod- or plate-shaped
Ch. 9, 44
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Microstiwcture
duplex structures, such as directionally solidified eutectics (for a review, see LIVINGSTON 119711). As the growth of large lamellae at the expense of smaller ones is associated with a decrease in the surface-to-volume ratio, lamellar structures are expected to coarsen. In the absence of substructural effects this process is expected to occur by motion of lamellar terminations. W o mechanisms have been put forward for this process. CLINE [1971] and GRAHAM and KRAFT [1966] proposed the curvature at the lamellar termination (fault) to be associated with a flux of atoms from the (Y phase to the p phase (fig. 16). The second mechanism of lamellar coarsening involves the diffusion of solute atoms from the finely spaced (A,) lamellae along a migrating boundary to the widely spaced (A3 lamellae on the other side of the interface (fig. 17). The theoretical analysis of the process (LIVINGSTON and CAHN[1974]) relates the boundary migration rate (V) to the spacing A, of the widely spaced lamellae. The coarsening rate increases with increasing temperature and decreasing spacing of the lamellae as both effects reduce the diffusion times required. As short-circuit diffusion along grain boundaries becomes dominant at lower temperatures, coarsening by boundary migration is expected to become more prominent than coarsening by fault migration with decreasing temperatures as was observed experimentally. In comparison to coarsening by fault recession (fig. 16), coarsening by boundary migration becomes more important at finer spacings of the lamellae. Rod-shaped microstructures
I
I
Fig. 16. Schematic of mass flux in the vicinity of a lamellar termination (fault). The curvature at the termination is proposed to induce mass flow of A atoms (from the a-phase) to the p matrix, leading to a recession of the termination with a corresponding increase in the thickness of the adjacent lamellae. (From CLINE[1971].) References:p . 935.
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A
Fig. 17. Idealized model of discontinuous coarsening process. The grain boundary AB moves at a velocity V, consuming fine lamellae with spacing A, and generating coarse lamellae with spacing A,. (From LIVINGSTON and CAHN [1974].)
are unstable with respect to shape and dimensional changes as they may decrease the surface-to-volume ratio by these processes. As was pointed out by CLINE[1971] and ARDELL[1972], the processes involved in dimensional changes are identical to Ostwald ripening ($4.2.2). In alloys produced by eutectic growth, the microstructure may be initially stabilized by the very uniform rod diameter. The time required for the steady state distribution of rod diameters to be built up during coarsening may be longer than for normal precipitate coarsening where a whole spectrum of particle sizes is present at the very beginning. In the present paragraph, attention will be focused on shape changes due to the coarsening of rod-shaped microstructures. The growth of a shape perturbation [1878]. However, it is on a cylinder was already analyzed a century ago by RAYLEIGH only relatively recently that quantitative models for the spheroidization of cylindrical precipitates have been put forward (CLINE [1971], Ho and WEATHERLY [1975], NICHOLS [1976]). The theoretical treatments indicate (fig. 18) that a long fibre (length I ) of radius d (with l/d>7.2) is eventually replaced by a string of spheres (Ruyleigh instability) where the sphere radii and spacing, A, depend on the active kinetic processes. When the aspect ratio (l/d) is less than 7.2, shape relaxation to a single sphere is predicted. For infinite fibres, Rayleigh instabilities are predicted to dominate, whereas in the case of
Ch. 9, $4
881
Wicrostrucfure
fa 1 infinite fibres
il
I c 1 finite
f ibms
1L7.2) d
INCREASNG TIME
-
Fig. 18. Schematic representation of different modes of spheroidization of cylindrical inclusions. (From MCLEANri97s1.)
fibres with finite length, drop detachment at the end of the fibres (fig. 18c) should be the faster process. Experimental observation for metallic (W-Na, Al-Pb, fig. 19) and nonmetallic systems (NaN0,-H,O, Ni-A120,) seem to support the view that progressive spheroidization from the ends of the fibres dominates. Yet another mechanism of spheroidization, which applies to both fibrous and lamellar inclusions, exists when grain boundaries in either phase intersect phase boundaries between the fibres and the matrix. At the points of intersection, grooves develop and progressively deepen with time to establish a local equilibrium configuration. Eventually, the grooves cause a division of one phase and result in spheroidization (fig. 18d). This process seems to be a serious limitation to many directionally solidified eutectic materials for high temperature operation. In addition to the instabilities mentioned so far, rod phases may also coarsen by fault migration. Faults are points of a rod at which additional rods form by branching References:p . 935.
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45
10 5
4 65
855
225
1 9 6 5 min
Fig. 19. Series of microradiographs showing the change in shape of Pb inclusions in A1 as a function of annealing time at 620°C. (From MCLEAN[1973].)
or at which a rod terminates. The termination of a rod is expected to shrink backward for the same reasons as apply to the shrinkage of a terminating lamella (cf. fig. 16). Because of the negative radius of curvature at the rod-matrix interface at a branching point, the branches are expected to fill in, i.e., to migrate in the growth direction, as was observed. Theories of the kinetics of fault migration have been presented by CLINE[1971] and by WEATHERLY and NAKAGAWA [19711. Instabilities were reported at the periphery of spherical cavities growing under stress along interfaces. If the stress was sufficiently large (WINGROVEand TAPLIN[ 1969]), finger-shaped instabilities were seen to develop. The critical parameter for the development of these instabilities instead of spherical growth seems to be the ratio of the diffusion coefficients at the cavity surface and a boundary supplying the vacancies (BEEF& [1978]). Instabilities cannot develop if this ratio is large (typically > 100). For materials with slower surface diffusion, cavities above a critical size become unstable. The critical size depends on the applied stress and the cavity spacing. Coarsening by Brownian motion The spontaneous random motion of gas-filled cavities, leading to a coarsening process by cavity coalescence when two cavities meet, has been deduced both by direct observation, and indirectly. The first observations were apparently made on UO, plates irradiated with neutrons to produce fission fraqments (krypton, xenon) which precipitated in the form of gas-filled cavities (GULDEN[1969]). These cavities were seen in the 4.2.4.
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Microstructure
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electron microscope to show Brownian motion, the rate of which was controlled by volume diffusion in the host crystal for cavities above 3.7 nm diameter. Similar observations were made for helium-filled cavities in gold and copper, krypton in platinum and xenon in aluminum (cf. GEGUZINand KRIVOGLAZ[1973]) and helium bubbles in vanadium (TYLERand GOODHEW [1980]). 4.2.5. Microstructuralchangesstimulatedby interfacial energy in the presence of
external potential fields The presence of a field of varying potential (e.g., due to a stress- or a temperature gradient, or due to electric or gravitational fields) modifies the driving forces for diffusion and, thus, may result in microstructural changes. 4.2.5.1. Temperature gradients. The theory of diffusional migration was first developed for volume diffusion in a temperature gradient. In the subsequent decade this work was extended by several authors to surface-diffusioncontrolled processes and other fields, such as electric, magnetic, stress or gravitational field gradients. For a review of this development, we refer to the book by GEGUZIN and KRIVOGLAZ[1973]. The dominant physical reason for an inclusion to migrate in a temperature gradient is the temperature-dependenceof the solubility. For example, let us consider a liquid inclusion in a solid. We assume that at the “front” side of the inclusion (where the temperature is highest), the liquid in contact with the solid has a lower solute content than at the (colder) ‘‘rear” surface. The different solute content results in a concentration gradient and, hence, in a flux of solute atoms from the rear to the “front” surface, which causes the “front‘ surface to melt or dissolve and the “rear” surface to freeze or migrate by crystal growth from solution. In addition to the atom flux resulting from this process, causing the inclusion to migrate up the temperature gradient, there is also a drag exerted on the atoms by a directional flux of phonons (phonon wind) which results from the temperature gradient. Similarly, in metals with aspherical Fermi surfaces, the diffusing atoms are dragged by an electron wind which appears under these circumstances in a temperature gradient. In the simplest case of a spherical inclusion in a temperature gradient, the velocity (v) of the inclusion of size R is found to depend linearly on the temperature gradient (grad 7). The velocity v is proportional to (grad T).R“, with n=O, 1, -1 if the rate-controlling process is diffusion through the matrix, the particle-matrix boundary or the particle, respectively. As all parts of the inclusion move with the same velocity, there is no shape change during migration. All of these results assume the matrix-particle interface to act as ideal sink and source for vacancies. If this is not so, the velocity is smaller or the inclusion is not mobile at all. The first experimental investigations on the motion of inclusions in a temperature gradient were apparently carried on the motion of aqueous solution droplets in sodium [1952]), although the motion of brine inclusions in a temperature nitrate (LEMMLBIN gradient was already invoked by WH~UAN[1926] to explain the fact that polar ice becomes purer at its cold upper surface. The motion of gaseous and liquid droplets in a solid in a temperature gradient has been studied in a variety of metallic and nonmetallic systems (e.g., He in Cu, Kr in UO,, W in Cu, water in NaNO,, water in KCL, water in NaC1, Li in LiF, NJ3,Cl bubbles in NH,Cl, gas-filled bubbles in KBr, NaCl, KC1, Pb in References: p . 935.
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Al; for a review we refer to the book by GEGUZMand KRIVOGLAZ[1973]). The observed migration velocities as well as the correlation between the velocity and the inclusion size was in all cases well accounted for by the theoretical prediction. Above a certain temperature gradient, the migrating droplets (e.g., voids or gas bubbles in nuclear fuel elements, metal droplets in semiconductors or water droplets in ionic crystals) start to break down by the growth of protrusions from the rear corner, releasing a thin trailing liquid veil. The physical reason for the breakdown is the difference between the thermal gradient at the edges and in the center of the droplets, resulting in different migration rates of the two regions. An additional factor comes in when the inclusion contains two phases, such as liquid and vapor. The additional factor is the change of the interfacial free energy with temperature, and it may cause the inclusion to migrate down the temperature gradient (ANTHONY and C L [1973]). ~ Consider a spherical gas-filled inclusion in potassium chloride subjected to a temperature gradient. The wall of this gas-filled inclusion is assumed to be covered by a liquid film or brine. In addition to the normal diffusive flux from the hot to the cold surface, there is a flow of liquid in the liquid film caused by the fall in the liquid-vapor interfacial energy as the temperature falls. The interfacial-energy-inducedflow in the liquid film is the essential step in the movement of the inclusion, as it carries potassium chloride that will be deposited at the hot side of the inclusion so that the inclusion moves to the cold end of the crystal. An analysis based on this model successfully predicted the observed velocities of two-phase inclusions in potassium chloride. Probably the technologically most important observations are those of thermal migration of nuclear fuels through protective coatings in the temperature gradients associated with nuclear reactors (MCLEAN[1982]). In recent years, thermal migration effects led to some concern over the microstructural stability of high-temperature alloys; in particular, in-situ composite materials, exposed to high-temperature gradients, in turbine blades. Investigations on the thermal stability of eutectic composites mi-, Al-, Pb-base alloys) provide evidence for thermal instabilities under the conditions used in modem aircraft engines (HOUGHTON and JONES [19781). The other problem of considerable practical relevance is the effect of temperature gradients on Ostwald ripening. The available evidence is controversial, indicating - often for the same system - that thermal gradients may increase, not affect or decrease the rate of Ostwald ripening (e.g., DAVISet al. [198Ob], and JONES and MAY[1975]). This controversy may be due to different rate-controlling processes in the various experiments. Ostwald ripening is enhanced if adjacent migrating particles collide and join up (JONES and MAY{1975]) or because the back and front of an inclusion migrate with different velocities owing to the different temperatures at those sites (MCLEAN[19781). However, temperature gradients can also decrease Ostwald ripening owing to the generation of a shape instability, as was discussed previously (veil formation). (See als ‘thermomigration’, ch. 7, Q 6). 4.2.52. Temperaturecycling. This may affect the microstructure of alloys by three effects (MCLEAN[19821): (i) a variation of the solubility of the phases; (ii) a differential thermal expansion leading to local strain gradients; (iii) capillary terms arising from the Gibbs-Thompson effect.
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In most alloys, the first effect seems to dominate and may result in accelerated Ostwald ripening or morphological chanqes, as was observed in composites that were remarkably [1980]). stable under isothermal conditions (COOPERand BILLINGHAM 4.2.5.3. Magnetic fields. The energy of a magnetic phase is altered by the presence of a magnetic field, and hence the microstructure of alloys that are magnetic will be changed by the application of a magnetic field. This subject has been extensively reviewed by CULLITY[1972]. Magnetic fields may affect both the atomic order in stable solid solutions and the precipitation from supersaturated solid solutions. In stable solid solutions, magnetic fields generate directimd order by altering the proportion of like pairs that are aligned in the field direction. Such an alignment causes no change from the random situation in terms of the total fractions of like and unlike nearest-neighbor pairs. During precipitation from solid solutions, magnetic fields favor the formation of those precipitates that are aligned with respect to the external field. The best known example is the preferred formation of magnetic rods aligned parallel to the magnetic field in Alnico alloys. A preferred orientation of precipitates may also be achieved during coarsening in the presence of a magnetic field. For example, if Fe,N precipitates coarsened in a magnetic field, a complete orientation of the disc-shaped particles normal and PrrScH [1971]). Owing to the to the field direction was achieved (NEUHAUSER smaller demagnetizationfactor, the spins of the precipitates parallel to the magnetic field (El) become aligned so that a single-domain structure is formed. This domain structure increases the energy of the Fe,N/Fe interphase boundary. Hence, precipitates oriented normal to H have the lowest free energy and, thus, grow at the expense of the other precipitates. In materials of anisotropic magnetic susceptibility, external magnetic fields induce phase or grain-boundary migration. The first observation on this effect, was reported in the classical paper by MULLINS[1956] on boundary migration in diamagnetic bismuth. Magnetic annealing has recently been applied to ceramic superconductors(DERANGO et aZ. [1991]). The objective was to align the crystallites in polycrystalline YBa,Cu,08. Alignment of the crystallites (by other methods) has been shown previously to increase the maximum critical current density of these materials. In fact, magnetic annealing may prove to be applicable to all paramagnetic, diamagnetic or ferromagnetic materials provided the driving force due to the aligning field is large compared to the thermal fluctuation at the anneaIing temperature. Naturally, the same should apply to electrically polarizable materials (e.g., ferroelectrics) when annealed in an electric field. 4.25.4. Stress field. By analogy with magnetic fields, external stresses can modify the atomic order in stable solid solutions and the precipitate morphology in two-phase alloys. Directional atomic order has been induced in Fe-AI alloys by stress-annealing (BIRKENBEIL and CAHN [1962]). In two-phase materials, external stresses may result in the alignment of precipitates andor in shape changes. Several examples for the first effect have been reported: FesN in FeN, Au in Fe-Mo-Au, 8 and 8’ in Al-Cu, ZrH,,5in Zr-H, Ti-hydride in Ti-H, y’ in et al. [1979]). Apparently, only a few observations of stressNi-Al alloys (MIGAZAKI induced shape changes have been published (MIGAZAKI et al. [1979], TIENand COPLEY [1971]). Owing to the elastic anisotropy of the matrix and the precipitate, the free energy References: p. 935.
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of a precipitate depends on the precipitate orientation and shape. The theoretical and KRIVOGLAZ[19731, SAUTHOFF [19761and WERT treatment of both effects (GEGUZIN [1976]) seem to be consistent with the experimental observations. Similar effects have been observed in alloys undergoing an order-disorder or a martensitic transformation. When a CuAu single crystal is ordered, it becomes subdivided into many domains, the tetragonal (c) axes of which are parallel to any of the three original cubic axes. In the bulk material and without an external stress field, the three possible cdirections are randomly distributed among the domains. The application of a compressive stress during ordering imposes a bias on the distribution of the c-axes such that the cube axis nearest to the compression axis becomes the preferred direction for the c-axes of ordered domains (HIRABAYASHI [1959] and ARUNACHALAMand CAHN[1970]). The microstructure of materials undergoing martensitic transformations (cf. ch. 16) depends on external stress fields. The following two factors contributing to this effect are and WARLIMONT [1975]): (i) The orientation variant on whose most important (DELAEY macroscopic shear system the maximal resolved applied shear stress is acting will grow preferentially, (ii) near certain symmetric orientations, individual plates and selfaccommodating groups will compete. Essentially, the same arguments apply to the microstructure of materials undergoing mechanical twinning. Effects of this type play an important role in shape-memory effects. Gas bubbles situated at interfaces represent a special case of microstructural changes caused by stress fields. Owing to the compressibility of gas bubbles, the binding energy between a bubble and the interface depends on external stresses. Compression stresses lead to a decrease of the binding energy and, hence, may result in microstructural changes resulting from break-away effects of the boundaries from the bubbles (GREENWOOD etaZ. [1975]). 4.2.5.5. Electric fields. In bulk metals, strong electric fields may induce complex interactions between thermomigration and electromigration due to Joule heating (ch. 7). In thin films, efficient heat removal ensures reasonable isothermal conditions. Under these conditions, two effects resulting from the presence of electric fields were observed: et al. (i) enhanced grain-boundary migration (LORMAND et al. [1974] and HAESSNER [19741 and (i) the growth of grain-boundary grooves which can eventually penetrate the film so that nucleation and growth of voids by grain-boundary diffusion occurs (Ho and KIRKWOOD [1974]). The theoretical understanding of the processes involved is still poorly developed. In fact, the driving force exerted by a dc current on grain boundaries in gold was found to be several orders of magnitude larger than the theoretically estimated value (HAESSNER et aZ. [1974]). A similar result was also found from the motion of rod-shaped tungsten inclusions in Cu and from the displacement of deep scratches on the surface of Ag owing to the passage of a current along wire-shaped samples. (See also ‘electromigration, ch. 7, 0 6). 4.3. Deformation
All forms of plastic deformation may result in irnportant changes of the microstructure of materials with respect to the distribution and density of defects as well
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Microstmcture
as with regard to the morphology, volume fraction and sometimes also structure of second phases. They are discussed in chs. 19-26. 4.4. Multiphase microstructures generated by migrating lattice defects 4.4.1. Moving grain boundaries If grain boundaries are forced (e.g., during recrystallization or grain growth) to sweep through a precipitate dispersion or a duplex structure, the following microstructures may result (DOHERTY [1982], HORNBOGENand K ~ S T E R [1982]):
0
0 0 0 0
.
0
.
0
.
0 0 0
.
..
0
0
0
O.
0
bl
cl References: p . 935.
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Fig, 20. Microstructural changes induced by a grain boundary migrating through a two-phase alloy containing dispersed precipitates: (a) transformation of coherent precipitatesinto incoherent ones as the boundary bypasses the precipitates; (b, c) dissolution of the precipitates, resulting a supersaturated solid solution, followed by continuous (b) and discontinuous (c) precipitation;(d) grain boundary passing through the precipitates without affecting the shape and size. The solid (open) circles in figs. 22a and 22d indicate solute (solvent) atoms.
(i) The boundary bypasses the precipitates which, therefore, retain their initial orientation and become incoherent (fig. 20a). Owing to the Gibbs-Thompson effect, the solubility of the precipitates rises so tfiat the smaller precipitates may dissolve, as was observed, for example, in the case of NbC in y-Fe. (ii) The coherent precipitates or the components of a duplex structure dissolve after contact with the moving grain boundary, so that a supersaturated solid solution is obtained. From this supersaturated solid solution, the solvent may reprecipitate either continuously (fig. 20b) or discontinuously (fig. 20c). Both cases have been observed experimentally. The dissolution as well as the precipitation processes seem to occur far more rapidly than anticipated, suggesting strongly enhanced diffusion in the migrating interfaces an in the lattice behind due to vacancy supersaturation (SMIDODA et al. 119781, GOTTSCHALK et al. [19801). (iii) The grain boundaries can pass through the coherent precipitates and, thereby, preserve the preexisting microstructure (fig. 20d). This precess seems rare. as it requires the formation of new precipitates to match the rate of boundary migration. (iv) The grain boundary is held by the coherent precipitates which then coarsen. This process occurs if the driving force for boundary migration is not sufficient to initiate one of the above processes. (v) The moving grain boundary drags the precipitates (SMITH[1948]), as was reported for a variety of alloy systems containing gaseous particles as well as solid inclusions, for
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example, for He in Cu, He in U, air in camphor, carbides in various metal transition alloys, GeO, in Cu, B,O, in Cu, SiO, in Cu, Ag in W, Ag in Sn, Al,O, in Ni, A1203in Ag. The experimental results have been reviewed by GEGUZINand KRWOGLAZ[1973] and GLEITERand CHALMERS [1972]. Partick drag results from the directional movement of atoms from one (%on”’) side of the inclusion to the other (“rear”) side. Hence, diffusional migration may occur by diffusion of atoms around or through the inclusion and diffusion of atoms along the particle-matrix interface. The kinetics of the particle drag may be controlled by the rate of one of these diffusion processes or by interfacial reaction at the boundary between the inclusion and the matrix. Examples of all cases mentioned have been reported and may be found in one of the reviews mentioned. Once particle drag starts, the boundary collects practicaIly all particles in the volume which is swept. Particles collected in the boundary usually coarsen rapidly. Therefore, particle drag may result in the following changes of the properties of the boundary: (i) boundary brittleness and/or corrosivity due to a high density of undeformable particles, the electrochemical properties of which differ from the surrounding matrix; (ii) different mechanical and chemical properties in the particle-free zone and in the rest of the material. At high driving forces (e.g., during recrystallization) particle drag seems negligible as it is possible for the mobile boundary to migrate past the inclusions. (See also ch. 28, 0 3.8). Moving dislocations The formation of colonies of precipitates in the vicinity of dislocations has been observed in a number of alloy systems, e.g., in iron, nickel, copper, aluminum and semiconductor materials. In the initial model of this process (SXLCOCK and TUNSTALL119641) and in the subsequent modification by NES [1974], a dislocation was proposed to climb so that precipitates can nucleate repeatedly in the moving stress field of the dislocation. During climb the dislocation emits vacancies. The incorporation of these emitted vacancies in the lattice of the precipitates is believed to reduce the precipitate-matrix mismatch energy if the precipitating phase has a larger atomic volume than the surrounding matrix. In the opposite case, a vacancy flux from the precipitate to the dislocation was also invoked (GUYOTand WINTENSERGER [19741). More recently, the experimentally observed coupling between dislocation climb and precipitate formation was re-interpreted by two other models. Dislocations were proposed to climb owing to the annihilation of quenched-in vacancies, while the precipitates form simultaneously by heterogeneous nucleation in the stress field of the moving dislocation. A study by transmission electron microscopy (WIRTHand GLEITER[1981a,b]) led to the conclusion that coherency strain relaxation by incorporation of vacancies in the precipitates may be only one of the processes involved. In fact, colony formation was found to occur (fig. 214 by the climb of a prismatic dislocation loop which emits vacancies and, thus, generates a region of enhanced diffusivity. The excess solute atoms of this region of enhanced diffusivity migrate rapidly to the climbing dislocation and precipitate in the form of a chain of small particles (fig. 21b). Owing to their large surface-to-volumeratio, these fine particles rapidly coarsen by Ostwald ripening to form rows (colonies) of coarse precipitates behind the climbing dislocation loop. The spacing of the rows of coarse precipitates is 4.4.2.
References: p . 935.
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controlled by the balance between the interfacial energy and the diffusion path. Under steadystate conditions, such systems are known to approach a constant precipitate spacing. Hence, if the dislocation loop expands during growth, a constant precipitate spacing can only be maintained by increasing the number of precipitates as the loop radius increases. This condition results in a spoke-like precipitate arrangement, as was observed (fig. 21).
4.5. Periodic microstructures in open, dissipative systems (“self-organization”) Dissipative processes in open systems are frequently associated with pattern formation (NICOLISand PRIGOGINE[1977], HAKEN[1978], MARTINand KUBIN [1988]). The following microstructures seem to be the result of pattern formation in dissipative systems (“self-organization”):
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Microstructure
Fig. 21. (a) Bright-field and dark-field electron micrographs of colonies of silver precipitates in a Cu-5 wt% Ag alloy. The colonies formed behind the dislocation loops surrounding them. The loops expands during colony growth. Two types of precipitates may be noticed. A chain of small precipitates along the dislocation loop, and large precipitates arranged radially in a spoke-like fashion. @) Schematic model for colony formation by a twostep process involving the nucleation of a chain of small precipitates along the climbing dislocation followed [1981a,bj.) by coarsening into large precipitates with a spacing S. (After WIRTHand GLEITER
precipitate lattices; void lattices; dislocation-loop lattices; dislocation lattices*; point-defect lattices, flux-line and magnetic-bubble lattices; long-period antiphase boundary structures; domain boundaries in ferromagnetic and ferroelectric materials**.
* For sub-boundaries see 0 2.2.5, for periodic structures during fatigue see ch. 27.
** The arrangement of domain boundaries in ferromagnetic materials is discussed in ch. 29.
References: p . 935.
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4.5.1. Periodic structures due to long-range interaction forces 4.5.1.1. Precipitate lattices. Several studies by means of X-ray diffraction and electron microscopy have revealed the existence of periodically arranged precipitates (fig. 22a) in alloys of Cu-Ni-Co, Cu-Ni-Fe, nickel-base alloys containing y’ (Ni,X) precipitates, Cu-Ti, Au-Pt, Au-Ni, Co-Fe, Co-Nb,Co-Ti, Al-Zn, Fe-Bi, FeBe, as well as in Alnico-Ticonel alloys. The models first proposed (ARDELL et al. [1966] and KHACHATURYAN[19691) to account for the formation of periodic precipitate arrays were based on (long-range) elastic interaction forces between the precipitates, due to coherency strain between the precipitates and the surrounding matrix. Both the precipitates and the matrix were assumed to be elastically isotropic. JOHNSON and LEE[19791 refined these approaches by including the strain fields induced by neighboring inclusions and by considering second-order terms. Elastically strained particles of arbitrary shape but identical moduli in an anisotropic medium were also shown (KHACHATURYAN[1969], JOHNSONand LEE[1979] and MORIet al. [1978]) to form periodic arrays. The arrays correspond to one of the 14 Bravais lattices. / In the particular case of spherical inclusions in a cubic datrix with a negative anisotropy parameter, a simple cubic lattice of precipitates was found to form the minimum-energy arrangement which is stable with respect to externally induced fluctuations. Precipitates positioned along directions of the matrix turned out to exhibit attractive interaction forces with a maximum value at 2-3 precipitate radii. This result may provide an explanation for the frequently observed alignment of precipitates along directions. For and alignments, the interaction forces depend on the anisotropy factor of the matrix. alignment in Mo is found to result in attractive forces, whereas the precipitates in the same arrangement in Cu and Ni repel. By applying these results to inclusions associated with a dipole-type strain field embedded in an iron lattice, a precipitate lattice with bcc structure was found to be stable. This arrangement corresponds approximately to the arrangement of N atoms in Fe,,N,. So far, the discussion of precipitate lattices has been limited to systems in which the precipitation process occurs by nucleation and growth. In systems decomposing by a spinodal process, periodic arrangements of precipitates result from the time-dependent growth of concentration fluctuations. The processes involved and the factors governing the periodicity are discussed in ch. 15. 4.5.1.2. Void lattices. EVANS’[1971] report on the creation of a stable bcc array (superlattice) of voids in irradiated Mo stimulated numerous studies on void lattices in other systems. Void lattices have been seen, for example, in Ni, Al, stainless steel, Mg, Mo, Mo-Ti,Nb, V, W and NbO, BaF,, SrF,, CaF, (fig. 22b). Two kinds of models have been advanced to explain the formation of void lattices. The observed symmetry of the void lattices has originally initiated an interpretation in terms of equilibrium thermodynamics assuming elastic interaction forces between the voids (STONEHAM[1971], TEWARY [1973], WILLIS[1975]). As these theories do not include the radiation damage explicitly, they cannot explain the observed influence of temperature and damage type. Moreover, this approach would not predict void lattice formation in isotropic crystals such as tungsten in contrast to the experimental observation. The second type of theoretical approach to explain void lattices, dislocation pattern-
n
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subsequent years, periodic arrays of dislocation loops have been detected in Al, Ni, U, BeO, Ti,Mg and Zr,In the case of Ni and Al, ordered clusters of loops were observed. Stereomicroscopy revealed that the clusters consisted of about six loops forming a fcc and FARRELL 119741). As the lattice regions between the clusters superlattice (STIEGLER were found to be elastically strained, elastic interaction forces were proposed to be the dominant factor for the formation of the loop lattice. Recent theoretical approaches to explain the formation of loop lattices are based on self organization principles. In fact, the dynamic models discussed in 8 4.5.1.2 seem to apply to dislocation loop and void lattices as well. 4.5.1.4. Point-defect lattices. By analogy with the formation of void lattices, point defects may be expected to form ordered arrangements.This idea is indeed confirmed by recent studies on vacancy and interstitial lattices in certain alloys (cf. HIRAGA[1973] and JTJNG and TRENZINGER[19741). In alloy forming vacancy lattices, e.g., vanadium carbides of the V,C, type, a variation in the alloy concentration between V,C, and V,C, did not result in an expansion (or contraction) of the spacing of the carbon vacancies, but rather caused the formation of a one-dimensional long-period superlattice structure consisting of enantiomorphic domains of the superstructure V,C,. For obvious reasons, the linear dimension of the enantiomorphic domains depends on the vacancy (carbon) content and increases with the increase in carbon vacancies. In vacancy lattices elastic interaction forces are believed to be the most important parameter. In pure metals, vacancy or interstitial lattices have not yet been revealed experimentally. However, calculations based on the vacancy-vacancy interaction potential showed that vacancy lattices may exist with a simple cubic structure aligned parallel to the identical axis of the host lattice (CHANG[1976]). The vacancy lattice constant was found to be about three times the lattice constant of the atomic lattice in the case of K and Na. 4.5.1.5. Long-period antiphaseboundary structures. Ordered alloys, mostly of fcc structure in the disordered state, exhibit in the ordered state a regular three-dimensional array of antiphase boundary (APB) structures. Ordered structures of this type, which are called “long-period antiphase boundary structures”, are treated in ch. 3, 0 11.2. An excellent review dealing with various aspects of orderldisorder phenomena in materials has been published recently (CAHN[1994]). 4.6. Microstructure in the vicinity of point defect sources and/or sinks 4.6.1. Enhanced precipitationand precipitate-freezones The significance of point-defect sourceslsinks for the precipitation of solute atoms from supersaturated solid solutions was first demonstrated by BARNES et al. [1958] for the enhanced precipitation of helium atoms in the vicinity of point-defect sources. Helium atoms were injected into metals (spectroscopicallypure Cu and Be) by bombardment with alpha particles. On subsequent heating, the He atoms have a tendency to precipitate within the metal in the form of gas bubbles and, to acquire the extra space necessary for this, they capture vacancies. Thus a blanket of bubbles forms in the vicinity of a vacancy source. For low He contents and large grain sizes, grain boundaries as well as dislocations are the most important vacancy sources. For small grains, grain bound-
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aries are the dominant suppliers. These results were confirmed later for a variety of other metals. Precipitute-free zones denuded of second-phaseparticles adjacent to grain boundaries in age-hardened alloys were originally attributed to the localized depletion of solute arising from preferential precipitation at the grain boundaries. However, it was soon recognized that local depletion of vacancies might be the more important factor, as a critical concentration of vacancies may be required for precipitate nucleation. In fact, this idea was discussed in terms of the thermodynamics of solute clustering (LORIMER and NICHOLSON [1969]) and in terms of the precipitation kinetics (PASHLEY et ul. [1967]). Evidence for the local depletion of vacancies by annihilation at the boundaries was obtained from electron microprobe measurements and energy-analyzing electron microscopy for Al-Ag and Al-Zn-Mg alloys. The results of 60th investigations showed no solute depletion in the vicinity of the boundaries. In this simplified picture no coupling between vacancy flow and solid distribution is assumed. However, if a binding energy exists between solute atoms and vacancies, the vacancy flow from or to vacancy sources/sinks is necessarily coupled with a solute flow and thus produces a solute gradient in the vicinity of vacancy sources/sinks, such as grain boundaries, dislocations, [1970]). pores, or free surfaces (JOHNSON and LAM[1976] and ANTHONY The solute segregation generated by vacancy flow involves two processes: the dragging of solute atoms by the moving vacancies and the reverse atom flow which is a consequence of vacancy flow. The first process dominates if the binding energy (E) between a vacancy and a solute atom is much greater than the thermal energy (kr). Under these circumstances, a solute atom is dragged to (from) the vacancy sink (source) so that solute enrichment (depletion) of the sink (source) regions will result. As a consequence, an enhanced density of precipitates forms in the vicinity of the sink. The opposite type of solute distribution may result in the second case (Eckr). For E c kT, solute atoms may be pumped in or out of the sink region depending on the relative diffusivity of solute and solvent atoms. When a vacancy flows into an enclosed sink region, an atom as a consequence must simultaneously flow out of this region. If the solute and solvent atoms in this region have identical mobilities, the ratio of solute to solvent atoms will remain the same as the original ratio. However, if the mobility of the solute atoms is greater than of the solvent atoms, proportionately more solute than solvent atoms will be moved out of the enclosed reqion by intruding vacancies, producing a solute-depleted sink zone. The solute depletion of the sink region will not continue indefinitely but will stop when the solute flow generated by the vacancy flux is balanced by the opposing solute flow produced by the solute gradient. "bo special solute pumping processes were proposed for hydrogen in stress gradients. The first mechanism (the Gorsky effect, for a review see VOLKL[1972]) arises because hydrogen dissolved in a metal expands the crystal lattice for the host material. Hence, if a crystal contains a gradient in dilatation, the hydrogen concentration is enhanced in the and dilatated region, e.g., in the vicinity of crack tips. The second effect (TILLER. SCH[RIEPFER [1974]) is due to the redistribution of the free electrons in strain fields. Owing to this redistribution, dilatational centers (into which electrons flow) become cathodic. Hence, H+ ions will migrate into the cathodic (dilatational) regions. Estimates for conditions typical for crack tips led to H+enhancements of up to ten orders of magnitude. References:p . 935.
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4.6.2. Irradiation-induced precipitation In irradiated materials, a high supersaturation of vacancies and/or interstitials may be present. It follows from the previous section that the condensation of these point defects at suitable sinks (e.g., grain boundaries) may induce solute segregation in the vicinity of the sink. If this segregation is sufficiently strong, a local transgression of a phase boundary and, hence, irradiation-inducedprecipitation processes may be obtained, as has been reported for many alloy systems. For comprehensive reviews on this subject, we and BENNET[19771and POIRIERand Dmow refer to conference proceedings (BLEIBERG [1979]), chs. 7 and 18. 4.63. Point-defect condensation The significance of point-defect sourceshks for the development of microstructures resulting from point-defect condensation was discovered by etch-pit studies. Etch pits were observed to form on electropolished surfaces of A1 crystals during cooling from elevated temperatures. The formation of the pits was attributed to the condensation of vacancies at the surface. In polycrystalline specimens, pits were not observed in the vicinity of high-angle grain boundaries, suggesting that the vacancies in the pit-free region had been drained by the boundaries. In regions far away from the free surface, supersaturatedpoint defects may condense in the form of dislocation loops, stacking-fault tetrahedra and/or voids which may be observed by transmission electron microscopy. The condensation process leads to a non-uniform microstructure in polycrystalline specimens in the sense that denuded zones exist near grain boundaries. As the condensation occurs by a nucleation and growth process, a certain supersaturation of point defects is required. Hence, the observation of denuded zones suggests a lower point-defect supersaturation in the vicinity of grain boundaries than in the perfect lattice, owing to the annihilation of point defects at the boundaries. With the exception of coherent twins and small-angle boundaries, the results suggest that high-angle grain boundaries are ideal vacancy sinks so that the width of the denuded zones is diffusioncontrolled(for a review see GLEITER[1981a]). 4.7. Microstructure due to lattice defects formed by migrating grain boundaries In recent years, it has become apparent that the microstructure of crystals growing by solid-state processes depends on the mode of crystal growth. The defect structures resulting from solid-state phase transformations and solid-liquid (glass) transformations, are discussed in chs. 15-19. In the present section, attention will be focussed on the microstructures developed due to the generation of vacancies, dislocations and twins by migrating grain boundaries. For a recent review of this field, we refer to the article by GLEITER [1981a]. Creation of vacancies. The creation of vacancies by migrating boundaries has been studied by means of the diffusion coefficient, the density, the electric resistivity and the morphology of the precipitates formed in the crystal region behind migrating boundaries (GORLIK etal. 119721 and GOTTSCHALK et al. [1980]). The observations reported suggest that behind a migrating grain boundary a high supersaturation of vacancies may exist. The high vacancy supersaturation observed was explained in terms of “growth accidents”
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occurring during grain-boundarymigration (GLEITER [1979]). A growth accident involves a jump of an atom of the growing crystal into the migrating boundary so that a vacant site is left behind in the lattice of the growing crystal. The excess vacancies retained in the lattice alter the properties of this crystal and exert a drag force on the migrating [1979]) which may dominate impurity drag under boundary (“vacancy drag”, GLEITER certain conditions (ESTRINand LOcm [1982]) (cf. also ch. 28, 5 3.4.1). Creation of dislocations. Indirect observations of dislocations created by migrating boundaries come from recrystallized materials. However, because of the high dislocation density ahead of the recrystallization front, the interpretation of these results is not unambiguous. A distinction between dislocations generated by the migrating interface and dislocations due to other processes is possible if the boundary migrates into a dislocationfree crystal or a crystal with low dislocation density. Studies of this type have been et al. [1980]). The results obtained support carried out in Cu, InP, InAs and Si (GLEITER the idea of dislocation generation by migrating boundaries. The generation process may be envisaged by growth accidents as well as the stress-induced dislocation emission (GASTALDIand JOURDAN [1979], GLEITER et al. [1980]). Creation of two-dimensional lattice defects. The most prominent lattice defects generated during boundary migration (e.g., during grain growth) are coherent twin bounduries (cf. ch. 28,s 4.2). In order to explain the formation of twin boundaries during boundary migration, several models have been proposed. According to the dissociation models twins are formed by dissociating a grain boundary (A) into a twin boundary (T) and a new grain boundary (B). The stimulation model proposes that a twin boundary is created if a growing recrystallized grain meets a dislocation-bearingfragment which lies accurately in a twinned orientation to it. Since the fragment has discharged its dislocations, it is now stress-free and able to grow at the expense of the surrounding deformed matrix. The coalescence model proposes twin boundaries to be formed if the orientation relationship between the impinging grains corresponds exactly to a twin orientation. The growth-accident hypothesis of twin-boundary formations follows the concept that twins are formed and terminated by errors of the stacking of the planes which happen in a random way. Studies by optical microscopy, thenno-ionic and photoemission microscopy, transmission electron microscopy, X-ray topography, grain-boundary migration experiments in bicrystals and polycrystals have been reported (for a review see GLEITER [1981a]). The results of these studies are inconsistent with the dissociation, the stimulation and the coalescence hypotheses. The observations so far available seem consistent only with the growth-accident hypothesis. In fact, in situ observations of twin formation in A1 by X-ray topography agree with the evolution, shape and growth and JOURDAN [19791). direction predicted by the growth-accident theory (GASTALDI 4.8. Microstructure of glasses
Historically, the observation and interpretationof microstructureshave only dealt with crystalline materials. Indeed, until the late 1950s noncrystalline materials, e.g., oxide glasses, were regarded as free of all microstructure. This viewpoint was fostered in part by earlier triumphs of the random network theory of glass structure (ZACHARIASEN References: p . 935.
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[1932], WARREN [1937]) and was strengthened by the implicit glass engineering goal of this era: produce a homogeneous, single-phases product through fusion. As will be discussed below, this concept of a microstructure-free vitreous state has been seriously questioned in the last decade (ROY[1972]). This challenge has progressed sufficiently to occasion a rethinking of .nearly all aspects of the fabrication and characterization of noncrystalline materials. To a large extent this more modern concept of glass materials evolved from the discovery of the ability to induce, by appropriate thermal treatment, a clearly discernible and well-controlled microstructure in glasses. This microstructure is now recognized to be either crystalline or noncrystalline. Moreover, the occurrence of the noncrystalline variety is so ubiquitous that it has been suggested that its occurrence may be an intrinsic characteristic of all glass-forming melts (ROY[1972]).
4.8.1. Microstructure of amorphouslyphase-separatedglasses There are two ways of producing amorphously phase-separated glasses. First, one may simply prepare a glass-forming melt, quench it to room temperature and assess the extent to which the process has taken place. A second and more controllable method involves the isothermal annealing of glass above its transformation temperature (ch. 7, Q 9.1). One thus extends the amount of time available for the process to proceed. By intermittent examination of samples for telltale opalescence at various times and temperatures, it is possible to construct a timetemperature-transformation diagram. The microstructure developed in this process depends on the curvature of the freeenergy vs. concentration curve. The loci of equilibrium compositions trace out the immiscibility gap. The loci of the inflection points delineate the region of spinodal decomposition. Between the equilibrium compositions and the inflection points, single, phase glasses transform into two-phase glasses by nucleation and growth. In this case, the microstructure is characterized by spherical glassy regions dispersed through a continuous glassy matrix of different chemical composition. If the decomposition is spinodal a microstructure characterized by a high interconnectivity of both glassy phases and irregularly shaped diffuse boundaries results. Glass-glass phase transformations are, however, not limited to immiscibility effects. For example, it is known that the viscosity of liquid sulfur changes by a factor of two thousand when heated over the narrow temperature range 158-166OC (BACONand FANELLI [1943]). The low-temperature low-viscosity melt is thought to be constructed of 8-member rings, whereas the high-temperature high-viscosity melt is regarded as made up of long sulfur chains. Both liquids are in thermodynamic equilibrium (POWELLand EYRING[1943]). Evidence for a second-order phase transformation between the two liquid fields has been presented, leading to a microstructure of finely dispersed clusters with different molecular structure. In addition to these considerations there is evidence for clustering effects in silicate glasses. Clusters have been defined as cooperative compositional fluctuations surrounded [1965]). These clusters are not necessarby melt of less organized structure (UBBELOHDE ily equivalent to the crystalline nuclei. It has been suggested that such clusters could be “frozen in” during quenching of a glass-forming melt (MAURER119561).
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4.8.2. Microstructureof partially crystalked glasses Figure 23 summarizes the major reaction paths for crystallization of glasses and the resulting microstructures. Path (1) represents the direct transformation of a pure singlephase glass into a more stable crystalline phase. This transformation involves both the creation of stable nuclei and the subsequent growth of the crystalline phase. In contrast to path (l), the development of an intermediate amorphous phase as in path (2,3) and also for path (2,4,5) is well documented for a number of glassforming systems. A good representative of the former is the crystallization of Al,O,-SiO, glasses (MACDOWELL and BEALL[1969]). The latter path has been observed in classical glass ceramic compositions, e.g., Li,O-SiO,-Al,O,-TiO, (DOHERTY et al. [19671). On cooling, glasses in this system show amorphous phase separation on a scale of about 5 nm. On reheating, this microstructure promotes formation of a nucleant phase, Al,Ti,O,, which in turn crystallizes a major crystalline phase of this system, p-eucryptite. Path (6,5) represents the case where small amounts of metals such as Ag, Au, Pt, Cu, Rh, Pd, etc. are incorporated into a glassy matrix and by suitable control of initial concentration, melting conditions, thermal history, and in some cases (Au, Ag) exposure to actinic radiation, nanometer-sized particles of these metals are precipitated in the glass. Since the process involves both a reduction to a metallic state and diffusiPon of the reduced species to form a particle, their mean size may vary over a wide range depending on the interplay of the above factors. MAURER[1959] has shown that the minimum size of a gold particle capable of catalyzing lithium metasilicate is about 8 nm (10,000 atoms). This is in contrast to the smallest, stable gold particle which may contain only three or four atoms. Thus, the need for microstructure control to achieve catalyzed crystallization is apparent. It has been recognized for some time that the presence of these metal particles induces a characteristic color in the bulk glass. Thus, gold and copper particles give rise to a magneta color, silver to a yellow cast, and platinum to a somewhat dull grey appearance. Similarly, selenium gives a characteristic pink color in soda-lime silica glasses (average particle size 5-20 nm). Since this pink color is complementary to the bluish green arising from ionic iron in these same glasses, this element is used extensively in the glass container industry as a decolorizer. It is here that the control of microstructure in glass is crucial because the precise shade of pink required to achieve decolorization depends on composition, furnace atmosphere, and thermal history (PAUL[1975]). The need to understand the origin and significance of microstructure in glass to render a product commercially acceptable is evident. Still another group of technologically important glasses which are partially crystalline is the photochromicglasses (ARANJOand STOOKEY [19671). Their photochromicbehavior arises from minute silver halide crystals which may be regarded as being suspended in an inert glass. Satisfactory photochromic properties are obtained when the average particles/cm3.This corresponds particle size is about 5-10 nm at a concentration of to an average particle separation distance of about 60 nm. This separation distance is crucial since the inert glass host prevents diffusion of the halogens freed by absorption of light by the halide crystal. Thus, recombination with free silver within the halide crystal is enhanced. It is this alternate decomposition and recombination that gives rise to the variable optical transmission of these glasses. References: p. 935.
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2
AMORPHOUS PHASE SEPARATION
4
1
7 CRYSTALLIZATION of MAJOR PHASE
5
COLLOIDAL CRYSTALS of "NUCLEANT PHASE
Fig. 23. Various paths associated with potentia1 CrystaIIiion of glass forming melts. (FromSTEWART[1972].)
5. Nanostructured materials 5.1. Materials with reduced dimensionality, reduced dimensions and/or high densities of defect cores The perfect single crystal of infinite size has been the model system of solid state physics for several decades. Remarkable progress was achieved in the physical understanding of solids by means of this idealized approach. However, about 40 years ago, scientists started to realize that the disorder present in most real materials cannot be treated as a weak perturbation of the corresponding ideally ordered crystals. In fact, a variety of new physical effects (e.g., new types of phase transitions, quantum size effects, new transport phenomena, etc.) were discovered which existed only in imperfectly ordered solids. In fact, if the characteristic dimensions (e.g., the diameters of small spheres or the thickness of a thin film) of the crystalline regions of'a polycrystal approach certain characteristiclengths such as an electron wavelength, a mean free path, a coherency length, a screening length, a correlation length, etc. one obtains materials the properties of which are controlled by their reduced dimensionality or their reduced dimensions and/or their high density of defect cores (e.g., grain boundary cores). As a kind of introduction, let us consider two specific examples. The first example is shown in fig. 24 in the form of a compositional superlattice characterized by a length scale, d, which characterizes the thickness of the layers. If d is equal or less than the mean free path of the conduction electrons, the electronic band structure and, hence the electric properties of this materials differ significantly from the ones of a superlattice with the same chemical composition but with a modulation length, d, that is much larger than the electronic mean free path (cf. section 5.2). The second example to be considered here is a crystalline solid with a high density of grain boundary (Le., defect) cores (fig. 36, below). About 50% of the atoms are located in the cores of these grain boundaries. In
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electron subbands
GaAs
a
c
hole subbands
AIxGal,As
I\ a
??
F
c
a, .-I P
.-Iz
8 -
0
m P n 0
P 0
0
a , 0
S al m .e
m .-c
([I
) .
g% ED
m
0. 0
spatial coordinate
-
c
0 .c 0 3 -0
c u
:: 9
m
9
0
electron energy
-
Fig. 24. Schematic illustrationof the layer sequence (left side) and of real-space energy-band profile (right side) of a GaAs/A&Ga,, as superlattice. Both, the GaAs and the AlxGal, as layers are assumed to be equally thick. E8is the energy of the band gap edge.
grain-boundary cores, the atomic structure (e.g., the density and the nearest neighbor coordination) differs from that of a perfect crystal with the same chemical composition. Hence the atomic structure and the properties of a material with a structure as shown in fig. 36 may deviate (in some cases by many orders of magnitude) from those of the corresponding single crystal (cf. sections 5.4, 5.5 and 5.6). Materials exhibiting reduced dimensionalities, reduced dimensions and/or high densities of defect cores are termed nanostructured materials, because the typical size of the crystalline regions or of another characteristic length scale in such materials are in the order of a few nanometers. In recent years, the following five types of nanostructured materials have attracted widespread scientific attention: 1) Thin metallic, semiconducting or polymeric films with clean or coated surfaces. 2) Man-made superlattices and quantum well structures. 3) Semicrystalline polymers and polymer blends. 4) Nanocrystalline and nanoglassy materials. 5) Nanocomposites made up of metallic, covalent, ionic and/or molecular components.
This paragraph will be limited to the materials mentioned in 2, 3,4 and 5. Concerning thin films and free surfaces we refer to chapters 28 and the numerous reviews in the literature and textbooks on surface science as well as thin solid films. References: p. 935.
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5.2. Man-made superlatticesand quantum-well structures In 1969 research on man-made superlattices was initiated by the proposal of Esaki and Tsu to engineer the electronic structure and properties of semiconductors by generating superlattices with a periodicity shorter than the electronic mean free path either by alternating the chemical composition (compositional superlattice) or the doping (doping superlattice) of consecutive layers in a multilayer structure. The first compositional superlattice was grown from the material system GaAs/A&Ga,,As. The layer sequence (chemical composition) and the real-space energy band structure of the electrons in such a superlattice are illustrated in fig. 24. The different energy levels of the bands of the two components at the heterointerfaces determine the potential barriers for the electrons and holes in the vertical direction (fig. 24), and thus define the periodic superlattice potential in the conduction and in the valence bands. The characteristic feature of this superlattice is that layers of a narrowgap semiconductor are sandwiched between layers of a wide-gap semiconductor. This structureresults in two square quantum wells: one for the electrons and one for the holes. Figure 25 displays the corresponding density of states in such a two-dimensional system in comparison to the parabolic curve of the classical (three-dimensional) free electron gas. If the electron mean free path exceeds the superlattice periodicity, resonant tunneling between adjacent subbands occurs. Technological applications of such superlattice structures lie primarily in the optoelectronic regime, comprising injection lasers, light emitting diodes, avalanche photo diodes and photoconducting detectors.
ENERGY ( eV ) Fig. 25. Comparison of density of states in the three-dimensional (3D) electron system with those of a superlattice (SL) assuming a quantum well and barrier width of 10 nm, and an effective electron mass 0.067 m, in the SL. The fmt three subbands are indicated (Mmmzand VON KLITZING[1989]. m, is the mass of the free electron).
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The term doping superZum'ces refers to periodic arrays consisting of layers of the same semiconductor doped in two different ways e.g., n- and p-doped layers. The unusual electronic properties of doping superlattices derive from the specific nature of the superlattice potential which, in this case, is the space charge potential of ionized impurities in the doping layers. This is in contrast to the compositional superlattices (fig. 24), in which the superlattice potential originates from the different band gaps of the constituents. The space charge potential in the doping superlattices modulates the band edges of the host material in such a way that electrons and holes become spatially separated (fig. 26). This separation can be made nearly perfect by the appropriate choice of the doping concentrations and the layer thicknesses. One of the attractive features of doping superlattices is that any semiconductor that can be doped in both n- and p-type in well controlled ways can be used as the host material. Another advantage of doping superlattices originates from their structural perfection. The relatively small concentrations of impurities used in doping superlattices (typically 10'7-10'9/cm~3),induce only minor distortions of the lattice of the host material. Thus, doping superlattices do not contain interphase boundaries as does compositional superlattice. The absence of any significant disorder or misfit strains leads to unprecedented electron and hole mobilities. Doping supperlattices have had profound impact on not only the progress made in recent years in the physics of two-dimensional electronic systems (quantum Hall effect, Shubnikov-de Haas oscillations) but also on device applications such as high-speed MODFETS.For further details we refer to some of the excellent reviews in this rapidly growing area of research (MENDEZand VON KLITZING [1989], FERRYet al. [l990], KIRKand REED [1992]). 5.3. Semicrystalline polymers
Semicrystalline polymers constitute a separate class of nanostructured materials. The remarkable feature of this class of polymers is that the nanostructured morphology is always formed if the polymers are crystallized from the melt or from solution unless crystallization occurs at high pressure or if high pressure annealing is applied subsequent to crystallization. If a polymer is crystallized from a dilute solution, isolated single polymer crystals or multilayer structures consisting of stacks of polymer single crystals result (fig. 27). Inside the crystals, the atoms forming the polymer chains arrange in a periodic three-dimensional fashion. The interfaces between neighboring crystals consist of both macromolecules folding back into the same crystal and tie molecules that meander between neighboring crystals. The typical thicknesses of the crystal lamellae are in the order of 10 to 20 nm. These relatively small crystals thicknesses have been interpreted in terms of one of the folIowing models. The first model hypothesizes the formation of the thin crystallites to result from nucleation kinetics. If the height of the energy barrier for the formation of a critical nucleus of a chain-folded polymer crystal formed in a supersaturated solution is computed by means of homogeneous nucleation theory, it is found that the energy barrier of a critical nucleus consisting of extended chain molecules is larger than the bamer height for a nucleus of folded chains. The physical reason for this energy difference is as follows. Extended chain crystallization References: p . 935.
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IONIZED DONOR IMPURITIES I
CONFINED ELECTRON
Fig. 26. Modulation doping for a superlattice and a heterojunction with a Schottky junction. E, indicates the position of the Fermi energy (MENDEZ and VON KLITZING [1989]).
r
Fig. 27. Schematic representation of the conformation of chain-folded polymer molecules in a semicrystalline polymer. One molecule belonging to adjacent crystals is indicated as a heavy line.
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results in a needle-shaped critical nucleus, the length of which is equal to the length of the molecular chains. Hence the system is left with only one degree of freedom to reduce the energy barrier for the critical nucleus. This reduction occurs by adjusting the diameter of the needle. However, if chain folding occurs, the energy barrier associated with the critical nucleus can be minimized by adjusting the size of the nucleus in all three dimension. Detailed computations reveal that the energy barrier for chain folded nuclei is in general significantly lower than for extended chain crystallization. The second group of models for chain-folding is based on the excess entropy associated with the folds relative to an extended-chain crystals. If the Gibbs free energies of an extended chain crystal and of a chain-folded crystal are compared, the chain folds are found to increase the internal energy of the system. However, the chain folds also contribute to the entropy of the system. Hence, at finite temperatures, a structure of lowest Gibbs free energy is obtained, if a certain concentration of chain folds is present in the crystal. In other words, chain-folded crystals have a lower Gibbs free energy at finite temperatures than extended chain crystals (cf. also ch. 32, 5 2.2-2.6). Polymers crystallizing from the molten state form more complex morphologies. However, the basic building blocks of these morphologies remain thin lamellar crystals. Figure 28 shows spherulitic crystallization of thin molten polymer film. The spherulites consist of twisted lamellae which exhibit radiating growth. If the molten thin film is strained during solidification, different morphologies may result, depending on the strain rate. However, all of these morphologies have in common that the macromolecules are
Fig. 28. Bright field transmission electron micrograph (defocus contrast) of a two-dimensional spherulite in isotactic polystyrene.The spatial arrangementof the lamellae formed by the folded macromoleculesis indicated on the left side (PETERMANN [1991]). References: p . 935.
906
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Fig. 29. (a) Stacked lamellar morphology in polyethylene (TEM bright field). (b) Needle-like morphology in polybutene-1 (TEM bright field). (c) Oriented micellar morphology in polyethylene terephthalate (TEM dark field micrograph). (d) Shish-kebab morphology in isotactic polystyrene (TEM dark field micrograph)
(PETERMANN [1991]).
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more or less aligned in the straining direction, High temperatures and small strain rates favour a stacked lamellar morphology (fig. 29a), high temperatures combined with high strain rates result in needle-like arrangements (fig. 29b). Low temperatures and high strain rates lead to oriented micellar structures (fig. 29c). The transition between these morphologies is continuous and mixtures of them may also be obtained under suitable conditions (fig. 29d). The way to an additional variety of nanostructured morphologies was opened when multicomponent polymer systems, so-called polymer blends, were prepared. For thermodynamic reasons, polymer blends usually do not form homogeneous mixtures but separate on length scales ranging from a few nanometers to many microns depending on the thermomechanical conditions of crystallization and the molecular structure of the costituents involved. So far the following types of nanostructured morphologies of polymer blends have been reported for blends made up by one crystallizable and one amorphous (non-crystallizable) component: Type I morphology: The spherulites of the crystallizable component grow in a matrix mainly consisting of the noncrystallizable polymer. Type 11morphology: The non-crystallizable component may be incorporated into the interlamellar regions of the spherulites of the crystallizable polymer. The spherulites are spacefilling. Type III morphology: The non-crystallizable component may be included within the spherulites of the crystallizable polymer forming domains having dimensions larger than the interlamellar spacing. For blends of two crystallizable components, the four most frequently reported morphologies are: Type I morphology: Crystals of the two components are dispersed in an amorphous matrix. Type 11morphology: One component crystallizes in a spherulitic morphology while the other crystallizes in a simpler mode e.g., in the form of stacked crystals. Type I11 morphology: Both components exhibit a separate spherulitic structure. Type IV morphology: The two components crystallize simultaneously resulting in so-called mixed Spherulites, which contain lamellae of both polymers. Morphologies of lower complexity than spherulites, such as sheaves or hydrides may also be encountered. In these cases, the amorphous phase, may be arranged homogeneously or heterogeneously depending on the compatibility of the two components. The morphology of blends with one crystallizable component has been studied for a variety of macromolecular substances e.g., poly(&-caprolactrone)/poly(vinylchloride), poly(2,6dimethyl-phenylene oxide)/isotactic polystyrene, atactic polystyrene/isotactic polystyrene blends. Block copolymers constitute a third class of nanostructured polymers. All macromolecules of a block copolymers consist of two or more, chemically different sections which may be periodically or randomly arranged along the central backbone of the macromolecules and/or in the form of side branches. An example of a block copolymer are atactic polytyrene blocks alternating with blocks of polybutadiene or polyisoprene. The blocks are usually non-compatible and aggregate in separate phases on a nanometer scale. As an example for the various nanostructured morphologies possible in such systems, fig. 30 displays the morphologies formed in the system polystyrene/ polybutadiene as a function of the relative polystyrene fraction. The large variety of nanostructured morphologies that may be obtained in polymers depending on the crystallization conditions and the chemical structure of the macromolecules causes the References: p . 935.
908
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polystyrene blocks
Fig. 30. Electron micrographs of the morphologies of a co-polymer consisting of polystyrene and polybutadiene blocks, as a function of the fraction of polystyrene blocks. The spacial arrangements of the polystyrene and polybutadiene in the solidified polymer are indicated in the drawings above the micrographs (PETERMAW [ 19911).
properties of polymers to vary dramatically depending on the processing conditions. An example of a polymeric material with novel properties originating from a special nanoscale microstructure is shown in figs. 31 and 32. Polyethylene with a nanostructured morphology consisting of stacked crystalline lamellae (fig. 3la) exhibits remarkable elastic properties (fig. 32) if strained in tension in the direction perpendicular to the plane of the lamellae. The strain causes the lamellae to separate so that fibres of extended tie molecules form between them (fig. 31b). Upon unloading, the surface-energy of these molecular fibres causes them to shrink and thus pull the lamellar crystals together again. In other words, one obtains a material that can be strained reversibly by more than 100%. The restoring force (contraction) of the material is driven by surface energy and hence the material may be termed surface-energy pseudoelastic. If the stacked morphology is replaced by, e.g., a spherulitic microstructure, no such effects are noticed. In recent years, the large variety of nanostructured morphologies that may be generated for example in polymer blends or block copolymers has caused a rapidly expanding research activity in this type of materials (MARTUSCELLI et al. [1980], PETERMANN [1991]). For further details. see ch. 32.
5.4. Nanocrystalline and nanoglassy materials 5.4.1. Basic concepts The difference between the atomic structure of nanocrystalline materials and other states of condensed matter ( e g , single crystals, glasses, quasicrystals) may be understood by considering the following two facts:
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Fig. 31. (a) Defocus electron micrograph showing the stacked-lamellae structure of a polyethylene fiber. The dark regions are the crystallites. The pattern of bright lines indicates the positions of the chain folds between the lamellae (cf. fig. 27). (b) Electron micrograph of a strained polyethylene fiber (cf. fig. 31a) showing complete separation of the lamellae interconnected by fibrils. The horizontal arrow indicates the straining direction (the strain is approximately 100%).
(i) The properties of a solid depend (in addition to its dimensions and its dimensionality) primarily on its density and the coordination of nearest neighbors. The physical reason is that interatomic forces are of short-range character and hence the interaction between nearest neighbors is the most dominant factor. The total interaction energy between nearest neighbors depends on the number (coordination) of nearest neighbors and their References: p . 935.
910
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STRAIN Fig. 32. Stress-strain curve for straining and destraining (in air) of a stacked-lamellae structure of polyethylene (cf. fig. 31) at 22" C at a strain rate of 0.0005/s. In the plateau region, the deformation occurs primarily by the separation of the lamellae and the formation of the fibrils between them (cf. fig. 31b).
interatomic spacing (i.e., the density). Perhaps, the best-known example demonstrating the significance of the density and the coordination of nearest neighbors is the phase transformation between diamond and graphite (fig.33). During this phase transformation, the density and coordination number change by about 30 to 40%. The resulting variation of the properties is well known: A hard, brittle, insulating material (diamond) transforms into a soft, ductile and electrically conducting substance (graphite). (ii) In the cores of incoherent interfaces (grain boundaries, interphase boundaries) and other lattices defects, the density and nearest-neighborcoordination deviates significantly from the surrounding perfect crystal lattice. For example, fig. 34 displays the computed
Fig. 33. Atomic structure of diamond (left side) and graphite. The carbon atoms are represented as closed circles.
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L c-
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r/a Fig. 34. Computed radial distribution function, 3g(r), for the atoms in the core of a 43.60' [lo01 (829) grain boundary in Au (cf. fig. 35). Arrows indicate the corresponding perfect crystal peak positions. The interatomic spacings between the atoms in the boundary core are strongly affected by the presence of the interface. In the computations, the interatomic forces were represented by an embedded atom potential corresponding to Au (PHILLPOT et al. [ 19901).
radial distribution function in a 229 (100) grain boundary in Au. The positions of the delta-function-like distribution peaks of the Au crystal lattice are indicated by arrows. Obviously, the boundary core structure is characterized by a broad distribution of interatomic spacings. In fact, the width of this distribution is also much larger than the one in glassy solids. Concerning the density reduction in the boundary core regions, the situation appears similar. For instance, in the case of the grain boundary shown in fig. 35 (deduced from a high resolution electron micrograph of this boundary between two misoriented NiO crystals) the density is about 80% of the cubic crystal far away from the boundary. ~
With these facts in mind, it was proposed (MARQUARDT and GLEITER[1980], GLEITER [1981b]) that if one generates a material that consists primarily of incoherent interfaces (i.e., a material that consists for example of 50 vol% incoherent grain boundaries and 50 vol% crystals), the structure (and properties) of such a material will deviate significantly from those of a crystal and/or glass with the same chemical composition. Materials with such a high density of interfaces were called nanocrystalline materials*. In order to specify this idea further, let us consider a hard-sphere model of a nanostructured material which may be generated in the following way: We start off with isolated nanometer-sized
* Other terms used in the literature in the subsequent years were nanophase materials, nanometer-sized materials, nanostructured materials or materials with ultra-fine microstructures. References: p. 935.
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Fig. 35. Atomic structure of the core of a grain boundary between two NiO crystals that are tilted relative to one another by 36,9" about a common [lo01 direction normal to the plane of the figure, The structure was deduced from the high resolution electron micrograph of 36,9" [loo] tilt boundary in NiO (MERKLEet al. 119871). The boundary core structure may be described as a two-dimensional periodic may comprising two different pentagonal polyhedra (indicated on the right side of the boundary by broken lines). The two NiO crystals forming the boundary have cubic structure.
crystals with uncontaminated surfaces and consolidate them at high pressure. If this is done, one obtains a structure that is represented (for the sake of simplicity) in the form of a two-dimensional hard-sphere model in fig. 36. The open and full circles represent atoms all of which are assumed to be chemically identical. Different symbols (operdfull circles) are used to emphasize the heterogeneity of the structure of nanostructured materials. Obviously, the structure consists of the following two components: (i) The crystallites which have all the same atomic arrangement. The only difference between the crystallites is their crystallographic orientation. The atoms in the interior of the crystallites are represented in fig. 36 as full circles. (ii) The second component of the nanocrystalline structure is formed by the interfaces (grain boundaries) between the crystallites. At the interfaces, two crystallites are joined together. Due to the different crystallographic orientations between adjacent crystallites, a region of misfit (called a grain boundary) results. In these grain boundaries, the atoms (indicated as open circles) are packed less densly than in the interior of the crystallites and the arrangement of nearest neighbors is changed relative to the crystal interior. The fraction of atoms situated in interfaces increases if the crystallites are made smaller. In fact, a straightforward estimate shows that (in the three-dimensional case) about 50% of the atoms are located in interfaces (open circles) if the crystal size is in the order of 5 to 10 nm. A crystal size
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Fig. 36. ’lhodimensional hard sphere model of the atomic structure of a nanocrystallinematerial. For the sake of clarity the atoms in the centers of the “crystals” are indicated in black. The ones in the boundary core regions are represented by open circles. Both types of atoms are assumed to be chemically identical.
in the order of a few nanometers has been the reason for calling materials that consist primarily of interfaces “nanocrystallinematerials”. Obviously, the structure and properties of nanocrystalline materials depend on the large fraction of interfaces with densities and nearest neighbor coordination numbers that deviate from the ones of crystalline and the glassy state and on the reduced size of the crystallites. In order to achieve a high density of interfaces, the size of the crystals forming a nanocrystalline material has to be reduced to a few nanometers. If the size of the crystalline regions approaches a few nanometers, size effects as well as reduced dimensionality effects may become important as well. In other words, in addition to the ej$Gectsresulting from the variation of the density and the nearest neighbor coordination in the intetj4aciaZ region, nanocrystalline materials are also expected to exhibit size and reduced dimensionality effects. The structural difference between a nanocrystalline and a glassy material (with the same chemical composition) may be seen by considering the physical origin of both structures. In a glass generated by quenching the molten state, the spacial density and coordination variations are controlled by the frozen-in thermal fluctuations. This is not so in nanocrystalline materials. The density and coordination numbers in the boundaries are controlled by the misfit between adjacent crystallites (fig. 36) with different crystallographic orientations. In other words, the density and the coordination are controlled by geometry (or crystallography) rather than by thermal fluctuations. References:p . 935.
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In the following section an attempt will be made to summarize our present knowledge about nanocrystalline materials. This section will be subdivided into the following five paragraphs: (a) The generation of nanocrystalline materials. (b) The atomic structure of nanocrystalline materials. (c) Nanoglasses. (d) Nanmmposites. (e) Technological applications. For obvious reasons, the summary given in this section is not intended to be encyclopaedic. In discussing the atomic structure and properties of nanocrystalline materials, we draw attention to those aspects of the experimental evidence that must be examined the most critically. The discussion of the properties will be limited to properties that appear relevant to technological application. For further details, the reader [1989], BIRRINGER [1989], is referred to one of the recent review articles (e.g., GLEITER SURYANARAYANA and FROES[1992], SIEGEL[1993], SHULL[1993]), GLEITER[19951. 5 . 4 2 Generation of nanocrystallinematerials In principle the following three routes have been used so far to generate nanocrystalline materials. The first one starts from a noncrystalline structure, e.g., a glass. The nanocrystalline materials are obtained by nucleating numerous crystallites in the glass e.g., by annealing. These nuclei subsequently grow together and result in a nanocrystalline material. The various modifications of this approach differ primarily in the starting material used. So far metallic glasses (e.g., produced by melt spinning, LU et al. [19911) and the sol-gel technique (CHAKRAVORTY [19921) have been successfully applied. The most important advantages of this approach are as follows. Low-cost mass production is possible and the material obtained exhibits little or no porosity. Obviously this approach is limited to chemical compositions which permit the generation of glasses or sols. The second route is based on increasing the free energy of a (initidly coarse-grained) polycrystal. The various modifications differ by the procedures that are applied to increase the free energy. Ball-milling, high-strain-rate deformation, sliding wear, irradiation with high-energy particles, spark erosion and detonation of solid explosives have been used so far. All of these techniques are based on introducing a high density of lattice defects by means of plastic deformations or local atomic displacements. The crystal size of the final product can be varied by controlling the milling, the deformation, the irradiation or the wear conditions (e.g., the milling speed, temperature, type of mill used, etc.). This group of methods have been scaled up successfully. For example, cryomilling has been applied to produce commercial quantities of nanocrystalline AVAl,O, alloys. The third route of production processes for nanocrystalline materials involves a twostep procedure. In the first step, isolated nanometer-sized crystallites are generated which are subsequently consolidated ifito solid materials. PVD, CVD, electrochemical, hydrothermal, thermolytic, pyrolytic decomposition and precipitation from solution have been used so far. The most widely applied PVD method involves inert gas condensation (fig. 37). Here, the material is evaporated in an inert gas atmosphere (most frequently He at a pressure of about 1 Wa). The evaporated atoms transfer their thermal energy to the (cold) He and hence, condense in the form of small crystals in the region above the vapor source. These crystals are transported by thermal convection to the surface of a cold finger. In addition to thermal evaporation, dc or ac sputtering as well as laser
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Fig. 37. Schematic drawing of gas-condensation chamber for the synthesis of nano-crystalline materials. The material evaporated condenses in the inert gas in the form of small crystallites which are subsequently transported via convection (mows) to the liquid-nitroqen-filledcold finger. The powder of crystallites is finally scraped from the cold finger, collected via the funnel and consolidated first in the low-pressure compaction device and then in the high-pr&sure compaction unit. Both compaction units are kept under UHV conditions (GLEITER 119891). Instead of an evaporation device, a sputtering sowe has been utilized as well.
evaporation or laser ablation have been used. Instead of evaporating the material into an inert gas atmosphere, bulk nanocrystalline materials may also be obtained by depositing the material in the form of a nanometer-sized polycrystalline layer onto a suitable substrate. PVD, CVD and electrochemical deposition has been applied successfully ( C F I A ~ and E ~ CHAKRAVORTY [1989], OSMOLAet al. [1992], VEITHet al. [1991, 19921, HASEEBetal. [1993]). In the special case of nanocrystalline Si, plasma-enhanced CVD using an rf or dc plasma source turned out to be a versatile method that permits one to generate thin Si films. Depending on the deposition parameters, any microstructure may be obtained between amorphous Si and coarse-grained crystalline Si (VEPREK and SARO-IT [1987]). The methods for generating small crystallites by precipitation reactions may be divided into processes involving precipitation in nanoporous host materials and host-free precipitation. In both cases a wide range of solvents (e.g., water, alcohol, etc.) as well as different reactions (e.g., addition of complex forming ions, References:p . 935.
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photochemical processes, hydrolytic reaction, etc.) have been utilized. A widely applied method for generating nanometer-sized composits is based on the sol-gel process. An interesting subgroup of sol-gel generated nanocomposits are organic-inorganicnanoscale ceramics, so called ceramers, polycemzs or o m c e r s (SCHMIDT[1992]). Following the et al. [1990]), these materials are prepared by ideas of Mark and Wilkes, (GARRIDO dissolving preformed polymers in sol-gel precursor solutions, and then allowing the tetraalkyl orthosilicates to hydrolyze and condense to form glassy SO, phases of different morphological stmctures. Alternatively, both the organic and inorganic phases can be simultaneously generated through the synchronous polymerization of the organic monomer and the sol-gel precursors. Depending upon such factors as the structures of the organic and inorganic components, the phase morphologies, the degree of interpenetration, and the presence of covalent bonds between the phases, the properties of these composites can vary greatly and range from elastomeric rubbers to high-modulus materials. Precipitation in nanoporous materials involves a large spectrum of host substances. Hosts providing a threedimensionalframeworkcontaining nanometer-sized pores include zeolites, microemulsions,organogels (microemulsionscontaining gelatin), porous glasses, protein cages, micelles, capped materials. Graphite tubes, urea channels, phosphazene tunnel, lipid bilayer vesicles are examples of one-dimensional tunnel hosts. Layered (twodimensional) hosts are provided by clays, graphite,halide layers and self-assembled mono- or multilayers. For a recent review of the fascinating developments in these areas we refer to the paper by OZIN[ 19921. One of the attractivefeatures of such nanosized confirzedprecipitates is that they allow one to produce bulk quantities of unagglomerated nanoparticles. During the second step of the third production method, the loose powder of nanometer-sized crystals is consolidated into a bulk material. Consolidation of the small crystals has been performed at low as well as elevated temperatures under static (e.g., uniaxial pressure, hydrostatic pressure, laser-sintering, laser reactive sintering) or dynamic conditions (e.g., sinter-forging, shear, explosive consolidation). Laser reactive sintering is applicable preferentially to nanocomposits. The strategy of this method is to select mixtures (e.g., a nanocrystalline ceramic and a metal) so that one Component has a significantlylower melting point than the other(s). The objective is to achieve enhanced sintering from the low-melting component during pulse heating by a laser beam that is scanned over the pre-consolidated material (MANTHDRAM et al. [1993]). In consolidatindsintering nanocrystalline powders it is noticed that the nanometer-sized powders densify at much lower temperatures (lower pressures) than coarse grained powders with the same chemical composition. In comparison to the first two methods described at the beginning of section 5.4.2, the main advantages of producing nanocrystalline materials by a two-step procedure (involving the generation of isolated nanometer-sized crystals followed by a consolidation process) are as follows: (i) Crystals with different chemical compositions can be cogenerated, leading to “alloying” on a nanometer-scale. (ii) The free surfaces of the small crystals may be coated prior to the compaction process by evaporation, sputtering, chemical reaction (e.g., surface oxidation) or in suspension. (iii) The interior of the crystallites may be modified by ion implantation before consolidation. Due to the small
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crystal size, the implantation results in materials that have the same chemical composition throughout the volume. In bulk materials, ion implantation is limited to surface regions. Due to the high density of interfaces, single-component nanostructured materials frequently exhibit crystal growth during sintering. One approach to minimise crystal growth is to limit the time spent at the consolidation temperature. This may be achieved by sinter-forging or explosive consolidation. Another approach is to use grain growth inhibitors such as pores, second phase particles, etc. (MAYOand HAGUE[1993]). The latter method has been utilized, for instance, to sinter nanostructured WC alloys (cf. 0 5.7.1). Grain growth was inhibited by transition metal carbides. 5.4.3. Atomic structure The hypothesized reduced density in the cores of the grain boundaries in nanocrystalline materials (cf. fig. 36) agrees with the following observations. In comparison to a single crystal of the same chemical composition, nanocrystalline materials exhibit: (i) A reduced bulk density and a reduced Debye temperature; (ii) a positron lifetime spectrum that indicates the presence of interfacial free volume which varies locally in size from a single vacancy to about eight vacancies (WURSCHUMet al. [1993]); (iii) an enhanced small-angle X-ray and/or neutron diffraction cross-section; (iv) an enhanced specific heat; (v) a modified isomer shift in the Misssbauer spectrum; (vi) a hyperfine field distribution that differs from that of the crystalline state; (vii) an enhanced solute diffusivity and substitutional solute solubility; (viii) an enhanced compressibility of the boundary regions and (ix) the observation of quantum size-effects in nanocrystalline semiconductors (e.g., in nanocrystalline ZnO) which indicates that the boundary regions act as barriers for electron propagation. In addition to these findings, high-resolution electron microscopy, wide-angle X-ray or neutron diffraction measurements and computer simulations of the structure of individual grain boundaries (in bicrystals) indicate a boundary core density that is in the order of 75% to 90% of the corresponding crystal density (cf. fig. 35). Depending on the method used to study the grain-boundary structure, core widths between about 0.5 and several nanometers were measured. The boundary core densities and widths in bicrystals seem comparable with the boundary-core densities in nanocrystalline materials deduced fromthe data of the various measurements listed above. However, at this point it should be noted that the structure of the grain boundaries in nanocrystalline materials may be expected to differ Erom the one in bicrystals if the crystal size is reduced to a few nanometers. If the crystal size approaches a few nanometers, the lengths of the boundary segments between adjacent triple junctions are just a few interatomic spacings. In other words, the boundary lengths are shorter than the boundary periodicity. This is not the case for boundaries in macroscopic bicrystals. Moreover, the elastic interaction between neighboring boundaries (spaced a few nanometers only) may result in significant structural changes of the atomic arrangements in the boundary cores. The significance of both effects has been demonstrated recently by means of computationsbased on elasticity theory and computer simulations (KING [19931, HAHN [19941). Computer simulations of the boundary structure indicate that the atomic arrangements in the boundary cores References: p. 935.
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change significantly, if the boundary lengths and spacings between adjacent boundaries approach the periodicity of the atomic structure of the boundaries. In fact, experimental evidence suggesting a different boundary structure in bicrystals and nanocrystalline materials has been reported. For example, the grain-boundary free volume (measured by positron lifetime spectroscopy, fig. 38a) and the specific electric resistivity per unit area of the boundaries of nanocrystalline materials was noticed to vary as a function of the crystal size (TONGefal. 119921). The crystal size dependence suggest that the atomic structure of the grain boundaries depends on the lateral dimensions of the grain boundaries and on the spacings between neighbouring boundaries. This result agrees with et al. [1989]). recent theoretical studies of grain boundaries of finite extent (GERTSMAN These studies suggest that boundaries of finite extent differ from infinite boundaries primarily due to the disclinations formed at the triple junction between these boundaries. As the crystal size decreases, these disclinations become more and more important for the boundary structure. Another explanation for the experimentally observed grain size dependence has been proposed by PALUMBO et al. [1991b]. They pointed out that the volume fraction of the triple junctions between three boundaries in a nanocrystalline material becomes equal or larger, than the volume fraction of the planar boundary segments, if the crystal size is reduced below about 5 nm. If the atomic structure of the triple junctions differs from that of the boundaries, a grain size dependence will be noticed. Experimental evidence supporting this view by means of hydrogen diffusion in et al. [1991b]). A correlation between nanocrystalline Ni has been presented (PALUMBO the atomic structure of the boundaries and the preparation procedure has been evidenced by positron lifetime as well as Mossbauer spectroscopy. Nanocrystalline iron was produced by ball-milling and inert-gas condensation. Although both materials had the same chemical composition and comparable crystal size, they exhibited different hyperfine parameters i.e., different atomic structures. Similar results were obtained by means of positron annihilation measurements for nanocrystalline Cu or Ni in comparison to nanocrystalline metals produced by crystallizing metallic glasses. The expected modified number of nearest neighbor atoms and the broad distribution of interatomic spacings in the boundary cores agree with computer simulations of the structure of grain boundaries, with the results of measurements by means of wide angle X-ray diffraction, E M S and the enhanced width of the distribution of the quadruple splitting in the Mossbauer spectrum of a nanocrystalline material in comparison to single et al. [1995]). A discussion of crystals with the same chemical composition (LOEFFLLER the various measurements mentioned in this paragraph and the relevant original publications may be found in recent review articles listed at the end of 0 5.4.1. Ample evidence for size effects and sects resultingfrom a reduced dimensionality has been reported for nanocomposits and will, hence, be discussed in paragraph 5.6. In single component nanocrystalline materials only rather few observations of this kind seems to be availableup to now. Figure 38b shows the blue shift of nanocrystalline (consolidated) ZnO resulting from quantum size effects. Apparently the grain boundaries act as potential barriers separating the individual crystals electronically. Size andor dimensionality effects have been noticed by positron lifetime measurements as a function of the grain size. With decreasing grain size, the free volume in the interfaces becomes larger (fig. 38a) (TONGet al. [1992]).
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350 Excitation wavelength ( nrn )
400
Fig. 38. (a) Variation of the mean positron lifetime as a function of the grain size in nanocrystalline Fq8B,,Si, (TONGet al. [19921). With decreasing grain size (at constanct chemical composition) the lifetime increases, indicating that the size of the sites of enhanced free volume in the grain boundaries changes as a function of grain size. @) Photoluminescence spectrum of bulk (crystalline) 2x10in comparison to the (blue-shifted) spectrum of nanocrystalline ZnO (crystal size 4 nm). The detection wavelength used was 550 nm. Tne nanocrystalline ZnO was generated by inert-gas sublimation. The resulting crystallites were consolidated at 0.9 GPa pressure (MCMAHON[1994]).
References: p . 935.
920
H.Gleiter
Ch. 9, $ 5
The presently available experimental observations indicate that the structural model outlined so far (cf. fig. 36) is oversimplifiedin several ways. The atomic structure in the boundary cores (e.g., the average density, coordination number, etc.) depends not only on the crystal size and chemical composition. It also depends on the type of chemical bonding, the presence of impurities, the preparation mode and the time-temperature history of the specimens. For example, Mbssbauer spectroscopy reveals that nanocrystalline Fe specimens prepared by ball milling or inert gas condensation exhibit different atomic structures in the boundary regions although the grain sizes and chemical compositions are comparable. The same applies to the structure of the boundary regions with and without impurity atoms. For instance, nanocrystalline Fe containing about 1% oxygen in the boundaries was noticed (by X-ray diffraction) to exhibit a different grainboundary structure from the same material without or with less oxygen. The effect of the production procedure on the structure and properties of nanocrystalline materials, generated by electroplating and other techniques, has been discussed by ERB et al. [1993]. Some of the differences revealed were attributed to the residual porosity. However also pore-free, high-purity nanocrystalline materials (e.g., prepared by rapid shear and electroplating) exhibit properties that differ significantly from those of bulk materials. The deviations were found to depend primarily on the grain size and the et al. [19931). This result agrees with the basic concepts preparation procedure (BAKONYI of nanocrystalliie materials discussed in section 5.4.1 and suggests that the details of the atomic arrangement in the boundaries depend (for a given chemical composition) primarily on the grain size, the preparation procedure and the time-temperature history of the material. The significance of the type of chemical bonding for the structure of nanocrystalline materials was recently demonstratedfor nanocrystalline diamond. Studies by means of Raman spectroscopy revealed a mixture of sp2 and sp3 bonds in nanocrystalline diamond. The sp2 (graphite like) bonds seem to be associated with the boundary regions. The crystalline regions exhibited pure (diamond-like) sp3 bonds. A change of the type of chemical bonding does not seem to occur in systems with nondirectional bonds such as in metals. In all metallic systems studied so far, the chemical bonding in single-crystalline and nanocrystalline materials seems to remain metallic. Nevertheless, the reduced atomic density, electron density and the modified nearestneighbor structure in the boundary regions seem to change the interatomic interaction. This is evidenced by the variation of the ferromagnetic properties of the boundary regions in comparisons to the interior of the crystallites (e.g., in Fe, Ni and Gd). Several other measurements (e.g., by X-ray diffraction, positron lifetime spectroscopy, EPRspectroscopy, Raman spectroscopy, IR-spectroscopy, the diffusivity, the thermal expansion or the the excess energy stored in nanocrystalline materials), support the results discussed so far. In other words, nanocrystalline materials are not fully characterized in terms of their chemical composition and grain size. Depending on the preparation procedure or the time-temperature history, the atomic arrangements in the interfaces, and hence the properties of nanocrystalline materials vary significantly even if the chemical composition and grain size are identical. In certain nanocrystalline ionic materials (e.g., AgC1, AgBr, TICI, LiI, p-AgI) space charge effects (MAIERet al. [1988]) are expected to play an important role in addition to the grain-boundary core effects discussed so far.
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Microstructure
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If the crystal size becomes comparable to the Debye length of the space charge region at the boundaries, the electric conductance was observed to be dominated by the electric conductivity of space charge region. 5.5. Nanoglasses
The considerations so far were limited to arrays of nanometer-sized crystallites (fig. 36). However, arguments similar to the ones advanced in $5.4.1,apply to so-called nanoglasses as well. Nanoglasses are solids that are generated by consolidating nanometer-sized glassy spheres (JINGet aZ. [1989]). A two-dimensionalmodel of the resulting structure is displayed in fig. 39. The similarity between a nanostructured material composed of nanometer-sized crystaflites and nanometer-sized glassy spheres may be apparent. In fact, the nanoglass may be considered as consisting of glassy regions (represented by full circles in fig. 39) joined together by a network of “interfaces” that are represented by open circles. The interfaces are created when the originally free surfaces of the glassy spheres are “welded” together during consolidation. Just as in the case when two crystals of different crystallographic orientations are joined together during consolidation, a region of reduced density and modified nearest-neighbor coordination may be expected to result in the regions where the glassy spheres are joined together. If this is so, a close analogy may be expected to exist between the microstructure of nanocrystalline materials and of nanoglasses. As a matter of fact, such an analogy has been suggested on the basis of studies by MiSssbauer spectroscopy
Fig. 39. Schematic (hard-sphere) mode1 of the atomic structure of a nanoglass. The nanoglass is assumed to be generated by consolidating nanometer-sizedglassy spheres. The atoms in the interior of the glassy spheres are represented by full circles. The atoms in the interfacial regions between adjacent spheres are drawn as open circles. Both types of atoms are assumed to be chemically identical (GLEITER [1992]). References: p . 935.
922
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(fig. 40). The quadrupole splitting distribution* of nanocrystalline materials indicates that these materials consist of two components (fig. 40b): The crystallinecomponent and the interfacial one. The crystalline component is characterized by a narrow quadrupole splitting distribution (fig. 40a), whereas the distribution of the interfacial component (fig. 40b) extends over a wide range along the horizontal (QS) axis. By analogy to the
2.9
2.1
1.3 0.5 0
1
2
Quadrupole Splitting , QS [ mm/s ] Fig. 40. Comparison of the quadrupole splitting distribution of a single crystal FeFz (fig. Ma) and of nanocrystalline FeFz (fig. 40b). Right side: Comparison of the quadrupole splitting distribution of a Pd,$i,,Fe, metallic glass produced by melt-spinning (fig. 4Oc) and in the foim of a nanoglass (fig. 4Od) produced by consolidating 4 nm-sized glassy spheres. Both glasses had the same chemical composition. The interfacial components in the Miissbauer spectra of the nanocrystalline and nanoglassy materials are the areas hatched horizontally (JING et al. [1989]).
* The quadrvpole splitting distributions shown in fig. 40 are essentially plots that display the probability,p(QS), that a particular atom is surrounded by nearest neighbors fonning arrangements of different degrees of asymmetry. Atoms surrounded by symmetrically arranged nearest neighbors appear on the left side of the QS axis (horizontal axis in fig. 40). The Mossbauer signals of atoms in asymmetric environments appear on the right side of the QS axis.
Ch. 9,$ 5
Microstructure
923
spectrum of the nanwrystalline material, tbe quadrupole splitting distribution of a nanoglass was found to exhibit the same features (fig. 40d): It consists of a (narrow) glassy (fig. 44k)and an (wide) interfacial component (fig. 4Od)as might be expected on the basis of the model suggested (fig. 39).
5.6. Nanocomposites Nanocomposites are compositionally modulated nanostructured materials. Basically the following three types of microstructures have been revealed experimentally in the case of cluster-assembled materials. (Compositionally modulated layer structures have been treated in section 5.2). 5.6.1. Nanocompositesmade up of crystalliteswith different chemicalcompositions Figure 41a displays schematically the expected atomic structure of this kind of a nanocomposite. Examples of real nanocomposites of this type are F e A g nanometer-sized alloys. Although Fe and Ag are immiscible in the crystalline and liquid state, the formation of solid solutions (probably in the interphase boundary regions) was noticed in Fe-Ag nanocomposites. The proposed structure of nanocrystalline Fe-Ag nanocomposites deduced from Mossbauer spectroscopy and X-ray diffraction is shown in fig. 41b. Similar alloying effects were reported for many other nanometer-sized composites produced by one of the methods discussed in section 5.4.2 (cf. SHINGUet aZ. [1989], RADLINSKIand CALKA[1991], OEFIRING and BORMANN[1991], MURTYer aZ. [19931). The formation of metastable nanocrystalline solid solutions and/or amorphous phases during the production process, e.g., by ball-milling seem to depend primarily on the free energy of the phases. A large free energy difference relative to the amorphous phase favors the nanocrystalline smcture. Nanocrystalline metastable solid solutions seem to result if these phases have a higher stability than the amorphous phase, as was demonstrated for W A l (OEHRINGand BORMANN [1991]). The significance of the interfacial structure between chemically dissimilar and immiscible materials was also emphasized by the results of thermodynamic studies (TURNBULLet aZ. [1990]). A reduction of the melting point and the melting enthalpy of finely dispersed Ge particles in a Sn or Pb matrix was interpreted in terms of a disordered interphase boundary structure between both materials. Remarkable reductions or exchange reactions in the solid state at the interfaces between chemically dissimilar componentsin nanometer sized materials, e.g., metals and sulfides or oxides have been reported during ball-milling or heavy mechanical deformation (mechanochemical reactions). The existing evidence indicates that the newly formed interfaces during milling play a crucial role. The reactivity of newly formed solidsolid interfaces has been demonstrated about two decades ago by co-extruding different materials, e.g., mixtures of metallic and ceramic powders or mixtures of metallic and polymer powders (e.g., AIfAl,O,, AVNaCl or Al/Teflon) (FROMMEYER and WASSERMANN [1976]). In all cases studied, the adhesion between the highly extruded components at the newly formed interphase boundaries was much stronger than the cohesion at interphase boundaries formed during conventional sintering. In order to explain the reactions and/or adhesion, the formation of free radicals, References: p . 935.
924
(b)
H.Gleiter
0 Fe atoms
Ch.9, 5 5
0 Ag atoms
Fig. 41. (a) Schematic (hard sphere) model of a nanocomposite consisting of crystals with different chemical compositions. The chemically different atoms are represented by different symbols (open, closed etc. circles). (b) Schematic model of a nanocrystalline Ag-Fe alloy deduced from Mossbauer spectroscopy. The alloy consisting of a mixture of nanometer-sized Ag and Fe crystals. In the (strained) interfacial regions between Ag and Fe crystals, solid solutions of Fe atoms in Ag, and Ag atoms in Fe are formed although both components are insoluble in the liquid as well as in the solid state. Similar effects may occur in grain boundaries between eral. [1990]). adjacent Fe and Ag crystals (HERR
Ch. 9, 5 5
Microstructure
925
deformed chemical bonds, ions and free electrons at the interfaces has been suggested. Reactions having positive enthalpies or even positive free energies are shown to occur. In fact, mechanochemical reactions seem to open new ways for processing, mining or generating materials avoiding high temperature reactions (CALKA [ 19931). Nanocomposites consisting of metallic crystallites embedded in an insulating matrix exhibit interesting electrical properties such as quantum size effects (HENGLEIN[19881, ~ L P E R I N[1986]), large dielectric permittivities as well as ac and dc resistivities. For example, the conductivity of the metallic particles embedded in an insulating matrix has been reported to decrease by several orders of magnitude if the particle size is reduced et al. into the nanometer regime (size Induced Metal Insulator T-ransition, MARQUARDT [198&]).The physical interpretations of this “SIMIT” effect is still a matter of controversy. Quantum size effects as well as a statistical approach of free electrons in small metallic particles based on a modified Drude model have been proposed. Nanocomposites made up of crystallites and glassy components with different chemical compositions An interesting example of this type of nanocomposites are nanometer-sized metallic or semiconductorparticles (e.g., Ag, CdS or CdSe) embedded in an amorphous dielectric matrix (e.g., SiO,) (fig. 42). If the embedded particles are metallic, the plasmon resonance of the particles is broadened (relative to the bulk material) by more than a factor 3. According to Me’s theory, this broadening results from chemical interfacial effects between the metallic clusters and the (dielectric) matrix. Electron mean free path and quantum size effects seem to be of limited significance in the metallic systems studied so far (HOVEL et al. [1993]. This is not so for semiconductorparticles. They are found to exhibit quantum size effects (blue-shifted band gap, discrete molecule-like excited electronic states, appearance of interface states in the band gap causing fast nonradiative recombinations, or luminescence with a large Stokes shift). If one encapsulates the embedded nanocrystals in a high-band-gap insulator, then resonances occur at very high energies. The nanocrystalline HOMO and LUMO states remain as confined bulk eigenstates inside the insulator shell. This is the ideal quantum dot case, where surface states are not important in luminescence. For example, CdS has been encapsulated inside a monolayer of Cd(OH),, CdSe inside ZnS, Si inside SiO,, etc. More complex structures have recently been made: a three layer “quantum dot quantum well” involving a monolayer of HgS grown on a CdS core and capped with four monolayers of CdS, shows luminescence from the HgS shell monolayer. The conceivable possibilities for generating new optical materials in this way are vast: experimental progress depends critically upon the development of advanced synthetis methods for highquality materials with closely specified dimensional tolerances. 5.6.2.
5.6.3. Nanocomposites with intercalated (doped) grain boundaries The thickness of the intercalate layers in the cores of the boundaries may vary from less than a monolayer to multilayers. In the case of thin intercalates (less than a monolayer), the atoms of the intercalate seem to enter specific, low energy sites. For example Bi atoms that are incorporated in the boundaries of nanocrystalline Cu,were References!p. 935.
926
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amdensation, addition of photoinitiators h-v
Fig. 42. Reaction and structural model of the incorporation of ligand stabilized CdS quantum dots into ormocer matrices (SCHMIDT 119921).
found by E M S studies (HAUBOLD[1993]) to enter sites that are characterized by three Cu atoms surrounding every Bi atom (fig. 43a). Multilayer intercalate boundary structures (fig. 43b) are apparently less well ordered. For example the structure of Ga in nanocomposites consisting of nanometer-sized W crystals separated by several layers of Ga was found (by PAC measurements) to exhibit a disordered structure. A special case of this type of nanocomposite structures are soft ferromagnetic nanostructured materials generated by partial crystallization of certain metallic glasses (cf. section 5.7.3). Another way in which nanocomposites with doped grain boundaries may be formed has been et al. [19911. The energy stored in the stress field in the vicinity discussed by TANAKOV of grain boundaries may be reduced by a spacial redistribution of the solute atoms. This effect was proposed first by Cottrell for the redistribution of solute atoms in the stress field of a dislocation (Cottrell cloud). Under suitable conditions, the stress-induced solute redistribution in the vicinity of grain boundaries may lead to nanocompositeswith soluteenhanced boundary regions. Stress-induced solute segregation to the boundaries may be important for the stability of nanostructured materials against grain growth. Stability against grain growth may be expected if the reduction of the strain energy by removing a solute atom from the lattice of the crystallites is equal or larger than the energy required to bring the solute atom into the boundary core. A potential model system of this kind are nanocrystalline Y-Fe solid solutions. Indeed, little grain growth was noticed in Fe-rich nanocrystalline Y-Fe alloys (WEISM~LLER et al. [1992]). (See also ch. 28, 0 4.3).
Ch. 9, $ 5
Micmsrrucrure
927
(b)
Fig. 43. 'ho-dimensional model of the atomic structure of a (doped) grain-boundary alloy. The doping atoms (full circles) are. confined to the boundary core regions between the nanometer-sized crystals (open circles). In the case of grain boundaries with less than a monolayer of doping atoms (fig. 43a, e.g., Bi in nanocrystalline Cu)the doping atoms occupy specific sites of low energy. If the doping concentration is increased, intercalate structures may result that are more than one monolayer thick, (fig. 43b, e.g., Cia in nanocrystalline W).
References: p . 935.
H.Gleiter
928
Ch. 9, 8 5
5.7. Technological applications Although nanocrystalline materials were proposed just about 15 years ago (MARQUARDT R a few technological applications have already and GLEITER[1980], G L ~ T E[1981b]), emerged.
5.7.1. Hard, wear-resistant nanocrystalline WC-Co materials Nanostructured WC-Co materials have been commercially generated and are presently applied as protective coatings and cutting tools. Both applications utilize the enhanced hardness, wear resistance and toughness of the nanostructured WC-Co in comparison to the conventional, coarser-grained materials of this kind. As may be seen from figs. 44 and 45, the replacement of coarse-grained WC-Co by nanostructured WC-Co alloys improves the mechanical properties by up to one order of magnitude. This improvement does not seem to be the upper conceivable limit because these new materials are not yet optimized in any way. The nanostructured WC-Co was produced and WILLBRAND [1992]) on an industrial scale either by high energy milling (SCHLUMP or by chemical processing (MCCANDLISH et al. [1990]). Chemical processing involves three coordinated steps: preparation and mixing of starting solutions; spray-drying to form chemically homogeneousprecursor powders; and, finally, fluid-bed thermochemical conversion of the precursor powders to nanocrystalline WC-Co powders. Both spraydrying and fluid-bed conversion are proven scalable technologies that have been used for producing bulk quantities of cemented carbide powders (KEARand MCCANDLISH [1993]). 5.7.2. Near net shape forming of nanocrystallineceramics/intermetallics The small crystal size of nanocrystalline ceramics suggests that superplastic deformation can occur in these materials at temperatures as low as ambient. This idea (KARCH [19901) has been recently tested for various ceramics as well as metallic and BIRRINGER nanometer-sized materials. In some of the fine-grained materials, large deformations (100% to>lOOO%) were obtained at strain rates in the order of l/s (e.g., HIGASHI [19931). Failure occured in ceramic nanocrystalline material preferentially by nucleation, growth and interlinkage of cavities if the specimens were deformed in tension. Superplastic deformation of nanocrystalline materials may be utilized for near net shape forming of brittle materials such as ceramics or intermetallics. Figure 46 reveals the replication of a surface step by low temperature plastic deformation of nanostructured TiO,. A control experiment was performed with polycrystalline TiO, (50 pm grain size) and BJRRINGER [1990]). The polycrystalline TiO, under the same conditions (WCH showed no measurable plastic deformation. The plasticity of nanostructured materials has also been suggested to improve the mechanical properties of non-oxide ceramics such as Si,N, or Sic. These materials exhibit poor high-temperaturemechanical properties owing to crack growth and boundary embrittlement which prevent their application in gasturbines and aerospace technology. Many attempts have been made to solve this problem e.g., by incorporating second-phase particles, platelets or whiskers. The success achieved has been very limited. Hence, it attracted attention when striking improvements were recently reported upon preparing these ceramics in the form of nanocomposites (NIIHARA [1991]).
a.9,95
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Microstructure
2000 1800
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24
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WC-Co alloys (circles, upper lines) as a function of the Co content (SCHLUMP and WILLBRANDT [1992]). (b) Comparison of the crack resistance of nanocrystalline and commercial grades of fully sintered WC-Co materials (KEAR and MCCANDLISH 119931). In physical terms, the crack resistance measures the mechanical energy required to displace the crack tip by one pm. In other words, the tougher the material the higher is its crack resistance.
References: p. 935.
Ch. 9, 55
H.Gkiter
930
10 8 -9 6 z 4 X
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31
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Q)
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Fig. 45. Comparisons of the measured wear rate (under identical wear conditions) of conventional (upper lines) and WILLBRAND [1992]).The wear and nanocrystalline WC-Co alloys as function of the Co content (SCHLUMP rate was measured by means of the rate of weight loss of the specimens during the wear experiment.
5.7.3. Soft ferromagnetic nanostructured materials (“Finemet”) Partial crystallization of certain metallic glasses with the composition Fe,,,Si,,,B,Cu,Nb, leads to nanometer-sizedFe-Si (B) crystallites (5 to 20 nm diameter) embedded in a residual glassy matrix (YOSHIZAWA et al. [1988]). The dopants (Cu, Nb) are found to be accumulated in the non-crystalline regions left between crystallites. The small crystal size is achieved by enhancing the nucleation rate of the crystallites dramatically. In fact, the numerous clusters of Cu atoms formed in the glassy matrix act as nucleation sites. Studies of the magnetic properties of these materials revealed high (-lo5)initial permeabilities,low (-lo-’ Ncm) coercitivities, high saturation induction (up to 1.7 T) combined with a low mean magnetostriction (-lo4). These properties are comparable or better than the best values achieved with (the more expensive) permalloys and cobalt-based amorphous magnetic alloys. The attractive magnetic properties of these “Finemet” materials are believed to result from a small mean anisotropy together with a drastically reduced mean magnetostriction and an enhanced saturation magnetization of the crystallites. The small anisotropy is explained in terms of the fact that the magnetic exchange length is larger than the grain size. Therefore, the domain wall width exceeds the size of the grains.
Ch. 9, $ 5
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93 1
a)
Fig. 46. Replication of a surface step by nanocrystalline Ti02. The experimental arrangement used is drawn in fig. 46a. A nanocrystalline Ti02 pellet is compressed between two WC pistons. The free surface of the lower piston contains a surface step (height 0.3 mm). Fig. 46b shows a top view of the piston with the surface step. Fig. 46c displays the Ti02 pellet which replicated the surface step. The deformation was performed at 800" C over 10 min at a pressure of 1 GPA. (KARCHand BIRRINGER [1990])
5.7.4. Magneto-caloric cooling with nanostructured materials The magnetic properties of nanostructured materials containing nanometer-sized magnetic particles dispersed in a nonmagnetic matrix differ from those of the corresponding bulk magnetic material. These differences affect the magnetocaloric properties of such materials. Upon the application of an external magnetic field, the magnetic spins in a nanostructured material of this kind tend to align with the field, thereby reducing the entropy of the spin system. If this process is performed adiabatically, the specimen's temperature will rise. This effect is called magneto-caloric heating. The incremental temperature rise during magneto-caloric heating, dT, is related to the degree of spin alignment and the magnitude of the magnetic spin moment of the individual particles. Creation of a nanostructured material with many small ferromagnetic regions provides an effectively enhanced magnetic moment which can result (in certain temperature and magnetic field ranges) in an increased dT over that provided by either paramagnetic or ferromagnetic materials, with potential application to magnetic refrigeration. It was shown that efficient magnetic refrigerators may then be operated at hiqher temperatures (above 20 K) and require relatively low magnetic fields (e5 T) in comparison to magneto-caloric cooling with Paramagnetic materials. Figure 47 shows an example for the temperature variation (magnetocaloric cooling and heating) of a nanostructured material due to a varying external magnetic field. The details of how the altered magneto-caloric effect is governed by composition and processing has recently been studied in detail by MCMICHAEL e l al. [1993]. References: p . 935.
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80 100 Time (sec) 60
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1
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5.7.5. Nanocrystalline magnetic recording materials In most video recorders used today, the magnetic material of the record/playback head is usually a crystalline ferrite. These materials are, however, not suitable, when substantially finer details on the magnetic tape are required, e.g., for high-definition television (HDTV) recording. The search for an alternative has resulted in a new type of nanometer-sized magnetic materials: multilayers in which crystalline iron layers of about 10 nm thickness are alternated with even thinner amorphous iron-alloy layers. The chemical composition of the layers used are F a e C r B , NiFeFeC, FeAVSiN and FeNbSiBEeNhSiBN. These nanocrystalline materials combine high saturation magnetization with a high permeability in the MHz frequency range, allowing video recording at much increased information densities (DE WIT et aZ. [1991]). 5.7.6. Giant magnetoresistance in nanostructured materials The recent reports of giant magnetoresistance (GMR)in a number of antiferromagnetically coupled multnayer systems have stimulated widespread research activity because the effect is of interest for basic research as well as for technological applications in the area of recording and data storage. Modeling of the GMR has emphasized spin-dependent scattering both within the ferromagnetic layers and at the interfacial regions between the ferromagnetic and non-ferromagnetic layers. The GMR phenomenon is different from the conventional magnetoresistance which is due to the effect of magnetic fields acting directly on the conduction electrons or on the scattering impurities. The GMR in multilayered structures comes from the reorientation of the single-domain
Ch. 9, 8 5
933
Microstmcture
magnetic layers. This is the reason why GMR is not usually seen in conventional bulk magnetic materials. GMR was observed first in both antiferromagnetically coupled and uncoupled layer structures, as a consequence of the fact that the relative orientation of the magnetization of successive ferromagnetic layers changes from antiparallel to parallel in an applied external magnetic field. The change in the magnetization of the layers by the external field d u c e s the scattering of the conduction electrons travelling through these layers and hence leads to a remarkable change of the dc resistivity of the material. This resistivity change is called GMR (BATBICHet al. [1988]). For systems with uncoupled magnetic layers, the orientation of the magnetization is random at the coercive magnetic field (H,)and there are many magnetic layers which are statistically arranged antiparallel. If these uncoupled layers are replaced by nanometer-sized particles in a nonmagnetic matrix, one may also expect GMR as well. This is indeed the case (fig. 48). The data indicate that a GMR effect in annealed Cu-Co samples is associated with the presence of appropriately sized and spaced Co precipitates in the Cu matrix (BERKOWITZ et aL [1992].
5.7.7. Luminescence from porous Si The recent discovery of visible photoluminescence from porous Si has captured considerable attention (CANHAM [1990], LEHMANN and GOSELE[ 19911). Prepared by electrochemical and chemical etching of single-crystal Si wafers in hydrofluoric acid solutions, porous Si consists of a network of pores that can range in size from micrometers down to a few nanometers. The origin of visible luminescence in porous Si is still controversial. Bulk Si, with a bandgap at 1.12 eV, is weakly luminescent in the nearinfrared region of the optical spectrum. For porous Si, the luminescence peak energy can range from 1.12 (infrared) to about 2.3 eV (green light), depending on parameters such as dopant concentration and type, current density during etch, duration of etch, and
25
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9 ,
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100 150 200 Temperature [ K ]
-=---I
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Fig. 48. Temperature dependence of the magnetoresistance (&/p) for 01-19 at% and Cu-28 at% Co samples, treated as indicated (BERKOWITZ etaL 119921).
References: p . 935.
934
Ch. 9, 5 5
H.Gleiter 0.5
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A Catalyst C
A Catalyst F
0
3
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Is
0.1
0 0
1
2
3
4
5
6
7
Time of Exposure [ h ] Fig. 49. Activity of nanocrystalline T i 4 (curve A) for H,S decomposition as a function of exposure time at 500OC compared with that from several commercial TiO, materials and a reference (A: 76 m2/g nanocrystalline rutile; B: 61 m2/g anatase; C: 2,4 mz/g rutile; D: 30 mz/g anatase; E: 20 m’/g rutile: F reference alumina) (BECKand SIEGEL t19921).
subsequent chemical treatments. These results have been qualitatively explained by invoking sizedependent quantum confinement effects. In fact, transmission electron microscopy (TEM)data support the existence of crystalline Si domains with dimensions small enough (-5 nm)to expect such phenomena. Indeed, recent studies on size-selected, surface-oxidized Si nanocrystals agree well with this idea (WILSONet ul. [1993]). Another proposed explanation for the luminescence of porous Si is that the electrochemical etch generates surface or bulk chemical species: for example, polysilanes or polysilylenes. Polymers containing only Si and hydrogen are well known as chemicals that can photoluminesce at visible wavelengths. Moreover, siloxene, a chemical which contains Si, oxygen and hydrogen, has emissive properties that are very similar to those of the luminescence of porous Si. Although the current emphasis for utilization of the unique properties of luminescent porous Si is in electroluminescence, other potential applications are beginning to emerge. The high sensitivity of the luminescence to chemical adsorbates, mentioned above, has been found to be readily reversible for a variety of molecules (BAWENDIet al. [1992]). This observation may lead to chemical microsensors that are easily incorporated into hybrid Si chips. In addition, photoelectrochemical and ion-irradiation techniques have been developed to allow the resistless
Ch. 9, Refs.
Microstructure
935
patterning of luminescent porous Si directly onto Si substrates, which may lead to a range of optical display or storage applications (DOANand SAILOR [1992]). 5.7.8. Catalytic materials Nanocrystalline cerium oxide, iron carbides and titaniumdioxidehave been studied for catalytic applications. The cerium oxide catalysts were generated by inert gas condensation (cf. section 5.4.2), the iron carbides by high-energy milling. Selective catalytic reduction of sulfur dioxide by carbon monoxide on nanocrystalline cerium oxide was investigated in a microreactor, and compared to that of conventional cerium dioxide synthesized by the decomposition of cerium acetate. The nanocrystalline cerium oxide catalyst enabled 100% conversion at 50O0C, about 100°C lower than the temperature and YING [1994]). The needed in conventional cerium dioxide catalysts (TSCHOEPE higher activity seems to be related to the oxygen deficiency of the nanocrystalline cerium oxide due to imcomplete oxidation. The nonstoichiometry of the nanocrystalline cerium oxide was confirmed by thermogravimetric analysis and X-ray photoelectron spectroscopy. The iron carbides are found to be active and stable catalysts for CO, hydrogenation. The catalytic properties for this reaction appear to be comparable or superior to those of more expensive catalytic materials, such as noble metals dispersed on porous supports. Figure 49 shows the enhanced catalytic activity of nanocrystallite TiO, for S removal from H,S via decomposition compared with commercial (coarser-grained)TiOp Obviously, the nanocrystalline sample with a rutile structure was far more reactive than the other ones. Similarly to the enhanced reactivity of nanocrystalline cerium oxide, it was found that the activity of the nanocrystalline TiO, results from a combination of features: the high surface area combined with its oxygen deficiency and its rutile structure (BECKand SIEGEL[1992]).
References AARONSON, H. I., C. L m and K. R. KINSMAN,1970, Phase Transformations (ASM, Metals Park, Ohio) p. 313. ABBRUZZESE, G., 1985, Acta Metall. Mater. 33, 1329. ABBRUZZESE, G. and K. L~CCKE, 1986, Acta Metall. Mater. 34,905. ABROMEIT, C. and H. WOLLENBERGER, 1988, J. Mat. Res. 3,640. ABROMEIT, C., 1989, J. Modem Phys. B3, 1301. ADAMS,B.L., 1993, Mat. Sci. and Eng. A l a , 59. ALLEN,S.M. and J. W. CAHN, 1979, Acta Metall. 27, 1085. ANDERSON, A. B., S.P. MEHANDRU and J. L. SMIALEK, 1985, J. Electrochem. Soc. 132, 1695. ANDERSON, A. B., C. RAVIM~HAN and S. P. MEHANDRU,1987, J. Electrochem. Soc. 134, 1789. ANDERSON, M. P., D. J. SROLOVITZ, G. S. GREST and P. S. SAHNIG, 1984, Acta Metall. Mater. 32,783. ANTHONY,T.R, 1970, Acta Metall. 18, 307. ANTHONY, T. R. and H. E. CLINE,1973, Acta Metall. 21, 117. ANTONIONE, L.BATTEZZATI, A. LIJCCI,G. RIONTIand M. C. TABASSO,1980, J. Mat. Sci. 15, 1730. ARANJO,R. D. and S. D. STOOKN,1967, Glass Ind. 48,687. ARDELL,A., 1972, Acta Metall. 20, 601. ARDELL,A., R. B. NICHOLSON and J. D. ESHELBY,1966, Acta Metall. 14, 1295. ARDELL,A., N. MARDBICHand C. WAGNER,1979, Acta Metall. 27, 1261.
936
H.Gleiter
Ch.9, Refs.
ARUNACHALAM, V.S. and R. W. C m , 1970,In: Pmc. 3rd Bolton Landing Conf., eds.B. Kear, C. Sims and N. S. Stoloff (Claitor’s Publ. Div., Baton Rouge, LA) p. 215. ASHBY, M.F., F. SPAEPEN and S. WILLIAMS,1978,Acta Metall. 26,1647. AUBAUER,H.P., 1972,Acta Metall. 20,165. Ausr, KT., 1981,Structure and Properties of Grain Boundaries, in: Chalmers Anniversary Volume, hog. Mater. Sci., eds. J. W. Christian, P. Haasen and T. B. Massalski (Pergamon Press, Oxford) p. 27. BACKHAUS-RICOULT, M. and H. SCHMALZRIED, 1985,Ber. Bunsengesellsch. Phys. Chem. 89, 1323. BACON,R. and FANELLI, 1943,J. Am. Chem. SOC.65,639. BAIBICH, M. N, J.M. BROTO,A. FERT,F. NGUYENVAN DAU, E PETROPF,P. ETE INNE, G. CREUZET, A. FREDRICHand J. CHAZELAS, 1988,Phys. Rev. Lett. 61,2472. BAKONYI, I., E. TOTH-WAR, T. TARNOCI, L. K. VARGA,A. CZIRAKI,I. GEROECS and B. FOGARASSY, 1993, Nanostructured Materials 3, 155. BALLUFFI, R W., 1980,Grain Boundary Structure and Kinetics (ASM, Metals Park, OH). BARNES,R. S., G. B. -DINS and A. H. COWL, 1958,Phil. Mag. $97. BAWENDI,M. G., P. J. CAROLL,W. L. WILSONand L. E. BRUS, 1992,J. Chem. Phys. 96,946. BECK, D.D. and R. W. SIEGEL,1992,J. Mat. Res. 7,2840. BECK,P. A., 1954,Phil. Mag. 3,245. BEER& W., 1978,Phil. Mag. A38,691. BERKOWITZ,A.E., J.R. MITCHELL,M. J. CAREY,A.P. YOUNG,S. ZHANG,F.E. SPADA,F. T, PARICER, A. 1992,Phys. Rev. Letters 68,3745. HUTTENand G. THOMAS, BHAITACHARYA, S . K. and K. C. RUSSELL,1976,Metallurg. Trans 7A,453. BIRKENBEIL, H. J. and R. W. C m , 1962,Proc. Phys. SOC.79,831. BIRRINGER, R, 1989,Mater. Sci and Eng. A117, 33. BISHOP,G.H. and B. CIIALMERS, 1968, Scripta Metall. 2,133. BJORIUUND,S., L.DONand M. HILLERT,1972, Acta Metall. 20,867. BLEIBERG,M. L. and J. W.BENNET,1977,in: Radiation Effects in Breeder Reactor Structural Materials (Met. Soc. AIME, Warrendale, PA) p. 211. BLOECHL,P., G.P. DAS, H.F. FISCHMEISTERand U. SCHOENJ~ERGER,1990.Metal-Ceramic Interfaces, eds. M. Riihle, A. G. Evans, M. F. Ashby and J.P. Hirth (Pergamon Press, Oxford) p.2. W., 1970,Crystal Defects and Crystal Interfaces (Springer, Berlin). BOLLMANN, BOLLMANN,W., 1974,Phys. Stat. Solidi 21,543. BOYD,A. and R.B. NICHOLSON,1971,Acta Metall. 19,1370. BRANDON,D.G., B. RALPH,S. RANGANATHAN and M.S. WALD, 1964,Acta Metall. 12,813. BRICXCENKAMP,W. and K.LOCKS,1983,F’h. D. Thesis RWTH Aachen. BURGER,M., W. M ~ E and R M. R~HLE, 1987,Ultramicroscopy 12,l. C m , J. W.and J.E. HILLIARD,1958,J. Chem. Phys. 28,258. CAHN,R. W., 1994,Springer Series in Materials Science 27, 179,Springer Verlag Berlin. C a m , A., 1993,Proceedings of the 122 TMS Annual Meeting and Exhibition, Denver Col., USA, 21-25 February 1993,TMS Commonwealth Drive Warrandale,PA 15086 USA, p. B 63. CANHAM,L.T., 1990,Appl. Phys. Lett. 57, 1046. C a R. and M. PARINELU),1985,Phys. Rev. Letters 55,2471. CHADDEKTON, L. T., E. JOHNSONand T. WOEKENBERG, 1976,C o r n . Solid State Phys. W, 5, 105. CHAKRAVORTY, D., 1992,New Materials, eds. J. K. Joshi, T. T s u N ~C. ~ N.R , Rao and S. Nagakura, Narosa Publishing House, New Dehli, India, p. 170. CHALMERS, B. and H. GLEITFR, 1971,Phil. Mag. 23, 1541. CHAT~EREE,A. and D. CHAKRAVORTY, 1989,J. Phys. D Appl. Phys. 22,1386. CHANG,R., 1976,Scripta Metall. 10, 861. C m , H. S. and S.Y.CHANG, 1974,Phys. Stat. Sol. 25,581. CLINE,H., 1971,Acta Metall. 19,481. COOPER,S.P. and J. BILLINGHAM, 1980,Met. Sci. J. 14,225. COTERILL,R.M. J ., 1979,Phys. Rev. Lett.42, 1541. CULLITY,B. D., 1972,Introduction to Magnetic Materials (Addison-Wesley, London) p. 357 and 565. DAMS,C.K., P. NASHand R. STSVENS, 1980a,Acta Metall. 28, 179.
.
Ch.9, Refs.
Micmstnrcture
937
DAVIS, J. R., T. A. COURTNEY and M. A. PRZYSTUPA, 1980b, Metallurg. Trans. llA, 323. DE FONTA~NE, D., 1973, Scripta Metall. 7,463. DELAEY,L. and H. WARLIMON,1975, in: Shape Memory Effects in Alloys, ed. J. Perkins, Plenum, New York, p. 89. DE RANGO,P., M. LEES,P. LEJAY,A. SULPICE,R. TOURNIER, M. INGOLD,P. GERNIEand M. PERNET, 1991, Nature 349, 770. DE WIT, H. J, C. H. M WITTMER and E W. A. DIRNE,1991, Advanced Materials 3,356. DOAN,V.V. and M. J. SAILOR,1992, J. Appl. Phys. 60,619. DOHERTY, P.E., D. W. LEEand R. S. DAVIS,1967, J. Am. Ceram. SOC.50, 77. Domxm, R.D., 1982, Met. Sci. J. 16, 1. DUFFY,D. M., J. H. HARDINGand A.M. STONEHAM,1992, Acta Metal. Mater. 40, 11. EDWARDS,S.E and M. WARNER,1979, Phil. Mag. 40,257. EISENBERGER, E? and W.C. MARRA,1981, Phys. Rev. ktt. 46,1081. ERB, U., W. ABELand H. GLEITER,1982, Scripta Metall. 16, 1317. Em, U., A.M. EL-SHERIK, G. PALUMBO and K.T. A m , 1993, Nanostructured Materials 2,383. ESTRIN,Y. and K. LUCKE,1982, Acta Metall. 30,983. EVANS,J.H., 1971, Nature 229,403. FECHT,H. J. and H. GLEITER,1985, Acta Metall. Mater. 33,557. FERRANTE, M. and R. D. DOHERTY, 1979, Acta Metall. 27,1979. FERRY,D. K., J. R. BAKERand C. JACOBINI, 1990, Granular Nanoelectronics NATO Advanced Study Series, Series B: Physics, Vol. 251. FINNIS,M., A.M. STONEHAM and P.W. TASKER, 1990, Metal-Ceramic Interfaces, eds.M. Ruhle, A. G. Evans, M. E Ashby and J. P. Hirth, Pergamon Press Oxford, p. 35. FINNIS,M. W. and M. ROHLE,1991, Structure of Interfaces in Crystalline Solids, in: Structure of Solids, ed. V. Gerold, Vol. 1 of Materials Science and Technology, eds. R. W. Cahn, P. Haasen and E. J. h e r (VCH, Weinheim), p. 553. FINNIS, M., 1992, Acta Metall. Mater. 40,25. FITZSIMMONS, M. R. and S. L. SASS,1988, Acta Metall. 36, 3103. FREEMAN, A. I., C. Lr and C. L. Fu, 1990, Metal-Ceramic Interfaces, eds. M. Riihle, A. G. Evans, M. F. Ashby and .I. P. Hirth, Pergamon Press Oxford, p. 2. FROMMEYER, G. and G. WASSERMA~W, 1976, Zeitschrift fiir WerkstofRechnik 7, 129, 136 and 154. GASTALDI, J. and J. JOURDAN,1979, Phys. Stat. Sol. (a) 52, 139. GAUDIG,W. and H. WARLIMONT,1969.2. Metallk. 60,488. GARRIDO,L., J. L. ACKERMANN and J. E. MARK,1990, Mat. Res. SOC.Sympos. Proceedings 171,65. GEGUZIN,Ya. E. and M. A. KRIVOGLAZ, 1973, Migration of Microscopic Inclusions in Solids (Consultants’ Bureau, NewYork) p. 157. R. Z. VALIEVand V. I. VLADIMIROV, 1989, Phil. Mag. A59, GERTSMAN, V., A. A. NAZAROV,A. E. ROMANOV, 1113. GLEITER,H., 1971, Phys. Stat. Sol. @) 45,9. GLEITER, H., 1979, Acta Metall. 27, 1754. GLEITER,H., 1981a. Chalmers Anniversary Volume, Prog. Mater. Sci., e&. J. W. Christian, P. Haasen and T. B. Massalski (Pergamon Press, Oxford). GLEITER,H., 1981b, Proceedings Second Rise International Symposium on Metallurgy and Materials Science, eds. N. Hansen, T. Leffers and H. Lilholt, Reskilde, Denmark, p. 15. GLEITER, H., 1982, Mater. Sci. Eng. 52, 91. GLEITER,H., 1989, Progress Materials Science, eds. J. W. Christian, P. Haasen and T. P. Massalski, Vol. 33 (4), 223. GLEITER,H., 1992, Nanostructured Materials 1, 1. GLEITER,H., 1995, Nanostructured Materials 6, 3. GLEITER,H. and B. CHALMERS, 1972, Prog. Mater. Sci. 16, 145. GLEITER,H., S. MAHAJANand K. J. BACHMANN, 1980, Acta Metall. 28, 1603. GORLIK,S. S., L. KOVALEVA and M. BLAUTER,1972, Fiz. Met. Metalloved. 33(3), 658. GOITSCHALK, C., K. SMIDODA and H. GLEITER,1980, Acta Metall. 28, 1653.
938
H.Gleiter
Ch.9, Refs.
GRAHAM,L.D. and R. W. KRAFT, 1966, Metallurg. Trans. 236,94. GRATIAS,D. and A. THALAL, 1988, Phil. Mag. Letters 57,63. GREENWOOD, G. W., 1956, Acta Metall. 4,243. GREENWOOD,G. W., H. JONESand J. H. WESTBROOK,1975, Phil. Mag. 31,39. GULDEN,M.E., 1969, J. Nucl. Mater. 30, 30. GUYOT,C. and M. WINTENBERGER, 1974, J. Mater. Sci. 9, 614. HAESSNER,E,S. HOFFMANN and H. SEKEL, 1974, Scripta Metall. 8,299. HAHN, W., 1994, private communication. HAKEN,H., 1978, Synergetics, (Springer Verlag), Berlin. HALPERIN, W.P., 1986, Rev. Mod. Phys. 58,566. HARASE,J., R. SHIMIZUand T. WATANABE, 1988, Eighth Int. Conf. on Textures in Metals, eds.: J. S. Kallend and G. Gottstein, The Metallurgical Society, Warrendale, PA, p. 723. HASEEB,A., B. BLANPAIN,G. WONTERS,J.P. CEIS and J.R ROOS,1993, Mater. Sci. and Eng. A l a , 137. HAWSOLD,T., 1993, Acta Metall. et Mater. 41, 1769. HENDERSON, D., W. JOST and M. MCLEAN,1978, Met. Sci. J. U,113. HENGLEIN, A., 1988, Topics in Current Chemistry, (Springer Verlag, Berlin), Vol. 143, p. 113. HERR,U., J. JING,U. GONSEXand H. GLEITER, 1990, Solid State Comm. 76, 197. HERRING,C., 1950, J. Appl. Phys. 21,301. HERRING,C., 1953, Structuce and Properties of Solid Surfaces, eds. R. Gomer and C. S. Smith, (University of Chicago Press), Chicago, p. 4. G., H. GLEITER and G. BAERO,1976, Acta Metall. 24,353. HERRMANN, HIGASHI,K., 1993, Mater. Sci. and Eng. A166, 109. HILLERT,M., 1965, ActaMetall. 13,227. HIRABAYASHI, M., 1959, J. Phys. Soc. Jap. 14,149. HIRAGA,K., 1973, Phil. Mag. 27, 1301. Ho, E and C. G. WEATHERLY,1975, Acta Metall. 23, 1451. Ho. P.S. and J. K. KIRKWOOD, 1974, J. Appl. Phys. 45,3229. HOEVEL,H., S. FRITZ, A. HILGER,U. KREIBIG and M. VOLLMER,1993, Phys. Rev. B48, 18178. HOLZ,A., 1979, Physica 97, A75. HONDROS,E.D., 1993, Mater. Sci. andEng. A l a , 1. HORNBOGEN, E. and U. K b m , 1982, Recrystallization of Metallic Materials, 4 . F . Haessner (Riederer Verlag, Stuttgart) p. 159. E., 1984, Acta metall. mater. 32,615. HORNBOGEN, HORNBOGEN, E., 1986, J. Mater. Sci. 21, 3737. HORNEIOGFN,E., 1989, Internat. Materials Rev. 34,277. HOUGHTON,D.C. and D. JONES,1978, Acta Metall. 26,695. HUNDERI,0. and N. RYUM,1979, Acta Metall. Mater. 27, 161. HUNDERI,0. and N. RYUM,1981, Acta Metall. Mater. 29, 1737. J m , S.C. and A.E. HUGHES,1978, J. Mater. Sci. W, 1611. JANG,H., D.N. SFIDMANand K. L. MERKLE,1993, Interface Science, 1,61. JING, J., A. KRAEMER,R. BIRRINGER, H. GLEITERand U. GONER, 1989, J. Non-Cryst. Solids, lU, 167. JOHNSON,K. H. and S. V. PEPPER,1982, J. Appl. Phys. 53,6634. JOHNSON,R A. and M.G. LAM,1976, Phys. Rev. B13,434. JOHNSON,W. C. and J. K. LEE,1979, Metallurg. Trans.lOA, 1141. JOHNSON,W. C., 1984, Acta Metall. Mater. 32,465. JOHNSON,W. C. and J. I. D. ALEXANDER, 1986, J. Appl. Phys. 59,2735. JONES,D.R. and G. J. MAY,1975, Acta Metall. 23,29. JUNG,P. and K. TRENZINGER, 1974, Acta Metall. 22, 123. KAHLWEIT, H., 1975, Adv. Colloid and Interface Sci. 5, 1. KARCH, J. and R. BIRRIIUGER, 1990, Ceramics International 16,291. KEAR,B. H. and L. E. MCCANDLISH, 1993, Nanostructured Materials 3, 19. KHATCHATURYAN,A.G., 1969, Phys. Stat. Sol. 35, 119. KHATCHATURYAN, A. and G. SHATAUIV,1969, Sov. Phys. Solid State JETP 11,118.
Ch. 9, Refs.
939
Microstmture
KJNG, A.H., 1993, Materials Science Forum 126-128,221. KIRCHNER, H.,1971,Metallurg. Trans. 3,2861. KIRK, W. P. and M. A. REED, 1992,Nanostrucmres and Mesoscopic Systems, Proceedings of an International Symposium held at Santa Fe, New Mexico 20-24 May 1991,(Academic Press, Boston, San Diego, New York), p. 1. KOHN,W. and L.J. SHAM,1965,Phys. Rev. 140,A1133. KRONBERG,M. L. and E H. WILSON,1949,Trans. AIME 185,501. KUHLMANN-WILSDORF, D., 1965,Phys. Rev. 140,A1599. KUNC, M. and R. M. MARTIN,1981,Phys. Rev. B24,3445. LANGER,J. S. and R. SEKERKA,1975,Acta Metall. 23, 1225. LARCHE,E C. and J.W. CAHN, 1973,Acta Metall. Mater. 21, 1051. LARCHE,F.C. and J. W. CAHN,1978,Acta Metall. Mater. 26, 1579. LEHMANN, V. and U. WSELE, 1991,Appl. Phys. Lett. 58, 856. LEMMLEIN, G.G., 1952,DOH.Adad. Neuk SSSR 85,325. LI, J.C.M., 1961,J. Appl. Phys. 32,525. LI, J .C. M ., 1969,Trans. Met. Soc. AIME 245, 1591. LIFSHITZ, I.M. and V. V. SLYOZOV,1961,J. Phys. Chem. Solids 19,35. LIVINGSTON,J.D., 1971,J. Mater. Sci. 7,61. LIVINGSTON, J. D. and J. W. CAHN,1974,Acta Metall. 22,495. LOEPFLER, J., I. WEISMUELLER and H. GLEITER, 1995,Nanostructured Materials 6,567. LORIMER,G. W. and R. B. NICHOLSON, 1969,Acta Metall. 13, 109. LORMAND,G., J. ROUAISand C. EYRAND,1974,Acta Metall. 22,793. LOUAT,N. P., 1974,Acta Metall. 22,721. LOUIE,S.G. and M.L. C o r n , 1976,Phys. Rev. B13,2461. LOUIE,S.G.,J. G. CHELKOWSKY and M. L. C o w , 1977,Phys. Rev. B15,2154. Lu, K.,J.T. WANGand W.D. WEI, 1991,J. Appl. Phys. 69,522. MACDOWELL, J.F. and G. H. BEALL,1969,J. Am. Ceram. Soc. 52, 117. MAIER,J., S. PRILLand B. REICHERT,1988, Solid State Ionics 28-30, 1465. MANTHIRAM, A., D.L. BOURELL and H.L. MARCUS,1993,Journal of Metals 11, November 1993,67. MARQUARDT, P., G. N m z and B. M-HLEGEL, 1988,Solid State C o r n . 65,539. MARQUARDT,P. and H. GLEITER, 1980,Vehandlungen der Deutsch. Physikal. Gesellsch. 15,328. MARRA,W.C., P. EISENBERGER and A. Y. CHO,1979,I. Appl. Phys. 50,6927. h ~ mG.,and L. P. KUBIN,1988,editors,Nonlinear Phenomena in Materials Science, Vols. 3 and 4 of Solid
.
State Phenomena (Trans. Tech. Aedemannsdorf, Switzerland). MARTIN,J. W. and R. D. DOHERTY,1976, Stability of Microstructure in Metallic Systems (Cambridge University Press) p. 154. (Second edition, 1996,in press). MARTUSCELLI, E., R. PALUMBO and M. KRYSZEWSKI,1980,Polymer Blends, Plenum Press, NewYork, p. 1. MASAMURA,R. A. and M. E. GLICKSMAN,1974,Can. Met. Quest. 13,43. MAURER, R and H. GLEITER,1985,Scripta Metall. 19, 1009. MAR.D., 1956,J. Chem. Phys. 25, 1206. MAW& R.D., 1959,J. Chem. Phys. 31,244. MAY,J.E. and D. TURNBULL, 1958,Trans. Metall. Soc. AIMF 212,769. MAYO,M. J. and D.C. HAGUE,1993,Nanostructured Materials 3,43. MCCANDLISH, L. E.,B. H. KEAR and B. K.KIM, 1990,Materials Science and Technology, 6,953. MCLEAN,M., 1973,Phil. Mag. 27, 1235. MCLEAN,M ., 1978,Met. Sci. J. 12, 113. MCLEAN,M.,1982,Met. Sci. J. 16,31. MCMAHONG., 1994,Ph.D. Thesis Universitl des Saarlandes, FB 15,66041 Saarbruecken, Germany. MCMICHAEL, R. D., R. D. SHULL,L.H. BENNEIT,C. D. FUERSTand J. E HERBST,1993, Nanostructured Materials 2,277. MWEZ, E. E. and K. VON KLITZING, 1989,Physics and Applications of Quantum Wells and Superlattices, NATO Advanced Studies Series, Series B: Physics, Vol. 170. MEXKLE,K.L.,J.F. REDDY,C.L. WILEYand D. J. SMITH,1987,Phys. Rev. Letters 59,2887.
940
H.Gleiter
Ch. 9, Refs.
MIGAZAKI, T., K. NAKAMURA and H. MORI, 1979,J. Mater. Sci. 14, 1827. MORI,T.,P. CHENG,M. Urnand T. MURA,1978, Acta Metall. 26,1435. M o m , J . E and N.P. LOUAT,1974,Scripta Metall. 8,91. MULLINS,W. W., 1956,Acta Metall. 4,421. MURTY,B.S.,M.M. RAo and S. RANGANATIWN, 1993,Nanostructured Materials 3,459. NE& E., 1974,Acta Metall. 22, 81. NEUHAEUSER, H.J. and W. PITSCH,1971,Z.Metalik. 62,792. NICHOLS,EA.. 1976,J. Mater. Sci. 11, 1077. NICOLIS,G.and I. PRIGOOINE, 1977,Self Organization in Non-Equilibrium Systems, Wiley-Interscience New York. NIIHARA,IC, 1991,J. Ceram. SOC.Japan, 99,974. OEHRING,M., Z.H. Y m ,T. KLASSEN and R. B o w m , 1992,phys. stat. solidi (a)131,671. OEHRING, M. and R. B o r n , 1991,Mater. Sci and Eng. A134, 1330. ORIANI, R A., 1964,Acta Metall. 12,1399. OSMOLA,D., P. NOLAN,U. ERB, G.PALUMBO and K.T. AUST,1992,phys. stat. sol. (a)131,569. OZAWA,T.and Y.ISHID&1977,Scripta Metall. 11,835. OZM,G.A, 1992,Advanced Materials 4,612. PALUMBO, G.,P. J. K~NG,K.T. Ausr, U. E m and P.C. LICHTENBLXGER,1991a,Scripta Metall. 25, 1775. P~UMBO G.,,D.M. DOYLE, A.M. EL-SHERM,U. ERBand K.T. Ausr, 1991b,Scripta Metall. et Mater. 25, 679. PASHLEY,D. W., M. H. JAKOBSand J. T. V m , 1967,Phil. Mag. 16,51. PAUL,A., 1975,J. Mat. Sci. 10, 415. PAYNE, M. C., P.D. BRISTOWEand J.D. JOANNOPOULOS,1985,Phys. Rev. Lett. 58,1348. PERKOVIC, V., C.R. PURDYand L. M. BROWN,1979,Acta Metall. 27, 1075. PETER MA^, J., 1991,Bulletin of the Institute of Chemical Research, Kyoto University 69,84. PHILLPOT, S.R.,D. Worn and S. YIP, 1990,MRS Bulletin 11,38. POI-, J. and J. M. DUPOUY,1979,Proc. Int. Conf. on IrradiationBehaviour of Metallic Materials for Reactor Core Compounds, Ajaccio, Corsica (publ. by CEA-DMCEN, 91190, Gif-sur-Yvette, France). POND,R.C. and V. V m , 1977,Roc. Roy. SOC.A357,453. Porn,R.C., D.Smm and V. VITEK,1978,Scripta Metall. 12,699. P o m r s , V., 1988,J. Physique 49,-327. POTAPOV,L.P., B. E GLOWINand P.H.SHIRYAEV,1971,Fin. Met. Metalloved. 32, 227. POWELL,RE.and H.EYRING,1943,J. Am. Ceran Soc. 65,648. R A D L A.P. ~ and A. C a m , 1991,Mater. Sci. and Eng. A134,1376. RAHMAN, A,, 1977,Correlation Functions and Quasiparticle Interactions in Condensed Matter, NATO Adv. Stud. Ser. 35,Plenum Press, New York, p. 37. RAYLHGH, LORD, 1878,Roc. Lond. Math. Soc. 10,4. READ,W.T. and W. SHOCKLEY, 1950,Phys. Rev. 78,275. RIVIER,N., 1986,J. Physique 47,C3-299. ROGERS, J.T., H.M. FLOWERS and R. RAWLINGS, 1975,Met. Sci. 9,32. ROY, R., 1972,Advances in Nucleation and Crystallization in Glasses, eds. L. L. Heck and S. Freiman, Am. Cer. Soc., Columbus, Ohio, USA, p. 57. ROHLE,M., K. BURGERand W. MADER, 1986,J. Microsc. Spectr. Electron. 11,163. ROHLE, M., M. BACKHAUS-RICOULT, K. BURGERand W. ~ E R 1987, , Ceramic Microstructure,Plenum Press New York, p. 295. ROHLE,M. and A. G. EVANS,1989,Mater. Sci. and Eng. A107,187. R ~ E M., , A. H. H E W A. G. EVANSand M. F. ASHBY, 1992, Proc. Intemat. Symp. on Metal-Cecam. Interfaces, Acta metall. mater. 40, 1. RYAZANOV, A. 1. and L. A. MAXIMOV, 1981,Rad. Effects 55, 165. SASS,S.L., 1980,J. Appl. Cryst. l3, 109. SASS, S.L.and P. D. BRISTOWE,1980,Grain Boundary Structure and Kinetics, eds. R. W. Balluffi, Metals Park Ohio, American Metals Soc., p. 71. SAUTHOFF, G.,1976,Scripta Metall. 10,557.
Ch. 9, Refs.
Microstmture
941
SCHLUMP, W. and J. WILLBRAND,1992, VDI Nachrichten 917,23. SCHMIDT,G., 1992, Mat. Res. SOC. Sympos. Proceedings 274, 121. SCHONBERGER, U., 0.K. ANDERSON and M. MEITDEWEL, 1992, Acta Metall. Mater. 40, 1. SEEGER,A. and G. SCHOITKY,1959, Acta Metall. 7,495. K. N. ISHIHARA and S. NASU,1989, in New Materials by Mechanical SHINGU,P. H, B. HUANG,J. KUYAMA, Alloying, eds. E. Arzt and L. Schultz (DGM Verlag Oberursel), p. 319. SHULL,R.D., 1993, Nanostructured Materials 2,213. SIEGE, R. W., 1993, Mater. Sci. and Eng. A168, 189. SILCOCK, J. M. and W. T. TUNSTALL, 1964, Phil. Mag. 10, 361. S ~ NC. J., , K. T. AWST and C. WINEGARD, 1971, Metall. Trans. 2, 987. SMIDODA,K., CH. GOITSCHALK and H. GLEITER,1978, Acta Metall. 26, 1833. SMITH,C. S., 1948, Trans. AJME 175, 15. SMITH,C. S., 1952, Metal Interfaces, Amer. Soc.for Metals, Cleveland, Ohio,USA, p. 65. SMITH,C.S., 1954, Met. Rev. 9, 1. SMITH, D. A. and R. C. Pond, 1976, Internat. Met. Rev. 205,61. STEWART,1972, Introduction to Glass Science, eds. L. D. Pye, H. J. Stevens, W. C. Lacourse. (Plenum Press New York, USA), p. 237. J. 0. and K. FARRELL, 1974, Scripta Metall. 8, 651. STIEGLER, STONEHAM, A.M., 1971, J. Phys. F1, 118. STONEHAM, A. M. and P. W. T,1988, Surface and Near Surface Chemistry of Oxide Materials, eds. J. Nowotny and L. C. Dufour (Elsevier Publ., Amsterdam), p. 1. SURYANARAYANA, C. and F.H. FROES,1992, Met. Trans A23, 1071. S ~ O NA. ,P. and R. W. BALLUPFI, 1987, Acta Metall. 35,2177. SUTTON, A. P., 1988, Interfacial Structure, Properties and Design, in: Mat. Res. SOC. Symp. Proc. 1 2 5 4 3 (see also Further reading). SUTTON,A.P., 1989, Phase Transitions 16/17,563. SUTTON.A. P., 1990, J. de Physique suppl no 1 51, C1-35. SUTTON,A. P. and R. W. BALLUFFI,1995, Interfaces in Crystalline Materials. Oxford University Press. TANAKOV, M. Yu., L. 1. TRUSUV and B. YA. m o v , 1991, Scripta Metall. et Mater. 25,413. TEWARY, V.K., 1973, J. Phys. F3, 1275. TIAPKIN,Yu.D., N.T. TRAVINA and T.V. YEVTSUSHENKO, 1976, Scripta Metall. 10,375. TIEN,J. and S. COPLEY,1971, Metallurg. Trans. 2,215. TILLER,W. A. and R. SCHRIEFFER, 1974, Scripta Metall. 8,487. TONG,H. Y., B. 2.DING,J. T. WANG,K. Lu, J. JIANGand J. ZHU, 1992, J. Appl. Phys. 72,5124. H., C. ABROMEIT and J. VILLAIN,1989, Phys. Rev. B40, 12531. TRINKHAUS, TSCHOEPE, A. and J. YING,1995, Nanostructured Materials, 6, 1005. TURNBULL, D., 1951, Trans. AIME 191, 661. TURNBULL, D., J. S.C. JANGand C.C. KOCH, 1990, J. Mat. Res. 5, 1731. TYLER, S.K. and P. J. GOODHEW, 1980, J. Nucl. Mater. 92,201. UBBELOHDE, A. R., 1965, Melting and Crystal Structure (Oxford Univ. mess). VANDER WALLE,C.G. and R.M. MARTIN,1985, J. Vac. Sci. Techn. B3,1256. VEITH,M. and K. KLJNZE,1991, Angew. Chem. 103,845. VEITH,M., D. KAEFER,J. KOCH,P. MAY, L. STAHL,V. Hum, 1992, Chem. Ber. 125, 1033. VEPREK,S. and E A . SAROTT,1987, Phys. Rev. B36,3344. VOLKL,J., 1972, Ber. Bunsen Gesell. 76,797. WAGNER,C., 1961, Z. Elektrochem 65,581. WARLIMONT, H. and G. THOMAS, 1970, Met. Sci. J. 4,47. WARREN,B. E., 1937, J. Appl. Phys. 8,645. WASYNCZUK, J. A. and M. RUTILE, 1987, Ceramic Microstructure ' 8 6 Role of Interfaces, eds.: J. A. Pask and A. G. Evans (Plenum Press, New York), p. 87. WATANABE, T., 1984, Res. Mechanica 2, 47. WATANABE, T., 1993, Mater. Sci. and Eng. A166, 11. WEATHERLY, G. C. and Y. G. NAKAGAWA, 1971, Scripta Metall. 5,777.
942
H.Gleiter
Ch.9, Refs.
WEISSMUELLER, J., W. KRAuss, T. HAUBOLD,R. BIRRINGERand H. GLEITFX,1992,Nanostructured Materials 1, 439 and ibid 3,261. WEINS, M., H. GLEITJB and B. CHALMERS, 1971,J. Appl. Phys. 42,2639. WERT,J., 1976,Acta Metall. 24,65. WHITMAN, W. D ., 1926,Arner. J. Sci., Ser. 5, 11, 126. WILLIS,J. R., 1975,Mech.Phys. Sol. 23,129. WILSON,W. L.,P.F. SZAJOWSKI and L.E. BRUS, 1993,Science 262,1242. WINGROVE, M.and D. M. TAPLIN, 1969,I. Mater. Sci. 4,789. WIRTH,R. and H. GLEITER, 1981a. Acta Metall. 29, 1825. WIRTH,R. and H. GLEITER, 1981b,J. Mater. Sci. 16, 557. WITTMER,M., 1985,Mat. Res. SOC. Symp. Proc. 40, 393. WOLF,D., 1985,J. Physique 46,C4-197. WOLF,D.,1990,J. Appl. Phys. 68,3221. WOLF,D.,1991, PMOS.Mag. A63, 1117. WOLF,D.and S. YIP, eds. 1992,Materials 1nterfae.s. Chapman and Hall, London, p. 1 and 139. WUERSCHUM, R,W. GREINERand H.-E. S e w 1993,Nanostructured Materials 2,55. YOSHIZAWA, Y.,S. OGUMAand K. YAEAAUCHI, 1988,J. Appl. Phys. 64,604.4. ZACHARJASEN, W.H., 1932,J. Am. Chem. Soc. 54,3841.
Further reading For further reading we refer to the review articles mentioned in the various sections. Since this chapter was written, a major text has been published: A.P. Sutton and R. W. Ballufi: “Interfaces in Crystalline Materials”. (Oxford University Press, 1995).
- SUBJECT INDEX 1st Volume: Pages 1-942 2nd Volume: Pages 943-1830 3rd Volume: Pages 1831-2740 “page number ff’signifies that treatment of the topic continues on the following page(s).
- -vacancy interaction, 1651ff --: CIOWpairs, 1653 --: spontaneous recombination, 1651ff - X-ray scattering from, 1136 Interstitial positions -: body-centred cubic structure, 1562 Invar alloys, 2540 ff Inverse melting, 1734ff Ion-beam mixing, 637 Ionic bond, 61 Ionicity, degree of, 61, 137 Ion microprobe analysis, see “Secondary-ion microscopy” Ion-probe techniques, 989 ff Ion-scattering spectroscopy, 1214ff Iron, 1556ff - allotropy, 20ff, 30ff, 1416 --: effect of substitutional solutes, 1566ff --: property changes at phase change, 1560 --: thermodynamics, 1558 --: role of entropy of demagnetization, 1558 ff - carbides, 1563ff -- precipitate microstructure, 2053 - -carbon phase diagram, 771, 1565 - -carbon solid solution, 1561ff, 2035 --: precipitation hardening, 2052ff --: discontinuous yield, 2052ff - -chromium-cobalt permanent magnet alloys, 2517ff -, cleavability, 2217 -: diffusion rates of interstitial and substitutional solutes compared, 1563 - dislocation density in deformed iron, in relation to flow stress, 1590ff -: fatigue behavior, 2334ff, 2382 -: interstitial alloys, 1561ff, --: fatigue, 2334ff --: flow stress, 2035, 2037 -: interstitial plus substitutional alloys, 1568ff - nitrides, 1563ff, 1571 - -nitrogen solid solution, 1561ff, 2037 -: octahedral and tetrahedral voids, 1562 -, phase transition, 1416 (see ulso “allotropy”) -, phosphorus in, 1208, 1214, 1215, 1237, 1272, 1582, 2271 -: properties of pure element, 1557 ff -: solubility of elements in, 1556, 1563 -: strength of ferrite, 1589ff -: substitutional alloys, 1566ff --: effect on form of gamma-field, 1566ff
-, sulphur in, 1224, 1582 -: vacancies in a-iron, 1558 -: yield stress, in dependence on temperature and grain size, 1583ff Iron aluminides, 2078 - as soft magnetic materials, 2533 ff Iron-chromium alloys, 2035 Iron-oxygen-sulphur system, 1312ff, 1316 Iron-silicon steels, see “Silicon steels” Irradiation (effects) -: amorphization, 1758ff - (in) amorphous alloys, 1804 -: atom redistribution, 64Off, 1708 -: biased point-defect annihilation, 1697 -: cavities, electron microscopy of, 1066 -: defect clusters, 1689ff -: dislocation wall lattice, 1703 - effects, miscellaneous, 1682ff --: collision (displacement) cascade, 1684ff --: displacement spike, 1684 --: intracascade defect reactions, 1688ff -: electron, see “Electron irradiation” -, fast heavy-ion, 1690ff - -enhanced diffusion, 635 ff - -induced creep, 1700ff - -induced Gninier-Preston zones, 1709 - -induced phase transformation, 643, 1709 - -induced precipitation, 640ff - -induced segregation, 640ff, 1708 -: loss of order, 1687 -: swelling, 1695ff --, reduction, 1698ff -: void formation, 1695ff (see ulso “cavities”) -: void rearrangement, 1706ff -: void lattice, 1701ff
Janecke coordinates, 5 18 Jellium, 861 Jogs, 1853ff, 1904 Johnson-Mehl-Avrami-[Kolmogorov] (JMA[K]) kinetic equation, 1435ff, 1788,2421,2674 -: relation to soft impingement, 1435ff -: necessity for a spatially uniform driving force, 1436 Jominy test, 1579 Jones theory of solid solubility, 151ff K a g o m b net, 234,246 Kauzmann paradox, 1731 Kerr effect, 957
R a d i a l (electron) probability density, 56 ff Radiation effects, see “Irradiation effects” Radius of gyration, 1164 Rafting, see “Superalloys” Random walk motion - in a crystal, 546ff - in a glass, 649ff Raoult’s law, 436 Rapid quenching from the melt, see “Solidification, rapid” Rapid solidification processing (RSP), see “Solidification, rapid” and “RSP crystalline alloys” -: pseudo-RSP, 1758 Rare earth metals, see “Lanthanides” Rayleigh instability, 880 Reaction equilibrium in solutions, 447ff Read-Shockley equation, 1870,2412 Reciprocal lattice, 68, 1043, 1097ff
S29
Subject index Recovery h m deformation, 2401 ff -: aluminum, 2403ff -: annealing textures, effect of prior recovery on, 2418 -: cell formation, 1978, 2412ff -: cell evolution, 1980, 2418 -: cell size in relation to flow stress, 1981 - (in) copper does it exist?, 2402ff, 2407 -: (role in) creep, 1973ff -: dislocation density reduction, 1978 dynamic, 1924, l929,2003,2030ff, 2127, 2408 --: dynamic secondary recrystallization, 2486 -: fatigue-strain enhanced, 2408 -: impurity influence, 2403 -: iron alloys, complete recovery, 2405 ff -: kinetics, 2405 ff --, theories of kinetics, 2417ff -, (of) mechanical properties, 2405 ff -, meta-, 2408ff -, oeho-, 2410 -: polygonization, 2410ff - (in) steels, 1587 -: stored internal energy and its recovery, 2401 ff -, stress-enhanced, 2406 ff Recovery of electrical resistivity after irradiation, 1692ff Recovery of electrical resistivity after quenching , 1634ff -: resistivity per interstitial, 1654ff -: resistivity per vacancy, 1629ff -: stage I, 1667ff -: stage II, 1674ff -: stage III controversy, 1622ff, 1636ff, 1640, 1670ff -, use to determine volume of vacancy formation, 1628 (and) vacancy concentrations, 1634 Recrystallization, 2419f -: annealing textures, see “Annealing textures” -: classification of phenomena, 2400 - diagram, 2421 ff -, directional, 1818, 2205 dynamic, 1999ff, 2453ff -: grain-boundary migration, see “Grain lboundary” -: grain growth, 870ff, 2474ff --, epitaxial, 2491 ff --: grain-size distribution, 2476 --: impurity influence, 2475
x - r a y absorption and scattering -: absorption coefficients, 1120ff -: absorption edge, 1121 -: angle of total reflection, 1131
539
Subject index
-: Bragg peak broadening, 1132ff -: Bragg peak intensity (and changes), 1118 , 1132ff -: Bragg peak shifts, 1130 ff -, Compton scattering, 112 -detectors, 1130, 1139 -: diffuse scattering, 1118ff, between Bragg peaks, 1139ff (see also “monotonic Laue scattering”) -- components, 1148ff --near Bragg peaks, 1134ff -- (due to) point defects, 1664ff -: extended X-ray absorption fine (EXAFS), 1183, 1777 -: fluorescence, 1121 -: Had CO~IEC~~OIIS,1121 -: Huang scattering, 1135ff, 1147, 1665, 1673ff - inelastic scattering, 1126ff --, coherent, 1126 --, incoherent, 1127ff -: isomorphous and isotopic substitution, 1771ff -: line broadening due to plastic deformation, 1924 -: pair distribution function, 1769ff -: phonon role in inelastic scattering, 1126 - (from) point-defect clusters, 1136ff -: peak shifts due to plastic deformation, 1924 -: radial distribution function, 1770 -: scattering, 1116ff -: single-particle scattering function, -: size-effect scattering, 1132 ff -: small-angle scattering, 1161ff (see also “Small-angle scattering...”) -: spurious radiation, -: surface sensitivity, 1131 ff -: thermal diffuse scattering, see “diffuse scattering” -: X-ray photoelectron spectroscopy, 1213ff -: X-ray S O U ~ C ~ S ,1121, 1128ff -: X-ray topography, 988 -: Zwischenreflex scattering, 1139
--
Y i e l d anomaly, see “Anomalous flow behavior in L1, phases” Yield, discontinuous eieid phenomenon) - in fcc solid solutions, 2028 ff - in lithium fluoride, 1938ff - in non-femous metals, 1939, 1941 - in polymers, 2695 ff
-
in semiconductors, 1938 -in steels, 1585ff, 15%ff, 1869, 1938, 2053 ff - (due to) strain softening, 1939ff Yield stress - (in terms of) continuum mechanics (yield criteria), 1946ff --: Mohr diagram, 2129ff --: (for) polymers, 2698 ff, 2709 --: von Mises condition, 1946, 2590, 2698 --: Tresca criterion, 2698 -: critical resolved shear stress for glide, 1885ff -, dependence on cell (subgrain) size, 1930ff, 1981, 1984,2416 -, dependence on dislocation density, 1925 -, dependence on grain size, 2168 ff (see also “Hall-Petch relationship”) -, dependence on mesh length, 1923 -: dependence on order, 2059 ff -, Fleischer-Friedel theory, 1903 -: forest dislocation cutting, 1903ff -, LabUSCh, 2020 -, mechanisms determining, 1894ff --, extrinsic, 1896ff --,intrinsic, 1894ff -, metal-matrix composites, 2584 ff -: particle resistance, 1897ff - of polymers, 2693 -: solute resistance, 1896 -: superposition of different resistances to plastic deformation, 1905ff -: threshold stress, 2185 ff Young’s modulus, 1880
z e n e r relationship, 1009,2159,2467ff. 2642 Zener relaxation, 567 ff Zero creep technique, 1210,2630 Zinc, recovery from deformation, 2401 ff Zinc group metals - crystal structures, 21 ff Zintl phases, 225 ff Zirconium -, purification by electromigration, 619 -: fast diffusion, 595 -: allotropy, 20, 24 Zone-hardened Al-Cu alloys, 205 1 Zone-melting (zone-refining), 719ff Zone-refined iron, 1588
The late Prof. Peter Haasen
Colour picture onfiont cover:
Simulation of an alloy dendrite growing into a supercooled liquid using the phase-field method. The colours show variation of composition (atomic fraction Cu) in the liquid and solid for parameters approximating a Ni-Cu alloy with 0.41atomic fraction Cu. See ch. 8, par. 7.5 (Courtesy of William J. Boettinger and James A. Warren).
ISBN
o 444 89875 I
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Robert W. Cahn and Peter Haasen
FOURTH,REVISED A N D
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PHYSICAL METALLURGY VOLUME I1
LIST OF CONTRIBUTORS
A.S. Argon E. Arzt H. K. D. H. Bhadeshia H. Biloni J. L Bocquet W. J. Boettinger G. Brebec R.W. Cahn G.Y.Chin? T. W. Clyne R.D. Doherty H.E. Exner R. Ferro D.R. Gaskell H. Gleiter A.L. Greer P. Haasen? J. P. Hirth S . Hofmann E.D. Hondros E. Hornbogen G. Kostorz
C. Laird I? LejEek W.C. Leslie Y.Limoge J. D. Livingston E E. Luborsky T.B. Massalski J. R. Nicholls AD. Pelton D.G. Pettifor D.P. Pope M. Riihle A. Saccone S . R. J. Saunders M.P. Seah W. Steurer J.-L. Strudel R.M. Thomson C.M. Wayman M. Wilkens A. H. Windle H. J. Wollenberger
PHYSICAL METALLURGY Fourth, revised and enhanced edition
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SYNOPSIS OF CONTENTS Volume 1 1. 2. 3. 4.
5. 6. 7. 8. 9.
Crystal structure of the metallic elements Electron theory of metals Structure and stability of alloys Structure of intermetalliccompounds and phases Appendh: Quasicrystals Metallurgical thermodynamics Phase diagrams Diffusion in metals and alloys Solidification Microstructure
Volume 2 10. Surface microscopy, qualitative and quantitative 11. Transmission electron microscopy 12. X-rayand neutron scattering 13. Interfacial and surface microchemistry
14. Oxidation, hot corrosion and protection of metallic materials 15. Diffusive phase transformations in the solid state 16. Nondiffisive phase transformations 17. Physical metallurgy of steels 18. Point defects 19. Metastable states of alloys
Volume 3 20. Dislocations 21. Mechanical properties of single-phase crystalline media: deformation at low temperatures 22. Mechanical properties of single-phase crystalline media: deformation in the presence of diffusion 23. Mechanical properties of solid solutions 24. Mechanical properties of intermetallic compounds 25. Mechanical properties of multiphase alloys 26. Fracture 27. Fatigue 28. Recovery and recrystallization 29. Magnetic properties of metals and alloys 30. Metallic composite materials 31. Sinteringprocesses 32. A metallurgist’s guide to polymers V
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Argon Argon Haaseni Pope Srmdel Thomson Laird C h
Livingston, Luborsly, Chin? Clyne h e r ;A m windle
CHAPTER 10
QUALITATIVE AND QUANTITATIVE SURFACE MICROSCOPY H.E. EXNER Technical University Darmstadt Department of Materials Science 0-64287 Darmstadt, Germany
R. W Cahn and P: Haasen?, eds. Physical Metallursy;fourth, revised and enhanced edition 0Elsevier Science BY 19%
944
H. E. Exner
Ch. 10, 0 1
1. Introduction In technical materials, the microstructure develops during processing. Apart from rare cases where microstructural features persist unchanged in nature and geometry from the raw material to the final product (e.g., hard nonmetallic refractory particles), each individual processing step influences the amount, the composition and the geometric appearance of the constituents and of defects in a material. This is due to the fact that microstructures usually are far from the ideal thermodynamic and geometric equilibrium predicted by thermodynamic (or rather thermostatic) considerations (see ch. 5 on metallurgical thermodynamics and ch. 6 on phase diagrams). The usual route -casting, plastic forming, heat-treating - leads to microstructures vastly different from those obtained after powder-metallurgicalproduction, for example. Vice versa, the mechanics of the individual processing techniques can be best studied by monitoring the microstructural changes as a function of processing conditions (compare, for example, ch. 8 on solidification; ch. 9, $2; chs. 15 and 16 on diffusive and nondiffisive phase transformation, respectively, or ch. 28 on recrystallization). Knowledge of the details of the formation of microstructures is essential in order to understand the relationships between processing parameters and the behaviour of materials in practical application. Since the most important technological properties are strongly influenced by the microstructure (see, for example, ch. 25 on the mechanical properties of multiphase alloys) this understanding is important for the development of metallic (as well as non-metallic) materials. Several definitions of the term microstructure have been proposed in the literature (see, for example HORNBOGEN and PETZOW [1970.1991], SCHATT[1991], HOUGARDY [1981], HORNBOGEN [1981, l984,1986a,b], LOCKE[1984], METALSHANDBOOK [1985], HEROLD-SCHMIDT[1988] or JEGLITSCH [1989]. For the purpose of this book, the following seems appropriate: The microstructureof crystalline materials is defined by the type, the structure and the number of phases, by the number, the geometric appearance (size, shape etc.) and the topological arrangement of the individual phase regions and their interfaces, and by the type, structure and geometry of lattice defects (which are in most cases not part of the thermodynamic equilibrium stsucture). The experimental study of metallic microstructures, and their qualitative and quantitative description is termed metallography. (Sometimes, this term has also been used for the preparation of ceramic materials and polymers for microscopic inspection. Eventually, this improper use will be substituted by muteriulogruphy or a similar term to be newly introduced to include metallography, ceramography and plastography.) Metallography dates back to the 17th century when English, French and German scientists first studied metallic objects by means of simple optical devices (see, for example, SMITH [19601, TENSI119681 or PuSCH [1979]). The birth of modem metallography took place 200 years later and is dated to 1863 when H. C. Sorby developed an incident-light microscope, or to 1865 when he first observed and described some microstructural elements of technical iron. Today, a large arsenal of devices and techniques HANDfor microstructural investigations has become available (see, for example, METALS BOOK, Vol. 9 119851 and Vol. 10 119861, LIFSHIN11992, 1994a1, CAJXN and LIPSHIN[1993], and the books and journals listed under Further Reading at the end of this chapter).
Ch. 10, $ 2
Sur$ace microscopy
945
This chapter deals with the techniques of microscopy and metallography as means for microstructural investigation. It focuses on qualitative and quantitative methods of optical, scanning-electron and scanning tunneling surface microscopies with short reviews of the other imaging and compositional analyzing techniques. A special chapter (chapter 11) is devoted to transmission electron microscopy, including analytical TEM.
2. Optical microscopy Metallic materials are usually opaque; therefore investigations of plane cross-sections by incident light prevail in metallography. However, the transparency of some metals and silicon to infrared light in thin sections has been effectively exploited. Optically, the individual components of a metallic alloy differ in their amplitude and phase characteristics. While amplitude objects become visible owing to differences in light absorption and thus appear in different grey shades or even colours, phase objects only differ in the refractive indices which cannot be recognized without additional provision. The preparation of cross-sections, the enhancement of contrast by etching and other methods, as well as the microscopic set-up must be carefully optimized for the material under investigation and adjusted to the purpose of the investigation in order to get maximum information from a microscopic study.
2.1. Metallographic specimen preparation The essential steps and techniques of metallographic sample preparation are shown in table 1. This large variety of methods has been described in handbooks, monographs and review articles, e.g., by PETZOWand EXNER[1968], SAMUELS [1971], in METALS HANDBOOK[1985, 19861, by ELSSNERand KOPP [1984], VANDERVOORT[1984a], SCHUMANN [1990], LLWTHAN [1992], n L L E and PETZOW [1992], ASH HANDBOOK [1992], or F’ETZOW [1994]. Details are discussed in a multitude of original papers in a variety of journals and conference proceedings (see also Further Reading at the end of this chapter). Though some systematic studies of the construction of metallographic devices (e.g., WASCHULL[1985], KOPPand MULLER[1987] or FUNDAL and GROSS [1993]) and of the consecutive steps of sample preparation (e.g., NELSON[1989], MULLERand KOPP[1989], TELLEand PETZOW [I9921 or WASCHULL [1993]) have been published, successful preparation of metallographic laboratories samples is still a matter of skill. Since accreditation of metallographic became a major issue, systematic evaluations of procedures and standard documentation are necessary additions to the empirically developed recipes (see, for example, RUCKERand BJERREGARD [1993] or WIELAND [1993]). In the following, a few of the more basic aspects of the present state of the art are reviewed. 2.1.1. Sampling The location from which a specimen is taken depends on whether the investigation is aimed (1) to give data for a specific area (systematic sampling), e.g., if the origin of a failure is clearly visible, (2) to characterize a larger piece (e.g., a laboratory sample) or (3) to characterize the quality of a large amount of material (as in quality control). In the References: p . 1016.
Table 1 Steps of metallographic preparation(after Pmmw and EXNER [1%8]). sawing
rod electrode acid sawing acid milling acid-jet cutting
electmhernid
clamping
sheet electrode
4
low-melting alloys sulphur, glass cement, plaster glue Mhual B i n elastomers duromers
galvanic
-G stamping
vibro-e.ngmving
electro-erosive
writing
enpving+ co-embedding of label electro-erosive
electrolytic
GRINLlING-1
mechanical
+ band
microtome cutting micromilling
POLISHING
IgyF
D R N Y *G I-
vibrational
mechanical chemical chemicdmmhanical electrolytic
electrolytic/mwhanical
CLEANING-fl
rotating disc
b
air jet Vacuum
heating
combined
I
alternating
manual
a.10, 92
Sugace microscopy
947
last two cases the statistical fluctuations due to unavoidable inhomogeneities must be considered, and usually more than one specimen is necessary to get a reliable result (statistical sampling). Since usually nothing is known as to the degree of homogeneity, statistical parameters (usually taking the arithmetic mean and the relative standard error, see, e.g., RTZOW and EXNER[1968]) should be determined from samples which are taken either at arbitrary or at specially defined locations. Furthermore, damaging the specimen during cutting it from a larger piece gives rise to erroneous results: Electroerosive cutting (“spark-machining”), for example, changes the composition near the cut faces to an appreciable depth: e.g., 0.9 and 0.3 wt% carbon (stemming from the electrolyte) and 0.8 and 0.2 wt% copper (from the electrode) were found in pure iron at 50 and 150 p m depth, respectively, below the electro-eroded surface. Careful work (slow and interrupted cutting) reduces the depth of influence to 10 pm. Mechanical cutting (usually by water-cooled wheels) does not change the composition but introduces stresses to a depth of 100 p m and more (WAVER[1973], WELLNER [1980], KIESSLERet al. [1982], VANDERVOORT[1984a] or TELLSand F’ETZOW [1992]). In spite of the disadvantages of these commonly used techniques, others, like chemical cutting by a fastmoving endless wire wetted by an aggressive liquid, available commercially as “acid saws”, are only used for special purposes (for single crystals, semiconductors, brittle intermetallics, etc.) because of the long cutting times needed (hours, instead of the minutes needed for mechanical cutting). The same considerations apply to ultrasonic erosion, electrochemical sectioning or laser cutting. In order to avoid artifacts, a careful choice of the sampling technique adjusted to the specific material and its conditions and a careful control of the result must be made.
2.12. Mounting Embedding or clamping are relatively uncritical operations. Some resins reach a temperature up to 150°C during curing, which may lead to annealing effects in the specimen; others are cold-setting. Galvanic deposition of a thin copper or nickel layer reduces edge-rounding during preparation to an acceptable level even for oblique sectioning (see below). Smearing and edge-rounding of open porosity during polishing can be avoided by infiltration of a low-viscosity resin under vacuum or by a well-wetting melt (solder for metals, glass for ceramics).
2.1.3. Grinding The surface of a cut cross-section usually shows a high degree of irregularity which is removed in successive steps of grinding with emery paper (paper covered with Sic particles closely graded from coarse to fine between 80 and 20 pm, see fig. 1). Heating can be limited to a tolerable degree using water-cooling, but deformation of the surface is unavoidable (SAMUELS [1971], PETZOWand EXNER[1968], WAVER[1973], KIESSLER et al. [1978], VANDERVOORT[1984a], TELLEand PETZOW [1992], and F’ETZOW [1994]). It was found empirically (LIHLand MEYER[1960]) that the deformation depth X, is a square function of scratch depth Xs (X,=ax,- bX,’,where a and b are material constants). Figure 1 shows the depth of scratches, the deformation depth and the total depth influenced in grinding of steel. In an oblique taper section, the deformed layer becomes visible after etching (fig. 2). References: p. 1016.
948
Ch. 10, 82
H.E. Emer
20
E
a .I5
-=
-
-
Diamond
Q,
0 0 L
v)
Grinding wheels
10 Emery p a p e r , m e s h size
3
0
25 J=
c
n
a, P O 0
10
20
30
LO
50
60
70
80
Particle size , Nrn Fig. 1. Depth of roughness, depth of deformation and total depth of damage as a function of particle size of the grinding and polishing medium (after ~ Z O and W EXNER [1968]).
2.1.4. Polishing In order to prepare a surface free of artefacts, the damaged layer is removed by polishing. While it is relatively easy to obtain scratch-free surfaces, preparation of an ideally undisturbed surface is difficult if not impossible by mechanical polishing. As shown repeatedly (e.g., by TURLEY and SAMUELS [1981] for copper) mechanically polished surfaces show all the signs of plastic deformation (shear-bands, recrystallization, subgrains, etc.) even after prolonged polishing with fine grades (< 1 pm). The depth of the remaining disturbance is small (< 1 pm) and is tolerable for most purposes. Chemical and electrolyticpolishing techniques do not cause deformation. Another technique leaving little surface damage is ion-milling or ion-polishing (LEHTINEN and MELANDER [1980]), i.e., sputtering of the surface resulting from the impact of energetic argon ions (used more frequently for preparation of TEM foils). Removal of the layer damaged during grinding by polishing is time-consuming. If sample preparation is not carefully controlled, deformation from the coarse grinding steps may resist and may influence not only the microscopic appearance (BUHLERand HOUGARDY[1979], SMUELS [1971] and POKORNY [19801) but also X-ray measurements (line broadening, blurring of reflections) or mechanical tests like indentation hardness and toughness (BERNST[19651 and EXNER C1969al). Figure 3 shows the influence of various polishing methods on reflectivity, which is sensitive to surface damage. Investigations under polarized light suffer from surface strain as well. T h e m 1 polishing, i.e., annealing in vacuo after mechanical polishing, was shown to produce smooth strain-free surfaces by uniform evaporation (IRAN1 and CAHN [1971]). Disopolishing using emulsions of carefully graded loose diamond or alumina
Ch. 10, 82
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powders on cloth is the usual way of mechanical polishing. For hard materials, a drilllike set-up using a wooden stick with diamond paste has advantages. Material removal is adequate and most materials can be polished this way. Microcutting by diamond blades has been used as a one-step, time- and labour-saving preparation mode for soft materials. Such ultrumicrofoming (for references see PETZOW and EXNER[I9681 and KLOCKENKAMPER et al. [1979]) has been shown to produce very large localized deformation (BOCKLE[1964]); nevertheless, if the cutting parameters are optimized, a surprisingly perfect surface quality of microcut metals can be achieved (PETZOWand KNOSP[1973] and PETZOW and EXNER[1975]). By micromilling cross-sections and serial sections of medium-hard metals and alloys can be prepared (KIESSLERand ELSSNER[1980] and PETZOW and EXNER[1975]). There is a nearly infinite number of recipes for chemical and electrochemical polishing. Reviews are to be found in early metallographic standard texts, e.g., TEGART [1957], PETZOWand EXNER[1968], BIGGS[1970] and SHIGOLEV [1974] while more modem references are scarce (METALS HANDBOOK [1985], F'ETZOW [1994]). Theoretical approaches are available (e.g., WAGNER[1954] and TOUSEK[1981], for reviews and early references see PETZOWand EXNER[1968] and BIGGS[1970]); however, they do not allow deduction of optimized polishing procedures for a given material. For electrolytic polishing, the form of the current density-voltage curve suggests which potentials should
100 4-
V
n
-. 95
polished
x
-1
c .-> c
V
2 90 0,
a
polished
85 Ni Fig. 2. Deformed layer in brass ( C a n 30) after cutting by a new diamond wheel. Oblique section (5' to surface which is on the left side). Etched. 200x. (From KIESSLER etal. [1982].)
Ag
Au
Cu
Fig. 3. Reflectivity (in percent of cleavage surface) of metal surfaces polished by different techniques (after PETZOW and KNOSP [1973]).
References: p. 1016.
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be used to avoid etching effects. (Usually a plateau is observed where polishing occurs.) It has been shown that these curves should be measured under practical conditions and not in special cells (R~SCHENBLECK and WOLTER[1979]). In spite of the limited exact theoretical knowledge on the mechanisms of mechanical, chemical and electrolytic processes, preparation of metallographic cross-sections is not a limiting factor for microstructural investigations since the state of the art allows the investigator to deal with even the most difficult materials.
2.1.5. Replica techniques Surface studies by transmission electron microscopy are possible by pressing a foil of a suitable substance tightly to the surface or to form a replicating foil by casting an organic replica. This technique was used extensivelyprior to the developmentof scanning electron microscopy for high-resolution surface studies. Sample preparation is well developed (see, for example, GOODHEW[19731). By the so-called double-replica technique, fine precipitates can be extracted from a matrix by chemical or electrochemical etching and embedded in a resin foil. This technique is a quick and reliable standard procedure for investigating the morphology and the spatial distribution of dispersed particles and, in particular, is useful for EDX analysis of small features avoiding signals from the matrix (see, for example, SCHR~DER et al. [1990] or CZYRSKA-FILEMONOWICZ et al. [19921). Replicas can also be studied by optical microscopy, scanning electron microscopy (5 3) and soft X-ray microscopy (§ 5.5). For remote sampling {e.g. for large engineering components which must not be destroyed, for taking samples at temperatures up to 120°C or for radioactive materials), replica techniques using movable preparation equipment are frequentlythe best if not the only way for microscopic inspection (see, for [1970], WENDLER and NEUBAUER [1979], BIRNERand L~HBERG example, BLLOUX [1980], or L ~ B E R[1982]). T Replication techniques have also been applied in optical and SEM fractography, e.g., for the study of stress-corrosion cracking (CONOR [1972]) and fatigue [1984], or BRUGELetal. [l988]). Serial crack initiation (BROWN and SMITH[1982], DIESER sectioning of plastic replicas is much easier than that of the sample itself, and was proposed [1981]). for three-dimensional reconstruction of fracture surfaces (BAUERand HALLER 2.2. Etching and other contrasting techniques The human observer is capable of distinguishing between different phases and lattice defects if these show a grey or colour contrast in microscopic viewing. Polished surfaces rarely provide sufficient contrast owing to the fairly similar reflectivity of metallic phases, and contrast enhancement is usually necessary. A number of metallographic techniques are available to reveal the microstructure.
2.2.1. Chemical and electrolyticetching When a polished surface is attacked by an etching medium, different phases and different lattice orientations usually show differing rates of dissolution. Crystal imperfections and grain boundaries are locations of increased dissolution potential. These differences in chemical and electrolytic attack are the basis for the most frequently used
ch. lo, 4 2
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metallographic techniques for optical contrast enhancement. In the monographs by TEGART[19571, BECKERTand KLEMM[19841 and PETZOW [1978,19941 or in handbooks as, for example, METALSHANDBOOK[1985], known recipes for technical metals and alloys are reviewed. Theoretical understanding of material removal (see, for example, ENGEL [1958], SCHAARWI~CHTER[1968], BIGGS [1970], HERBSLEB and SCHWAAB [1978]), though well-established in corrosion science, is rarely used for finding the optimum etching conditions for a new material as, for example, in potentiostatic etching (WORCHet al. [1994]). Usually these are established empirically, aided by educated guesses. Local changes in reflectivity and shadows produced by the rough topography of the specimen surface give rise to grey contrast when viewed in the microscope. Thus, grain boundary grooves and facets, height differences between grains of different orientation and between phases, or etch pits at points where dislocations penetrate the cross-section, are typical contrast features created by attack-etching. Sometimes, deepetching by chemical or electrochemical attack may favourably be used to show the spatial geometry of microstructural features and to make them accessible to stereometric measurement as shown, for example, by PAULand M~ZRRLE[1981], FEIJOO and EXNER [1990], or FELToo etaZ. [1990] (see also $2.4.1).
23.2. Thermal etching The thermodynamic instability of a polished surface will lead to effects similar to those mentioned in $2.2.1 when material transport is activated by heating. The basic mechanism is surface diffusion (rather than selective evaporation), and the kinetics of thermal grain-boundary grooving and facetting are well understood (see, for example, MULLINS[19611). Thermal etching is advantageouslyused for chemically stable materials such as ceramics (compare fig. 7c below).
22.3. Ion-etching The basis of the well known but infrequently used technique of ion-etching, reviewed [1980a,b], GRXF et aZ. [199?], POHL by WECHSUNG[1977], POHLand BURCHARD [19941, PETZOW [19941, PECHENYAKOV and KOVACHEVA [19951, is cathodic atomization (sputtering) by bombardment of the surface with chemically neutral (e.g., argon) or reactive (e.g., oxygen) ions. The physics of sputtering has been surveyed by OCHSNER [1975] and by PIVIN[198?]. The rate of material removal depends on the atomic weights of the material and the ions (the highest rates being observed when these are approximately equal), on the energy and density of the ions hitting the surface, and on the atomic bonding in the material. By adjusting the sputtering parameters (voltage, gas pressure), selective material removal can be made to produce a clear topography and clean surfaces. Ion-etching is advantageously used for composite materials as shown in fig. 4 and coated metals (STAPFet al. [1986], GRAF et al. [1993], POHL [1994]). However, artifacts are easily produced (fig. 5). Ion etching is also useful, if the microstructure is prepared for the application of surface-sensitive analytical techniques as well as interference-layer contrasting (see Q 2.2.5).
References: p. 1016.
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2.2.4. Staining (tinting) and anodic oxidation A large number of so-called etching techniques do not produce a surface relief by dissolving the surface but produce a surface layer by a chemical reaction (JEGLITSCH [1968]). These layers vary in thickness as a function of composition and orientation of the microstructural components. They are transparent and rather than having a specific colour themselves, produce interference colours varying with thickness, d. Light waves reflected at the surface and at the layer-substrate interface interact, causing extinction of a specific wavelength A, according to the equation (for normal incidence)
2n A, = - d , m where m determines the order of interference ( m =1, 3, 5 ... corresponds to 0, 1, 2 ... order) and n is the refractive index of the deposited layer; for a more detailed discussion see $2.2.5. Reaction layers of which the thickness varies with composition of the substrate can be deposited by chemical attack, by electrolytic processes, e.g. by potentiostatic oxidation (anodizing), or by oxidation when heating a metallic specimen in air (thermal tinting). Tinting techniques have been extensively discussed in the literature (see, for example, JEGLITSCH [ 19681, GRUTZNERand S C ~ L L E[R 19691, BERAHA[ 19701, YANKOVITH [1970], BERAHAand SPIGHLER [1977], HERBSLEB and SCHWAAB[1978], GAHM and JEGLITSCH [1981], GAHMet al. [1982], WECKand LEISTNER[1982-19861, VANDER VOORT[1984a,b, 1985a1, ZHOUetal. [1993], F’ETZOW [1994], and many others). Anodic oxidation is another possibility to produce layers with a thickness varying with orientation and composition of the microstructural features resulting in an orientation- or phase-specific colour contrast. Anodic oxidation is carried out by immersing the sample in an acid solution of carefully adjusted pH and applying a voltage in the order of 100 V. Intermetallic phases have been identified in this way (SEEGER et al. [1990]) and the grain structure of aluminium can be revealed (YANG[ 1990]), among numerous other applicat-
Fig. 4. Ionetched cross section of a graphite fibrereinforced polymer. Scanning electron micrograph, 300 x (courtesy I. Wf).
Fig. 5. Artefacts produced by ion-etching of a Nibase superalloy. (Cones formed owing to the presence of non-conducting inclusions). Scanning electron micrograph, 10,OOO x (courtesy I. Graf).
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ions (DANIELSON [1985]). In its early days, the colour contrast obtained after depositing interference layers in the presence of oxygen (reactive sputtering) was also attributed to varying thickness (ONDRACEKand SPIELER[1973]); in fact it depends on a different optical principle, as discussed in detail in the next section.
23.5. Interference-layer contrast A plane-parallel layer of a non-absorbing or weakly absorbing substance acts like an optical reflection-interference filter which, by multiple reflection at the metal-layer and layer-air interfaces, causes contrast enhancement between neighbouring phase regions, provided these differ in their optical constants. The optical principles of these effects have been reviewed by PEPPERHOFF and ETTWIG[1970], ZOGGet al. [ 19771,B ~ Eand R HOUGARDY[1979, 19801, BUHLER[1981] or PETZow [1994], and a large number of applications in metallographic practice have been published (for references see B ~ E and R [198l], GRAP HOUGARDY [1979, 19801, EXNERand ROTH[1980], GAHMand JEGLITSCH [1981] and Wu et al. [1982]). The important factor in interference-layercolour contrasting is the phase shift of the light wave reflected at the layer-metal interface. In normal brightfield microscopy, differences of this phase angle are much too small for metallic phases to be detectable. The transparent or semitransparent layers enhance these differences dramatically, revealing a pronounced colour contrast if two conditions are fulfilled: (i) The p h s e condition relates thickness of the layer d, the optical properties of the metallic phase (phase shift of the reflected wave through the angle 8 ) and of the layer (refractive index a), and the order of interference (defined by m as above) to the wavelength A ~ for , which interference causes maximum reduction in intensity:
&, = 47rnd/l7r(m - 1) -I-SI.
(2)
Usually, the first-order interference (m= 1) yields the best results. (ii) The amplitude condition describes the relative intensity R,, of the reflected light with wavelength A& For non-absorbing layers,
(
intensity of reflected wave = q - 4, %' = intensity of incident wave 1- q . q,
)
2
'
(3)
with q = (n - I)/(n
+ I),
4 , = J(nm
2
2
- n) + k:/(n, + n) + k i .
n, and k, are the refractive index and the absorption coefficient of the metallic phase, R HOUGARDY [1979], BUHLER respectively. Much effort has been undertaken ( B ~ E and and KOSSEL119811, AYJXNand BUHLER[1981, 19841, AYDINetal. [1983] and B ~ H L E R [1986]), to determine n,,, and k, values for phases occurring in metallic materials in order to be able to calculate the required n, A,, and d values for maximum contrast between phases present in a material. The contrast is defined by K = (R,-R J / R , , where R , and R2 (R, > RJ are the relative reflectivities of two phases, and reaches a maximum if R2= 0, i.e, when interference causes complete extinction of the colour under consideration in one References: p. 1016.
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of the two phases (K=1). If, then, a filter for the corresponding wavelength A is used, this phase appears black. If white light is used, maximum colour contrast will be obtained. Plane-parallel interference layers can be deposited by evaporation or by sputtering. In order to fulfill the amplitude condition R,, = 0, a non-absorbing layer must have a high refractive index if the substrate has a high reflectivity (as do all metallic phases). ZnS, ZnSe, TiO, and ZnTe (n=2.4-3.5) are used for evaporation. Absorbing layers can be deposited by reactive sputtering. This technique, first described by BARTZ[1973], uses a low gas pressure (-lo6 bar, usually oxygen) in the sputtering chamber. A commercially available device has proved very useful in practical application ( B m m and HOUGARDY [1979], E m and ROTH[1980]) and, at least in principle, allows layers to be produced with widely varying optical properties. A disadvantage is the fact that calculating the optimum contrast conditions becomes somewhat more difficult than for nonabsorbing layers (Zow et al. [19771). A general difficulty is that the quality of surface preparation before applying the interference layer is crucial for the result. For example, scratches or contamination by polishing liquids will produce artefacts. Ion polishing, therefore, is preferable compared to electropolishing or mechanical polishing techniques [1984]. Also, the accuracy of as shown, for example, for steels by GAUDIGand SCHECK measurement of the optical constants depends on the reproducibility of surface preparation (KRONER[1986]). Since calculation of optimum contrast conditions can be applied and HOUGARonly if the optical constants are known to an accuracy of 2% (SCHRODER DY [1985]), the trial-and-error approach can often be quicker and more practicable. The reason that interference contrasting is described here more fully than other methods is that this technique is still not widely familiar. It has excellent reproducibility and versatility and could substitute many of the classical contrasting techniques. The extreme sensitivity of the human eye for colour hues and the possibility of using filters makes colourcontrasting highly attractive. In addition, quantitative evaluation with respect to phase composition and exact phase identification (see, for example, ZOCGet al. [1977]) are possible. Care must be taken in the latter cases in photographic reproduction which may change the original colours appreciably (CROUSE et al. [1977] and EXNERet al. [ 19801). 23. Principles of light microscopy and optical contrast enhancement The highly developed state of the mechanical and optical design of microscopes used in metallographic work makes it impossible to come near to an adequate description in the context of a book on physical metallurgy. A large number of monographs (e.g., PAYNE [1957], 0ETl"EL [1959], MALES[1959], F R E W [1960, 19691, KINGLAKE [1965], BIGGS[1970], LOVELAND[1970], PHILIPS[1971], GALOPMand HENRY [1972], MODIN and MODIN[1973], BEYER[1974], ROST[1981], BRADBURY[1991], SCHADE [1993]) as well as articles in handbooks (e.g., VANDERVOORT[1985b], LOUTHAN [1987], Scr-ruMANN [1990], TELLE and PETZOW [1992] or HOLIK[1993]) are available which treat the basic as well as the practical aspects of optical microscopy and photography comprehensively. Accordingly, the optical fundamentals (e.g.. wave optics, properties of lenses and correction for aberrations) or the various components of the optical microscope (illumi-
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nation systems and light sources, objectives and eyepieces, polarizers, interferometric attachments, phase contrast equipment, stages etc.) need not be discussed in detail here. Important features are the resolution limit, depth of focus, and the different ways to enhance contrast by optical manipulations. 2.3.1. Resolution and depth of focus Figure 6 shows the limiting resohtion (minimum distance between two points and maximum number of lines per unit length, seen as separate features) and the depth of focus as a function of the objective’s numerical aperture n sina (where n is the refractive index of the medium between the objective’s front lens, e.g., 1 for air, 1.25 for cedar oil, and a is half the opening angle of the objective; thus, n sina is a quantitative measure for the amount of light reaching the objective). The resolution limit is, in theory, d=0.6 A/nsinar, where A is the wavelength used (Abb6 limit). For green light (A = 500 nrn) using an immersion oil between specimen and high-power objective (resulting in a numerical aperture n sina = 1.25), we get d=0.24 pm. In practice, however, the resolution limit is close to 1 p m even if the illumination system as well as all the lenses and apertures are optimally adjusted. Magnifications between 500 and 1000 times the numerical aperture, i.e., up to lOOOx, are useful while higher magnifications yield no additional information and therefore are called empty magnification. The depth of focus is inversely proportional to the square of the numerical aperture and is extremely limited at high magnifications (approx. 0.1 pm).
2.33. Bright-field illumination By far the majority of microstructuralinvestigations by optical microscopy are carried out with vertical illumination, usually called bright $eZd, and most optical micrographs of metallic materials are taken this way. Regularly reflected light is used and no additional manipulations of the light beam are necessary. Thus, high light intensities are obtained. The instrumentation is rather simple, and the use does not require great skill. The contrast is a result of locally varying light intensity due to one of the pretreatments of the specimen section discussed above. The human eye can differentiate between two grey levels if the contrast K is approximately 0.2, Le., five grey levels between black and white are easily distinguishable. Photomultipliers and television cameras are more sensitive, and more than a hundred grey levels can be registered. If higher contrast is needed, one of the special techniques described in the next four sections (sometimes improperly called “optical etching”) can be useful.
2.3.3. Oblique illumination, dark field and stop contrast If the direction of the incident light is changed from vertical to oblique, the contrast can be reversed with a gain in contrast which, for suitable specimen surfaces, is often striking. This can be achieved by simply moving the condenser aperture slightly off the optical axis, which produces a shadow-like contrast. While such oblique illumination is only applicable for low magnifications (long-working-distance objectives), a very useful alternative is dark-$eld illurnination: The light from the light source does not pass through the objective but is reflected to the surface by a ring-shaped mirror or lens References: p. 1016.
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0
0.5
1#O Numerical aperture
1.5
Fig. 6. Theoretical resolution (resolved lines per mm and distance between two distinguishable points) and depth of focus as a function of the objective’s numerical aperture (theoretical for green light).
around the objective so that only stray (diffusely reflected) light reaches the objective. Rough surfaces, fissures, pores, grain boundaries and other surface irregularities are revealed, appearing bright on a dark background. This technique also lends itself for checking the quality of polish since scratches clearly show as bright lines. Opaque-stop microscopy produces images similar to dark field. Instead of changing the illumination, a ring stop is placed between the light source and the condenser lens. By moving the stop, different areas may be illuminated. Tilt angles with respect to flat portions of the specimen surface have been measured, for example during studying tilt and twist boundaries (BIGGS[1970]).This is possible to a high degree of accuracy (depending on magnification, of the order of 1’ to 60’ of arc). For qualitative inspection, opaque-stop microscopy provides a sensitive type of dark-field contrast at no loss in resolution.
Polarized-light microscopy Plane-polarized light (produced by placing a polarizer in front of the condenser lens) vibrates in one plane only. When reflected from an optically isotropic surface, the direction of polarization does not change and will be transmitted by an analyzer placed behind the eyepiece and set parallel to the polarizer. If the analyzer is rotated, the transmitted light intensity is reduced and ideally drops to zero at crossed position of polarizer and analyzer. If the plane-polarized beam is reflected by an optically anisotropic surface it is sub-divided into two components vibrating at right angles to each other. The intensities of the components vary as a function of crystallographic orientation and its relation to the plane of polarization. Therefore, the amount of light transmitted by the analyzer is a 2.3.4.
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function of the orientation of a crystal which causes changes in brightness (degree of extinction) when the specimen stage is rotated or, for a polycrystalline material, for the various crystals when viewed with crossed polarizers. Furthermore, some optically anisotropic substances (e.g., nonmetallic inclusions such as cuprous oxide) show distinctly different tints in white polarized light due to an optical effect called rejection pleochroism, owing to a variation of reflectivity with wavelength or degree of extinction. The tint and its change when rotating the analyzer or the stage are characteristic for such materials. Polarized light is particularly useful in metallography for differentiating between optically isotropic and anisotropic components of the structure and for revealing the grain structure and twins in anisotropic metals and alloys, such as Zn, Mg, Ti and V. Though it is difficult to achieve a polished surface which is wholly strain-free, some metals and CAHN which are hard or impossible to etch can be effectively examined (IRANI [1971]). Even the cubic crystals can become optically active if etch pits or grooves are [1971]) or by coating with an anisoproduced by etching (for references see PHILLIPS tropic film e.g., by anodizing or by other epitaxially grown films. Comprehensive reviews on the use of polarized reflected light as an aid in metallography (and mineralogy) are available (CONNand BRADSHAW [1952], HARTSHORNE and STUART[1952], Mom [1952], PHILLIPS[1971], GALOPINand HENRY[1972], MODINand MODIN[1973], MCCRONE etai. [1978], SCHUMANN [1990], TELLEand PETZOW [1992], among others). The quantitative use of polarized light has been restricted mainly to transmitted light in the fields of petrography and biology, though various measurements are possible in reflection on metals as well. A special application is the imaging of magnetic domains in metal crystals making use of the Kerr effect, reviewed by BOWMAN and BOOTH [1971]. The contrast which results from a rotation of polarization direction by only I' to 20' of arc can be improved by interference layers (see 5 2.2.5).
23.5. Phase contrast and interference contrast Before some of the modern sample preparation techniques (e.g., interference-layer deposition or reproducible electrolytic etching) were fully developed, optical techniques transforming phase-angle or height differences into grey or colour contrast gained some interest in metallography (JEGLITSCH and MITSCHE [1967], BEYER[1974], MODINand and PETZOW [1992] MODIN[1973], VANDER VOORT[1985b], SCKUMANN [1990], TELLE or HOLIK[1993]). Though they are developed more than 30 years ago, they are now available with some higher-priced metal microscopes, and have been described in detail in numerous publications (e.g., OETTEL[1959], JEGLITSCH and MITSCHE[1967] and BEYER [19741) their applications in studying metallic microstructureshave remained few. Phase contrast (used extensively for transmitted-light studies in biology, see, for example, YAMAMOTO and TAIRA[19831) transforms the invisibly small phase-angle shift caused by a small difference in height (or in optical properties) of an object and its surrounding into an amplitude (light-intensity) difference visible to the human eye. This is effected by retarding (positive phase contrast) or accelerating (negative phase contrast) a portion of the directly reflected wave by half a wavelength by inserting a phase platelet. Interference of this modified reflected wave from the phase object with the References:p , 1016.
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unmodified wave yields a grey contrast, i.e. the image of the phase object, in the field of view. Extremely small height differences (1-5 nm) become visible whereas in bright field a step height of approximately 100 nm is necessary for detection. Interference contrast uses polarized light and a prism arrangement which splits the polarized light into two beams of equal intensity, the directions of which diverge by a small angle. These two beams hit the surface at slightly different positions and, after reflection, interfere with each other after passing a crossed polarizer. Several technical arrangements are possible; those used in practice are the differential interference contrast due to Nomarski (generally known as Nomarski contrast) and Michelson interference ( C o w and BRADSHAW [1952], JEGLITSCHand MITSCHE[1967], PHILLIPS [1971], ROSENBERGER [1977] or HOLIK[1993]). If height differences exist at a specimen surface, the different levels appear in different colour hues which can be changed by shifting the prism. JEGLITSCH and MITSCHE[19671 have demonstrated applications of interference contrast with steel and cast iron as examples; other applications covering a multitude of materials are interesting as well (BENESOVSKY and I-GER [1965], SPES [1965], [1990]). BEYER[1974] and ROSENBERGER [1977] or SCHUMANN 23.6. Filters By the appropriate choice of illumination wavelengthsthe contrast between differently coloured constituents may be enhanced. Alternativelyto coloured glass, a movable-wedge interference filter allows monochromaticlight of any wavelength to be selected. The contrast obtained is especially suitable for black and white photography (where green light is usually used) and for automatic image-analyzing equipment with a black and white camera as the detector. 2.4. Special optical devices and accessories 2.4.1. Stereomicroscopy Optical stereomicroscopes (see, for example, WATTS[19821) are useful for rough surfaces up to lOOx magnification. A recent version allows vertical as well as 45" side viewing without tilting the object. Today, the larger depth of focus and wide range of magnification available with scanning electron microscopy has nearly completely replaced optical stereomicroscopy for studies of rough surfaces. Optical stereomicroscopes have kept their place in the metallographic laboratory for quick inspection and quality control purposes. If a higher resolution or a larger depth of focus than available with optical stereomicroscopes is needed, scanning light macrography or confocal microscopy (see 5 2.4.2) and particularly scanning electron microscopy (which is presently clearly the best choice, see 5 3.4.1) can be used. 2.4.2. Laser scanning and confocal microscopy The light scanning microscope dates back to an invention of YOUNGand ROBERTS [1951] called the Jlying spot microscope. A similar principle is used in scanning light macrography developed in 1968 and applied for imaging of fracture surfaces with a high depth of field (CLARKE[1987]). With the development of small gas lasers producing
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monochromatic and virtually ideal parallel light rays, a light spot of high intensity can be focussed on tiny areas of a polished or rough surface. The first laser scanning microet aZ. [1979], SHEPPARD[1987]) were soon followed by a scopes (BRAKFNHOFF commercial instrument built by Carl Zeiss, first in 1982. A major advantage is the application of the confocdprinciple first described and patented in 1961 (see MINSKY [1988]). By reducing the depth of focus to a very small dimension with an effective depth resolution of 50 to 20 pm, rough surfaces can be optically sectioned plane by plane. The consecutive images can then be put together to produce a micrograph with a high depth of resolution orders of magnitude larger than in the conventional microscope. The information can also be stored in a computer and, using available software, can be used for a thee-dimensional reconstruction of rough surfaces. The optical principles and instrumental design have repeatedly been discussed (see, for example, WILSONand SHEPPARD [1984], BOYDE[1985, 19901, VANDERVOORTetaZ. [1989], or TURNER and SZAROWSKY[19931). Applications of the laser scanning microscope in materials science et al. E19911 and TURNER and and other fields have recently been reviewed by ELSSNER SZAROWSKY [19931.
2.43. Scanning near-field optical microscopy As discussed in 5 3 and 4, the scanning principle is used in many ways to increase the resolution up to the atomic level. With the upcoming of nanoscale science and technology, nano-optics, Le., microscopes which collect light from extremely localized sources by fibre optics and produce images with a resolution in the range of 20-100 nrn with a theoretical limit of approx. 5 nm, were developed. Though the principle idea dates back to 1928, the systematic development started only in the 80’s resulting in the socalled scanning near-feZd optical microscope (SNOM) which can be regarded as a [1992a,b]) described in member of the super-resolutionmicroscopes (WICKMARASINGHE 0 4. The resolution of this new type of microscope is not limited by the AbbC limit, i.e., it is not determined by the wave length of light but by the size of the probe aperture. A review on near-field microscopy, nano-optics and SNOM was recently published by POHL[19921. 2.4.4. High-temperature microscopy Heating stages for high-temperatureoptical microscopy need a long-working-distance objective (e.g. a reflecting objective) that will offer the necessary oxidation protection. They are also on the way to obsolescence since stages for scanning electron microscopy are now available and used much more often than optical hot stages. Reviews of earlier work which often resulted in important new information on the transformation behaviour of metallic alloys, on melting and on other kinetic processes have been published by REINACHER[1965], L~ZMSKIJ [1961], MITSCHE et al. [1964,1969], MODM and MODIN [1973], KULMBURG et al. 119741, METALS HANDBOOK [1985] or SCHUMANN [1990]. Besides the technical problems which now seem under better control (MUGGLIand MC CRONE[1977], HOEKSTRA and BRUIS[1979]), the question remains unsettled whether processes observed on the polished surface take place in the bulk qualitatively or quantitatively in the same way. References:p . 1016.
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2.4.5. Television cameras First attached to the optical microscope for demonstration purposes, television cameras, as was soon realized by an English company, could be adapted for quantitative analyses of microstructural images by adding a voltage threshold and a few electronic components. This started a rapid development of quantitative television microscopes. The modern versions are now the most powerful automatic devices for image analysis, details of which will be discussed below (07.1). 2.4.6. Microphotometry and ellipsometry Only twenty years ago, microscope photometers were introduced in metallography after having been used for some time in ore microscopy. Several highly sensitive commercial instruments are now available which use photomultipliers to register the intensity of reflected light (usually compared to a standard). Identification of small microstructural components (e.g. nonmetallic inclusions), detection of surface damage, orientation and concentration analysis, and sensitive detection of phase boundaries in quantitative image analysis were prominent early applications (MITSCHEand SCHEIDL [1964] and PETZOWand KNOSP [1973]). Measurement of optical constants, e.g., for contrast calculations in interferencdayer contrasting, have become an important domain of photometry (see 0 2.2.5). Differential reflectometry is an interesting but seldomly used surface analytical technique filling the gap between X-ray diffraction and the ion- and electron probes discussed in 0 6. Numerous applications (study of ordering in solid solutions, of corrosion phenomena and of lattice defect) have been reported by HUMMEL [19831. Another sensitive reflection technique is eZlipsometry (see, for example, VISSCHER [19731 and AZZAMand BMHARA [19771) which uses the phase shift of linearly polarized light for unambiguous determination of optical constants and for following the growth kinetics of thin surface films down to atomic dimensions. 2.4.7. Interferometry One of the most useful techniques for measuring the height of surface steps and other topographic features with high resolution (routinely 10-100 nm in reflection) is interferometry (KELLER [1977]). For double-beam int@erometry, the same set-up as for interference contrast (8 2.3.5) is used. The prism arrangement is set in such a way that interference of the two beams after reflection at an oblique plane surface causes a linear sinusoidal fluctuation of brightness which to the human eye appears as a series of parallel black stripes. If the surface is not plane, the stripes are distorted or set off (at steps). The width of the stripes can be reduced by a more difficult multiple-beam inter$erence arrangement and their distance from each other, corresponding to half the wavelength in height, can be suitably chosen by adjusting the prisms. (For interference contrast as described in 0 2.3.5 infinite stripe distance, Le., only one brightness minimum, is used for illumination.) A large variety of microinterferometers are available (see, for example PHILLIPS [1971], MODINand MODIN[1973], BEYER[1974] and KELLER[1977]). By measuring stripe distances and set-offs, height differences are determined. Also, angles of surface tilt can be measured with high accuracy. Among the applications in metallography have been studies of crystal growth kinetics (e.g., NANEV[1981]), of
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grain-boundary grooving (MYKURA [1955]) and of surface films and surface roughness (for references see BEYER[1974]). A recent application of interferometry is the characterization of targets for inertial-confinement fusion (FOREMANet al. [1993]), and special devices have been developed for this purpose.
2.4.8. Microhardness As a local mechanical test, indentation hardness measurements using low loads (5-500 mN) are carried out in microscopically small regions. Micrometer eyepieces or, more often, specially designed arrangements which keep the image and the scale sharp simultaneously are used. Detailed specifications for microhardness testing are found in and BERGMANN [1969], GAHM literature (Mom [1956], BUCKLE[1965], SEMLITSCH 119691, DENGEL[1973], MORNHEIM[1977], SCHUMANN [1990], TATE[1993], among others). Microhardness indentations are often used as markers for measuring distances and for accessing surface deformations induced subsequent to the indentations. More recently, ultra-hardness testing with loads in the range of 0.5 to 5 mN was developed. However, the resolution of optical microscopy is at its limits as the indentation size drops below 1 p m and the scanning electron microscope (see, for example, ZAHN[19921) or depth measurements (as with instruments called mechanical microprobe or nanoindenter; see, for example, OLIVER[1993]) must be used.
3. Scanning electron microscopy One of the most versatile instruments for microstructural investigations is the scanning electron microscope (SEM). Compared to the optical microscope it not only expands the resolution range by at least one order of magnitude (useful magnifications beyond 104x) and the depth of field by two orders of magnitude (ranging from 1 p m at lo4x to 2 mm at 10X) but also offers a series of possibilities for image formation which are more or less easy to interpret and reveal a clear picture of plane cross-sections as well as three-dimensional surfaces such as, for example, fracture surfaces, deep-etched or corroded surfaces and surfaces of porous materials.
3.1. Basic features of scanning electron microscopy Excellent monographs and handbook articles on the physical fundamentals, the various techniques, the instrumental details and the application of scanning electron microscopy are available (e.g., SEILERE19681, HEYWOOD[1971], HEARLEet al. [1972], OATLEY [1972], REIMER and PFEFFERKORN [1977], LORETTO[1984], EXNER[1985], VERHOEVEN [1986], NEWBURY et al. [1986], GABRIEL [1987, 1992a1, CHESCOE and GOODHEW 119901, LYMAN [1990], LYMANet al. [1990], JOY [1992], WELLS[19931, LIFSHIN[1993b], GOLDSTEINet al. [1994]), and a large number of various types of commercial instruments are in daily use in almost every materials research and development laboratory around the world. The principle of the scanning electron microscope is rather simple: An electron beam scans the surface of the sample in the same way as in a cathode-ray tube which is used References: p. 1016.
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to display the image. The electrons are emitted usually from a heated tungsten cathode. For higher electron currents (about 10 to 100 times that of a W electron gun) yielding a higher brightness, a lanthanum hexaboride (LaB6)cathode may be used. The beam is focussed at the surface to a small diameter (approx. 10 to 100 nm). More recently, fieldemission cathodes became commercially available which provide high-resolution images by reducing the probe diameter further and increasing the brightness by another 100 times needing a higher vacuum (lo-’ Pa as compared to 10” Pa for the thermionic guns). The acceleration voltage ranges between 1 and 50 kV while the current through the surface is in the order of 10 PA. The magnification is increased by reducing the current in the deflection coils; the normal range is 1-50 OOO x. The SEM image is written on a high-quality screen (cathoderay tube) with an image resolution typically of 0.1 mm which, at 10 000 times magnification, corresponds to 10 nm resolution on the specimen. The brightness of each picture point is determined by the detector signal which is of the order of a few PA and must be amplified by direct electron multiplication or by conventional amplifiers. Noisy background due to high amplification is the major cause of unclear images. The dramatic improvement of image quality and resolution during the three decades of commercial development is due to improvement of all components of the microscope (more sensitive detectors, more powerful cathodes and more precise scanning devices) and to noise reduction in all electronic devices. Figure 7 shows the interaction between the primary electron beam and the sample: Various kinds of radiation are emitted which, when collected in a suitable detector, can be used for image formation. Table 2 gives some data important for imaging metallic microstructures. By far the most popular techniques are the secondary electron mode (SE) revealing surface topography and, in advantageous situations, also atomic-number and crystal-orientation contrast with high resolution and information from a small depth below the surface, and the backscattered electron mode (BE) which gives topographic as well as pronounced materials contrast from a much larger depth and width and therefore reduced resolution. Figure 8a shows a typical SE image, fig, 8b a typical BE image of polished metal surfaces. Detector electronics can be adjusted so as to respond to either SE or BE electrons with their quite different energy distributions. Back-scattered electrons (BE) are produced by single large-angle as well as by multiple small-angle elastic scattering events. In the 10-20 kV range of incident beam voltage, approximately 50% of each type leave the surface with a wide spectrum of energies in a range higher than 50 eV. As the atomic number of the material hit by the incident beam decreases, a smaller number of electrons are back-scattered and more energy is lost. In materials with high atomic number, a large number of electrons is back-scattered by atoms close to the surface, with little change in energy. Thus, yield, energy spectrum and depth of escape of back-scattered electrons are directly related to the atomic number of the material (fig. 9, see also Q 3.3.2). Low-energy secondary electrons (SE) are formed by interaction of the primary electrons with loosely bound atomic electrons. The energy distribution of secondary electrons depends on the primary energy of incident electrons, the number of outer-shell electrons, the atomic radius and, most pronouncedly, on the surface barrier of the
Fig. 7. T y p e s of electron-beam-excited electrons and radiation (schematic) and depth of information in the scanning electron microscope. (The fluorescence emission can come from as deep as 100 pm.)
material. The energy of secondary electrons is in the range of 0 to 10 eV with a pronounced maximum at around 3 eV. At 50 eV the frequency of SEs approaches zero, i.e., there is no overlap with the energy spectrum of back scattered electrons. However, high-energy BEs excite SEs as well which contributes to noise and decreases lateral as well as depth resolution (see also $3.3.1). The probability that low-energy secondary electrons will escape from the surface decreases exponentially with the depth of their generation. More than half of the total yield is emitted within a depth of about 0.5 nm. The yield is not strongly dependent on atomic number (for Au it is higher only by a factor of two than for C, see fig. 9). The main factor for secondary electron yield is the angle between incident beam and sample surface. Since the envelope of the excited volume (see fig. 7) moves closer to the surface when the beam hits the surface at a small angle, SE yield increases. Thus, variations of surface inclination cause pronounced changes of SE yield (fig. 9). This fact is used for obtaining information on surface topography (3 3.3.1). A series of techniques are available to modify the signals in order to obtain better or additional information, as, for example, black-level suppression (i.e. differential amplification which distributes the contrast over the full range of the cathode-ray tube or the photographic film) or nonlinear amplijication (contrast enhancementby preferential contrast expansion at either end of the gray scale, improving the visibility of features in otherwise dark holes). In Y-modulation,the CRT (cathode-ray tube) beam is deflected proportionately to the detector RHerences: p , 1016.
Table 2 Physical effects producing radiation, detector types and detected signals used for imaging and analyzing metal surfaces in the scanning electron microscope. Detected signal
Type of detector
Information
Basic effects
Secondary electrons (SE)
Scintillator photomultiplier with Faraday cage
Surface topography, material contrast, crystal orientation constrast
Back-scattered electrons (BE)
Solid state or scintillator photomultiplier
Material composition, topography, crystal orientation
Maximum resolution
Minimum depth of information
Remarks
SE yield depends 5-20 nm strongly on surface tilt and weakly on atomic number and crystal orientation
1-2 nm
Background due to SE excited by BE reduces resolution and enlarges depth of information. Material contrast can be suppressed by superimposing the inverted BE signal
BE yield depends on atomic number, increasing for heavier elements
0.1-1 p m
0.1-1 p m depending on primary electron energy (acceleration voltage)
Topological contrast can be suppressed by a ring-shaped detector. Higher resolution can be obtained by using an energy filter
Specimen current No external detec(absorbed or target tor necessary current)
Complementary contrast BE yield results in to BE corresponding electrical current
0.1-1 p m
As with BE
Conventional amplification difficult and noisy. Used in early development of SEM, may revive with better amplifiers
X-rays
Semiconductor detector
Element distribution
Emission of characteristic radiation by electron bombardment
-1 p m
>lpm
Element analysis by spectrometers, X-ray intensity images with point density corresponding to element concentration
Cathodeluminescence
photomultiplier with mirror
Detection of nonmetallic and semiconducting phases
Emission of photons by electron bombardment
03-100 p m
0.5-100 urn
Applicable to metallic materials in rare cases only
r
*
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I
!
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‘u I
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30 kV , normal incidence
0
I
I
20
LO
I
60
Atomic number
I
80 100 90
60
30
0
Angle of tilt , deg.
Fig. 9. Backscattering coeficient and secondary e l m n yield as a function of atomic number of tht material hit by the dectron beam (normal incidence). and as a function of inclination of surface with respect to he incident beam (schematic. aftcr Goand YAKOWTTZ [ 19773).
signal (fig. 8c) allowing detection of low contrast not apparent to the eye in the intensitymodulated image. If one of these signal transformations is uqed, its limitations and deficiencies must be considered in interpreting the image, otherwise artefacts may provide erroneous information. Some practical aspects of the more frequently used techniques for investigation in the materials laboratory have been reviewed by HILLMER[1979a], EXNER [ 19851, LYMAN et af. [ 19901 or GABRIEL [1990b], GOLDSTEIN et af. [ 19941, among others.
32. Specimen preparation The major advantage of scanning electron microscopy is that in many cases little effort is necessary for specimen preparation. Practical experience has been reviewed for example by NEWBURYand Y ~ ~ o w r r[1977a], z by HILLMER[1979b], EXNER[1985] or NEWBURYet af. [1986]. and, more recently. by G,mweL [ 1992bl and GOLDS~EIN er af. [1994]. Clean, deformation-free cross-sections and clean ftacture surfaces of metallic materials can be investigated directly. Nonconducting materials must be coated by a thin conducting layer (10-100 nm, usually gold applied by sputtering or carbon and metals applied by high-vacuum evaporation) to avoid charging effects. Oxides caused by longtime exposure to the atmosphere or by high-temperature reaction can be removed electrolytically (YU~AWCHand HUGHES [1978]) or by hydrogen reduction (MADESKI [ 19801). Cross-sections are prepared in exactly the same way as for light microscopy but
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care must be taken in cleaning since residual polishing liquids or etchants trapped in pores or cracks cause contamination of the surface when putting the sample into the vacuum chamber. Organic mounting resins and any other hydrocarbons (grease etc.) must be removed in order to keep contamination by carbon and cracking products as small as possible. The shape of microstructural features in multiphase alloys can advantageously be studied after deep etching to remove some of the matrix metal (see, for example, et al. [1980], MADESKI [1980], HUN-DE and JING-Ym MCCALL[1973], FREMUNT [1980], or PAUL and MORRLE[198l]). Figure 8a shows an aluminium-siliconalloy with part of the aluminium matrix removed by etching. The octahedral shape of the primary silicon crystals and the complicated arrangement of the eutectic silicon lamellae would not show up as clearly in a plane cross-section. Etching is not necessary when material (atomic-number)contmst is utilized for image formation: Differences of atomic number show up as differences in brightness, the phase containing lighter elements appearing darker. Unavoidable height differences of hard and soft phases after mechanical polishing result in additional and usually unwanted topographic effects. However, differences in surface roughness after polishing have been shown recently to produce pronounced contrast between hard and soft phases if contamination is avoided (see Q 3.3.1). Weak contrast mechanisms, as for example channelling electron contrast or magnetic contrast (see Q 3.3.3 and $3.3.4 respectively), are impossible to detect in the presence of a deformed layer or of topographic features. Therefore, deformation-free and plane cross-sections must be prepared by careful polishing when studying microstructures by means of these contrast mechanisms. Sample preparation by electron polishing or by ion polishing is advantageous for achieving maximum contrast [1987]). and avoiding artefacts in these cases (NAKAGAWA 3.3. Typical forms of contrast 3.3.1. Topographiccontrast The most pronounced contrast effects result from the dependence of secondary electron (SE) yield, back-scattering coefficient and detector collection efficiency on the angle between the surface element and the primary electron beam (see $3.1) or the detector position, respectively. The resulting contrast (see fig. 8a) is analogous to an optical image where the light comes from the detector and the observation direction is that of the incident electron beam (NEWBURY [1977], REI= and PFEFFIBKORN [1977] or JOY[1992]). The stereoscopic effect is enhanced by shadows in regions which are hidden from the detector and by enhanced emission at edges. These latter effects are disadvantageous when deep holes and cracks or transparent edges are present. On the other hand, the edge effect can be advantageouslybe used to differentiate between phases of similar atomic weight for which material contrast ($ 3.3.2) is not pronounced. This has been demonstrated for an aluminium silicon alloy (fig. Sd), e.g., by PAULand BAUER [1983], see also EXNER[1985]. The high spatial resolution (both transversal and in depth) obtained with secondary electrons excited by the incident beam is disturbed by secondary electrons excited by the back-scattered electrons coming from deeper in the specimen; thus the clear topographic References: p. I016
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image can be obscured by atomic-number contrast and the resolution is reduced pronouncedly if special measures an? not taken. There are several ways to reduce these effects (BLASCHKE[1979]). Superposition of the SE signal and the inverted BE signal seems to reduce interference from deep specimen layers almost completely (VOLBERT [I9811 and PFEPPERKoRN and BLASCHKE [1982]). The vast majority of applications of the SEM in materials science (as well as other fields) rely on this mode while all other types of contrast together are used much less (estimated to less than 10 percent). The major reason for this fact is the high resolution obtained with SE micrography ranging from 5 to 10 nm in conventional instruments to 1 nm and below in the currently most advanced (ion-emission gun equipped) scanning electron microscopes.
33.2. Material (atomic number) contrast As discussed in 03.1, the yield of back-scattered electrons as well as that of secondary electrons is a function of the atomic number Z (fig. 9). Material contrast is useful for qualitative identification of phases and is especially suitable for quantitative evaluation of microstructural geometq by stereological techniques (see 0 7). However, the atomic number contrast is usually obscured by topographic contrast (fig. 9) and, for secondary electrons, is sufficiently strong only in favourable cases, Le., for ideally flat surfaces. For backscattered electrons, the topographic contrast can be greatly reduced even if the surface is rough by using ring-shaped detectors. Edge effects may still be a problem, and ideally flat polishing is necessary for difficult specimens in which the phases have similar compositions or consist of elements which are near neighbours in the periodic table. A difference of average atomic number, AZc1 (i.e., even mixtures of nearest neighbours in the periodic table) is sufficient for slight material contrast, at least in the lower range of atomic numbers where the contrast is more pronounced (see fig. 9). Figure 8b shows an example of strong BE material contrast. The limitation for many problems in physical metallurgy is the lack in resolution due to the large depth of information (see fig. 7) which, for highly dispersed phases, results in blurred edges and detection of features not intersected by the cross-section but close below it. However, if phase interfaces are normal to the cross-section, excellent resolution is obtained (fig. 8e). By reducing the energy of the primary beam, disturbing effects can be reduced. Table 3 shows the depth of information as a function of primary voltage. Even at low voltages, the depth of information from metal surfaces is at least one order of magnitude larger for backscattered electrons than for secondary electrons. Material contrast is reduced at lower voltages and additional contrast-enhancementtechniques by surface treatment have been worked out for individual cases. A large number of problems in physical metallurgy and materials science have been solved using the BE mode, particularly since the early semiconductor-based detectors were substituted by optimized scintillator-light guide-photomultiplier (SLP) detectors (Robinson or YAG type). Some useful applications of these types of detectors were reported by BAUERand EGG [1984], for example.
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Surface microscopy
Table 3 Information depth obtained with secondary and backscattered electrons for a few elements as a function of acceleration voltage of the primary electron beam, and with secondary electrons (according to SEILER[I9681 and REIMER and PF!SFERKORN[1977]). Element
C Al
cu Ag Au
Atomic number
6 13 29 47 79
Information depth (nm)
Density (Mg/m3)
2.3 2.7 8.9 10.5 19.3
back-scaaered electrons 5 keV
10 keV
20 keV
30keV
50 keV
330 120
970 360 110 90 50
2800 1050 320 270 150
5300 1950 590 500 270
11 600 4290 1300 1100 600
40
30 20
secondary electrons 10
1.2
0.5 1.0 1.8
3.3.3. Electron channellingcontrast, electron channellingand Kossel patterns Electron channelling pattern (ECP) arise because of the fact that the primary electrons penetrate into the crystal to a depth which depends on the atomic packing density along different crystallographic directions. If the electrons follow the channels between rows of atoms, their re-escaping probability becomes low. A large number of Bragg conditions are satisfied with quite small angufar variations. By rocking the primary electron beam about a point in the sample, diffraction lines from a small area (e10 pm) are obtained. These selected-area electron channelling pattern (SACP) are similar to Kikuchi patterns obtained in transmission electron microscopy (ch. 11, 0 3.2) and are therefore often called pseudo-Kikuchipatterns. Information on the crystal orientation and crystal perfection, grain boundaries, twins and other crystallographic features is obtained from minute regions of a surface layer less than 50 nm thick. An example is shown in fig. 8f. Extensive discussions of electron channelling contrast show its usefulness for and YAKOWITZ[1977b], PAYNE materials investigations WCCALL[1973], NEWBURY [1986], NEWBTJRY et al. [1986], JOY[1992], RIEDLEet al. [1994]). [1982], VERHOEVEN In the investigation of fracture surfaces, bend contours were observed by ECP @AVIDSON [1974]). The study of orientation relationships of twins and of local textures are prominent examples of successful application for SACP. Though more often used in transmission or with higher primary energy in the electron microprobe analyzer, Kossel pattern (named after the explorer of this effect) are another efficient means to analyze the structu~of small volumes in the order of pm3 in the scanning electron microscope. Kossel lines arise by interference of X-rays excited by the focussed electron beam at the surrounding lattice planes. On the film, lines with bright-dark fine structure are obtained which are easily interpreted and give useful information on orientation, lattice parameter and lattice deformation after distortion (BR~MMER and NIEBER[1975], DINGLEYet al. [1982] or NEWBURY et al. [1986]). An interesting study of orientation relationships of new grains formed by recrystallization of aluminium, using the transmission Kossel technique in a modified electron and CAHN[1972] is discussed in ch. 28, § 3.3. microprobe analyzer by DOHERTY References: p. 1016.
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A new type of microscopy called “orientation image microscopy” (AD- et al. 119931 and WRIGHT[1993]) uses backscattered Kichuchi patterns to determine lattice orientation in localized regions (approximately0.2 pm). Rapid indexing of the diffraction patterns by computer analysis is combined with computer-controlled movement of the electron beam or the sample stage. Thus, a computer image of the surface is obtained displaying orientation differences of the grains in polycrystalline material with a resolution of 1 degree misorientation. This technique was shown to yield information exceeding that by X-ray texture analysis and stereological grain size analysis as demonstrated for rolled aluminium and iron-silicon alloys ( h m s et al. [19931). The above-mentioned methods have recently been discussed and critically compared by RANDLE[1993].
33.4. Magnetic contrast Magnetic fields of ferromagnetic crystals can affect the interaction of the primary beam or the resulting emission. Type-Zmugneticcontrast uses the deflection of the highly directional electrons by the leakage field which may amount up to 20% for materials with strong fields such as cobalt. The resolution with which the boundaries of magnetic domains can be picked up is only of the order of several pm owing to the diffuse nature of leakage fields. Type-II magnetic contrast arises from the deflection of primary electrons by the Lorentz force inside the crystal. Magnetic domains appear in light-dark contrast due to differences in backscattering coefficient, with a strong tilt-dependence, and can be enhanced by filtering using high-energy BE only, with a resolution in the and order of 100 nm. (For details and applications of magnetic contrast see NEWBURY Ymowrrz [1977b], REIMER and F’FEWERKORN [1977], ELSBROCKand BALK 119821, VANDER VOORT[1985c], NEWBURY et al. [1986], VERHOEWN [1986] or JOY [1992]). 33.5. Charge collection microscopy and electron-beam-induced current measurements Secondary electron emission is sensitive to surface potentials and electric field gradients. Therefore, in samples where local variations of potentials appear, regions of differing brightness will show up in the SE image. This effect can be enhanced by a positive or negative bias of a few volts which impede or enhance SE emission, respectively. Usually (though incorrectly) called voltage contrast, this technique provides another way to produce images in the SEM. EBIC (electron-beam induced current) and EMF (electromotive force) measurements depend on similar phenomena (creation of excess electrodhole pairs by the electron beam) and give useful information on diffusion length and lifetime of minority carriers in semiconductors. The term “charge collection microscope” (LEAMY[1982]) has been coined for these techniques which make the scanning electron microscope one of the most versatile tools for the characterization of semiconductor materials and failure analysis in the electronic industry (see, for example, NEWBURY et al. [1986], SCHAPERand B ~ S E [1985], L or VERHOEWN [1986]). 33.6. X-ray mapping Emission of fluorescent X-rays can be picked up by a solid-state detector and
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analysed with respect to its energy. For producing an image (i.e., a so-called X-ray map), an energy window is set which selects X-ray quanta typical for a specific element. The efficiency of the X-ray process is small, typically lo4 photonshncident electron, and only a few thousand photons reach the detector per second. The beam of the display cathoderay tube receives an intensity pulse whenever a certain preset number of X-ray quanta of preset energy reaches the detector. In this way, regions with a high concentration of the corresponding element appear bright while others appear dark. If this is done consecutively for several elements, X-ray maps of the distribution of these elements in the region scanned by the electron beam are obtained. Owing to the large depth of X-ray emission (see fig. 7) and the large background noise due to Bremsstrahlung, the resolution of X-ray maps is limited to a few pm. Nevertheless, X-ray maps are used extensively for demonstrating element distributionsin metallic materials allowing a quick and unambiguous identification of phases and concentration gradients caused, for example, by segregation during solidification or by diffusion at the interface between phases which are not at thermodynamic equilibrium.
3.3.7. Cathodoluminescence The recombination of electrodhole pairs causes a release of energy which is emitted from the sampie. Part of the emission is in the visible or infrared part of the spectrum and can be collected and amplified to form an image. The ability of some materials to emit this long-wave radiation is known as cathodoluminescence(CL). Only few examples have been reported with metallic materials where radiation is caused by surface plasmon effects (e.g., SIEBER[1982] or RICHARDS and TRIGGS [1982]). Ceramic materials and semiconductors have been occasionally studied by CL, for example A1203-Zr02,MgO and 230, ceramics. S i c fibres or diamond (CZERNUSKA and PAGE[1985], BROWNet al. [1988] and PAGE[1993] and GaAs (DAVIDSON[1974], see also PFEFPERKoRN etal. [1980], HOLTand SABA[1985], NEWBURYet a!. [1986] or JOY [1992]). Mineralogical and particularly biological applications are frequent (see, for example, REIMERand PFEFTERKORN [19771, GOLDSTEIN and YAKOWITZ [ 19771, GOLDSTEINet al. [1994]). 3A. Accessory equipment
The large specimen chambers of most commercial instruments allow special stages to be mounted by which various experiments can be carried out to yield additional information on the nature of metallic microstructures and their development. 3.4.1. Stereomicroscopy Simple tilting stages or goniometer stages allow stereopair micrographs to be taken which give an excellent three-dimensional impression of rough surfaces when viewed in a stereoscope, by the “anaglyphe” method using a red and a green filter for the two images or by viewing with polarized Light. The tilt angle and the viewing distance determine the subjective impression of depth. Quantitative evaluation of the parallax yields accurate data on the z coordinates at regularly spaced or arbitrarily chosen x-y positions. These data can then can be combined to construct height profiles or height References: p. 1016.
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maps (e.g., BLASCHKE and WALTINGER [1971], BOYDE[1973], ARZT and FISCHMEISTER [1979], BAUERand EXNER[1981], BEATONand FILSHIE [1983], GABRIEL [1987, 1992a1, LYMANet aZ. [1990], LIENKAMPand KUNAVER[ 19941). This technique can be used to obtain height differences destruction free and with high resolution which will often compare favourably with stylus-type instruments) and characteristic parameters (roughness indices, distribution of tilt angles for surface elements, etc.) for fracture surfaces and other rough surfaces. Figure 10 shows the profile map of a ceramic fracture surface together with the tilt-angle distribution. Further applications for solving problems in materials science have been reported for example by EXNEXand ERIPAN [1985], SIGLand EXNER [1989], FEIJOO and EXNER[1991], LIENKAMP and EXNER[1994], and by many others.
Dynamic and non-ambient-temperatureSEM Large depth of focus and the possibility of rapidly changing the magnification in combination with mechanical or low- and high-temperature stages are prerequisites for continuous observation of specimens subject to applied stress, magnetic or electric fields, chemical reaction, and various effects due to cooling or heating. A multitude of phenomena in physical metallurgy have been studied, including fatigue crack growth in L crack propagation and delamination in hardferrous alloys (SCHAPERand B ~ S E[1985]), layer coated steel during cyclic bending (WETZIGet al. [1984]), cracking of surface layers during heat treatment (GRABATINet aZ. [1983]), deformation of superplastic lead-tin eutectic, the effect of stress and magnetic field on the configuration of magnetic and YAKOWITZ[ 1977b]), solid-state domains in an iron-silicon alloy (NEWBURY [1972]), or liquid-phase sintering (RIEGGW et al. [1980]). Videosintering (FULRATH recording is an ideal way to register events of interest, and special devices have been developed to record fast processes such as cracking or martensitic transformation with much higher resolution than is possible with optical microscopy. 3.4.2.
4. Scanning tunneling, atomic force and related microscopies Only a little more than a decade ago, the application of the concept of electron tunneling known since the early 1920's led to the development of a new type of microscope, the etaZ. [1982], BINNIGand ROHFER [1982]. scanning tunneling microscope (STM)by BINNIG STM is not just another type of high-resolution electron microscopy. A whole class of novel techniques based on an astonishingly simple concept allows the detection of almost any kind of interaction on a nanometer or even subnanometer scale. Some of these techniques (scanning tunneling microscopy, scanning tunneling spectroscopy, atomic force microscopy and magnetic force microscopy) have rapidly evolved into routine tools for surface characterization, and new techniques based on similar principles are invented each year. In the decade of its development, dramatic impacts in fields as diverse as materials science, biology or electro-chemistry (among many others) have been made, and the papers published exceeded the thousand some time ago with a rapidly increasing rate of several hundreds each year. Review papers and monographs became available (e.g., HANSMAand TERSOFF[1987],
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Fig. 10. Fracture-surface analysis by instrumented stereoscopy: (a) scanning electron micrograph evaluated by stemmetry; (b) line profiles; (c) distribution of profile line length as a function of tilt angle of surface profiles perpendicular to crack direction. (After BAUERetal. [1982].)
BINNIGand ROHRER[1987], GARCIA[1987], WICKRAMASINGHE [1989, 1992a,b, 19931, BONNELL[1990, 1993a,b], SARID[1991], G ~ H E R O Dand T WIESENDANGER [1992], WIESENDANGER and G ~ H E R O D[1992], T STROSCIOand KAISER [1993], DINARDO [1994a, b], see also Further reading). The rapid development makes it difficult to follow up all issues of interest, even in a single field like physical metallurgy, and only some basic principles and a few examples of typical applications will be presented in the following. References: p . 1016.
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4.1. Basic principles and capabilities An infinite potential energy barrier exists between two surfaces separated by a large distance. This barrier becomes finite when the surfaces are in close proximity, typically in the order of a few nanometers, and, when a voltage is applied, a statistically significant number of electrons can penetrate the energy barrier. This effect was called electron tunneling and has been theoretically very well understood and quantitatively described on the basis of the classical wave particle dualism concept of quantum mechanics though a microscopic quantum-mechanicaltheory taking geometric effects into account is needed in the case of atomic resolution (see, for example, TERSOFF and HAMANN [1985], TERSOFF[1993], TERSOFF and LANG[1993]). The overall scheme of a scanning tunneling microscope is shown in fig. 11. To produce images of a surface at very high resolutions, a sharp metal tip, often made of tungsten or platinum, is scanned over the surface of the sample under investigation. The motion of the tip is controlled by piemelectronic elements (usually lead zirconium titanate) in x and y directions by applying simple linear voltage ramps. A third element controlling the z position of the tip can be operated in two modes: in ‘%onstantcurrent” imaging, the tip follows a constant charge density contour representing the surface profile by keeping tunneling current constant. This is achieved by a computer-controlled feedback mechanism. The signal required to alter the vertical tip position produces the image. Altering the level of constant current (or the voltage applied between the tip and the surface) produces contours of different charge densities. In the second mode, “constant height” operation, the tip is held at a fixed z position and, while scanning the surface, changes its relative position to the sample surface. Thus, the sample/tip separation and, as a consequence, the tunneling current vary which is used as signal. As the tunneling currents are in the range of 10+ A, the signal must be amplified by 7 to 10 orders of magnitude. Effective shielding, avoiding mechanical vibrations by combining a series of damping strategies, careful sample preparation and a clean and sharp tip are critical for obtaining optimum resolution and unambiguous interpretation of the images. In contrast to conventional electron microscopy, scanning tunneling microscopy can not only be performed in vacuum but also in air or even in liquids.
42. Atomic force microscopy Only a little later than STM, in 1986, BINNIGet aZ. [1986] developed the atomic force microscope (AFM)capable of measuring or imaging the forces between the sample surface and the tip. This is achieved by attaching the tip to a cantilever. Forces acting on the tip produce a deflection which can be measured down to m corresponding to N. (At the time this article was written, commercial instruforces in the order of ments have force and depth resolutions of approx. N and 4 x lo-” m, respectively). Since these forces can be measured locally with lateral resolutions in the order of 0.1 nm, the forces between groups of atoms or even single atoms can be measured, and force microscopy has rapidly developed to a quantitative probe of all kinds of surface forces as electrostatic, capacitive, magnetic, adhesive, capillary, frictional or van der Waals (SARID[1991], MEYERand HEINZELMANN [1992], HUES et aZ. [1993], BURNHAM
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Fig. 11. Overall scheme of the scanning tunneling microscope (after BONNELL[19!93c]).
and COLTON[1993]). The theoretical treatment of AFM is a complex problem and is still in progress involving empirical potentials, molecular dynamics (ABRAHAM et al. [1989]), References: p . 1016.
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continuum elasticity theory (TOMANEK et al. [ 19891) and ab-initio calculations (for a recent review see MEYERand HEINZELMANN [1992]). Though there are a number of open questions in interpreting the major features of force distance curves and the nature of the imaging mechanisms as discussed by BURNHAMand COLTON[1993], atomic force microscopes are now commercially available at reasonable prices, capable to measure surface forces and nanomechanical properties of materials as well as to modify surfaces.
4.3. Tunneling spectroscopy Tunneling spectroscopy (TS) is another source of information available with the scanning tunneling microscope (see, for example, HAMERS[ 19931, STROSCIO and FEENSTRA [1993], BONNELL[1993b]) providing data relating to the local electronic structure of the surface. Compared with other surface spectroscopy techniques such as photoemission microscopy (UPS and IPS), electron energy loss spectroscopy (EELS) or infrared spectroscopy (IS and IRRAS) which provide information averaged over a large surface region (see 8 6), with TS it is possible to measure the electronic structure on an atom-by-atom basis. Surface irregularities as impurities, steps, defects or electronically inequivalent atoms in ordered structures can not only be seen by STM but also be correlated with their effect on local changes of the electronic structure. Point spectroscopy involves moving the tip to a feature of interest, modulating the tip voltage and recording the resulting change of tunneling current. In a second method for obtaining spectroscopic information, the tip voltage is modulated at high frequency with respect to the time constant of the feedback controller and the current is recorded at discrete values of the tip voltage. Though well funded by bulk tunneling (see, for example, HANSMA [1982] or WOLF[1986]), proper analysis of the data is crucial to the use of this powerful technique and is presently still in the state of development.
4.4. Related scanning techniques The same scanning and feedback principles as in scanning tunneling microscopy can also be applied to use other types of interactions between the sample and the tip for characterizing the properties of surfaces and to produce images. In addition to atomic force microscopy (AFM), magnetic force microscopy (MFM), after the first successful attempts in 1987, has been developed as a technique of its own with commercial instruments becoming available and interesting results being reported (for a recent review see GR~~TTNER et al. [1992]). The principle is the same as that of the atomic force microscope, but the tip is made by electrochemical etching of ferromagnetic foils or wires, most often nickel, but also iron, cobalt or amorphous FeBSiC, or by coating the tungsten tip with a ferromagnetic thin film. The probe senses the magnetic stray field (which is not easily related to sample magnetisation and, therefore, may be a disadvantage of this technique compared to electron probe based techniques like scanning electron microscopic polarisation analysis (SEMPA)). Also, lateral resolution is routinely limited to 40 to 100 nm with optimum values of 10 to 20 nm. On the other hand, MFM can be operated in air and requires little or no sample preparation. Another modification of the force microscope is the electrostatic force microscope.
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A family of techniques yielding information on the structure of buried interfaces, is ballistic electron spectroscopic emission microscopy (BEEM), useful for studying the transport and scattering of electrons and holes in multilayer structures (see reviews by KAISERet al. [1993] or BELL et al. [1993]). Furthermore, numerous STM related techniques have been invented and demonstrated to yield specific information on materials [1992a,b, 19431, surfaces. Among those described in recent reviews (WICKRAMASINGHE POHLI:1992]), are the scanning noise microscope, the scanning tunnelingpotentiometer, the scanning photon microscope, the inverse photoemission microscope, near field thermal microscopy, scanning optical and acoustical probe microscopes, with more to come up in the future (for a list of more than 20 different techniques of scanning microscopy see WICKRAMASINGHE [1993], for details see the most recent publications on STM conferences). 4.5. Applications
The major reason for rapid acceptance of scanning tunneling microscopy is the fact that three-dimensional images of the real surface are obtained at unprecedented levels of lateral and vertical resolution. Under optimized sample and tip conditions, quantitative information about topography and electronic structure can be obtained for individual surface atoms as well as for groups of atoms, defects, etc. Thus, STM is not just another microscopic technique but, as discussed above, comprises a new class of surface analysis techniques not available when the last edition of this book was published. Applications are manifold, and, though the majority of research papers are aimed to demonstrate the opportunities of STM and related techniques under extreme conditions, applications to actual problems of physical metallurgy and materials science have been reported. Obviously, scanning tunneling microscopy is the ideal technique to study the arrangement of atoms on the surface of crystalline solids. This arrangement influences processes like epitaxial growth of thin films or catalytic reactions as well as surface properties like surface energy or electronic behaviour (see the recent review by UNERTL [1993]). Since surface atoms have dangling bonds, they rearrange in order to minimize 119881). The energy and form a structure which is said to be reconstructed (ZANGWILL reconstructions of the (111) and (100) surfaces of Si are probably the ones most often studied (see, for example, GIESSIBL[1995]. The (100) and the (110) surfaces of Ir, AM and Pt have similar reconstructions. Reduced atomic densities are observed for the (110) surfaces called missing row structure because every second row of atoms has been removed. Surprisingly, the atoms in the (111) surfaces show a higher density than in the et bulk. These and many other results have been reviewed by UNERTL[1993], OPIELKA aZ. [1993] or by STROSCIO and KAISER[1993]. An example of an STM image is shown in fig. 12. The local symmetry reflects the local electronic density of states. The defects in the periodic structure (upper left) are due to impurities. For the study of amorphous materials or where large single crystals do not exist, STM and related techniques are presently the best structural methods available. Their major limitations are that the subsurface structure cannot be detected and the atomic position cannot be determined with the same degree of precision as with diffraction techniques. These and other limitations of STM and AFM in surface crystallography are References: p. 1016.
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deposition and long-range ordered structures on the surface of etched Si, like those obtained in ultra-high vacuum, were observed. Images of magnetic recording media were produced by MFM and their recording behaviour was investigated. Laser-written domains in magnetically hard and soft materials as well as natural domains and Bloch wall segments in a wide range of magnetic materials were imaged, and magnetic surface forces were measured. Again, there are a number of limitations, some of which will certainly be overcome in future developments. The various spectra, obtained with tunneling spectroscopy, particularly tunneling currendapplied bias curves, have been analysed for their information on the correlation between the position and the electronic structure of surface atoms. With the development of nanotechnology more and more questions will seek answers which can only be given on the basis of techniques reading atomic resolution. With the exception of atomic force microscopy, the use of these techniques is rather specialised at present, and it remains to be awaited which of them will contribute most to solve materials science problems.
5. Other special techniques of suvace microscopy In addition to light microscopy and transmission and scanning electron microscopy used routinely in all fields of materials research, development and control, microstructures can be investigated by several more exotic image techniques. While some of these, such as scanning tunneling, atomic force, photoemission or field-ion microscopy, are of high interest for various advanced studies of material surfaces, others are still in the stage of experimentation, have been substituted by other techniques or are more useful in other fields of application like biology or mineralogy. Instruments capable of analyzing the chemical nature and the electronic state of surface atoms have been developed at a rapid rate in the last few years, utilizing all kinds of interaction with incident photons, electrons and ions. In $ 6 , a summary of these techniques in a classification with respect to exciting and emitted species is presented. The information obtained stems nearly exclusively from regions close to the surface and thus is decisively distinct from that furnished by bulk chemistry. If the lateral resolution is sufficiently good, the signals can also be used for surface imaging. A full account of all available techniques cannot be given in this context but short surveys of those being used to some extent in the investigation of metal surfaces in physical metallurgy will be given in this section.
5.1. Scanning acoustic and thermal wave microscopy Though based on early discoveries and inventions (for reviews on historical developments see WICKRAMASINGHE [1983], SOMEKH [1990] or BRIGGS[1992]), acoustic and thermal-wave microscopy are rather new developments (KORPEL et al. [1971], LEMONS and QUATE[1974], WICKRAMASINGHE etal. [1978], KOCH [1979], ASH [1980], ROSENCWAIG [1982] and MURPHY[1993]). References: p . 1016.
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Various types of signals and detectors have been used to form magnified images and, depending on the physical principle, the instruments have been called acoustic, thermalwave, photoacoustic, or electronacoustic microscopes or thermal-wave and thermographic microscopes, or thermoacoustic probes. Either a piezoelectric crystal is attached to the surface which emits acoustical signals, or alternatively, a laser or an electron beam are focussed on the surface resulting in periodic surface heating as the beam intensity is modulated and thermal and acoustic waves are sent through the sample. After reflection or scattering, the signals are picked up by a suitable acoustic or thermal detector (ultrasonic transducer, gas microphone, infrared detector) and transformed back into electrical signals. By applying the scanning principle, the efficiency of these types of microscopes can be fully utilized, The laser or electron beam is scanned across the field of view, and the signal received is used to modulate the brightness of a cathode-ray tube. The information furnished is different from that obtained with optical microscopes or with scanning electron microscopes in that it reveals sub-surface features such as, for example, pores and other defects, grain boundaries and microstructural components which differ from the surrounding matrix by their acoustic or thermal properties (density, specific heat, thermal conductivity).
5.1.1. Scanning laser acoustic microscopy The scanning laser acoustic microscope (SLAM), also termed scanning photoacousfjc microscope (SPAM) has now been developed to a commercially available instrument owing to its high versatility. In the range of high frequencies (1 to 10 GHz) the laser scans typically a field of 200 p m square, and the typical magnification is 1OOOx. The penetration depth is in the order of 100pm. Since at these high frequencies the wavelength is similar to that of visible light (3 GHz corresponding to 520 nm), the maximum resolution of the acoustic and the optical microscopes are comparable. At the other hand of the spectrum, at 1 Hz to 100 MHz, the resolution is low but the depth of information increases to the size of the sample, e.g., to 50 mm at 10 MHz where, at a typical magnification of 2 to 3 x , the scanning area is approx. 100 mm square. With the possibility of changing the frequency, the depth from which the information is obtained can be varied and, as with the confocal principle, three-dimensional images can be produced. Also, using stereographic or holographic type of image processing is useful not only to get 3D images but also to reduce the blemishing effects of superimposed features. Quantitative information on the depth of features below the surface, the thickness and composition of surface layers or the depths of cracks can be extracted (for a recent review see BRIGGS[1994]). Various useful applications of scanning acoustic microscopes have been reported in the field of materials as, for example, by WICKRAMASINGHE [1983], TAM[1986], DIESER and HERRMANN[1988], DIESERand MATTHAEI[1988], MATTHAEIet al. [1988a,b], OPIELKAet al. [1990], ARNOLD[1991], GILMORE [1993], MURPHY[1993] or BRIGGS [19941. With the commercially available instrumentation and higher resolutions to be achieved in the future by decreasing the noise and wavelength with extensive work under [1993]), acoustic way further, at research institutes as well as at industrial firms (GILMORE microscopy is a promising alternative for materials characterization and product testing.
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5.1.2. Thermal-wave microscopy With the electroacoustic principle excitation by pulsing the beam in a scanning electron microscope, a thermal-wave microscope is available by just attaching an additional detector to the SEM. The spatial resolution of such a device (also called thennoacoustic probe) is determined by the spot size of the incident beam, the modulation frequency and the thermal conductivity ranging from one p m or less for insulating materials at high frequencies (several GHz) to a few mm at 100 Hz. Owing to the fact that thermal waves are more sensitive to local variations and have a better resolution than acoustic imaging, there are numerous potential applications for the thermal-wave microscope as well. Nevertheless, this technique has not been routinely applied in materials investigations. Some applications of thermal-wave microscopy were reported by WmsCHm et al. [1989, 19931 and MURPHY [ 19931 among others.
5.2. Field-ion and field-electron microscopy Very high gradients of electric fields at the surface of a metal cause emission of electrons and ions. This is the basis of field-ion and field-electron microscopy.
5.2.1. Field-ion microscopy Extremely high resolutions, down to the atomic dimensions ( ~ 0 . 1 5nm) are achieved in field-ion microscopy (FIM) and the imaging atom probe (IAP). A large number of monographs (e.g., MULLERand TSONG[ 19691, BOWKETTand SMITH[ 19701, WAGNER [1982, 19851, SAKURAIet al. [1990], TSONG [1990] or MILLER et al. [1992] and handbook articles (e& EDELMANN 119801, SMITH119861, SEIDMAN [I9931 or CEREZO and SMITH[1994]) give a full account of these and the related spectrometric techniques. In field ion microscopy, noble gas atoms (usually helium, but also neon or argon) or hydrogen are ionized by tunneling of electrons at the cooled surface of a pointed metal tip. The ions are extracted from the surface by a high voltage and accelerated to hit a channel-plate converter. The secondary electrons are multiplied and emitted radially to a fluorescent screen. A high resolution image of the tip is obtained showing individual atoms and their arrangement. As a typical example, the image of a tungsten tip is shown in fig. 13. Ionization occurs more readily at terrace steps which appear bright and outline facets of low-surface-energy planes. Lattice defects cut by the tip surface such as dislocations, stacking faults, grain boundaries or antiphase boundaries in ordered structures are revealed. Vacancies and interstitials can be observed and their movement studied by taking photographs at certain time intervals. At the field strength produced at a pointed tip in the field ion microscope (approx. 5 x 10” Vlm at an applied voltage of 10 kV) it is also possible that atoms of the specimen material are ionized and are emitted from the surface and accelerated toward a detector the light output of which is recorded photographically. This mode of image generation is called field ion evaporation microscopy (removal of specimen material) in contrast to the field desorption microscope (removal of absorbed gas ions) described above. Field evaporation can be controlled with great precision by applying the potential as short high-voltage pulses. By using pulse times in the order of a few milliseconds, two to three References: p . 1016.
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Fig. 13. Field-ion micrograph (reduced in size for printing) of a single-crystal platinum tip, (100)-oriented with facets of low-index planes. Magnification approx. 35 x lo6. (From DOERRand OWNBY[1975].)
atoms can be removed at a time. The amount of the material analysed by the pulse technique is in the order of one million atoms per day, i.e. a volume of m3. The resolution is further increased by cooling the tip. With sharp tips, and at low temperatures, it is possible to image individual atoms but typically much larger areas with a higher ion output are observed in order to get a larger volume analysed, though at a decreased resolution.
5.2.2. Atom-probe field-ion microscopy The chemical nature of the ions pulled off from the tip surface by the high-voltage pulse can be identified by attaching a time-ofjlight (TOF) mass spectrometer. By positioning the tip with respect to an aperture hole in the fluorescent screen it is possible to focus each individual surface atom. In the detector system specially designed for this purpose, each ion arriving produces a single voltage pulse. A minicomputer controls a digital timer and analyses the data. Combining the time of flight information with the information on the location where the ion had been removed, it is possible to obtain an
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atomic resolution image of microstructural features and to measure the mass-to-charge ratio of single atoms from preselected regions of the specimen. The lateral resolution of this instrument called atom probe is in the order of a few tenths of a nanometer while the depth resolution depends on the interplanar spacing which, for high index planes, is in the order of hundreds of a nanometer. In another mode, called imagingatomprobe (IAP)maps can be produced for different kinds of elements by switching the detector on during the time when the species of interest arrive at the screen after the pulse was applied. These maps are similar to the Xray maps produced in the SEM with two decisive differences: The resolution of the atom probe maps in orders of magnitude better than that of X-rays. However, since in the imaging atom probe the surface atoms are removed by the pulsed voltage and time gating can only be performed for one species at a time, the maps observed for different elements can not be obtained from exactly the same parts of the specimen. For one element, three-dimensional information is obtained by analysing layer by layer, and the element distribution in the sample is obtained which offers unique opportunities in advanced studies of microstructure and its evolution. Finally, the pulsed-laser atom-probe field-ion microscope or, in short, pulsed laser atomprobe (FLAP) must be mentioned which uses a laser pulse instead of the voltage pulse on top of the steady-state dc voltage (KELMGG and TSONG[1980]). Pulsed-laser atom-probe field-ion microscopy has been used to characterize semiconductors and conducting ceramics where a voltage pulse may lead to fracture due to thermal stresses.
5.23. Field-electron microscopy Field-electron microscopes (FEM)are non-commercially made laboratory equipment in which an etched single-crystal tip is heated in high vacuum. The emitted electrons are accelerated by an anode and produce an image on a fluorescent screen. The intensity of electrons emitted (field emission current) depends on the voltage and the work of emission; the lattice structure and local geometric structure of surfaces can be studied with high resolution down to a few nanometers (MULLER[1936], EDELMANN [1980]). The crystallographic structure of clean surfaces and (if by chance a grain boundary was located in the tip) the structure and the movement of grain boundaries as well as changes of the tip geometry and allotropic transformations of the tip material during heating have been studied; by measuring the energy distribution of the field electrons the electronic structure of the single-crystal tip was investigated (for references see EDELMANN [1980]). Absorption of gas from the vacuum chamber or of evaporated substances (metals or oxides) changes the image drastically, which has been used for studying the sites of adsorption, the migration of adsorbed species along grain boundaries and the formation of compounds. It Seems that this technique has been overrun and replaced by other developments since no mention is made in more recent reviews. 5.2.4.
Applications of field-ion microscopy In contrast to the statement made in the last edition of this book, the disadvantages connected with the high effort of preparing suitable tips and the extremely small volume References: p . 1016.
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\J
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Fig. 14. Distribution of cobalt atoms in a very small volume (approx. IO-= m3)of a copper-lat.% cobalt alloy at different stages of annealing at 450'C. The annealing times are (a) 10 min,(b) 30 min, (c) 120 min and (d) 440 min (from CEREZOand SMITH[1994]).
analysed (requiring long analysis times and a large number of tips for obtaining statistically significant results) do not prevent the application of field ion microscopy, and many interesting studies of materials have been reported. The identification of small carbides in HSLA steels (KNEISSLand BRENNER [1990]) and particularly the study of the et ai. composition of carbides in secondary hardening in high speed steels (RSCHMEISTER [1988]) are examples where field ion microscopy has contributed to the quantitative understanding of technically important materials. Extensive studies of radiation effects in metals have been carried out (for references see SEIDMAN [1993]). In a recent review by CEREzo and SMITH[ 19941, a large number of case studies are discussed including secondary precipitates and segregation at y/-y'-interfaces in superalloys, phases and Bloch walls in Alnico permanent magnets, ordering and site occupation in various alloys, spinodal decomposition of iron-chromium alloys, segregation at
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grain boundaries and surfaces, oxidation processes and investigations of nanostructured and multilayer materials. Numerous field-emission microscopic studies have been made in the field of nucleation and growth where experimental results were needed to understand the morphology, crystallography and chemistry of particles in statu nascendi and growth for example of Guinier-Preston zones in various aluminium alloys (HONO et al. [1986, 19891) and particularly of cobalt particles in copper cobalt alloys (WAGNER and BRENNER [1978], HAAEN and WAGNER[1980, 19851, WAG^ [1985], HAASEN [1985], and CEREZO etal. [1992]). Figure 13 shows, as an example for the kind of threedimensional information obtained in such studies, the evolution of particles from a homogeneous Cu-1 at.%Co solid solution (2-0 et al. [1992], CER~?ZO and SMITH [19941). as analysed by the position-sensitive atom probe jeld-ion microscope (POSAP). For further examples of applications of the various types of field-ion microscopy as well as for details and references see the monographs by WAGNER[1982, 19851, MILLERand SMITH [1989], TWNG[1990], SAKURAI et al. [1990], MILLERet ai. [1992], the overviews by BENNER[1978], EDELMANN [1980], SMITH[1986], SEIDMAN [1993], or CEREZO and S m [1994] and the proceedings of various conferences (e.g., those of the International Field Emission Symposia) and current and special issues of journals devoted to surface analysis (e.g., those of Surface Science and Journal de Physique).
53. Photo-electron emission microscopy Microscopes using electrons excited by ultraviolet light were developed in the late fifties and made commercially available at the end of the sixties (for reviews see WEGMANN[ 19721, SCHWARZER [19811). In these photo-electron emission microscopes (PEEM), a high-intensity beam of ultraviolet light is focussed by means of quartz lenses and mirrors on a small area of a surface (area in the order of 0.2 nun2,energy density several W/cm2) which activates emission of relatively slow electrons. A high voltage is applied between the sample and a pierced anode facing the sample surface. These electrons, after passing the axial aperture of the anode and the magnetic lenses of a three-stage electron microscope, produce a highly magnified image on a fluorescent screen or a photographic film. Heating the sample up to temperatures of beginning thermionic emission allows in-situ high-temperature studies. Contrary to scanning electron microscopy, the light-beam cannot be focussed to a sufficiently small size, and integral image generation is used which has the advantage of small exposure times, varying with material and magnification from a tenth to a hundred seconds. The sample surface is cleaned by ion bombardment, but contamination is sometimes a problem in spite of the high vacuum in the chamber (typically lo-'' bar). The intensity distribution of photo-electron emission is essentially determined by the variation of electron emission with material and orientation (material and orientation contrast), with the orientation of a surface element with respect to the optical axis (relief or topographical contrast) and with distortions of the magnetic field in the vicinity of tips and edges. The topographical contrast is stronger than in SEM images but the stereoscopic impression is usually less pronounced since the surface of the sample is always normal to the axis of projection and cannot be tilted. Material contrast is usually very References: p . 1016.
986
H.E. h e r
Ch. 10, ($5
clear in unetched plane surfaces if the work-functions of different phases in a material are sufficiently different, which is usually the case. Orientation contrast is only pronounced in cubic metals but not in non-cubic metals and alloys, in oxides or in carbides. Thus, material- and orientation contrast can be easily separated in most practical cases. Studies in materials research have provided much interesting information in all kinds of highquality metallographic work (for early references see WEGMANN[I9721 and SCHWARZER [1981]). Owing to the very small depth of information (IO nm), the high lateral resolution (< 10 nm), the excellent phase separation and the possibility for in-situ heating, photo electron microscopy is excellently capable for quantitative kinetic studies of changes in microstructural geometry (DANN~HL et al. [1971], WEGhimN [1972] and M I D D L ~ Oand N EDMONDS[1977]). For example, a study by TAYLOR and POLLARD [1982] has revealed the bonding sequence (grain-boundary movement and annihilation) during diffusion-bonding of steel under load at temperatures up to 1000°C. The major draw-back of PEEM has been the very high price of instrumentation. The production of the commercially built Metioscope was discontinued and recent application reports are scarce. The photo-electric effect using ultraviolet light excitation is now utilized in various kinds of spectrometers, like ESCA (electron spectroscopy for chemical analysis), and U P S (ultraviolet light-excited photoelectron spectroscopy), see 0 6. 5.4. Scanning Auger-electmnmicroscopy
So-called Auger electrons are emitted when an electron beam hitting the surface creates electron holes in a lower shell (K, L or M) and when this hole is filled by an electron from a higher shell. The excess energy of this jump is transferred to another electron which is emitted. An electron detector and an electron spectrometer are used to register the number of electrons as a function of kinetic energy which depends exclusively on the energy levels of the electron shells involved and thus is specific for the element. By scanning the electron beam, successive points and line profiles can be analysed and used to form an image by modulating the brightness of a cathode-ray tube by the signal intensity of the Auger electrons. Auger maps very similar in appearance as X-ray maps (see 0 3.3.6) are obtained in this way, but, since emission of Auger electrons is limited to a depth of a few ( < 5 ) nanometers (owing to the short mean free path of the low-energy Auger electrons) compared to a few micrometers of the characteristic X-rays, see fig. 7) with a much better depth resolution. Three-dimensional information can be obtained by consecutively sputtering off layers by argon (with sputtering rates in the order of micrometers per hour). Lateral resolution is also much below 1 pm, down to 10 nm in favourable cases depending mainly on the spot size of the beam. In experimental microscopes, 5 nm resolution has been obtained more recently (VENABLES and HEMBREE [1991]), and a resolution in the order of 1 nm seems possible (BAKERand CASTLE [1994]). In principle, Auger electron microscopy could be performed in the scanning electron microscope with an Auger electron detector attached. However, since ultrahigh vacuum is usually not available in an SEM, the scanning Auger microscopes (SAM) have become instruments on their own. Recent reviews are available which give a full account of the basic principle and the electronic systems of the SAM (SIGNORELLI et a!. [ 19881, R T V ~ R E
Ch. 10,$ 5
Surface microscopy
987
[1990], BAKERand CASTLE[1994]). Sample preparation is not critical as long as clean surfaces are produced. This is possible by sputtering off any adsorbed layers and other contaminants inside the UHV chamber or by fracturing the sample in the specimen chamber. An extension is angular distribution Auger microscopy (ADAM)in which the angles at which the Auger electrons leave the surface are measured and analysed to produce images providing information on the structure of thin films, single crystals and monolayers (FRANK and HUBBARD[1993]). Applications of scanning Auger electron microscopy in physical metallurgy, microelectronics and corrosion are manifold, particularly in combination with Auger electron spectroscopy (AES) which is discussed in detail in Chapter 13 (see also P6). The following case studies have been reviewed by BAKERand CASTLE[1994]: pit formation at inclusions in steel and pit propagation laterally under the passivating oxide film (BAKER[1993]), embrittlement by segregation (SEAH[1975]), in-situ fracture of sprayformed aluminium alloys (BAKERand TSAKIROPOULOS [1993]), wetting during vacuum brazing of a gold-nickel alloy (BROOKER et ul. [1984]), among others. With some of the present problems relating to difficulties in imaging insulating materials or rough surfaces solved, SAM will increase its potential for materials characterization owing to its sensitivity for chemical composition, good lateral and extremely high depth resolution.
5.5. X-ray microscopy, topography and fluorescence The concept of X-ray microscopy was developed early, at the beginning of this century (see reviews by COSSLETT and N1~0~’[19601 or DUKEand MICHETTE[1990]). The lack of sufficiently intense sources was overcome by the development of microfocus X-ray sources and plasma sources, and, particularly, by the availability of synchroton sources (F’EDER etal. [1977], ASH [1980], DUKE[1981], KOCH[1983], MICHETTE [1990], BUCKLEY and RARBACK119901). Amplitude and phase contrast can be used for imaging (see, for example, RUDOLPHet al. 119901). Scanning X-ray microscopes (BUCKLEY and RARBACK[1990]), devices for integration of the low-intensity projected or reflected X-rays by means of special TV cameras, microchannel plates or image intensifiers (CHENG et aL [1990]) and holographic techniques for three-dimensional imaging (HOWELLS et al. 119901) have been developed. Owing to the high absorption of X-rays in crystalline materials, applications of s o f t - r u y microscopy has been more or less limited to biological objects (for details and references see, for example, DUKEand MICHETTE [1990]). X-my tupogruphy is a useful technique to study defects in single crystals such as dislocations, twins and stacking faults, subgrain boundaries, interfacial defects and strains, domain structures and other substructural entities (TANNER[1976], HOCHE and B R W R [1980], PANGBORN [1992]). The penetration depth of 5 p m and a lateral resolution of > 1 ,urn restricts application to relatively perfect crystals (defect density < 10”/cm2) but owing to its high selectivity for different types of defects and their location (subgrain boundaries, stacking faults, structure of ferromagnetic domains, dislocations) X-ray topography has become a standard technique for monitoring crystal quality, especially in the semiconductor industry. Imaging uses a Bragg reflection, the intensity of which varies with local variations of orientation, lattice parameter and References: p . 1016.
988
H.E. h e r
Ch. 10. 56
structural factor. It is not possible to magnify the image directly, owing to the lack of X-ray lenses. High-resolution film and photographic magnification has been widely used, typical exposure times ranging from 10 min to 2 h with a I kW X-ray source. More recently, digital image storage and accumulation have become available, providing better resolution and higher speed. X-ray topography has been applied mainly in the following principal areas of research (WEISSMANet al. [1984], PANGBORN[1992]): development and quality control of electronic devices, kinetics of transformation, crystallisation and crystal growth, and deformation and fracture. Classical studies are the investigation of the internal magnetic structure of nontransparent ferromagnetic crystals (BOWMANand Boom [1971]), and the investigation of damage due to ion implantation (TANNER 119761, DE NODE and SMITS[1981]). Of particular interest are in-situ dynamic studies like the movement of dislocations in a silicon single crystal under external stress (CHANG et af. [1982]), the evolution of the plastic zone during crack propagation in a silicon crystal (MICHOTand GEORGE[ 1982]), and recrystallisation of deformed aluminium (GASTALDIand JOURDAN [19841). With portable X-ray generators, advanced detection systems and computer analysis of results, X-ray topography is also useful in nondestructive evaluation of deformation states, microscopic flaws and cracks in parts outside the laboratory (PANGBORN [19921). Finally, fluorescence microscopy must be mentioned. If a fluorescing substance is irradiated by photons (X-rays or light of short wavelength, usually ultraviolet), some of the energy is re-emitted as light of longer wavelength which is typical for the substance. This effect is called fluorescence and is used in mineralogy for identification purposes and, after suitable staining with fluorescent substances, in biology (ROST[I9811 and AXELROD et al. [ 19831). Extremely small amounts of fluorescent nonmetallic phases can be detected. However, since very few phases in metallic alloys are fluorescent, this technique is rarely used for materials characterization. 5.6. Imaging by other types of spectroscopicinformation
There is a very large number of spectroscopic techniques in which photons, electrons or ions interact with surface near regions of the sample and some kind of radiation is analysed (see 0 6). In addition to the microscopic techniques discussed up to now, there are presently a few more using the signals reflected or emitted from the surface to form images. Examples are nuclear magnetic resonance (NMR)microscopy (PETTEGREW [19901, MALLARD[19933, WACHTMAN[19931) presently only used in biomedical applications, or imaging in secondary ion mass spectroscopy (SIMS). No doubt several other microscopic techniques will be developed in the future. It will be interesting to see to what extend these techniques will be useful in physical metallurgy.
6. Topochemical techniques and suflace spectroscopy Surface analytical methods have become available to materials researchers for nearly three decades. Two of the techniques which have become most popular, ESCA (electron spectroscopy for chemical analysis) and AES (Auger electron spectroscopy) were
Ch. 10, 46
989
Surface microscopy
described in the late sixties (SIEGBAHN et ai. [ 19671, HARRIS[ 19681). The number of additional spectroscopic techniques for chemical analysis increases at an even higher rate than that of microscopies and often combinations of topochemical and imaging methods are realized in a single instrument. Excellent reviews and books are available which present the progress and state of the art of the fundamentals and the applications of photon, electron and ion probes as well as electric field probes (for example CARLSON [1975], CZANDERNA [1975], IBACH[1977], MCINTYRE[1978], BROMMERet af. [1980], METALSHANDBUOK [1986], FISCHMEISTER119881, SIBILIA [ 19881, WALLS [1992], R I V [1990], ~ BRIGGSand SEAH[1990,1992], LIFSHIN[1990, 19941, ASM HANDBOOK [1992], CAHNand L ~ H M [1993], among many others). Some of the techniques based on these physical principles are of special interest to materials scientists and engineers while others are used only occasionally. No attempt will be made here to give a full coverage of these methods. Rather, a recent overview by ORTNERand W l L H A R n r Z [1991] based on an earlier compilation by GRASSWBAWER [1989] is used here to present the relevant information on the large variety of methods in a systematic way (Tables 4 to 6). For references, the reader is referred to the books and reviews mentioned above. In addition, reviews on the application of surface analysis in materials science and technology, for example that by FISCHMEISTER 119881, and the relevant chapters in the encyclopedia edited by CAHN and LIFSHIN [1993] and in Volumes 2A and 2B of Materials Science and Technology edited by Lmm [1992] and [1994], respectively, should be consulted. Combining the most effective techniques for characterizing the chemical and structural properties of materials (see fig. 15) will result in deeper insights into the nature of materials as well as improvements in the development of processing and properties of products needed for modem technologies.
-
1
-
AAS, OES, XRF, MS Activation analysis Chemical methods
In-situ microanalysis Surface and interface analysis
-
EPMA, SEM,TEMP (EDS, E E L S ) SIMS,LAMMS XPS,AES, RBS, ISS
-
Bulk composition
- Bulk analysis
Distribution of elements and phases
I
Geometric Stn€ctUre
-
Structure. d y s i s
Electronic stn~cturc
-
Electronic struciure analysis
XRD,TEM, THEFD IR, Raman, NMR EXAFS, NEXAFS, STM, AFM
UPS, XPS, STM X-rayvalence band specaoscopy
Refemnces: p. 1016.
Table 4 Ion probe techniques (from ORTNERand W I L H A R ~ T[1991]). Z
Technique
Resolution Lateral
Depth
Process
Information content
Techniques bused on spurrering
SIMS (Secondary ion mass spectroscopy)
(a) Static
first monolayer
Imml (b) Dynamic 50&1000 nm 50 nm (with Cs+)
1-5 AL (atomic layers)
1u)nm (with 0)
Sputtering with Ar*, 0;.0-, Ga', Cs' at 0.001 to 0.1 n d s ; TOF-SIMS (timeof-Right S M S ) As above, but with typically 1 n d s Double focussing ( m a s spectrometry) with M/AM betwccn IO00 and IOU000
Surface studies in catslysis, gas-surface reactions, polymer analysis, only method with trace and ultratrace detection power at high lateral and depth resolution (detection limit from ppt to high ppb-levels) Isotopic specificity High resolution and high density depth profile possiblc Imaging resolution 1 p m Three-dimensional distribution analysis
LRI-SIMS (Laser resonance ionization SUIS)
IO00 nm
1-5 AL
Very complex instrumentation, single element at a timc method; selective ionization of one atomic species in sputtered atom cloud by double or triple photon absorption. T O P M S (time-of-flight mass spectrometer)
Very high detection sensitivity at ppb- and sub-ppblevels, e.g. 2 ppb "Fc in Si with consumation of 0.8 AL = 40 Fe atoms sputwred!
LNRI-SIMS (Laser non-resonant ionization SIMS)
I O nm
1-5 AI.
Multiphoton ionization with intense pulsed UV-laser beam (2 IO9 W cm") in sputtered atom cloud
As SIMS, but with detection limit presently still in pg/g-range
SNMS
loo0 nm
1-5 AL
Secondary ionization of sputtered particles in HF-plasma above sample surface Quadruple MS-detection
As SIMS, but with detection limit presently still in P6/g-mnge
[mml
10-100 nm
Ar-glow discharge for sputtering. Ionization in negative glow region Detection with double-focussing mass spectrometer, M/AM = 1000-1oooO
Advantages more on bulk-ultratrace analysis side than on topchemical analysis side Detection limit in the ppt to ppb range for pin samples only!
(Sputtered neutrals mass spectrometry) GDMS (Glow discharge mass spectrometry)
Technique
Resolution ___Lateral Depth
Process
Information content
As for GDMS, but with detection of atomic emission in UVNIS-range
Detection limit in 1 ppm range Routine method for p m layer characterization in metallurgy
RBS (HEIS) (Rutherford backscattering spcctroscopy, or high energy ion scattering spectroscopy)
Imml
2-3 nni Measurements up to IO00 nm
a or 'He*-scattering E,:1-2 MeV Measurement of energy distribution of back-scattercd u or 4He'-ions
Qualitative and quantitative elemental analysis; depth profiling by deconvolution of measured energy distribution with energy loss functions possible, however. generally poorer depth resolution than SIMS, but non-destructive depth profiling! Additional structural information by channeling and blocking phenomena: study of formation of epitaxial films
Lmml
0.34.4 nm
200-600 keV
Same as above
Imml
First monolayer analysis
500-600 eV He', Ne'
Same as above. Adsorption results
lmml
First monolayer up to some Pm
Elastic recoil detection of forward scattered light target atoms by bombrrrdment with typically 3 MeV 4He+' up to 30 MeV Si+
Depth profiling for 'H and 'H with a,or H, C, N, 0 with Si'
Techiiiyues bused on srurrering
MElS (Medium energy ion scattering spectroscopy) ISS (LEIS) (Ion scattering spectroscopy, or low energy ion scattering spectroscopy)
ERD (Elastic recoil detection)
-
Table 5 Electron probe techniques (from ORTNERand WlLHARmz [1991]). Technique
-
EPMA (EPXMA) (Electron probe microanalysis, or electron probe X-ray micro-analysis)
EELS (Electron energy loss Spectrometry)
UREELS (High resolution electron energy loss spectrometry)
Resolution Lateral
Depth
Primary electron-beam: 5-50 keV, diameter 1-100 nm Excited volume: 1-5gm3 (pear shaped) 1-5 p m
1-5 g m
Resolution same as as for X-ray nano-analysis Detection limit 3-5% accuracy: IO-20% relative Especially suitable for light elements due to lacking sensitivity of EDS. High electron yields with electron spectrometersbeneath thin specimen. However low signaVnoise ratios and high background due to multiple elastic scattering.
-
poor (mm to cm)
-
Process
Information content
Scanning primary electron-beam excites Characteristic X-rays in flat, polished samples. With conventional detectors the elements Na-U, with windowless proportional flow counters the elements B to U can be detected
Quantitative topochemical analysis: WDS (wavelength dispersive spectroscopy): 10-100 Pplp EDS (energy dispersive spectroscopy): 100-5000 PPk Optimized for quantitative X-ray analysis after correction for absorption and enhancement (computer programmes)by use of standards possible with an aaxu;lcy of - 1% rel., standardless with an accuracy of 3-10% rel. as a function of composition
In connection with transmission electron microscope
Qualitative and semiqualitative information of elemental composition Position and structuTe of absorption edges of valence band or near valence band core levels contain bonding information Fine structure evaluation in combination with high spectral resolution (0.1 eV at 100 keV) allows to gain stoichiometric information for extremely small domains Sensitive detection of structural changes in nano domains possible by study of position and width of plasmon peaks
Imaging possible by beam scanning (STEM) or stigmatic electron sptWometer
monolayers
Monoenergeticbeam focussed onto the surface of a solid, measurement of reflected electrons ' b o electron spectrometers necessary (one for monochromatizationof primary electron beam, typical Eo=500 meV, and one for energy analysis of reflected electrons)
Useful for studies of adsorbates on solid surfaces (surface coverage with adsorbates, information on bonding geometry, studies of catalytic phenomena, thin film characterization)
Resolution Technique AES (Auger electron spectrometry)
THEED (Transmission hiEh energy electron diffraction)
LEED (Low energy electron diffraction)
.~
-
- .
Lateral Primary elecuon beam: 1-10 keV poor (low signal to noisc-ratio, very high continuum electron background due to multiple scattering requires registration of 1st derivative of spectrum. hut very high surface sensitivity)
Depth
Process
Information content
0.5-5 nm
Measurement of kinetic cncrgy of emitted Augerelectrons UHV of 10 I" mhar mandatory In situ sample preparation: fracturing, annealing: thin film deposition
Micro surface analysis (elemental identification, qumtative elemental analysis. direct compound identificationby evaluation of chemical shifts and line shaps possible. depth profiling in combination with Ar*-sputccring) Relative detection limit depends on heam diameter and current density. c.g. 10-20 mass% at 100 nm and InA. 1-2 mass% at I p m and I nA or 100 nm and 100 nA. or 0.1-0.2 mass% at lfim and 100 nA Imaging resolution 200 nm
Electron diffraction (coherent interference effects)
TEM diffraction paltcrns (point patterns for
A =0.1 nm at 150 e V A =0.01 nm at 15 keV (High energy of primary electron beam yields high resolution of imaging) Minimum phase sirs For structure analysis of precipitutes: - 20 nm Primary electron beam 1-5 keV Poor
RHEED (Reflected high energy clectron diffraction)
Primary elemin bcam > 10 keV poor
XNA (X-ray nano-analysis)
5-30 nm at beam diameter 3nm and ssmple thickness 100-20 nm g Absolute detection limit: Elemental distribution imaging: lateral resolution ea. 1 nm (signal enhancement by digital image
~
monolayers
Diffraction at solid surfaces Measurement of reflected electrons of a mono-energetic PE-beam on a fluorescent screen
monocrystallincspecimens, ring patterns for polycrystallinespecimens, swipe fonned patterns for large grained or textured specimens, and diffuse circular halos for amorphous materials) useful for determination of lattice constants and identification of phases LEED patterns reveal the periodicity of atoms of the surface and the overall symmetry, but not the
individual atomic position (periodicity is needed for reflection) Useful for determination of crystalline order of the surface of a single crystal or study of adsorption and .segregation processes on surfaces or ovcrlayers
thin tihns
In connection with TEM h g e area detectors (EDS) in small distance from specimen F'rincipal disadvantages are low collection yields (10.' lo IO4) and low X-ray yield for light elements (6, C. N, 0. )
...
Important for the identificationof pmipitates at grain boundarics or interfaces and for sub-Km phaws and inclusions
Table 6 Photon probe techniques (from ORTNERand W I L H A R[IWl]). ~ Resolution DeDth
Process
Information content
0.1 p m
lnelaslic p h o m scsltaiog (Stokes and antiStokes)
Molecular and structural microprobe; complementary info to IR ahout functional gmup vibrations, evaluation of polarization effects
SERS (Surface enhanced Raman
0.1 p m
As above
As above, but enhancement up to 106, limited to Ag, Cu. Au 88 substrates
1R (0.75-1000 pm) absorption by excitation of vibrational and rotational states Molecular and functional group information
Technique
Lateral
LRMA (LawRaman microanalysis)
)-
FT-IR-M aansrorm infrared
I-lOpm
microscopy) ATR-LR-S (Attenuated total reflection infrared spectrometry)
1-lOpm
Internal IK-reflection
As above: SA of polymers, catalysis, electrochemical reactions. By use of p o l a r i d light: information on orientation of adsorbates on surfaces
IRRAS (Infrared reflectance
I-IOFm
Angle resolved IR-reflectance measurements
&termination of complex optical pmpcrtiw and identification of c h i s o r b e d species
AbEorption of monochromatic X-rays (synchrotron radiation), s e v d keV
"Local atomic S(NCIU~ pmbe" Mcasunmot of atomic distaoas berweem absorbing atom and mighbours (range 0.6 nm)
As above, but up to 1 keV
Accurate determinations of bond lengths of adsorbate. surface molecular studies. The polarization of synchrotron radiation allows to probe atomic distances in spccific dinctions
As above, but keV
Determination of valence states of very Rm p i p i t a t e s , not measurable by 'I'M.studies of crystal growth, nucleation, formation of interfaces
or specrrn
(Fourier
absorption spectrometry)
EXAFS (Extended X-ray
abswption fine structure Spec~~OmWy)
SEXAPS (Surface EX-)
a.2 0 p m
NEXAFS (Near edge X-ray absorption h e ~ t
ca. 20 p m
spectromc(ry)
~ c t ~ n
1 nm
-
-___..___ Resolution
Technique
Lateral
Depth
ESCA Electron spectronietry for
Process _-.___
Information content
~-
Two techniques, X P S and UPS. see helow
chemical analysis XPS (X-ray photoelectron
100 p m (crystal focussing) 5 lOpm
spectroscopy)
(EXASCOPE of VG,FT-lens)
UPS (Ultraviolet photoelectron
Qpically S-10 atomic layers
Absorption ol monochromatic X-rays and measurement of the kinetic encrgy of ejected photo-electrons
Qualitative and quantitative elemental analysis of surfaces (Z> 1: sensitivity limit a.0.1 mass% hut 0.01 monolayers! plus information on chemical bonding (by chcmical shifts of binding energies) and binding energies. Depth profiles with sputtering installation possible Imaging resolution < IO p m
In the order of mm
Several atvmic layers
Absorption of monochromatic UV-radiation and mcasurement of the kinetic energy of ejected photoclectrvns. (Excitation with noble gas discharge lamps or synchrotron radiation)
Study of electrvnic structures (of the valence band of solids or of bonding electrons in gasmolecules (even vibrational exciled states can be resolved) Extremely good vaccuum (pS IO-" mhar) necessary. Study of chemisorption phenomena. adsorption, desorption (esp. with respect to catalysis)
Same as for UPS. strunursl sensitivity enhancement by use of polarized radiation
Measurement of valence band emission spectra for "density of state"-calculations: very weak adsorption phenomena. e&. of noble gases. Variation of detector angles allows to exploit directional effects in photoclectroncmission
Microanalysis for clemmts, somc bonding and molecular information from fragmentation puttcms. Detection limit in thc order of l-lOpg/g
sprctroscopy)
TVLE 1-2 p m
In the order of pin
Detection by TOF-MS (time of flight mass
species or from surface contaminants
996
Ch. 10, $ 7
H.E. Exner
7. Quantitative interpretation of microstructural geometry Quantitative methods for characterizing the geometry of microstructures are an important means for understanding the relationship between production parameters and behaviour of technical materials in application as well as for providing the experimental basis for theoretical studies of microstructural genesis and its effect on mechanical and physical properties. The important elements of quantitative microstructuralanalysis (also termed quantitative metallography) are image analysis, i.e.. the quantification of features visible in metallographic cross-sections or transmission images; stereology, i.e., estimation of three-dimensional geometry from two-dimensional image data;and the interpretation of the three-dimensional parameters. Generally, more than one single geometric aspect is of relevance in characterizing the microstructural geometry of multiphase materials, and a combination of parameters will be needed to give a full description. Such combinations have been proposed for studying the evolution of microstructure or for empirical correlations with mechanical properties. A large number of techniques have been developed and employed for characterizing microstructural geometry. Table 7 shows an overview over important analysis methods. This section aims to give a short introduction to techniques of image analysis and stereology, focussing on applications to problems relevant in materials science.
7.1. Image analysis Simple comparison of microstructure with standard test charts is extensively used in quality control of materials (for example, ASTM standards for determination of grain size or of slag inclusions). The accuracy of these comparison methods is rather limited compared with counting techniques (EXNER[1969b, 19941 and E m and HOUGARDY [ 19881). These semiquantitative methods should not be used for research work and will not be discussed any further here. When scanning an image for collecting data, counting with point and line grids, measurements along straight lines, or area measurements can be used. Each of these strategies has its merits, according to the purpose of the investigation (parameters to be determined) and available instrumentation (manual, semi-automatic or fully automatic). The manual acquisition of image data is carried out directly at the microscope, using inserts in the ocular or ground plates with arrays of points or lines, or with micrographs and suitable grids or rulers (GAHM [1971], EXNERand HOUGARDY [1988] or Ex[ 19941). Figure 16a shows a typical point grid superimposed over the microstructure of a two-phase material. The number of points falling within the black phase and the number of intersections between the outer circle and the perimeter lines of the black phase are counted (yielding, as discussed below, estimates for volume fraction of the black phase and its area of interface with the white phase). Figure 16b shows linear analysis of the grain size in a single-phase material. A number of fields must be evaluated in this way in order to get statistically significant numbers. Simple electronic counters and step stages are useful means to speed up these counting procedures. More complex data (size distribution and elongation of planar features) can be determined by References: p . 1016.
Fig. 16. Qpical grid-, line- and graticule-counting procedures in manual image analysis: (a) point-counting for volume fraction analysis (circles can be used for interface density measurement, see text); (b) line-counting for determination of mean linear grain size of a single-phase material (numbers on the right margin show number of grain boundaries intersected); (c) graticule-counting for size distribution measurements (area of circles and ellipses of varying aspect ratio increase by a factor of two in each higher class).
H. E. Exncr
998
Ch. 10, $7
Table 7 Techniques for the geometric characterization of three-dimensional microstructures and two-dimensional images (“fingerprinting”). 3D Description
I.
2.
Stereology Global parameters (volume-, surface- and line densities, orientation) Distributions (size, shapc) Stereometry Nonplanar surfaces (fracture surfaces, deepetched interfaces), Transparent materials (3D microscopy, TEM foils)
2D Description 1.
Image comparison Standard charts, graticules
2.
Simple geometric parameters Perimeter, area, shape factor, size and shape distribution
3.
Serial sectioning Topological parameters (number dcnsity, genus) Shape and arrangement
3.
Mathematical morphology Manipulation of digital images (opening, closing, skeletonizing) Morphological parameters
4.
Model structures (simple shapes like spheres, cubes) Regular lattices, statistical arrangements
4.
5.
Computer gcnerated structures (dendritic patterns, fractal structures etc.) Point processes and mosaiques
5.
Complex special techniques Fourier analysis (periodicity) Fractal analysis (selfsimilarity) Pair correlation functions, graph theory (arrangement) Geometric abstraction Circles, hexagon, planar lattices (analytical modelling) Poisson point process, Voronoi mosaiaue fcommter modellin.-)
simple graticules, an example of which is shown in fig. 16c. Manual methods for microstructural analysis have been reviewed in detail by FISCHMEISTER [1965, 19811, GAHN[1971], EXNER[1972a, 19941, UNDERWOOD[1985] or EXNERand HOUGAKDY [ 19881. Exact procedures for practical assessment of the most important stereological parameten: (volume fraction, interface density, size and distance, orientation, elongation, proximity and contiguity, see below) have been worked out recently by a committee of the German Society for Materials including estimates of experimental accuracy (EXNER [ 19931). The first step towards automation was realized in lineal scanners which were popular during the sixties and seventies (EXNER[1972a] and GAHM[1975]). More recently, semiautomatic instruments have been provided with digitizers as input device to a calculator or computer (see, for example, HILLJEand REDMANN[1976], EXNER[1978b] and EXNER and HOUGARDY [ 19881). Digitizers are tablets by which the exact coordinates of points touched with a special pencil are registered at a high rate (up to lo4 per second) with high precision (0.1 mm). The operator indicates the intersections of the scanning line
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with the perimeter or outlines the perimeters of the microstructural features using his ability to differentiate between features of different kind (composition, structure) on the basis of grey or colour shading and of shape. Thus, the operator keeps steady track of what is measured and is able to exclude artefacts by simple judgement. Modem instruments compute all relevant microstructural parameters from these data (see, for example, RUSS [19951). Disadvantages are the high time-expenditure and concentration required, in addition to some danger of introducing human bias. Nevertheless, this type of instruments is often the best choice for practical work due to their reliability and versatility. Fully automatic instruments are available with a wide efficiency range (FISCHMEISTER [19811 and EXNWand HOUGARDY[1988]). Most of them use videocamera (CCD) type scanning modules with electronic phase detection. Simple low-price instruments allow determination of a limited number of parameters for high-contrast situations. High priced instruments, usually modular devices with fast hardware modules or, more recently, flexible software modules and full grey-level image-storage with up to 1024 x 1024 picture-point lateral resolution and 256 steps of grey-level resolution combine high speed, versatility and convenience for any type of automatic image data acquisition. Scaling, shading correction, scanning-table steering and focussing are all carried out automatically, and all kinds of image manipulation (filtering, edge contrast enhancement, halo correction, etc. and image modification techniques based on mathematical morphology the principles of which are outlined in 8 7.3 below, etc.) can be implemented if required. Figure 17a shows the grey-level histogram and fig. 17b demonstrates the effect of a special filter (pseudoplast filter) on the appearance of a light micrograph of a three-phase alloy. Phase discrimination (either by dedicated hardware modules or by software algorithms), parameter selection and evaluation, statistical treatments and many other routines can be activated by push-button commands or by a simple computer program designed individually for any specific problem. As an example, fig. 17c shows the appearance of a microstructural evaluation on the monitor or screen of an automatic image analyzer. "Interactive" devices - usually using a mouse or a light-pen to be used on the display monitor - allow the operator to use his judgment for excluding artefacts or adding missing details (e.g., parts of a grain-boundary network). In principle, any imageforming instrument can be used as input device; only light microscopes and scanning electron microscopes have been used in on-line configurations. Transmission electron micrographs and other printed images have been evaluated only occasionally for solving problems in materials science and technology. Software for automatic image analysis is now offered with most highly computerized scanning instruments mentioned in this chapter, particularly with energy dispersive X-ray analysers (electron microprobe). The combination of image analysis and energy-dispersive X-ray analysis was used for determining geometric parameters and composition of individual particles simultaneously (EKELUND and HERTZMANN[19811). Instrumentation for image analysis has developed rapidly over the past two decades (for historical and recent reviews see FISCHMEISTER [1965], GAHM[1971], EX= [1972a], HOUGARDY[1976], FISCHMEISTER 119811, E m and HOUGARDY 119881, or EXNER [1994]) and is still advancing at a rapid rate. In this context, the dramatic References: p . 1016.
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B
Fig. 17. Typical procedures of television-equipped fully automatic image analyzers: (a) Microscopic image of a three-phase copper-cadmium alloy with superimposedgrey-level distribution; (b) Pseudoplast filter producing a plastic appearance by showing gradient variation only; (c) Size-distribution analysis of nonmetallic inclusions in a dirty steel with superimposed counting flags indicating proper detection and area distribution histogram.
developments in digital image processing must be mentioned, which not only are useful for quantitative image analysis but also for contrast enhancement and image restoration (Russ [1990], J m [1993] or TALBOT[1994]). The state of the art has now reached a level which allows only minor improvements with respect to practical applications.
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Though recent developments (e.g., processing of real colours and grey levels) may widen the scope to the treatment of even more complex problems in image analysis, the present rapid decrease of price levels at a given performance is certainly the best reason for a much more general utilisation of automatic image analysers.
7.2. Planar characteristicsand stereology Image analysis yields a number of geometric characteristics describing the geometry of planar images of cross-sections or thin-film projections, the most important of which are listed in table 7. They usually can be qualitatively interpreted in terms of the threedimensional microstructures from which they originate. Quantitative three-dimensional characteristics are obtained by a body of mathematical methods based on integral geometry and statistics which is called stereology. There is an enormous number of original papers and formulae spread over the scientific literature of fields as far apart as mathematics, geology, materials science and biology. Useful monographs and standard textbooks (DE HOFFand RHINES[ 19681, UNDERWOOD [ 19701, SALTYKOV [ 19741, WBIBEL [1988] [1979, 19801, RHINES [1986], Russ [1986, 1990, 19921, EXNERand HOUGARDY or COSTERand CHERMANT [1989]) review the literature and should be consulted whenever a quantitative evaluation of spatial microstructures is attempted, in order to avoid misinterpretations and redundant derivations. A few of the more important aspects of microstructural geometry (table 8), their experimental evaluation from surface images and some applications in physical metallurgy are discussed in the following sections. Standard nomenclature for the stereological terms as outlined in the classical textbooks by DE HOFFand RHINES[1968], UNDERWOOD[1970] and WEIBEL[1980], see also EXNBR[1994] and as standardized by the International Society for Stereology (EXNER[19871) is used in the following and is recommended for general use.
7.2.1. Volume-fraction analysis Simple plausibility proofs as well as sophisticated mathematical treatments show that, for a homogeneously dispersed phase, area fraction, lineal fraction and point fraction are statistically unbiased estimates for the volume fraction, i.e., in standard nomenclature (UNDERWOOD[1970], WEIBEL[1980] and EXNERand HOUGAKUY [ 19881): V, = A, = & = P,.
(4)
The experimental error of this estimate decreases with an increasing number of points counted or lines and areas measured and, at a 5% level of error probability, for pointcounting using a wide-spaced regular point grid, is approximately (HILLIARD [ 19761):
where P is the total number of points of the test grid. A more stringent evaluation of errors includes the variation of area fractions between different test fields (NICHOLSON [ 19781). Volume fraction has been the parameter most frequently used in quality control of References: p. 1016.
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Ch. 10, 57
H. E. h e r
Table 8 Some important geometric characteristics of planar images and three-dimensional microstructural features". Planar characteristic'
Example
Comsponding bedimensional characteristic'
Micmsuuclural example
Area fraction, A,
Particle intersects
*Volume fraction. Vv
Amounts of phases in multiphase material
Line density, LA
Intersect perimeters
'Surface fraction, Sv
Grain-boundary or phaseinterface area
Point density, PA
Triple points in inlefiected grainbouadary Dctwork
'Line density.
Number density, N,
Number of intersected particles
Numerical density, Nv
Distribution by area
Intersect area
Distribution by number
Number of spherical particles as
A4
distribntion
and tiocar Sizefir)
a function of diamtier
&*
Length of dislocation lines or grain-boundary triple l i i Number of particles
Distribution by length Chord length distribution
Distribution by volumeflv) Number of grains as a function of volume
Mean Size
Meanintasectareaor Ineanliiinkrcept
*Mean linear size L Mean volume, V
Mean linear grain size
Distances
Nearest-neighbour distance
*Mean linear distance D
Surface-to-surfacedistance of panicles in one direction Distance of particles in dispersion-hardened materials
Ab
Nearest-neighbor distance
Mean particle volume
Shape characteristics
Adperimerer ratio. lengthhuidth ratio
Planar curvature
Local and mean curvature
*Mean cumatum
Curvature of pore-solid interfaces
Arrangement in plane
Random, mgular, clustend
Arrangement in space
Clustering of d iin highs@ stal Numbcr of closed loops in the pore space of a sintered material Amount of surface shared with other grains of h e same phase in a multiphase material
Shape changes of particles or ports in processing
Connectivity "Contiguity
Orientation
' Threedimensional
Preferred orientation of intersection lines
*Orientation of grain boundaries and interfaces
Anisotropy of poreholid interface, grain boundary area in rolled shccts or drawn w k s
parameten marked with an asterisk can be calculated from data obtained with planar images without any assumptions but randomness of sampling. All other thresdimensional characteristics are restricted to model geometries, to known shapes o r known size distribution or both, or can only be assessed by three-dimensional measurements (stereometry, serial sectioning or chemical desintegration). Subscript A indicates that quantity is per unit area. Subscript V indicates that quantity is per unit volume.
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materials and has found many useful applications in materials research. It is obviously closely related to composition and exerts a decisive effect on the properties of multiphase materials. In equilibrium phase diagrams, the lever rule allows phase boundaries to be determined to a high accuracy with a minimum number of samples. For example, LIND&N[1972] has redetermined the two-phase region in the copper-aluminium system, and ALJXNGER119691 has shown excellent agreement of results obtained by classical and by image-analyzing methods in the four-phase region of the Ag-Cu-Cd-Zn diagram. Modem phase-diagram studies employ computer calculations to a high degree which are facilitated if accurate quantitative data are available. A typical field for volume-fraction analysis is the determination of nonmetallic inclusions, of carbides, and of the time-temperature dependence of phase transformations in steels. Figure 18 shows a typical isothermal time-temperature-transformation 0 diagram for a chrome-molybdenum steel which makes it possible to read not only the nature of phases obtained after a certain holding time at constant temperature but also the fraction of austenite transformed (LAFONDand MOL= [1975]). Studies of transformation kinetics of discontinuous precipitation or recrystallization are further examples. A very interesting aspect of volume-fraction analysis is the possibility of determining local composition, which is straightforward if the equilibrium compositions of the individual phases present in the microstructure are known. Thus, segregation and inhomogeneity phenomena and other systematic compositional variations can be quantitatively determined, sometimes with as good or better accuracy than by microchemical methods. Good examples can be found in the investigation of surface and heat treatment of steels, e.g., the analysis of the carbide distribution in the surface of a hardened steel part
0
600 . I
0,
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200
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1
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1
1
1 5 3 0 1
2
I
5
min
l
,
,
15301
I
l
2
5
1
1530
h
Time Fig. 18. Isothermal time temperature transformation diagram for a chromium-molybdenum steel. (A = austenite, F = femte, C = carbide, M = martensite). The heavy lines give start and end of a reaction. the numbers at the thin curvts give the volume fraction of austenite transformed. (After LAFONDand MOLIEXE119751.) Refemnces: p . 1016.
1004
H.E. Emer
Ch. 10.47
(FIscHmm [1972]), or the scanning electron microscopy study of the influence of heat treatment on the composition of the matrix in high-speed steels (FISCHMEISTFR et al. [1986], KARAG~Z and FISCHMEISTER [1987]).
7.2.2. Interface density Dispersity of a microstructural component (isolated particles as well as complexshaped continuous or semi-continuous phases) can best be quantitatively described with a single number by measuring the density of its interface area per unit volume. Stereology (SALTYKOV [1974], UNDERWOOD[1970] and WEIBEL[1980]) tells us that:
S" = 2PL, (6) where S, is the interface area per unit volume of the material and P , is the number of intersections @er unit length) between a scanning line and the intersections of a plane cross-section with the interface as seen in a metallographic image. If we relate the interface of a microstructural component (phase a) to the volume of this particular component, we get the specijic inrelface density of the a-phase: Sr = &/V, = 4NdLa = 2PL/Pp,
(7)
where L, is the length of the line inside the a-phase (L, = LV,, L is the total length of the scanning line) and N, is the number of a-regions transversed by the scanning line. Of course, the density of grain-boundary area can be determined in the same way. An analysis of statistical errors shows (HILLIARD[ 19761) that the standard error, AS,, is approximately
AS,, = k S V / y F ,
(8)
where P is the total number of intersections and k is approximately 1.2 for contiguous grains and 4 for randomly dispersed particles. The range Sv*ASv corresponds to 5% error probability (95% significance level). Prominent examples of application of surfacedensity measurements are found in powder metallurgy, where the pore-solid interface provides the driving force for sintering processes (ch. 33). in the study of coarsening processes (grain growth and particle growth, ch. 28) and in finding correlations between microstructure and properties. Figures 19a and b show two simple examples of the latter, where linear relationships are observed between the hardness of various metals and their grain-boundary density (SALTYKOV[1974], RHINES[1976, 19861 and ONDRACEK[1978]) and between the coercive force and the interface per unit volume of magnetic cobalt phase in WC-Co hard alloys used in cutting and wear applications (FISCHMEISTER and EXNER[1966]). The concept of surface density is less popular for characterizing microstructural geometry in physical metallurgy than might have been expected considering its versatility combined with extreme ease of evaluation. While for the separate determination of number and size of particles simplifying assumptions about shape are necessary (see below), the only requirement for obtaining significant numbers for S, is statistical sampling, while shape effects as well as variations of the size of the individual features
Ch. 10, $ 7
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, pm 10
2
8
Mean linear intercept in binder , J 1.5 1.2 1 0.8 0.7 0.6 0.5
J
~
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100 -
80 -
0
J
0.1 OJ5 0.2 0.25 Specific grnin boundary surface, m Y c d
0.05
2
3
4
5
6
7
8
9
Specific surface of Lo-binder, m2Icrn3
Fig. 19. Simple relationships between properties and microstructural geometry: (a) hardness of some metals as a function of grain-boundary density (data from SALTYKOV [19741 and RHth2s [1976]); (b) coercive force of the cobalt phase in tungsten carbidecobalt hardmetals as a function of interface density (after F r s c ~ ~ n s n a and EXNER[1966J).
have no influence. Surface densities can also be used for describing the contiguity, i.e., the degree to which grains of a phase are connected in space or the degree of preferred orientation (see below). In modelling microstructure-property relationships or processes of microstructural genesis during casting, heat-treatment, deformation, etc., interface densities are most useful quantities for averaging the complex geometric appearance of phases in most technical materials. The use of interface densities (instead of size) was suggested in theoretical modelling of coarsening by EXNERand RSCHMEISTER [1966a] and by GLICKSMAN er al. [ 19921, of sintering by AIGELTINGER and EXNER[ 19771, EXNER and GIESS[I9891 and Lru and PATTERSON[1993], and of recrystallisation by DE Horn [1972, 19891. The latter author developed the framework for stereological theories of kinetic and dynamic evolution of microstructures (DE HOW [ 19721). 7.2.3. Sue and distance In contrast to volume fraction and surface density or specific surface, three-dimensional size and separation (distance) are not related to two-dimensional data in a simple manner. Only if the shape of an object is known can its size be defined by a single parameter. The assessment of nearest-neighbour distances requires knowledge of the spatial arrangement of objects, which is often not available from planar images. Thus, models are needed to estimate size and distance and their distribution. The most useful model relates to specific surface and surface. density. The size and distance can be described by the average linear distances between the surface points inside (mean Linear size) and outside (mean linear distance) the volume covered by a References: p. 1016.
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H. E. Emer
ch.10, 6 7
phase a.For multiphase materials we get the mean linear intercept (mean linearparticle size for a dispersed phase):
Ea = LJN,
= 4-,VV
(9)
SV
and the mean linear distance between the surface elements of phase a: 0, =(L-La)/Na=4-.
1 - v, SV
For space-filling grains (V, = l), the mean linear grain size (mean linear grain intercept, Heyn's grain size) is:
L
= l/NL = 2s".
(11)
This concept makes it possible to characterize size and distance independently of shape, arrangement, or size distribution. Mean intersect area (Jeffries' grain size) A is an indirect measure related to the length of grain edges per unit volume (RSCHMEISTER and ARZT[1981]). A has no concise meaning with respect to the size of isolated particles if shape is not uniform (and simple). Rather, as can be shown using derivations by DE HOFF11967, 19801and CAHN[1967], see also EXNER[1994], there is a direct correlation between the mean intersect area and the integral mean curvature C,
C,
= 2 r N , = 2rrV,/x
(12)
where NA is the number of intersect area per unit area of the cross section. C, thus is another important global stereological quantity useful for describing microstructural changes (see fig. 21 below, and EXNER[1994] for references). There are numerous other measures for characterising mean size andlor distance. For example, a measure of size frequently used is i? = 3 U 4 , derived by simply relating the radius of a sphere to its mean linear intercept. It is obvious that the result obtained does not give much more useful information than do E or Sv since R is simply the radius of equally-sized spheres with the same specific surface and volume fraction (but not average volume or number) as the (irregular) particles or grains. For space-filling polyhedra, similar size measures can be derived (THOMPSON [ 19721) which, however, are similarly restricted in describing the geometry of real grain structures. Other size parameters discussed in literature (for references and derivations see RHINESand DE HOFF119681, THOMPSON [1972], uNDERwooD [1970], WEIBEL [1980], RSCHMEISTER and ARZT [19811, COSTER and CHERMANT [19891 or OHSERand L ~ R [Z19941) rely on well-defined geometries as well. The assumptions made are usually clearly stated in the original papers but often neglected by users of the final results, which may lead to erroneous conclusions. Measures of distance other than that defined in eq. (10) usually are based on model assumptions concerning the spatial arrangement. For example, nearest-neighbour distances in three dimensions can be calculated from two-dimensional counts if the shape and size of all particles is identical and if the volume fraction is not too high (see, for
Ch. 10, 5 7
Surface micmscopy
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example, BANSALand &DELL [1972]) or if parallel platelets are considered (Roosz et al. [1980] and T ~ R R ~ N [1980]). EN However, if the nearest-neighbour distances of irregularly shaped and distributed particles is to be estimated, no straightforward methods are available. As discussed recently by OHSER and L o u [1994], the transformation of pair correlation functions (see e.g., STOYANet af. [ 19871) from two to three dimensions seems to be a possibility which, however, may prove too complicated for practical applications. Thus, computer simulations of sections through three-dimensional arrangements (for example based on the Poisson-Voronoi mosaique (see OHSERand L o u [1994]) may be a viable alternative to serial sectioning. The situation becomes even more complex when information on the three-dimensional size distributions of grains or particles is required. Direct stereological procedures to derive the 3D distributions from dispibutions of linear intercepts or from intercept areas are available for idealised models in which the assumption is made that all features have constant shape and that this shape is extremely simple (sphere, cube, ellipsoid). Since this situation is rare or at least not typical in real microstructures, these models are not useful in most practical cases. The same is true for other analytical approaches. For example, KING[19821has derived a method for assessing three-dimensional size distributions (cumulative particle mesh size distribution) for irregular particles from planar data, Le., area distribution of sections or projections, or from linear intercept distributions. No a priori assumptions on regular geometric shape, convexity, congruency or particular functional form of the size distributions are made in this widely neglected paper. However, a shapedependent constant must be known or determined experimentally for a uniformly sized sample. Due to this complication, this or other unfolding procedures have not found much interest outside the stereologicalliterature. Nevertheless, solutions to the so-called “tomato salad problem” have been published continuously over the years. A recent approach concerns polyhedric grains and may prove useful under some conditions for grain size characterizationin single-phase materials (ZHAO and CHEN [ 19931, OHSERand MUCKLICH[ 19951). As a general rule, rather than standard text books or original papers, special reviews (e.g., EXNER[1972b], SIMPSON and STANDISH[1977], CRUZ-ORIVE [1978], R ~ I S T Eand R ARZT [1981], EXNER[I9941 or OlmfE~and L o u [19941) should be consulted for representative conversion procedures, their applicability and restrictions, and the related derivations should be followed up before deriving (or even worse, publishing) more of the already abundant solutions. In reality, microstructural elements vary not only in size but also in shape. The shape variation may be determined by growth processes and is often limited by crystallographic conditions. In such cases, simplifying assumptions may be justified and special solutions may be found to solve a particular problem. An interesting example was published el al. [1994], OH~ER and L o u [1994]) where a numerical solution recently (M~CKLICH was presented for establishing the size and shape distribution of hexagonal prisms as present in silicon nitride from the size and shape of the intersect areas. In view of these difficulties, serial sectioning has been used for establishing threedimensional grain and particle size distributions. Alternatively, chemical disintegration or removing of the embedding matrix can be applied to separate grains (e.g., by attacking the grain boundaries of aluminium by liquid gallium as shown by WINES and PATI-ERReferences: p. 1016.
1008
H.E. h e r
Ch. 10, 87
SON [1982]) and to size the isolated features microscopically or by weighing. It is interesting to note that logarithmic normal size distributions have been observed in most cases (see, for example, EXNER[1972b] and RHINESand PATTERSON [1982] or RHINES [1986]) which make possible to describe the size distribution by only two parameters (e.g., the arithmetric mean and the standard deviation). There are a large number of problems in materials research where size, distance and size-distribution are relevant. One of the best-known relationships between microstructural geometry and properties is the Hall-Petch relationship predicting a linear dependence of the yield stress of polycrystalline materials on the square root of reciprocal grain size. However, size distribution has a significant influence on this relationship (KDHLMEYER[1978], G~KHALE and DE Horn [1984], RHINES [1986], see also EXNER [1994]) since deformation starts at the largest rather than the average grain (ESSMANN et al. [1968]). Experimental results confirm the simple form of the equation owing to the fact that the shape of grain-size distributions in most annealed single-phase materials is fairly similar. Examples for the predicted linearity between the yield strength and the reciprocal square m t of the grain size or the particle size are shown in figs. 20a and b. It is interesting to note that prestraining changes the parameters but not the general form of the Hall-Petch relationship (WANGand MURR[1980], see fig. 20a). However, geometric similarity is not a general rule and can not be assumed a priori but needs experimental proof for every material and processing schedule. Large deviations from geometric similarity are frequent. For example, the width of grain-size distributions in recrystallized metals depends strongly on deformation prior to annealing, even at long annealing times (OKAZAKI and CONRAD [ 19721 and RHINESand PATTERSON [19821). This has consequences for the relationships between properties and microstructure which must not be neglected. For aluminium samples, it has been reported by G~KHALE and RHINES [1993] (see also DE Horn [1984], RHINES [1986] and EXNER[1994]) that the variation in size distribution has a much larger influence on the yield strength than the variation in mean grain size by a factor of 2 and that the same is true for the tensile strength. This example clearly demonstrates the importance of a full appreciation of microstructural parameters in theoretical modelling and the ambiguity of well-accepted relationships between microstructure and properties. Another typical example is the dependence of transverse rupture strength of WC-Co alloys on carbide particle size, spacing and contiguity (see EXNER[ 19941). Other useful equations involving mean size and distance are derived for simplified geometries, often taking a model in which grains or particles are assumed to be spheres of uniform size. For instance, the Zener relationship for dispersed particles limiting the grain size (ch. 28),
E . = k&/V, (13) (where Z is the equilibrium grain size in a material with particles of mean linear size Z, and volume fraction V,), is usually derived in this way (PORTER and EASTERLWG [1981]). k is a constant depending on the shape, the size distribution and the arrangement of the particles. The value usually taken (k=4/3) relies on the assumption that the
Ch. 10, 57 LOO 200
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Surface micruscopy Grain size , pm 100 50 LO 30
20
t C L5
0.05 0.1 03 5 ;quare root of reciprocal gmin size, J O e 2
J ''2~
io
15
23
25
30
Reciprocal square root of gmin size ond particle spacing, mm-+
Fig. 20. Experimental correlations between size and distance of microstructural features and thc yield strength of metallic alloys. (a) yield smngth of some metals as a function of square root of mean lincar grain size. CR = cold reduction in percent (data from WANGand MURR [1980] and RAZA[1982], (b) yield strength of unalloyed low-carbon steels a$a function of the reciprocal square mot of the mean particle spacing and of the mean linear grain size (data from LIU and GURLAND [1968] and ANANDand GURLAND[I9761 according to Ehmm and [19881). HOUGARDY
particles are regularly dispersed single-size spheres. Therefore, deviating numbers for k are obtained experimentally. A more sophisticated treatment is needed including size distributions and it seems that Monte Carlo simulations provide a means for assumption-free modelIing. A very promising approach for a generalisation of the Zener equation is to treat the interaction between grain boundaries and matridparticle interfaces by stereological modelling (AIGELTINGER and EXNER[1977], EXNER [1994]. Equations describing the kinetics of grain growth and particle growth by Ostwald ripening or coalescence, the dependence of dendrite arm spacing and lamellar spacing on freezing rate, and several other processes determining the size of microstructural features have been derived on the basis of simplified model geometries some of which are discussed in more detail in other chapters (e.g., ch. 9, 5 15). Experimental proofs given on the basis of quantitative microscopy will be statistically significant and free of systematic errors only if the geometric details of the theoretical models are compatible with the real microstructure. As an example, fig. 21 shows the comparison of size distributions of vanadium carbide particles of spherical shape with predictions made by the Wagner-Lifshitz theory for Ostwald ripening (ch. 15, 0 2.4.2). Perfect agreement for reaction-controlled growth is obtained for long annealing times while significant deviations exist for the as-sintered @e., briefly annealed) state (EXNER119731). This comparison is made on the basis of linear intercepts (EXNER and LUKAS[1971]) rather References: p . 1016.
Ch. 10, Q7
H. E. Exner
-
Annealed
\
Diffusion
\
Reaction
, Normalised intercept LIT Fig. 21. Size distributionof spherical VC particles in a nickel matrix sintered and heat-treated at 1500°C (liquid matrix). Histograms: experimental results; solid curves: theoretical prediction (after EXNER 119731). (a) Sintered 1 h, mean linear intercept = 17.5 pm; (b) annealed 8 h, mean linear intercept =45.5 Fm.
than of sphere diameters, since unfolding of the experimental data generally increases the experimental scatter pronouncedly (NICHOLSON [ 19781). 7.2.4. Orientation, contiguity, shape and other complex parameters In addition to volume fraction and interface density (and the related parameters size and distance for separate features in a microstructure) there is a variety of other, more complex aspects of microstructural geometry. l k o of these aspects, both rather important in many practical problems, can be quantitativelycharacterised using measurements made in plane cross sections, namely orientation and contiguity (degree of neighbourship and skeleton formation) using the concept of weighted interface densities. The degree of orientation is quantified by relating the oriented part of the interface to the total interface of a microstructural component yielding
F, = S, (oriented)/S, (total)
(14)
which, as outlined in early literature (e.g., UNDERWOOD [1970], SALTYKOV [1974], see also EXNER[1994]) can be calculated from directed point counts along perpendicular lines for linearly as well as planarly oriented interfaces (for counting recipes see EXNER [19931). Similarly, the degree of skeleton formation of a microstructural component can be derived by relating the grain boundary density to the total interface, i.e.,
C" = S, (grain boundary)& (total)
(15)
as suggested by GURLAND [1958, 19661, and the degree of neighbourship between two microstructural components, a and p, by relating the interface density between component a and (in a two- or a multiphase alloy) to the total interface of component a , i.e.,
Ch. 10, 87
Surface microscopy
cab= s,, (interface CIP>/S,,(totala>
1Q11
(16)
Useful extensions of the concept of contiguity are the continuous volume and the separated volume (LEEand GURLAND[1978]). Recent reviews of these aspects, their assessment including practical recipes, and a review of applications in materials science are available (EXNER[1993, 19941, FANetal. [1993]). Particularly for characterizing the microstructure of cemented carbides, the concept of contiguity proved indispensible (see, for example, EXNERand RSCHMEISTER [1966], GURLAND 119661, EXNERand GURLAND [1970], or LEEand GURLAND[1978]). Most recently, a new parameter for characterising the degree of connectiveness in two-phase microstructures was proposed by POECHand RUHR [1994]. Based on the skeletonizing procedure provided by mathematical morphology (see 0 7.3) and available with most modern image analysers, the “matricity” of the two phases is defined by the length of the skeleton line in one of the phases divided by the sum of the length of skeleton lines in both phases. The application of this parameter was successfully demonstrated for nickel-silver and martensihaustenite model alloys and for modified aluminium-silicon castings (POECHand RUHR 119941 and POMPE[1994]). The most difficult problem in quantification of microstructural geometry is shape. Though numerous attempts have been made to find parameters for “average” shape in three dimensions, a general solution is not available, and some proposed parameters (see, for example UNDERWOOD [1970, 1976, 19801, RSCHMEISTER [1974], MEDALIAI: 19801, EXNERand HOUGARDY [1988], or EXNBR[1994]) are very limited, while others are of not much use at all since they either cannot be assessed experimentally or are sizesensitive. There are two ways out: one is to use planar shape descriptors for sections or projections as a “fingerprint” for three-dimensional shape, the other is serial sectioning in order to get the three-dimensional reconstructions or to isolate individual grains and particles chemically (RHINES and PATTERSON [1982], DE How [1982]). While the latter techniques are time consuming and only usehl for demonstration purposes or very specific applications, e.g., for proving or disproving a theoretical concept, planar shape characteristics are in widespread use. Classical shape parameters to describe various aspects such as elongation, bulkiness, waviness, or symmetry can be defined on the basis of combining metric properties like area, perimeter, Ferret’s diameter, moments of inertia, etc. (BEDDOWet al. 119801, S C ~ A R Z[1980], EXNERand HOUGARDY [1988], COSTERand CHERMANT [1989], see also EXNER[1994]). Applications of planar shape descriptors to problems in physical metallurgy have remained scarce, but now - with the availability of shape parameters in most automatic instruments - they can be used for descriptive purposes or for establishing empirical correlations for shape-sensitive properties of materials. As an example, a close correlation between fracture strength of eutectic aluminium-silicon alloys and shape of the silicon precipitates has been demonstrated (PAULet al. [ 19821). Other concepts of shape characterization are based on clear mathematical concepts, for example those using Fourier or Walsh coefficient (see, for example, MELOY119801 or FLOOK[19871), those based on mathematical morphology (COSTERet al. E19811, COSTERand CIIERMANT [ 19891, see 5 7.3) or, as a derivative of the presently fashionable References: p . 1016.
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fractal analysis, fractal dimension. Provided that the experimental procedure for the assessment of fractals is sufficiently precise, a unique value characterising the irregularity of interfaces and nonplanar surfaces is obtained. Applications of this concept to problems in materials science are abundand and some interesting examples have been published including fracture surfaces, grain boundaries in deformed metals, phase interfaces after martensite transformations, among other (see reviews by WRIGHTand KARLSSON[ 19821, HORNBOGEN [1986a,b, 19871, or EXNER[1994]). Considering the fact that these concepts are rather abstract from human perception and even more their limitation to planar geometry, it seems rather doubtful that they will be successful in the fields of materials science and technology (see also EXNER [1994]). Serial sectioning has been used to evaluate topological parameters, mainly in connection with studying the grain shape in metals (RHINEs etal. [1974] and RHINESand PATTERSON [1982]) and the development of microstructure during sintering (DE HOFFet al. [1972] and AIGELTINGER and EXNER[1972]). In this way, the number of separate parts (or, in other words, the number of pores or particles) per unit volume can be assessed. Stereological equations relating numbers per unit area or unit length to numbers in three dimensions are usually not reliable since they depend on information concerning shape- and size-distribution. Another useful topological parameter is connectivity (or genus) which describes the number of connections (channels) between nodes (larger regions). Figure 22 shows the results of a study of sintered copper on which the geometric properties of the pore space have been followed up as a function of porosity, i.e., pore volume fraction (AIGELTINGER and EXNER[19721). If chemical disintegration for isolating microstructural features is not feasible, serial sectioning, though extremely time-consuming even with automatic sample preparation, automatic instrumentation and computer processing, may be the ultimate answer to geometric analysis of irregularly shaped microstructural features in three dimensions (DE HOFF[19821). Another important aspect closely related to both shape and size is curvature, Variations of local curvature provide the driving forces for shape changes and coarsening of microstructural features (e.g., fibers, pore systems, precipitations, etc.) but are difficult to quantify. Integral mean curvature, on the other hand, can be measured quickly and unambiguously from cross-sections using equation (12), see 87.2.3. As a sensitive parameter for monitoring the path of microstructural changes during annealing (DE How and ISAWARAN [1982]), it should find much more widespread application. In fig. 22, the variation of curvature during the sintering process can be clearly related to the qualitative and quantitative changes of pore geometry. Curvature distribution, on the other hand, cannot be assessed from planar cross sections, rather, stereometric techniques, must be applied (FEIJOOet al. [1990]). Though very tedious, the distribution of curvatures of dendrite surfaces was measured with interesting results on the development of dendrites during solidification of aluminium alloys, and a comparison with theoretical predictions was made (FEUOOand EXNER[1991]). Finally, there remains the problem of characterizing the arrangement of microstructural features. Again, three-dimensional characterization must rely on simplifying assumptions or on serial sectioning. The classification (random, regular, clustered) and quantitative description of planar arrangement of features defined by a characteristic
a.10, 5 7
1013
Surface microscopy
Pore volume fraction . ~ c t . I
3
LO
20
c , \ electrol tic
spherical
d
electrolytic /
3
60
\
I
80
1
Percent of theor: density ,pct Fig. 22. Changes of metric and topological properties of pore space in sintered copper powder as a function of porosity varied by sintering at 1OOO"C.The spherical and the electrolytic (dendriteshaped) powders show different pore stNctures at comparable porosity. The right part of the figure clearly shows the formation of new contacts (increasing genus) at the early sintering stage and the closure of pore channels (sharply decreasing genus) and formation of isolated pores and their disappearance by shrinking in the later stages. These topological changes are also reflected by the metric properties shown in the left part of the figure: Surface density and integral mean curvature decrease due to smoothening of the pore-solid interface but increase for a short period when the channels start to close and at the very late stages when isolated pores shrink. (After AIGELTINGER and EXPI-ER [ 19721.)
point (e.g., centre of gravity) on the basis of nearest-neighbour distance distributions has been proposed (SCHWARZ and EXNER[1983]) and a number of statistical procedures became available from other fields, e.g., from pattern recognition and cluster analysis. Up to recently, application of arrangement parameters to materials is near to nonexistent. A very interesting exception is a study on the arrangement of carbides in high-speed steels, which has shown the feasibility of simple statistical concepts and the value of arrangement analysis (WERLEPORS et al. [1979]).Mathematical morphology (see $7.3) and, particularly, techniques based upon pair correlation functions and other statistical approaches based on stochastic geometry (see, for example, STOYAN et af. [1987], STOYAN [1990], OHSERand L ~ R Z [1994] or EXNER119941, BERTRAMand WENDROCK [1995] or WENDROCKet al. [1995]) supply a wealth of concise descriptors which, however, are only developed for planar arrangement. As mentioned above (3 7.2.3). most of these approaches seem not a practicability for 3D arrangements and seem rather insensitive to changes of arrangements which are often easily detected by visual References: p. 1016.
1014
H. E. h e r
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inspection, as, for example, lamellar patterns or particles lined up in rows which have a certain degree of irregularity.
73. Mathematical morphology In addition to classical geometric parameters, a special methodology based on “structuring elements”, termed mathematical morphology has been developed, largely in France, into a consistent and practically useful framework for image analysis. A full theoretical account of this field, including many pertinent references, has been published by SERRA [1982, 19881. There is no space here to describe these sophisticated concepts in detail. Attempts to give a comprehensible treatment of image analysis and of image manipulation based on mathematical morphology which, for the materials scientist, are easier to understand than Serra’s mathematical theory, have been published by FISCHMEISTER [1981], COSTERand CHERMANT [1989], Russ [1990, 19921, CHERMANT and COSTER[I9941or CHERMANT [19!94], among others. Some procedures which have been proven useful in practical work are outlined in the following. The microstructure or, rather, its binary image is treated as a set of points which can be transformed by various operations in order to extract characteristic parameters of this set, which then are used to describe the geometry of the microstructure. The effects of the most useful operations, erosion and dilatation, opening and closing are demonstrated in fig. 23. Erosion is achieved by moving the structuring element from picture-point to picture-point and removing all those points which do not fulfill a specified criterion, e.g., that all neighbour points are inside the dispersed features shown in fig. 23a, which, in other words, means removing all points at the edge of the features. By this operation, the two small features 1 and 2 and the bridge connecting two parts of the large particle are eliminated. The area of the large feature has obviously become smaller as shown by the dashed line in fig. 23a. By dikztarion, i.e., by adding all those matrix points which have at least one neighbouring point inside the features, the original size is very closely reconstructed but the two small features are still gone as is the bridge between the two large features (fig. 23b). Therefore, this combined operation is called opening. Similarly, if dilatation is carried out first, followed by erosion (figs. 23c and 23d) the result is reversed and a bridge is formed by this combined operation called closing. It is obvious that this manipulation very effectively allows the elimination of small features or contacts arising from detection noise in automatic instruments or to separate relevant information from the image. Another operation for segmenting connected features is the grussjre transformation (NAWRATHand SERRA [ 19791) allowing identification and counting of separate particles and grain comers by alternating or repeated erosions and dilatations. Another transformation procedure called skeletonizing reduces each feature (e.g., irregular intersect area) to a pattern of lines which then can be analysed with respect to the number of branches and nodes. The ratio of line lengths of the skeletons of two interdispersed phases was suggested by POECHand RUHR [ 19941 to characterise the character of two-phase materials. The concepts of covuriogrums and star funcrion, and a large number of other parameters were suggested for characterizing size, spacing, shape, arrangement and many other aspects of microstructural geometry by
Ch. 10, 57
Surface microscopy
1015
Fig. 23. lko-dimensional image transformations by operations of mathematical morphology (EXNERand HOUGARDY[1988]). E is the element of 9 picture-points (with the reference point in the centre) by which the following operations are carried out: (a) Original features and eroded outline (dashed) of feature 3; (b) dilated outline of eroded fig. 23a; (c) dilated outline of fig. 23b; (d) eroded outline of fig. 23c. Combined operations are called opening (a and b) and closing (c and d).
SERRA[1982, 19881 and COSTERand CHERMANT [1989] and a large number of authors which tried to apply mathematical morphology to practical problems. Most information obtainable in this way should be strictly considered as fingerprinttype information on microstructural geometry ( F I S C H M E T ~ J [X19811) owing to the lack of quantitative relationship with three-dimensional characteristics of the microstructure and to uncontrolled effects of image distortion. The stereological aspects of mathematical morphology have been discussed by SERRA[1982, 19881 and COSTERand CHJZRMANT [1989], among others. In general, mathematical morphology is very useful in all those cases where changes of microstructural geometry need to be monitored without seeking 3D information. In cases where physical modelling of microstructural changes is attempted, stereological parameters should be preferred. This statement seems to remain valid in spite of the later attempts to clarify the stereological aspects of mathematical morphology culminating in the mathematically correct but unpracticable fact that all stereological equations could be derived from mathematical morphology and thus are a sub-set of this concept. More recently, mathematical morphology has been extended to the analysis of greytone pictures and certainly will further proceed to colour images and other applications not directly relating to microstructures. As mentioned above (0 7.1), image modification by morphological processing (filtering, cleaning and many other amendments) are indispensible applications of mathematical morphology used in most modem image analysers. Refemnces: p. 1016.
1016
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Ch. 10, Refs.
7.4. Further aspects
Generally, more than one single geometric aspect is of relevance in characterizing the microstructural geometry of multiphase materials, and a combination of parameters will be needed to give a full description. Such combinations have been proposed for studying the evolution of microstructure (for example, during sintering, see fig. 22 and ch. 31) or for empirical correlations with mechanical properties (for example, those of cast and annealed aluminium alloys (PAULer al. [1982]), of cemented carbides (EXNERand GURLAND [ 19701) or of sintered steel (EXNER and POHL[ 19781). In addition to these and other examples cited above, successful applications of quantitative characterization of microstructural geometry have been reported in the literature; some of these have been reviewed in overviews, for example by FISCHMHSTER [ 19721, SALTYKOV [1974], HOUGARDY [ 19751, LAFOND and M o m [19751, RHINE~[1976, 19861, MATHY[19773, E x m [1978a, 1994, 19951, ONDRACEK [1978, 19821, Russ [1986, 1990, 19921, EXNER and HOUGARDY 119881, or C O ~ and R CHERMANT [1989], among others. Quite often, however, data are published not so much to aid the solution of a materials problem but rather to demonstrate the feasibility of a new technique or a new instrument. Considering the high standard of methodology now achieved, it seems timely to utilize quantitative analysis of microstructure more directly as a clue for understanding the behaviour of materials during processing and in application.
References ABRAHAM, E E. 1. P. BATRAand S. Cnuc~,1989, Phys. Rev. Lett. 60, 1314. AOAMS,B. L., S. 1. WRIGHTand K. Kmze, 1993, Metall. Trans. 24A, 819. AIOELTINGER,E. H., and H. E. EXNER,1972, 2.Werkstofftechn. (J. Mater. Tech.) 3, 425. AIOELTINGBR, E. H., and H. E. EXNER,1977, Metatl. Trans. 8A, 421. ALDINGER,E, 1%9. Metallography 2,363. ANAND.L., and J. GURUND, 1976, Metall. Trans. 7A. 191. ARNOLD,W.,1991, h p h y s . News 22, 171. ARZT, E., and H. E FISC: , 1979, Pract. Metallogr. 16, 547. -, E. A., ed.,1980, Scanned Image Microscopy (Academic, London). ASM HANDBOOK,1992, Vol. 10, Materials Characterization (ASM International, Metals Park). AXELROD,D., N.L. THOMPSON and T.P. BURG~URDT,1983, J. Microsc. 129, 19. AYDIN,I., and H.E. BOHLER,1981, in: Metallographie - Techniken der Gefiigebeschreibung bei metallischen and keramischen Werkstoffen, F’ract. Metallogr. Special Issue 12, eds. H.E. Biihler, K. Dieser and W.U. Kopp (Dr. Riederer Verlag, Stuttgart) p. 35. AYDIN. I., H.E. BCIILER,G. EL~SNERand I. STAPF,1983, Pract. Metallogr. 20, 105. A Y D ~I.,. and H. E. B ~ H L E R1984, , Pract. Metallogr. 21, 243. AZZAM,R. M. A., and N. M. BASHARA, 1977, Ellipsometry and Polarized Light Microscopy (North-Holland, Amsterdam). BAKER, M. A., 1993, Surf. Interf. Anal. 20, 535. BAKER,M.A. and P. TSAKIROWULUS, 1993, Surf. Interf. Anal. 20. 589. BAKER, M. A., and J. E. CASTLE,1994, in: Materials Science and Technology, Vol. 2B, ed. E. Lifshin (VCH, Weinheim) p. 219. BAN~AL, R. P., and A. J. ARDELL,1972, Metallography 5.97. BARDA,J., and E F. FAN,1993, in: Scanning Tunneling Microscopy and Spectroscopy,Theory, Techniques and Applications, ed. D.A. Bonnell (VCHPubl., New Yo&) p. 287.
Ch. 10, Refs.
Surface microscopy
1017
BARTZ,G., 1973, Pract. Metallogr. 10. 311. BAUER,B., and A. HALLER.1981, Pract. Metallogr. 18, 327. BAL% B., and H. E. EXXER,1981, Stereolog. Jugosl. 3, Suppl. 1, 255. BAUER.B., M. FRIPANand V. SMOLEI, 1982, in: Fracture and the Role of Interfaces, vol. 2. Fatigue, eds. K.L. Maurer and F.E. Matzer (Engineering Materials Advisory Services, Warley) p. 591. BAUER,B.. and B. EGG, 1984, Pract. Metallogr. 21,461. BEATON,C., and B. FUSIE, 1983, Pract. MetaJlogr. 20, 31. BECKERT.M., and H. KLEWV, 1984, Handbuch der metallographischcn Atzvcrfahren (Deutscher Vcrlag Nr Grundstoffindustrie, Leipzig. Stuttgart). BEDDOW,J. K., M. D. NASTAand G. C. PHILIP, 1980, in: .Testing and Characterization of Fine Particlcs. eds. J.K. Beddow and T. Meloy (Heyden, London) p. 44. BELL, L. D., W.I. KAISER, M.H. Hem~ and L. C. DAVIS, 1993, in: Scanning Tunneling Microscopy, eds. J.A. Stroscio and W.J. Kaiser (Academic Press, Boston) p. 307. E, and A. I ~ B E R G E R 1965, , Pract Metallogr. 2, 31. BENFSOVSKY, BERAHA,E., 1970,Pract. Metallogr. 7. 131 and 242. BIXAHA, E., and B. SPIGHLER,1977, Color Metallography (ASM, Metals Park, OH). BERNST, R. 1965. h t . MetaJlogr. 2, 162. BERTRAM, M., and H. WENDROCK. 1995, in: Fortschritte in der Metallo,gaphic, Pract. Metallogr., Spec. Issuc 26, ed. A. Kneissl (Carl Homer Verlag, Miinchen) p. 375. B-, H., 1974, Theorie und Praxis der Interferenzmihskopie (Akademische Verlagsgesellschaft Geest und Portig, Leipzig). Biaas, W.D., 1970, in: Physical Metallurgy, 2nd Ed., ed. RW. Cahn (North-Holland, Amsterdam) p. 655. BINNIG,G., H. ROHRER,C. GERBERand E. WEIBEL.1982, Phys. Rev. Lett. 40,178, and 49, 57. BINNIG,G., and H. ROHRER.1982, Hclv. Phys. Acta 55, 726. BINNIG,G., C.F. QUAIXand C. GERRER,1986, Phys. Rev. Lett. 56, 930. BISNIG,G., and H. ROHRER,1987, Rev. Mod. Phys. 59, 615. BIRNER,E., and R. WHBERO, 1980. Pract. Metallogr. 17,14. .,R. and H. WALTINGER, 1971, Beitr2ge elektronenmikroskopischeDirektabbildung von Obertllchen, BLASCHK~ vol. 4/2 (Verlag R.A. Remy, MUnster) p. 425. BLASCHKE, R., 1979, Beitrtige elektrunenmikroskopischeDirekiabbildung von Oberfkhen, vol. l U l (Verlag R.A. Rcmy, MUnster) p. 359. BONNELL, D. A., ed., 1990, Scanning Tunneling Microscopy, Theory and Practice (VCH, Pub]., New York). BONNELL, D.A, ed., 1993% Scanning Tunneling Micmscopy and Spectroscopy (VCH Publ., New York). BONNELL. D. A,, 1993b. in: Concise Encyclo@a of Materials Charactcrization, eds. R.W. Cahn and E. Lifshin (Pergmn. Oxford) p. 427. BONNELL,D.A., 1 9 9 3 ~ .in: Scanning Tunneling Microscopy and Spectroscopy, Theory, Techniques and Applications, ed. D.A. Bonnell (VCH Publ., New York) p. 7. B o w ~ e r r ,K. M. and D. A. SMITH. 1970, field Ion Microscopy (North Holland, Amsterdam). BOWMAN. M. J., and A. D. BOOTH,1971, Metallography 4, 103. BOYDE, A., 1973, J. Micmsc. 98,452. BOYDE.A., 1985, Science 230, 1270. BOYDE,A,, 1990, in: Mtdern Microscopies: Techniques and Applications, eds. P. J. Duke and A. G.Michette (Plenum Press, New Yo& London) p. 185. BRADBURY, S., 1991, An Introduction to the Optical Microscope (Oxford University Press, Oxford). G.J., P.BUM, and P. BAREND$1979, J. Microsc. 117,219. BRAKENHOW, S.J., 1978. Surf. Sci. 70, 427. BRENNER, BRIOGS,D., and M.P. %AH. 1990, Practical Surface Analysis (Wiley, Chichester). BRICOS,A., 1992, Acoustic Microscopy (Oxford University Press, Oxford). BRIGGS,A,, 1994, in: Materials Science and Technology, Vol. 2B, ed. E. Lifshin (VCH, Weinheim) p. 241. BROOKER. A.D., J. E. CASTLE, J. M. COIEN and M.B. WALDRON,1984, Met. Technol. 2, 66. BROWN.L. M., M.Y o m &IAN and M. M. Chaudri, 1988, Philos. Mag. ASI, 187. BROWN,R., and G. C. SMITH,1982, Metallography 15.269. BROOEL,H.,J.R. Ross and G. KONIG, 1988, Pract. Metallogr. 25,578.
1018
H.E. h e r
Ch. 10, Refs.
BR0.. and J. NIEBER,1975, Microchim. Acta, Suppl. 6.345. B R O M ~O., , J. HEYDENREICH, K.H.KREBSand H.G. SCHNEIDER, e&., 1980, Handbuch Festk6rperanalyse n i t Elektronen, Ionen und Rontgenstrahlen (Vieweg, Braunschweig). BOCKLK, H., 1964, Berg- und HUttembn. Monatsh. 109,72. BOCKLE,H., 1965. Mikrohtlrtepriifung und ihre Anwendung (Verlag Berlin Union, Stung&). BUCKLEY,C. J. and H. RARBACK, 1990, in: Modem Microscopies, Techniques and Applications, eds. P.J. Duke and A.G. Michette (Plenum Press, New York) p. 69. BOHLER,H. E., and H. P. HOUGARDY. 1979, Atlas der Interferen~~hichten-Metallographie @GM Informationsgesellschaft, Oberursel). 1980, Atlas of InterferencbLayer Metallography (DGM InformationsBOHLER, H.E., and H. P. HOUGARDY, gesellschaft, Oberursel, and 'Ihe Metals Society, London). BUHLER,H.E.. 1981, in: Microstructural Science. vol. 9, eds. G. Petzow, R. Paris, E.D. Albrecht and J.L. McCall (Elsevier North-Holland, New York) p. 19. BWLER, H.E., and D. KOSSEL,1981, Pwct. Metallogr. 18,385. H. E., 1986, h c t . Metallogr. 23, 309. BOHLER, BURNHAM, N. A.. and R. J. COLTON,1993, in: Scanning Tunneling Microscopy and Spectroscopy, Theory, Techniques and Applications, ed D.A. Bonnell (VCH Publ., New York) p. 191. CAHN.J. W., 1967, Trans. Metall. Soc. A M 239, 610. CAHN,R. W., J. E. E v m J. PATTERSON, R. E. SOhlEKH and C. K. JACKSON,1980, J. Matcr. Sci. 15, 702. CAHN, RW., and E LIPSRM,eds., 1993, Concise Encyclopedia of Materials Characterization (Perpamon, Oxford). CALVO, EA., A. J. CRIAW.J. M. GOEZ DE SALAZAR and E MOLLEDA, 1987, Metallography 20,213. CARLSON,T. A.. 1975, Photoelectron and Auger Spectroscopy (Plenum, New York). CATALDI, T.R.L. I.G. BLACKHAM. 0.A. D. BRIGGS,J. B. ~THICA and H.A.O. HILL, 1990, J. Electroanal. Chem. 290, 1. CEREZO, A., J. M. HYDF,M. K. MILLER, S. C. PETTS, R P. S t m A , G. D. W. SMITH,1992, Phil. Trans. Royal Soc. A341, 313. C m u ) , A., and G. D. W. SMITH,1994, in: Materials Science and Twhnology, Vol. 2B, ed. E. Lifshin (VCH, Weinheim) p. 513. CHANG,S.L., H. J. QUEKSER, H. BAUMGART, W. HAGENand W. HARTMANN, 1982, Phil. Mag. 46. 1009. C m a , P.C., S. P. NEWEIURRY, H. G. Klhf and I. S. HWANG,1990, in: Modem Microscopies, Techniques and Applications, eds. P.J. Dukc and A.G. Michette (Plenum Press, New York) p. 87. C H E R M AJ.L.. ~ , 1994, Rev. Mktallurgie 91, 201. CHER~~ANT, J. L., and M. C m , 1994, Acta Stereol. 13, 125. CIIERNUSKA, J.T., and T. F. PAGE.1985, J. Am. Ceram. SOC. 68, C196. C~rescOE,D., and P. J. GOODHEW, 1990, The Operation of Transmission and Scanning Electron Microscopes (Oxford University Press, Oxford). CLARKE, T. M.. in: Metals Handbook, 9th ed., Vol. 12, Fractography (American Society for Metals, Metals Park) p. 78. CONN,G. K.T., and F. J. BRALWAW, 1952, Polarized Light in Metallography (Buttcnvorths, London). CONOR,P. C., 1972, Metallography 5, 301. COSSLETT,V. E.. and W. C. NIXON,1960, X-ray Microscopy (Cambridge University Press). C m R , M.,and J.L. CHER~MNT,1989, W i s &Analyse &Images (Resses du Centrc National de la Recherches Scientifique, Paris). CROUSE, R. S., R. J. GRAYand B. C. LESLIE,1977. in: Interpretive Techniques in Microstructural Analysis, eds. J.L. McCall and P.M. French (Plenum, New York) p. 43. C R U Z - ' O R ~LM., , 1978. J. Micmc. 112, 153. CZAhDERNA, A.W., ed., 1975, Methods of Surface Analysis (Elsevier, Amsterdam). and K. SPYRADEK, 1992, Pract. Metallogr. 29.35. CZYRSKA-FILE~~ONOWICZ. A.. S. GORCYCA DANIWN. P. E., 1985. in: Metals Handbook, Vol. 9, Metallography and Microstructures (American Society for Metals, Metals Park) p. 142. DANN~HL, H.D., R. GRABER and L. WEGLIANN, 1971, Microstructures 2, 15. DAVIDSON, D. L., 1974, in: SEM 1974, Proc. 7th Annual SEM Symp. (a Research Institute, Chicago) p. 927.
Ch. 10, Refs.
Surface microscopy
1019
DE HOFF, R.T., 1967, Trans. Metall. Soc. AIME 239. 617. eds., 1968, Quantitative Microscopy (McGraw-Hill, New York). DE HOFF, R. T., and F. N. RHI~FS, DE HOW, R.T.. 1972, in: Treatise on Materials Science and Tcchnology, Vol. 1. ed. H. Herrrnann (Academic Press, New York, London) p. 247. and K. R CRAIG,1972, J. Microsc. 95.69. DE HOFF, R.T., E. AIGELTINGER DE HOW, R.T., 1980, Mikroskopie 37.32. DE HOFF,R.T., 1982, paper presented at Stereology 82, Symp. Int. Soc. for Stereology, Shefield. DE HOFF,R.T., and C.V. ISAWARAN,1982, Metallurg. Trans. l3A, 1389. DE HOW, R. T.. 1984, in: Practical Applications of Quantitative Metallography, eds. J.L. McCall. J.H. Steele (American Society for Testing and Materials, Philadelphia) p. 146. DE HOFF, R. T., 1989, in: Science of Sintering: New Directions for Materials Processing and Microstructure Control. eds. D.P. Uskovic, M. Palmour and R.M. Spriggs (Plenum Press, New York, London) p. 55. DE MODE,W.H., and J. W. S m , 1981, J. Appl. Phys. 52, 3969. DENGEL. D., 1973, Z. Werkstofftechn. (J. Mater. Techn.) 4, 292. DIESER,K., 1984. Z . Werkstofftechn. 15, 177. 1988, Pract. Metallogr. 25,374. DIESER,K., and E. HERRYANN, DIESER,K., and E. MATI’HAR.1988, Microstnrctural Science, Vol. 16, Metallography of Advanced Materials, eds. H.J. Cialone. G.W.E.Johnson, M.E. Blum and G.F.Vander Voort (ASM International, Metals Park) p. 305. DINARM, N. J., 1994a. Nanoscale Characterization of Surfaccs and Interfaces (VCH, Weinhcim). DINARDO,N. J., 19944 in: Materials Science and Technology, Vol. 2B. ed. E. Lifshin (VCH, Weinheim) p. 1. DINGLEY,D. J., C. HARPER and S. LONG, 1982, in: Electron Microscopy and Analysis 1981, Conf. Series 61, ed. M.J. Goringe (The Institute of Physics, Bristol and London) p. 63. DOERR,R. M., and P. D. OWNBY,1975, F’ract. Metallogr. 12, 78. DOmn, R. D., and R. W. C m , 1972, J. Less-Common Met. 28, 279. DOYEN, G., and D. DKAKOVA, 1995, The Physical Principles of STM and AFM Operation (Akadcmic-Verlag, Berlin). DUKE,P. J., 1981, Proc. Roy. Microsc. Soc. 16, 186. DUKE, P. J.. and A.G. MICHETTE(eds.), 1990, Modem Microscopies: Techniques and Applications (Plenum Press, New York, London). EDELMANN,C., 1980, in: Handbuch Festk6rperanalyse mit Elektronen, Ionen und RGntgenstrahlen, eds. 0. Blammer, J. Heydenreich, KH. Krebs and H.G. Schneider (Vieweg, Braunschweig) p. 263. FKELUND, S., and S. HERTZMANN, 1981, in: Computers in Materials Technology, ed. T. Encson (Pergamon Press, Oxford) p. 153. ELSBRDCK,J. B., and L. J. BALK, 1982. BeiMge elektrunenrnikroskopische Direktabbildung von Obcrffkhen, vol. 15 (Verlag R.A. Remy, Mhster) p. 17. ELSSNER, G.. G. KILSLEK and H.G. KAPITZA, 1980, Pract. Metallogr. 28, 443. ELSSNER, G., and W. U. KOPP, 1984, Pract. Metallogr. 21, 633. E N G ~H., I., 1958, Arch. Eisenhtittenw. 29, 73. ESSMANN, U., M. RAW and M. WILKENS,1968, Acta Metall. 16, 1275. E ~ N E RH., E., and H. F.FISCHMEISTER, 1966, Z. Metallkde. 57, 187. EXNER,H. E., and H. E FISCHMEISTER, 1966b, Arch. Eisenhiittenw. 37, 417. EXNER,H. E., 196% Trans. Met. Soc. AIME 245,677. EXNER,H.E., I%%, Pract Metallogr. 6,639. EXNER. H. E., and J. GURLAND,1970, Powder Metallurg. 13, 13. EXNER, H.E., and H.L. L w , 1971, Metallography 4, 325. EXNER,H. E., 1972a, in: Stereology and Quantitative Metallography, ASTM Spec. Tech.Publ. 504 (American Society for Testing and Materials, Philadelphia). EXNER,H.E.. 1972b. Int. Metallurg. Rev. 17, 111. EXNER, H. E., 1973, Z. Metallk. 64. 273. EXNER,H. E., and D. PoIiL, 1978, Powder Metallurg. Int. 10. 193. EXNER,H. E., 1978a, in: Quantitative Metallography (Associazione Italiana di Metallurgia, Milano) p. 313. EXSER, H.E., 1978b. Pract. Metallogr. 15, 15.
1020
H.E. Emer
Ch. 10, Refs.
EXNER,H.E., and J. ROTH, 1980. Pract. Metallogr. 17,365. EXNER,H. E., H. BACKand J. R o m , 1980, Pract. Metallogr. 17, 344. EXNER,H. E., and H. P. HOUGARDY, 1983, Quantitative Image Analysis of Microstructures@GM Infoxmationsgesellschaft. Oberursel). EXNER,H.E.. 1985, in: Metals Handbook, 9th ed., Vol. 9; Metallography and Microstructures (American Society for Metals. Metals Park) p. 89. EXNER,H.E., and M. FRPAN,1985, J. Microscopy l39, 161. EXNER,H.E., 1987, Acta Stereol. 6, Suppl. 11, 179. EXNER,H. E.. and E A. GIESS, 1989, in: Science of Sintering: New Directions for Materials Processing and Microstructure Control, eds. D.P.Uskovic, M. Palmour and R.M. Spriggs (Plenum Press, New York, London) p. 55. EXNER,H. E., ed., 1993, Pract. Metallogr. 30. 216, 287. 322 and 389. EXNER, H. E., 1994, in: Materials Science and Technology, Vol. 2B, ed. E. Lifshin (VCH, Weinheim) p. 281. EXNER,H. E.. 1995, in: Fonschritte der Metallographie, Pract. Metallogr., Spec. Issue 26, ed. A. Kneissl (Carl Hanser Verlag. Miinchen) p. 49. FAN,Z.. A.P. MIODOWNKand P. TSAKIROPOUU)~, 1993, Mater. Sci. Technol. 9, 1094. FEDER,R,E. SPILLER,J. TOPALIAN, A. N. BROW W. GUDAT,B. J. PANESSA, Z. A. ZADUNAISKY and J. SEDAT, 1977, Science lW, 259. FELJOO, D., B. BAUERand H. E. EXNER, 1990, J. amp.-Assist. Microsc. 2, 3. Moo, D., and H.E. E m , 1991, J. CIyst. Growth. 1l3.449. FEUOO, D., and H.E. EXNER,1992, Pract. Metallogr. 27, 605. FISCHMEXS~ER, H., 1965, Pract. Metallogr. 2, 257. R S C H M E I ~H., R , and H. E. EXNER,1%6, Arch. Eisenhiittenw. 37, 499. FISCHMELSTKR, H., 1972, J. Microsc. 95, 25. R S C H M E I ~H., F., 1974, Z. Metallk. 65, 558. FISCHMEISTER, H., 1981, in: Computers in Materials Technology, ed. T. Ericson (Pergamon Press, Oxford) p. 109. FISCAMEISTER, H. F., and E. ARZT, 1981, in: GefUge der Metalle (DGM Informationsgesellschaft, Obenmel) p. 177. FIScHbfEm, H.F., S. KARAGOZ,I. LE IM, J. PAULand M.H. POECH, 1986, Acta Stereol. 5,287. FWZHMEIS~W. H.F., 1988, Fresenius Z. Anal. chem 334,421. FISCH~~EISTER, H.F., S. KARAG~Z and H.D. AND&, 1988, Acta Metall. 36,817. F ~ KA. ,G., 1987, Acta Steml. 6, Suppl. III, 1009. FOREMAN, L. R., N. E. ELLIOTTand D. M. STUPIN,1993, in: Concise Encyclopedia of Materials Characterization, 4 s . R.W. Cahn and E. Lifshin (Pergamon Press, Oxford) p. 238. FRANK,D.G.and A.T. HUBBARD, 1993, in: Concise Encyclopedia of Materials Characterimtion, eds. R.W. Cahn and E. Lifshin (Pergamon Press, Oxford) p. 4. FREMIJNT, P., J. SWCARand J. VARHANICEK,1980, h c t . M d o g r . 17,497. WD, H., ed., 1960, Handbuch der Mikroskopie in der Tcchnik (Umschau Verlag, Frankfurt) Band 1, Die optischen Cirundlagcn, die Instrumente nnd Nebenapparate fiir die Mikroskopie in der Technik, Teil 2, Allgemeines Instrumentarium der Auflichtmikmskopie. FREUND, H., ed., 1969, Handbuch der Mikmskopie in der Technik (Umschau Verlag. Frankfurt) Band 3, Mikroskopie der rnetallischen Werkstoffe, Teil 2. Qualitative und quantitative Untersuchungsverfahrenin der Mctallkunde. FULRATH,R. M., 1972, in: SEM 1972, Proc. 5th Annual SEM Syrnp. (IIT Research Institute, Chicago) p. 17. FUNDAL,E., and S. GROSS,1993, Pract. Metallogr. 30, 294. GABRIEL, B.L., 1987, in: Metals Handbook, Vol. 12: Fractography (American Society for Metals, Metals Park) p. 166. GABRIEL,B. L.. 1992a. SEM: A User’s Manual for Materials Science (American Society for Metals, Metals
Park). GABRIEL. B. L., 1992b. in: SEM: A User’sManual for Materials Science (American Society for Metals, Metals Park) p. 137. GAHM,I., 1%9, Zeiss-Milteilungen 3, 40.
Ch. 10, Refs.
Surface microscopy
1021
G m , J., 1971, Zeiss-Mitteilungen 5, 249. m I., 1975, in: Quantitative Gefdggeanalyse in Medizin, Biologie und Materialentwicklung, Rad. Metallogr., Special Issue 5, ed. H.E. Exner (Dr. Riederer Verlag, Stuttgart) p. 29. GAHM,H., a d E JEGLITSCH, 1981, in: Microstructural Science. vol. 9, eds. G. pebow. R. Paris, E.D. Albrecht and J.L. McCall (Elsevier North-Holland, New York) p. 65. GAHM,H., E JEOLITSCH and H. HORL, 1982, Pract Metallogr. 19,369. GALOPIN,R., and N. E M. HEh'RY, 1972, Microscopic Study of Opaque Minerals (Heffer. Cambridge,). GARCIA, N.. ed., 1987, Proc.First Intern.Conf. on Scanning Tunneling Microscopy, Surf. Sci. 181, 1. GASTALDI, I. and C. JOLTRDAN, 1984, in: Applications of X-Ray Topographic Methods, eds. S. Weissmann, E Balibar and J.F. Petroff (Plenum Press, New York) p. 273. GAUDIG,W.. and R. SCHECK,1984, Pract. Metallogr. 21, 624. GIESSIBL,J., 1995, Science 267,68. GILMORE, R., 1993, in: Concise Encyclopedia of Materials Characterization, eds. R.W. Cahn and E. Lifshin (Pergamon Press, Oxford) p. 4. GLICKSMAN, M. E, R.N. SMITH, S. P. MARSH and R. KULIK,1992. Metall. Trans. A, A23,659. GOKHALE, A.M., and E N . RHINLS, 1983, in Microstructural Science, Vol. 11, eds. RT. De Hoff, J.D.Braun and J.L. McCall (Elsevier Publ., Amsterdam) p. 3. GOLDSTEIN,J. I., and H. YAKOWITZ, eds., 1977, Practical Scanning Electron Microscopy (Plenum, New York). GOLDSCEZN,J. I., D. E. NEWBURY, P. ECHLIN, D. C. JOY,A. D. ROMIG, C. E. ZYMAN,C. FIORIand E. LTFSHIN, 1994, Scanning Electron Microscopy and X-Ray Microanalysis (Plenum Press, New York). GOODHEW, P. J., 1973, Specimen Preparation in Materials Science (North-Holland, Amsterdam, and American Elsevier, New York). GRABATIN,H., H.D. KUNZE and J. RICKEL,1983, Pract. Metall. 20,609. GI&, I., M. POHLand W.G.BURCHARD,1993, Pract. Metallogr. 30,469. G R D , I., 1981. in: Microstnrctural Science. vol. 9, eds. G. Petzow, R. Paris, E.D. Albrecht and J.L. McCall (Elsevier North-Holland, New York) p. 45. GRASRBAUER,M., 1989. Surface. and Interface Analysis (Technical University, Vienna), cited from ORTNER and WILIIARTITZ [1991]. GROP., H. J. MAMINand D. RUGAR.1992, in: Scanning Tunneling Microscopy II, eds. R. Wiesendanger and H.J. GiIntherodt (Springer, Berlin) p. 151. G R ~ N E R G., , and H. J. SCH~LLER, 1969, Pract. Metallogr. 6,346. GUNTHEKODT,H. J. and R. WESENDANGER. eds., 1992, Scanning Tunneling Microscopy I, (Springer-Verlag, Berlin). GLXLAND,J., 1958, Trans. Metall. SOC. AIME 212, 452. GURLAND,J., 1966, Trans. Metall. Soc. AIME 236,642. HAASEK,P., 1985, Metall. Trans. A, 16A, 1173. HAASEN,P. and R WAGNER, 1985, AM. Rev. Mater. Sci. 15,43. HAMERS,R. J., 1993, in: Scanning Tunneling Microscopy and Spectroscopy, Theory, Techniques and Applications, ed. D.A. Bonnell (VCHPubl., New York) p. 51. HANSMA,P. K., ed., 1982, Tunneling Spectroscopy: Capabilities, Applications, and New Techniques (Plenum Press, New York). HANSMA, P. K., and J. TERSOFF, 1987, J. Awl. Phys. 61. R I . HARRIS,L.A., 1968, J. Appl. Phys. 39, 1419. HARTSHORNE, N. H., and A. STUART,1952, Crystals and the Polarising Microscope (Edward Arnold, London). HEARLE,J. W. S., J. T. SPARROW and P. M. CROSS.1972, The Use of Scanning Electron Microscopy (Pergamon Press, oxford). HERBSLEB.G., and P. SCHWAAB. 1978. Pract. Metallogr. 15, 213. HEROLD~CHMIDT, U., 1988, Pract. Metallogr. 25, 3. HLYWOOD,V. H., ed., 1971, Scanning Electron Microscopy - Systematic and Evolutionary Applications (Academic, London). HILLIARD, J. E., 1976. in: Fourth International Congress for Stereology, NBL Spec. Techn. Publ. 413, eds. E.E. Underwood, R De Wit and G.A. Moore (National Bureau of Standards, Gaithenbnrg) p. 59. HILLJE.G., and G. R L D ~ W ,1976, Pract. Metallogr. 12. 629. G
1022
H. E. Exner
Ch. 10, Refs.
Hnurm, T.. 1979a. Pract Metallogr. 16,476. HILLLIER, T.. 1979b. W t . M d o g r . 16, 521. HILLMER,T., 1982, Pract. Metallogr. 19, 509. H&HE. H. R., and 0. BRBMMKR, 1980, in: Handbuch Festktkperanalyse mit Elektronen, Ionen und R6ntgenstrahlen, eds. 0. BrUxnmer, J. Heydenreich, K H . Krebs and H.G. Schneider (Vieweg, Brdunschweig) p. 57. HOEKSTRA, S., and W. H. J. BRUIS, 1979, Pract. Metallogr. 16.583. HOFYAXN.S.. 1980, Surf. Interf. Anal. 2, 148. HOLIK, A. S., 1993, in: Concise Encyclopedia of Materials Characterization, eds. R.W. Cahn and E. Lifshin (Pergamon Press, Oxford) p. 286. HOLT,D.B. and E M . SABA, 1985, Scann. Electr. Microsc. 3, 1023. HONO,K., T. HASHIZUME, Y. HASEGAWA, K. H ~ N and O T. SAKURAI,1986, Scripta Metall. 20,487. HONO,K., T. SAKURAI and H. W. PICKERING, 1989, Metall. Trans. A, 2OA, 1585. HORNDOGEN. E., 1968, Pract. Metallogr. 2, 51. HORNBOGEN,E., and G. Pmzow, 1970.2. Metallkde. 61.81. HORNBOOEN, E., 1981, Z. Metallk. 72, 739. HORNBOOEN. E., 1984, Acta Metall. 32, 615. HORNBOOEN, E., 1986a, Pract. Metallogr. 23, 258. E., 1986b, J. Mater. Sci. 21, 3737. HORNBOGEN, HORNBOOEN,EL, 1987,Z. Metalllrde. 78,622. HORNBOGEN, E., 1989, Intern. Mater. Rev. 34, 277. HORNBOGEN, E., and G. Pmzow, 1991, Pract. Mctallogr. 28, 320 and 383 (reprint of HORNB~GEN and -W [ 19701 with English translation). HOUGARDY, H.P., 1975, Research Film 8, 444. HOUGARDY,H. P., 1976, in: 4th Int. Congr. for Stereology,NBL Spec. Techn. Publ. 431, eds. E.E.Underwood, R. De Wit and G.A. Moore (National Bureau of Standards, Galthersburg) p. 141. HOUGARDY.H. P., 1981, in: Metallographie: Techniken der Gefllgebeschreibung bei metallischen und kenunixhen Werkstoffen, Pract. Metallogr. Special Issue 12, eds. H.E. BUhler, K. Dieser and W.U. Kopp (Dr. Riederer Verlag, Stuttgart) p. 9. HOWELIS, M. R, C. JAWWN, J. Kmz. K MCQUAID and S. S. ROTHMANN,1990, in: Modem Microscopies, Techniques and Applications, eds. P.J. Duke and A.G. Michette (Plenum Press, New York)p. 119. HUB, G.M., R J. COLTOX, E. MEYERand H. J. GCNTHEKODT,1993, Mat. Res. Bullet. 18, HUMMEL,R. E., 1983. phys. stat. sol. (a) 76, I I . HUN-DE.2.. and W. JING-YUN, 1980. Pract. Metallogr. 17, 608. IBACH, H., 4.. 1977, Electron Spectroscopy for Surface Analysis (Springer. Berlin). IRANI, R. S.,and R. W. CAHN,1971. Metallography 4.91. J A W . B., 1993, Digital Image Processing: Conccpts, Algorithms and Software Applications (Springer Publ., Berlin). JEGLITSCH, F., and R. MITSCITE,1967, Radex Rundschau, 587. JEGLITSCH, E, 1968, in: Handbuch der Mikroskopie in der Technik, Band 3. Teil 1, ed. H. Freund w m x h a u Verlag, Frankfurt), p. 187 and 247. JEGLITSCH,E, 1989, Pract. Metallogr. 26, 389. JWr, N., and E.HORNBOGEN, 1988, Pract. Metallogr. 25, 157. JOY,D.C., 1992. in: Materials Science and Technology, Vol. 2A; 4. E. Lifshin (VCH, Weinheim) p. 221. KAISER,W. J., L. D. BELL, M. H. HEcm and L. C. DAVIS, in: Scanning Tunneling Microscopy and Spectroscopy, Theory. Techniques and Applications, ed. D.A. Bonnell WCH Publ., New York) p. 251. K A R A G ~ ZS.,, and H. FISCIIMEISTER,1987, Steel Research 58, 46. I ~ L L X , H.E., 1977, in: Interpretive Techniques for Microstructural Analysis, eds. J.L. McCall and P.M. French (Plenum, New York) p. 105. KELLOG, G. L.. and T.T. TSONG, 1980, J. Appl. Phys. 51, 1184. KG., L. G ~ N Eand R G. ELSSWER,1978, in: Metallographie und Keramographie, hact. Metallogr.. Special Issue 9, eds.W.U. Kopp and H.E. Biihler o r . Riederer Verlag, Stuttgart) p. I 13.
Ch. 10, Refs. ~ S L E R G., ,
Surface microscopy
1023
and G. ELSSNER, 1980, in: Metallographie, Anschliff- und Dilnnschlifftechnik an Metallen, Keramikcn und Kunststoffcn, Pract. Mctallogr. Special Issuc 1 I. eds. H.E. Biihlcr and W.U. Kopp (Dr. Riederer Vcrlag, Stuttgart) p. 21. KIESSLER, G., H. RAPPand G. ELSS~TR, 1982. in: Metallographie. Pract. Metallogr., Spccial Issue 13, eds. H.E. Blihlcr, K. Dicser and W.U. Kopp (Dr. Riederer Verlag, Stuttgart) p. 9. KING, R. P., 1982, Powder Technol. 32,87. KINGLAKE,R.. ed., 1965, Applied Optics and Optical Engineering (Academic. New York). KLEISER.T., and M. BOCEK,1986, Z. Mctallkde. 77, 582. KLOCKEWKKMPER, R., A. BEYERand M. MONES,1979, Pract. Metallogr. 16, 53. KNEISSL.A., and S. S. BRENNER,1990, Pract. Metallogr. 27, 159. KOCH,R., 1979, Pract. Mctallogr. 16, 11. KOCH. E. E., (ed.), 1983: Handbook of Synchrotron Radiation (North Holland, Amsterdam). KOPP. W. G., and G. MULLER,1987, Pract. Mctallogr. 24, 336. KOKPEL,A,, L. W. KBSSLER,P.R. PALERMO,1971, Nature 232, 100. KRONER,B., 1986, Pract. Metallogr. 23, 400. KUHLMEYER, M.. 1978, Einfluss der statistischen Komgriissenvertcilung auf die Streckgrenze von Stahl (Verlag Stahleisen, DUsseldorf). KUK, Y., 1993, in: Scanning Tunneling Microscopy, eds. J.A. Stroscio and W. J. Kaiser (Academic Press, Boston) p. 277. KULMBLXG, A., F. KORNTHEUER and P. SCIiIMhtEL. 1974, Pracf. Metallogr. 11, 183. LAFOND,C.,and E MOLIEXE,1975, in: Quantitative GeHIgcanalyse in Medizin, Biologie und Materialentwicklung, Pract. Mctallogr., Special Issue 5, ed. H.E. Exner (Dr. Riederer Verlag, Stuttgart) p. 200. LEAMY,H.J., 1982, J. Appl. Phys. 53. R 51. LEE. H.C., and J. GURLAND,1978. J. Matcr. Eng. 33, 125. LEKTINEN.B., and A. MELANDER,1980, Metallography 13, 283. LEMONS,R.A., and C.F. QUATE,1974, Appl. Phys. Left. 24, 162. LIENKAMP,M., and H. E. EXNER,1994, Mater. Sci. Technol. 10,526. and H.E. EXNER,1994, J. Comp. Assist. Micro.%. 6, 103. LIENKAMP, M. U. KIJNAVER LIFSHIN,E., ed., 1992, Characterization of Materials, Part I, Materials Science and Technology, Vol. 2A (VCH, Weinheim). LIFSHIN,E., 1993, in: Concise Encyclopedia of Materials characterization, eds. R.W. Cahn and E. Lifshin (Pergamon Press, Oxford) p. 286. LIFSHIN,E., ed., 1994a. Characterization of Materials, Part 11, Materials Science and Technology, Vol. 2B (VCH, Weinheim). LIFSHIN,E.. 1994b. in: Materials Science and Technology, Vol. 2B, cd. E. Lifshin (VCH, Wcinheim) p. 351. LIHL, E. and H. MEYER,1%0, Z. Metallk. 51, 186. LIND~N,G., 1972, Pract. Metallogr. 9, 3. LIU, C.T.. and J. GUKLAND,1968, Trans. Amer. Soc. Metals. 61, 156. LIU,Y.,and B. R. PATTERSON, 1993, Metall. Trans. MA,1497. MBERT, P., 1982, Der Maschinenschaden 55, 136. LORETO, M. H., 1984, Electron Beam Analysis of Materials (Chapman and Hall, London). LOUTITAN, M. R., 1992, in: ASM Handbook, Vol. 10: Materials Characterization (ASM International, Metals Park, OH) p. 299. LOVELAND,R. P., 1970, Photomicrography - A Comprehensive Treatise (Wiley. New York). LOZINSKII, M.G., 1961, High-Temperature Microscopy (Pergamon Press, Oxford). LUCKE,K., 1984, Z. Metallkdc. 75, 948. LYMAN, C. E., 1990, Scanning Electron Microscopy, X-Ray Microanalysis and Analytical Elcctron Microscopy (Plenum Press, New York). LYMAN,C.E., J.I. GOLDSTEIN,A.D. ROMIG, P. ECHI.IN.D.C. JOY, D.E. NEWBURY,D. WILI.IAMS, J.T. ARMSTRONG, C. E. FIOHIand E. LIFSIIIN,1990, Scanning Elcctron Microscopy, X-Ray Microanalysis and AnaIytical Electron Microscopy - A Laboratory Workbook (Plenum Press, New York). MADESKI, A., 1980, Pract. Metallogr. 17. 598. MALIES, H. M.. 1959, Applied Microscopy and Photo-'Micrography (Fountain Press, London).
1024
H. E. Exner
Ch. 10, Refs.
MALLARD,J.R., 1990, in: Modem Microscopies: Techniques and Applications, eds. P.J. Duke and A.G. Michette (Plenum Press, New York,London) p. 133. MATHY,H., 1977, Microsc. Acta, Suppl. 3, 3. m and P. fin, 1988% Pract Metallogr. 25,361. MATIXAEI, E., H. V W m m , E., A. SCHULZ,H. VETTERSand V. SCHWARZMANN, 1988b, in: Microstructural Science, Vol. 16, Metallography of Advanced Materials, eds. H J . Cialone, G.W.E. Johnson, ME. Blum and G.F. Vanda Voort (ASM International, Metals Park) p. 465. M c CRONF,W.C.. L. B. M c CRONEand J.G. DEUY, 1978. Polarized Light Microscopy (Ann Arbor Science, Ann Arbor). MCCALL, I., 1973, in: Microshuctural Analysis: Tools and Techniques, eds. J.L. McCall and W.M. Mueller (Plenum, New York) p. 93. MCINTYRE,N.S., ed., 1978, Quantitative Surface Analysis of Materials, ASTM Spec. Tech. Publ. 6?4 (American Society for Testing and Materials, Philadelphia). MEDALIA,A.I., 1980, in: Testing and Characterization of Fine Particles, eds. J.K. Beddow and T. Meloy (Heyden, London) p. 66. ~&LOY, T.P., 1980. in: Testing and Characterization of Fine Particles, eds. J.K. Beddow and T. Meloy (Heyden, London) p. 1. METALSHANDBOOK,1985, 9th ed., Vol. 9: Metallography and Microstructures (American Society for Metals, Metals Park). METAU HANDBOOK,1986, 9th ed., Vol. 1 0 Materials Characterization (American Society for Metals, Metals
Pd). IMEYEK. E., and H. HBINZELMANN, 1992, in: Scanning Tunneling Microscopy, cds. R. Wiesendanger and H.J. G i l n t h d (Springer, Berlin) p. 99. MICHETTE,A.G., 1990, in: Modem Microscopies, Techniques and Applications, cds. P.J. Duke and A.G. Michette (Plenum Press, New York) p. 41. MICHOT,G., and A. GEORGE, 1982. Script. Metall. 16,519. MIDDIETON,C. J., and D. V. EDMONDS, 1977. Metallography 10.55. MILLER,M. K., and G. D. W. SMTH, 1989, Atom Probe Microanalysis: Principles and Applications (Materials Research Society, Pittsburgh). MILLER,M.K., A. CEREZO, M.G. HFTHERINCTONand G.D.W.SMITH, 1992, Atom Probe Field-Ion Microscopy (Oxford University Press, Oxford). MINSKY,M., 1988, Scanning 10, 128. MITSCHE, R., E JEGLITSCHand E GABLER,1964, Berg- und Hiittenmbn. Monatsh. 109, 110. R., and H.SCHEIDL,1964, Berg- und Hiittenmhn. Monatsh. 109, 82. M TSC IXE, R, F. GABLERand F. JEGLITSCH,1969, in: Handbuch der Mikroskopie in der Technik, Band 3, Teil M~~SCHE, 2, 4. H. Freund (Umschau Verlag, Frankfurt), p. 269. MODIN,H., and S. MODIN,1973, Metallurgical Microscopy (Buttenvorths. London). MORNHEIM, A.F., 1977, in: Interpretive Techniques for Microstructural Analysis, eds. J L . McCaIl and P.M. French (Plenum, New York) p. 117. Mom, B. W., 1956, Micro-Indentation Hardness Testing (Butterworths, London). MC'CKLICH,E, J. OHSER, S. HARTMANN,M. J. HORXANN and G. ~ Z O W 1994, , in: Tayloring of High Temperature Properties of Silicon Nitride (Kluwer Academic Publ., Dordrecht) p. 73. MUGGLI,R. Z., and W. C. MC CRONE, 1977, in: Interpretative Techniques for Microstructural Analysis, eds. J.L. McCall and P.M. French (Plenum, New York)p. 127. M ~ L L E RE., W.. 1936, Z. Phys. 102, 734. M I ~ L E RE,.W., and T. T.TSONC.1%9, Field Ion Microscopy: Principles and Applications (American Elsevier, New York). MULLER,G., and W. U. KOPP,1989. Pract. Metallogr. 26,640. MULLINS,W.W., 1%1, Phil. Mag. 6, 1313. MURPHY, I. C., 1993, in: Concise Encyclopedia of Materials Characterization, eds. RW. Cahn and E. Lifshin (Pergamon Press, Oxford) p. 515. Xand ~D. STOCKEL,1980, Metall 34,617. MORRLE,U., H.E. E MYKURA,H., 1955, Acta Metall. 3, 436.
Ch. 10, Refs.
Surface microscopy
1025
NAKAGAWA, S., 1987. Pract. Metallogr. 24, 431. NANEV,C., 1981, Jenaer Rundschau 5, 219. R.. and J. SERRA,1979, Microsc. Acta 82, 101. NAWRATH, Nasoh-, J. A., 1989, Pract. Metallogr. 26, 225. NEWBURY, D.E., 1977, in: Practical Scanning Electron Microscopy, eds. J.I. Goldstcin and H. Yakowitz (Plenum, Ncw York) ch. 4, p. 95. NEWBURY,D. E., and H. YAKOWIIZ,1977% in: Practical Scanning Electron Microscopy, ds.J.I. Goldstein and H.Yakowitz (Plenum, New York) ch. 6, p. 21 1. NEWBURY,D. E., and H. YAKOW, 1977b. in: Practical Scanning Electron Microscopy, eds. J.I. Goldstcin and H. Yakowitz (Plenum, New York) ch. 5, p. 149. NEWBLXY.D. E., 1986, D. C. JOY, P. E W N , C. E. FIORIand J.I. GOLDSTEIN,1987, Advanced Scanning Electron Microscopy and X-Ray Microanalysis (Plenum Press, New York). NICHOLSON, W. L., 1978, J. Microsc. ll3,223. OATLEX, C. W., 1972, The Scanning Electron Microscope (Cambridge University Press). ~CHSNER,H., 1975. Appl. Phys. 8, 185. OETTEL,W. O., 1959, Grundlagen der Metallmikroskopie (Akadcmische Verlagsgesellschaft Gccst und Portig, Leipzig). OHSER. J., and U. URZ,1994, Quantitative Gefiigeanalyse, lkoretische Grundlagen und Anwcndung (Deutschex Verlag f i r Grundsmffiidustrie, Lcipzig. Stuttgart). OHSER.J., and E MOCKLICH,1995, Adv. Appl. h b . (SGSA) 27, 384. OKAZAKI, K.,and H. CONRAD,1972, Trans. Japan. Inst. Metals 13, 198. OLIveR W. C., 1993, in: Concise Encyclopedia of Materials Characterization, eds. R.W. Cdhn and E. Lifshin (Pergamon Press, Oxford) p. 232. ONDRACEK,G., and K. SPIELER,1973, Pract. Mctallogr. 10, 324. ONDRACEK, G., 1978, in: Quantitative Analysis of Microstructures in Materials Science, Biology and Medicine, Pract. ,Mctallogr. Special Issue 8, ed. J.L. Chermant (Dr. Kiederer Verlag, Stuttgm) p. 103. ONDRACEK, G., 1982, Acta Stereol. 1, 5. OPIELKA.H., H. LAHODNY. S. BRASD and E. BISCHOFF,1990, Pract. Metallogr. 27, 171. 1993, Pract. Metallogr. 30,399. OPIELKA.H., P. MCKAY and B. RZEHAK-~PIELKA, ORTNER,H. M., and P. WILHARTITZ, 1991, Mikrwh. Acta 11. 177. PAGE!, T.F., 1993, in: Concise Encycloptdia of Materials Characterization. eds. R.W. Cdhn and E. Lifshin (Pergamon Press, Oxford) p. 43. PANGBORN,R. N., 1992, in: ASM Handbook, Vol. 1 0 Materials Characterization (ASM International, Metals Park) p. 365. PAUL,J., and U. MORRIX, 1981, Pract. Metallogr. 18,418. 1982. Z. Metallk. 73,50. PAUL,J., H. E. EXNBRand D. MOLLER-SCHWEI.I.ING, PAUL,J., and B. BAUER,1983. Pract. Metall. 20, 213. PAYNE,B. O., 1957. Microscope Design and Construction, 2nd Ed. (Cooke, Troughton and Simms. York, UK). PAW, S.M., 1982, in: Electron Microscopy and Analysis 1981, Conf. Series 61, cd. M.J. Goringc (The Institute of Physics, Bristol, London) p. 287. PECHENYAKOV, 1.. and R. KOVACHEVA, 1995, Pract. Metallogr. 32, 17. PELLOUX,R.M.. 1970, in: Applications of Modern ,Metallography Techniques, ASTM Spec. Tcchn. Publ. 480 (American Society for Tcsting and Materials, Philadelphia) p. 127. PEPPERHOI.I',W., and H.H. ET-WIG, 1970. Intcrferenzwhichten-Mikroskopie o r . Steinkopf Verlag. Darmstadt). PETTEOREW, J. W., ed., 1990, NMR Principles and Applications in Biomedical Research (Springer-Verlag, Berl in). PETZOW,G., and H.E. EXNER,1968. in: Handbuch der Mikroskopie in der Technik, Band 3, Tcil I , ed. H. Freund (Unischau Verlag, Frankfurt), p. 37. PETZOW,G., and E. HORNBOGEN. 1970, Z. Metallk. 61, 81. PETZQW, G., and H. KNOW, 1973, Metallography 6, 249. PErzow, G., and H. E. EXNER,1975, in: Microstructural Science, eds. P.M. French, R.J. Gray and J.L. McCall (American Elsevier, New York) vol. 3. p. 291.
1026
H. E. Emer
Ch. 10. Refs.
Pnzow, G., 1978, Metallographic Etching (ASM,Metals Park, OH). Psrzow, G., 1994, Metallographisches. Keramogaphisches, Plastographisches A w n , (Gebr. Bomtraeger, B d i , shrttgart). P ~ ~ R K O R G., N , and R. BLASCHK~, 1982, BeiMge elcktronenmikroskopische Direktabbildung von ObedLhen, vol. 15 (Verlag R.A. Remy, MUnster). p. 1. PHILLIPS,V. A., 1971, M o d h Metallographic Techniques and their Applications (Wiley-Interscience, New York). PIVIN,J. C., 1983.1. Mater. Sci. 18, 1267. POECH,M. H., and D. RUHR,1994, Pract. Metallogr. 31, 70. POHL, M.. and W.G. BURCHARD, 1980a, in: Metallographic. Anschliff- und Diinnschlifftechnik, Pract. Metallogr., Special Issue 11, eds. H.E. BUhler and W.U. Kopp (Dr. Riederer Verlag, Stuttgart) p. 42. POHL,M., and W. G. BLXCHARD.198%. Scanning 3, 251. POHL, D.W., 1992, in: Scanning 'hnneling Microscopy 11, eds. R. Wiesendanger and H.J. GUntherodt (Springer, Berlin) p. 233. POIIL,D. W., 1994, 2. Metallkde. 86, 22. POKOKNY, A,, 1980, Pract. Metallogr. 17. 23. POMP&0.. 1994, Pract. Metallogr. 31, 274. PORTER, D.A., and K.E. EASTERLING, 1981, Phase Transformations in Metals and Alloys (Van NostrandReinhold, Wokingham, New York). P~SCH, R., 1979, hact. Metallogr. 16,26 and 79. RAND- V., 1993, m e Measurement of Grain Boundary Geometry (Institute of Physics Publ., Bristol, Philadelphia). RAZA,S. M., 1982, Scripta Metall. 16, 1325. REIMER,L., and G. ~ R K O R N 1977, . R a s t e r e l e b n e n m i k k o p i e (Springer, Berlin). RHNACHER.G., 1965, Pract. Metallogr.2,45. RHINES, EN., K. R. CRAIGand R.T. DE HOW, 1974, Metallurg. Trans. 5,413. RHINES, EN., 1976, in: 4th Int. Congr. for Stereology, NBL Spec. Techn. Publ. 431. eds. E.E. Underwood, R. De Wit and G.A. Moore (National Bureau of Standards, Gaithersburg) p. 233. RHINES,EN., and B. R. PATTERSON, 1982, Metall. Trans. 13A. 985. R H I N ~E, N., 1986, Microstructurology: Behaviour and Microstructure of Materials (Dr. Riederer-Verlag, Stuitgart, now Carl Hanser Verlag, MUnchen), also published in Pract. Metallogr. 30,216, 287. 322 and 389.
RICHARDS, B. P., and A. D. TRIGO, 1982, in: Electron Microscopy and Analysis 1981, C o d . Series 61. ed. M.J. Goringe (The Institute of Physics, Bristol, London) p. 227. RIEDLE,J., R. DENNOCHWEILER and H. OPIELKA, 1994, Pract. Metall. 31. 580. RIEGGER,H., J. A. PASICand H. E. EXNER,1980, in: Sintering Processes, ed. G.C. Kuczynski (Plenum, New York) p. 219. RIVI~RE,J. C.. 1990, Surface Analytical Techniques (Clmndon Press, Oxford). R o o s z A., Z. GACZIand M.K. BAAN, 1980, Metallography U,299. R~SCHHNBLECK, B., and K. WOL'IHR,1979, in: Fortschritte in der Metallographie, Pract. Metallogr., Special Issue 10, eds. F. Jeglitsch and G. petzow (Dr.Riederer Verlag, Stungm)p. 95. RWNBERGER, H.E., 1977, in: Interpretative Techniques for Microstructural Analysis, eds. J.L. McCall aod P.M. French (Plenum, New York) p. 79. ROSENCWAIG,A., 1982, Science 218, 223. R o n , F.W.D., 1981, Roc. Roy. Microsc. Soc. 16.44. R ~ C K E R TM., , and L. BJERREOMRD, 1993, Pract. Metallogr. 30,579. RUDOLPH,D., G. SCHMAHL and B. NIEMANN,1990, in: Modcm Microscopies: Techniques and Applications, eds. P.J. Duke and A.G. Michette (Plenum Press, New York, London) p. 59. RUS, J. C., 1986, Practical Stereology (Plenum Press, New York). RuSS. J. C., 1990, Computer-Assisted Microscopy: The Measurement and Analysis of Images (Plenum Press, New York). RUS, J. C., 1992, The Image Processing Handbook (CKC Press, Boca Raton, E). RUS, J. C., 1995. J. Comp. Assist. Microsc. 7, 35.
Ch. 10, Refs.
Surface microscopy
1027
T., S. SAKAI and H. W. PICKERMG.1989, Atom Probe Field-Ion Microscopy and its Applications (Academic Press, Ncw York). SALTYKOV, S. A.. 1974, Stereometrische Metallographie (Dcutscher Verhg ftir Grunstoffindusbie, Leipzig). S m T L S , L. E., 1971, Metallographic Polishing by Mechanical Methods (Pitman, Melbourne, London). SARID, D., 1991, Scanning Force Microscopy with Applications to Elcctric, Magnetic and Atomic Forces (Oxford University Press, New York). ScHAARWACiim?t, W., 1968, in: Handbuch der Mikroskopie in der Technik, Band 3, Teil I. ed. H. h u n d (Umschau Verlag, Frankfurt), p. 291. SCHADE,K. H., 1993, Lichtmikroskopie: Technologie und Anwendung (Verlag Moderne Industrie, Landsbeg). SCIMPER. M.,and D. BOSEL, 1985, Pract. Mctallogr. 22, 197. SCHAIT, W., 1991, EinfIihrung in die Werkstoffwissensch& (Deutscher Vdag fiir Orundstoffindustrie, Leipzig). SCHR~DER B.,, and H. P. HOUGARDY,1985, Pract. Metallogr. 22,587. SCHR~DEK, J., H. OPIELKA, B. HAUKEand E. ARZT, 1990, kact. Metallogr. 27, 17. SCHUMANN, H., 1990, Metallographie (Deutscher Verlag ffir Grundstoffindusbie, Leipzig, Stuttgart). SCHWARZ, H., 1980, Mikroskopie 37,Suppl., p. 64. SCHWAKZ,H., and H. E. Exhm. 1983, J. Microsc. 129, 155. SCHWARZER,R. A.. 198 1, Microsc. Acta 84, 5 I. SEAH,M. P., 1975, Surf. S c i 52, 186. SEEGER,J., J. K L m and H. MECKING.1980. Pract. Metallogr. 27,236. SEIDMANN,D.N., 1993, Mater. Sci. Eng. A137,57. SEILER,H., 1968, Abbildung von Oberllkhen mil Elektronen, Ionen und R&intgenstrdhlen, HochschultaschenWcher 4281428a (Bibliographisches Instituf Mannheim). S ~ H M., ,and B. BERGMANN,1969. in: Handbuch der Mikroskopie in der Technik, Band 3, Teil 2, cd. H. Freund (Umschau Verlag, Frankfurt), p. 347. S m , J., 1982, Imagc Analysis and Mathematical Morphology (Academic, London). SERRA, J.. 1988, Image Analysis and Mathematical Morphology, Vol. 2 Theoretical Advances (Academic Press, London). SHEPPARD,C. J., 1987, Optica Acta 25,269. S ~ ~ L EP.V., V ,1974, Electrolytic and Chemical Polishing of Metals (Freund, Tel Aviv). SIBILIA, J. P., 1988, A Guide to Matcrial Characterization and Chemical Analysis (VCH, Weinheim). Smm, B., 1982, in: Electron Microscopy and Analysis 1981, Conf. Series 61, ed. MJ. Goringe (The Institute of Physics, Bristol, London) p. 223. SIEGBAHN,M., C.N. NORDLING,A. FAHLMAXV, R. NORUBERG,K. HAMKIN,J. HEDMAN, G. JOHANSSON, T. BERHARK, S . E KARLSSON,I. LINDGREN and B. LINDBERG,1967, ESCA: Atomic, Molecular and Solid State Structxre Studied by Means of Electron Spectroscopy (Almquist and Wicksell, Uppsala). SIECENTHALER H., 1992, in: Scanning Tunneling Microscopy, eds. R. Wiesendanger and H.J. Ciintherodt (Springer, Berlin) p. 7. SIGL L.and €LE. EXNER, 1989, Mater. Sci. Eng. Aim, 121. SIGNORELLI, A. J., E. A. LEONEand R. L. CHIN, 1988. in: A Guide to Materials Characterization and Chemical Analysis, ed. J.P. Sibilia (VCH, Weinheim) p. 167. SIMPSON, 1. D., and N. STANDISH, 1977, Metallography 10, 149 and 443. Smm, C. S., 1960, A History of Metallography (University of Chicago Press). SMW G.D. W., 1986. in: Metals Handbook, 9th ed.,Vol. IO, Material Characterization (American Society for Metals, Metals Park) p. 583. SWTH,R., and J. M. WALLS,1992, in: Methods of Surface Analysis: Tcchniques and Applications. ed.J. M. Walls (Cambridge University Press,Cambridge) p. 20. So-, M.G., 1990, in: Modem Microscopies: Techniques and Applications, cds. PJ. Duke and A.G. Michctte (Plenum Press, New York, London) p. 205. SONNENFELD,R., J. S c m ~ m and P. K. HANSMA,1990, in: Modem Aspects of Electrochemistry, Vol. 21, eds. R.E. White, 0. Bockris and B.E. Conway (Plenum Press, New York) p. I . SPIESS,H. J., 1965, Freiberger Forschh. B111,87. STAPF,I., U. KOPACZand P. S C H L ~ E 1986. R , Pract Mctallogr. 23,297. SAKURAI.
1028
H.E. Exner
Ch. 10, Refs.
STOYAN, D., W. S. KENDALLand J. MECKE.1987, Stochastic Geometry and Its Applications (Akademie-Verlag, Berlin). STOYAN,D., 1990, Intern. Statist. Rev. 58. 227. 1993, in: Scanning Tunneling Microscopy, eds. J.A. Sttuscio and WJ. STROSCIO,J. A., and R. M. FEENSTRA, Kaiser (Academic Press, Boston) p. 96. STROSCIO, J. A., and W. J. KAISER,eds.. 1993, Scanning Tunneling Microscopy (Academic Press, Boston). SYNGE,E.H., 1928, Phil. Mag. 6, 356. S., 1982, Teilchenwachstum durch Koaleszenz warend des Flllssigphasensinterns von Fe-Cu und CuTAKAJO, Ag, Ph. D. thesis (University of Stuttgart). TALBOT,H.. 1994, Rev. Mbtallurg. 91, 211. TAM,A. C., 1986, Rev. Mod. Phys. 93.384. TANNER. B., 1976, X-ray Diffraction Topography (Pergamon Press, Oxford). TAT%,D.R., 1993, in: Concise Encyclopedia of Materials Characterization, eds. RW. Cahn and E. Lifshin (Pergamon Press, Oxford) p. 183. 1982, Metallography 15, 225. TAYLOR,D.S.. and G. POLLARD, TEGART,W. J. McG., 1957, The Electrolytic and Chemical Polishing of Metals in Research and Industry (Pergamon Press, London). TELLE.R., and G. PFTzow, 1992, in: Materials Science and Technology,Vol. 2A, ed. E. Lifshin (VCH, Weinheim) p. 357. TENS, H. M., 1968, in: Handbuch der Mikroskopie in der Technik Band 3, Teil 2, ed. H. Freund (Umschau Verlag, Frankfurt), p. 1. TERSOFF, J.. and D.R. HAMANN, 1985. Phys. Rev. B31, 805. TERSOFF, J., 1993, in: Scanning Tunneling Microscopy and Spectroscopy, Theory, Techniques and Applications, ed. D.A. Bonnell (VCH Pub]., New York) p. 31. TERSOFF, J., and N.D. LANG,1993, in: Scanning Tunneling Microscopy, eds. J.A. Stroscio and W.J. Kaiser (Academic Press, Boston) p. 1. THOLIPSON, A. W., 1972. Metallography 5, 366. H. MIYAZAKI, S. D. MAIIATIKI and H. J. G~NTHERODT, 1989, Phys. Rev. Lea. 63, T O M A ~ ~D.,C G. , OVERKEY, 876. T ~ R R ~ N EK., N .1980, Metallography 13,329. TOUSEK,J.. 1981, Pract. Metallogr. 18,471. TSONG,T. T., 1990, Atom Probe Field-Ion Microscopy (Cambridge University Press, Cambridge). TURLEY, D. M., and L.E. SAMUEL$ 1981, Metallography 14, 275. TURNER, J. N., and D. H. SZAROWSKY, 1993, in: Concise Encyclopedia of Materials Characterization,eds. R.W. Cahn and E. Lifshin (Pergamon Press, Oxford) p. 68. UK’DERWOOD.E.E., 1986, in: Applied Metallography. ed. G.F. Vander Voort (Van Nostrand Reinhold, Amsterdam) p. 101. UNDERWOOD, E.E. (1987). in: Metals Handbook, 9th ed., Vol. 9: Fractography (American Society for Metals, Metals Park) p. 193. UNDERWOOD, E. E., 1970, Quantitative Stereology (Addison Wcsley, Reading, MA). UNDERWOOD, E. E., 1976, in: 4th Int. Conf. for Stcrcology. NBL Spcc. Techn. Publ. 431, eds. E.E. Underwood, R. De Wit and G.A. Moore (National Bureau of Standards, Gaithersburg) p. 91. UNDERWOOD,E.E., 1980, in: Tcqting and Characterization of Fine Particles, eds. J.K. Bedow and T. Meloy (Heyden, London) p. 77. UNBRTL, W.. 1993, in: Scanning Tunneling Microscopy and Spectroscopy: Theory, Techniques and Applications, cd.D.A. Bonnell (VCH Publ., New York) 107. VAKDERVOORT,G. E, 1984a, Metallography: Principles and Applications (McGraw Hill, New York). VAKDERVOORT,G. E, 1984b. Metal Progress 127, March, 3 1 and 36. VANDERVOORT,G.F., 1985a, in: Metals Handbook, 9th ed., Vol. 9: Metallogrdphy and Microstructures (American Society for Metals, Metals Park) p. 139. VANDER VOORT. G.F., 1985b. in: Metals Handbook, 9th ed., Vol. 9 Metallography and Microstructures (American Society for Metals, Metals Park) p. 71.
Ch. 10, Refs.
Surface microscopy
1029
VANDERVOORT,G. E , 1985c, in: Metals Handbook, 9th Edition, Vol. 9, Metallography and Microstructures (American Society for Metals, Metals Park) p. 531. V m m VOORT.H.T. M.,G. J. BRAKENHOFFand M.W. BAASLAG,1989, J. Microw. 153, 123. VENABLES, J.A., and G.G. HEMBREE, 1991, in: Inst. Phys. Conf. Scr. No. 119 (Institute of Physics, Bristol) p. 33.
VERHOEVEN, J. D., 1986, in: Metals Handbook,9th ed.. Vol. 10, Materials Characterization (American Society for Metals,Metals Park) p. 490. VI-, W., 1973, Z. Werkstofftechn. (J. Mater. Techn.) 4. 320. VOLBERT.B., 1981, B e i m elektronenmhskopische Direktabbildung von Oberflaehen. vol. 14 (Verlag RA. Remy, MUnster). WACHTMAN,J. B., 1993: Characterization of Materials (Butteworth-Heineman. Stoneham, USA). WAGNER, C., 1954, J. Electrochem. Soc. 101,225. WAGNER,R., and S.S. BRENNER,1978, Acta Metall. 26, 197. WAGNER,R.. 1980, Phys. Blgtter 36, 65. WAGNER,R., 1982, Field-Ion Microscopy in Materials Science (Springer, Berlin). WAGNER, R., 1985, Field-Ion Microscopy (Springer, Berlin). WALLS, J. M. (ed.).1992. Methods of Surface Analysis (Cambridge University Press, Cambridge). WANG,S., and L. E. MURR, 1980, Metallography 13.203. WASCHULL, H., 1993, Mparative Mctallographie (Deutscher Verlag fUr Grundstoffindustrie, Leipzig, Stuttgart). WAITS, J.T., 1982, Proc. Roy. Microsc. Soc. (London) 17, 192. WAVER, G., 1973, Z. Werkstofftechn. (J. Mater. Techn.) 4, 298. WECHSUNG,R.. 1977, Vacuum-Techn. 26. 227. WECK,E., and E. L m m , 1986, Metallographic Instruction for Colour Etching by Immersion (In German) Fachbuchreihe Vol. 77 (Deutscher Verlag f i r Schweisstechnik. DUsseldorf). WEGMAN~. L., 1972, J. Microsc. 96, 1. WEIEEL, E. R., 1979, Stereological Methods, vol. 1, Practical Methods for Biological Morphometry (Academic, London). W ~ W LE. , R., 1980, Stereological Methods, vol. 2, Theoretical Foundations (Academic, London). S., F. BALIBARand J. E m o m , 1984, Applications of X-Ray Topographic Methods (Plenum W~SSMANN, Press, New York). WELLNER,P., 1980, Pract. Metallogr. 17. 525. WELLS,0.C., 1993, in: Concise Encyclopedia of Materials Characterization. eds. R.W. Cahn and E. Lifshin (Pergamon Press, Oxford) p. 423. WENDLER,B., and B. NEUBALTR,1979, Pract. Metallogr. 16, 3. WENDROCK,H., R. HOBn and M. BERTRAM,in: Fortschritte in der Metallographie, Pract. Metallogr., Spec. Issue 26, ed. A. Kneissl (Carl Hanser Verlag, MUnchen) 365. WERLEFORS, T., C. ESKIUSONand S. E m m , 1979, Scand. J. Metallurgy 8.221. WETZIO, K., A. MASLOVand J. ELELMANN, 1984, Pract. Metallogr. 21, 161. WICKRAMASINGHF, H. K., R.C. BRAY,V. JIPSON. C. E QUATE,J. R. SA-. 1978, Appl. Phys. Lea. 33,923. WICKRAMASINGHE, H. K., 1983, J. Microsc. 129, 63. WICKRAMASMGHF~ H. K., 1989, Scanned-Probe Microscopes, Sci. Amer. 261,74. WICKRAMASINGHE, H. K. cd, 1992a, Scanned Probe Microscopies, STM and Beyond (AIP. New York). WICKRAMASNOHE, H.K., 1992b, in: Scanning b e l i n g Microscopy, eds. R. Wiesendanger and H.J. GUnthcrodt (Springer, Berlin) p. 109. WICKRAMASINOHE, H. K., 1993, in: Scanning Tunneling Microscopy, e&., J.A. Stroscio and W.J. Kaiser (Academic Press, Boston) p. 77. WIELAND.H.J., 1993, Pract. Metallogr. 30,261. W ~ ~ A N G R., E Rand , H. J. GCNTHERODT.eds., 1992, Scanning Tunneling Microscopy 11 (Springer, Berlin). WILSON,T., and C. J. R. SHEPPARD,1984. Theory and Practice of Scanning Optical Microscopy (Academc Press, London). W m s c m ~ E., , P.VETTERLEIN and M. KRONING,1989, DGZP-Berichte, Vol. 18/1, Deutsche Gesellschaft fur Zersthngsfreie PrUfung, Berlin) p. 186. W m s c ~ u ~E., r , D. SKUDLIKand J. HIMMELMANN, 1993, Pract. Metallogr. 30, 595.
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WOLF,E. 1986, Electron Tunneling Spectroscopy (Oxford University Press, Oxford). WORCH, H., K. NOCKE,C. BUNK, W. O E B m and E BERTHOLD,1994, Pract. Metallogr. 31,245. WRIGHT, K., and B. KARLSSON, 1982, J. Microsc. 129, 185. WRIGHT,S. I., 1993, J. Comp. Assist Microsc. 5, 207. WU, W.T., I. A Y D and ~ H. E. BCHLER, 1982, Pract. Metallogr. 19, 322 and 347. YAMAMOTO, K., and A. TAIRA. 1983, J. Microsc. 129, 49. YANG,H.S.,1990, Pract. Metallogr. 27. 539. YANKOVITH, H., 1970, in: Application of Modem Metallographic Techniques, ASTM Spec. Techn. Publ. 480 ( A m . SOC. Testing Mater., Philadclphia) p. 49. YOLJNC,J.Z., and E ROBERTS,1951, Nature 167, 231. YUZAWICH, P.M., and C. W. HUGHES, 1978, Pract. Mctallogr. 17, 607. ZAM, I.. 1992, Pract. Metallogr. 29, 158. ZANCWILL,A., 1988, Physics at Surfaces (Cambridge University Press, Cambridge). ZHAO, X., and Z CHEN,1993, J. Comp.-Assist. Microsc. 5, 257. ZHOU, J., F. Z m N G , W. SCHMITZ and S. ENGLER,1993, Pract. Metallogr. 30, 122. ZOGG, H., S. WEBERand H. WARLIMO~T,1977, Pract. Metallog. 14, 553.
Further reading Most of the standard journals in thc field of physical metallurgy and materials science report frequently on metallographic techniques, surface microscopy, surface analysis, and stereology, with emphasis on the application of these techniques.
Journals The following journals are specifically devoted to aspects of metallographic sample preparation and microscopic investigation of materials: Practical IMetallography, bilingual EnglisWGerman (Carl Hanser Verlag, MUnchen). Materials Characterization (Elsevier Science Publishing Co., Ncw York). Journal of Computer Assisted Microscopy (Plenum Publ. Corp., New York). Journal of Microscopy (Royal Microscopical Society, London). Ultramicroscopy (North-Holland Publishing Co., Amsterdam). Acta Stenologica (International Society for Stereology, Ljubljana).
Books Ash, E. A., ed., Scanned Image Microscopy (Academic Press, London. 1980). ASM Handbook, Vol. 10, Matwials Chantcterization (ASM International. Meials Park, 1992). Beyer, H., Theorie und Praxis der Interferenzmikroskopie (Theory and Practice of Interference Microscopy) in German (Akadcmische Verlagsgesellschaft Geest und Portig, Lcipzig. 1974). Bonnell, D. A., (ed.), Scanning Tunneling Microscopy: Theory and Practice (VCH Publ., New York, 1990). Bonnell, D. A., (4. Scanning ). Tunneling Microscopy and Spectroscopy (VCH Publ.. New York, 1993). Briggs, D., and M. P. Seah (eds.), Practical Surface Analysis, Vol. I: Auger and X-Ray Photoelectron Spectroscopy (Wiley, Chichester, 1990). Brigs, D., and M. P. Seah (eds.), Practical Surface Analysis, Vol. 11: Ion and Neutral Spectroscopy (Wiley, Chichester, 1992). Brlimmer, O., J. Heydenreich, K. H. Krebs and H. G. Schneider, Handbuch der Festkorperanalyse mit Rontgenstrahlen (Handbook of Solid State Analysis by Means of X-Rays) in German, (Vieweg, Braunschweig, 1980). BUhlcr, H.E., and H. P. Hougardy, Atlas of Interference-Layer Metallography @eutsche Gesellschaft fUr Metallkunde, Obetursel, and The Mctals Society, London, 1980). Cahn. R. W., and E. Lifshin, (eds.),Concise Encyclopedia of Materials Characterization (Pergamon Press, Oxford, 1993).
Ch. 10, Refs.
Surface micmscopy
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Coster, M., and J. L. Chermant, W i s d’Analyse d’Image (Intrcduction to the Analysis of Images) in French (Presses du Centre National de la Recherche Scientifique, Paris, 1989). Czandema, A. W., ed., Methods of Surface Analysis (Elsevier Scientific, Amsterdam, 1975). Doyen, G., and D. Drakova, The Physical Principles of STM and AFM operation (Akademie Verlag, Berlin, 1995). Duke, P.J., and A. G. Michette (eds.), Modcrn Microscopies: Techniques and Applications (Plenum Press, New York, London, 1990). Exner, H. E., and H. P. Hougardy, Quantitative Image Analysis of Microstructures (DGM Informationsgesellschaft-Verlag, Oberursel, 1988). Fcldrnann, L. C.. and J. W. Mayer, Fundamentals of Surface and Thin Rlm Analysis (PTR Prentice Hall, Ehglewood Cliffs, 1986). Flewitt, P.E., and R. K. Wild, Microstructural Characterization of Metals and Alloys (The Institute of Metals, London, 1985). Freund, H., ed.,Handbuch der Mikroskopic in der Technik, (Handbook of Microscopy in Technology) in German, 8 Volumes (Umschau Verlag, Frankfurt 1960-1969). Fuchs, E., H. Oppolzer and H. Rehme, Particle Beam Microanalysis: Fundamentals, Methods and Applications (VCH. Weinheim, 1990). Gabriel, B. L.,SEM: A User’s Manual for Materials Science (American Society for Metals, Metals Park, 1992). Galopin. R, and N. F. M. Henry, Microscopic Study of Opaque Materials (W. Heffer and Sons, Cambridge, 1972). Goldstein, J.I., and H. Yakowitz, eds., Practical Scanning Electron Microscopy (Plenum, New York, 1977:. Goldstein, J. I., D. E. Newbury, P. Echlin, D. C. Joy, A. D. Romig, C. E. Lyman, C. Fiori and E. Lifshin, Scanning Electron Microscopy and X-Ray Microanalysis (Plenum Press, New York, 1994). Goodhew, P. J., Specimen Preparation in Materials Science (North-Holland, Amsterdam, and American Elsevier, New York, 1973). Gi)pel, W., and C. Ziegler, Stxukntr der Materie: Grundlagen, M i s k o p i e und Spektroskopie (Structure of Matter Fundamentals, Microscopy and Speatpscopy) in German (Teubner Verl. Ges., Stuttgart, Leipzig, 1994). GUntherodt, H. J., and R. Wiesendanger, (eds.). Scanning Tunncling Microscopy I (Springer-Verlag, Berlin, 1992). Hamann, C., and M. Hietschold, Raster-Tunnel-Mikroskopie(Akademie Verlag, Berlin, 1991). Hearle, J. W.S., J. T. Sparrow and P.M.Cross, TIE Use of Scanning Electron Microscopy (Pergamon Press, oxford, 1972). Hornbogen, E., and B. Skrotzki, Wexkstoffmihkopie (Microscopy of Materials) in German (Springer-Verlag, Berlin, 1993). Lifshin, E. (ed.),Characterization of Materials Part I. Materials Science Engineering, Vol. 2A (VCH Publ.. Weinheim. 1992). Lifshin. E. (ed.). Characterization of Materials, Part II, Materials Science and Engineering, Vol. 2B (VCH Weinheim, 1994). Loretto, M.H., Electron Beam Analysis of Materials (Chapman and Hall, London, 1988). Loveland, R. P., Photomicrography: A Comprehensive ’kcatisc (Wiley, New York, 1970). Lyman, C. E. (with 10 authors), Scanning Electron Microscopy, X-Ray Microanalysis, and Analytical Electron Microscopy: A Laboratory Workbook (Plenum Press, New York, London, 1990). Metals Handbook, 9th ed.,Vol. 9: Metallography and Microstructures (American Society for Metals, Metals Park, 1985). Metals Handbook, 9th ed., Vol. 10: Materials Characterization (American Society for Metals. Metals Park, 1986). Metals Handbook, 9th ed.,Vol. 12 Fmtography (ASM International, Metals Park,OH, 1987). Modin, H., and S. Modin, Metallurgical Microscopy (Butterworths, London, 1973). Muller, E. W. and T. T. Tsong, Field Ion Microscopy: Principles and Applications (American Elsevier, New York, 1969). Mulvey, T.,and R. K. Webster, Modern Physical Techniques in Materials Technology (Oxford University Press, 1974).
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Ch. 10, Refs.
Newbury, D. E., D. C. Joy, P. Echlin. C. E. Hori and J. I. Goldstein, Advanced Scanning Electron Microscopy and X-Ray Microanalysis (Plenum Press, New York, 1986). Pepperhoff, W., and H. H. Ettwig, Interferenzschichtenmikroskopie (Interference Layer Microscopy) in German (Dr. Steinkopf Verlag, Darmstadt, 1970). Peaow, G.. Metallographic Etching (ASM, Metals Park, OH, 1978). Petzow, G., Metal1ographische.s Atzen (Metallographic Etching) in German (Gebr. BomtrZger Verlag, Stuttgart, 1994).
Phillips, V. A., Modem Metallographic Techniques and their Applications (Wiley, New York, 1971). Reimer, L. and G. Pfefferkom, Rasterelektronenmilkopie (Scanning Electron Microscopy) in Gennan (Springer, Berlin, 1977). Rhines, E N., Microstructology: Behaviour and Microstructure of Materials (Carl Hanser Verlag, Mllnchen. 1986).
Rivike, J. C., Surface Analytical Techniques (Clarendon Press, Oxford, 1990). Rnss, J.C., Practical Stereology (Plenum Press, New York. 1986). Russ, J. C., Computer-Aided Microscopy: The Measurement and Analysis of Images (Plenum Press, New York, 1990).
Russ, J. C., The Image Analysis Handbook (CRC Press, Boca Raton, 1992). Saltykov, S. A., SrereOmetrische Metallographie (Stereometric Metallography) in German (Dcutscher Verlag fiir Grundstoffmdustrie, Leipzig, 1974). Samuels. L. E., Metallographic Polishing by Mechanical Methods (Pitman, Melbourne, London, 1971). Sad, D., Scanning Force Microscopy with Applications to Electric, Magnetic and Atomic Forces (Oxford University Press, New York, 1991). Schatt, W., EinfUhrung in die Werkstoffwissenschaft (Introduction to Materials Science) in German (Deutscher Verlag fur Grundstoflindustrie, Leipzig, 1991). Schwann, H.. Metallographie (Mctallography) in German (Deutscher Verlag tUr Grundstoffindushe, Leipzig, stuttgart, 1990). S C R ~ J., , Image Analysis and Mathematical Morphology, V01.2: Thereotical Advances (Academic Press, London, 1988). Stroscio, J. A., and W. J. Kaiser, (eds.),Scanning Tunneling Microscopy (Academic Press, Boston, 1993). Sybilia, J.P., A Guide to Materials Characteriultion and Chemical Analysis (VCH Publ, Weinheim 1988). Underwood, E. E., Quantitative Stereology (Addison-Wesley, Reading, MA, 1970). Vander Voort, G.F., Metallography: Principles and Applications (McGraw Hill, New York, 1984). Walls, J. M. (ed.), Methods of Surface Analysis (Cambridge Univenity Press, Cambridge, 1992). Waschul. H., Mparative Metallographie (Preparative Metallography) in German (Deutscher Verlag fiir Grundstoffindustrie. Leipzig, 1993). Weibel. E. R.. Stereological Methods, vol. 2, Theoretical Foundations (Academic, London, 1980). Wells, 0.C., Scanning Electron Microscopy (McGraw-Hill, New York, 1974). Witsendanger, R., and H.J. Glinthercdt, (ais.), Scanning Tunneling Mimscopy II (Springer-Verlag, Berlin, 1992).
CHAPTER 11
TRANSMISSION ELECTRON MICROSCOPY M. RUHLE and M. WILKENS Max-Planck-Institut fur Metallforschung 7000 Stuttgart, FRG
R. M? Cahn and I? Haasen?, eds. Physical Metallurgy;fourth, revised and enhanced edition 0 Elsevier Science BV; 19%
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1. Introductory renaarks In the fifties, when transmission electron microscopesbecame commercially available, their potentialities for enabling applied and fundamental research in physical metallurgy and materials science were realized soon, Within a few years the resolution limit for a direct imaging of structural details of solids - up to that time limited by the lightwavelength in optical microscopy to some fraction of a p m at best - was reduced to about one nm; and in the course of this rapid development the horizon was opened for completely new aspects of research with a spatial resolution to nearly an atomic level. However, at that time it was quickly realized that transmission electron microscopy (TEM) differs from the classical optical microscopy significantly in the sense that TEM, in particular when applied to crystalline specimens, requires a much more profound knowledge of the interaction of the imaging (electron) waves with matter than in the case of optical microscopy. This holds for the imaging of crystal inhomogeneities (lattice defects, precipitates etc.) by TEM via elastic interaction of the imaging electrons with the specimen atoms. But it holds also for a local material analysis via various processes of inelastic interactions. The present article attempts to present a brief, and necessarily incomplete, introduction (i) to a number of methods for imaging different kinds of lattice defects and precipitates by diffraction contrast, constituted mainly by elastic interaction, and (ii) to problems of the new field of analytical electron microscopy, based on inelastic interactions. Because of the restricted space available, the references quoted are necessarily incomplete and, perhaps, not altogether free of arbitrariness. Other, more special aspects of TEM such as, e.g., Lorentz microscopy of ferromagnetic domains and special aspects of high-voltage electron microscopy (e.g., the critical voltage effect) are not dealt with at all. In order to facilitate the step into the topics of this chapter the reader is recommended to study first the Appendix, Elements of kinematical diffractiontheory, which may provide him with the basic knowledge necessary for an understanding of the content of the present chapter. This appendix may also be useful in connection with ch. 12.
2. The instrument The technology of modem electron microscopes has been developed so far that nowadays commercially available instruments and the sophisticated attachments fulfill practically all requirements necessary for studying crystalline and non-crystalline thin films. Some essential properties of the instrument are sketched in this section. The resolution of an electron microscope is governed by errors of the magnetic lenses, in particular by the spherical aberration of the objective lens. The ultimate resolution r,, can be reached for an optimum objective aperture angle a,pf,with
where A is the wavelength of the incident electrons and C, the constant of spherical
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aberration. A and B are constants, their actual values depend on the way in which different contributions to lens errors are combined (HIRSCHet ai. [1977]). High-resolution instruments (with accelerating voltages of 200 kV or 300 kV)possess a point-to-point resolution of S 0.25 nm. If ultimate resolution has to be reached then stringent requirements are necessary for the stabilities of the high voltage and the lens currents. The pole piece of the objective lens possesses a narrow gap and small borings, so that in such instruments the specimens can be manipulated only in a very limited range. Usually, ultimate resolution is not necessary for electron microscopy studies in materials science. For these studies it is more important that the specimen can be shifted and tilted over large ranges and that different signals of scattered electrons and X-rays can be detected in analytical microscopy studies. The lenses of the standard instruments possess pole pieces with larger boring which allow specimen manipulations and the mounting of detectors for analytical purpose. Thereby, the resolution of the instrument is reduced to 0.4 nm. This resolution is, however, sufficient for most TEM studies in materials science. An electron microscope can be split into three components according to its function: (i) the illumination system composed of the electron gun and the condensor lenses produces a fine electron beam to “illuminate” the specimen; (ii) the objective lens, which immediately follows the specimen, produces the diffraction pattern and a first magnified image of the specimen; (iii) the magnification system produces the final image. In addition to the optical components the microscope encloses also a specimen chamber (specimen handling system) and a recording system (HIRSCHet al. [1977] and HRENet al. [1979]). The illumination system provides a beam of adjustable size, intensity and convergence angle at the specimen, resulting in a limited coherency. The term coherence refers to the range of phase differences in the illuminating beam as it approaches the specimen. If the electrons come from a single point source, then all the waves in the incident beam are in phase with each other and the illumination is coherent. On the other hand, if the source of electrons is so large that there is no phase relation between the incident waves, the illumination is completely incoherent. In reality, the filament in an electron microscope is somewhere between these two extremes and the incident illumination is defined as partially coherent (SPENCE[1981]). In TEM the dimensions of the region on the specimen, a, over which the illumination appears coherent is related to the angular aperture of the illumination, ai,by (GEM [19791):
-
a = -.A
2ffi
Field emission guns possess a very small ai which results in a large coherence length. Charged electrons interact strongly with the transmitted specimen, the scattering cross-section a,for electrons is rather large compared to the cross-section of neutrons or X-rays (see appendix). Specimen thicknesses are required in the range of 10 nm to 1 p m References: p . 1105.
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depending on the imaging mode and on the voltage of the TEM. Recipes exist for electrochemical thinning of conducting materials (THOMPSON-RUSSEZL and EDINGTON [1977]) as well as for ion beam etching of insulators (TIGHE[1976]). Different kinds of electrons and electromagnetic waves are emitted from a specimen which is irradiated with high-energy electrons. The different waves result from elastic or inelastic scattering processes. Different signals (fig. 1) are used for different imaging modes. Information on the crystal structure and on defects in the specimen can be obtained by studying the elastically scattered electrons (in micrographs inelastically scattered electrons are also present and contribute to the background intensity) whereas investigations of inelastically scattered electrons and of other waves leaving the specimen allow the determination of chemical composition and topology of the specimen surface. The microscope can be operated in different modes as described briefly in the following. For more details the reader is referred to textbooks on electron microscopy given in the list of references. In standard or conventional transmission electron microscopy ("EM) mode the microscope is operated to form images by bright field (BF), dark field (DF), or lattice image (phase) contrast, see fig. 2. A BF (DF)image is formed when only the direct (one diffracted) beam is used for image formation. The objective aperture prevents all other beams to pass to the recording system. Usually, the specimen is oriented so that the Bragg condition is nearly fulfilled for a set of lattice planes. Then one reflected beam is strongly excited besides the incident beam (see 04). A weak-beam dark-field (WBDF) [1978]). image is produced if a weakly excited DF beam is used for imaging (COCKAYNE A lattice image is formed by the interference of at least two beams in the image plane of the objective lens. Lattice fringes can be observed if a row of systematic beams (reflected at the lattice planes in question) is used for imaging, while a structure image is formed by using many beams present in a low-indexed Laue zone. Special adjustments of the microscope are required for the formation of the high resolution electron microscope images (SPENCE[19811 and THOMASand GORINGE[1979]), see § 9. Incident beam
-I!/
Diffracted electrons-
Secondary electrons Auger electrons
Transmitted electrons -Energy
Loss electrons
Fig. 1. Electrons and electromagnetic waves emitted from a transmitted specimen as a result of elastic and inelastic scattering or diffraction of the incident electron waves.
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Illumination s)
0
Right. Field
Dark Field
weak Beom
Lattice
IrnagLq
Dark Field
. OUO 0-. . hkl
. Mg-k.000 .
043 000 hkl
.
e
Diffraction Fbttern
Fig. 2. Ray diagrams (including Ewald sphere construction) for: (a) a conventional two-beam bright-field (BF) image; (b) dark-field (DF) image: (c) weak-beam dark-field (WBDF) image; (d) lattice image. The gun of the electron microscope is tilted by the appropriate angle in going from (a) to (b) or from (a) to (c).
The objective lens produces a diffraction pattern of the specimen in its backfocal plane (see fig. 3). The first image of the object is rotated by 180' against the diffraction pattern. The diffraction pattern and the first image are magnified by the subsequent intermediate lenses and projector lenses. The information obtainable from the diffraction pattern is summarized in § 3. In the scanning transmission electron microscopy (STEM) mode, the electron beam is focused as a fine probe on the specimen by the prefield of the objective lens. The beam probe is scanned over the specimen (by scanning coils) and the transmitted intensity is recorded. The STEM mode is usually applied in materials science for analytical microscopy (see $ lo), where the probe is fixed on a selected small area and either the energy losses of the transmitted electrons are studied (EELS)or the X-rays emitted from the specimens are investigated (usually by EDS) for the determination of the chemical composition of the specimen. The use of a very small probe size (e10 nm) is often limited by a strongly enhanced contamination rate (HREN [19793). In the STEM mode backscattered electrons can be collected as well as secondary electrons. These scanning micrographs are similar to those obtained by regular scanning electron microscopy (SEM). Information on the topography of the specimen surface can be obtained.
Rgerences: p.
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I inckient beam
backfocal plhne F of objective lens
Fig. 3. Ray diagram in the area of the objective lens of an electron microscope. A diffraction pattern is formed in the backfocal plane F of the objective lens, whereas the (first) image I of the object 0 lies in the plane I. The first image of the object is rotated by 180' relative to the diffraction pattern.
3. Informutionfrom the diflaction pattern 3.1. Diffraction spot pattern The diffraction pattern provides crystallographic information on a qualitative phase identification and on the orientation-relations between crystals and the direction of the incoming electron beam. Kikuchi patterns (see $3.2) can be used for the exact determination of the orientation, whereas the convergent-beamtechnique allows statements on crystal symmetry and determination of the foil thickness. The possibilities and the accuracy of analysis of diffraction patterns are discussed in many textbooks on electron microscopy and special papers (e.g., ANDREWS e t d . [1971]). Important factors for the accuracy are: (i) the shape factor described in the appendix which determines the intensity distribution in the reciprocal space; (ii) instrumental alignment and beam divergence; (iii) specimen perfection; (iv) curvature of the Ewald sphere and its orientation with respect to the foil; (v) double diffraction.
3.1.1. Double diffraction From the structure factor it follows that certain reciprocal lattice points are not present (zero intensity) for certain crystal symmetries. However, each diffracted beam (within the crystal) can act as an incident beam and can diffract electrons to a reciprocal lattice point forbidden by the structure-factor rules, especially in orientations where
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several different reflections are excited simultaneously. For example, in the diamond cubic structure the (002) reflection is not allowed, but this reflection can be excited in a [ 1101 foil via double diffraction: if a (lil) reflection is excited then this reflected beam can act as a primary beam for ( i i i ) planes; this gives a total reflectiong,+g2=(1i1)+(i11)=(O02). 3.1.2. Patterns from ordered crystals The symmetries of ordered crystal structures are often changed compared to the disordered crystal structure. This results in the appearance of superlattice reflections at positions that are forbidden for the disordered structure. The intensities of the superlattice reflections correspond to the difference between the atomic scattering factors of the different atoms, in contrast to the intensities of the fundamental reflections, which are related to the sum of the scattering factors. As an example, the B2 superlattice is selected. It is based on the bcc structure of the CsCl lattice with one kind (A) of atoms at O00 and other (B)at f ). The structure factors are (for complete ordering):
F = f, + f, for h + k + 1 = even: fundamental reflections, F = f, - f, for h + k + 1 = odd: superlatticereflections.
(3)
The corresponding diffraction pattern is shown in fig. 4a. In general, superlattices can be identified from their diffraction pattern, either by comparison with structure-factor calculations for different possible superstructures, or by an analytical method which can be viewed as the reversal of the structure-factor calculations (KHACHATURYAN[1974]). In this method the ordered lattice is represented by a superposition of concentration waves. Theoretically, it allows unequivocally the determination of the real space lattice from the complete set of experimentally determined superlattice diffraction vectors. This method by Khachaturyan can be used to identify superlattices in substitutional (DASetal. [1973]) and interstitial solutions. However, it must be carefully observed that superstructures cannot always be identified by diffraction patterns alone. If, as in many
Fig. 4. Diffraction patterns from ordered crystals: (a) diffraction pattern of an ordered P-AlNi alloy (B2 superlattice), [lo01 zone axis, superlattice reflections are visible; (b) Diffraction pattern of an ordered Ta-0 alloy, [110] zone axis, variants of different orientations contribute to the pattern. References: p . 1105.
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a.11, § 3
disordered cubic alloys, an ordered structure has non-cubic symmetry a number of orientational variants may exist. Figure 4b shows, as an example, a (110) pattern of an ordered Ta-0 interstitial phase. The superlattice spots arise from very small ordered domains. If the selected area contributing to the diffraction pattern is large compared to the domain size, then the different domain variants contribute to the diffraction pattern. It is then impossible to determine the superstructure from the diffraction pattern alone. The size, shape, and the number of the different variants of the ordered domains can be observed, however, from different DF images taken with superlattice reflections.
3.2. Kikuchi lines Electrons can be scattered inelastically by interaction with the atoms of the specimen. Those electrons lose energies in the range of about a hundred eV. The inelastically scattered electrons can subsequently be diffracted coherently when Bragg’s law is fulfilled at a suitable set of reflection planes. Since the (primary) inelastic scattering process occurs in different directions, the loci of the different subsequent coherent scattering are cones with semivertex angles of (90” -0) to each side. The two cones are bisected by the reflection plane. The lines are, therefore, produced in pairs which are in contrast to the background. A dejiciency Zine of less intensity than the background occurs nearer the origin of reciprocal space than its associated excess line with intensity above the background. Two conditions must be fulfilled so that these Kikuchi lines are observed. One is that the crystal is thick enough, so that enough inelastic scattering processes occur,the second condition is that the crystal must be nearly perfect, especially not bent. The width of the lines indicates the curvature range of the crystal planes over the thickness traversed. If this becomes excessive, the lines disappear into the background as their intensity is spread over a larger angle. The Kikuchi line pattern can be used to determine the orientation of the crystal with respect to the incoming electron beam. There exist two ways for doing this. Either three pairs of &chi lines have to be indexed and then the orientation can be calculated, or the observed Kikuchi pattern has to be compared to Kikuchi maps (THOMASand GORINGE[1979]). The orientation of the specimen with respect to the electron beam can be determined with an accuracy of better than 0.3’. The excitation error s (cf. 84) can also be determined from the relative position of Kikuchi lines compared to the diffraction spots as demonstrated in fig. 5.
33. Convergent-beamdiffraction The size of the area giving rise to the diffraction pattern can be substantially reduced
if
- similarly as in STEM - the electron beam is focused onto the specimen. Under
this condition it is practically impossible to maintain the nearly parallel illumination condition. Some convergence of the beam is introduced. As a result, the diffraction ‘‘spots” become “discs”. For a convergence angle a O E
= Vg
C Fig. 5. Kikuchi lines on a diffraction pattern: (a) excitation error S = 0; @) s>O. (c) Sketch showing shift in Kikuchi lines produced by a tilt E .
References: p . 1105.
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(i) Each diffraction disc is produced from exactly the same small area (microdiffraction) without any shift introduced by lens errors (HIRSCH et ul. 119771). Therefore, these diffraction patterns can be used for the determination of the orientation and the crystal structure of the small illuminated area. (ii) Each disc contains the intensity diffracted by this same area of crystal for a range of incident orientations. That means that diffraction discs (convergent-beam electron diffraction pattern - CBED) are two-dimensional maps of diffraction intensities as a function of inclination between the incident electrons and a particular crystal direction. The lattice parameters of a crystal can be determined very accurately from the fine structure of CBED patterns. A change in Bragg angle A0 results in a change of the positions of lines within a pattern. The method is very strong in measuring rekztive changes in lattice parameters. Absolute numbers do not have a high accuracy. If the change of lattice parameters of an alloy or compound is directly and unambiguously related to its chemical composition, then the chemical composition may be deduced from lines observable in CBED patterns. As an example, for dual-phase steels the local concentration of carbon could be determined with an accuracy of cO.1 at% (STEEDS[1981]). The spatial resolution (including beam broadening) is governed just by the geometry of the electron probe. Local strains (STEEDS[1981]) and also the foil thickness (KELLYetal. [1975]) can be measured from CBED patterns. Furthermore, CBED patterns allow the determination of the symmetry and of the space group of the crystals (STEEDS[198l]).
3.4. Moir6 pattern Moir6 patterns (see fig. 8c for an example) occur from overlapping crystals as in composite films or in two- (or more) phase systems. Two general cases must be considered (i) Parallel maid fringes are formed if two parallel planes of different spacings are reflecting. The lattice spacings (d,= l/g, and d2= l/gJ differ only in magnitude. Fringes can be observed with distances d = l/Ag= l/lg,-g21. (ii) A rotational moir6 pattern is formed when planes with equal spacing d, but rotated through an angle a,diffract together. For this situation the moid spacing 4 is given by 4 = l/gsin a. Sometimes moik patterns have to be distinguished from other periodic defects in the crystals, e.g., a set of parallel dislocations. This can always be done best by imaging the same area with different diffraction vectors. The direction of the moire lines is usually perpendicular to the diffraction vector.
4. Theory of dieaction contrast 4.1. Introduction Diffraction contrast of defects in crystalline specimens is mainly a problem of highenergy electron diffraction in non-perfect crystals. It will turn out later that, by intrinsic
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reasons, for the imaging of lattice defects (dislocations, stacking faults etc.) an electronoptical resolution of about 1 nm is in general sufficient. Accordingly, lens errors of the objective lens can be neglected. We assume that the objective lens is focused onto the lower specimen surface through which the electrons leave the specimen. Then it suffices for the purpose of this section to calculate the electron wave function at the lower specimen surface since the electron-optical imaging system is assumed to be perfect. In a first part, we deal with the diffraction in a perfect crystal specimen. In a second part, it will be outlined how the diffraction theory must be extended for specimens containing lattice defects. Electron refraction effects, due to the mean inner potential of the crystal, are neglected throughout.
4.2. Specimen,reciprocal lattice and excitation error We consider a specimen of constant thickness f of the order of 100 nm. A Cartesian coordinate system is introduced with its origin in the upper specimen surface and with the z-axis (unit vector e3 perpendicular to the specimen plane and pointing downwards. The lateral dimensions L,, 4 of the specimen are orders of magnitude larger than t. According to the appendix, this means that the intensity distributions IF(K)I* at the reciprocal lattice points g are rod-shaped or spike-shaped with the spike axis parallel to e, and a spike length of the order of U t . This is indicated in fig. 6, where also the wavevector k,, of the incident wave and the Ewald sphere are inserted. Note that this figure is not to scale: assuming a modulus of (0.2 nm)-' for the low-order diffraction vector g,,and k,= l / h = (3.7 pm)-' (100 keV electrons), we have kdg,= 54. Having this in mind, one realizes that near the low-order reflections the Ewald sphere is so flat that, if the sphere cuts through the spike of one of the reciprocal lattice points, this will happen also for the spikes of other adjacent reciprocal lattice points. Accordingly, in high-energy electron diffraction, in general several Bragg reflections are excited
Fig. 6. The Ewald sphere and the reciprocal space. The intensity distributions IF(K)I' around the reciprocal lattice points are spike-shaped with spike lengths inversely proportional to the specimen thickness. The excitation error s is positive for g=g, and negative for the other g. References: p , 1105.
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simultaneously; besides the primary wave a number of diffracted plane waves leave the crystal. However, in general the specimen can be orientated so that only one diffracted wave is strong. Then, we speak about a two-beam case (primary plus diffracted beam). Images taken with either of these two beams are called strong-beamimages. On the other hand, dark-field images taken with an extremely weakly excited beam are called weakbeam images. The direction of incidence of the primary beam with respect to the specimen can be characterized by the excitation e m r sg, or simply s, which is given by the distance between the reciprocal lattice point g considered, and that point on the intensity spike where the Ewald sphere cuts through it. s is positive (negative) if the point g lies inside (outside) the Ewald sphere. If 8 is the glancing angle between the direction of k, and the lattice planes belonging to g and if 8, is the corresponding Bragg angle, then s = gAe,
with A e = e - e,.
(4)
43. Outline of the dynamical diffraction theory The essential points of the dynamical diffraction theory, in contrast to the kinematical theory as outlined in the appendix, are that diffractions between all plane waves involved have to be treated as being equivalent and that, in the absence of absorption effects, conservation of intensity must be fulfilled. There are several ways for formulating the dynamical theory for a perfect crystal. Here we refer very briefly to the classical way first developed by BETIIE[1928]. Another way will be outlined later when crystals with lattice defects are considered. Follow-ingBethe, the time-independent Schrodingerequation with a periodic potential is solved in terms of independent “eigen solutions” or Bloch waves h.Generally, Bloch waves can be written in the form
&(r) = b(r)exp(2~iK* r),
(5)
where the wavevector K has to be found for a given electron energy and a given direction of K.The function b(r)is periodic with the crystal periodicity. In the two-beam case, to which we restrict ourselves, there are two independent Bloch waves belonging to a given tangential component of K parallel to the electron entrance surface of the crystal. We assume that the two-beam diffraction vector g is perpendicular to e, and that the origin of the coordinate system lies on a reflecting lattice plane. Before the solutions are presented we introduce some terms which will be used repeatedly later. One important diffraction parameter is the extinction length &,
where V, is the volume of the elementary cell and F, is the structure amplitude, see appendix. For low-order reflections 6, is of the order of some 10 nm. Besides tgwe use the “wavenumber” notations
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112
l/s, = a,, a = a(w)= a, . (1 -k w’) , w = s&.
(7)
w is the ‘‘normalized”excitation error and u is the reciprocal of the effective extinction length as a function of w. The wanted solutions, expressed in these terms, are (e.g.,
r
7112
r
1“’
The Ako are the “eigen values” and the cf (n=0, g) are the components of the “eigen vectors” of the Schriidinger equation. The relation between eqs. (5) and (8) is easy to realize if the phase factor exp(27rK3 r) is put in front of the two terms of @ ,I in eq. (8). If the are known, a linear combination of the @ ,I ,
eB
*=cp
(1) (1) *B
+rp
(2)
(2)
(10)
*B9
has to be found which satisfies the boundary conditions at the specimen surface at z = 0 where both partial waves with the wave vector KOhave to add up to unity so that at z=O they fit to the incident primary wave:
#o
- (bo ex~2ni(ko-r ) )
Cpo = 1 for z e 0.
(11)
Further, both partial waves with wave vectors Kghave to cancel at z = 0. In our case, this leads to
(p= co(1)
3
= cp).
(12)
At the lower specimen surface at z=t, the Bloch waves decompose into their partial waves, and those having the same subscript, 0 or g, interfere and constitute a plane wave below the specimen. Thus, we obtain for z> t: @o =
40(t)exp[27ri(ko*r)],
+g
= 4&)exp[2vi(kg. r)].
References: p. 1105.
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M.Riihle Md M.Wilkens
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with
k, = k, + se, + g, lkgl = Ikol, and
+&) = cos m~ - i
W
(1 + w’)
1,2
sinmot, +(t)= i
1
112
(1 + w’)
sin ?rut.
4.4. Normal and anomalousabsorption High-energy electrons, when passing through a crystal, experience, besides the elastic scattering at the atom potential, also inelastic scattering, e.g., by interaction with the thermal vibration of the crystal atoms (phonons) or with the crystal electrons (plasmons, inner-shell excitation). By these events energy is transferred between high-energy electrons and the crystal, which leads to a loss of coherency of the wave fields of the elastically and inelastically scattered electrons. This effect can formally be described as an “absorption” (although, of course, the inelastically scattered electrons are not really absorbed). This “absorption” is different for different Bloch waves, depending on the high-energy electron density distribution l#,I2 with respect to the atom positions (H~smom etal. [1960]). From eqs. (8) and (9) it is easy to derive that I+~)l’ reveals a maximum electron density at the lattice planes characterized by g and a minimum in between; for i#(;l* the reverse is true:
Accordingly, #$) interacts more strongly with the crystal atoms and is thus more strongly absorbed than #E). We subdivide the absorption into normal absorption, which a high-energy electron would experience when travelling through the crystal far away from any Bragg reflection, and anomalous absorption, which takes the structure of +i$!I’ into account. Normal absorption is accounted for by adding a common factor exp(-p,,z/2), where pois usually expressed by the “normal” absorption length with po= 2n/5,,’. The anomalous absorption is introduced by adding a positive imaginary part to w , ,
e,,’,
l/s, +
1
1
-+-9
e;
sg
(16)
e,,’
where typically &‘ = (10-20)5, (RADI[1970]). Accurate values of are not well known since they depend, e.g., on the size of the objective aperture. Normal absorption acts only as a scaling factor, independently of the actual diffraction conditions. Therefore, accurate values of are not required, and often 6: is used for intensity calculations.
=si
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Working through the abbreviations introduced in eq. (7) leads in first order of &,/g to the substitution o+a+ia’,
1
a’=
g(l+w’)1/2 This finally gives: )#,t
= exp[-+p‘”zJ, I = 1,2,
with p(’)= po -F Zlra’,
p2 = & - 21ru1,
(18)
i.e., the absorption of &)(&))is enhanced (reduced). (tgis usually left uncorrected where entering into the c;’.)
4.5. Dynamical bright-field and dark-field intensities The same substitution as eq. (17) must be applied to the argument 1rm in eq. (14). After some algebraic operations, we find for the bright-field (ZJ and dark-field (Ig) intensities:
I
I, = exp(-p,r) cosh 2 ~ u ’ r+
g‘
= exP(-Pot)
w sinh 2m’r (1+ w’)”’
si&’ w’t + sin’ 1rm +
- sinh’ m ’ r + sinh’ rut l+wz
i
(19)
w2
In fig. 7a, b we have plotted lo@) and I&) as a function of r for w=O and w = 1 (with E =fg g= 105,). We see that the total intensity oscillates between I,, and Igr a maximum in I, corresponds to a minimum in Ig and vice-versa (“Pendellosung”).These oscillations are due to the fact that the z-componentsof the wavevectors of the Bloch waves $$) and$$) differ by a, which leads to a beating of the partial waves constituting $, and r,bg, respectively. The full oscillation period At= =a-’and the oscillation amplitude decreases with increasing Iwl. Further, the oscillations are damped with increasing r which is a consequence of the anomalous strong absorption of $:); if b,t): has decayed, a beating between partial waves is no longer possible. On wedge-shaped specimens the Pendelltisung-oscillationsgive rise to “thickness fringes” or “thickness contours” along lines of constant specimen thickness. An example is shown in fig. 7c,d. In fig. 8%b we show Zo and Zg as a function of w for some values of r. Both terms show oscillations with varying w (“bend contours”), which decrease in amplitude with increasing t. Whereas Zg is symmetric in w,this is not true for I,: the maximum in I, (i.e., best transmittivity!) occurs at w >0. A complicated system of bend contours is visible in fig. 8c.
References: p. 1105.
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M.Riihle and M. Wlkens
a
3
2
1
4
-
tit,
b
3
2
1
0
1
2
5
4 3
tlfg
Fig. 7. (a, b) l h - b e a m thickness contours calculated for (i =(A = lO& as a function of specimen thickness t: solid line = bright-field intensity Io, dashed line = dark field intensity Ig,with (a) excitation e m r w = 0, and (b) w = l . (c, d) Thickness contours in copper, g = ( l l l ) , w-0, in (c) bright fields and (d) dark field. Amws indicate equivalent points on the images. The lower arrows point to the specimen edge.
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-3
-2
-1
0
1
2
-3
-2
-1
0
1
2
3
3 W -
a I
I
Fig. 8. (a, b) TWO-beambend contours calculated for 6; =E = lggas a function of w, with (a) t = 2.gg. and (b) z= yg. (c) Bright field bend contours in copper. The dark lines correspond to specimen orientations where a certain set of lattice planes is in Bragg orientation. The moire patterns in the centre of the figure are due to a thin oxide layer on the specimen surface.
References: p . 1105.
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4.6. The column approximation We consider in fig. 9 the point C at the lower specimen surface. The lines through A and C and through €3 and C are parallel to the wavevectors k, and kg of the primary and diffracted wave, respectively. Triangle ABC is known as the Tukugi-triangle.It has been [1962], H o w and BASMSKI[1968]) that the wave functions $, and shown (T.AKAGI at point C are determined in very good approximation only by the specimen parameters within this triangle. With specimen parameters we mean local specimen thickness, local orientation, i.e., local excitation error, and local displacements of atoms from their positions in a perfect lattice. For high-energy electrons the Takagi triangle is very narrow: The angle at C is 28, with the Bragg angle 8, being about 1” or less. Taking t=100 nm, it turns out that the distance A-B is some nm at most. Accordingly, if variations of the specimen parameters over such a distance are sufficiently small, then and $, at point C can be calculated under the assumption that the specimen parameters along the “column” above C, Le., along the z-axis in our case, are the same for the entire specimen. This means that for calculation of t,bo and the “column” coordinate is the only intrinsic variable, whereas the lateral coordinates (x, y in our case) act only as parameters. This is the cohmn approximation introduced by HIRSCHet aZ. [1960].
+,
+,
+,
4.7. Diffraction at imperfect crystals
4.7.1. The displacement field Lattice defects cause displacementsR(r,) = Rrz of the atoms from their positions rn in the defect-free reference lattice”. In general, IRA is of the order of, or smaller than, the interatomic distances. Here we assume for simplicity that the displacements do not vary appreciably over the atom positions within the elementary cell so that the structure-
Fig. 9. The Takagi triangle. The wave functions (Clo and t+4g at point C are determined by the specimen parameters within triangle ABC.
* Regarding the notation R,cf. the footnote in 55.2.
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amplitude F, is unaffected by the lattice defects. Further we confine ourselves to cases where R is a continuous function in space, giving thus rise to “strain contrast”. Contrast due to stacking faults and other planar defects will be considered briefly in 6 8. 4.72. The kinematical approach A first insight in the intensities Z,,(x, y) and Zg(x,y) for perfect and imperfect crystals may be obtained by the kinematical diffraction theory outlined in the appendix. We start from eq. (A.46), where we neglect, as usual, the term SORin the exponent. Thus, the kinematical diffraction amplitude F(s) follows as:
exp(-2mi[g
- R(r)+ s
r])d3r,
where the integration runs over the specimen volume Vc=LxLytwith t 1. Equation (22) can be rearranged in order to separate the defect-induced part of the contrast integral. By partial integration we get
9, = 4; + 9;s
(25)
with
If goR(t) and goR(0) are neglected in eq. (26a), then 4; represents the “background amplitude” with identical to (24). 4,” is the defect-induced part of 4g.Since the integrand of eq. (26b) is proportional to the displacement derivative d(g*R(z))/dz, the integration limits can in general be extended to k 03. In the dark-field intensity I, = 14J2 the interference term between 4; and 4,” is often neglected (cf. e.g., HIRSCHetal. [1960] and WEKENSand HORNBOGEN [1964]): lg(.9
Y) = 14f + Iq..Y)l
2
-
(27)
Then adjusting to conservation of intensity gives the bright-field intensity:
i.e., the kinematical contrast in dark field (bright field) is always bright (dark). Later we will see that this is not true if Iwl is not > 1. 4.7.3. Dynamical diffraction theory in terms of plane waves In 84.3 we have outlined that one of the prerequisites of a dynamical diffraction theory is that all plane waves involved must be treated equivalently, i.e., diffraction between all plane waves must be taken into account. Equation (22) suggests intuitively one way by which this condition can be met: Both sides of eq. (22) are differentiated
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with respect to t, then we set t=z and concede that also +o may be z-dependent. In a next step we construct an equivalent equation, describing transition from the diffracted into the primary wave, i.e., k, and k,, change their roles. This requires a change of the signs of g and s. Then we end up with a set of coupled differential equations:
These equations are indeed one form of the differential equations established by H o w and WHELAN[1961] in a more detailed way for describing strain contrast by dynamical diffraction in imperfect crystals. These equations have to be integrated down to z =t with the boundary condition &= 1, +,=O at z=O. For R = 0 (perfect crystal) they can be integrated analytically; the result is identical to that given in eq. (14). 4.7.4. Dynamical diffraction theory in terns of Bloch waves A different approach for deriving dynamical diffraction in imperfect crystals starts from the Bloch-wave solution of the perfect reference lattice as given in Q 4.3 (WILKENS [1964], H~~USSERMANN etal. [1973], WILKENSet aZ. [1973]). A trial solution, in the twobeam case given by (6 = qP(z)+:)
+ cp(2)(z)+:),
(30)
with zdependent Bloch-wave amplitudes c p O is inserted into the SchrMinger equation which now contains a potential, the periodicity of which is perturbed by the atomic displacement field R(r). This leads in the column approximation to a set of ordinary differential equations for the cp". In the two-beam case one possible representation is given by:
These differential equations describe the defect-induced transition from Bloch-state (2) into Bloch-state (1) (first equation) and in the opposite direction (second equation). This transition is proportional to the displacement derivative d(g.R)/dz and is essentially controlled by the phase factors exp(k27riaz). Sufficiently far away from the defect, where d(g.R)/dz vanishes, the differential equations obey trivial solutions, p(')and p(2) are constant. Once eq. (31) is integrated down to z = t with po,(0)=c:' the wanted amplitudes cp,(t) (n=O,g) are obtained as outlined in $4.3 with cppo(t)inserted into eq. (10). Anomalous absorption is taken into account by substituting cr in the exponents of eqs. (8) and (31) according to eq. (17). Normal absorption can be added subsequently. Both sets of differential equations, eqs. (29) and (31), are in principle equivalent Refewmes: p . 1105.
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(WILKENSet ai. [19731). However, one consequence of the special form of the equations (31) is that they are especially suitable for deriving semiquantitative analytical solutions (e.g., WILKENS[1964], CHIKet al. [1967], WILKENS and R M E [1972] and KATERBAU [1981]).
4.7.5. Properties of strain contrast in strong-beam images Figure 10 shows strong-beam images in bright and dark field of dislocations traversing the specimen from the top to the bottom surface. Near the surface the dislocation contrast reveals characteristic brightdark oscillations, which are “in phase” in bright and dark field near the top surface and which are “in anti-phase” near the bottom surface. Further, the bright-dark oscillations are reversed if the sign of g is changed. In the middle of the specimen the contrast is mainly dark in both bright and dark field. These phenomena, which are of quite a general nature, will be explained by a qualitative discussion of eq. (31). For this purpose we re-define the Bloch-wave amplitudes cp(’) into @(’) by incorporating the absorption terms of eq. (18), which are incorporated in the usual procedure when going from the @(t) to cp,(t),
doz, ~ ( ” ( z ) = cp(”(z) exp- 2
1 = 1,2,
and restrict ourselves to dynamical diffraction, i.e., w = 0. The results, however, are valid also for IwI e 1. In a perfect crystal, where p(’)and p(’) are constant, @(”(z) decreases rapidly with z, whereas @(*) is only weakly damped ( , $ e ~ We ~ ) .assume a fairly thick specimen, so that @(I) has already decayed significantly in the middle of the specimen, cf. fig. 11. We consider the three regions (l), (2) and (3), indicated in fig. 11, separately. Now concerning contrast formation by a defect (e.g., a dislocation segment in fig. lo), we consider one particular column close to the defect, and the depth position z, of the latter is varied over the regions (l), (2), and (3). If z, lies in region (l), where @(I) and @(’) are still of about equal strength, the
i
Fig. 10. Diffraction contrast of a single dislocation (S) and a narrow dislocation dipole (D) in copper traversing the specimen from the top (T) to the bottom (B) surface g= (220). (The black-white dots in the background are due to ion damage (cf. $6.2.3) inside the microscope, as caused by an insufficient vacuum in the microscope column.) (a) Bright field (g-b) = 2; (b) dark field (gob)= 2; (c) bright field (g-b) = 0; (d) dark field (g-b)= 0.
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Fig. 11. hoperties of strain contrast in strong-beam images taken with w=O, schematically. Shown are the moduli of the Bloch-wave amplitudes &z), I = I, 2, for defect depth-positionsz,, in either of the regions (l),(2) or (3). for details see text. (a): @@)(t)I, z,, in (1); (b): l@e)(t)l,,g;(c): l@e)(t)l,p in (3); (d): l@'2'(t)l,z,, in (2); (e): t4")(t)t,z,, in (3).
scattering within a narrow interval near z,, will increase (or decrease) @(')I and at the same time decrease (or increase) depending on the sign of d(g.R)/dz and on the phase factors exp(*2niuz,,) on the right-hand sides of eq. (31). The phase factor is periodic in z,, with period &efF. Below the defect, the qdl) in eq. (31) are constant again and the @(O follow the decay as described by eq. (32). Then, going down to z= t, @(') has decayed irrespective of what happened to @(') at, or near to, z,,. Thus, the amplitudes +o(t) and &(t) at z = t are both determined solely by 4("(t).And since the modulus of the latter oscillates with the period we get bright-dark depth oscillations in both bright field and dark field, which are "in phase": bright (dark) contrast in bright field corresponds to bright (dark) contrast in dark field. In region (2) we have already @(')I I@(%Accordingly, for defects with z,, within this region the scattering goes essentially only in one way from @c2) to @('I, since there is (nearly) nothing to scatter from @(I) to @(2). So @(*)I is decreased irrespective of the actual value of z,,. On the way down to z = t, @(l) has again decayed, and the amplitudes +o(t) and +&) are, as in region (l), solely determined by @(2)(t). Since now I@~(~)(t)l e I~$~)(t)l~~, where the subscript bg refers to the corresponding value in the background far away from the defect, we obtain a dark contrast in both bright field and dark field, in striking contrast to the kinematical approach, cf. eqs. (27) and (28). If zo lies in region (3), the scattering occurs even more than in region (2) only from @(') to @(*). However, now @('I, on its enhanced level below q ,, can "survive" during the short distance from z,, to t, along which absorption may be negligible. Consequently, the amplitudes and 4gat z = t are determined by the interference of partial waves as described in $4.3.The mutual phase shifts of these partial waves are given by exp
+*
References: p . 1105.
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[k2?ria(t-zo)], and since absorption can be disregarded over the distance t-z,,, the depth oscillations in bright field and dark field must be complementary or in “antiphase”. Of course, in practice, these three regions are not sharply separated. Depending on the actual specimen thickness, we expect depth oscillations, which are “in phase” in bright field and dark field for defects at positions close to the top surface of the specimen. These depth oscillations are damped out with increasing z,,. For thick, or moderately thick, specimens they disappear, or nearly disappear, for zo in the interior of the specimen, resulting in only a dark contrast in bright and dark field. Finally, they reappear in “antiphase” for ~0 close to the lower specimen surface, see fig. 10.
4.7.6. Structure-factor contrast Coherent precipitates may reveal a mismatch between the lattice parameters in the precipitate and the surrounding matrix. This mismatch gives rise to a spacedependent displacement field and, accordingly, to strain contrast as described by eqs. (22), (29) or (31). If, in addition, the structure-amplitudeF, inside the precipitate differs from that of the matrix, e.g., by a change of the chemical composition, an additional contrast, named structupe-factorcontrast may be caused. The reason is that now the extinction length lg(proportional to 1/FJ becomes space-dependent, &=i$(r).This can be incorporated into eqs. (22) and (29) by setting 1/&= l/&(x,y; z) and taking 1/& in eq. (22) under the integral (cf. ASHBYand BROWN[1963] and WILKENS[198l]). For eq. (31) an approp nate extension for including structure-factorcontrast was given by WILKENSet al. [1977].
4.8. Practical applications of the differential equations Standard programs are available for the numerical integration of the differential equations (29) or (31) by electronic computers. The user has only to take care for subroutines for computing the displacement field R(x,y; z ) or its derivative aR(x,y; z ) / & If the diffraction contrast of a particular defect is to be calculated as a function of the specimen thickness t and the depth position z,, of the defect centre, considerable computer time can be saved by making use of the method of “generalized cross-section” (HEAD et al. [1973]), referring to eq. (29), or of the scattering matrix method (HAussmmxw et al. [1973], and KATERBAU [1980]), referring to eq. (31). In many cases computer time can additionally be saved by application of symmetry relations inherent in the particular diffraction contrast problem or in the displacement field of the defect ( H o w and WHELAN [1961], BALL[1964], POGANY and TURNER [1968] and KATERBAU[1980]).
5. Dislocations 5.1. Introduction Dislocations and dislocation distributions, e.g., in deformed crystal, are generally best investigated in the two-beam bright-field mode operated at a sufficiently positive excitation error, which ensures best transmittivity, cf. 54.5. This means a maximum specimen volume accessible by TEM and avoidance of pronounced black-white depth
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oscillations for dislocations close to the specimen surfaces (0 4.7.3, which may perturb the image. On the other hand, if details of the dislocation structure, e.g., splitting into et partials etc., are of interest, the weak-beam technique IwIB 1 is superior (COCKAYNE al. [1969]), at the cost of transmittivity. 5.2. The displacement field
We assume elastic isotropy and introduce the displacement field R of a straight dislocation in its own (right-handed)coordinate system x‘, y’, z’. The dislocation line runs along the y’-axis with its line unit vector u pointing from y’ e0 to y’ >O. The Burgers vector b lying in the 2-y’ plane can be split according to b =b, +be, where the screw component b, is parallel, and the edge component be is perpendicular to u. Then R is given by (e.g., HIRSCH et d.[1977]) *:
R=R+%,
(33d Z‘
with v=Poisson’s ratio, P=x’2+zy, rO=innercut-off radius. Note that R, is antisymmetic and R, is symmetric in x‘,z’. Further, R changes its sign if the sign of either b or u is changed. This is obvious for R, but must be noticed also for R,. 5.3. Contrast profiles of single perfect dislocations
We assume that b is a translation vector of the crystal structure (b is a “perfect” Burgers vector) so that g=b is integer. Regarding the contrast profiles of such dislocations we have to distinguish between the cases gob # 0 and g*b= 0.
5.3.1. The g=b# 0 contrast In case of g=b# 0 the contrast is mainly governed by the component R, of eq. (33b). This component causes an S-shaped bending of the reflecting lattice planes as schematically indicated in fig. 12. We assume s # 0 and recall the fact that diffraction contrast is especially strong (weak) where the local excitation error skz) is decreased (increased) as compared to the background value s, cf. $4.7.2, in other words where the reflecting lattice planes are bent by the dislocation displacement field towards (away from) the exact Bragg orientation. Consequently, for s # 0 the centre of gravity of a dislocation-
* In the TEM literature it is customary world-wide to denote the displacement field vector by R and the dislocation line unit vector by u. Here we adopt the same notation, although in ch. 12 the symbol u is used for the displacement vector and R is used for denoting particle radii in various connections. References: p . 1105.
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I
I
T
Fig. 12. Diffraction contrast of an edge dislocation for (g.b)s#O, s>O, schematically. Solid arrows represent the direction of incidence of the primary beam. Dashed lines represent local directions of incidence which would fulfill the Bragg condition. The angle between the two types of arrows is a measure of s,(z). Bright field intensity is denoted by IBp
contrast profile does not coincide with, but is shifted with respect to the image position of the dislocation line. Assuming a given direction of the line unit vector u the direction of this lateral contrast shift depends on the sign of (gob)s,as will be demonstrated below in connection with the images of dislocation dipoles. cf. eqs. (26)-(28), was evaluated for different The kinematical contrast integral types of dislocations by HJRSCHe t d . [1960] and GEVERS [1962]. It turned out that the width of the kinematical dislocation contrast is about (1/3-1/2)eK, where f K = s-' denotes the kinematical extinction length. If dynamical diffraction has to be taken into account (lwl< 1) this result remains essentially valid, with tKsubstituted by &eff=61 [eq. (17)] (How and WHSLAN[1962]). It should be added that the contrast of a dislocation line is always single-lined for Igobl= 1 but may be (asymmetrically)double-lined for Ig&l=2 and O in a fcc lattice. In this case three images taken with three diffraction vectors g = { 111} are sufficient for indexing all dislocations with b=), since the six significantly different vectors b of this kind give rise to a different contrast-extinctionbehaviour for the three g = { 111) (HIRSCH et al. [ 19771). The series of micrographs shown in fig. 16 stems from a deformed and subsequently annealed Ni crystal and was taken from a specimen parallel to the primary slip plane (111) with ga=(lli), g b = ( i l l ) , g,=(lil), where the subscripts a, b, c refer to figs 16a,b,c, References: p . 1105.
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*
-?%#
n/
Fig. 16. Dislocation network in nickel. no-beam bright-field images of the same specimen area, specimen normal (111). Three types of segment directions (see text) of the network are indicated by open circles (I), crosses (2), and dots (3). (a) g = ( I 1i), (b) g = (i1 I), (c) g= (1 1). The small arrows point to the same position on the three images.
respectively. In the central part of figs. 16b,c a network is visible constituted by three types of dislocation segments, as indicated in the figures. The segments (1) give rise to a gob # 0 contrast on fig. 16c and to a residual (gob = 0) contrast on fig. 16b and (scarcely to see) on fig. 16a. Accordingly, we have b, =e[101]. The segments (2) show gob # 0 contrast on figs. 16b and c, but not on fig. 16a. This leads to b,=e[liO]. The segments (3) are in g&#O contrast only on fig. 16b, thus b,=G[011]. Taking always the upper sign, we have b,+b,=b, as it is required for dislocation networks. Other dislocation segments in figs. 16a-c can be indexed in the same way. For a full indexing of a dislocation a discrimination between +b and -b (for a given choice of the line direction u ) is required. This can be achieved by making use of the fact that the lateral shift of (gob)$# 0 contrast lines is indicative for the sign of the S-shaped bending of the reflecting lattice planes from which the signs of u and b can be determined by physical arguments. In this context, certain contrast peculiarities due to surface-stress relaxation effects may also be helpful, cf., e.g., SIEMSet al. [1962], TUNSTALLet al. [1964], WILKENS et al. [ 19671 and HAZZLEDINE et al. [ 19751. The dislocation density p is defined as the (average) dislocation length per unit volume or, in a somewhat misleading way, as the number of dislocations per unit area. A full and reliable evaluation of p requires rather extensive experimental work. Here we stress the attention to one aspect only: On a given set of micrographs taken with a given two-beam diffraction vector g dislocations of some slip systems may be fully in contrast (gob # 0). Dislocations of other slip systems may be visible in residual contrast (gob = 0) only or may be completely invisible. This fact suggests immediately that any reliable determination of p requires a careful evaluation of different sets of micrographs of the same specimen area taken with different g-vector, cf., e.g., STEEDS[1966], ESSMANN [1966]. 5.6. Elastic anisotropy
Experience has shown that regarding the fundamental properties of the dislocation contrast of dislocations, elastic isotropy is a useful approximation even for moderately
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anisotropic crystals like copper. However, for strongly anisotropic materials such as &brass the contrast rules, described above on the base of elastic isotropy, break down. This holds in particular for the discrimination between g=b# 0 and gab = 0 contrast. In such cases a Burgers vector analysis requires the comparison of observed contrast figures, in particular of those parts showing depth oscillations, with computer-simulated images for modelled dislocation types, cf. AD et al. [1973] and HUMBLE[1978].
6. Point-defect agglomerates, radiation damage 6.1. Introduction Irradiation of crystals with energetic particles (e.g., electrons, neutrons, or ions) leads to the formation of interstitials and vacancies. Point defects may also be created by plastic deformation or by quenching-in of thermal vacancies. If these point defects are mobile, they may cluster together, forming point-defect agglomerates such as dislocation loops. In fcc metals they may also form stacking-fault tetrahedra. This holds especially for vacancies, but in principle also for interstitials. Vacancies may also agglomerate into cavities (bubbles, voids). One of the problems to be solved by TEM is the determination of the crystallographic nature of such agglomerates and the determination of their type, ),whether they are produced by an agglomeration of vacancy (V)or interstitial (Ii.e., vacancies or of interstitials. In this section some of such methods are briefly outlined. For a review see WEKENS [1978].
6.2. Dislocation loops
6.2.1. Formation of loops Dislocation loops are formed by agglomerationof point defects into plates on densely packed lattice planes. For instance, a monolayer (diameter D) of agglomerated vacancies on a { 111}-plane in a fcc metal gives rise to an intrinsic stacking fault surrounded by a dislocation loop (diameter D) with a partial Burgers vector b,=f of Frank type, perpendicular to the loop plane. Agglomeration of interstitials on { 111) gives rise to an extrinsic stacking fault over the loop area. If the loop size exceeds a critical value, it becomes energetically more favourable to eliminate the stacking fault area by sweeping a Shockley partial bs=b112> over the loop area to convert the Frank partial into a perfect Burgers vector b, e.g., b, + bs = b, (34d +[111]++[112] = 4[110]. Thew is evidence that similar two-step mechanisms for the formation of loops with perfect Burgers vectors are existing also for bcc (EYREand BULLOUGH[1965], JAGER and WILKENS1197.51) and for hcp metals (FOLLand WILKENS119771).
References: p. 1105.
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6.23. Analysis of large dislocation loops If the loop diameter D is sufficiently large compared to ,fg,ea then the loop nature is clearly visible on the micrograph and the Burgers vector b of the loop can be determined by the contrast-extinctionrules outlined in Q 5.5. Approximate information about the habit plane of the loop can be obtained by following the change of the projected shape and width of the loop when tilting the specimen around an axis parallel to the operating g-vector (for details see, e.g., MAHERand EYRE[1971]). Segments of a dislocation loop which are opposite to each other have the same direction of b, but opposite line direction u. Their contrast behaviour is therefore similar to that of a dislocation dipole, cf. 55.4. Accordingly, if the loop is imaged with @b)s # 0, the loop is imaged either as inside contrast (loop contrast inside the true projected loop position) or as outside contrast, depending on the sign of (g&)s. The question, inside- or outside contrast, can best be clarified by comparing images taken with g and -g and leaving the sign of s unchanged, cf. fig. 17 where dislocation loops of interstitial type in electron-irradiated molybdenum are shown; the apparent sizes of the loop contrasts change remarkably when going from g to -g. Then, the loop type, V or I, can be determined if the inclination of the loop plane with respect to the primary beam (direction of k,,)and the operating g-vector is known (HIRSCH er al. [1977]). In application to practical cases the inside outside method suffers from a number of pitfalls which may lead to the wrong answer regarding the loop type. Recipes have been worked out in order to avoid these pitfalls (WR and EYRE[1971], and FOLLand WILKENS119751). 6.2.3. Small dislocation loops dislocation loops are no longer If the loop diameter D becomes smaller than imaged “as loops”. On kinematical strong-beam images the loop contrast degenerates to
Fig. 17. Dislocation loops of interstitial type in electron-irradiated molybdenum. B o - b e a m bright-field images taken with w>O. Note the apparent change in the loop size when changing the sign of g. For details see text.
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a black-dot contrast containing very little information on details of the loop (R~ZHLE [19671). Even under weak-beam conditions the inside-outside contrast method breaks down for D < 7 nm. (HAUSERMAWef al. [19731). For dislocation loops below this limit the black-white contrast method becomes applicable when the specimen is imaged under two-beam dynamical conditions with w = 0, in bright or dark field ( R ~ [ 19671). E If under such an imaging condition a small dislocation loop is located within the surfacenear regions of depth-oscillation (§ 4.7.5) it gives rise to a characteristic black-white contrast figure. An example is shown in fig. 18 referring to a Cu specimen containing mainly small Frank dislocation loops of V-type (produced by irradiation with 30 keV Cuions) in a very surface-near layer close to the bottom surface of the specimen: Such black-white contrast figures may be characterized by a black-white vector 1, pointing from the centre of the black to the centre of the white lobe. Because of the very nature of the depth oscillation contrast, the black-white contrast of a loop is inverted, i.e., bright is changed into dark and vice versa, if either the loop type, V or I, is changed or if the Accordingly, in order to discriminate loop depth-position z,, is shifted towards z,,f between loops of V or I type we need information about the sign of @ I ) and of the depth position z,, of the loop. The latter information can be obtained by careful stereo measurements (e.g., R ~ [1967] E and R ~ andEWILKENS[1967]). Additional information about the direction of the Burgers vector and the loop plane normal n can be obtained from the fine structure in the centre of a black-white contrast figure (KATERBAU[1976] and EYREe l al. [1977]) and from the outer shape of the black-white contrast figures by comparison of experimentally obtained and calculated contrast figures. For elastically isotropic, or nearly isotropic, materials the shape of the contrast figure and in particular the angle between 1 and g depend in a characteristic manner on the direction of b and n with respect to k, and g, respectively. These dependencies have been worked out by numerical integration of the differential equations r--
------k
r !
4
Fig. 18. Black-white contrast of small dislocation loops @rank loops) in a copper specimen irradiated with 30 keV copper ions. Dynamical dark field image taken with w = 0, specimen normal (1 10). g = (002). The inserted triangles indicate small stacking-fault tetrahedra. References: p . 1105.
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(29) or (31) (e.g., R ~ [1967], E KATERBAU[1976], EYREetal. [1977] and ENGLISH et ai. [1980]) or by an analytical first-order perturbation integration of eq. (31) (e.g., WEKENS and ROHLE[1972]). The latter was successfully applied to the analysis of small dislocation loops in ionirradiated tungsten (HXUSSERMANN et al. [1972] and JAGERand WILKENS[1975]) and cobalt (FCILLand WILKENS [1977]). More recent analytical calculations by WILKENSand K~RCHNER[1981] have shown that the shapes of the black-white contrasts of small loops are sensitive to elastic anisotropy.
6.3. Stacking-faulttetrahedra Frank dislocation loops in fcc metals can dissociate into stacking-fauit tetrahedra (SFT') constituted by stacking faults on the four (111)-type tetrahedra faces interconnected by stair rod dislocations with b=i at the edges of the tetrahedra. This transformation is favoured in particular in metals of low stacking-fault energy such as gold, silver or copper. One consequence of the small strength of the stair-rod dislocations bounding the SFT is that their strain contrast is rather weak. Accordingly, the contrast of a SFT is predominantly determined by the stacking-fault areas (which may give rise to a fringe pattern if the SFT is larger than the extinction length (CHIK[1966]), i.e., the contrast shape of a SFT is in general close to the shape of the SFT projected onto the image plane. Some SFT, as produced by ion-damage in Cu, are indicated in fig. 18 by a symbol reflecting their outer shapes, as expected for a specimen orientation close to (110). In spite of the weakness of the strain field, S l T may reveal black-white contrast. For the black-white contrast large SFT with edge lengths of the order of, or larger than, degenerates to the so-called Ashby-Brown contrast (ASHBYand BROWN[1963]) where the depth oscillations are suppressed due to surface-stress relaxation (MCINTYREand BROWN[1966] and CHIKet al. [1967]). Under suitable conditions even small SFT may reveal black-white contrast similar to that of small dislocation loops (SALDMet ai. [19781). However, in contrast to small loops, black-white vector 1 points always parallel or anti-parallel to g (because the strain field does not show up a pronounced preferential direction). The latter property is sometimes helpful in order to discriminate between small loops and small SFT. 6.4. Cavities
Cavities (voids, bubbles) are best imaged under two-beam conditions with s=O. Then cavities act like a local reduction of the specimen thickness t. Thus, in a specimen with foil thickness t in that range where thickness fringes are observable the contrast of a cavity is dark on the front flank of a bright thickness contour and bright on the rear side (VANLANDWT et ab [19651). If t is beyond the region of thickness fringes, the contrast is bright but decreases rapidly with decreasing diameter d of the cavity. Finally, if d is below a critical value of some nm, cavities are no longer visible on in-focus images. Then the contrast is considerably improved by an appropriate defocusing of the objective lens (RUHLE and WILKENS[1975]).
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6.5. Displacement cascades and disordered zones Energetic ions (in the case of fast neutron irradiation the primary knocked-on atoms) deposit their kinetic energy in a crystal in so-called cascades, within which a large number of atomic replacements takes place. In an ordered alloy these replacement events lead to a zone of reduced long-range order which reflects size and structure of the cascade. Such disordered zones can be made visible on dark-field images taken with a superlattice diffraction vector g (JENKINS and WILKENS [ 19761). An example of disordered zones in Cu,Au irradiated with Cu+-ionsis shown in fig. 19.
7. Precipitates The precipitation of a second phase can be pursued by TEM, which was done successfully since the early days of TEM of materials (HIRSCHet al. [ 19771). Depending on the structure of the interface, we talk about coherent, partially coherent or incoherent precipitates. A partially coherent particle may possess one coherent interface and one where coherency is lost (Le., is semi-coherent or incoherent). This can be caused either by different structures in the precipitate and the matrix or by a large displacement along the interface (e.g., OLSONand COHEN[1979]). An incoherent particle has a crystal structure different from the matrix. Often there exist no orientation relationships between the two phases. A precipitate can give rise to TEM contrast for two reasons: (i) due to the alteration of the electron waves passing through the particle (precipitate contrast) and (ii) due to alteration of the electron waves passing through columns near the particle where the crystal has been distorted due to the presence of the precipitate (matrix contrast or strain contrast).
Fig. 19. Dark-field image taken with a superlattice reflection g =(110) of a fully ordered Cu,Au specimen irradiated with 30 keV copper ions. The dark dots represent zones Of reduced long-range order as produced by the displacement cascades of the incident ions. The dark bands are due to antiphase boundaries.
References: p . 1105.
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The interpretation of the matrix contrast is straightforward and can be done in a similar way as for radiation-induced defects or dislocations. The strain contrast has to be observed for different foil orientations (and different diffraction vectors) under welldefined dynamical two-beam or well-defined kinematical conditions (see 0 4). If a strain contrast can be observed then contrast calculations have to be performed for different strain distributions surrounding the precipitate in a thin foil. Matching the results of the calculations with the observations gives a qualitative and sometimes even quantitative model of the precipitate (ASKBY and BROWN[1963]). MCINTYREand BROWN[1966] and CHIK et al. [1967] showed that black-white contrasts (BW contrasts) can be observed if certain conditions for the size and magnitude of the strain are fulfilled. The strain is thereby expressed in terms of a dimensionless quantity, 1 Q=Er:-gg--,
8.
(35)
where r, is the radius of the particle, g the modulus of the diffraction vector, tgthe extinction length, and E a parameter describing the constrained strain of the particle (ASHBYand BROWN[1963]). BW contrasts are expected for certain combinations of r, and Q (CHIK et al. [1967]). Small values of r, result in BW contrasts with depth oscillation similar to those of radiation-induced defects (5 6), whereas for large values of r, and Q the depth oscillations of the BW contrasts are suppressed by the stress relaxations at the foil surfaces (Ashby-Brown contrast). In this case the sign of the displacement field can be determined by DF images unambiguously (ASHBYand BROWN [1963]). Very large defects produce no BW contrast at all (CHIKet aZ. [1967]). The calculation of TEM contrast was performed for inclusions of different shapes and included also elastic anisotropy (see e.g., LEPSKI[1974]). The main problem in doing such computer simulations is the determination of the displacement field of the precipitate placed in a thin foil. Spherical precipitates form in Cu-Co alloys (fig. 20) and the sizes depend on the annealing treatment (MCINTYRE and BROWN[19671). The typical coffee-beancuntrast can be observed. A “line-of-no-contrast” is perpendicular to the diffraction vector g. The width of the lobes measured perpendicular to g is a measure of the size of the precipitate. The small plate-like precipitates in AI-Cu also produce a strain contrast. The visibility depends on the operating diffraction vector since the displacement field is strongly anisotropic. Coherent Ag precipitates in AI-Ag alloys do not form a strain contrast, since there exists (nearly) no constrained strain around the particle (see fig. 21a). Semicoherent precipitates may produce a strain contrast which can be evaluated similarly as for coherent precipitates. This was demonstrated for small Mo2Cprecipitates in Mo by LEPSKIand BURCK[1981, 19821. HIRSCH et aZ. 119771 showed that coherent, semicoherent, or incoherent precipitates can be studied by utilizing the strain contrast @F and BF techniques). If the symmetry of the precipitate is different from the symmetry of the matrix, then additional reflections in the diffraction pattern can be
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Fig. 20. Contrast from spherically symmetrical strain fields due to precipitations in a Cu-Co alloy. The coffeebean type contrast is visible.
observed (fig. 21e). The shape of the precipitate is revealed by DF images taken with such a reflection belonging to the precipitate (fig. 21c, d). The precipitates themselves can be made visible (i) if difference in the structure factor in the two phases (“structure-factor contrast”), of which an example is shown in fig. 21, (ii) if the orientation of the foil is such that certain planes in the precipitate are near the exact Bragg condition (strong reflection) while no planes in the matrix are in such a position (“orientation contrast”), (iii) if the matrix displacements induced by the precipitate cause an abrupt phase-change at the precipitate (“stacking-fault contrast”), or (iv) if a moire pattern is formed due to different lattice parameters in the matrix and precipitate. The contrast caused for the different cases is already extensively described by HIRSCHet al. [ 19771. Particles of a second phase change the diffractionpatternof the pure matrix because: (i) there are extra reflections due to the diffraction pattern from the precipitate itself, (ii) the second-phase particles exert an influence on the matrix diffraction spots, (iii) there is the possibility of double diffraction (if the particle is embedded in the matrix), and (iv) since the precipitates are usually small, the shape of the diffraction spots is reciprocal to the shape of the (small) particles.
8. Structure of grain boundaries and inter3caces Internal surfaces determine many properties of materials. For an understanding of the properties it is essential to know the structure of the defects, which are actually the regions in space at which two crystals meet. Interfaces between dissimilar materials are called hererophase boundaries (or just interfaces). Interfaces between crystals differing only in relative orientation and/or translation are called homophase boundaries. Homophase boundaries include grain boundaries, stacking faults, twins, and antiphase References: p . 1105.
. Fig. 21. Contrast of precipitates. (a) Structure-factorcontrast of small coherent precipitates in A M % Ag alloys; no strain contrast is visible. (b) Structure-factor contrast of incoherent (large) precipitates in A M % Ag alloys annealed for loo0 min at 400°C (0" precipitates); foil orientation - (110); interface dislocations are visible between the AI matrix and the Ag precipitates. (c) Contrast of incoherent A M % Cu alloys; structure-factor contrast; foil orientation (106), BF image. (d) DF image with a reflexion of the Cu precipitate. Only Cu precipitates of one variant are in contrast. (e) Diffraction pattern of figs. (c, d).
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boundaries in ordered alloys. In the first part of this section the possibilities for revealing the structure of homophase boundaries will be described. The second part deals with some observations on heterophase boundaries. KALONJI and CAHN[1982] manifested that ten parameters are required for the geometrical description of an interface, see fig. 22. In this concept one assumes two interpenetrating crystals which are then misoriented (four parameters) and shifted against each other (three parameters). Three additional parameters are required to specify the location of a plane interface in the interpenetrating crystals. Finally, atoms of one crystal are removed from one side of the surface and those of the second crystal from the other. In this way, the symmetry of the interface can be described very elegantly by means of group theory (GRATIASef al. [1979] and KALONJI and CAHN[1982]). Stacking faults and antiphase boundaries are pure translation interfaces, whereas grain boundaries represent general homophase boundaries. The geometric parameters are potentially capable of undergoing relaxation.
Superpostion of 2 crystals
Rigid body dation 9
Rbid body translation I
Insertion of gmin bound~ryn
Rejection of atoms of inappropriate colcur
Fig. 22. The idealized construction of a crystalline interface: (a) superposition of the two crystals; (b) imposition of rigid body rotation (four parameters); (c) imposition of rigid body translation (three parameters); (d) insertion of surface S of the interface (three parameters); (e) rejection of inappropfiate atoms at the interface. References:p. 1105.
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8.1. Transmission electron microscopyof pure translationinterfaces At a pure translation interface (stacking faults, antiphase boundaries) two perfect crystal parts are shifted against each other. Usually, the plane of the interfaces coincides with a close-packed plane. For example, the translation vector is T =$ for a stacking fault (STF) in fcc materials, whereas T is a lattice vector of the disordered crystal structure for antiphase boundaries in an ordered structure. TEM contrast is expected in the projected area of the (usually) inclined translation interface. The depth position of the interface may be t,. Then the crystal part lying below the interface is shifted against the upper reference crystal by the translation vector T which defines also the sign of T. The diffraction contrast can easily be calculated by utilizing the so-called “matrixmethod” introduced by AMELINCKXand VAN LANDTJYT[1978]. With this method the contrast can be calculated for a general n-beam situation. We restrict ourseIves to the two-beam case which takes into account only one scattered beam 4g besides the transmitted beam 4,,. The amplitudes of the scattered beam 4gand of the transmitted beam +o are represented by a column vector
J/
=
(t).
AMELINCKX and VANLANDWT[1978] described the changes of the amplitudes after passing through a perfect crystal of thickness t [normalized excitation error w,cf. eq. (7)] by:
where the subscripts “in” and “out” refer to the incoming and outgoing waves for a perfect crystal, respectively. The initial values are of course +,,= 1, #g =O. M(t, w) represents a “response matrix” given by:
with [cf. eq. (14)]: &(t,w)=cosn-ut-i
w sin mt i sin mrt 4,”(t,w) = (1 + w,),,, ’ (1 + w y ’
(39)
anomalous absorption is included, which means (T is complex, whereas normal absorption in neglected. The normalized excitation error w has to be replaced by -w in eq. (39) to form 4;- and 4:- of eq. (38). The influence of a planar defect included in a transmitted specimen on the amplitude distribution of the different waves can be described by a response matrix of the defect Md. This matrix is called M* for a pure translation interface, with
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and a =2vgw (g=diffraction vector, T =translation of the perfect crystal below the defect with respect to the perfect crystal in which the electrons enter). The amplitudes of the waves transmitted through a specimen (total foil thickness t) containing such a translation interface (depth position tl(x), see 94) are determined by three contributions: (i) the contribution of the perfect crystal [foil thickness t,(x)] lying above the planar defect, described by a response matrix M(t,,w), [eq. (38)l; (ii) the contribution of the planar defect itself [response matrix M", eq. (40)]; and (iii) the contribution of the perfect crystal (thickness t2= t - t l ) lying below the defect [response matrix M(t2,w)]. The amplitudes of the waves at the lower foil surface are just expressed by
The contrast caused by an inclined defect can be calculated by an evaluation of eq. (41) for different depth positions t,. The following results can be obtained for an inclined STF (for details see AMELINCKX [1978]): and VAN LANDUYT (i) Bright and dark fringes are expected in the projected area of the STF. (ii) With increasing thickness new fringes are created in the centre of the foil. (iii) The fringes are parallel to the closest surface. (iv) The BF fringe pattern is symmetrical with respect to the foil centre, whereas the DF image is similar to the BF image close to the top surface but complementary close to the bottom surface. Figure 23 shows an example of the contrast of a stacking fault in a Cu-10 at% A1 alloy. The image of a stacking fault (STF) is characterized by the value of a =27rgur. In fcc materials stacking faults are predominantly produced on { 111} planes by a shear of ~=&112> or by removal or insertion of a plane of atoms. This may happen when vacancies or interstitial atoms condense on close-packed planes. In fcc materials the STF are classified as intrinsic or extrinsic. The determination of the type of STF is identical with the determination of the sign of a.We know from the symmetry rules that the sign of the contrast fringe in BF and DF at the top of the transmitted foil is positive (bright fringe) for a >0 (thick foil). From the BF and Df images of a STF (see fig. 23) we can determine the top surface and the bottom surface. Since we know the sign of a from the edge fringe of the BF image, we have the information necessary to determine the sense of 7. This method was first given by HASHIMOTO et al. 119621. GEVERSet al. 119631 pointed out that the type of the fault can be determined from a DF image alone; the sense of inclination need not be known. GEVERSet ai. showed that T can be determined by the following simple method. The diffraction vector g is drawn as an arrow with its origin at the centre of contrast fringes on the DF image. The nature of the stacking fault is intrinsic (extrinsic) if the limiting References: p . 1105.
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Fig. 23. Images of a stacking fault in Cu-10 at% Al, intrinsic stacking fault: (a) BF, g = ( i l l ) ; (b) DF, g = ( i i i ) ; (c) BF, g = ( i i i ) ; (d) DF, g = ( i i i ) . The rule of GEVERSet al. [1963] is fulfilled. (Courtesy A. KORNERand H.P. KARNTHALER.)
fringe on the side of the arrow of g is dark (bright). This rule is valid in fcc materials for { 1111, {220) and {400)reflexions while it is reversed for {200), (222) and (440} reflexions. The contrast of the STF vanishes for g w = integer. The direction of T can be determined from two images taken with different diffraction vectors on which the STF is out of contrast. The contrast of domain boundaries, twins, antiphase boundaries can also be calculated with the matrix method, see AMELINCKX and VAN LANDUYT [1978].
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83. Transmissionelectron microscopyof grain boundaries Grain boundaries (GBs) have been the object of intensive TEM studies for many years, and in that time many papers and review lectures have appeared (eg., CHADWICK and SMITH[1976], CLAREBROUGH and FORWOOD [1978] and BALLUFFI [1980]. It is well recognized that the structure of GBs is important for the understanding of the problems of segregation,recrystallizationtexture and intergranularembrittlement,and of course the bonding across the boundary also governs those properties. In the papers quoted above it is shown that much information can be obtained by TEM diffraction contrast studies, diffraction studies and recently also by direct lattice imaging. By TEM diffraction contrast studies, information on the intrinsic structureof GBs can be obtained; mainly dislocations can be analyzed. For special situations the geometrical parameters can be determined by TEM work. Diffraction studies allow the determination of the "width and symmetry of GBs, whereas direct imaging by means of highresolution 'EM allows an insight into the atomic structure of the boundary. In ch. 9 it is explained that GBs may be described in different ways: the dislocation model, the plane-matching model, and the coincidence-site model. All models will summarize the relaxation phenomena at the GB in different configurationof defects. The models assume that there exist certain misorientations which possess a low GB energy, and these misorientations are characterized by a low I: value of the coincidence-site lattice *. If the symmetry of a boundary and the included dislocations are determined, then the GB is completely specified. The parameters included in the description can be determined by TEM in principle by comparing experimental micrographs taken under well defined diffraction conditions with results of computer simulations carried out using the dynamical theory of electron diffraction. For the computer simulations the displacement of the atoms (lattice planes) due to relaxation as well as many experimental parameters such as the thickness, orientation of the interface, and extinction- and absorption lengths must be known. In addition, experimental diffraction parameters, such as the number of beams excited and their s value, must be established accurately. The contrast calculations can be very complicated. Each beam excited in the upper crystal is incident on the lower crystal and can give rise to further beams in that crystal. Thus, the total number of beams propagating in the lower crystal can be substantial, and the coupling of the beams is strongly influenced by the crystallographic relationship of the two crystals. There are three experimental conditions which allow a more straightforward interpretation: (i) Two-beam diffraction in one crystal and negligible diffraction in the other. Either the upper or lower crystal can be oriented for two-beam diffraction, and the nondiffiacting crystal is regarded as a block in which normal but not anomalous absorption
*The CSL can be obtained by allowing the two misonented crystal lattices adjoining the boundary to interpenetrate and translate so that lattice points of each crystal coincide. The space lattice made up of the coincident lanice points is called the CSL. The fraction of lattice points (in one crystal) in good coincidence is defined as 1/8. Coincidence models of GBs are discussed in ch. lOB, 52.2.1.3. References: p . 1105.
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occurs. The contrast behaviour is similar to thickness fringes. This diffraction condition can only be obtained approximately, since weak beams are inevitably excited in the “non-diffracting” crystal, and these may be coupled to some extent with the beam selected for image formation. This was demonstrated by HUMBLE and FORWOOD [19751 who compared microdensitometer traces taken across experimentally obtained GB thickness fringes to computer profiles. Humble and Forwood observed that the “nondiffracting” crystal does not behave simply as an absorbing wedge, and that no simple function could be found which would adequately describe the attenuation in this grain. (ii) Simultaneous two-beam diffraction in the two crystals. In this mode the specimen is exactly oriented so that only one diffracted beam is strongly excited in the upper crystal - and in such a way that this (excited) beam does not excite further strong beams in the lower crystal. At the same time, only one beam is strongly excited in the lower crystal by the incoming beam. Usually, only BF images are taken (HUMBLEand FORWOOD [1975]). (iii) In cases where the crystallography permits, two-beam or systematic diffraction by sets of planes with identical spacing and orientation in the adjacent crystals may occur. Such sets of planes are referred to as common, and this case is a special instance of condition (ii). This orientation has been used for the measurement of the relative shift of the two adjacent crystals. Experimentally, the orientation relation between the two adjacent grains of an interface and the normal on the interface can be obtained by three micrographs and accompanying diffraction patterns taken under different, well established orientations. The lateral shift T can be measured for special configurations (MATTHEWS and STOBBS [19771 and POND[19791). The TEM contrast of GB dislocations can be treated similarly as for lattice dislocations. However, for the GB dislocations further parameters, including the geometry of the interface and the diffraction conditions pertaining to both crystals must be considered. The direction of the Burgers vector 6 of the GB dislocation can be determined by using the gob = 0 criterion. However, the accuracy of this semi-quantitative method is small, since the magnitude of the Burgers vector of secondary GB dislocations is usually small. Boundaries between two crystals are assumed to have a low-energy structure for certain misorientations (e.g., BROKMAN and BALLUFFI [19811). This low-energy structure can be thought to exist of finely spaced arrays of dislocations, the so-called “primary” dislocations. Any deviation from such low-energy orientation relations is accommodated by a network of dislocations, the “secondary” dislocations. Experimental observations on those dislocations were carried out by Balluffi and coworkers (e.g., SCHOBERand BALLUFFI[1969]) mainly on “artificial” (001) twist boundaries in Au. The TEM studies reveal that a grid of undissociated secondary dislocations is present, see fig. 24. The distance of dislocations with the same Burgers vectors depends on the deviation A0 from the exact coincidence position. The observations are in agreement with the predictions of the theory (BROKMAN and BALLUFFI[1981]). Secondary dislocations can split into partial secondary dislocations and stacking-fault-like structures. This is observed not only for metals but also for semiconductors (BACMAN et al. [198l]). Lattice dislocations can interact with grain boundaries during yielding, creep and
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!
Fig. 24. “Artificial”twist boundary in a gold bicrystal, twist angle near 2 = 5 (36.9’)’. Square grids of secondary screw grain-boundary dislocations can be observed, Burgers vector b=(a/10). The dislocations accommodate the deviations from the exact B = 5 misorientation. (Courtesy T. SCHOBERand R. W. BALLUFFI.)
recrystallization (SMITH [1976]). Many TEM observations on reactions of lattice dislocations with GB dislocations can be explained on the basis of the dislocation model for GB structures. For the reactions of the lattice dislocations with grain boundaries one has to observe that Burgers vectors are conserved and dislocation lines can end only on other dislocations or at a free surface.
83. Dif’€raction studies on the structure of grain boundaries
SASSand BRISTOWE[ 19801 demonstrated that diffraction techniques also in the TEM can answer questions concerning the atomic structure of GBs. Each GB represents a periodic arrangement of strains (misfit) and good coincidences. Electrons and X-rays are scattered by this periodic grid which possesses a unit cell which corresponds to the CSL. If the (periodic) displacements at the GBs are known, then the scattering factor of such boundaries (mostly twist boundaries) can be calculated for the different reflections caused by the GBs. The reciprocal lattice, e.g., of a twist boundary, exists of thin rods, the integral intensity in the rod depends on the scattering factor (see appendix). The length of the rod is proportional to the inverse of the “ thickness” of the disturbed region near the GB. This grain-boundary “ thickness” approximately equals the distance of the dislocations present in the GB. 8.4. Direct imaging of grain boundaries High-resolution electron microscopy (HREM) allows the direct imaging of certain GBs. In HREM the structure of the specimen must be periodic in the direction of the transmitted electron beam. Therefore, only tilt boundaries can be studied where the tilt References: p. 1105.
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axis is parallel to the beam direction. Since the strong periodicity is disturbed in the core region of the boundaries, the point-to-point resolution of the instrument must be good enough so that information on the relaxation of atoms is transferred through the objective lenses despite the still rather high spherical aberration of the best lenses available to date. and DESSEAUX [1979]), germanium (KRIVANEK Tilt boundaries in silicon (BOURRET et al. [1977]) and molybdenum (PENISSON et al. [1982]) were studied by HREM. For these studies the specimen thickness has to be below 15 nm. Figure 25 shows a HREM micrograph of a tilt boundary in molybdenum. For the determination of the positions of the atoms close to the GB,contrast simulations have to be performed. In the simulation work one assumes a certain atomic configuration near the tilt boundary. The simulated micrographs have to be compared with the experimentally observed pictures for different defocusing values. The atomic arrangements have to be modified until a complete fit between the observed and calculated images is reached.
8.5. TEM contrast of heterophase boundaries Heterophase boundaries are formed when two materials of different structure andor different chemical composition meet. If, for example, any type of phase transformation occurs then a heterophase boundary is present where the different phases meet. This is true for the interface of a martensitic phase transformation where two materials with the same chemical composition but different structures meet, as well as for an interface of thermally grown precipitate.
i
Fig. 25. High-resolution image of a = 1 pure tilt boundary in Mo. Foil orientation [Ool] (for both grains). (Courtesy A. BOURRETand J. M.PENISSON.)
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The importance of the structure of these heterophase boundaries for the nucleation and growth process of the precipitate is recognized. However, only few systematic TEM studies were performed investigating the structure of such interfaces. The studies would require experimental imaging of the inclined interface under different diffraction conditions under which reflections from both crystal structures are excited simultaneously or one after the other. The observations have to be compared with contrast simulations. General expressions were derived even in the framework of anisotropic elasticity theory for the elastic displacement and stress fields by BONNET[1981]. The results, however, have as yet not been applied to contrast simulations of an interface. The geometry of the heterophase interfaces can be described by a generalization of the geometry of grain boundaries (BALLUFFI et al. [ 19821). These models are essentially fit-misfit models in which the regions of a good “fit” are patched where partial lattice matching across the boundary is achieved, and the regions of ‘misfit’ are boundary line defects which possess the character either of a dislocation or of a boundary step (facet). (See also ch. 9, 0 2.3 for a discussion of heterophase [“interphase”] boundaries). The model has been applied to interfaces between two cubic crystals - copper and silver (LAFFONTand BONNET [1982]) - and for hexagonal materials (BONNETet a!. [ 1981 I). A metal-metal-carbide interface was studied by different authors (DAHMEN e?af. [1981] and FLORJANCIC et al. [1982]). FLORJANCIC etaf. studied in detail the interface between Mo and Mo2C precipitates which forms after cooling from a supersaturated solution. In TEM micrographs (fig. 26) a dislocation-type contrast can be observed. The determination of the Burgers vector by means of the simple g 3 = 0 rule is not unambiguously possible. The contrast behaviour suggests that the Burgers vector is of type tcl11>,diffraction patterns yield the orientation relation between Mo and Mo,C. Dense packed planes of the Mo lattices are about parallel to dense-packed planes of Mo,C (Burgers relation). Careful diffraction studies were performed for an edge-on configuration of the interface, fig. 26. It was found that besides the reflections of Mo and Mo,C, streaks can also be observed lying between them. The formation of such additional streaks can simply be explained with the assumption that an interface crystal is present between the (pure) Mo and MoJ. From the maximum streak length the minimum thickness of the interface crystal can be determined to within nine lattice planes of the Mo lattice.
9. High-resolution TEM 9.1. Introduction By high-resolution TEM we understand all attempts of utilizing modem electron microscopes up to the ultimate resolution limit. The latter is determined by the wave aberrations of the objective lens - spherical aberration, chromatic aberration, astigmatism etc. - under optimal adjustments of size and position of the objective aperture, of the direction of the incident electron beam with respect to the optical axis and of the actual focusing of the objective lens, cf. also $2. Under such optimum conditions a point-to-point resolution of about 0.2 nm and a lattice-fringe resolution of 0 (S e 0) means overfocusing (underfmusing) with respect to the lower specimen surface (z=O). We denote the electron wave function in the plane z=O by JI,(P) *. By propagating from z=O to the plane z = 8 , JI,(P) is modified by Fresnel diffraction to $,(i). Note that for 6 >O, the wave function &(P) is really existing at the plane z = 6, whereas for 6 e 0, &(P) means a virtual wave function, which would merge into @,(i) at z = 0 when propagating from z = S a, with the optical axis. With a = p / k , see above, we introduce an aperture function A@) by A(p) = 1 forp 5 p,,
= ka,,
A(p) = 0 forp > p,. Thus, immediately behind the objective aperture we are left with
&dP) = $(P).
W
9
T(P)= A(P) * exp - iX(P).
(49)
T(p) is called the optical transfer function. The subsequent imaging system (intermediate lenses, projector lens) can be assumed to be aberration-free. Thus the final image amplitude on the viewing screen (or on the photographic plate) is found from $,@) according to eq. (49) by an inverse Fourier transformation,
where the index “m” stands for “magnified” image, M = elecmn+ptical magnification. [The same sign of the exponents of the first, eq. (42), and of the inverse, eq. (50), Fourier transformations accounts for the 180’ rotation between the diffraction pattern and the image, cf. $2.1
9.3. Consequencesof the wave aberration By the objective aperture all partial waves $,@) with p > p , are cut off. These partial waves are produced by structural details of Go(?) of “wavelengths” sl/p,,. Accordingly, these structural details are completely suppressed in $#). Moreover, the phases of the partial waves $,@) with p Ip,, are shifted by the wave-aberration term x(p) so that, after recombination of these partial waves in the image plane, the “structure”of qm(3)may be significantly different from &(i)even for structures of wavelength > l/p,,. For illustration, we calculate x(p) assuming C,=3 mm, A = 3.7 pm, S = O (exactly focused obiective lens), and p = p i = (1 nm)-’ and p2= (0.7 nm)-‘.From eq. (47) we find xbl)= 13.7” and x(p2)= 57’. Accordingly, in this case the wave aberration is probably negligible for the imaging of structures of dimension of 1 nm and above, but it becomes important near 0.7 nm or below. Obviously, the deteriorating influence of x(p) on the image formation can be reduced for a certain p-interval if both terms of x(p)
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act against one another: Since C, is always > 0, it is expedient to operate the microscope with 6 1, is proportional to the electrostatic potential V(i,z) in the specimen projected along the electron beam direction (=z-axis, origin in the lower specimen surface!): 0,(3) =
joV(P,t)dz. -I
(52)
Then, the formalism outlined in the preceding section leads to: $,(P) = 1 + u:)(f)+ iu:)(P)
(53)
with
(54) For a(,!,! the same equation is applied with sin,y substituted by cosx. From x(p)=x(-p), A@) =A(+), and w,(P) real it follows by symmetry arguments that u‘: and W: are real as well. Accordingly, in linear approximation of the “perturbation” term u,(i) we get the image intensity I&)
+
= 1 2wi)(P),
(55)
i.e., the only phase-modulated wave function &(i)at the specimen [eq. (51)], which would give rise to no contrast in a perfect microscope (lt+bO(?)l2=l), is converted by the wave aberration term sinxy(p) into an amplitude-modulated wave function t,b(?) at the image plane, eq. (55). In order to keep an utmost similarity between the observable structure of Z,(?) and the unknown structure of +b0(P), we have to adjust the focusing parameter and the size of the objective aperture in such a way that the partial waves of a maximum range of space frequencies p are transferred through the imaging system by the sinxb) term with a reasonable value close to 1 or -1 and with the same sign of sinxb). This is achieved by the socalled Scherzerfocus. 8, which correlates S to C, and A by:
References: p . 1105.
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.
3 2
1
0,5
43
0,4 p-’ [nml
Fig. 27. The transfer function sinX(p) for C,=3 mm. Solid curve: 6 =So= Schemer focus; dashed curve:
6 =0.86,,, dashed-dotted curve: 6 = 1.28,.
6, = -($ C,A)
li2
.
Figure 27 displays the term sinxy(p)for three values of S assuming C, = 3 mm and h = 3.7 pm. In the case of S =So (solid curve) we find a broad band-pass with sinxb) close to -1 (the shallow dent near p = 1.8 (nm)-’ is tolerable) and with the first zero at p = p l =2.5 (nm)-’. This means that all partial amplitudes of space frequencies with p < p l are transferred with the same sign, i.e., they may contribute to a correct reconstruction of the image. The partial amplitudes associated to space frequenciesp > p l change in sign due to the sin&) term more and more rapidly with increasing p. So their contribution to the final image may be confusing rather than constructive (e.g.. they may be washed out by a finite beam divergence). Therefore, the size of the objective aperture should be chosen so that pmax(eq. 48) coincides approximately with p l , thus cutting off the oscillating part of sinxy(p). In this way a resolution limit of l/p, (= 0.4 nm in fig. 27) is achieved. If S is reduced by 20% to 0.86, (dashed curve) we find a reduction of the first zero to p1= 2.2 (nm)-’ which corresponds to a reduction of the resolution limit to 0.45 nm. If, on the other hand, 6 is increased by 20% to 1.2 6, (dashed-dotted curve) the shallow dent near p = 1.8 (nm)-’ becomes so deep that it crosses the abscissa. As a consequence, the broad band-pass of the solid curve is completely disturbed and the resolution becomes drastically worse. We learn from these examples that fully utilizing the resolving power of an electron microscope requires a precise knowledge of the spherical aberration constant C, and a precise control over the actual defocusing value 6. In the meantime several methods have been developed in order to make the partial waves of space frequencies beyond p 1 also useful for the image interpretation. Here the key-word “image reconstruction” must be sufficient, cf., e.g., HAWKES[19801.
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9.5. Some remarks to the high-resolution images of crystalline specimens The weak-phase object approximation, originally developed for the study of noncrystalline specimens, is valid also for crystalline specimens, provided they are sufficiently thin. The space frequency spectrum $@) of a perfect crystalline weak-phase object consists of a discrete set of sharp Bragg diffraction spots. If such a specimen is carefully orientated with the incident beam along a low-order zone axis and if the lattice constant is large compared to that of pure metals or alloys, a large number of diffraction spots may pass through the objective aperture. In the frame of the weak-phase object approximation they form an image which reveals many details on an atomic scale of the (projected) crystal structure. An example is shown in fig. 28. In particular, complicated stacking or shear variants of otherwise comparatively simple crystal structures, which are hardly, or not at all, determinable by X-ray or neutron diffraction techniques, may become easy to analyze on high-resolution TEM. Regarding stacking variants of ordered and A ~ L I N C K X alloys (antiphase structures) we refer as an example to VANTENDELOO [1977, 19781. Various applications of high-resolution TEM to crystal-structure research in mineralogy and inorganic chemistry may be found in a book edited by WENK[1976]. Silicon is a substance especially suitable for high-resolution TEM because of its simple structure and because of its comparatively large lattice constant. Detailed information on an atomic scale of split dislocation configurations and of the structure of and KOLAR[1979] and BOURRET dislocation cores have been obtained, cf., e.g., SPENCE er al. [ 19821. One point of special interest in physical metallurgy concerns the imaging of a local variation of lattice-plane distances as a consequence of a local variation of the chemical composition, e.g., in the course of a spinodal decomposition. SINCLAIR er al. [1976],
Fig. 28. High-resolution image of BaJ%,O,,. (hexagonal unit cell, a=0.572 nm, c = 1.172 nm). Projection along .The insert shows the projected structure: solid circles indicate positions of Ba atoms, hatched diamonds indicate Nbo, octahedra. (Courtesy J.L. HUTCHISON;inset from SHANNON and KATZ [1970].) References: p. 1105.
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studying a spinodally decomposed Ni,Au alloy, found indeed a periodic modulation of the lattice plane fringes, in accordance with the expectation. However, subsequent model calculations of COCKAYNE and GRONSKY[19811 have the spatial modulation of the clearly demonstrated that, due to the wave aberration lattice fringes observed on the image may be significantly different from the spatial modulation of the lattice plane spacings in the specimen. Even the number of lattice fringes per modulation period may be different from the corresponding number of lattice planes in the specimen. In conclusion it is worthwhile to emphasize that at the present state of the art all kinds of high-resolution work, in particular that concerning lattice defects, must be accompanied by extended theoretical work, Le., by corresponding model calculations. This holds even more if one crosses the border of the restricted regime of applicability of the weak-phase object approximation.
x@),
10. Analytical electron microscopy 10.1. Basic considerations Electron scattering can be divided into two categories: elastic and inelastic scattering. When elastic scattering occurs, the direction of the electron velocity Y is changed, but the magnitude IvI remains constant, so that the kinetic energy, E =imp2(me= electron mass) is unchanged. During an inelastic scattering event, energy is transferred to the target atoms and electrons, and the kinetic energy of the beam electron decreases. Inelastic scattering can happen by: (i) plasma excitation; (ii) excitation of conducting electrons leading to secondary electron (low-energy) emissions; (iii) deceleration of the electron beam in the Coulomb field of an atom (Bremsstrahlung); (iv) excitation of phonons; (v) ionization of the inner shells of atoms. Signals arising by the latter process can be used for analytical studies (fig. 29). .In the ionization process of inner shells, a sufficiently energetic electron can interact with an atom and cause the ejection of a tightly bound inner-shell electron, leaving the atom in an ionized and highly energetic state. The energy transferred to the bound electrons must be at least the energy of the bound state of the electron in a shell of the
(EELS) Fig. 29. Inelastic scattenng of incoming electrons at electrons of inner shells (schematic drawing). The characteristic X-raysand the energy losses are used for analytical purposes.
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atoms. Subsequent decay of this excitation state results in the emission of characteristic X-rays and Auger electrons. The X-ray yield is plotted in fig. 30 for different elements. Qualitative and quantitative information on the presence of the different elements can be obtained from the wavelength of a characteristic X-ray and from its intensity. A detailed treatment of the properties of characteristic X-rays is found in the literature (e.g., BERTIN [ 19751 and GOLDSTEIN et d.[ 198I]). In electron energy loss spectroscopy (EELS) the inelastic scattering of the incident beam is studied directly. The intensity of the inelastically scattered beam is measured as a function of energy loss by spectral analysis. At characteristic energy losses the intensities of the inelastically scattered electrons increase due to the ionization of inner shells. From the position of the edge and the increase in intensity, information is again available on the chemical composition. EELS is most powerful for light elements (atomic number Z< 12), whereas the analysis of the characteristic X-rays is possible for Z > 9. The cross-section for the inelastic processes (i) to (v) has been calculated (see GOLDSTEIN et ai. [1981]) for different metal targets as a function of energy. All of the cross-sections are observed to decrease with increasing energy. Considering a range of atomic numbers, inelastic scattering is favoured at low atomic numbers and elastic scattering at high atomic numbers. Inelastic scattering occurs by a variety of discrete processes, with a variable amount of energy transferred to the solid, depending on the strength of each interaction. Crosssections for the individual processes are difficult to obtain for all targets of interest. It is useful in many calculations to consider all inelastic processes grouped together to give a “continuous energy loss” (BETHE[1933]). It is found experimentally that an appreciable fraction of the beam electrons which strike a target subsequently escape where the electrons enter the solid (backscattered
---------~
lo
=-. u
09
c
3
2 08
-- 0 7 c
0
P x )G
~~~
\
\
\
\
\
\
\
\
\
06
I 2
z 05 T2
0 c
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5 02
=
ac
A 01 1 2 3 I 5 6 7 8 9 10 11 12 13 1L 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 333L 35 H He ti Be B C N 0 F N e N a M g A t SI P S CL A K C a Sc TI V Cr Mn FeCo Ni Cu ZnGaGe AsSe Br
Fig. 30. X-ray yield and Auger-electron yield for K-shell excitation for different elements. References: p . 1105.
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electrons). Backscattering electrons provide an extremely useful signal for imaging in scanning electron microscopy and scanning transmission electron microscopy. For proper image interpretation, it is necessary to understand the properties of the backscattered electrons as a fmction of the parameters of the beam and the characteristics of the specimen (for a detailed review see NIEDRIG[1978]). It is well established by experiments and by Monte Carlo calculations that the volume of interaction between a high-energy electron beam and a solid possesses the shape of a pear (fig. 31). The width of the neck of the pear is mainly determined by the shape of the electron beam, while the length of the neck depends on the energy of the electrons as well as on the material investigated. The diameter of the pear can exceed several microns 0.m). Therefore, studies with high spatial resolution can only be performed in thin foils. Those studies are preferentially performed in a conventional TEM fitted with a STEM unit, appropriate detectors for X-rays and energy loss spectra. 10.2. Quantitative analytical electron microscopy of thin foils: analysis of X-rays
In X-ray microanalysis the detector is as close as possible to the specimen. Usually, the detectors are energy-dispersive devices, which offer the advantages of speed of operation and ease of attachment to the system. Crystal spectrometers (wavelengthdispersive devices) are usually preferred for quantitative studies. However, it is very cumbersome to attach those to a transmission electron microscope. An example of an EDS X-ray microanalysis spectrum in the STEM mode of a TEM is shown in fig. 32.
electron
electron
bulk samDle ZA F corrections
Fig. 31. Schematic representation of the interaction of a high-energy electron beam with a bulk sample and a thin foil, respectively To the first approximation absorption and fluorescence effects can be ignored in the thin specimen.
Ch. 11, 10
Transmission electron microscopy
counts
’
L T r wos
HSS
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H6
I
lrn
-1 Fe
500 -.
Mo
10.2.1. Cross-section for inner-shell ionization Numerous cross-sections for inner-shell ionization can be found in the literature; this has been reviewed by POWELL [1976]. The number of X-ray photons produced per incident electron in a thin foil of thickness Ax can be predicted from 1
n, = OWN, -PAX, A
(57)
where CT is the ionization cross-section, w the fluorescence yield (see fig. 30), No Avogadro’s number, A the atomic weight, and p the density of the material. Experimental measurements of the X-ray yield (absolute efficiency) give a range of 0.11 x lo4 for Mg to 0.933 x lo4 for Ni in units of X-ray photons per electron and per angular unit.
-
-
10.2.2. Thin-film approximation If the sample is sufficiently thin to carry out quantitative transmission electron microscopy at 2 100 kV, the activated volume is approximately a cylinder equal to the beam diameter (see fig. 31); that means a high lateral resolution can be achieved. This high lateral resolution, however, results in a disadvantage in the use of thin specimens in that activated volume for X-ray production is small compared to a bulk-sample; x-ray counting rates are low and hence the potentioal accuracy is inferior to that generally accepted for bulk specimens in scanning electron miscroscopy. CLIFFand LORIMER [19751 showed that for a thin foil, to a first approximation X-ray absorption and fluorescence in the specimen can be neglected. Therefore, the ratio of two
References: p . 1105.
1090
Ch. 11, 0 10
M. Riihle and M.Wilkens
observed X-ray intensities ZA/ZB can be related to the corresponding weight fraction ratio cA/cBby the equation
where kAB is a constant at a given accelerating voltage and is independent of specimen thickness and composition. A normalization procedure, e.g., Zc,= 1, must be used to convert the ratios of the weight fractions into weight percentages. In mineral specimens, which are frequently used for calibration, assumptions must also be made concerning oxidation states, e.g., it is impossible to differentiate between Fe304and Fe,03 if ratios are measured, and oxygen cannot be detected (LORIMER et al. [1981]). For quantitative analysis the parameters kAB in eq. (58) have to be determined experimentally and also compared to theoretical results. There are discussions in the literature on the reliability of different kAB values available (e.g., GEISS 119811). Furthermore the effects of absorption of the X-rays and fluorescence in the specimen have to be considered for a reliable quantitative analysis (see 0 10.2.4).
10.23. Beam-spreading in the specimen Beam-spreading in thin foils is currently an active field of research in both experimental investigations and theoretical analysis (LORIMER ef al. [1981] and GEISS[ 19811). A simple approach to the problem of beam-spreading has been proposed by GOLDSTEIN et al. [19771 who assumed that a single elastic scattering event occurs at the centre of the foil, and defined the X-ray source as that volume in which 90% of the electron trajectories lie. Experimental observations surprisingly show good agreement with the simple theory by GOLDSTEINet al. [1977]. The beam-spreading limits the spatial resolution of analytical TEM,as shown in table 1. 10.2.4. Errors limiting the data of X-ray analysis X-ray counting statistics are usually assumed to obey Gaussian behaviour and at the 20 confidence level the relative error in the number of counts I is 2fi. Using eq. (58)
Table 1 Spatial resolution for X-ray microanalysis at 120 kV for incident electron probe sizes defined by d=O nm etal. [1981]). and d=5 nm (after LQRIMER Foil thickness (m) 40 100
Spatial resolution
d
(nm) 0 5 0
5 200
0
400
5 0 5
C
AI
22 4 22 11 24 30 35
1 22 6 22 18 21 50 53
cu 4 22 17 26 47 50
132
134
Au 10 23 38 42 107 108 302 302
Ch. 11, 9: 10
Transmission electron microscopy
1091
a ratio of two counts is converted into a weight-fraction ratio via the constant km. Thus it is necessary to add the total relative errors in Z, I , and kAB to obtain the relative error in the weight fraction cA/cB would be 6%. Unfortunately, X-ray counts often a premium in thin specimens, and it may be necessary to accept only a few hundred counts for one element. If, for example, IA contained only 900 counts and I , stilled equalled lOO00, then the relative error in the cA/cB ratio would be f 10%. The statistical error was often not taken into account in quantitative analytical electron microscopy (AEM). Therefore, materials-science interpretation of AEM work is bases on weak or incorrect foundations. In order to make corrections for absorption, fluorescence and beam spreading, it is necessary to know the thickness of the sample. Various parallax techniques can be used, including the contamination spots formed on the top and the bottom of the foil during the analysis. However, it seems that the most accurate technique is that of convergent-beam diffraction as described by AMELINCKX [1964] and KELLYet al. [1975]. This enables specimen thickness to be determined with an accuracy of k 2%. Many investigators have been aware that the measured X-ray spectrum may contain X-rays generated from other than the sample region excited by the focused electron beam. These additional X-rays are called “spurious” or “extraneous” radiation. This problem has received much attention (cf. GOLDSTEIN and WILLIAMS[1981]). The spurious radiation can be avoided by a correct experimental arrangement of the specimen in the analytical electron microscope. The absence of spurious radiation can be checked by a hole count. 102.5. Examples Since about 1979 reliable results obtained by AEM appeared. C H A M P ~ EetSal. S [1982] determined the partitioning of various alloying additions during the austenite pearlite transformation in eutectoid steels. The composition of different precipitates in a variety of steels was obtained and first results on segregation of impurities at grain boundaries and interfaces were reported. More results are published in the proceedings of recent workshops and conferences on AEM (HRENef al. [ 19791, LORIMER et al. [ 1981] and GEISS[1981]).
10.3. Quantitative analytical electron microscopy of thin foils - electron energy loss spectroscopy A transmission microscope fitted with an electron spectrometer can be used to produce energy-filtered images (KRAHL et al. [1981]) or for electron energy-loss spectroscopy (EELS) (JOY 119811). The latter can be used for measurements of innershell loss for quantitative microanalysis of light elements. There are two instrumental principles used nowadays. They are based on (i) the use of a magnetic analyzer below the camera chamber of a conventional TEM or a TEM with STEM attachments (KRWANEK and SWANN[1981]) and (ii) the use of an analyzer between the objective and the intermediate lens in a conventional TEM (ZANCHIet al. [ 19771). Figure 33 shows schematically the energy spectrum of 100 kV electrons after transmission through a specimen of about 25 nm thickness. The elastically or (by References: p . 1105.
M.Riihle and M. Wilkens
1092
Ch. 11, 8 10
Fig. 33. Schematic electron energy-loss spectrum of the signal intensity T(E) versus energy loss E. The hatched area close to the energy .EKis proportional to the number of atoms present in the specimen.
phonons) quasi-elastically scattered electrons give rise to a zero-loss peak whose width provides an indication of the instrumental energy resolution. For small energy losses (< 100 eV) surface and bulk plasmon losses are observed (JOUFFREY et al. [1978]). For higher energy losses E, inelastic scattering is caused at electrons of the outer atomic shells (e.g.. valence electrons). The intensity of the inelastically scattered electrons decays rapidly, approximately as AE', where A is a constant and r is between 2 and 6. At an energy loss equal to the ionization energy Ek of an inner atomic shell k (where k = K , L, M) the energy loss intensity increases rather abruptly, followed by a slower decay. By determining A and r from a region just preceding the ionization edge, the outer-shell background can be subtracted (by a computer, or manually using a log-log plot). The area I&, A) above the background is measured up to an energy loss E,+ A (see fig. 33). The number of atoms per unit area of the corresponding element is then given by (JOY [19811)
N = [Ik(a,A)/l, (a,
(a,A),
(59)
where uk(a,A)is a partial cross-section for the ionization of shell k within an energy range A and for scattering angles up to a. The angle a is determined by the collecting angle of the spectrometer. For a quantitative analysis uk(cy,A) can be determined experimentally or by calculations (EGERTON [19791 and JOY [19811). Unlike X-ray or Auger microanalysis, energy-loss spectroscopy therefore provides a convenient standardless method for determining the absolute number N of atoms of light elements. In addition, the ratio of two (or more) elements may be determined from the appropriate ionization edge, provided an equal integration range A is chosen for both elements. For that case Z,(a,A) will cancel in eq. (59) so that the low-loss region needs not to be measured. As the specimen thickness increases, multiple-scattering contributions to the background beneath an inner-shell edge cause the background to increase faster than the inner-shell "signal", so that the signalbackground ratio falls, which reduces the visibility of an edge and may make the background-fitting and subtraction process insufficiently accurate (JOY[1981]). Therefore, judging by the criterion of signalhackground ratio, the specimen should be as thin as possible within the analyzed region.
Ch. 11, $10
Tmmission electron microscopy
1093
However, by making the sample very thin we reduce the amount of valence-electronand inner-shell scattering, so that the relative noise level in both signal and background is increased. Consequently, the signdnoise ratio (due to the finite number of electrons detected within the energy window chosen for analysis) initially increases with thickness and then goes through a maximum at a sample thickness of the order of Ai, the total mean free path for inelastic scattering. This last statement however ignores the practical problem of separating the signal from the background (i.e., background-fitting and subtraction) which generally becomes more difficult as the thickness increases and the signalhackground ratio deteriorates. In practice it is generally found that the optimum [1979, specimen thickness is closer to AJ2, i.e., 60 nm for 100 keV electrons (EGERTON 1981)), 50-100 nm for 200 keV, and 100-200 nm for 1000 keV incident energy (JOUPPREYet al. [19781). The sensitivity of a microanalytical technique is an important factor in determining its usefulness. For EELS, as for EDS, the limiting condition will be the statistical requirement for a minimum signal-to-noise ratio between the edge and the background. Because of the dominance of the background in the spectrum the. sensitivity will depend very much on the experimental parameters such as beam current, recording time and acceptance angle. Consequently, detailed calculations are necessary in order to obtain realistic results. JOY and MAHER[1980] calculated the sensitivity for different atoms distributed in a carbon matrix, under a variety of experimental conditions typical for STEM operation. g, this repreAlthough the absolute detection limits are very good, typically sents, however, a mass- or atomic fraction of only between 0.5 and 5% in most cases. This indicates that EELS is best suited for studies of precipitates, segregation effects and similar situations where the local mass fraction is high or can be made so by choosing the probe diameter. It can also be seen that much of the potential gain in microanalytical sensitivity derived from EELS operation is sacrificed when the normal sequential mode of spectrum recording is used, since the analysis time is only of the order of a few hundred milliseconds per channel compared with 100 s or more for EDS. However, recent developments in parallel detector arrays which acquire data simultaneously in many channels will ultimately remove this disability and yield a substantial gain in sensitivity.
10.3.1. Examples EELS is applied mainly to ceramic specimens since it is the only technique for the determination of the distribution of light elements; an example is shown in fig. 34, below. Grain-boundary phases could be detected and analyzed chemically (GEISS [ 19811, ROHLE and PETzow [1981]). Ion implantation is sometimes employed to introduce light-element dopants into semiconductors. EELS could be used to measure local dopant concentrations, if the latter exceed -0.1% (JOY and MAHER [1980]). Light elements occur in certain metal specimens in the form of nitride or carbide precipitates; these materials have also been analyzed by the EELS technique (ROSSOW et al. [1976], GEISS [19811). References: p . 1105.
Ch. 11, Appendix
M.Riihle and M. Wilkens
1094
250
350
1400
150
1500
550
'
1600
650
1700
850 AEleVl
750
1800
1900
dE lev1
Fig. 34. Electron energy-loss spectrum of a 25 nm thick amorphous area in a silicon nitride ceramic. Edges in the spectrum are found at characteristic energy losses. The concentration of the different elements can be determined from the hatched areas: a) Energy losses 250-850 eV. b) Energy losses 1350-1950 eV. (From BISCHOFF [1983].)
Acknowledgement The authors would like to express their sincere thanks to all their colleagues in and outside Stuttgart who have helped them in preparing the present article by critical comments and/or by providing them with suitable micrographs. The latter holds in particular for their colleagues Dr. Bourret, Dr. Hutchison and Dr. Karnthaler.
Appendix. Elements of the kinematical digraction theory A.1. Introduction In materials science, scattering or diffraction experiments using some kind of waves of wavelengths X comparable to, or smaller than, interatomic distances play a key-role
Ch. 11, Appendix
1095
Transmission ehctmn microscopy
for the study of the atomistic structure of solids *. This holds in particular for solids of more or less perfect crystalline structure. Such experiments are carried out either by Xrays, as provided by conventional X-ray generators or, more recently, by synchrotron radiation of electron- or positron storage rings, or by the de Broglte waves of thermalized neutrons, as provided by nuclear reactors om more recently, by spallation sources or by the de Broglie waves of high-energy electrons as used in a transmission electron microscope ("EM). One of the basic theories which describe the interaction of the above-mentioned radiation with matter is the so-called kinematical diffraction theory which can be derived as a first Born approximation, for X-rays from the Maxwell equations, and for neutron and electron waves from the Schrodinger equation. The kinematical diffraction theory is applicable under the following conditions: (i) The scattering at individual atoms is elustic, which means that the wavelength of the scattered wave is the same as the wavelength of the incoming wave. (ii) The scattering at atoms occurs coherently, i.e., the phase relationship between the scattered wave and the incoming wave is always the same for a given kind of atoms. (iii) The scattering at individual atoms is so weak that a second scattering of the already scattered wave at other atoms can be neglected. (iv) The total difracrion at an ensemble of atoms remains so weak that the intensity of the incoming wave is practically not affected, i.e., all the atoms of the ensemble (specimen) are irradiated with the same intensity. Under these assumptions a fairly sophisticated kinematical diffraction theory has been developed and applied - using X-ray and neutron waves - to various problems in crystallography and materials science, as set out in ch. 12. However, it turns out that this kind of theory is only of restricted use for TEM investigations of crystalline material: Conditions (iii) and (iv) are in general strongly violated. Therefore, in TEM the theory must be extended to the so-called dynamical diffraction theory. Nevertheless, a number of elements of the kinematical diffraction theory will be needed also for understanding TEM of crystalline materials. Therefore, we give here a brief introduction to the general concept of the kinematical diffraction theory, which should be understood as an introduction to the more detailed reports in this chapter and ch. 12. The amplitudes of X-rays are vectors, because X-rays are electromagnetic waves, whereas the amplitudes of neutron and electrons are scalar. This gives some marginal differences in the formulation of the kinematical diffraction theory for these two kinds of radiation. For simplicity we consider here waves of scalar amplitudes only. A.2.
Fundamental equations
The incident plane wave A(r, t) is given by
* In the following the term scattering is used for interaction of individul scattering centres, as e.g., individual if the coherent superposition of scattcrcd atoms, with radiation. On the other hand, we speak about di'aclion waves of more than one scamring centre is considered. werences: p.
1IO5.
1096
Ch. 11, Appendix
M.Riihle and M.Wilkenv
A(r, t) = A, exp[2ri(k0r - w)],
(A. 1)
where r is the space vector, t the time coordinate, k, the wave vector with Ikd = k = 1/A, A the wavelength and v the wave frequency. Further, A and A, are scalar terms with IA,I*=intensity per unit area perpendicular to k, From
k, - r - vt = const.
(A.2)
we obtain the phase velocity v in direction of k,, u = v/k = Y .A.
63)
Since we consider only elastic scattering, where the wavenumber k and the frequency v remain unchanged by scattering events, we can omit the timedependent part of A. Thus, in the following we start with A(r) = A,, exp(2aiko r )
(A;4)
as the primary wave incident on a number of N atoms. For the moment these atoms are assumed to be point-like. They are characterized by atomic scattering amplitudesf, and are located at positions r, near the origin r=O; n = 1 N. At all these atoms, the primary wave excites the emission of spherical waves in coherency with the primary wave. Thus the sum of the scattered waves can be represented by A,(r), with e-.
On the righthand side the term A@,) represents amplitude and phase of the incident wave at the positions of the atoms. The term behindf, represents a spherical wave emerging with phase T from the atom positions. It has the dimension of a reciprocal length. Thus, since A, and A must be of the same dimension, the atomic scattering amplitudef, must have the dimension of a length. Accordingly, in neutron diffraction f, is called the ‘‘scattering length”, b,,. A detector for receiving the resultant intensity of the scattered, or diffracted, waves may be placed at a position r = R so far away from the N atoms that in the denominator of eq. (AS) the approximation
lR - rl -- 1 4= R
(A.6)
is applicable. This is the condition for Fruuenhofer dijkzction. Further, we assume that near the detector all spherical wave fronts can be approximated by plane waves with a common wave vector k, parallel to R. Thus, near the detector, we have k , * ( R - r,) = klR - r,l cos y,
bsI= k,
(A.7)
where y is the angle between k, and R -m.Since y + 0 for R + 00, we set cosy = 1. Then we can rearrange eq. (AS) with eqs. (A.6) and (A.7) to the form
Ch. 11, Appendix
1097
Transmission electron microscopy
1 A,(R) = A,, - F ( K )exp(2rrik, R
R),
with K
= k, - k,,
IKI
= 2k0 sin 8
(A.9)
and F(K) =
Cf, e x p ( - ~- r.).~
(A. 1Oa)
K is called the diffraction vector (in neutron diffraction, K is often used for 4rrk sin e). 0 is half the angle between k, and k,,and F(K) may be called the di@-actionamplitude of the ensemble of scattering atoms. Note that information about this ensemble (rn,fn) is only contained in F(K) and that F depends only on K but not explicitly on the moduli and directions of K,, and K,. Equation (A.lOa) may easily be extended to a continuous distribution of the atomic scattering amplitude. If pf(r) denotes the scattering amplitude per unit volume at the point r then eq. (A.1Oa) is modified to
F(K)= p,(r) exp(-2rri~* r)d3r.
(A.lOb)
The detector may receive a diffracted intensity I, via a window of area Q, with Q / R 2e 1. Then 1, is given by:
(A. 11) Q/R2 = A a is the solid-angle interval under which the detector receives radiation from the scattering atoms. Thus, IF(K)I’, which has the dimension of an area, will be called the dzferential difraction cross-section of the ensemble, i.e., the cross-section per unit solid angle, &/day which is a function of K only. Accordingly, the significance of IF(K)I’ is rather general, and it is in particular free of “instrumental” parameters of the experiment such as the actual values of A,, R and Q. Therefore, in the following we concentrate mainly on a more detailed evaluation of F(K) or lF(~)l’.
A.3. Real space and reciprocal space, description of perfect crystal structures We assume now that the atoms are periodically distributed, i.e., they form a perfect crystal lattice. The elementary cell of the lattice is given by three non-coplanar basis vectors a,, a2,a3,or briefly a j , j =1, 2, 3,. For simplicity we assume here that the crystal is shaped like a parallel epiped, Le., in the direction of ai the elementary cells are repeated Nj times so that the lattice consists of Nl Nz N3 unit cells. The elementary cell contains M atoms, associated with atomic scattering amplitudes f,, m = 1 M. Then the positions of the lattice atoms are given by: e-.
i,,,,, =
r,, + r,,
(A.12)
where r, denotes the position of the elementary cell,
References: p . 1105.
1098
M.RUhle and M. Wlkens
Ch. 11, Appendix
(A. 13) and rm denotes the position of the mth atom within the elementary cell. We express r, in dimensionless coordinates 8, related to the ai as: (A. 14) Besides the basis vectors aj we introduce so-called reciprocal basis vectors which constitute the reciprocal lattice: (A.15) where x means the vector product and V, the volume of the elementary cell. The bj are related to the ai by:
a, -bit = Sr.
(A.16)
where
Sj, = 1 fori = j‘, = 0 forj # j’.
Note that the b, have the dimension of a reciprocal length, as is also true for the wave vectors k,,,k, and the scattering vector K. So K can be expressed by a linear combination of the 6,. (A.17) where the hj are dimensionless numbers. Within the K-space, or reciprocal space, the reciprocal lattice points given by h, = integers play a special role. We denote them by g or, if necessary, by g,,, with g = g,
= hb,
+ kb, + lb,,
h, k, 1 = integers.
(A.18)
The integers h, k, I are the well-known Miller indices of a given set of lattice planes. In standard text books of crystallography it is shown that g,,, is perpendicular to the set (hkl) of lattice planes and that lghtl =
W4kD
(A.19)
where dw is the repetition distance of the lattice planes (hkl). In the following we will denote reciprocal lattice points, depending on the context, by g,,,.,, or g, or simply by (hkl). If the h, k, I refer to the ensemble of crystallographically equivalent reciprocal lattice vectors we use the symbol {hkl}.If a particular one is meant we write (hkl). For instance, concerning a crystal of cubic symmetry, { 100) comprises six g-vectors, namely +(loo), f(010),and f(001).In quite a similar way we will denote vectors in real space in an abbreviated form by r = [uvw]with the meaning
Ch. 11, Appendix
1099
Transmission electron micmscopy
r = ua,+ ua2 + wa,.
(A.20)
If the ensemble of all crystallographically equivalent vectors r is meant, we write r=. For instance, in a cubic crystal the symbol means all vectors of the kind f[100], f[010], and f[001]. Sets of different lattice planes (hkf) with a common direction in real space, e.g., rz, are called a zone, and rz is the zone axis. The corresponding vectors g, constitute a plane in the reciprocal lattice with g d r z = 0. In a cubic crystal, rz= [1001, for instance, is the zone axis for all lattice planes given by (Okl). Note that for crystals of cubic symmetry a real space vector r = [uvw]is parallel to a reciprocal lattice vector g = (hkl) with u = h, v = k, w = 1. For crystals of lower symmetry this is, in general, not the case. Considering, for instance, a crystal of tetragonal symmetry (kzJ = c # t , l =la,l=a), it turns out that the real space vector r=[101] is not parallel to the reciprocal lattice vectOrg= (101). Accordingly, if crystallographicelements of crystals of lower than cubic symmetry have to be denoted, a careful distinction between vectors t in real space and lattice plane normals g in reciprocal space is necessary.
A.4. The kinematical diffraction amplitude F(K) of a perfect crystal Applying the coordinates of the crystal atoms as given by eqs. (A.12)-(A.14), eq. (A.lOa) can be written as F(K)
= 444. F , ( K )
(A.21)
with F,(K) =
C exp(-riu
(A.22)
rn),
n
where the summation extends over all n,, n2, n3, i.e., over all unit cells, and with (A.23) The subscript “L,” stands for “lattice amplitudes”. Accordingly, F,, reflects the outer shape of the crystal and is independent of details of the structure of the elementary cell. The subscript “s” stands for “structure amplitude” by which the structure of the elementary cell is taken into account. Both terms have to be considered separately. First we derive FL.For simplicity we assume an orthogonal elementary cell where the a, are parallel to b, and lb,l= l/la,l. (This restriction is not necessary, but facilitates the interpretation of the final result.) We consider the environment of a particular diffraction vector g and write: K
= g +s,
s = cslbj/lb,l = C s j a j b , .
(A.24)
I
I
Then, with (A.13)-(A.18), K *
r, = [ q h -k tI.$
+ 511 4-
n,
Uj
.
(A.25)
j
References: p . 1105.
1100
M. Rahle und M.W1ken.q
Ch. 11, Appendix
The term in square brackets is integer and can be omitted in the exponent of eq. (A.22).
Thus: (A.26) For s,a,a 1, the argument in the exponent varies smoothly with varying the integers n,. Thus, the summations can be replaced by an integration
FL(s)=
v, JL’
exp(-2rix sjx,)dx, dx, dx,
(A.27)
x1=0
(A.28) with Lj=Npj denoting the length of the crystal in the direction of uj and V,=a,ap,. In eq. (A.28) a phase factor was omitted, since only IFL12is accessible by experiment. An indication of the “structure” of the lattice factor IFL12in the reciprocal space is obtained by considering the square of one of the three factors of FL,e.g., (A.29) The following properties of Zl can be recognized: Z&,) is sharply peaked at s,= O where I, reaches the value N:. In the tails Z, oscillates and decreases proportional to s,.~. The first zero on both sides occurs at lsll = 1/L, = l/N,a,. Accordingly, the half-width of I, is about l/Nla,. Thus, one may suppose that (A.30) which is indeed the case. (If eq. (A.26) is summed exactly, it turns out that the terms m,uj in the denominator of eq. (A.28) have to be replaced by sin rs,u,. This small “error”
in (A.28) is, however, without any practical relevance in general.) Having understood the structure of Z,(sl), it is easy to understand the threedimensional structures of IFLI2, (A.31)
Around every reciprocal lattice point g the intensity of IFL12is sharply peaked, and the shape of the peak is reciprocal to the shape of the crystal: parallel to a direction where the crystal is thin (Lj small), IF,l2 is broad, and parallel to a direction where the crystal is large (L, large) FLI2is narrow. For instance, a plate-like crystal gives a rod-like structure of the peaks of IFLIzand vice versa. Now we proceed to the term F,(K),cf. eq. (A.23). We recall that for N,, N,,N 3 s1, the lattice factor IFL12is essentially # O only for K close to g, or for (h,h,h3) close to (hkl).
Ch. 11, Appendix
Transmission electron microscopy
1101
Further, we realize from eqs. (A.14) and (A.23) that F, is, comparative to FL, a smooth function of K [or (h,h2h3)].Therefore, considering K close to g it is allowed to approximate F, by (A.32) We consider F,(hkl) for some simple cubic lattices. (i) face-centred cubic (fcc) lattice; four equivalent atoms in the unit cell are located at r=[000],+[011], 3[101], 3[110]. Thus: F,(hkI) =f( 1 + exp[-i.rr(k+ l)] + exp[-im[h + l)] -I-exp[-i.rr(h + k)]} =4f for h, k, 1 unmixed, i.e. all even or all uneven = O for h, k, 1 mixed. (ii) body-centred cubic @cc) lattice; two equivalent atoms at r = [OOO], $[ 1111. Then: F,(hkI) = 2f for h + k + 1= even, = O for h+k-1= uneven. For these two types of Bravais lattices, F, takes either full value (fmultiplied with the number of atoms per unit cell) or vanishes. (iii) Ordered lattice of type L1, (e.g. Cu,Au); one atom of kind A in r = [OOO], three atoms of kind B in r=$[011], ;[loll, +[110]. This gives: F,(hkl) =f, + 3fB for hkl unmixed, =f A-fB for hkl mixed. Lattice plane of the first category are called fundamental lattice planes and those of the second category are superlattice planes. The F,-values of the latter are in general much smaller than those of the former.
AS. The Ewald sphere and Bragg’s law In order to decide whether a crystal gives rise to a diffracted beam for a given direction and a given wavelength of the incident radiation we introduce the so-called Ewuld Sphere in the reciprocal lattice. Figure 35 gives a two-dimensional representation. A reciprocal lattice is schematically drawn and the wave vector k, of the primary beam is inserted in such a way that k, ends at the origin of the reciprocal lattice. A sphere, the Ewald sphere, is drawn around the starting poine of k, (point C) with Ik,l = k as radius. All vectors starting from C and ending on the Ewald sphere represent possible wave vectors k, of diffracted waves of intensity IF(K)I~,with K =k,-k,. If the Ewald sphere cuts a reciprocal lattice point g where ~F(K)?takes high values a strong intensity will be diffracted in the direction of the particular wave vector k,, named k,, pointing from C to g. We denote the angle between k,, and k, by 28,. Then we have: lgMil= 2k sin,,e,
(A.33)
or with eq. (A.19): References: p. 1105.
1102
M.Ruhle and M.Wlkens
Ch. 11, Appendix
Fig. 35. The Ewald sphere and the intensity distribution IF(K)I’in the reciprocal space.
A = 2dhk,sin &.
(A.34)
This is Brugg’s law with 6hU being the Bragg angle of the set of lattice planes (hkl). These Bruggpeuks (or “Bragg reflections”) are not infinitely sharp but rather reveal a certain finite width (reciprocal to the crystal dimensions). Therefore, appreciable intensity will be diffracted not only for one discrete direction of k, but for a (more or less) small interval of directions. In the same way the wave vectors k, of the diffracted beam will cover a similarly small interval of directions. Consequently, for a particular direction of k,,the diffracted intensity Z,, is given by a two-dimensional integration,
(A.35) over the Ewald sphere in the neighbourhood of ghu. df,/P is a differential solid angle dfl with dfE= differential area on the Ewald sphere. In a practical diffraction experiment the crystal will be rocked with respect to k, over an angular interval so that the Ewald sphere is swept over the peak at the reciprocal lattice point under consideration. If effective source size and detector aperture are sufficiently large to capture all the intensity belonging to g,the diffracted intensity Zhpl is proportional to the integral over the diffraction spot,
(A.36) cf. 5 A.4, where the integration runs over the neighbourhood of,,g N is the number of unit cells and lFs(hkl)12is the structure factor of the Bragg reflection (hkl). A.6. The atomic Scattering amplitudes and the Debye-Waller factor
In 5 A.l we had assumed that the atoms are point-like scattering centres and that, consequently, the atomic scattering amplitudes, or atomic scattering lengths, are constants
Ch. 11, Appendix
1103
Transmission electron microscopy
which are characteristic for the kind of atoms considered. This assumption is a very good approximation for nuclear scattering of thermal neutrons having wavelengths of the order of 0.1 nm. Nuclear scattering occurs at the atomic nuclei the diameters of which are of the order of some m, Le., smaller by a factor of lo4 than the neutron wavelengths. In neutron diffraction physics it is customary to denote the atomic scattering length by the symbol b. In general h is positive m; however, for some isotopes b may be negative. Considered and of the order of as a function of the atomic number Z or the atomic mass A, no smooth dependence of b on Z or A exists. The values of b may even vary considerably for different isotopes of a given element. However, apart from these variations the general trend is that h increases roughly proportional to A”3, i.e., to the size of the nuclei. X-rays are (nearly) exclusively scattered by the electrons constituting the electron cloud of an atom. From Maxwell’s equations it follows that a single electron scatters Xrays of wavelength of the order of 0.1 nm like a point-like scattering centre with a scattering amplitudefo with
-
f, - 2/(%2)= 2.8 x io-%,
(A.37)
where e is the elementary charge, m, is the rest mass of the electron and c is the velocity of light. Around an atomic nucleus, located at r=O, the electrons are distributed according to a density function p,(r). If the atom of atomic number Z is neutral, jp,(r)d3r = Z.
(A.38)
Applying eq.(A.lOb) we get for the atomic scattering amplitude
fx =fofa(K). with &(K)
=
IP,(K) exp(-rc. r ) d3r.
(A.39)
The index “X” stands for X-rays. The term& is named the afomicfomfacforand is tabulated in general as a function of l~1/2=sinB/h. Obviously, we have f , = Z for sin 8/A = 0. With increasing sin O/A,fdecreases monotonously. In a very rough approximation it decreases to about half its maximum value for sin B / A l / d A where dAis the atomic diameter. Transmission electron microscopes operate at accelerating voltages U of 100 kV or higher. Taking U = 100 kV, we get the corresponding de Broglie wavelength from J
h
(A.40)
giving A = 3.7 pm (h =Planck’s constant). The wavelengths of such electrons are thus smaller by a factor of about fifty than the wavelengths normally used in X-ray and neutron diffraction work. Because of Bragg’s law [eq.(A.34)] the Bragg angles e,,tlare References: p . 1105.
1104
M. Riihle and M. Wilkens
Ch. 11, Appendix
also smaller by about the same factor. Electrons are scattered at an atom by the atomic electrostatic potential V(r) built up by the charge +Ze of the atomic nucleus and by the charge distribution -ep,(r) of the surrounding electron cloud. Since charge distribution and potential are linked by Poisson’s equation,
V2V(r)= 47re pe(r),
(A.41)
(V2 = d2/d2 + 8 / / a y 2+ d2/&2), the atomic scattering amplitudes for electrons are closely related tof,. The result is (A.42) with m0e2/2h2= 2.39x los/,. f,decreases rapidly (more rapidly thanf,) for small values of sinB/A but only slowly, = Z/(sinB/A)2, for large sinB/A where f, becomes small compared to Z. The valuefe(0) will not be discussed here. Diffracted intensities are proportional to the square of the atomic scattering amplitudes. In order to compare the different scattering strengths for neutron-, X-ray- and electron waves, we choose as a representative example the 200 Bragg reflection of a copper crystal. With sin B,odA = 1/2dm we get for copper sin 02JA = 2.76 nm-’.Taking tabulated values for b, f, and f, we obtain: neutron waves: b2= 5.8 * m2, X-rays: f; = 3. lo-’’ mz, mz. electron waves: f,’ = 8.7 * It is obvious that neutrons are least effectively scattered and that electron scattering is orders of magnitude stronger than X-ray scattering. For Bragg reflections of high order (Ig,l large or dRRlsmall), the difference between neutron and X-ray scattering becomes less spectacular, because f, decreases, but not b, with increasing sin B / A . There is another effect which reduces the intensities of Bragg reflections with increasing lgl (or decreasing d) even for “perfect” crystals: The atoms are not at rest because of thermal vibrations. This leads to an additional “smearing-out” in space of the scattering power of the individual atoms. If {u:) is the mean square displacement of an atom from its ideal lattice position by thermal vibration the atomic scattering amplitudes f have to be multiplied by a factor e-M, with
M = 8wz(u:)-
sin2 e A’
(A.43)
Of course, (u:) increases with temperature T. For T>O,, where 0,is the Debye temand thus M , increases about linearly with T. perature (for copper, 0,= 320 K), a:>, The diffracted intensities are damped by the factor e-? Because M = 2.this damping is particularly effective for high-order reflections. The factor e-Mis named Debye-Waller factor.
A.7. Imperfect crystals The Debye-Waller factor as an unavoidable consequence of thermal vibrations is but
Ch. 11, Refs.
Transmission electron microscopy
1105
one example for the effect of displacements of atoms from ideal lattice positions. Apart from thermal displacements, “static” displacements are usually encountered with lattice defects, e.g., stacking faults, point defects, dislocations (including solute atoms), clusters and coherent precipitates. Displacements will generally vary with the position in the crystal. Let r, denote the position for the nth unit cell of the ideal reference lattice and R,=R(r,) the displacement of the first atom in this cell, R,, are the corresponding displacements from the other ideal positions r,, in the nth cell. A factorization of IF(K)I~ as in eq. (A.21) is now no longer possible. Instead, we can write: F(K) =
C 4,
(K)
-
-
exp(-2ri~ r,) e x p ( - h i ~ R,),
n
with
-
F,,( K > = C f,, e x p ( - 2 r i ~ rfim) exp[-2riv * (R,,, - R,)]
(A.45)
m
for the structure amplitude of the nth cell. Equation (A.45) shows that positional disorder alone leads to different structure amplitudes for different cells. Furthermore, eq. (A.45) also shows the general effect of compositional disorder (chemical disorder), as f,, reflects exactly the occupation of the various sites by different atoms. In the special case, where compositional disorder is not considered and where the displacement vector R,, can be regarded as constant for all atoms of the elementary cell, the structure amplitude F,(K)remains unchanged, and using eqs. (A.24) and (A.25) the lattice amplitude FL can be written in the integral form: 1
v,vs exp(-2ri[g
F,(K) = -
-
R(r)+ s r
+ s - R(r)]}d3r,
where the integration runs over the sample volume V,. In many cases the term s.R can be neglected. These general expressions are the basis for more detailed evaluations in this chapter and ch. 12, where several special cases are discussed.
References AMELINCKX, S., 1964, The Direct Observations of Dislocations (Academic, New York). S., and J. VANLANDWT,1978, in: Diffraction and Imaging Techniques in Material Science, eds. AMELINCKX, S. Amelinckx, R. Gevers and J. Van Landuyt (North-Holland, Amsterdam) vol. I, p. 107. AMELINCKX, S., R. GEVERSand J. VANLANDWT,eds., 1978, Diffraction and Imaging Techniques in Material Science, vols. I and I1 (North-Holland, Amsterdam). K. W., D. J. D Y ~ and N S. R. ICEOWN,1971, Interpretation of Electron Diffdtion Patterns (Adam ANDREWS, Hilger, London). ASHBY,M. E, and L. M. BROWN,1963, Phil. Mag. 8, 1093 and 1649. and M. PETIT,1981, Phil. Mag. A43, 189. BACMAN, J.-J., G. SILVE~TRE BALL,C. J., 1964, Phil. Mag. 9,541. R. W., ed., 1980, Grain-Boundary Structure and Kinetics (ASM, Metals Park, OH). BALLUFFT, BALLUFFI, R. W., A. BROKMAN and A. H. KING, 1982, Acta Metall. 30, 1453. BERTIN,E. P., 1975, Principles and Practice of X-Ray Spectrometric Analysis (Plenum, New York). BE-~HE, H.A., 1928, Ann. Phys. (Leipzig) 87, 55.
M. Riihle and M. Wilkens
1106
Ch. 11, Refs.
BETHE,H. A,, 1933, Handbuch der Physik, vol. 24 (Springer, Berlin) p. 273. BISCHOFF,E., 1983, unpublished. BONNET,R., 1981, Phil. Mag. A43, 1165. BONNET, R., E. COUSINWUand D. H. WARRINGTON.1981, Acta Cryst. A37, 184. BOURREX, A., and J. DESSEAUX,1979, Phil. Mag. A39.405. BOW, A., J. D f f s ~ ~ and u x A. RENAULT,1982, Phil. Mag. A45, 1. BROKMAN, A, and R. W. BALLUFFI,1981, Acta Metall. 29, 1703. CHADWICK, G. A,, and D. A. SMITH,eds., 1976, Grain-Boundary Structure and Properties (Academic, .ondon). CHAMPNBSS, P. E., G. CLIFFand G. W. LORIMER,1982, Ultramicroscopy 8, 121. CHIK,K.P., 1966, Phys. Stat. Sol. 16, 685. CHK, K.P., M. WILKENSand M. R m m , 1967, Phys. Stat. Sol. 23, 113. CLAREBROUGH, L.M., and C. T. FORWOOD, 1976, in: Proc. 9th Int. Congress on Electron Microscopj ed. J.M. Sturgess (Microscopical Society of Canada, Toronto) vol. 11, p. 38. CLIFF,G., and 0.W. LOR1975, J. Microsc. 103, 203. ~OCKAYNE, D. J. H., 1978, in: Diffraction and Imaging Techniques in Material Science, eds. S. Amelinckx, R. Gevers and J. Van Landuyt (North-Holland, Amsterdam) vol. I, p. 153. C O C K A D. ~ ,J.H., and R. GRONSKY,1981, Phil. Mag. A44, 159. COCKAYNE, D.J. H., I. L. F. RAY and M. J. WHELAN,1969, Phil. Mag. 20, 1265. DAHMEN, U., K. H. WESTMACOTT and G. THOMAS, 1981, Acta Metall. 29,627. DAS,S. K., P.R. OKAMOTO, P.M. J. FISHOR and G. THOMAS, 1973, Acta Metall. 21,913. EGERTON,R. E, 1979, Ultramicroscogy 4, 169. EGERTON,R.F., 1981, in: Analytical Electron Microscopy 1981, ed. R.H. Geiss (San Francisco Press, San Francisco) vol. I, p. 154. ENGLISH,C., B. L. EYREand S. M. HOLMES,1980, J. Phys. F10, 1065. ESSMANN, U., 1966, Phys. Stat. Sol. 17, 725. EYRE,B. L., and R. BULLOUGH,1965, Phil. Mag. 12, 31. EYRE,B.L., D.M. MAHERand R.C. F’ERRIN, 1977, J. Phys. lV,1359 and 1371. FLORJANCIC, M., M. R-LE and S. L. SASS, 1982, in: Proc. loth Int. Congress on Electron Microscopy, (Deutsche Ciesellschaft fiir Elektmnenmikroskopie,Frankfurt) vol. 2, p. 359. F ~ L LH., , and M. WILKENS, 1975, Phys. Stat. Sol. (a) 31,519. F ~ L LH., , and M. WILKENS,1977, Phys. Stat. Sol. (a) 39, 561. GEIS R H., 1979, in: Introduction to Analytical Electron Microscopy, eds. J.J. Hren, J.I. Goldstein and D.C. Joy (Plenum, New York) p. 43. GEISS,R. H., ed., 1981, Analytical Electron Microscopy 1981, vols. I and I1 (San Francisco Press, San Francisco). GEVERS,R., 1962, Phil. Mag. 7, 59 and 651. GEVERS,R., A. ART and S. AMELINCKX,1963, Phys. Stat. Sol. 3, 1563. GEVERS, R., J. VAN LANDUYT and S. AmrmcKx, 1965, Phys. Stat. Sol. 11,689. GOLDSTEIN,J. I., and D. B. WILLIAMS,1981, in: Quantitative Microanalysis with High Spatial Resolution, eds G.W. Lonmer, M.H. Jacobs and P. Doig (The Metals Society, London) p. 5. GOLDSTEIN, J.I., J.L. COSTLEY,G. W. LORIMER,and S. J.B. REED,1977, in: Scanning Electron Microscopy 1977, ed. 0. Johari (IIT Research Inst., Chicago, IL) vol. I, p. 315. GOLDSTEIN, J.I., D.E. NEWBURY, P. ECHLIN,D.C. JOY,C. FIORI and E. LIFSHIN,1981, Scanning Electron Microscopy and X-Ray Microanalysis (Plenum, New York). GRATIAS,D., R. PORTIER, M. FAYARD and M. GUYMOT,1979, Acta Cryst. A 3 5 8 8 5 HussWaNw,F., M. R ~ andEM. WILKENS,1972, Phys. Stat. Sol. (b) 50,445. HXUSSERMANN, F., K. H. KATERBAU,M.RDHLEand M. WILKENS,1973, J. Microsc. 98, 135. HASHIMOTO,H., A. HOW and M. J. WHELAN,1960, Phil. Mag. 5, 967. HASHIMOM,H., A. HOWIEand M. J. WHELAN,1962, Proc. Roy. Soc. A269, 80. HAWKES,P. W., ed., 1980, Computer Processing of Electron Microscope Images, Topics in Current Physics, vol. 13 (Springer, Berlin). HAZZLEDINE, F! M., H. P. KARNTHALER and E. WINTNER,1975, Phil. Mag. 32, 81. HEAD, A. K., P. HUMBLE,L.M. CLAREBROUGH, A. J. MORTONand C. T. FORWOOD, 1973, Computer Electron M i i g r a p h s and Defect Identiiication (North-Holland, Amsterdam).
.
Ch. 11, Refs.
Trunsmission electron microscopy
1107
HIRSCH,P. B., A. HOWE and M. J. W ~ L A N1960, , Phil. Trans. Roy. Soc. A252, 499. HIRSCII.P.B., A. HOWIE,R. B. NICHOLSON, D. W. PASHLEY and M. J. WIIELAN,1977, Electron Microscopy of Thin Films (Kneger, Huntington, NY). Howls, A., and Z. S. BASINSKI, 1968. Phil. Mag. 17, 1039. HOWIE,A., and M.J. WHELAN,1961, Proc. Roy. S a . ,4263, 217. Howl& A., and M.J. WHELAN,1962, hoc. Roy. Soc. ,4267,206. HREN,J. J., 1979, in: Introduction to Analytical Electron Microscopy, eds. J.J. Hren, J.I. Golstein and D.C. Joy (Plenum, New York) p. 481. H W ,J. J., J. 1. COLDSTEIN and D. C. JOY, eds., 1979, Introduction to Analytical Electron Microscopy (Plenum, New York). HUMBLE. P., 1978, in: Diffraction and Imaging Techniques in Material Science, eds. S. Amelinckx, R Gevers and J. Van Landuyt (North-Holland, Amsterdam) vol. I, p. 315. HUMBLE,P., and C.T. FOKWOOD.1975. Phil. Mag. 31, 1011 and 1025. JAGEK,W.. and M. WILKENS,1975, Phys. Stat. Sol. 32, 89. JENKISS,M. L., and M. WILKENS,1976, Phil. Mag. 34, 1155. JOWPREY,B., Y. KIHN.J. P. PEREZ,J. SWELYand G.ZANCHI,1978, in: Proc. 9th Int. Congrcss on Electron Microscopy, cd. J. M. Sturgess (Microscopical Socicty of Canada, Toronto) vol. Ill, p. 292. JOY,D.C.. 1981. in: Quantitative Microanalysis with High Spatial Resolution, eds. G.W. Lorimer, M.H. Jacobs and P. Doig (The Metals Socicty, London) p. 127. JOY,D.. and D. M. MAAER,1980, Ultramicroscopy 5, 333. KALONJI,G., and J. W. CAHN,1982, J. Physique C6. 25. KATERUAU, K. H.. 1976, Phys. Stat. Sol. (a) 38, 463. KAIERBAU,K.H., 1980, Phys. Stat. Sol. (a) 59, 211. KATERBAC;. K. H., 1981, Phil. Mag. A43, 409. KELLY, P.M., A. JoSrsOXs, R.G. BLAKEand J. G. NAPIER, 1975, Phys. Stat. Sol. (a) 31, 77. KHACIIATURYAN, A. G., 1974. in: Order-Disorder Transformations in Alloys, ed. H. Warlimont (Springer. Bcrlin) p. 114. KORNER,A., and H. P. UNTIIALER, 1980, Phil. Mag. A42, 753. KORNER, A., and H. P. UKKTHAI.F.R, 1981, Phys. Stat. Sol. (a) 68, 19. KRAHI., D., K.-H. HERMANN and W. KUSATH, 1978, in: Proc. 9th lot. Congress on Electron Microscopy, ed. J.M. Sturgess (Microscopical Society of Canada, Toronto) vol. I. p. 42. KRIVANEK,0.L., and P. R. SWANN,1981, in: Quantitativc Microanalysis with High Spatial Resolution (The Metals Society, London) p. 136. KRIVAKEK,0.L.,S. ISODAand K. KOSAYASHI, 1977, Phil. Mag. 36, 331. LAWONT,A,, and R. BOXNET,1982, Acta Mctall. 30, 763. LEAPMAN, R. D., S. J. SANDEKSON and M. J. WHELAK.1978, Mct. Sci. 23, 215. LEPSKI,D., 1974. Phys. Stat. Sol. (a) 24, 99. LEPSKI,D., and P. BUKC!K.1981, Phys. Stat. Sol. (a) 64, 625. UPSKI, D., and P. BURCK,1982, Phys. Stat. Sol. (a) 70, 571. LOKIMER, G.W.,M.H. JACOBS and P. DOIG,eds., 1981, Quantitative Microanalysis with High Spatial Resolution (The Metals Society, London). MAHER,D. M., and U.L. EYRI.. 1971, Phil. Mag. 23,409. MATHIWS,J. W., and W. M.STOBBS.1977, Phil. Mag. 36, 373. MCINTYRE,K. G., and L.M.BROWN.1966, J. Physique 27, C3-178. NIEDRIG, H., 1978, Scanning 1, 17. OLSON,G. B., and M. COIEN,1979, Acta Metall. 27, 1907. PENISSON, J. M., R. GRONSKYand J.B. BKOSSE,1982, Scripta Metall. 16. 1239. POGANY,A. P., and P. S. TURNER,1968, Acta Cryst. A24, 103. POND,R.C., 1979, Phil. Mag. 37,699. F’OWELI., C. J., 1976, in: National Bureau of Standards, Spec. Publ. 460,cds. K.K.J. Heinrich. D.E. Ncwbury and H. Yakowilz (National Bureau of Standards, Washington) p. 97. RADI, G., 1970. Acta Cyst. A26,41. Kosouw, C. J., R. E EGERTONand M. J. WIELAN, 1976, Vacuum 26. 427.
1108
M.Riihle and M.WiUTens
Ch. 11, Refs.
RUHLE, M., 1967, Phys. Stat. Sol. 19,263 and 279. R ~ EM.,, and G. F’ETZOW, 1981, in: Surface and Interfaces in Ceramic and Ceramic-Metal Systems, eds. J. Pask and A. Evans (Plenum, New York) p. 167. R ~ EM.,, and M. WILKENS,1967, Phil. Mag. 15, 1075. R~MLE, M., and M. WILKENS,1975, Cryst. Lattice Defects 6, 129. SALDIN,D. K., A.Y. STATHOPOULOS and M. J. WHELAN,1978, Inst. Phys. Conf. Ser. No. 41,350. SASS,S. L., and P. D. BRISTOWE,1980, in: Grain-Boundary Structure and Kinetics, ed. R.W. Balluffi (ASM, Metals Park, OH) p. 71. SCHOBER, T., and R. W. BALLUFPI,1969, Phil. Mag. 20,511. SHANNON, J., and L. KATZ, 1970, Acta Cryst. B26, 102. SIEMS, R., P. DEI AVIGNETTE and S. AMI LINCKX, 1962, Phys. Stat. Sol. 2,421. SINCLAIR, R, R. GRONSKYand G. THOMAS,1976, Acta Metall. 24,789. SMITH,D.A., 1976, in: Grain-Boundary Shucture and Properties, e&. G.A. Chadwick and D.A. Smith (Academic, London) p. 321. SPENCE, J. C. H.,1981, Experimental High-Resolution Electron Microscopy (Clarendon Press, Oxford). SPENCE, J. C.H.,and H. KOLAR,1979, Phil. Mag. A39,59. STEEDS,J. W., 1966, Proc. Roy. Sac. A292, 343. STEFJDS, J. W., 1979, in: Introduction to Analytical Electron Microscopy, eds. J.J. Hren, J.I. Goldstein and D.C. Joy (Plenum, New York) p. 481. STEEDS,J. W., 1981, in: Quantitative Microanalysis with High Spatial Resolution, eds. G.W. Lorimer, M.H. Jacobs and P. Doig (The Metals Society, London) p. 210. STOBBS,W.M., and C.H. SWORNE,1971, Phil. Mag. 24, 1365. TAKAGI,S., 1962, Acta Cryst. 15, 1310. THOMAS, G., and M. J. GORINGE,1979, Transmission Electron Microscopy of Materials (Wiley, New York). THOMPSON-RUSSELL, K. C., and J. W. EDINGTON, 1977, in: Practical Electron Microscopy in Matenals Science, vol. 5 (Mac Millan, London). TIGHE,N. J., 1976, in: Electron Microscopy in Mineralogy, ed. H.R. Wenk (Springer, Berlin). TUNSTALL, W. J., P. B. HIRSCH,and J. W. STEEDS,1964, Phil. Mag. 9, 99. VANLANDUYT,J., R. GEVERSand S. AMELINCKX, 1965, Phys. Stat. Sol. 10, 319. VAN TENDELOO, G., and S. AMELINCKX, 1977, Phys. Stat. Sol. (a) 43,553. VANTENDELOO, G., and S. AMELINCKX, 1978, Phys. Stat. Sol. (a) 50, 53. WEN& H.R, 1976, Electron Microscopy in Mineralogy (Springer, Berlin). W m s , M., 1964, Phys. Stat. Sol. 6, 939. WILE;IsIS, M.,1978, in: Diffraction and Imaging Techniques in Material Science, eds. S. Amelinckx, R. Gevers and J. Van Landuyt (NO~-HOlland,Amsterdam) VO~.I, p. 185. WILKENS,M., 1981, Phys. Stat. Sol. (a) 69, 123. WILKENS,M., and E. HORNBOGEN,1964, Phys. Stat. Sol. 4,557. WILKENS, M.,and H. 0.K. KIRCHNER, 1981, Phil. Mag. A43, 139. WILKENS, M.,and M. RUHLE, 1982, Phys. Stat. Sol. (b) 19,749. 1967, Phys. Stat. Sol. 22, 689. WILKENS,M., M. R ~ andEF. HAUSSERMANN, WILKENS,M., K.H.UTERBAU and M. RUHLE,1973, Z. Naturf. %a, 681. WILKENS,M., M. L. JENKINS and K. H. KATERBAU, 1977, Phys. Stat. Sol. (a) 39. 103. ZANCHI,G., J. SEVELY and B. JOWREY,1977, in: Proc. 5th Int. Conf. on High Voltage Electron Microscopy, Kyoto, eds. T. Imura and H. Hashimoto (Japanese Society for Electron Microscopy, Tokyo) p. 117.
General bibliography for transmission electron microscopy Textbooks Amelinckx, S., R. Gevers and J. Van Landuyt, eds., 1978, Diffraction and Imaging Techniques in Material Science, vols. I and Il (North-Holland, Amsterdam). Bethge, H., and J. Heydenreich, eds., 1982, Elektronenmikroskopiein der Festkorperphysik (Springer, Berlin).
Ch. 11, Addendum
Transmission electron microscopy
1109
Cowiey, J. M., 1975, Diffraction Physics (North-Holland, Amsterdam). Head, A. K., P. Humble, L. M. Clarebmgh, A. J. Morton and C. T. Forwood, 1973, Computer Electron Micmgraphs and Defect Identification (North-Holland, Amsterdam). Hirsch, P. B.. A. Howie, R. B. Nicholson, D. W. Pashley and M. J. Whclan, 1977, Electron Micmxopy of Thin Crystals (Krieger, Huntington, NY). Hornbogen, E., 1971, Durchstrahlungs-Elektroncnmihskopiefester Stoffe (Verlag Chcmie, Weinheim). Reimer, L.. 1967. Elektronenmikroskopische Untersuchungs- und Wparationsmethoden (Springer, Berlin). Thomas, G.. and M. J. Goringe, 1979, Transmission Electron Microscopy of Materials (Wiley, New York).
Evaluation of diftraction patterns Andrew?., K. W., D. J. Dyson and S. R. Kcown. 1971, Interpretation of Electron Diffraction Patterns (Adam Hilger, London).
Observation of dislocations Amelinckx, S., 1964, The Direct Observation of Dislocations (Academic, Oxford). Amelinckx, S.. 1979, Dislocations in Particular Structures. in: Dislocations in Solids, ed. E R. N. Nabarro (Noh-Holland, Amsterdam) vol. II. ch. 6.
TEM of grain boundaries Balluffi. R. W., 1980, Grain-Boundary Structure and Kinetics (ASM, Metals Park, OH). Chadwick, G. A., and D. A. Smith. eds.,1976, Grain-Boundary Structure and Properties (Academic, London).
Analytical electron microscopy Geik, R.H., ed.. 1981, Microbcam Analysis 1981, vols. I and I1 (San Francisco Press, San Francisco). Goldstein, J. I., D. E. Newbuty, P. Echlin, D. C. Joy, C. Aori and E. Lifshin, 1981, Scanning Electron Microscopy and X-Ray Microanalysis (Plenum, New York). Hren, J. J., J. I. Goldstein and D. C. Joy, eds., 1979, Introduction to Analytical Electron Microscopy (Plenum, New Yak). Lorimer, G. W., M. H. Jacobs and P. Doig, eds.. 1981, Quantitative Microanalysis with High Spatial Resolution (The Metals Society, London).
High-resolution electron microscopy Hawkes, P. W.,ed., 1980, Computer Processing of Electron Microscope Images, Topics in Current Physics, vol. 13 (Springer, Berlin). Spence, J. C. H., 1981. Experimental High-Resolution Electron Microscopy (Clarendon Press, Oxford).
Microscopy of Minerals Wenk, H.-R., 1976, Electron Microscopy in Mineralogy (Springer, Berlin).
Addendum Since the release of the third edition of Physical Metallurgy (R. W. Cahn and P. Haasen, eds.) enormous progress has been made regarding instrumention as well as further development in transmission electron microscopy of materials. The description of the basic physics of the scattering processes of electrons within a specimen still holds as described in the third edition. The main applications of transmission electron microscopy in materials science are still concentrated mainly on conventional transmission electron microscopy; however, high-resolution transmission electron mciroscopy and analytical electron microscopy are becoming powerful tools, too. This addendum will focus on
1110
M.RUhle and M. Wilkens
Ch. 11, Addendum
progress made in different areas of transmission electron microscopy. Transmission electron microscopy (TEM) is becoming more specialized and can be divided into conventional transmission electron microscopy (CTEM), analytical electron microscopy (AEM) and high-resolution transmission electron microscopy (HRTEM). In CTEM the morphology of materials as well as the distribution of different phases can be determined as described in sect. 4. Information on structure can be obtained with great success. In addition, the analysis of defects such as interfaces (phase boundaries, etc.), dislocations is possible (sect. 5 to 8). Recently, major progress has been made in determining the dislocation structure in intermetallics with emphasis on detailed analysis of the splitting of superdislocations into partial dislocations using the weak beam technique (Yu and KARNTHALER [1993], KARNTHALERet al. [1993]). In addition, in situ experiments in the microscope allow heating, cooling and deformation of the specimen. The latter technique revealed dynamic processes (Loucm and S m [1!393] and FUJITAetal. [1991]). Interesting results could be obtained for an understanding of deformation mechanisms in metals and ceramics, radiation damage, crystal growth and corrosion. In AEM the main emphasis of present studies is on extracting information on chemical composition with high spatial resolution (EGERTON[1986], JOYetal. [1986] and REIMER [1993]). The technique is applied especially for the determination of the composition at or close to the interface and gradients in materials (KRIVANEK[ 19911 and DISKOet ai. [19921). Recently, special techniques of electron energy loss spectroscopy (EELS) have come to allow the determination of bonding and electronic structure of materials. HRTEM allows the determination of the structure of materials, especially the analysis of specific defects in materials (SPENCE[1988], BUSECKet al. [1988], COWLEYet al. [1992] and HORIUCHI [1994]). These defects are dislocations in specific orientations in the thin film and interfaces. With great success the structure of complex high-temperature superconductors could be revealed by "EM (HORIUCHJ [ 19941, MITCHELL and DAVIES [1988], MATSUIand m o w [1995]). This progress was possible because of major advances in instrumentation as described in the first section below.
B.l. Instrumentation Over the last ten years enormous progress has been made towards better instrumentation and auxiliary parts. In conventional TEM it is now quite common that near ultrahigh vacuum conditions can be reached close to the specimen in a microscope. The conditions of the specimen environment are extremely well-defined and the resolution power of conventional TEMs (operating at 200 kV to 400 kV) approaches 0.2nm or better depending on the high-voltage of the instrument. This enormous progress was made through middle-voltage instruments where the high voltage is in the range of 300 kV to 400 kV resulting in a resolution of better than 0.2 nm. The specimen stages of the different instruments are mechanically extremely stable. New developments in fine mechanics allow a stability of the specimen which often results in a drift rate smaller than 0.001 d s . Large progress has therefore been made in the analysis of
Ch.11, Addendum
Transmission electmn microscopy
1111
defects as described in sect. 5 since extremely long exposure times are now possible. The acquisition of data is now also possible with slow-scan CCD (charge-coupled device) cameras (REIMER[1993]) which possess a large linear detection range. Field emission guns are available that operate at high voltages between 100 kV and 300 kV. These guns possess an extreme brightness and coherence (REIMER[1993] and Spence [ 19881). In AEM, the high-voltage stability and sensitive X-ray detectors (Newsmu et al. [1986]) together with very sensitive energy parallel detection systems for energy loss spectroscopy, result in shorter exposure times (by an order of magnitude compared to ten years ago). Of course, the same physical principles apply; however, the detection limit in AEM was drastically reduced by more sensitive EDS detectors and by parallel EELS detection systems. Since 1990, energy-filtering electron microscopes or imaging filters are available which allow imaging with elastically scattered electrons only (REIMER[1995]). All inelastically scattered electrons are filtered out of the image, resulting in micrographs of much higher contrast. The filtering can be done either by a system of magnetic sector field segments built into the microscope column or by an imaging filter mounted at the bottom of the instrument (REIMER[1995]). In HRTEM the resolution power of the instruments has been pushed towards 0.1 nm. This resolution is reached by using high-voltage high-resolution electron microscopy instruments operating at 1 MeV or beyond. A resolution of 0.1 nm can be reached (ROHLXet al. [19941).
B.2. Conventional transmission electron microscopy including weak beam In conventional TEM the resolution of weak beam techniques has reached its theoretical limits. The limitation is caused by the width of the contrast lines which in turn is correlated to the extinction length of the operating reflection. The observation of splitting of dislocations down to the 1 nm level could be achieved and valuable information on the splitting, especially, of superdislocations in intermetallics could be obtained (LIUet al. [ 19921).
B3. Analytical electron microsopy In AEM the energy-filtered convergent beam electron diffraction (CBED)(SPENCE and Zuo [1992], REIMER[1995]) has become an important tool for the analysis of crystal structures. A convergent beam with a converging angle in the order of lo-’ is falling on a specimen and scattering occurs in different directions. From the structure within the diffraction disc of a CBED information on the structure of the material can be obtained with high spatial resolution. Scattering factors for different reflections can be calculated by comparing the calculated intensity distributions within a disc of the CBED with corresponding experimental energy-filtered micrographs. Using this method, the scattering factors of reflections in silicon could be reproduced with high accuracy. Recently, the scattering factors of intermetallic alloys could also be determined and the Debye-Waller factor could be determined for the different elements existing in the ordered alloy.
1112
Ch. 11, Addendum
M.Riihle and M. Wilkens
The electronic structure of the material with high spatial resolution can also be determined by spatially resolved valence electron energy loss spectroscopy (SREELS) (MULLUANSet ai. [1994],BRULEY[1993]). Recently, it was shown that the spatial variation of the electronic structure of interfaces can be determined from valence spectroscopy. This value is critical to both interatomic bonding at atomically abrupt interfaces such as grain boundaries and also for the development of van der Waals attraction forces at wetted interfaces (HOFFMANN and PETZOW119941).The physics for the quantitative analysis of SREELSrequires a series of corrections of transformation of data including single scattering deconvolution, convergence correction and KronigKramer analysis (MOLLMANSet al. [1993] and FRENCH et af. [1995]). With this technique the bonding across clean interfaces and wetted interfaces could be determined. The near-edge structure of EELS gives some information on bonding across the interfaces and the atomic environment at defects. The experimental technique is conducted as follows. First a spectrum is taken at the bulk material (next to the interface) and then another spectrum at the interface itself. The difference of both spectras results in a interface specific contribution. For this analysis a linear addition of the different components is assumend. This technique was applied for grain boundaries in alumina (MULLEJANSet af. [1993]) as well as meWceramic interfaces (BRULEYet af. [1994]). The type of bonding and the environment of the bonding could be determined with high precision.
B.4. High-resolution transmission electron microscopy Similar advances have also been made in the analysis of micrographs obtained by HRTEM. Methods and programs have been developed that allow the simulation of HRTEM images of any given atom arrangement. These simulated images are used to interpret experimentally obtained micrographs. The recent developments enable us to use HRTEiM as a method for solving problems in materials science. Atomic structures of different lattice defects, such as phase boundaries, grain boundaries and dislocations, can be determined by HRTEM (FINNIS and RUHLE[1993],SWON and BALLUFFI[1995]). The difficulties in the interpretation of HRTEM are similar to those in conventional transmission electron microscopy of lattice defects in crystalline materials: the experimentally obtained micrographs do not usually present a direct image of the object. The extraction of information on the structure of lattice defects from HRTEM micrographs is complicated (SPENCE [1988]and BUSECKet d . [1988]).Despite this, very useful information has been obtained on the structure of lattice defects using HRTEM, particularly for semiconductors. The new generation of instruments also allows the characterization of defects in ceramics and metals by HRTEM.
Addendum References BKLZEY, J., 1993, Micros. Microanal. Microshuct. 4, 23. BR~~LGY, J., R BRYDSON. H.MOLLWANS,J. ~ ~ A Y E RG. , GLTEKUNST,W.~ 1994, J. .Mater. Res. 9, 2574.
E
R
D., KNAUS and M.RCKLE,
Ch. 11, Addendum
Transmission electron microscopy
1113
BUSECK,P., J. COWLEYand L. EYRING,cds., 1988, High-Resolution Transmission Electron Microscopy (Oxford University Press, Oxford). COWLEY, J.M., ed., 1992, Electron Diffraction Techniques, vol. 1 (Oxford University Press, Oxford). COWLEY, J. M., ed., 1992, Electron Diffraction Techniques, vol. 2 (Oxford University Press, Oxford). DISKO, M.M., C.C. A m and B. FULTZ, eds., 1992, Transmission Electron Energy Loss Spectrometry in Materials Science (”MS, Warrendale, PA). EGERTON, R. E, 1986, Electron Energy-Loss Spectroscopy, (Plenum Press, New York). FINXS, M. W., and M. R&JL& 1993, in: Materials Science and Technology, eds. R. W. Cahn, P. Haasen and E. K&ner, vol. 1, ed. V. Gerold (VCH, Weinheim), p. 533. FRXNCH, R. H., C. SCHEU,G. DUSCHER,H. MULLEJANS,M. J. HOFFMANX and R M. CANNON,1995, in: Interfaces in Ceramics, eds. D. Bonnell, U. Chowdhry and M. Riihle, vol. 357 (MRS, Pittsburgh, PA). FUJITA,H., K. URA and H. MORI,eds., 1991, Proc. 2nd Int. Symp. on High-Voltage Electron Microscopy, Osaka, ultramicroscopy 39, 1 4 2 . HOFFMANN,M. J., and G. PETzOW, eds., 1994, Tailoring of Mechanical Properties of Si,N, Ceramics (Kluwer Academic Publ., Dordrecht). Ho~rvcHI,S., 1994, Fundamentals of High-Resolution Transmission Electron Microscopy (North-Hollmd, Amsterdam). JOY, D. C., A. D. ROMIGJr. and J. I. G O m m i i N , eds., 1986, Principles of Analytical Electron Microscopy (Plenum Press, New York). KARNTHALER,H. P., C. RENTENBERGER and E. M~HLBACHER, 1993, in: High-Temperature Ordered Intermetallic Alloys V, eds. J. D. Whittenberger, R. Darolia and M. H. Yoo, vol. 288 (?vfRS, Pittsburgh, PA) 293. KRIYANEK,0.J., ed., 1991, Proc. Electron Energy Loss Workshop, Tahoe City, CA, Microsc. Microanal. Microstruct. 2, 143-411. LIFSHIN,E. (ed.) Characterization of Materials, Part I, Vol. 2A of Materials Science and Technology, ed. R. W. Cahn, P. Haasen and E. J. Kraamer (VCH, Weinheim). Includes chapters by S. Amelinckx (“Electron Diffraction and Transmission Electron Microscopy”, p. 1) and by E.L. Hall (“Analytical Electron Microscopy”, p. 147). LRT, C.T., R. W. CAHEi, S. SAIJTHOFF, eds., 1992, Ordend Intermetallics - Physical Metallurgy and Mechanical Behaviour (Kluwer Publ., Dordrecht). LQUCHET,E, and H. SAKA, eds., 1993, Proc. French-Japanese Seminar on In Situ Electron Microscopy, Nagoya, vol. 4, Micmsc. Microanal. Microstmct. 4, 101-346. MATSUI,Y., and W. KRAKOW, eds., 1995, Microscopy of High Temperature Superconductors, Part I, Microscopy Research and Technique 30, 101-192. eds., 1988, Electron Microscopy of Ceramic Superconductors,J. of Electron MITCIIELL, T. E., and P. K. DAVIES, Microscopy Technique 8, 245-341. H., J. BRULEY,R. H. FRENCH and P. A. MORRIS,1993, in: Pruc. Electron Microscopy and Analysis M~LLFSANS, Conference, Liverpool, vol. 138 (IOP Publ., Bristol). M O L ~ A N S Et, , J. BRULEY,R. H. FRENCH and I? A. MORRIS, 1994, in: Determining Nanoscale Physical Properties of Materials by Microscopy and Spectroscopy, eds. M. Sarikaya, H. K. Wickramasinghe, M. Isaacson, vol. 332 (MRS, Pittsburgh, PA). NEWBLJY, D.E., D.C. JOY, P. ECHLIK,C.E. FIORIand J.I. GOLDSTEIN, eds., 1986, Advanced Scanning Electron Microscopy and X-Ray Microanalysis (Plenum Press, New York). REIMEKL., 1993, Transmission Electron Microscopy (Springer Verlag, Berlin). REIMER, L.,ed. 1995, Energy-Piltering Transmission Electron Microscopy (Springer Verlag, Berlin). M., eds., 1994, Proc. Int. Workshop on High-Voltage and R~HLE , E PHILLIPP,A. SEEGER and J. HEYDENREICH, High-Resolution Electron Mkroscopy, Stuttgart, Ultramicroscopy 56, 1-232. SPENCE,J. C. H., 1988, Experimental High-Resolution Electron Microscopy (Oxford University Press, Oxford). 1992, Electron Microdiffraction (Plenum Press, New York). SPENCE,J. C. H., and J. M. ZUO, SLITTON,A. P., and R. W. BALLUPFI, 1995, Interfaces in Crystalline Solids (Oxford University Press, Oxford). Yu,H., and H. P. KAR~THALER, 1993, Materials and Design 14, 69.
CHAPTER 12
X-RAY AND NEUTRON SCATTERING G. KOSTORZ Institut fir Angewandte Physik Eidgenossische Technische Hochschule Ziirich, Switzerland
R.W! Cahn and I! Ffaasent, eds. Physical Metallurgy; fourth, revised und enhanced edition 0 Elsevier Science St: 19%
1116
G. Kostorz
Ch. L2. 8 1
I . Introduction There are many textbooks and monographs on X-ray and neutron scattering (see the selection at the end of this chapter), and the number of publications is immense. This short chapter cannot give a complete account of the history or the current state of the field. The basic theory, standard methods and many of the more classical applications are well described in various textbooks (a brief compendium of fundamentals of scattering and crystallography is given in the Appendix to ch. 11). The main purpose of the present chapter will be to demonstrate that there is much more we can “learn from scattering experiments besides the average structure” (SCHWARTZ and C~EIEN [1987], ch. 7). One immediate question concerns the range of stability of a given structure, and scattering experiments are helpful in locating phase transformations. There are classical methods (e.g., powder diffraction) and very sophisticated recent techniques (e.g., quasielastic neutron scattering) to study transformations and phase separation in metallic systems. Some examples will be discussed. Much space, however, will be devoted to studies of inhomogeneities, Le., deviations from the average structure (point defects, clusters, short-range order, precipitates, etc.). In 0 2, some remarks on scattering from crystalline material that contains defects are followed by a discussion of X-rays and neutrons as the two types of radiation now commonly used for scattering studies of essentially bulk materials. Scattering of lowenergy electrons, important in the investigation of surfaces and surface phenomena, will not be covered. High-energy electron diffraction is discussed in ch. 11. Throughout this chapter, scattering intensities are expressed in terms of scattering cross-sections (differential cross-sections without energy analysis, double-differentialcross-sections for scattering experiments with energy analysis), mostly applicable to both X-rays and neutrons. The reader should not underestimate the difficulties in converting measured intensities to absolute cross-sections. Calibration, absorption, background, polarization and other corrections may introduce considerable errors. These problems and questions of instrument design and optimization can only be mentioned occasionally. $9 3-6 cover recent applications, and the subject matter has been divided according to simple criteria. Diffraction and elastic scattering at and near Bragg peaks, between Bragg peaks, and near the incident beam are treated in 003-5, and $ 6 gives a few examples for inelastic and quasi-elastic scattering. Theoretical expressions are kept to a minimum but some are required to interpret the main effects. There are several systematically elegant and quite general schemes to describe the same scattering phenomena (starting, e.g., from lattice sums, correlation functions or convolutions). No formal elaboration of this type will be attempted. Results of the kinematical theory and of firstorder Born approximation adapted to simple but manageable cases will mostly be relied upon. Dynamic diffraction theory as the basis of transmission electron microscopy is discussed in ch. 11, and related X-ray and neutron applications can be found there.
Ch. 12, $ 2
X - r q and neutron scattering
1117
2. Scattering from real crystals 2.1. General predictions of the kinematical theory As can be seen in ch. 11, eq. (A.29), the kinematical scattering theory for an otherwise perfect, but finite crystal yields Bragg peaks whose width A K equals ~ l/LK where L, is the linear dimension of the crystal parallel to K . As A K is~ independent of the magnitude of g, this “finite size effect” broadening can be recognized by comparing the width of different Bragg peaks (at Bragg angles OM!). For powders, we thus have a method of determining crystallite sizes, since a constant AK means:
with L as the average crystallite diameter. However, experimental conditions will severely limit the range over which peak widths are controlled by the crystallite size according to eq. (1). If L is too large (2 1 pm), we may reach the resolution limit of the experiment, and if L is too small (5 100 A = 10 nm), it becomes difficult to separate the diffraction lines (see, e.g., BARRETTand MASSALSKI [1980] for a detailed discussion). For large bulk samples, the size broadening may be ignored, and the differential crosssection for Bragg scattering may be written as [cf. eq. (A.36), ch. 111:
where 8 ( -8) ~ is the (three-dimensional) delta function and V, is the unitcell volume. Nevertheless, a line or peak broadening is frequently found in real crystals - caused by imperfections. We discuss the consequences of imperfections on the scattering pattern starting from the general expressions given in $ A.7 of ch. 11. If defects are present in a crystal, the displacements un and the structure amplitudes F,, will be different in different cells. For specific defects it is useful to express these quantities in terms of the properties of individual defects and their mutual arrangement. If t denotes all possible positions for a defect, and u , , is the displacement vector at position n related to the defect at f, the total displacement u, may be expressed as I
where if a defect is present at site t, = { 0 if no defect is present at site f. 1
ct
(4)
This simple superposition is certainly justified for low defect concentrations but may be questionable in the case of concentrated alloys. The corresponding expression for the structure amplitude of the nth cell reads References: p, 1188.
1118
Ch. 12, 5 2
G. Koston
where F,, is the structure amplitude of the defect-free crystal. Replacing u, and F, in eq. (AM), ch. 11 by eqs. (3) and (5), the mean value for lF(~)l~, Le., the scattering cross section d g / d n of the scattering ensemble, calculated for a random distribution of defects of concentration c, is (see KIUVOGLAZ 119691):
f
where
and
E is the average structure amplitude of the crystal with defects: E =Fa
+ccdFsm. f
It is now possible to split d a / d n into two terms, a Bragg-like term with sharp maxima at the Bragg positions, and a second term describing scattering for all other scattering vectors. Taking into account that for the Bragg peaks, du/Ctn from the double sum of eq. (6) is controlled by terms corresponding to large distances p = Ir, -rJ, we may write:
where 2M is the real part of 5 according to eq. (7) for p
+
00:
Equation (9) defines a generalized Debye-Waller factor e-M. The reciprocal lattice vectors g occurring in eq. (9) are defined in the average lattice that contains defects. Defects may cause a shift of Bragg peaks because the average lattice constant changes, but also a reduction in intensity due to the factor e-2M[eqs. (9) and (lo)]. As long as 2M remains finite, Bragg peaks will still be present, but there are cases where 2M tends to infinity. lf we subtract the Bragg intensity according to eq. (9) from the total scattering crosssection, eq. (6), we obtain the d i f i s e scattering cross-section
Ch. 12, $ 2
X-ray and neutron scatrering
1119
The cross-section (dg/dn), contains no 8-function but varies smoothly with K , even for an infinitely large crystal. KRIV~GLAZ [ 19691 distinguishes two types of defects depending on whether or not the Bragg intensities are reduced to zero [eq. (9) with 2M+ -]. The limiting behaviour of 2M [eq. (lo)] may be discussed by considering a displacement u, that decreases rapidly towards zero with increasing distance between t and n. Then, only one of the displacements u and one of the AF, in eq. (10) are markedly different from zero, and:
M = C C [ l - COS(2TK.UI,)] 1 +
(
t
2) .
(12)
The convergence of the s u m depends on the contributions from large distances between n and r. There, u,,, is small and the cosine can be expanded, so that 1-cos(2.rrrc*u,) = (27r~*u,)~/2,and if one writes u, as a continuous function of r' =r,-r,,, the convergence of the integral j ( ~ * u ~ ) ~ d 2~ f / , > r'o (>> lattice constant), will assure a finite value of M.We see that for a large class of defects where u = (f)-2 (see ch. 17), M remains finite. Others, e.g., straight dislocations, small-angle grain boundaries and stacking faults, cause e-2Mto decrease to very small values for large crystals, and the total scattering is diffuse although it will be concentrated, but not in a Mike fashion, near the original Bragg positions (line broadening). The distinction between Bragg intensity [eq.(9)] and diffuse intensity (line broadening and scattering far away from Bragg peaks), discussed here for systems with a random distribution of equivalent defects, will not always be possible experimentally, as the width of the Bragg peak is also affected by the resolution function of the insbument. Correlations in the arrangement of defects may reduce long-range displacements and modify the expected effect, as will the presence of different types of defects. Figure 1 shows schematically, how the scattering of an ideal crystal is modified by defects.
-
2.2. X-rays and neutrons
Apart from electrons (see ch. Il), X-rays (and y-rays) and thermal neutrons are frequently used for structure determination and the study of defects as they provide the appropriate range of wavelengths for such investigations. Table 1 gives some of the important properties of both types of radiation. For the wavelengths of interest in normal diffraction work (0.5 to 20 A), corresponding photon energies are in the range of about 1 4 0 keV whereas neutron energies are between 0.85 and 400 meV. Excitations in condensed matter (phonons, magnons, etc.) are in the range of a few meV and above. The relative energy change of X-rays scattered inelastically (with energy loss or gain) by any sample is then very small (say < 10") and cannot easily be resolved (except if MSssbauer sources or synchrotron radiation combined with backscattering techniques are used). In contrast, neutrons can experience an appreciable relative change in energy, so that elastic (no energy change) and inelastic scattering can be distinguished (see below, $2.4). Another important difference arises from the magnetic moment of the neutron which interacts with the local magnetization density. This leads to magnetic scanering which has very important applications as a sensitive probe in the study of magnetic References: p . 1188.
1120
Ch. 12, 9 2
G. Koston
K
Fig. 1. Scattering from a crystal containing defects. Bragg positions of a perfect crystal are indicated by thin vertical lines.
substances (see 52.3). Finally, absorption differs appreciably for the two types of radiation. (Measurementsof absorption can also yield considerable insight in the structure of matter, e.g., in EXAFS = extended X-ray absorption fine structure, see 0 5.4.). The linear absorption coefficient 2,is defined by
where Z, and I are the intensities of incident and transmitted beam, respectively, and D, is the sample thickness. The absorption coefficient has the dimension of l/length or aredvolume, and can be understood as a total macroscopic removal cross-section, as is Table 1 Some properties of X-rays and ncutro11s. Value'
mass m
Fgl
momentum energy scattering length absorption speed ~ [ m / s ] ~ wavelength A [Alb magnetic moment fresuency vls-'lb
For X-rays (photons)
For neutrons
0
1.675 x lo-'' h/A = mv h2/2mA2 b(nuc1ear) = h (mostly weak) 431 E"'
Nh hC/A
zfo(for
K+ 0) = 24h3 (strong)
c
3956/v -1.913 pn 2.42 x 10" E
12.4/E 0
2.42 x 10'' E ~~
'h = Planck's constant, c = velocity of light,& = classical electron radius, pLn= nuclear magneton. For photons, E in key for neutrons, E in meV.
Ch. 12, $ 2
X-ray and neutron scattering
1121
common in neutron scattering. In the X-ray literature, a mass absorption coefficient is defined by &‘p with p =density. Its dimensions are aredmass, and values are independent of the thermodynamic state of the sample material. All beam-attenuating processes (including coherent and incoherent scattering) are included in S,, but for X-rays the excitation of fluorescence radiation can be singled out as the most important true absorption mechanism. When the e n e r u of the incident X-rays approaches a resonance energy in the electronic states, an absorption edge is observed, corresponding to the excitation of electrons in the K, L, etc., levels (see fig. 2 where the mass absorption coefficient is shown as a function of wavelength). The electrons associated with the absorption move out of phase from the others and a destructive interference results that reduces the atomic scattering factor by the equivalent of twice the number of resonance electrons (e.g., at least a total of four at the K edge). The atomic scattering factor has to be corrected according to
f, = Lo + f’+ if”,
(14)
where f’ and f” are the real and the imaginary part of the Honl corrections. The wavelength dependence off’andf’is shown schematically in fig. 3 (see International Tables [1952-741, vol. 111. pp. 214-216 and vol. IV, p. 148, and SASAKI[1989, 19901 for tabulated values). As synchrotron radiation sources now provide strong X-raybeams with continuously tunable wavelength, diffraction work in the neighburhood of absorption edges can take advantage of the variation of specific scattering factors.
L
c
h Fig. 2. Mass absorption coefficient z , / p for X-rdys as a function of wavelength near the K and L edges of an element (schematic). References: p . 1188.
1122
ch. 12, § 2
G.Koston
Fig. 3. Variation of real and imaginary cornponentsfandf”of the Htinl corrections as a function of X-ray wavelength near the K absorption edge (at h K )of an element. (After COWLEY[1981].)
Table 2 gives a comparison of linear (and mass) absorption coefficients for a few metals. According to eq. (13), Zp,= 1 corresponds to a reduction in intensity by a factor l/e which gives an estimate for the typical thickness of a sample. Table 2 shows that even Cd, a material fnquently used for shielding purposes in thermal neutron work, has a smaller absorption for neutrons than most common metals for X-rays. On the other hand, lead, a good shielding material for X-rays, is almost transparent to neutrons. (See BACON[1975] and SCHWARTZ and C O ~ [I9871 N for more details on the different properties of X-rays and neutrons.) For high-energy X-rays (about 100 keV or more), the absorption by matter is very
Table 2 Linear absorption coefficients of some metals, for neutrons (true absorption, 8,)and X-rays (&, total removal) and corresponding mass absorption coefficients (values from BACON[1975], and KOSTORZand Lovmy [ 19791). Metal
Be AI
cu Cd
W Au
Pb
Neutrons
x-rays
Neutrons
x-rays
2,
2,
W P
-VP
(an-’) 1.8 A
(cm-’) 1.54 A
On2/kg) 1.8 8,
(m2/ kg) 1.54 8,
0.001 0.0014 0.326 119 1.22 5.72 0.005
2.7 131 474 2000 3320 4170 2630
0.00006
0.150 4.86 5.29 23.1 17.2 20.8 23.2
0.0005 0.0036 1.29 0.0061 0.029 O.ooOo5
Ch. 12, 5 2
X-ray and neutron scattering
1123
weak, and the incident beam is mainly attenuated by Compton scattering. Diffraction experiments with the 412 keV y-radiation from radioactive gold were initiated in Grenoble in the sixties (see SCHNEIDER [I9811 for a detailed account) as a means to study the mosaic structure of large single crystals, of sizes comparable to those used for monochromators of thermal neutrons or in neutron scattering experiments. As the y-wavelength is small (0.03 A in this case), the Bragg angles are also correspondingly small (- lo). Thus, lattice tilts may be monitored directly for large crystals, also, owing to the easy penetration of windows, etc., during crystal growth (see BREUERand BLEICHERT [1993] and references therein for further applications). Diffractometers in the 100-150 keV range may now also be found at suitable synchrotron radiation facilities (see, e.g., BOUCHARDet al. [1993]).
23. Magnetic scattering Although there is a measurable interaction between photons and electronic magnetic moments, it is too small (cross-sections about six to eight orders of magnitude smaller than for Bragg scattering, see DE BERGEVIN and BRUNEL[ 19721) to be used conveniently in the study of magnetic structures. The magnetic moment of the neutron has a value of ypn (see table 1) with y = -1.913. Examination of the dipole interaction between neutron and local magnetic induction (see GUREVICHand TARASOV[1968], L ~ V E ~[1984]) E Y shows that of the Fourier transform of the local magnetization density, M(K),only the component M'(K), perpendicular to the scattering vector K , contributes to the scattering. Therefore, a magnetic interaction vector may be defined by qM =
M1( K ) / I M M( K )
1.
(15)
The magnetic scattering length, commonly denoted by p. is a vector:
P=wM,
(16)
with p proportional to IM(K)I. If we consider magnetic scattering due to unpaired spins (without orbital moments), p is given by
vo
Here, e2/& is the classical electron radius fo known from X-ray scattering = 2.8 x m, see eq. (A.37), ch. 111, S is the total number of unpaired spins, andfm,(K) 1. As the number p is the Fourier transform of the spin density, normalized tOf,,(O)= of Bohr magnetons (JL~= eh/moc) equals p = 2S, we can write p(~)[IO-'~rn] = 0.27dmg(~).
(1 8)
For not too large values of K , p is comparable to the values of the nuclear scattering amplitude b (for the pure elements Fe, Co, Ni one has b = 0.96, 0.28, 1.03 and p ( 0 )= 0.6, 0.47 and 0.16, respectively, all in units of m). The magnetic form factor&,.&) Referolces: p. I188.
1124
Ch. 12. Q2
G. KOSIOIZ
falls off more rapidly with sin 8/A than the atomic form factor for X-rays(ch. ll), as only a few electrons in an outer shell contribute to f,,,. The total scattering of a magnetic material will show a combination of nuclear and magnetic scattering and will also depend on the polarization of the incident beam. Figure 4 shows the scattering geometry, assuming a fully polarized beam with the polarization vector parallel to the z-axis (up: +, down: -). If the scattered beam is analyzed along the same direction, four different scattering cross-sections can be defined (+ +, + -, - + and - -). In an otherwise perfect crystal, the nuclear unit-cell structure factor, F&) [see eq. (A.32). ch. 1l]), can be combined with the magnetic structurefactor, m
where qMm[eq. (15)] is defined according to the direction of the moment at site m, and p at each site is given by eq. (18). This yields lFs+Fizf for (+ +), lF&12+lFM,,l I?
Fk12
for (+ -) and (- +), and IF,for (- -), where the FMi are the Cartesian compoI nents of FM. An unpolarized beam can be assumed to be composed of 50% positive and 50% negative polarization, and the scattering cross-section (per atom) without polarization analysis is d u / d n = IF,I2 + lFM12. For a simple ferromagnet,qh is the same for all sites, I and with F M =q,F,,,, [see q.(19)J:
----- I
i
-.
k0
Fig. 4. Scattering geometry for magnetic scattering. The incident neutrons are polarized along the z-axis.
Ch. 12, $ 2
X-ray and neutron scaftering
1125
where cy is the angle between K and M (see fig. 4). By varying CY (via the external magnetic field or the scattering geometry), nuclear and magnetic contributions can be separated. Equation (20) also applies to antiferromagnets if we define cy relative to the direction of the moment at site one and write Fmag =Zm&,,pmexp(-2wi K V ~ )where am is either +I or -1 depending on whether or not the moment at site in is parallel or antiparallel to 4,.Figure 5 shows the consequences of ferro- and antiferromagnetic order in the bcc lattice (a# 0). Additional peaks (“magnetic peaks”) occur for antiferromagnetic order whereas in ferromagnets the nuclear peaks are enhanced. If a polarized beam is used, the cross-section for scattering without polarization analysis is the sum of, e.g., (+ +) and (+ -) scattering and contains an interference term between nuclear and magnetic scattering with Fs and F A appearing unsquared;
where Po is a unit vector indicating the direction of polarization. “Flipping” Po from +1 to -1, we have a very sensitive method to measure FMI and consequently Fmag(tc), P,(K) orfmag(K). Equation (21), if generalized as indicated by eq. (8) for structural disorder, is also the basis for a separation of structural and magnetic disorder in alloys. Finally we mention the original result of HALPERNand JOHNSON [1939] for paramag-
Fig. 5. Schematic neutron diffraction patterns from a bcc poylcrystallinc (a) fcrromagnet and @) antiferromagnet. The shaded areas represent the magnetic scattering contributions which decrease with 0 becausefm, decreases. (Mer SCHWARTZ and COHW [1987].) References: p . 1188.
1126
G. Kostorz
ch. 12, $ 2
netic scattering, i.e., for a randomly oriented ensemble of identical magnetic moments (spins S).The paramagnetic scattering cross-section (per atom) reads
This scattering is similar to the Laue scattering term in diffuse scattering. Deviations from the simple monotonic Kdependence are of interest in both cases as they relate to correlations in the atomic or the spin arrangement. There are many more special features of magnetic scattering than this brief discussion can indicate (see BROWN[1979], HICKS [1979] for more details) and many complex magnetic structures have been studied in recent years. 2.4. Inelastic and quasielastic scattering
As mentioned in 92.2., the relative energy gain or loss of X-rays scattered from a sample with lattice vibrations (phonons) is very small. Near an absorption edge resonant Raman scattering (see SPARKS [1974]) occurs just below the edge, and incoherent fluorescence radiation is emitted above the edge. Compton scattering is another inelastic scattering process of X-rays. From the conservation of energy and momentum for the scattering of a photon from an individual electron, the wavelength shift of Comptonmodified radiation is (for electrons assumed at rest):
AA [A] = 0.024qi
- cos 2 e > ,
(23)
independent of the incident wavelength. The scattering is incoherent as there is no fixed phase-relationship between the different inelastic scattering events. The relative contribution of Compton scattering to the total scattering is given by 1 -Z,f:/Z]. wherefai is the form factor (normalized to one) of each of the 2 electrons of an atom. Compton scattering increases the background in X-ray diffraction experiments, but it can be eliminated experimentally (e.g., with a monochromator in the diffracted beam) or by calculation (see SCHWARTZ and COHEN[ 19871). Detailed study of the energy distribution of Compton intensities provides information on the momentum distribution of electrons in condensed matter (see, e.g., COOPER[1977]), one of the few techniques also applicable to alloys. Much smaller (absolute) energy transfers can be detected by neutron scattering, either by analyzing the change of wavelength with a single crystal or the change of neutron momentum by time-of-flight methods. Here, we consider one-phonon scattering only (for a complete account see, e.g., LOVESEY[1984] and BACON[1975]). Figure 6 shows two possible scattering configurations in reciprocal space. The scattering vector K can be written as K=k,-ko =g+q
(24)
where 4 is the phonon wavevector, counted from the nearest reciprocal lattice point. If the wavevector after scattering is k,,,the neutron has lost energy (ks, ck,), and a phonon
Ch. 12, $ 2
1127
X-ray and neutron scattering
./-000
.
.
.
Fig. 6. Two possible inelastic scattering events in a plane of reciprocal space involving the creation (K = k,, -ku) and the annihilation (K = k,, -ko) of a phonon.
has been created. For ks2,a phonon has been annihilated. As the neutron momentum is h / h , eq. (24) states the conservation of momentum in the scattering process (A4= l/q is the phonon wavelength). Simultaneously, the conservation of energy, h2/2rn,% - h2/2m,A: = h v P ,
defines the energy of the phonon participating in the scattering (Ao and A , denote the wavelength of the incident and scattered neutrons, respectively). There are only discrete values of phonon frequency vp for a given q, and appropriate scans can be designed to obtain directly the phonon dispersion curves of a crystal. Apart from coherent inelastic scattering processes, analysis of the incoherent inelastic scattering of neutrons m a y often be interesting. True incoherent scattering processes are due to the interaction of the neutron spin with nuclear spins I # 0. The scattering length of the compound nucleus depends on its total spin which is I + 1/2 or I- 1/2. The tabulated values of coherent scattering lengths and incoherent cross-sections for individual isotopes represent properly weighted averages (for unpolarized nuclei). Natural elements are frequently a mixture of different isotopes, each with its own nuclear spin. Coherent scattering lengths and spin-incoherent cross-sections are simply arithmetic averages, but owing to the lengths over the sites of the random distribution of nuclei with different in complete analogy to sample, another K-independent term, vi= References: p . 1188.
1128
G. Kostorz
Ch. 12, $ 2
the monotonic h u e scattering term in diffuse X-ray scattering (see $4). For neutrons, this part is included in the total value of incoherent scattering of an element. Incoherent neutron scattering cross-sections are quite large in several cases, and for coherent scattering experiments (see $2.5), they may impose severe limitations. However, similar to the case of Compton scattering and electrons, energy analysis of the incoherently scattered neutrons will reveal details of the motion of nuclei. The incoherent scattering function can be calculated for different processes (see, e.g., BACON[1975], L~VESEY [1984]). If the energy transfer is centred around hv =0, Le., is small, the scattering is called quasi-elastic. In condensed matter, the motion of atoms is restricted by the environment, and quasi-elastic neutron scattering has become a widely used technique to study atomic and molecular motion, especially if hydrogen with its high incoherent crosssection is involved. Especially in crystalline solids, a study of the quasi-elastic line width (coherent or incoherent) may help to identify diffusion mechanisms (see 0 6.2). 2.5. Some experimental considerations
Incoherent scattering as a source of background and absorption was already mentioned in $2.4. Extensive discussions on general experimental problems in X-ray and neutron scattering can be found in the book by SCHWARTZand COHEN[1987]), and, more specifically oriented towards defects in crystals, in the articles by SCHMATZ [1973, 19781. Some remarks will be found in subsequent chapters, relating to particular applications. Here, we only state a few quite general points. Laboratory X-ray sources with rotating anodes have been built, the maximum power of 100 kW (see HAUBOLD(19751) yielding 4 x 10l6 quanta/s cm2 sr (CuK,). Higher luminosities are obtained (originally as a by-product, now from dedicated sources) from electron or positron synchrotrons and storage rings. Depending on the energy of the electrons or positrons, synchrotron radiation emerges in a wavelength range from infrared to - 0.2 A. Synchrotron radiation is continuous and very intense (1012-10'4 quantds mrad per 0.1 % bandwidth, integrated over the vertical direction). The radiation is highly collimated perpendicular to the orbital plane. In the GeV region, the divergence is -lo4 rad. In the plane of the ideal orbit, synchrotron radiation is 100% polarized with the electrical vector parallel to this plane. There is a well-defined time structure of the beam with pulse durations as short as lo-'' s with repetition rates of 1 MHz or more. The use of synchrotron radiation for materials studies is currently experiencing a very rapid growth (see e.g. BARUCHEL et al. [1993]). Compared even with a classical sealed X-ray tube, neutron sources are not very powerful. Figure 7 shows a comparison of several X-ray and neutron sources according to SCHMATZ [19731. As suggested by MAIER-LEIBNITZ [1966], the momentum space density p(ko) (hk,,where h is Planck's constant, is the linear momentum of a particle) is an adequate quantity for comparison, as the count rate Z at a detector is proportional to p(k,) multiplied by the momentum space elements d3k0 and d3ks that can be optimized instrument parameters (within the bounds of k,. ky,k, fixed by the properties of the source), 2 p(k,-Js(~, v)d3k,d3k,, where S(K,v) is the scattering law to be studied. We see that p(k,) for neutrons is several orders of magnitude lower than the values of all 0~
Ch. 12, 5 2
1129
X-ray and neutmn scattering
t
\
Mossbauer source A
b UI .-
10'6
ba
10'4
Q,
Fixed anode
\
1
I
Cu K,
V c
h
0
2
10'2
(u
v
a 10' 0 108
0.1
0.3
1.0
2rck0,
3.0
10
30
.8-'
Fig. 7. Momentum space density p ( 2 w Q for different X-ray and neutron sources (after SCHMATZ 119731).
X-ray sources. On the other hand, restrictions on the choice of parameters of the incident beam ( A k d k , or vertical collimation), which are common for X-rays, are less severe for neutrons where d3k0 can be chosen more flexibly to match the resolution requirements for a given S(K,v). This may compensate in part for the lower values of p(ko). Traditionally, thermal neutrons are produced by fission and subsequent moderations i.e., in a nuclear reactor. A steady-state thermal flux of about l O I 5 d c m 2 is virtually impossible to surpass as heat removal is a limiting factor. With hot and cold sources, the Maxwellian spectrum of neutrons m a y be shifted to smaller or larger wavelengths (see fig. 7). Pulsed neutron sources may provide a higher peak flux. If the pulsed structure of the neutron beam is maintained after moderation, time-of-flightexperiments are advantageous at pulsed sources because the relevant flux at the sample will be higher than the average flux. One concept for the production of pulsed neutrons is based on the use of charged particles from accelerators (spallation). A variety of spallation sources are now operational. References: p . 1188.
1130
G. Koston
Ch. 12, $ 3
Apart from film techniques (with a converter foil for neutrons), the detection of the scattered radiation is based on electronic counting circuits attached to gas or solid state detectors. For X-rays, gas-filled proportional “counters” and solid-state scintillation “counters” detect the incident quanta via the ionization of a gas or the production of photoelectrons by scintillation photons. The energy resolution is poor for both detector types, typically 20% in the 10 keV range for gas detectors and -50% for scintillation detectors. Solid-state detectors based on electron-hole pair production in doped (Ge or Si with Li) or intinsic (Ge) semiconductors have a theoretical resolution of about 1% at 10 keV, about 200 eV in practice. Another important aspect for many experiments is the spatial resolution of large detectors. Linear position-sensitive detectors with a resolution of about 30 ,um employ a resistive wire, and peak heights or pulse shapes are analyzed as a function of position of the detected event. Two-dimensional gas-filled detectors for X-rays employ (sets of) mutually perpendicular wires and different electronic techniques to locate the detected events. The simultaneous measurement of scattering over a large solid angle is of particular importance for weak scattering signals but also for in-situ kinetic studies. Neutrons can only be detected after they have participated in a nuclear reaction with the emission of charged particles or y-rays. As beam dimensions are usually much larger in neutron scattering than in X-ray scattering, the use of large arrays of individual detectors is feasible without loss of resolution (and with time-of-flight resolution, i.e., energy resolution for each desired detector). Position-sensitive detectors have been developed for diffraction work, planar detectors for small-angle scattering and curved ones for powder patterns.
-
3. Bragg peaks and vicinity Scattering from real crystals may occur anywhere in reciprocal space but Bragg peaks, though modified, will remain a predominant feature as long as an average lattice can be defined. A diffraction pattern, after all instrumental corrections, reflects the distribution of scattering matter within the (average) unit cell of a substance, and the atomic coordinates (location of the centres of scattering objects) as well as the scatteringlength density distribution (electrons with X-rays, nuclei and magnetic moments with neutrons) can be determined from an analysis of Bragg peaks (see, e.g., LIPSONand COCHRAN119531, WARREN[1969], SCHWARTZ and C o w [1987]). Particular Bragg peaks occur for different phases in a sample, and diffraction methods are thus essential in the study of phase diagrams and phase transitions. The orientation distribution of Bragg peaks for a polycrystalline sample reveals its texture (see, e.g., BARRETTand MASSALSIU[ 19801). 3.1. Peak shifts
Changes in peak position may be caused by residual stresses, by faulting on certain crystallographic planes (WARREN [ 19691, WAGNER[1966], COWLEY [1981]), or by lattice parameter changes as a function of alloying or of defect concentration. Whereas a lattice
Ch. 12, $ 3
X-ray and neutron scattering
1131
parameter change affects all crystallographically equivalent Bragg peaks in the same way, the other two causes may lead to different shifts depending on the orientation of the reflecting planes relative to a preferred axis of the disturbance. Figure 8 illustrates this for a polycrystal with a surface under compression. As residual stresses and applied stresses combined determine load caving capabilities, X-ray methods in this field have received considerable attention (e.g., BARREIT and MASSALSKI [1980], CULLITY[1977]). The book by NOYANand C O ~ [1987] N gives a detailed account of procedures and applications. As the penetration depth of X-rays is limited and strongly depends on wavelength, only near-surface regions are accessible. This requires some caution in the interpretation, but may also be used to obtain some information on stress gradients perpendicular to the surface (see, e.g., O m [1989]). The preferred Bragg diffraction from near-surface regions leads to important applications in the study of epitaxial layers and heterostructures (see, e.g., CHU et al. [1985], Z ~ G et G al. [1994]), as accommodation stresses, their inhomogeneity and their relaxation may be determined. With the availability of powerful X-ray beams at synchrotron radiation sources, the surface sensitivity for Bragg scattering may be considerably improved (several nm instead of pm) by grazing incidence. For an incident beam which impinges on the sample at a glancing angle at or below the limiting angle of total reflection, diffraction peaks may arise from the evanescent wave travelling in the crystal parallel to the surface and decaying exponentially perpendicular to the surface (see MARRAet al. 119791). tor
CI
C
28
Fig. 8. Residual stress measurements with an X-ray diffractometer for a surface of a polycrystal under compression (schematic, not to scale). (a) Bragg's law is satisfied for lattice planes parallel to the surface. These planes are further apart than in the stress-free state (Poisson's effect). Their spacing is obtained from the position of the Brag peak. (b) The specimen has been tilted, and other grains now present suitable lattice planes for Bragg scattering. As these planes are more nearly perpendicular to the compressive stress, they are less separated than in (a), and the Bragg peak moves to higher angles. (c) The direction of the meawced stress is given by the intersection of the circle of tilt and the surface. (After M.R. JAMESand COHES[1980].)
References: p . 1188.
1132
G. Kostorz
Ch. 12, $ 3
Evanescent scattering of X-rays and neutrons offers many new experimental possibilities in surface science (see FEIDENHANS‘L [1989], &BEL and ROBINSON [1992], and Dosc~ [1992, 19931). As examples for evanescent X-ray Bragg scattering we quote the studies on depth-resolved near-surface ordering of Cu,Au (ZHU et al. [1988], Dosc~et al. [1991a] and on surface-melting of an AI single crystal (DOSCHetal. [1991b]). Gradients of lattice constants may also occur in diffusion couples, e.g., when a plated sample is annealed. The (nondestructive) analysis of such diffusion profiles using X-ray diffraction starts from similar ideas (see HOUSKA[1980]). Neutron diffraction averages over larger volumes, and several nondestructive measurements of internal stresses in “real” components have been reported (see e.g., ALLENet al. [1985], PMEYER et al. [1994]). The method has a special value for “hidden” stress concentrations.
3.2. Peak broadening and intensity changes In 0 2.1 we have already mentioned a broadening due to sample size that could be relevant for powders [eq. (l)]. However, for bulk samples we have tacitly ignored size broadening and written the Bragg cross-section as a delta-function. Thus, for a well collimated X-ray beam of characteristic radiation, the width of a Bragg peak should be controlled by the natural linewidth, since Bragg’s law yields (0 being a Bragg angle):
Ah A(28) = 2-tan@. h
(26)
With A h / A = 5 x lo4, A(20) is about one minute of arc at 28 = 30” but increases dramatically for 6 + 90°, and eq. (26) can be confirmed for large perfect crystals. However, quite frequently, the Bragg peaks are much broader (many minutes at moderate Bragg angles), and only crystals with such rnosaicify will actually approach the predictions of the kinematical theory as may be judged from the integrated reflecting power (which should be proportional to lFJ2).According to DARWIN ([1922], see ZACHARIASEN [I9451 and R.W. JAMES [1963]), the kinematical theory is valid for an ideal mosaic crystal consisting of small perfect domains which are tilted against each other by small angles (a few minutes). According to this concept, the effect of coherent multiple reflection of the same beam can be reduced to just the planes within one mosaic block and finally neglected (Le., primary extinction plays no role). The dynamical theory of diffraction shows (see ch. 11) that the tolerable size of the blocks depends on wavelength. As the extinction length (for Bragg reflections) is proportional to (AlFJ)-’, a sample of given thickness or mosaicity will approach the case of the ideal mosaic crystal with decreasing wavelength. Mosaicity in a real crystal is a merely formal concept for peak broadening which may be caused by various defects, e g , dislocations. Referring to crystals containing dislocations in connection with Bragg peaks appears contradictory since eq. (12) and the subsequent discussion imply that dislocations cause Bragg peaks to disappear as their strain fields decrease proportionally to ( 4 - l . However, this would hold only for a [I9691 has shown, random arrangement of dislocations in a large crystal. As WILKENS a random distribution is not a very suitable model for the calculation of X-ray line
Ch.12, $ 3
X-ray and neutron scatfering
1133
broadening or of the elastic energy. A so-called restrictedly random distribution of dislocations was used to calculate the line broadening and the peak profile more realistically (WILKENS[1970, 19751) from the kinematical theory. The linewidth AK is proportional to ghWpt(’ lnMe, where p,, is the dislocation density and Me is a parameter influencing the lineshape. It is given by M,=p?R, where Re is an effective outer cutoff radius, indicating the range over which a random dislocation arrangement can be admitted. Beyond this distance, long-range stresses should compensate each other (see UNGAR et aZ. [I9841 and MUGHRABIet al. [19861 for an application to deformed Cu single crystals). For peak profiles with neutrons, see SWALLOW and OSBORN[1992], who studied in situ the effects of elastic and plastic deformation in Cu. Racking-curve broading, on the other hand, is only very indirectly related to dislocation densities. It rather reflects the spread of lattice rotations around the rocking axis. An analysis of rocking curves taken at different angles with respect to the tensile axis of a deformed crystal may provide useful details on the slip geometry (for [001]-Cu, see WILKENS et al. [1987]). An analysis of the “static Debye-Waller” factor may be useful to determine displacements around defects (see KRIVOGLAZ 119691, DEDERICHS [1973], TRINKAUS [19751, DIETRICHand ~ N Z [1989a]). L In a dilute alloy, (c' , 3K5 + 4Gq,
where K is the solute bulk modulus, G is the solvent or matrix shear modulus and N is Avogadro's number. Two examples of the values for segregants in copper computed by MCLEAN[ 19571 are shown in table 1. The final column shows the experimental values. Values estimated by this approach are generally correct to within a factor of two. However, much higher accuracy is required for the quantitative prediction of segregation 119731. Using the behaviour. This was provided in the treatment by SEAHand HONDROS et aZ. [1940], they write the solid-state (BET) gas adsorption theory of BRUNAUBR analogue as
where X,",the solid solubility, is the important parameter. For slightly soluble substances, X,"= exp(AG,,JRT) so that the Langmuir-McLean and Truncated BET theories are in fact identical in the dilute limit with
AG, = AGf -I-AG,,,
(14)
References:p . 1284.
Ch. 13, $ 4
E.D. Hondros et al.
1222
C
Temperature
2000 I
1800 I
I
1600 I
1
I
h
x"x" I v c
9.9
9.8
9.7
I
4.3
I 4.5
I
4.7
I 4.9
10' (
I
I
I
5.1
5.3
5.5
5.7
K-' )
T Fig. 13. Langmuir-McLean plot for oxygen grain-boundary segregation measured by AES in molybdenum (after KUMARand EYRE[1980]).
as shown in the energy level diagram of fig. 12. The importance of the Truncated BET form is that one may define as a useful parameter the enrichment ratio, & which represents the ratio XJX, in the dilute limit. Since X,"generally turns out to be unity, we find pb
=x,"
where
K = ea(-=). A C
Figure 14 shows a compilation of measurements of grain-boundary enrichment ratios, P b , (estimated by the grain-boundary energy and surface-analysis approaches), in relation to
Ch. 13, 44
1223
Interfacial and su$ace rnicmchemishy Table 1
Values of the elastic distortion energy, E, for Sb and Bi in Cu.
K
r1
(GN/m2)
(nm)
Sb
41
0.1528
Bi
31.4
0.157
Element
(rL- W r l
+ 0.194 + 0.266
Ed
AGb
(kT/mol)
W/mol)
37
65
66
100
the solubilities of the segregant at the measurement temperature. Thus the grain-boundary enrichment for systems in which no measurements are available may be accurately estimated from handbooks of phase diagrams (MASSALSKI et al. [19861). It is clear that, in the dilute limit, eq. (15) provides the theoretical description of the experimental data of
-
lo6*Steel - S t *Steel -SI *Steel- P
n 01
-
0 " c
.-E
-
aFe-Ca ;
t
a r( w
-*Steel-Mi *Steel- Ni *Steel -CI
--
1 1
10-2
Atomic solid solubility,Xf
10-4
Solubility unknown
Fig. 14. Correlation of measured grain-boundary enrichment ratios with the atomic solid solubility (after SEAH [1980dl). References: p. 1284.
1224
Ch. 13, 54
E.D.Hoiulms et al.
fig. 14 quite accurately. Indeed it is seen that whilst AG,,, ranges from zero to -80 kJ/mol, AG' has a mean value of -10 kJ/mol with a standard deviation of only 6 kJ/mol. Thus,
AG, = AG,, - 10 f 6 kTjmo1.
(16)
Equation (13) is valid only for X,e X,".If excess solute is present, so that a second phase appears, the solute content is limited to X,"and eq. (13) becomes b' ~
xb"- Xb
= exp(
AG' - =).
The translation from eq. (13) to eq. (17)as the solubility limit is reduced through the solute level, is illustrated by the measurements for sulphur in iron by SEAH and HONDROS [1973], shown in fig. 15. Equation (13) is shown for the higher temperatures, where the solubility is high, and eq. (17) at the lower temperatures. WATANABE et aZ. [1980] proposed to refine the dependence of grain-boundary enrichment ratio on solubility limit by considering the effect of grain-boundary orientation (see also 0 4.5) and suggested the idea of constructing a grain-boundarysegregation diagram as a three-dimensional function of Pb in dependence on X," and on the misorientation angle of adjacent grains. The first experimental grain-boundary segregatand LEJCEK [1991] for Si, P and C segreion diagram was constructed by HOFMANN gation at [loo] symmetrical tilt grain boundaries in a-Fe. In this diagram, however, the Sulphur at grain boundary,
0.2
0.4
p moles m-*
0.6
0.8
1.0
1.2
1.4
Sulphur monolayers at the grain boundary
Fig. 15. AES measnrements of sulphur grain-boundary segregation in iron as a function of temperature for two bulk compositions (after HONDROS and SEAH[1977b]).
Ch. 13, $ 4
1225
Intelfmial and sulface microchemistry
grain-boundary enrichment ratio was replaced by a more general and independent thermodynamic quantity, @. Recently, LEJ~EKand HOFMANN[19931 theoretically derived the dependence of the segregation enthalpy on both the grain-boundary orientation and the bulk solid solubility. It was shown that this dependence can be expressed as a sum of two mutually independent terms
m0(a, x:) = AH* (a)+ V
~ InTx:]
(18)
where Ah?(@) is the enthalpy of segregation of a solute with unlimited solid solubility (i.e., X,"=1) in a given matrix at a given interface of orientation CP, and v is the parameter characteristic for the matrix element. Expression (18) was derived under two basic assumptions, first,
T In a,* = const
(19)
where a,* is the activity of a solute at the solid solubility limit, and second, a," = (x:)'
which relates the activity u,' and concentration X,"of any solute at the solid solubility limit in a given matrix. In fact, eq. (18) represents an extension of the model of SEAH and HONDROS[19731 by taking additionally into account (i) the structural dependence of interfacial segregation (Le., A.H*(@) f const), and (ii) the non-ideal behaviour of the system (i.e., v # 1). The grain-boundary segregation diagram for the [lo01 symmetrical tilt grain boundaries in a-Fe is shown in fig. 16.
4.2.2. Prediction of the free energy of segregation to surfaces The theory for the free energy of surface segregation, AGs, has developed along less phenomenological lines than that for grain boundaries although both are merely special cases of one general effect. Each of the two approaches leads to a reasonably accurate prediction for the case to which it pertains. The reader may blend the two approaches to gain a greater insight into the nature of the contributions to the free energy, however, at the present time it does not lead to a greater accuracy in calculation. The treatment for free surfaces is based on the derivation of WILLIAMS and NASON [1974] who use a layer-by-layer model of the surface region, with solute A atoms replacing the solvent B atoms substitutionally at the surface. They find that this model gives the Langmuir-McLean adsorption eq. (10) for the fractional coverage of A on the surface, which allows the calculation of AG,. The basic model adopts the quasichemical approach, shown in fig. 17, where the nearest neighbour bond energies are given by E ~ sBB , and between AA, BB and AB neighbours respectively. AG, is calculated as the energy released on removing an A atom from a lattice site and exchanging it for a B atom on the free surface. The energy required to remove A from the bulk is:
References: p . 1284.
E.D. Hondros et a[.
1226
Ch. 13, 94
60
0
I
4 20
0
eo[l001 Fig. 16. Grain-boundary segregation diagram for [lQO] symmetrical tilt grain boundaries in a-Fe showing the dependence of enthalpies of Si, P and C segregation on misorientationangle B of both adjacent crystals and on the bulk solid solubility data (after LEX% and HOFMANN[1993]).
where Z, is the atomic coordination number in the layer and 2, is the atomic coordination to one of the adjacent layers. The energy to replace A at the surface is similarly:
z,[xs.,
+ (1 - x s ) 4+ Z,[X,%A + (1 - &)&AB].
(22)
Equivalent terms occur for the B atom moving in the reverse direction so that by using the regular solution approximation in which Em
=
+
(CM
+ E B B )+ w,
(23)
we find that the free energy of surface segregation per atom, A G / N , is given by
In Williams and Nason's theory, successive layers away from the surface are considered and for nonzero values of o the composition of the second and further layers can also differ from that of the bulk. For instance, if w is negative, there is a perceptible
Ch. 13, 94
Intelfacial and surfnee microchemistry
1227
Fig. 17. Schematic substitutional model for the quasi-chemical segregation theory.
oscillation of composition decaying away from the surface in addition to the surface segregation. Here,however, to simplify the analysis and because it has not been possible to observe such small oscillations in practice, it will be assumed that all of the compositional variation occurs in the topmost atom layer. Numerical evaluation of eq. (24) may proceed in several ways. Following WYNBLATT and Ku [1979] the bond strengths and sBB may be derived from the molar surface energies of the two constituents. At absolute zero these energies may be expressed as follows: y; = - 3 Zv&,N,
3
yi = - Z p B B N .
(25)
The regular solution parameter, w [eq. (23)], may then be calculated from tabulated values (HULTGREN et al. [1973]) of the enthalpy of mixing, H":
a = No = Hm/zxc(l - xc),
cw
where Z is the total coordination number. Williams and Nason in their original paper evaluated eAAand eBB from the molar sublimation enthalpies, F*,where
H,Yb = - 3ZsiiN,
(27)
where i stands for A or B. The above terms now give some insight into the kinds of elements that will segregate to free surfaces. If the regular solution parameter can be ignored, it is seen from eqs. (24) and (25) that, other things being equal, the element with the lower surface energy enriches the surface of a binary system. Alternatively, from eqs. (24) and (27), this will also be the element of lower sublimation enthalpy. The above is particularly true for liquid binary metallic systems as shown by References: p . 1284.
1228
E.D. Hondms et al.
Ch. 13, $4
HONDROS [198Oa]. However, for solids McLean’s strain term given in eq. (12) is also important. The strain term is only significant for excess-sized solute atoms. TSAI et al. [1977] point,out that, as there is little relaxation around vacancies, the strain energy associated with ro< rl, must be much less than the converse. Using the above approach, SEAH[1979] gained an overall View of surface segregation by combining eqs. (24) and (25) and adding the strain term of eq. (12). From what we have seen in 54.2.1, this term will be of the order of 2.6 x x ro(rl - rJ2. In fact, using regression analysis of published data, Seah found for dilute binary systems:
-AGs = (7;
- 7;) + 1.8Q + M2.66 X
ro(q - Po)2 ,
where NI is unity for rl > ro and zero for rl < r,. The r values are the atom sizes given by ~lV(2r)~ =A, where p is the bulk density and A the atomic weight of the element. Figure 18 shows the resulting correlation for an improved version of eq. (28), using the relation p, =X/X,= exp (-AG/RT), and the experimental data for the surface enrichment ratio. This analysis confirmed the correctness of simply adding the bond and strain energy terms directly and that the latter is significant only for r, > r,. Thus we may write:
AGs = [bond term, eq.
(%)I
+ M[strain term, eq. (12)]
- T (entropy terms discussed in 4 4.2.4). Several other models have been developed to estimate thermodynamicparameters of surface segregation. On the basis of a simple tight-binding theory of surface segregation in alloys of transition metals, M u m m and MORAN-MPEZ [19871 proposed a method to predict surface segregation assuming a rectangular shape of the d-band density of states. This method enables the equilibrium surface concentration to be calculated as a function of temperature. Another technique for calculating the heat of segregation is et al. [1992]). The energy of segregation which is based on the BFS model (BOZZOLO composed of two basic contributions, the strain term and the chemical term, can be obtained using a zero-temperature Monte Carlo calculation. As the input data, pure et al. elemental parameters and only two alloy properties have to be used (BOZZOLO [1993]). In contrast to these two models, a phenomenological approach was used in modem thermodynamic calculation of integace properties (MEZEYet al. [19901). In this approach, the free energy of segregation of a solute I at a surface in a binary M-I alloy can be expressed as a sum of two terms, the dangling bond contribution:
AG,d = (7I - YM)A,,
(30)
and the real mixture contribution: E AGY = qApF - ApM.
Here ‘yi are the surface energies of pure elements i, q=AM/A1,Ai are the partial molar areas of species i, and ApY are the excess chemical potentials of mixing related to the elements i. The total free energy of segregation, AGI= AG; + AG,”’,can be determined by means of individual factors evaluated from elemental data, and from experimental values of
Ch. 13, 34
Inte$acial and su$me microchemism
Predicted
1229
Bs
Fig. 18. Correlation of measured surface enrichment ratios with the predicted values from eq. (28) (after SEAH [1979]).
interfacial free energies and molar enthalpies of mixing. It was also shown that this approach can describe anisotropy of surface segregation of F’t in Ni (MEZEYand HOFER [1992]). 42.3. Segregationwith adsorbate-adsorbate interactions In the above we have concentrated on evaluating AG in the limit of dilute X,,X, and X, to compare the overall behaviour with experiment. To study the effect of higher levels of segregation we may rewrite eq. (24) simply:
where AGP =~VZ,,(E~~-E,,~) does not depend on the segregation level. This equation is based on the substitutional segregation model shown in fig. 17, which is a good description for many free surfaces and grain boundaries. A completely different description, valid only for free surfaces, is the Fowler theory in which adsorbed atoms on top of the free surface have, simply, the interaction energy E ~ In. this case: References: p. 1284.
1230
E.D. H o n d m et al.
AG, = AG:
+ Z,NcMXs.
Ch. 13, 54
(33)
This is seen to be the same as the result for eq. (32) if the top atom layer B atoms are replaced by vacancies. Using either of the above equations, if the presence of a segregated A atom enhances the probability of the next site being occupied by an A atom, AG becomes more negative as the segregation proceeds. As E~ becomes more and more negative, the segregation shows progressively sharper rises as the temperature falls until eventually the rise in segregation becomes discontinuous at a temperature given by the solution of C = -4RT,
(34)
where C is the coefficient of X,in eqs. (32) and (33). Measurements illustrating this form of segregation are shown in fig. 19 for Se and Te segregation in pure iron. There is also shown the simple Langmuir-McLean case with C=O and a case in which a discontinuous rise occurs from a low to a high segregation level. As seen from the labelling of the ordinate, the lowest segregations occur at low bulk solute contents, X,, and high temperatures, T. The Fowler type of discontinuous grain-boundary segregation with C=-5RT was observed by MENYHARDet al. [1989] for Bi in Cu. 4.2.4. Temperature dependence of the free energies of segregation The temperature dependence of segregation occurs mainly through the explicit temperature term in eq. (13). As the temperature rises the segregant “boils off’, leading to a lower value. However, the free energy itself is temperature dependent through the customary relation:
AG = AH- TAS (35) where AS is the entropy change on segregation to a grain boundary or free surface. SEAH and LEA[1975] write AS as the sum of three terms, AS,,, AS, and AS,,,. AS,, is associated with the changes in the vibrational behaviour on segregation, ASdwith the anharmonicity of the potential of the segregant atom site and ASmwith site multiplicity. The vibrational entropy term is given, in the Einstein model, as S, = 3R [1 + ln(kT/hv)],
(36)
where kT>>hv, v being the Einstein frequency. hv may be rewritten k8, or 0.775k8, where 8, and 8, are the Einstein and Debye temperatures (KI-ITEL [1986]) so that
As,,= 3RIn(B,/ a),
(37)
where 8, and 8’ are the Debye temperatures for the solute atom in the matrix and at the distorted site of the grain boundary or free surface. Owing to the greater vibrational amplitude of atoms at surfaces, &/8, has been calculated for tin surface segregation in iron to be approximately 2 / 3 . In this way, taking into account the matrix atom exchanging places with the segregant atom, ASv is about 3.5R for the free surface and 3.3R at grain boundaries.
Ch. 13, 54
1231
Intelfacial and su$ace microchemistry I
I
0.5
x,
I
1.0
1x0,
Fig. 19. Grain-boundary Fowler isotherms with the AES measurements of PICHARDet al. (after HONDROSand
SEAH [1977a]).
In a similar manner, the two further entropy contributions, AS,, due to the anharmonicity of the atomic potential at the termination of a lattice, either by the surface or by a grain boundary @WING [1971]), and AS,,,,due to the possible site multiplicity for the segregating atoms, may be calculated. Both terms are much smaller than AS,, and so will not be treated here. The correlation of the above full prediction of the temperature dependence of AG with the experimental data for tin in iron is shown in fig. 20 for both surface and grainboundary segregations. This figure also serves to emphasize the generally observed fact that surface segregation at moderate temperatures is generally much greater than segregation at grain boundaries.
References:p. I284.
Ch. 13, 84
E.D. Hondms et al.
1232 -25
-50
-0
-75
c
E
1
3
0-
Q -1 00
-125
-1 50 250
500
750
1000
1250
1500
Temperat ure,OC Fig. 20. The predicted temperature dependences of the free energies of surface- and grain-boundary segregations of tin in iron, together with the experimental data at 823 K (550'C) and 1693 K (1420'C) (after SBAHand LEA [ 19751).
4.3. Segregationin simple ternary systems: site competition
Site competition in a ternary system occurs where there are two segregants which can fill the same sites at the free surface or grain boundary. Each segregant obeys the appropriate adsorption theory with the available empty sites reduced by the segregation of the other species. At saturation segregation, this leads to the situation where the sum of the segregants is constant at one monolayer. This has been observed on the free surface of iron for Sn, Sb and 0 in competition with S. Generally the Sn, Sb or 0
Ch. 13, 94
Interfacial and surface micmchemistry
1233
segregate rapidly to an equilibrium level but, with time at temperature, S (which may be present at only a few ppm in solution), will replace much of the Sn, Sb and 0. Such a competition is not generally expected at grain boundaries since the bond energy terms can continue to be reduced until two total monolayers are segregated, and this latter level of segregation is rarely reached. Grain-boundary site competition between N and S has and GRABK~ [1978]) in iron but not under all conditions (JONES been reported (TAUBER e t d [19Sl]). More importantly, P is displaced by C (ERHART and GRABKE[19Sl]). This is of great significance in steels, as discussed in P 6.3. 4.4. Segregationin complex metallurgical systems
In 1975 G m m extended the Fowler theory to allow for coupling interactions between two co-segregating species in multicomponent systems. This development, vital to explain the segregation behaviour resulting in intergranular failures in engineering materials, gives the analogue to the Langmuir-McLean equation. xbi --
X, exp(-AGJRT)
where Xbl and x b z are the molar fractional monolayers segregated by a noxious impurity and by an alloying element of bulk contents X,,and Xczrespectively. The AG's are given by equations of the form of eqs. (32) or (33) but depend not on the level of selfsegregation but on each other through the equations:
where AG; and AG; are the free energies of segregation of the impurity and alloying elements separately in the matrix, and C,, and C,, are the respective Fowler-type interaction coefficients (cf. eq. (32)). The interaction coefficient alzrefers to the changes in nearest neighbour bond energies cli in forming the alloy-impurity bonds,
and can be obtained from measurements of the effects of the alloying elements on the solubilities of the impurities (GUTIUANN[1976]). The stronger the reduction in solubility, the stronger the coupling of the alloy element to the impurity and the more the alloying addition enhances the impurity segregation. In a similar manner to that of the Fowler isotherm, as a;z becomes more and more negative the curves exhibit a more pronounced S shape and eventually attain a region in which the segregation shows a discontinuousincrease from a low to a high level. The general effect of Gwttmann's coupling can be seen by reference to fig. 14. Alloy element additions with a negative valwe of a;z cause the impurity element solubility to fall. Simultaneowsly, through eqs. (38) and (39), the segregation level rises. Thus, the addition of the alloying element causes the point for the original binary system in fig. 14 to move diagonally upwards to the right but to stay within the general correlation (GUTTMANNand MCLEAN[1979]). The general behaviour of this model is now well accepted and allows an accurate References: p . 1284.
1234
E.D. Hondros et al.
Ch. 13, 84
description of some of the behaviour observed in embrittlement probIems. The increase in the free energy of segregation of one species by a second, described in eq. (39), has been validated in careful experiments on grain-boundary segregation of Sb and Ni in steel by GASet al. [1982]. Although high -a{2values lead to a high enrichment ratio, the result in practice often will be a low segregation. This is because the reduction in solubility and hence the number of solute atoms available to segregate make the rate of build-up of the segregant at the surface or grain boundary very slow indeed so that, with practical heat treatments, a negligible segregation occurs. This is precisely the case, for example, with the addition of manganese to remove the sulphur embrittlement in steels. The problem of the kinetics of segregation is very important in practice and this is considered in the next section. Guttmann’s original theory was modified by SEAH [1977] to remove the site competition between alloy elements and impurities which had not been observed in practice. In eq. (38) the sum over the j terms is simply replaced by the i”’ term. This simplifies the calculations and gives a better description of the experimental results as discussed in $6.3. Several more complex theories are detailed by GUTTMANNand MCLEAN[1979] and by GUTTMANN[1980]. Solute segregation in multicomponent systems can alternatively be described in terms of chemical equilibrium (BRIANT[1990]). In this approach, the reversible chemical reaction r between two segregating elements I and J at the boundary (b) and in bulk (c), I(C)
+j(b),
I@)+ jb),
(41)
can be characterized by an equilibrium reaction constant K,,
where ubiand aCiare the activities of individual solutes at the boundary and in the bulk, respectively. In contrast to the Guttmann’s approach in which the solute interactions [1975]), the modify the value of segregation free energy (eqs. (38) and (39); GUTTMANN value of AG, is constant for the reaction r at a given constant temperature in the Briant’s approach (eq. (42)) and only the changes in bulk activities and solid solubilities are responsible for segregation behaviour of any element in the system, i.e., no solute atom interactions are considered. By means of the latter model, BRIANT[1988a] interpreted the grain-boundary segregation of Sb in a-Fe enhanced by additions of Ni as observed by GASet al. [1982], by increasing the bulk solid activity of Sb in a-Fe by Ni. Strictly speaking, both Guttmann’s and Briant’s approaches lead to rather similar expression (compare eqs. (38) and (39) with eq. (42)), supposing the activity ai to be expressed as the product of concentration X,and activity coefficient A, and the product of activity coefficients to be represented by combinations of interaction coefficients in eq. (39). 4.5. Anisotropy of segregation
In the above we considered that all sites are equivalent prior to segregation. This is generally not the case either at free surfaces or at grain boundaries. The simplest case to
Ch. 13, $ 4
Interfacial and surface micmhemistry
1235
understand is that of free surfaces. The driving force for segregation is the lowering of the overall system free energy. If the surface energy is already low as in the case of the low-index surfaces (lll), (110) and (loo), we should expect that the segregation here would be lower than that for a high index surface, and lowest for (111) surfaces in an and HOFMANN [1977]). Little direct work has been reported, but fcc lattice WEWIN measurements by JOHNSON et aZ. [19781for Au segregation on Ni, by ZHOUet aZ. [19813 for C and Sn segregation on Fe, and by SUZUKIand OKU 119901 for Cr segregation on Fe generally confirm this although Si segregation at Fe (111) was detected to be rather high (ZHOU etal. [1981], SUZUKI and OKU[1990]). Considerablymore work has, however, been completed on the influences of the anisotropy of surface segregation on the anisotropy of surface energy, because of the technological significance, and this is discussed in 8 5.1. 4.5.1. Segregation at symmetrical grain boundaries In contrast to free surfaces which can fully be described by the orientation of surface normal (Le., two degrees of freedom in a chosen reference system), five variables are necessary for a complete characterization of a grain boundary: Three of them determine misorientation of lattices of the two adjacent grains (rotation axis and misorientation angle) and two of them describe the orientation of the boundary plane (WOLF[1992]). Therefore, the orientation dependence of grain-boundary segregation cannot be displayed in the same simple manner as that for surface segregation. In principle, the complete anisotropy of grain-boundary segregation for a binary system should be constructed in an eight-dimensional space as an interdependence of interfacial composition (or grainboundary enrichment ratio) on five grain-boundary parameters, and on temperature and N To simplify the problem, orientation dependences of grainpressure ( S E I D ~ [1992]). boundary segregation are displayed as plots of grain-boundary composition vs. one grainboundary variable keeping the other four ones constant, in addition to constant temperature and pressure. In this way, solute segregation was found to increase with increasing misorientation angle for both tilt (e.g., BISCONDI[1982]) and twist (e.g., SEIDMAN [19911) low-angle grain boundaries. Among the high-angle grain boundaries, two types of interfaces can be distinguished: the majority of general grain boundaries with relatively high values of &, and a few special grain boundaries characterized by a low enrichment factor in comparison to the general ones (cf. fig. 21). From the viewpoint of atomic structure, the special grain boundaries are equivalent to so-calledfavoured or delimiting ones, the structures of which are created by single structural units whereas the structures of the general (or nonlfavoured) ones consist of a combination of structural [1985], see also units of the two neighbouring favoured pain boundaries (FISCHMEI~TER ch. 9, 0 2.2). To characterize the special grain boundaries, various bulk geometric parameters are used. One of the most widespread geometric characterizations of grain boundaries is the reciprocal density of coincidence sites 8 which can be determined for cubic crystals by
8 = S(h2 + k2 + 1’)
(43)
where h, k and 1 are the Miller indices of the symmetrical grain boundary corresponding + 12 is odd and S = 0.5 if hz + kz + 12 is
to the given misorientation 8, and S = 1 if hz +kz
References: p . 1284.
1236
Ch. 13, $ 4
ED. Hondms et al.
d
cl
0
20
40
60
80
100
eo (01 I) Fig. 21. Orientation dependence of the Re enrichment factor for (011) twist grain boundaries of a W-25at%Re alloy (after SEIDMAN[1991]).
even. Other geometric parameters are the inverse planar coincidence site density r defined as the ratio of the unit-cell area A, and the smallest possible area per atom A,
and the interplanar spacing d(hkl) defined for cubic lattices by
d(hk2) =
-d
Ea
(45)
where a is the lattice parameter and 8 has the value of either 0.5 or 1 depending on the particular combination of odd and even Miller indices and on the type of the cubic lattice (WOLF[1992]). Usually, it is accepted that special grain boundaries possess low values of 2 and r, or high values of d(hkZ); however, because these parameters are bulk and geometric characteristics, there is no one-to-one correlation between them and the structure of grain boundaries. Therefore, these parameters can serve as rough estimates only (SUTTON and BALLUPPI [ 19871). Owing to the great difference in the behaviour of special and general grain boundaries and because of the great importance of grain-boundary properties in the complex behaviour of a whole polycrystal, a large effort has been spent in the last two decades to learn the character of all possible special grain boundaries, their atomic structures and
Ch. 13, 54
Interfacial and surface microchemistry
1237
properties. As regard grain-boundary segregation, low values of Pb were found at I: = 11, 50.5" [110], { 113) and 2 =3, 70.5' [110], {112} symmetrical tilt grain boundaries in a Mo-O system (BISCONDI[1982]) or at the 2 =3, 70.5" (011) twist grain boundary in a [1991]). A low tendency to P, C and Si segregation was W-Re alloy (fig. 21; SEIDMAN detected f o r 2 = 13,22.5" [loo], {OlS},Z =5,36.9" [loo], (013) and2 =5,53.1' [loo], { 012) symmetrical tilt grain boundaries in an Fe-Si alloy (HOFMANNet al. [1992]). In this case, grain-boundary segregation was represented by segregation enthalpies AH: of individual elements as shown in fig. 16. This parameter is more general than the enrichment ratio &: for example, it is independent of temperature and bulk composition of the alloy. This is very important in the case of complex multicomponent systems in which site competition and solute interaction can modify the grain-boundary composition to such an extent that &, loses the ability to characterize grain-boundary segregation in a binary system (see also $94.3. and 4.4). The above mentioned interfaces represent special grain boundaries selected on the basis of geometric classification of symmetical tilt grain boundaries (PAIDAR[1987]). 4.53. Segregation at asymmetrical grain boundaries For asymmetrical grain boundaries, the relative orientations of both crystals to the grain-boundary plane have to be considered and therefore their description is more complex. In a neat study by SUZUKIet aZ. [1981] of phosphorus grain-boundary segregation in iron, they show that after fracture, the level of phosphorus observed may be associated with the Miller indices of the fracture surface, as shown in fig. 22a. The measurements of the segregation in relation to the stereographic triangle are shown in fig. 22b. It is clear that the lower segregations are associated with the lower index poles since, as one may expect, the boundaries involving these surfaces are more ordered than the general high-index boundaries. The preponderant role of the orientation of the grainboundary plane in solute segregation was also experimentally observed for sulphur in nickel. No segregation effects were detected at symmetrical { 111) and { 112) facets of a 2 = 3 twin boundary but some sulphur enrichment was observed at asymmetrical ones [1986, 19871). Similarly, a such as, for instance, (111)/(113) (BOUCHET and PRIESTER symmetrical { 113) grain boundary did not exhibit any S segregation in a nearly C = 11 Ni bicrystal whereas asymmetrical boundary facets with orientations close to e.g. (1i2)/(116) and (114)/(225) (i.e. with a deviation of the order of -2" from ideal I:= 11) er al. [1990]). In contrast to the were found to be enriched with sulphur (SWJATNICKI { 113},the other Z = 11 symmetical grain boundary { 332) was found to be very unstable and to split into numerous asymmetrical parts considerably enriched with sulphur (JUHAS et al. [1994]). The conclusions of SUZUKIet al. [198l] were directly supported by systematicmeasurementsof grain-boundary segregationin carefully prepared I:= 5 , 36.9' [lo01 tilt bicrystals of an a-Fe-Si alloy. Low tendency to Si, P and C segregation flow absolute values of segregation enthalpy) was observed at the boundaries created by at least one low-index plane, i.e., symmetrical {013} and {012}, and asymmetrical (001)/(034) and (Oi7)/(011) ones. On the other hand, high solute enrichment was detected at the high-index asymmetrical boundaries (018)/(047) and (0311)/(097) (LEJCEKet al. [1992]). En addition, different amounts of segregants were found at both Refemces: p. i284.
1238
Ch. 13, 94
E.D. Xondms et ai.
7' 7 7 Grain boundary
('
Low indices
F i n d i c e s
Low indices
High ( 3 )indices
indices
% Monolayer of P on fracture surface 0 c
A
40
40-50 50-60
A >60
F i n d i c e s
High indices
(a)
(b)
Fig. 22. The anisotropy of phosphorus grain-boundary segregation in Fe-l%P showing (a) the segregant atoms (dark squares) in relation to the fracture process and (b) the level of segregation in relation to the relative orientation of each grain to the grain boundary (after SUZUKI etnl. [1981]).
halves of asymmetrical grain boundaries, suggesting an asymmetry of solute segregation in the boundary core (LEJCEK [1994]). A good correlation exists between the segregation enthalpy and the ejgrective interplanar spacing d& This form of spacing was introduced by D. Wolf for boundaries where two well defined planes join, e.g., (OOl)A/(Oll)B.d@ is then given by d d a = (dA+ dB)/2.The boundaries characterized by high values of dtlr possess low absolute values of segregation enthalpy for a particular solute as is demonstrated in fig. 23 (HOFMANN and LEJCEK[1991], LEJCEKand HOFMANN [1995]). 43.3. Computersimulation of grain-boundary segregation Measurements of chemical composition of individual grain boundaries in wellcharacterized bicrystals by AES represent a substantial step forward in understanding the anisotropy of grain-boundary segregation. However, the information provided by such measurements is still averaged over a relatively large interfacial area (at present, the lateral resolution limit of AES is about 20 nm) as compared to individual boundary sites which may possess different ability to solute adsorption. Due to the present limitations of experimental techniques (although methods such as high-resolution transmission electron microscopy, HREM, combined with electron-energy loss spectroscopy, or AP FIM, approach atomic resolution), the only method of evaluating solute segregation at individual boundary sites remains computer modeling. Computer modeling has been successfully applied to the determination of atomic structure of grain boundaries in pure solids and to the development of the structural unit model (cf. ch. 9, P 2.2.7). Later, techniques such as molecular statics and dynamics
Ch. 13, 54
0
0
40
-
0
*.
AA A ......... ... A ....
E
\
3
1239
Inreflacia1and sur$ace microchemistry
30-
rn ................... .n.............. 0
'..
hk
W
A
0
I
P
-
2000
IO
-
0
................0..
%
60
'*
A ...........................................
.--..
A
A
A
........................................... e o A 0
0
I
I
0
P
Si
-
0 I
1
or Monte Carlo calculations using many-body potentials of either the embedded-atom or Finnis-Sinclair types, and molecular orbital models, were extended to theoretically simulate equilibrium atomic structures of segregated grain boundaries. Computer modeling of the structure of the 2 =5,36.9" [lo01 (012) grain boundary in Cu-Bi system revealed large differences in the segregation energy for particular sites. Because Bi tends to occupy the sites of the highest segregation energy, a two-dimensional (2-D) ordered structure occurs in the boundary layer where each Bi atom is surrounded by Cu atoms and vice versa ( V ~ and K WANG[1982]). The values of segregation energy of phosphorus and boron at 1013) and { 114) grain boundaries in a-Fe also show a high site sensitivity; however, their values calculated by HASHIMOTO et aZ. [19841 are much higher than those expected from experiments. More realistic values of the segregation energy of Mg, Al, Si, P, S and C1 at,the sites at the exact plane of a 8 =5, (013) symmetrical tilt grain boundary in a-Fe were calculated using a tight-binding type electronic theory of s-, p-, and d-basis orbitals by MASUDA-JINDO [1986]. Segregation energies of P and Si obtained in this way (-14.6 kJ/mol and -9.8 kJ/mol, respectively) are in excellent agreement with the values of segregation enthalpy of these elements experimentally determined at the same grain boundary by LEJCEKand HOPMANN[1991] (-13.3 kJ/mol and -8.5 kJ/mol, respectively). This agreement suggests that both these solutes segregate in a dilute Fe-Si(P) system at a specific type of sites at the boundary at all temperatures and therefore are characterized by specific segregation enthalpies. References:p . 1284.
1240
E.D.Hondros et al.
Ch. 13, 54
A very efficient and straightforward approach is based upon a point approximation for the configurational entropy, an Einstein model for vibrational contributions to the free energy with respect to atomic coordinates and composition of each site (NAJAF~ADI et al. [1991]). The approach was applied to the study of Cu segregation at 2 =5, (001) twist boundary in Ni-Cu alloys and revealed an oscillatory depth distribution of Cu in the equiatomic alloy (WANG et al. [1992]). Computer modeling can also reveal an orientation dependence of grain-boundary segregation as is demonstrated in fig. 24 for the example of Au segregation at (001) twist grain boundaries of R.Absolute values of segregation enthalpy were found to increase with increasing rotation angle 0 from 1.1 kJ/mol for 0 = 5" to 11.7 kJ/mol for 6 = 36.9' (Z =5). The average grain-boundary enrichment ratio Pb can be expressed by
p,( T,e) = 1 + m( T)sin(e,'2)
(46)
where the slope m(T) depends only on temperature (SEKIetal. [1991]). There are numbers of experimental studies on grain-boundary segregation on the one hand and computer simulations of segregated boundaries on the other hand. Nevertheless, there is still a lack of direct correlations between these two approaches. The first attempt to make a direct comparison of HREM observation of a segregated grain boundary and its computer-simulated structure was performed very recently to interpret the atomic structure of the 2 =3, (111)/(11T) grain boundary in the Cu-Bi system (LUZZI[1991]). In a non-segregated state, this boundary is the symmetrical coherent twin boundary with the stacking ABCAB~BACBA
(47)
where A, B and C represent individual { 111} atomic planes in an fcc lattice andE denotes the boundary layer. Usually, this boundary stays intact during solute segregation. However, Bi segregates at other grain boundaries and its segregation at non-coherent twin and X = 9 boundaries induces their transformationto (111)/( 11I) facets in which Bi atoms remain involved. The Bi-containing grain boundaries can be represented in the same way as the pure boundary but in the boundary layer C one third of the Cu atoms is replaced by hexagonally arranged Bi atoms. However, the Bi atoms are centered outside this atomic plane so that the structure of this boundary can be represented more appropriately by
where C denotes a plane of Cu atoms in which one third of atoms is replaced by hexagonally arranged vacancies, and C@i) the plane of Bi atoms positioned above these vacancies. Due to the splitting of the { 111} boundary plane C into two planes C and C(Bi) containing Cu and Bi, respectively, this boundary is not symmetrical in the sense of mirror symmetry across the geometrical boundary plane (YAN and et al. [1993]).
4.5.4. Correlation between grain-boundaryand free-surface segregation Owing to the similar nature of solute segregation to grain boundaries and to free surfaces, measurements of free-surface segregation have often been used to estimate
Ch. 13, $ 4 I
0
1241
Interfacial and suf a c e micnxhemistry
I
I
I
I
I
I
I
1
10
20
30
40
50
eo [1001 Rg. 24. Orientation dependence of the Gibbsian interfacial excess r, at (001) twist grain boundaries in an Pt-lat%Au alloy at 1900 K (0). 1500 K (0,1300 K (A), loo0 K ( 0 )and 900 K (.) (after S m ernl. [1991]).
solute segregation at buried grain boundaries qualitatively or semiquantitatively. Because of the similar nature of solute segregation to both types of interfaces, the same formalism is generally used for a thermodynamic description of solute segregation. Since a highangle grain boundary can roughly be considered as consisting of a complicated mixture of free surfaces and bulk binding states, the average free energy of general grain boundaries is about 1/3-1/2 of that for the free surface. Because solute segregation is associated with a lowering of the interfacial free energy, it is clear that grain-boundary segregation enthalpies are generally lower than those of surface segregation for the same system (HONDROSand SEAH[1977b]). On the basis of this similarity, SEAHand LEA [1975] proposed a method of quantification of grain-boundary segregation in a binary system. This method consists of the construction of a conversion diagram from the temperature dependence of surface segregation and a single measurement of grainboundary segregation at suitable temperatures (see fig. 20, § 4.2.4.). However, significant differences in segregation behaviour of the two types of interfaces can occur in three-or multicomponent systems resulting from site competition and solute interaction. For example, boron segregates significantly at grain boundaries in Ni,Al whereas no sulphur is observed there. On the other hand, extensive segregation of et aE. [ 19841). sulphur but low segregation of boron occur at its free surfaces (WHITE Qualitatively more pronounced differences between surface and grain-boundary compositions can be observed in systems with strong attractive interaction of two segregants resulting in the formation of two-dimensional surface compounds. In an References: p. 1284.
1242
E.D. Hondros et al.
Ch. 13, 94
Fe-6at%Si alloy containing P, S, N, and C as trace impurities, cosegregation of Si and N results in a stable Si& surface compound at the (013) free surface thus preventing significant segregations of other elements. In contrast, enrichment of P and C, and depletion of Si characterize the segregation behaviour at the (013j grain boundary while N was practically not detected there. In the denitridid and decarburized alloy, phosphorus enrichment and silicon depletion were again observed at the grain boundary but pronounced sulphur segregation, accompanied by weaker phosphorus and silicon enrichments, dominates at the free surface (LFJCEKetul. [1993]). From a phenomenological point of view these qualitative differences may be considered as a geaerdized anisotropy of integaciul segregation. 4.6. The kinetics of segregation
Most models of the kinetics of segregation follow MCLEAN'S[1957] approach. Solute atoms are assumed to segregate to a grain boundary from two infinite half crystals of uniform solute content or to a surface from one infinite half crystal. Diffusion in the crystals is described by Fick's laws and the ratio of the solute in the grain boundary to that in the adjacent atom layer of the bulk is given by the enrichment ratio, B, assumed to be constant and independent of the actual values of X, and X,. The kinetics of the segregation are thus described by:
where F = 4 for grain boundaries and 1 for the free surface, X,(t) is the boundary content at time t, D is the solute bulk difhsivity, and f is related to the atom sizes b and a of the solute and matrix, respectively, by f=a3b-'. For short times, eq. (49) approximates to X,(t)-X,(O) 2b2 FDt X,(") - X,(O) =
6-47] .
Equations (49) and (50) are, in fact, limiting extremes of a general problem. In practice
p is only constant for dilute systems with low segregation levels. As segregation proceeds fi generally falls as a result of saturation in X,. If /3 starts high and falls rapidly as the segregation saturates, eq. (50) is valid up to saturation (LEAand SEAH[1977]). A detailed analysis for the saturation occurring in the Langmuir-McLean adsorption theory [1979]. Their analysis, reinterpreted has been presented by ROWLANDSand WOODRUTT in fig. 25, shows how the time dependence of the segregation changes from eq. (49)to eq. (50) as the final equilibrium segregation level X,(m) approaches the saturation level X: . Calculations for particular systems in relation to temper brittleness are given in 5 6.3. Measurements of X,(t) at surfaces by A E S at different temperatures allow the determinand ation of the bulk diffusion constant through eq. (50), as shown by HOFMANN ERLEWEIN [1976] for Sn in Cu. The more complex problem of the kinetics of segregation in ternary systems has also been studied by T Y ~ [1978]. N Tyson uses McLean's approach but incorporates Gutt-
Ch. 13, $ 4
1243
Interjacial and surface microchemistry
Time t ~ I C ~ ~ ~ * I ~ D ) I ~ - ( X ~ I ~ ) I ~ ~ ) ~ ~ )
Fig. 25. The kinetics of segregation in binary systems for varying degrees of saturation in the final equilibrium level (after ROWLANDSand WOODRUFF[1979]).
mann's theory to define the grain-boundary concentration. For a ternary system with a dilute impurity and a non-dilute alloying element he finds that eq. (50) is often a good description if the final segregation level is near saturation and similarly eq. (49) is valid if the final level is low. However, for certain critical cases, in which the two interacting species have similar values of DP-*, the segregation curves exhibit an intermediate plateau. The effect of site competition and repulsive interaction between solutes on segregation kinetics in multicomponent alloys was analyzed by M I L I T ~and R WIETING[1986]. This effect results in a replacement of a quickly segregating element by another, slowly diffusing but more surface active element. As a result, a non-monotonic course of segregation kinetics appears. Under strong repulsive interaction, the replacement can reach such an extent that an interface miscibility gap occurs. The time necessary for nucleation of new phases is generally only very short and does not delay the kinetics of segregation (MLITZERand WIE~NG[1987]). The linear relationship between X,,and t"* (eq. (50)) was derived assuming the bulk diffusion to be a controlling mechanism of segregation kinetics. In some cases, however, deviations from this behaviour have been observed. The diffusion can be accelerated e.g. via pipe diffusion. Then, the kinetics of interfacial segregation is composed of several contributions, e.g., the bulk diffusion term, X,(t), the pipe diffusion term, XJt), and the term reflecting the enrichment of dislocations, X&),
References: p . 1284.
1244
E.D.Hondros et ai.
Ch. 13, 54
In this way, IV~ILITZER et al. [1992] explained a complex segregation behaviour of sulphur at the (001) surface of an Fe-Si alloy. The process of solute segregation generates composition changes in the vicinity of the interface, The generally encountered situation is illustrated in fig. 26, where this in-depth distribution was calculated for surface segregation of Sn in Cu for different annealing times. The course of the depicted curves was experimentally confirmed by sputter depth [1978]). profiling (HOFMANNand ERLEWEIN The kinetics of desegregation have, in the past, been considered as much faster than the kinetics of segregation. This should not be so. The relationships are as given in eqs. [1990]). (49) and (50), with X,(O) and X,(=) transposed (DUPLESSIS
4.7. Non-equilibriumsegregation The above segregation process leads to a well defined equilibrium value of the segregation as time approaches infinity if the temperature and constituents are kept constant. This equilibrium is approached at a rate defined by the relevant diffusion processes as discussed in the previous section. In non-equilibrium segregation phenomena, the enrichment disappears as time approaches infinity but it may be very significant at intermediate times. These nonequilibrium segregations arise through a number of discrete processes but all depend ultimately on the fact that changes in the materials environment cause the chemical potential of one the constituents to vary from place to place at a given concentration. This constituent then redistributes at a rate governed by diffusion theory, concentrating in certain regions in order to attain a constant chemical potential everywhere. In the meantime the spatial dependence of the chemical potential generally weakens also at a rate governed by diffusion theory, so that the constituent, having segregated, disperses again to become homogeneous throughout the material. This type of segregation is thus transient and is termed non-equilibrium but, in fact, includes many different separate mechanisms. (See also ch. 7, 9 8.3.2). Another important point is that each of these mechanisms leads to segregation in a zone of considerably greater width around the appropriate interface than occurred with the equilibrium mechanism discussed earlier. Depending on the mechanism, this zone may range from nm to pm. It should now be recognized that, having established a zone of enrichment, equilibrium partitioning between this zone and the interface will inevitably occur so that the real situation will incorporate contributions from both equilibrium and non-equilibrium segregations. Below, the non-equilibrium aspects are discussed but the above caveat should be borne in mind. 4.7.1. Solute pile-up at growing precipitates An early form of non-equilibrium segregation concerns the solute pile-up at the face of a growing carbide in steels. Owing to their low solubility in carbides, certain solutes such as Sn and Sb will be rejected as the carbide grows and will pile up on the matrix side of the matridprecipitate interface. Immediately this happens, the excess atoms start to diffuse into the matrix. Thus, if the carbide grew a constant rate, the solute pile-up would grow asymptotically to a steady level. However, the carbide precipitate growth
Ch. 13, 8 4
Interfacial and surface microchemistry
1245
Fig. 26. Calculated time dependence of concentration-depthdistribution near the surface for Sn (X,= 5 x 10”) at a Cu (111) surface (after HOFMANN and ERLEWEIN [1978]).
rapidly slows down and stops so that the pile-up disperses eventually to zero. In a practical situation, with the carbides growing as the material cools, the pile-up may be “frozen in”. Precise calculations are difficult since the extent of the pile-up has been shown, by RELLICKand MCMAHON[1974], to be critically dependent on the cooling rate, the carbide growth rate and the solute partitioning between the carbide and the matrix. With large, rapidly grown carbides, enrichments as high as a factor of 100 may be encountered although much weaker effects are likely to be observed in practice.
4.73. Quench-induced segregation Another form of non-equilibrium segregation which also occurs at interfaces has been analyzed by WILLIAMSet al. [1976] on the basis of a model proposed by AUST and WESTBROOK[1965]. In this model, cooling of the material causes the equilibrium vacancy concentration to fall so that vacancies flow to interfaces where they may be readily annihilated. Because solute atoms in the crystal lattice have an associated strain energy, as discussed in 53.2.1, which may be relieved by pairing the solute atom with a vacancy, the vacancy flow has an associated solute flow. In their calculations for boron segregation during the quenching of 316 steel (17% Cr, 11% Ni stainless steel) from solution temperatures in the range 1173-1633 K, WILLIAMS et al. write the vacancy concentration as: v = 4 exp(-Ef/kT)
(52)
where k(=4) contains geometric and entropy terms and Ep the vacancy formation energy is 1.4 eV (1 eV per atom = 96.5 kJ/mol). The concentration of boron-vacancy comReferences:p . 1284.
1246
E.D. Hondros et al.
Ch. 13, 4 4
plexes, vB, obeys a similar relation but also depends on v and the boron bulk concentration, B,
where kc(= 12) contains geometric and entropy terms and EB, the vacancy-solute binding energy, is 0.5 eV. In this system a significant population of complexes occurs at 1623 K but not at 1173 K and, since the diffusivity of the boron-vacancy complexes is higher at all temperatures than that of the vacancies or the boron atoms, significant boron segregation from the bulk level of 90 at ppm could be observed by autoradiography, as shown in fig. 27, for the higher temperature quenches at 50 Ws. These results were confirmed later by KARLSON et al. [1988], who found in a similar material (17% Cr-13% Ni, 316L stainless steel) that the amount of segregated boron increases with increasing starting temperature and is larger for lower cooling rates (-10 Ws).It was also detected after the highest cooling rate 530 U s . This was also confirmed by computer simulations (KARLSSON [19881). Non-equilibrium segregation is observed after cooling from temperatures above 1350 K whereas equilibrium segregation dominated after rapid cooling from 1073 K. Effects of non-equilibrium segregation seems to be independent for the majority of grain boundaries but no segregation was detected at coherent twin boundaries (KARLSSON and NORDBN[19881). To differentiate between equilibrium and non-equilibrium segregation at grain boundaries of a Ni-In alloy, MUSCHIK et al. [1989] found AES depth profiles of the shape for intermediate time in fig. 26 to be indicative for non-equilibrium segregation above 1OOO-1100K quenching temperatures.
Fig. 27. Optical autoradiograph of austenitic 316 stainless steel containing 90 ppm boron solution heated at 1623 K for 1/2 hour, furnace-cooled to the temperature Ti in the range 1173 to 1623 K, held for 1/2 hour and cooled at 50 Ws to mom temperature. Strong boron non-equilibrium segregation is visible for the higher et al. [ 19761). temperature quenches (after WILLIAMS
Ch. 13, 5 4
Interfacial and surface microchemistry
1247
Detailed calculations for Cr segregation in 2;CrlMo steel by DOIGand FLEWITT [1981] show that, in that case, narrow distributions for Cr enrichment are obtained around the grain boundary for quench rates in excess of 2000 K/s with significant segregation occumng for quenches from temperatures in the range 1300-1400 K. Experiment and theory agree in indicating an excess Cr level of some 2% over a zone of 5 nm on either side of the boundary. The limiting conditions for the quench-induced segregation are presented in fig. 28 as a function of the initial temperature before quench and the quench rate. At quench rates above line 1 there is insufficient time for the solute-vacancy pairs to diffuse an adequate distance, whereas at quench rates below line 2 there is sufficient time for the solute to diffuse away from the segregation region and so outside these lines little segregation will be observed. Again, at initial temperatures below line 3 there is neither sufficient kinetics or sufficient vacancy population for the segregation to be significant. No high temperature limit is appropriate if the quench goes through the temperature range to line 3. Observations of significant quench-induced segregations are reviewed by HARRIESand MARWICK[1980].
Quench rate too lowvacancies or solute homogenise
ench rate tocI highcouple cannot diffuse Vacancy population too low
/ Log quench rate
Fig. 28. Diagram of the limiting conditions for appreciable non-equilibrium segregation as a function of quench rate. and temperature prior to quenching. References: p . 1284.
1248
E.D. Hondms et al.
Ch. 13, 94
4.7.3. Stress-induced segregation A third form of non-equilibrium segregation occurs when the vacancy flux is not produced from a change in temperature but from an applied stress at constant temperature. The vacancy flux is directed by the stress to relieve the strain and consequently it in turn directs excess solute atoms, by the vacancy-solute coupling discussed above, to grain boundaries parallel to the stress axis. The source of the solute atoms is thought to be the other grain boundaries rather than the lattice, since the activation energy for grainboundary diffusion is much lower than that for lattice diffusion. In steels the problem is complicated by the stress-assisted dissolution and reprecipitation of grain-boundary carbides which produces the first non-equilibrium segregation mentioned above due to pile-up. Changes in grain-boundary phosphorus segregation as a function of tensile and and compression stresses in 2iCr steels have been observed and interpreted by SHINODA NAKAMURA [1981] although, at present, the interpretative model is not fully complete.
4.7.4. Segregation at moving grain boundaries It has been known for a long time that an atmosphere of solute atoms exerts a drag on a moving grain boundary (ch. 28, $3.4.1). However, up to now only very little is know on the reverse effect, Le. on the non-equilibrium segregation of solutes at a moving grain boundary during recrystallization. This is despite the fact that the solute enrichment effects can be much stronger than those of equilibrium segregation. For example, a nonequilibrium segregation of boron was observed on moving grain boundaries in an Fe-3 mass%Si alloy V b =1.6) although no equilibrium segregation was detected at static boundaries under the same conditions (ZHANGe l al. [1994a]). Boron segregation at moving grain boundaries is intensified with continuing recrystallization and persists with the new boundaries moving forward. With decreased velocity of moving boundaries at the end of the recrystallization process boron enrichment at grain boundaries decreases and eventually disappears. Obviously, a property responsible for segregation changes due to grain-boundary movement is the width of the boundary. The non-equilibrium segregation is facilitated by increasing width of a moving interface. The broadening of the boundary results from its interaction with lattice dislocations disappearing there during recrystallization. The width S of the moving boundary can be expressed by:
S = do + Vrd,Ap
(54)
where do is the width of the static boundary, V is the boundary velocity, T is the relaxation time for dislocation annihilation by moving grain boundary, d, is the increment of average width of unit area of the boundary during 7,and Ap is the difference of the dislocation density in deformed and new grains. The enrichment ratio at moving boundaries is then
where W is the width of grain-boundary area of increased solute concentration, and U, is the potential of solute atoms near the grain boundary. According to eqs. (54) and ( 5 3 ,
Ch. 13, $ 5
Intelfaciai and surface microchemistry
12.49
one obtains Pb= 1.7 for boron segregation at moving grain boundaries in Fe-3mass. %Si et aZ. [1994b1). during recrystallization at 1273 K after 20% deformation (ZHANG 4.7.5. Radiation-induced segregation This topic is treated in chapter 7, Q 8.3.2 and chapter 18, 8 4.7.
5. Segregation-related physicochemical properties The influence of the interfacial microchemical state on physical metallurgical properties, such as creep-rupture life and temper embrittlement, operates through associated effects on certain basic physicochemical properties which in turn determine the metallurgical behaviour. The main physicochemical properties which are influenced by segregation processes fall into three categories: interfacial energetics, kinetics and cohesion. These are considered in turn below. Wherever these basic interfacial parameters are found to be involved with metallurgical phenomena, clearly these phenomena are influenced by segregation even though no explicit mention of the fact may occur in this chapter or, indeed, elsewhere.
5.1. Interfacial energetics The close association between interfacial free energies and interfacial composition is discussed in 8 2 in terms of the Gibbs Adsorption Theorem. We note that the thrust of much of the effort on the determination of interfacial free energies was originally intended to derive the values for interfacial composition. Both this thermodynamic approach and the new direct spectroscopy techniques can now be used interchangeably by way of parameters such as the interfacial activities or enrichment ratio, through which patterns of enrichment behaviour for a wide range of systems and for various interfaces may be compared and interrelated. However, there are situations where the treatment of a microstructural or metallurgical problem requires a knowledge of the exact interfacial energy values relevant to the conditions which prevail. Examples of this are the nucleation of precipitates or second phases at grain boundaries, where a determining parameter is the actual interfacial free energy between the boundary plane and the second phase that is being nucleated. Again, in the study of the nucleation and stabilization of cavities formed along grain boundaries during creep, an important quantity is the value of the surface free energy pertaining to the cavity wall. The surface and grain-boundary free energies of most pure metals have been measured experimentally by the techniques discussed earlier and fair assessments have been made for others, on the basis of a knowledge of bulk thermochemical properties with which empirical relationships exist. These have been summarized elsewhere (HONDROS [ 1969, 19781). We note, as expected from eqs. (25) and (27), that ys= 0. 16Hs,, at around 1500 K, where ys and Hsh, the heat of sublimation, are measured per mole (OVERBURY etal. [1975]). At 0 K ysincreases to 0.25 Hsub,as expected (TYSON[1975]). In multicomponent crystalline solids, the main determinants of the interfacial energies are the presence of segregants, the temperature, and the crystalline orientation. The Refeiferences:p. 1284.
1250
E.D. Hondms et al.
Ch. 13, 8 5
evaluation of the interfacial free energy relevant to the modelling of a metallurgical event is difficult because of its sensitivity to the above parameters, in particular, adsorption of solute species. The direct experimental determination of that quantity is technically often very difficult and indirect. For binary dilute alloys, reasonable experimental data exist for several systems on the change in absolute surface free energies with increasing bulk content of the alloying element: these are summarized in fig. 29 (HONDROSand MCLEAN[1974]). The depression in ys up to'the limit of solid solubility is system dependent, however, the estimate usually adopted is about 40%. Furthermore, we note that the data were measured on polycrystalline samples and therefore they refer strictly to the average surface free energies. The anisotropy of surface energy is small for pure metals, but it may be quite significant for alloys in which surface enrichment is involved. In some cases, extremely low quantities of very surface-active elements in the vapour phase may produce significant effects on the interfacial free energies. Figure 30 shows as an example, the sensitivity of the surface free energy of Fe-3mass%Si at two temperatures to the potential of oxygen in the vapour phase. In order to obtain and hold such low values of oxygen potential, it is required to work with complex gas phase mixtures involving H,O in H,. Furthermore, these experiments were conducted on foils showing a (100) texture so that the values refer substantially to this crystal plane. (See also ch. 28, 54.4.2). In a subsequent study aimed at providing fundamental data on the surface-energyet al. [1973] showed that the driven secondary recrystallization in sheet alloy, MILLS surface energies of crystals exposing the principal poles differed in a complex but significant manner as a function of temperature and oxygen adsorption, so that by manipulating these variables, each of the textures (lll), (110) or (100) could be controllably grown. For the effect of temperature on the interfacial free energy, considering a single component isotropic solid, eq. (68), ch. 5 reduces to:
--dy
= S",
dT
where 5" is the interfacial entropy. Thus for pure elements, the temperature dependence of the interfacial free energy is negative. Theoretically,this is computed to be about -0.5 mJ/m2K which is close to the value conventionally adopted from experimental data, -0.33 mJ/m2K (HONDROS[19691). In contrast to the above, fig. 30 clearly shows that in a system of more than one component the temperature dependence of the surface free energy can be positive. This is consistent with the thermodynamics of surfaces. By differentiating eq. (68), ch. 5 with respect to T, a complex expression may be obtained containing two terms which describe the temperature dependence of the interfacial free energy in a binary system. The first term, relating to entropy, is always negative, while the second term relates to adsorption and is positive. (The latter term refers to the effect of "boiling off" of a segregate with increasing temperature, where it is expected that the amount of adsorption would decrease with increasing temperature, i.e., dr/dTis negative.) This shows that in a system of more
Solute concentration, atom *le Fig. 29. Dependence of the surface free energy on bulk composition for a number of dilute systems: the host element is shown first in each pair.
1252
Ch. 13, 45
E.D. Hondms et aL
!OOO
800 N
E
5
7
1600
Y Q c
0 Q
1400
2 L
J
cn
1200
I
-13
I I -12 -11 log pOZ(Pa)
I -10
-9
Fig. 30. Dependence of the surface free energy of the (100) plane in Fe-3%Si on oxygen potential and temperature (after HONDROS and STUART [1968]).
than one element, the interfacial energy may increase or decrease with increasing temperature¶ depending on the relative strengths of the two terms of entropy and adsorption. The above discussion indicates the complexities involved in assessing the precise interfacial free energy value for a given physicochemical situation. There are very few systems for which this quantity has been determined at the level of ngour shown in fig. 30. For more complex interfaces, such as those between a second phase and the matrix, the experimental problems are even more severe. As a general guideline for binary systems, a knowledge of the amount of surface excess of an identified species from a know bulk concentration provides an estimate of the associated change in interfacial energy, through the application of the Gibbs Adsorption Theorem in a reversed manner. Where the form of the adsorption is known, an approach that has been used for both free surfaces and grain boundaries is to integrate the isotherm. Thus, if the Langmuir isotherm holds, this may be combined with the Gibbs Adsorption Theorem, which for a dilute system yields the equation for grain-boundary free energy:
Ch. 13, $ 5
1253
Interfacial and surfme microchemistry
1100'
1000'
800'
900"
600"C
700"
-4 -5 c
)- 6 E Y
m-7
P
0 01
2
-8 - \
---+--
-'-
-9
-10
7
8
9
10
11
io4/f( K) Fig. 31. Surface self-diffusion in Cu, Au and Ag (solid curves) and the effects of adsorption of a number of species (broken curves) (after DELAMARE and RHEAI) [1971]).
where the quantities in J? are defined as above and yi is the grain-boundary energy of the pure metal. This approach was used by PETcH [19561 in a similar form (but retaining the gas pressure term), in order to determine the effect of adsorbed hydrogen in lowering the surface energy of iron in a theory of hydrogen-induced fracture, and also by SEAH [198Oc], in a study of impurity-aided creep cavitation embrittlement. The enrichment quantity required for eq. (57) may be obtained directly by spectroscopic measurements or assessed through the enrichment factor Pb and its dependence on the inverse of the solid solubility (see eq. (15) and fig. 14). This yields a change in free energy Ay which in many modelling studies is often the quantity required an absolute value of y for the appropriate conditions is assessed by referring to the standard state value for the energy of the pure component.
rb
References:p . 1284.
1254
E.D. Hondros et al,
Ch. 13, $ 5
5.2. Surf‘ace and grain-boundary kinetics The free surfaces of solids offer rapid diffusion paths for the transport of matter. For example, the grooves that are formed at the intersection of grain boundaries with free surfaces when metals are held at high temperature (commonly referred to as “thermal etching” (see fig. 5)), owe their origin to the interchange of matter by surface diffusion. In general, this mechanism predominates in solid state phenomena where the microstructural features involve dimensions of the order of pm. Thus surface diffusivity in an important mechanism in the early stages of sintering, where the necking of the spherical particles takes place. Considering the effects of surface-adsorbed elements, a good body of evidence affirms that adsorption will generally accelerate the surface diffusivity, while in some specific cases suppression will occur (GJOSTEIN[1966]). The interesting data in fig. 31 for example, compiled by DELAMARE and RHEAD [1971], show the effects of a number of impurities, either arriving from the vapour phase or segregating from the bulk, on the surface diffusivities of copper, gold and silver. The Arrhenius plots for the pure metals are shown as solid curves, whereas each of the broken curves refers to the metal substrate containing an adsorbed layer of impurity under controlled conditions. The second element of each pair denotes the adsorbed species. Clearly, in some cases such adsorbed layers induce spectacular increases in the self-diffusivity of the substrate metal and at the highest temperatures, these values can approach a factor lo4 above that of the clean substrate. As yet there is no agreement on the mechanism for this remarkable effect: one theory proposes that the uppermost atomic layer of the surface consists of a mixture of adsorbate and adsorbent, constituting in effect a two-dimensional compound with liquid-like properties. From this theory, it follows that adsorbates which form compounds with low melting points should lead to this accelerated surface diffusivity. An earlier theory proposed a “skating” mechanism in which the substrate atom is thought to climb on top of the adsorbed layer, where it may migrate freely with long jump distances of the order of 100 atom spacings, a hypothesis which could explain increases in surface diffusivities of the order of lo4. This overlayer model of adsorption diffusion now appears to be superseded by the mixed layer or two-dimensional liquid model, which is seemingly supported by Low-Energy Electron Diffraction (LEED) studies on adsorbed monolayers of lead and bismuth on copper. Whereas the copious studies on surface self-diffusion (ch. 7, 57.4) fall within the context of a fundamental understanding of mechanisms probably relevant to the important process of catalysis, the somewhat fewer studies of the effect of segregation on grain-boundary self-diffusivity have a more immediate applicability to physicalmetallurgical phenomena. We now turn to consider the latter. Grain boundaries are paths for rapid self-diffusion and it is commonly understood that the transport of matter along such interfaces should be rate-controlling in a number of important metallurgical processes, in particular in temperature regimes of about 0 . 9 , or below. In general, grain-boundary diffusivity data which refer to the pure elements have been employed in the modeling of various processes in metallurgy. Clearly, the use of such readily available data could introduce serious distortions if transport along grain boundaries
Ch. 13, 0 5
1255
Interfacial and surfme microchemistry
t
-
\A
0
I
I
20
30
I
10
Experiment
Grain boundary Sn content, pmol rn-' Fig. 32. The measured and predicted dependence of the grain-boundary self-diffusion in Fe-Sn alloys at 888 K on the amount of tin segregated at the boundary (after BERNARDINI et nl. [1982]).
were to be sensitive to the microchemical aspects of the interface. Below we examine the theory and evidence for the role of solute segregation on grain-boundary diffusivities. A pure grain boundary is characterized by a high free energy, being a structural zone with a low volume density of matter compared with the bulk lattice. From such simplistic considerations, we might expect that this should lower the activation energy barrier for substitutional self diffusion, so that atom transport is easier along these paths than in the bulk crystalline lattice. This concept has been expressed in a semi-quantitativemanner by G ~ B and S HARRIS119693 who related the activation energy for grain-boundary tmnsport Qb to that for volume diffusion Q,by the expression References:p. 1284.
1256
E.D. Hondms et al.
Qb
=
a - Nff'Yba , 2
Ch. 13, 95 (58)
where a2 is the cross-sectional area of an atom*, N is Avogadro's number and a' a structure factor which depends on whether diffusion is substitutional or interstitial. This predicts sensibly that increasing 3/b will lower the activation energy for grain-boundary transport and that for a highly coherent interface such as a twin boundary, with a low free energy, the activation energy for diffusion would be little different from that of the lattice. Consider the presence of a solute in the base solvent with accompanying high grainboundary enrichment of solute. If the segregation-associated change in grain-boundary energy is Ayb, it follows that the activation energy for grain-boundary diffusivity in the is related to that for the pristine boundary by: presence of segregation, QFg, Qrg = Qb + N~x'u~hy,.
(59)
Hence, according to this general model, segregation increases the activation energy for grain-boundary diffusion compared with the pure state. In effect, the segregation permits the sites of high compression and tension to be relaxed, reducing the net free energy of the boundary and increasing the volume density of the boundary by the occupation of vacant sites. Thus, from the diffusional point of view, the material in the boundary zone approaches the character of the material in the bulk lattice. The above approach allows estimates of the magnitude of the effect. Thus, when phosphorus segregates to the grain boundaries of pure iron, the grain-boundary energy is reduced by about 400 d / m 2 at its maximum (HONDROS[1965]); substituting in the above equation and including a value of 2 for the structure factor appropriate to vacancy diffusion, the activation energy for grain-boundary diffusion in pure iron is raised from 175 kJ/mol to 200 kJ/mol. The above account is probably satisfactory for rough estimates of expected effects, and supports the view that the consequence of segregation is the elimination of the boundary sites having a high degree of disorganization, thereby decreasing grainboundary transport with respect to that in a pure boundary. The subject has been developed analytically by BERNARDINIet al. [1982] who derived a more rigorous expression linking the change in diffusivity to the amount adsorbed at the grain boundary. This treatment takes as its point of departure of the theory of BORISOVet al. [1964] who derived the basic relationship between grain-boundary energy and grain-boundary diffusivity, given by the expression
where Db is the grain-boundary diffusivity, 0,is the volume bulk diffusivity, a,, is the size of the solvent atom, S is the width of the grain boundary, and 6 =ma,,.This equation
* In this chapter the atom size, a, is derived from pNa3 = M,where p is the bulk density and M the atomic weight of the solid.
Ch. 13, 9 5
Inteflacial and surface microchemistry
1257
is transformed by expressing Y b in terms of r b , the amount of segregation. By combining eqs. (58) and (60), we effectively extend the Borisov treatment to a binary system in which grain-boundary segregation occurs. This yields:
where D; is the grain-boundary selfdiffusion coefficient in the presence of segregation,0,: is the coefficient due to the presence of the solute in the bulk, Db and D, are the corresponding values in the pure solvent, a,, is the atomic size of the solute in the particular solvent, and rn is the quantity, retained from the Borisov equation (60), which relates to the number of effective atom layers in the boundary. The above equation has now been confirmed experimentally in iron containing tin, by directly measuring the diffusivity constant using radio-tracer techniques and also by determining the amount of grain-boundary segregationusing Auger electron spectroscopyon exposed grain-boundary surfaces. Figure 32 shows how the measured grain-boundary self-diffusivity of iron decreases with increasing levels of tin present at the boundary, at 888 K. Similar data were obtained for a number of temperatures and for both solvent and solute diffusivities along the boundary. In the same figure, the straight line indicates the effect predicted by the application of eq. (61), the treatment necessarily restricting the prediction to the dilute approximation. Equation (61) may be simplified further by expressing Dv* as a linear function of the bulk concentration of solute, X,, following a number of theoretical and experimental investigations on systems involving substitutional self-diffusion, as discussed in ch. 7, $4.1:
D: = Dv(l + bvX,), where b, is a temperature dependent coefficient representing the variation of Obwith the bulk solute concentration X,.Thus eq. (61) can now be expressed as
where a is a concentration ratio. This defines the quantity actually measured in boundary diffusion measurements in the presence of segregation as CUD,&and not the standard Obi? where 6 is the boundary width. To express eq. (63) in a more accessible form, we introduce the grain-boundary enrichment factor defined earlier, Pb (eq. (15)), which is closely related to a! by:
Therefore, the overall effect may be expressed by:
[ (
z;)xc].
D,*=D, 1 + bv--
(65)
References: p . 1284.
1258
E.D.Hondros et al.
Ch. 13, 5 5
In summary, this equation expresses the effect of segregation (contained in the enrichment factor &) on grain-boundary diffusivity. This is offset by the effect of the solute on volume self-diffusivity. The equation then predicts that grain-boundary transport of both solvent and solute atoms may be reduced or increased depending on the balance between the opposing effects implicit in &, and b,. Since b, is generally of the order of 10-100 and Pb for highly surface active species is of the order of 1000-10000, in general, the effect of the high enrichment factor will dominate in the presence of strongly segregating species and this will reduce the grain-boundary diffusivity. An enhancement of the solute diffusivities is caused by the increased matter transfer due to accumulation of the solute atoms in the grain boundary. It is only an apparent enhancement due to a being greater than unity and is not due to a real increase in D,. The magnitude of the enhancement depends on the interaction between grain boundary and solute (Urnand GUST[1988]). The ability to predict the effect of segregation on grain-boundary diffusivities is a development which permits the quantitative examination of a number of metallurgical phenomena in which these kinetics are known to be rate-determining: some of these applications will be considered in 5 6.
5.3. Grain-boundarycohesion As was noted in the Introduction, historically the most important manifestation of grain-boundary segregation has been the effects on grain-boundary cohesion as expressed in a number of well known forms of intergranular fragility. This continues to provide much of the focus for work in this field. The central problem is the role of solute atoms on atomic cohesion at the interface. This is now recognized to be a wider issue, extending to aspects of ad-hesion (i.e., cohesion between dissimilar pieces of matter), as in that between a metal and an oxide or another phase such as a precipitate or even a polymer. The theoretical and experimental framework for understanding adhesion in such systems and the role of interfacial microchemical interactions is hardly developed and this section will be restricted to grain-boundary cohesion which is now quite well advanced. However, in connection with adhesional phenomena in metals, HONDROS [198Obl has attempted to systematize the data available for metals adhering to oxides and other phases, and it appears that the classical concept of work ofadhesion, expressed in the formulation of Young and Dupr6 as the algebraic sum of the two free surface energies and the interfacial energy, may be applied to these solid state systems. This would predict that segregation to metaVoxide interfaces, by decreasing the interfacial energy should increase the work of adhesion but scant data are available to test this hypothesis. Because of its technological importance, much of the work on grain-boundary cohesion has been concerned with effects of additions in iron, and it is now well established that the elements Cu, Zn, Si, Ge, Sn, P, As, Sb, Bi, S, Se, Te, and Mn all cause intergranular weakness in iron. A similar set also weaken copper and nickel. We now consider the two questions “why do certain elements weaken grain boundaries?” and “what is the relative embrittling potency of these elements?”. A number of theories have been proposed to account for the effect of segregants on grain-boundary cohesion. All have a common approach but differ in their interpretation
Ch. 13, 8 5
Intei$acial and surface microchemistry
1259
of the way to carry out the numerical calculations. In 1957 M c L m first proposed that in low-temperature intergranular fracture, the total work of fracture is the sum of the ideal work of fracture, y, representing the energy to disrupt the bonds across the interface, and the irreversible plastic work of deformation, y,, which is absorbed in the crystal lattice as dislocations and phonons over a region well away from the boundary. Although y, is much larger than y it is easy to see that if y is increased, the forces at a crack tip must also be increased to propagate the fracture and hence y, will increase. Thus, for a given grain boundary, y and y, are directly related. The precise relation will depend on the orientations of the two grains and the grain boundary to the stress axis as well as on the microstructure. Indeed, analyses by MCMAHONand VITEK[1979] and by JOKL etal. [1980] show that y, is proportional to where n is in the range 2-5. Thus, to answer the above questions we must understand the effects of segregants on the ideal work of fracture of the grain boundary. The ideal work of fracture of a clean grain boundary with energy y: was given by MCLEAN[1957] as:
where y: is the fracture surface energy of the clean material. An analysis of 7: andy: values for many metals by MCLEANand HONDROS(SEAHand LEA[1975]) shows that, typically, y:==-y:/3 so that the ideal work of fracture of clean grain boundaries is approximately 5 / 6 of that of an average plane in the crystal. This does not mean that the grain boundaries will always fail in preference to cleavage, since the orientation of the slip planes and the surface energy of the low-index cleavage planes make cleavage generally more likely than intergranular failure in clean bcc metals (HONDROS and MCLEAN[19761). More important than the choice of brittle fracture modes is the balance between brittle and ductile failure. The fracture behaviour of a 0.2% carbon steel, as a function of the test temperature, is shown schematically in fig. 33. At high temperatures, where dislocations move easily, the yield stress is low and the steel fails in a ductile manner. As the test temperature is reduced, the dislocation movement, governed by a term of the form exp(-Q/RT), becomes more difficult and the material eventually fails in a brittle manner. The temperature at which this occurs is the ductile-brittle transition temperature (DBTT). If the grain boundaries contain the embrittling segregants, the boundary cohesion may be sufficiently lowered so that the transition temperature is raised and the otherwise ductile material becomes brittle. An analysis of the ideal work of fracture of grain boundaries with segregants, y, has been made by HIRTHand RICE [1980] using thermodynamic arguments. They show that the reduction in the ideal work of fracture, where there is no redistribution of the segregant species, is given by:
where pb,Jr)are the chemical potentials of the solute species in equilibrium with a level segregation I? rnoVm2 at the grain boundary and the free surface, respectively. This relation may be re-expressed: References: p. 1284.
1260
Ch. 13, 5 5
E.D. Ho~droset al. I
I
1200
1000
I
I
I
I
\
Britt le strength
\
Lowered boundary cohesion \
---
-----\
N
‘E z
r
2
800
I \
2
L
v,
-Yield
stress
600 Raised transition temperat ure
Transition temperature
L/
\ 400 I
- 250
I
II
- 200
I
Temperature,
I
-150
I
- 100
O C
Fig. 33. The effect of t e m p h u e on the tensile properties of a 0.2%carbon steel showing, schematically, the lowering of grain-boundarycohesion and the resultant increase in the ductile-brittle transition temperature (after SFAH [ 19761).
where r b is the segregation level at the grain boundary, are the maximum segregation levels at the boundary and surface, respectively, and the terms in AG refer to the binding energies of segregants, which, it should be noted, are negative quantities (RICE and WANG [1989]). In the dilute limit, the last right-hand term is equal to -RT ln2 and usually, it can be neglected (RTIn 2 = 5.76 kJ/mol at 1000 K). Thus we see that the ideal work of fracture falls linearly at first with the level of segregation, and that elements which segregate more strongly to free surfaces than to grain boundaries reduce the ideal work of fracture and vice versa. By measuring AH, AHb and rb under identical conditions in Ni,Al bicrystals with artificial cavities at the boundary, OTTERBEIN et al.
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[1993] applied eq. (68) to obtain the decrease of the grain-boundary fracture strength by segregation of sulphur. An alternative approach which gives the same result in the quasi-chemical pair bonding approximation has been given by SEAH[198Ob]. In this approach the actual bond energies across the boundary before and after fracture are calculated in the manner described in 0 4.2.2. In this way:
where the terms are as described in 0 4.2.2. and, in addition 2, corresponds to Z,across the grain boundary (approximatelyiZJ, is the value of r, at one monolayer, a, is the atomic diameter of the segregation atom A and Hisub*is the sublimation enthalpy of A atoms in J/m2. Calculations may be easily carried out for eq. (69) in the regular solution approximation, but only small errors are involved if R is ignored. In this case it is clear that the lower the value of H:b* for the solute, the more will a segregant embrittle. Figure 34 shows a compilation of flub*values for all elements. It is clear that the well recognized segregants, Sb, Sn, S, P, Mn, Si and Cu all with lisub*values below that of iron shown by the dashed horizontal line, will all embrittle iron, whereas Mo and C will improve the cohesion; this is confirmed experimentally (SEAH[1981,1990]). Detailed studies of the fracture properties of Ni,Al (LIU et al. [1985]) show that B has a strong effect inhibiting intergranular failure. The results of fig. 34 show that, if B segregates to grain boundaries, this is exactly what should occur. WHITEet aZ. [I9841 show that B does segregate to the boundaries and that they are strengthened, The theory of H I R and ~ RICE [1980] predicts the same result if AGs has a smaller magnitude than AGb- This too is shown by WHITEet d ' s [1984] data where no surface segregation is observed. The effect of individual segregants on boundary cohesion can also be considered from the viewpoint of atomic bonds at an interface. Presence of a foreign element at a grain boundary leads to a redistribution of electron charge density. In the case of electronegative sulphur or phosphorus atoms the charge is drawn from the surrounding metal atoms to the impurity atom and consequently, the adjacent metal-metal bonds are weakened (BRIANTand MESSMER[1982]). The bond parallel to the boundary is stronger than that perpendicular to the interface. This results in a reduction of cohesion across the boundary and thus, in grain-boundary embrittlement. In the case of the less electronegative elements such as boron and carbon, relatively strong covalent-like bonds exist with metal in direction perpendicular to the boundary whereas much weaker bonding occurs with metal atoms in parallel direction thus enhancing grain-boundary cohesion (WU et al. [1992]). As will be shown in the next section, it is not sufficient to consider solely the effect of segregants on the ideal work of fracture, as described above. At a crack tip it is the choice between dislocation emission and bond breaking that decides whether ductility or brittle fracture occurs. It can be shown that embrittling segregants increase the ease of dislocation emission at the crack tip, so offsetting the total embrittling effect discussed
ri
References: p . 1284.
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above. The correction is small but allows the correct prediction of the absolute embrit[1983]). tlement of cooper by bismuth (SEAHand HONDROS
30
l
I
I
I
I
I
Ideal solution
i
OC
Increasing grain boundary cohesion
25 OB
\\
0 Re
ON
20 MOO N
oNb
ORh
E -r 7
4 15 n 3 VI
I
0
1c
0
OU
cu
OPd .si 0Th
OAu
Increasing grain boundary embr ittlement
E
'Hg I
0.2
I
I
oBa
NaO I
,
I
0.4
0.3
OK
ORb I
cs 0
I
5
a,n m Fig. 34. The general embrittlement4uctility plot for matrix and segregant elements in the ideal solution approximation. Any element may be chosen as matrix and then those elements with higher values of €T@ increase the grain-boundary cohesion and vice versa. As an example iron may be taken as the matrix material (after SEAH[198Ob]).
Ch. 13, $ 6
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4. Metallurgical phenomena aflected by segregation Interfacial segregation influences metallurgical phenomena through its impingement on several basic physicochemical properties, involving the energetics, kinetics or the cohesion of the interfaces, which were treated in the previous section. This is illustrated schematically in fig. 35. For example, grain-boundary segregation will in general retard the boundary diffusivity and this in turn will affect the rate of diffusion creep or the rate of grain-boundary cavity growth involved in creep embrittlement. Again, because it retards grain-boundary ionic mobility in polycrystalline oxides, this should also inhibit the rate of surface oxide growth. In some instances, the particular metallurgical property may be affected through changes in more than one of the basic physicochemical properties considered for example, the rate oxidation of nickel alloys containing aluminium may be affected through the kinetics of ion transport across or through the boundaries, and in addition the integrity of the oxide layer is enhanced by improved interfacial adhesion resulting from the presence of segregated species at the mewoxide interface. As fig. 35 clearly indicates, numerous metallurgical and microstructural phenomena in alloys are influenced by interfacial microchemistry: in some cases this may be a determining factor while in others it may be simply one of a number of contributory factors. In the space of this chapter, we cannot consider all of these phenomena in detail and so some key references are cited in the caption of fig. 35. Below we consider for each of the important basic physicochemical properties treated in 0 5 (i.e., the first three columns in fig. 35) a significant example of its impact in physical metallurgy, following which we mention briefly a number of other metallurgical phenomena for which evidence is accumulating of effects related to microchemistry,
5.1. Surface free energy change: role in creep cavitation In microstructural terms, the creep-rupture life of alloys serving at high temperature under stress may be controlled by the nucleation, growth and coalescence of microcavities which develop at grain boundaries, in general those normal to the stress axis. A number of contending micromechanisms have been proposed for the nucleation of these cavities, and while there is very strong evidence that in certain engineering alloys these are initiated at sites where slip planes intersect second phase particles such as carbides or manganese sulphide inclusions, cavities have nevertheless been observed in quite pure metals. Many observations in the literature strongly affirm that impurity adsorption at both the grain boundary and the surface of the cavity wall will affect creep-rupture properties. Depending on the temperature and stress regime, grain-boundary segregation may affect the process of cavity growth through its influence on the grain-boundary diffusivity, but in addition, because segregation to the inner surface of the cavity reduces its surface energy, it will also affect the nucleation propensity. Among the many examples, we cite the work of TIPLERand MCLEAN[1970] who showed metallographically and by creep measurements that small quantities of antimony in cooper increase the amount of grain-boundary cavitation and also lower the creep [19641 showed rupture life compared with that of the pure material. KRAAIand FLOREEN References:p. 1284.
1264
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E.D. Hondms et al.
I
G r a i n Boundary Mobilily
I
Fig. 35. Some metallurgical phenomena affected by physicochemical parameters influenced by surface- and grain-boundary equilibrium segregation. Key references for the interested reader are: (1) BATTEet al. [1980]; MCMAHON[1992]; (2) TIPLER [1980]; GEORGEand POPE [1989]; (3) MILLSeral. [1973]; (4) NICOLET [1978]; HAYES and GRIEVESON [1975]; (5) WHII-I-LEand STRINGER [1980]; (6) HONDROSand HENDERSON [1983]; MATS~KO et al. [1991]; (7) BRIANT[1988b]; MCMAHON[1991]; (8) NICHOLAS and OLD [1979]; LYNCH [1989]; (9) LATANISION and OPPERHAUSER [1974]; LYNCH[1988]; (10) S ~ etal.H [1981]; SUNDARSHAN and LOUTHAN[1987]; (11) KUPPERetal. [1981]; WEIand GRABKE [1986]; (12) L w and HONDROS[1982]; BRUEMMER [1989]; (13) NANNIetal. [1976]; GERMANand D'ANGELO[1984].
that the ductility of nickel is reduced by a factor of ten by the presence of sulphur of up to 20 ppm in the bulk, while THOMAS and GIBBONS[1980], and OSGERBY and GIBBONS [ 19921 demonstrated the remarkable effects of trace impurities on the creep performance of nickel-base superalloys. Figure 36 shows the substantial reduction of creep rupture ductility in these alloys caused by ppm levels of lead and tellurium. At the maximum impurity level shown in this figure, the deleterious effect of each impurity was associated in every case with a much higher level of grain-boundary cavitation, of the order of 5-10 times greater than that observed in the pure alloy. In Nimonic 105 containing 34 ppm of lead, the amount of grain-boundary damage as measured by the number of cavities per mm2 was ten times greater than in the alloy with the base composition. Similar effects were observed under presence of bismuth. In addition to the considerable metallurgical evidence in which impaired creep life is associated with the presence of impurities, which presents circumstantial evidence for an
Ch. 13, $ 6
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10 20 Impurity content, w t ppm
30
Fig. 36. Reduction in creep rupture ductility in nickel-base superalloy induced by trace levels of impurities.
effect through the interfacial microchemistry, there are important observations using Auger Electron Spectroscopy on the creep grain-boundary fracture surface which show a considerable accumulation of the suspected contaminant. An elegant example of selective segregation through the application of the high resolution (0.2 pm) Auger spectroscopy in shown in fig. 37 (FRANZONI etal. [1981]) for a commercial 2iCrlMo steel, creep tested at 873 K and subsequently fractured at liquid nitrogen temperature for examination. The figure shows the high magnification scanning electron image of a grain-boundary area and the corresponding Auger spectroscopy maps for the elements Sn and Sb, indicating a simultaneous segregation of these elements only on the cavity surfaces. The non-cavitated areas of the boundary showed only P which was in turn significantly lower within the cavity. In considering the role of interfacial microchemistry on cavitation, we assume that the cavities have been successfully nucleated while the precise mechanism for nucleation will not concern us. We consider an applied tensile stress, (T, which performs an amount of work $r?a in generating a spherical cavity of radius r. The associated surface energy of the cavity is 4&y at it is simple to show that as a cavity grows from a very small size, the net energy that has to be supplied first increases and then, after reaching a balance point, decreases, at which critical point the radius is given by r,, where At this value, the competition between the surface energy which tends to sinter the pore and the applied stress is in balance. When the pore radius reaches a value greater than r,, energy will be saved as it grows to a microscopical cavity. If the surface segregation of the active species is sufficiently rapid, a pore nucleus with a radius smaller than r, References: p. 1284.
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Fig. 37. SEM image of a creep cavitated grain-boundary area in 2;CrlMo steel and the corresponding AES et al. [1981]). maps for Sn and Sb showing segregation on cavity surfaces (after FRANZONI
may survive according to the above equation, because of the reduction in ys,typically by a factor of two. The net effect of this impurity segregation is that a greater density of nuclei per unit area of grain boundary will achieve stability and hence they will be able to grow: the experimental evidence cited above in connection with the incidence of segregation to cavity walls strongly affirms this view. The detailed mechanism of surface-energy induced cavitation damage has been analyzed by MCLEAN[1981], specifically for the alloy Nimonic 80A. The steep function of the curve reproduced in fig. 38 shows the quantity N,the number of grain-boundary cavities in a cubic millimetre of material per 1% creep strain, as a function of r, the cavity radius. For the pure alloy, substituting for ys at a stress of 150 MPa, the critical value of r is 27 nm. At this stress, the cavities which are stable and which will contribute to the creep damage are those which fall to the right of r, under the solid line, shown as
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r , nm
Fig. 38. Theoretical distribution for the density of cavities, N, in Nimonic 80A as a function of the cavity radius, r indicating the areas (hatched) in which cavities will contribute to the creep damage. (Note, there are approx. 100 nun2of grain boundary per mm3 of material here (MCLEAN,private commun.)).
the cross-hatched area ABC in the diagram. McLean integrated the equation representing the curve between rc=27 nm and r+ 00, and found that the number of cavities in this area is about 1 . 6 1013/m3 ~ per 1% strain. In the first instance, segregation to the cavity wall surface affects this process as we have seen by reducing yo thereby reducing the critical stable pore size to r,'. This is marked in the diagram at 15.6 nm, corresponding to a 42% reduction in ys. There is also a second effect to be considered, associated with segregation to the grain boundary itself. The consequent reduction in grain-boundary energy leads to a reduction in intergranular cohesion (eq. (66)),and, irrespective of the precise mechanism of nucleus generation, a lower stress concentration is required to nucleate a cavity in a segregated boundary than in a clean boundary (HONDROS and MCLEAN[19741) so that more cavities will be generated for a unit strain. Thus the curve References: p . 1284.
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lo-' Dislocation glide
10-2
m
In
E!
c In 0, .-
In C
al c
400
600
800
1000
1200
1400
1600
Temperature, K Fig. 39. Theoretical deformation map as a function of tensile stress and temperature where Coble creep dominates and the effect of low levels of sulphur on the boundaries (broken lines), for a grain size of 32 pm, and strain rate of lo4 s.
for nucleation of cavities in the presence of grain-boundary segregation lies at a high value and is shown as the dashed curve in fig. 38. The two effects of segregation are cumulative with respect to cavity stabilization, and all cavities which fall in the hatched area A'B'C to the right of r: now contribute to the creep damage. It is difficult to ascribe accurately the position of the dashed curve representing the role of grainboundary segregation and the ease of decohesion. For an order of magnitude increase in the creep cavity density in the presence of a segregant, the dashed curve corresponds to N values about three times those associated with the solid curve. Thus, while at this stage the model appears somewhat qualitative, the quantities are plausible and consistent with some experimental observations. 6.2. Grain-boundary diffusivity: role in diffusion creep
It is now well established that at lower temperature regimes and under light stresses, the predominant mode of deformation of polycrystals is dominated by diffusional creep, in this case, by mass transport along the grain boundaries. In the course of studying sintering mechanisms in alumina, COBLE[19631 recognized this deformation mode and
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accordingly modified the equation for diffusional creep associated with the names of
NABARRO[1948] and HERRING[1950] in which volume diffusivity dominates: this is analogous to the Hemng equation and is of the form & = 4dsDb
d3kT Here C is the strain rate, LT the tensile stress, SZ the atomic volume, d the grain size, and B, a numerical constant which varies with grain geometry. It is appreciated that both lattice- and grain-boundary diffusion contribute independently to diffusional creep, and the general equation involving contributions from both volume- and grain-boundary diffusion is given by:
. = -[4D,, d
E
dZkT
+ 3%). d
Clearly, because the activation energy for boundary diffusion is less than that for lattice diffusion and because of a greater sensitivity to grain size, the Coble creep mechanism predominates at the lower temperatures and in fine grained material. ASHBY[1972] showed that a generalized deformation mechanism map may be produced by plotting stress as a function of homologous temperature (see ch. 22, § 11). Applying a version of the above constitutive equation, the Ashby map indicating the fields of predominance for various deformation mechanism is reproduced in fig. 39 and shown as solid lines, for nickel. This indicates clearly the important tracts where Coble creep predominates. Such maps have been computed on a basis of diffusional data derived from pure metals. If impurity species in the bulk alter the lattice diffusivity, or more importantly, if by enriching at the grain boundaries they affect the boundary transport, this should be reflected in significant shifts in the boundaries on such maps. We examine here to what extent the Coble creep mechanism is affected by the presence of interfacially active impurities and how this is reflected in the positions of the boundaries on the Ashby map. The predictive eq. (65) developed earlier shows the dependence of grain-boundary diffusivity on grain-boundary segregation as expressed in the enrichment factor Pb.We insert this into eq. (72) above and limit it to the deformation regime where grainboundary transport is rate-determining. From these equations we compare the strain rates C and C* resulting from a change of Db to D i and D, to Dv*;hence: &* -
&
+
= 1 b,Xc
- y2a:&
ma,,
X C ,
(73)
where the symbols are as defined in 8 5.1. This may be simplified further: noting that in general b, has a value of 10, and considering very dilute concentrations of highly surfaceactive species (X, of the order of lo"), the second term containing b, may be ignored in this instance. Hence in a binary system in which the second component is very dilute and also very surface-active, the grain-boundary diffusion creep rate at a given temperature is related to the diffusional creep rate of the pure solvent in the first approximation by: References: p. 1284.
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There exist practical systems in which the dilute approximation implicit in the above equation should apply. For example, consider the effect of small impurity additions of sulphur on the nickel deformation map shown in fig. 39 in connection with the boundaries of the Coble creep mechanism. For the application of eq. (74), it is required to know the solid solubility of the impurity in the parent lattice at the temperature of interest, and also the relevant atomic sizes. Thus the quantity Pb is derived by substitution in the solubility law of eq. (15) described earlier. In the case of nickel, a number of dilute impurities at the ppm level may introduce significant effects on Coble creep rates. The computed effect of trace additions of sulphur on the extent of the Coble creep regime in nickel is superimposed on the Ashby map in fig. 39 as the dashed lines, here specifically for a grain size of 32 p m and a critical strain rate of 104/s. This shows that by introducing quantities as low as 2 and 10 ppm of sulphur in nickel, the boundaries of the Coble creep regimes are shifted substantially. These large effects predicted for and HENDERSON [19821) sulphur, magnesium and other dilute impurities (HONDROS demonstrate the important role of interfacial microchemistry on grain-boundary diffusion phenomena and how in this particular case, the impurities induce a beneficial effect by inhibiting grain-boundary transport. 63. Interfacial cohesion: role in temper-brittleness
Temper-brittZemssis one of the oldest known forms of brittle intergranular failure. It occurs in low alloy steels incorporating alloying elements such as Cr, Ni and Mn when they are slowly cooled from temperatures above 773 K. High-purity alloys do not exhibit this embrittlement and it has been shown that the cause can generally be associated with the adventitious levels of P, Sn, Sb and As impurity, which are typically in the 50-200 ppm range. It has been recognized that Cu is also deleterious and this is important since much present day low alloy steel has copper contents as high as 0.15%. Analyses of commercial steels show that temper-brittleness is, in practice, mainly caused by the grain-boundary segregation of phosphorus whereas in nickel-containing steels, tin also contributes. The phosphorus comes from the original steel making, and is expensive to remove to very low levels. The tin arises from recycled scrap plated steel. The role of individual elements in temper embrittlement is summarized by BRIANT[1988b] and MCMAHON[1991]. It is instructive to consider the analysis of one very well documented instance of temper brittleness investigated at the Watertown Arsenal ( C m and et aZ. 119531). As we have mentioned earlier, as segregation increases in a steel, the ductile-brittle transition temperature (DBTT)also increases until it rises above the ambient temperature and brittle failure will occur in service. Studies of the DBTT through Charpy impact measurements therefore provide a characterization of the heat treatments that will cause temper brittleness. Figure 40 shows the DBTT measurements for S A E 3140 steel (1.26% Ni, 0.77% Cr, 0.39% C, 0.015% P) heated to 948 K and water-quenched, followed by iso-
Ch. 13, 86 65C
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I
I
--
600-
'j .B
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A-
Corre+d transition tempemtu'e,%
E,
c.
0-
55 50 65
*-
10 5 +-0 00-
.. 0.1
,
I 1.0
I 100
I
10 Embrittling time. h
Fig. 40.The time-temperature diagram for embrittlement of SAE 3410 steel corrected to constant hardness (after CARRet al. [ 19531).
thermal heat treatments at increasing times in the critical temperature range 648-848 K. In fig. 40 we observe several of the characteristic features of temper-brittleness. The embrittlementoccurs most rapidly around 773-823 K, and reheating above 873 K rapidly removes the embrittlement. Material quenched through the temperature range 673-873 K will not be embrittled, whereas large-section castings or forgings necessarily cooled
I
I
1
I
I
1
1
1
350
0.01 Embrittlinq time, h
Fig. 41. The time-temperature diagram for the segregation of P in the SAE 3410 steel of CARRetal. [1953]. The numbers against the curves denote the P segregation occurring as a result of the embrittling treatment, Le., in excess of the 0.06 monolayers segregated during 948 K (675°C)temper. (after SWH [1977]). References: p . 1284.
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slowly through this range will be severely embrittled. The embrittlement is often called reversible temper-brittleness since samples embrittled trough slow cooling can be d e embrittled by reheating above 873 K followed by a quench, and then again embrittled by a slow cooling treatment. The reconstruction of fig. 40 has been made by a calculation of the segregation of phosphorus in the steel using Guttmann’s modified theory as described at the end of 54.4. SEAH[1977] used this approach with AGli =-11.5, AG; =-44.0 and a&=-26.0 kJ/mol to predict a low phosphorus equilibrium segregation above 800 K but rapidly rising to near the saturation of one monolayer at 748 K. Combined with this prediction are the kinetics of segregation discussed in 0 4.5 with D p = 1.58 x 104xexp(-219000/RT) m2/s. These permit the generation of the segregation contours of fig. 41 which correlate very closely with the DB?T curves. This correlation shows the dramatic DBTT shift of 275 K per monolayer of segregated phosphorus. An alternative to the above interpretation of temper-brittleness via the alloy-impurity coupling model of Guttmann,has been proposed. Studies of phosphorus segregation in pure iron, and alloy with up to 0.1% carbon and 2.2% chromium by ERHART and GRABKE[1981] show that the phosphorus segregation in pure iron is as high as customarily observed in steels but that carbon, if present, will also segregate, reducing the phosphorus segregation through site competition, and improving the cohesion as described in § 5.3. The action of added chromium is shown to precipitate the carbon, so that it no longer segregates and hence the phosphorus segregation returns to the initial high level. Thus, in steels, the presence of chromium appears to enhance the phosphorus segregation, but in reality it is the carbon which the chromium influences and this, in turn, affects the phosphorus segregation. This view is supported by BRIANT[1981] who also finds that the alloying elements have no effect on phosphorus segregation in 0.3% carbon steels. If this model is correct, the lower part of fig. 41, below 723 K, should remain largely unaltered but the top part above 773 K, should arise as a result of carbon competitive segregation deriving from, say, dissolution of carbides at temperatures above 773 K. This is clearly not the situation in figs. 40 and 41 since the Cr can only precipitate a small fraction of the C in the steel. We conclude that, for the S A E 3140 steel, we must be concerned with Guttmann’s alloy-impurity coupling. However, it is possible that in higher-Cr steels such as 2iCrlM0, with lower carbon contents, the Erhart and Grabke mechanism may operate. In other steels both mechanisms may be important. Finally, such steels as the Mo-containing 2iCrlMo and 3CriMo alloys incur additional effects in that Mo and P react readily to form (FeMo),P precipitates. Thus, in untempered steels quenched from above 1123 K, there is little P segregation and most of the Mo is in solution. As a function of time in the tempering range 723-923 K, the M o that is free in solution couples with the P and according to the Guttmann theory this leads to high equilibrium segregation. However, the coupling is strong, causing the formation of Mo,P microprecipitates so that free P level is low, thereby considerably retarding the rate of build-up to high segregation at grain boundaries. This is clear from eqs. (49) and (50) where the initial rate of segregation is proportional to the free solute content. Mo retards the diffusivity of P in pure iron by an order of magnitude at these temperatures. Thus Mo initially retards the segregation of P. With time at temperature,
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Interfncial and sugace microchemistry
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14000 P)
:12000 (1
L
al
z 10000 E
.-0 W
0,
8000
.-Nul 01
t 6000 .-a .-U E CI
4000
C
0
s"
2000
40
80
Time. h Fig. 42. The effect of increasing bulk tin content on the rate of coarsening of nitride precipitates in iron at 643 K.
the Mo nucleates Mo2C carbides which precipitate and release the locked-up €? Thus, as time evolves the P segregation may eventually return to the level expected in Mo-free steels, as observed in practice (MCMAHON et al. [1977]). The initial retarding effect of Mo improves as the Mo content of the steel increases but reaches an optimum in the range 0.4-0.7%, after which the embrittlement deteriorates again (DuMOULIN et al. [1980],Yu and McMAHoN [1980]). Part of the initial improvement is, as discussed above,. due to the reduced P segregation. An additional effect arises through the improvement in cohesion caused by Mo segregation predicted in fig. 34 and observed by DUMOULIN etal. [1980]. However, as the Mo content increases, the propensity to Mo2C precipitation increases and above 0.7% content the remedial effect disappears. Later experiments of MENYHARD and MCMAHON[1989] suggest that the entire molybdenum effect could be that of cosegregation with carbon which offsets the embrittling effect of phosphorus, and the role of molybdenum as acting as traps for phosphorus seems to be only additional. References: p . 1284.
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6.4. Further examples of metallurgicalphenomena influenced by microchemical processes The case studies of metallurgical properties affected by microchemical processes presented above are being increasingly supported by a growing body of theoretical and experimental studies. Some of these are now fairly mature examples. As fig. 35 (above) indicates, there exist many other examples of metallurgical phenomena at various stages of understanding and different levels of significance, which are being progressively scrutinized in terms of the contribution of the interfacial microchemistry. Below we consider, in brief, a selection of case studies which are being currently expanded by work in several laboratories and which represent, to a certain extent, fresh directions in the understanding of traditional metallurgical problems.
6.4.1. Microchemical barrier layers There is a category of metallurgical phenomena in which interfacial microchemistry induces its effect through an impedance layer across the interface. The phenomena refer mostly to ripening or microstructural shape modification processes (ch. 9, $4.2.2.; ch. 15, 0 2.4.2), which depend on the ease with which atoms can transfer across the interface between solid phases, but they may also operate at a free surface (as in nitridation or carburization from gas phase equilibria), involving atom transfer across the free surface from the vapour to the interior of the solid. Consider second phase particles or precipitates in a solid matrix at service or processing temperatures. There exists an inherent tendency for such particles or precipitates to coarsen in order to reduce the total interfacial free energy associated with them. The rate-determining step in normal ripening processes is the volume diffusivity, however in the presence of segregants at this type of interface, the atom transfer step may become rate-determining. This may lead to a significant retardation in the kinetics of ripening. There exists much indirect evidence for the segregation of impurities to these interfaces. BOYD and NICHOLSON[1971] showed that the addition of 0.1% Cd to an Al-4%Cu alloy produced a remarkable reduction in the ageing kinetics of 6' precipitates at 473 K as indicated by the rate of growth of the mean particle size with time. In addition, they were able to measure by isothermal calorimetry a considerable reduction in interfacial free energy, from 1530 mJ/m2 to 250 mJ/m2, produced by the cadmium additive. This is clear evidence for the segregation of the cadmium to the intedace. This subject has been considered in most detail for the ripening of y'-Fe,N precipitates in an Fe-N alloy. Low bulk additions of antimony, oxygen, phosphorus and tin produce a remarkable inhibition in the rate of coarsening of the nitride. This is illustrated in the work of MILLER[1979] in fig. 42 in which the increasing bulk tin content is seen to reduce the mean nitride particle coarsening rate by an order of magnitude compared with the behaviour in pure iron. In considering the theory of the segregation-induced retardation of ripening, the starting point is the Lifshitz-Wagner equation (LIFSHITZ and SLYOZOV [ 19611; WAGNER [ 19611) for volume-diffusion controlled growth of a dispersion of particles of initial mean radius r,:
Ch. 13. 56
Interfacial and sulface microchemistry
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where T is the mean particle radius after a time (t - to),yi the interfacial free energy, V the molar volume of precipitates, C, the molar concentration of solute in equilibrium with a particle of infinite size and B, is a numerical parameter which can vary according to the shape of the precipitates. Segregation to such interfaces will reduce yitypically by about 50%. However, much larger inhibitions in coarsening rates have been observed, and a reduction in interfacial energy cannot in itself account for the observed behaviour. It is believed that the most likely mechanism is that the segregate inhibits the rate of transfer of solute atoms across the interface. In this situation the relevant ripening equation becomes:
where Q is a reaction rate parameter associated with the ease of transferring a species across the interface. Thus, according to this theory, the mean particle size varies as t”’ for volume-diffusion control and with t’” for interface-reaction control. In studies of this nature it has not been easy to make a clear distinction in the mechanisms on the basis of this time-dependent growth behaviour because these functions both lie within the experimental scatter. Thus, although these studies do not give a clear confirmation on this mechanism, neither do they contradict it. MILLER [1979] has re-examined the theory and has shown that it predicts a time-invariant size distribution profile depending on the above growth mechanisms and that the data shown in fig. 42 can be fitted more closely to the particle-size profile predicted for interface control. The surface-active elements which inhibit the ripening of precipitates are now shown to be able to form barrier layers at iron surface, thereby strongly retarding nitridation and carburization reactions. The inhibition can be quite significant as demonstrated by GRABKE et aZ. [1977] with sulphur impurity adsorption up to a level of a monolayer. Zn this type of atom transfer process, the alloying species is initially present in a molecular form, as CO,, CH, or NH,,in the vapour phase, and for its entry into the metal the basic mechanism is the stepwise catalytic dissociation of the surface-adsorbed molecule, allowing the C or N ions to penetrate into the interior. The strongly bound impurities at the surface impair the dissociation step, as in a poisoning reaction. In the reverse process, the surface denudation of nitrogen or carbon in iron alloys has also been shown to depend strongly on the existence of chemical barrier layers which inhibit the formation of the molecular species at the free surface. The precise mechanisms appear to be strongly related to the catalytic poisoning of surfaces by impurities, which operate in these instances in a manner that effectively retards the rate of ingress or expulsion of the elements. 6.4.2. Creep-embrlttlement It was noted earlier that as a consequence of surface segregation, the rate of grainboundary creep cavity generation is increased, leading to a deterioration in creep rupture life. This pertains to typical weldable creep-resistant steels of the Cr-Mo-V series where References: p. 1284.
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the impurities present in the bulk exercise their role in the heat-affected zone of the weld, following a stress-relief heat treatment at temperatures of the order of 973 K. Thus in a stress-relief operation in the average commercial UK steel of the type &r$Mct$V (impurities in ppm: P, 170; Sn, 120; Sb, 30; As, 160; Cu, 1200; Ni, 1200), the most active embrittling impurities are Sn, Cu, As, Sb and P in that order with relative fragility potencies given by the ratios 3:2.5:2:1:1, respectively (LEAet al. [1980]). We now consider these alloys in service conditions, where stress and temperatures are lower than in the above, and under longer-term creep durations. Analysis of existing data by SEAH[198oC] reveals a different effect. Here it appears that the rate-determining mechanism is the transport of matter or vacancies along the grain boundaries in order to feed the growing cavities. For a given strain associated with cavity growth, a high density of small cavities is much more harmful than a lower density of large cavities. The formed condition may be promoted either by increasing the cavity nucleation rate (which we have seen is induced by surface segregation to the cavity walls) but it may also be promoted by retarding the growth rate. These effects are noted in the analysis of [1975], and are described by the equation creep cavitation by SKEXTON 0.65 2?rDbSd
“ = T NkT [
]’
(77)
where E, is the strain to rupture, d is the average grain diameter, D, is the grain-boundary diffusivity and N the continuous rate of creation of cavities; the other symbols have been defined earlier. A regression analysis on the effects of various impurities on the rupture ductilities during long-term creep at 823 K on ;Cr$Mct$V commercial steels has shown a decrease in the rupture ductility with increasing impurity content as indicated in fig. 43 (SEAH [198Oc]). The dependence of rupture ductility upon impurities is given by: -A(rupture ductility) oc 1.5P t 0.36Cu + 1.6% t 1.2S,
(78)
where the element symbols refer to solute concentrations in the bulk. For the above typical commercial-purity steel, the relative damaging potencies of the impurities are different from that of the stress-relief case of embrittlement mentioned above: the order is now Cu,P, Sn and S with relative fragilities of 2.5:1.5:1:1, respectively. Thus, in both forms of creep embrittlement pertaining to the generation and growth of grain-boundary cavities, such as those illustrated by stress-relief cracking or long-term creep-embrittlement, the impurity sensitivity arises from different modes of interfacial segregation. This difference appears to be associated with the very different strain rates are experienced. Intergranular stress-corrosion cracking The notions of interfacial microchemistry and the effects on interfacial cohesion are being applied currently to explain various forms of intergranular cracking in the presence of both stress and a corrosive environment. Thus, pure iron is resistant to stress-corrosion cracking in nitrate environments, however in commercial carbon steels, failure is 6.4.3.
Interfacial and surface microchemistry
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1
1.5P + 1.25+ 0.36Cu+1.6Sn Fig. 43. Regression analysis of creep ductility in $CrjM+V steels as a function of a combined impurity parameter involving P, S , Cu and Sn (after SWH [198Oc]).
invariably intergranular. Many attempts have been made to trace the role of alloying elements, for example carbon, aluminium, and chromium, but there is seemingly little systematic behaviour. The importance of the impurity contents of these commercial steels has been recognized. LEAand HONDROS[19821 prepared samples from high-purity mild steel stock, each sample containing a single impurity at a dilute level comparable with that found in commercial material. Here the free residual sulphur, which is very active, was first rendered harmless by the addition of manganese. Each impurity element was added separately at a bulk level which was predicted to give rise to a grain-boundary segregation content of about 20% of a monatomic layer following an isothermal heat treatment. The samples were tested to failure in a controlled environmental cell containing ammonium nitrate, at a constant strain rate. In each case, a control sample was tested under exactly the same conditions except for a neutral solution. Here it was seen that every specimen exhibited stress corrosion in the nitrate environment, failing in a shorter time than that in the control environment. The copious data shown diagramReferences: p . 1284.
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matically in fig. 44 refer to the time for intergranular failure for alloys stress-corroded at a constant strain rate, noting that the data on the right-hand side of the diagram refer to the tests in the controlled solution, here paraffin. In the nitrate solution the fracture was almost always entirely intergranular, while in the control solution ductile fracture predominated. The figure shows that in the aggressive environment, with all other conditions identical, the reduction in life depends on the impurity present. Taking the time to failure under constant strain-rate testing, normalized to the value in a benign environment as the measure of the reduction in the resistance of the steel to cracking, the data can be presented in the form -AR = 20P + 1.9Cu + 1.OSn t 0.9Sb + 0.9(others).
(79)
Here the element symbols represent the bulk content in wt%. The changes in R thus provide a ranking order of the potent segregating impurities which affect the stresscorrosion propensity. We note that in commercial steels S, Ca and A1 are mainly present as precipitates. To optimize the performance of carbon steel in nitrate environments, the clearest metallurgical guideline from this is a reduction in bulk impurity levels, particularly that of residual phosphorus. Although the detailed micromechanism for this effect awaits elaboration, the principal observations can be described fairly adequately by a dissolution model in which the grainboundary material is preferentially removed, the driving force for this being the divergence of the segregant’s electrochemical potential from that of iron. In an oxidizing
Tests in hostile solution
0
I
I
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I
Tests in control solution I
30 40 Time to failure (hours)
I
I
I
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60
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Fig. 44. The times to failure of 11 specimens of carbon steel, each containing a different impurity element, stress-corrosion tested in ammonium nitrate solution at a constant strain rate. The control tests in a neutral environment are shown to the right (after LEAand HONDROS[1982]).
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environment such as aqueous ammonium nitrate, the driving force for atomic dissolution is related to the potential difference between the matrix and the segregants, which form a galvanic cell. Thus for segregants which have a more positive potential than iron, these atoms themselves are dissolved more rapidly and a very sharp grain-boundary crack is exposed for further dissolution. On the other hand, for segregants having a more negative potential than iron, atoms adjacent to the boundary are dissolved, thus enhancing the intergranular corrosion but with a different crack tip profile. Grain-boundary segregation during stress-corrosion cracking furthermore promotes dissolution by reducing the rate of repassivation after slip has ruptured protective oxide films and crack tips (LYNCH[19891).
6.4.4. Intergranularhydrogen-embrittlement An aspect of the ubiquitous problem of the hydrogen-embrittlement of steels is of interest here. This is the intergranular mode of failure, which has been considered recently in relation to the microchemical nature of the grain boundaries. Thus LATANISION and QPPERHAUSER [1974] proposed that hydrogen penetration along the grainboundary regions should be promoted by hydrogen-recombination poisons which are present as grain-boundary segregants. These elements, such as P, S, As and Sb, which have a strong tendency to segregate to grain boundaries in steel also have a stronger intrinsic binding to hydrogen than does iron (LYNCH[1989]). A probable effect of this is that following the dissociation of a hydrogen molecule which arrives from the vapour or from solution, the stronger bonding available at the solute atom sites causes the hydrogen to reside at these sites for much longer times than at normal iron sites, and therefore there is a greater chance for the hydrogen to be absorbed into the lattice. Hence the role of these solute species is effectively to increase the amount of hydrogen occluded along the grain-boundary zones. If the mechanism of hydrogen-embrittlement requires the concentration of hydrogen in the metal just ahead of the crack tip, the above grain-boundary segregants should therefore enhance this embrittlement. 6.4.5. Inhibition of surface oxidation on alloys In certain alloys, especially those containing aluminium, surface oxidation can be inhibited by the introduction of dopants which both reduce the rate of grown of the oxide and also inhibit “breakaway” oxidation by improving the scale adhesion at the metal/ metal-oxide interface. This complex subject has been reviewed in detail, for example by WHITTLEand STRINGER [19801 who assessed various mechanisms for the improvement in performance. Several microchemical phenomena have been proposed to account for this. One proposal, which requires experimental verification but which is consistent with the analysis given in 8 5.2, is that the dopant segregates to the oxide grain boundaries, thereby reducing the boundary diffusivity. In their study of oxidation behaviour of 3N)/-Cr,C, eutectic composite containing additions of yttrium at a low level, BULLOCK et al. [1974] observed a remarkable improvement (Le., reduction) in oxidation weight gain at llOO°C and in addition showed by AES that the yttrium level increased with distance in the oxide bulk and reached a peak level at a depth which suggested that the interface between the oxide and the alloy was enriched in yttrium. In addition to the probable improvements in growth kinetics by segregation to the boundaries, it was References: p , 1284.
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considered that the formation of a microchemical barrier layer at the metaumetal-oxide interface improved the adhesion of the alumina in the mixed metal-oxide scale to the metal substrate and hence increased the resistance to oxide scale sparling. A third possibility for inhibited high-temperature corrosion in through the formation of a microstructural barrier layer at the metal-oxide/vapour interface. The presence of boron in the vapour phase is known to improve greatly the oxidation resistance of chromium-containingsteels by increasing the time to “breakaway” oxidation. LEA[19791 showed by AES that the boron was present essentially only at the outer oxide surface layer and it was also accompanied by a high chromium enrichment. The mechanism for this type of inhibition thus has similarities with the processes considered above ($6.4.1). In particular, it was considered that the introduction of 0-B-O covalent bonds in the outer oxide surface structure would impede the transfer of 0 ions onto the lattice, thus forming a film which in turn allows slowly diffusing protective chromium from the alloy to reach the free surface (ch. 14, $3).
7. Inte#acial microchemistry and materials design theory All the important themes of physical and chemical metallurgy contribute collectively to the theory of materials design - including the thermochemistry of phase equilibria, crystal defect theory, fracture theory, oxidation and corrosion resistance and so on. This chapter has placed emphasis on the importance of the relatively new subject of interfacial microchemistry for a wide diversity of metallurgical properties, and the insight and information now made available should be viewed in terms of its contribution to materials design. Here we mention two basic approaches to the materials design theory, the alloy design and the grain-boundary design. In the former one we distinguish between deliberate additions of specific elements which procure beneficial effects on certain properties, and the planned control of known elements which are detrimental. In the latter approach, effort is focused on tailoring the structure of a polycrystalline material with improved properties using a specific arrangement of appropriate grain boundaries. The advanced knowledge in these fields provides many opportunities to manipulate material properties, often in the context of “fine tuning” but equally often to gain significant improvements in service life. In the past two decades, several lines of advance have been made in this subject matter which are valuable to alloy design: (i) the identification of potent impurity species (both harmful and beneficial); (ii) the basic mechanisms (sometimes only partly understood) by which they operate; (iii) the enrichment levels as well as bulk contents at which the effects are significant. As for grain-boundary design approach, progress has been made in: (i) identification of the types of grain boundaries characterized by different properties: (ii) processes for controlling the grain-boundary character distribution in polycrystalline materials. This chapter provides several guidelines. Thus, on the basis of a retardation of grainboundary transport, or on improved interfacial cohesion, certain types of elements can be beneficial - as in the improvement of oxidation characteristics or in the rate of Coble creep. Certain elements can be predicted to offer an improvement in sintering behavior
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through a segregant stabilization of fine grain structure. In an example of benefit to mechanical properties, WHJTEet aZ. [198l] demonstrated that the room-temperature strength and ductility of a WRh/W alloy can be improved by the addition of 75 wt ppm of B which was shown to enrich, in a typical manner, the grain boundaries. Where the impurities produce deleterious effects, clearly the alloy design aspects involve the elimination of the impurities or, more practicably, their control to within tolerance levels. The identification of potent surface-active species and the assigning of bulk tolerance levels have been the specific objectives of many studies in this field. The information is couched often in formulae of the type presented above for intergranular stress-corrosion cracking (eq. (79)) or creep rupture ductility (eq.(78)). This permits the design of alloys of closely controlled compositions for improved performance in the defined conditions of use (HONDROS [ 19861). In a previous discussion (HONDROSand SEAH[1977b]), we presented a number of procedures which might offer a control over the effects of harmful impurities including the judicious use of the principles of competitive segregation, the selection of remedial segregants, the thinning out of the embrittling species by spreading them out over a large interfacial area, Le., a very fine grain size, reducing the kinetics of the embrittling species, and the used chemical fixation or precipitation reactions. In the meantime, at least one of the above approaches has been explored in depth, in connection with affording a simple and economical control over temper-embrittlement. SEAH et aZ. [1979] demonstrated that temper-embrittlement induced by P and Sn in 2:CrlMo and %Ni-Cr-M+V steels can be reduced by additions of La or Ce at a predicted level. The selection of La, Ce and mischmetal (approx. 50% Ce, 25% La, 25% other rare-earth metals) are based on a comprehensive thermochemical assessment of the possibility of forming stable microprecipitates in order to reduce the bulk levels of active P and Sn. The optimum addition of La required to remove the embrittling effects of P and Sn is given by the formula La = 8.7s + 2.3Sn + 4.5P
(80) where the element symbols represent the wt% of the element in the steel. Figure 45, taken from this work, shows the high DBTT for a pure 2iCrlMo steel with 0.017% P added. The further addition of various levels of La lower the transition temperature significantly, to the limit of a high purity material. In the future, the increasing knowledge of structure/property relationships for grain boundaries (including anisotropy of grain-boundary segregation and thus, for instance, anisotropy of brittle intergranular fracture) may be practically used to control materials properties. The differences in grain-boundary character and geometric configuration enhance the heterogeneity of all types of deformation and fracture in polycrystals. This effect is schematically shown in fig. 46. The presence of low-energy grain boundaries a-boundaries in the figure) in a polycrystal will not contribute much to the deformation and fracture while the presence of general grain boundaries @-boundaries) will exert a significant contribution. Therefore, a heterogeneity of the fracture process occurs which can either lead to an easy fracture (figs. A, B, D or F) or to a difficult fracture (figs. C and E) under the same stress. On the basis of these considerations, WATANABE [1984] References: p. 1284.
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I
8t 0
I
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- 100
+0.017/.’
P *0.21% La
I
I
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Temperature,’ C Fig. 45. Improvement in the ductile-brittletransition temperature in a steel susceptible to temper embrittlement, resulting from low level additions of lanthanum which combat the effect of phosphorus (after SFAHetal. ~791).
pointed out the importance of the effect of the type and frequency of grain boundaries to the behaviour of a polycrystal and proposed the concept of grain-boundary design. The basic idea of grain-boundary design is to produce polycrystalline materials with grain boundaries of such character and distribution as will suppress a “detrimental effect” of the grain boundary and conversely, increase its “beneficial effect” as much as possible. In this way, the properties of a polycrystal could be improved to a significant extent. To fabricate a polycrystalline material by means of grain-boundary design, two fundamental steps have to be performed. The first step consists in prediction of the performance and thus, in the proposal of character and distribution of grain boundaries for the given purpose. The other one represents the processing method to achieve the predicted grain-boundary character distribution and configuration in order to obtain the proposed structural material. Practically, of course, the latter step seems to be the limiting one because no method has commonly been developed to produce such structures till now. Thermomechanical processing based on working and annealing has widely been applied to control microstructure of polycrystals such as grain size, second phase dispersion, and texture. Unfortunately, these processes are not developed to such an extent at present that they completely control the character and distribution of grain
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(A)
(C)
Random:*
I
Dwmeratc Con1Iguration
RR
L
Random :f
6-
k
Fig. 46.Intergranular fractureprocesses in polynystals with different grain-boundary character distributions and boundary configurations (after WATANABE [1993]).
boundaries. From this point of view, magnetic annealing of ferromagnetic materials seems to be promising to introduce specific types of grain boundaries with a characteristic frequency in a given material. Of course, novel procedures have to be developed in future (WATANABE [1993]). There is an aspect of the general subject matter of this chapter which has wider ramifications in the context of metals recycling, and which implies that it will be a subject of growing importance in the future. It is clear that because of geo-political, economic and social factors there will be in the future a greater emphasis on the recycling of used materials in industrialized countries. This will reduce their vulnerability to external sources of raw materials and also, there are inherent energy savings in reclaiming metals from used articles rather than obtaining them from raw supplies of ore. Because of the complex mechanical systems used by industry and society and the wide combination of metals involved, inevitably metals derived from recovered scrap will contain certain unwanted residuals which bypass the refinement process (HONDROS [198Ob]). This chapter has underlined that such small quantities of noxious elements can provoke remarkable effects on physical and mechanical properties, and more specifically the accumulation of residual elements, such as copper and tin in steel from scrap, will have effects on the performance. Thus, an increased effort devoted to the recycling of metals will, in general, exacerbate the effects of unwanted tramp elements on the properties of alloy aimed for particular end uses. On the basis of a better metallurgical understanding, there will be effort devoted to a control tramp element effects, involving remedial measures, of the type discussed above. However, we envisage the need for a broader perspective, leading to a new metallurgical or engineering subject matter encompassingthe theory of design involving recycling. In this global approach, we consider the final end use of the material as the primary References: p . 1284.
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Ch. 13, Refs.
parameter, and recyclability should be built in to the design of the final material. Thus we may plan for the evolution of secondary alloy products which may not be adequate for the original property requirements, but which may be suitable for other applications. An interesting example of this is a carbon steel with high levels of tin and cooper arising from the use of scrap containing much tin plate - this has been demonstrated to have an improved corrosion resistance associated with the presence of these elements in sub-scale layers. Such an alloy may not have adequate mechanical properties for the original purposes, however it can be used in situations where the improved corrosion resistance is advantageous. Thus, in this theory of design with recycling we can envisage recovery epicycles in which certain grades of alloy might be blended together to produce new alloys which are more suitable for specific applications. The science of interfacial microchemistry will make basic contributions in the future in connection with material design in the context of recycIing, improved reliability and advances in performance.
References Asmy, M. F., 1972, Acta Metall. 20, 887. 1965, Lattice Defects in Quenched Metals (Academic Press, London) p. Ausr, K. T.,and J. H.WESTBROOK, 771. BATTE,A.D., J.M. BEAR,S.R.HOLDSWORTH, J. MEYERS and P. R. REYNOLDS,1980, Phil. Trans. Roy. SOC. A295, 253. BENNINGHOVEN, A., J. GIBER,J. mu), M. RIEDELand H.WERNER, eds., 1982, Springer Series in Chemical physics, 19, Secondary Ion Mass Spectroscopy, SIMS In (Springer, Berlin). BERNARDMI,J., P. GAS,E.D. HONDROSand M.P. S m , 1982, Proc. Roy. Soc. A379, 159. BISCONDI, M., 1982, J. Phys. France 43, C6-293. and G.V. SHCHERBEDINSKIY, 1964, Phys. Met. Metallogr. 17, 80. B o m v , V.T.,V. M. GOLIKOV BOUCIIET,D., and L. PRmmx, 1986, Scripta Metall. 20,961. B o u c m , D., and L. FWmmt, 1987, Scripta Metall. 21,475. 1971, Acta Metall. 19, 1379. BOYD,J. D., and R. B. NICHOLSON, G., J. FERANTEand J. R. SMITH,1992, Phys. Rev. B 45,493. BOZZOLO, BOZZOLO,G., B. GOOD and J. FERANTE, 1993, Surf. Sci. 289, 169. BRIANT,C.L., 1981, Scripta Metall. 15, 1013. BRIANT,C. L., 1988a, Mater. Sci. Technol. 4, 956. BRIANT,C. L., 1988b, in: Auger Electron Spectroscopy, eds. C.L. Briant and R.P. Messmer. Treatise on Materials Science and Technology, Vol. 30 (Academic Press Boston) p. 111. B m , C.L., 1990, Metall. Trans. 21A, 2339. BRIANT,C.L.,and R.P. hllEssMliR, 1982, Acta Metall. 30, 1811. BRIGGS,D., 1982, Surf. Interface Anal. 4, 151. BRIGGS,D., and M. P. SEAH,eds., 1990, Practical Surface Analysis. 2nd ed., Vol. 1, Auger and X-ray Photoelectron Spectroscopy wiley and Sons, Chichester). BRICGS,D., and M.P. SEAH, eds., 1992, Practical Surface Analysis. 2nd ed., Vol. 2, Ion and Neutral Spectroscopy (Wiley and Sons, Chichester). BRUEMMER, S.M., 1989, Mater. Sci. Forum 46,309. J., I. TANAKA, H.4. KLEBEand M. RUHLE,1994, Anal. a i m . Acta 297,97. BRULEY, S., L. S. DEMING,W.E. DEMING and E. J. TELLER,1940, J. Am. Chem. Soc. 62, 1723. BRUNAUER, BULLOCK,E., C. LEAand M. MCLEAN,1979, Met. Sci. 13, 373. CARR, EL., M. GOLDMAN, L. D. JAPPE and D. C. BUFFUM,1953, Trans.AIME 197,998. COBLE,R. L., 1963, J. Appl. Phys. 34,1679.
Ch. 13, Refs.
Interfacial and surface microchemistry
1285
DFPAY, R L . , I. PRIGOGINE, A. BELLOMANS and D.H. EVERE~T, 1966, Surface Tension and Adsorption. (Longmans, London). E, and G.E. RHEAD, 1971, Surf. Sci. 28, 267. DELAMARE, DOIG, P., and P.E.J. FLEWFIT, 1981, Acta Metall. 29, 1831. DOIG, P., D. LONSDALE and P. E. J. FLEWITT, 1981, Metall. Trans. 124 1277. DUhfOmIN, RH., M. GUTTMANN, M. FOUCAULT, M. PALMIER, M. WAYMAN and M. BISCONDI,1980, Met. Sci. 14, 1. DU PLESSIS, J., 1990,Sol. State Phenom. 11, 1. EOWT, B.,,and 0. PANZNER,1982, Surf. Sci. 118,345. ERHART,H., and H. J. GRABKE,1981, Met. Sci. 15, 401. ERLEWEIN, J., and S. HOFMANN,1977, Surf. Sci. 68, 71. EWING,RH., 1971, Acta Metall. 19, 1359. FISCFIMJE.~, H., 1985, J. Phys. France 46,C4-3. FRANZONI, U., H. GOmzKI and S. STURLESE, 1981, Scripta Metall. 15,743. FUKUSHMA, H., and H. K. BIRNBAUM, 1984, Acta Metall. 32, 851. GAS, P., M. GUTTMANNand J. BERNARDINI, 1982, Acta Metall. 30,1309. GEORGE,E. P., and D. P. POPE, 1989, in: Interfacial Structure, Properties and Design, eds. M.H. Yoo, W.A.T. Clark and C.L. Briant. MRS Symp. Proc. Vol. 122 (MRS, Pittsburgh) p. 391. G U , R.M., and K. A. D’ANGELO,1984, Int. Met. Rev. 29,249. GIBBS,G. B., and J. E. HARRIS,1969, In: Interfaces, ed. R.C. Gifkins (Butterworths, London) p. 53. GIBBS,J. W., reprinted 1948, Collected Works, Vol. 1 (Yale University Press) p. 219. GJOSTEIN,N.A., 1966, Proc. 13th Sagamore Army Materials Research Conference, Raquette Lake, NY,eds. J.J. Burke, N.L. Reed and V.Weiss (Syracuse University Press). G ~ K EH., J., W. PAULITSCHKE, G. TAUBERand H. VIEFHAUS,1977, Surf. Sci. 63, 377. GIJWENHEIM, E. A., 1950, Thermodynamics, 2nd Ed. (North-HoIland, Amsterdam). GUTTMANN,M., 1975, Surf. Sci. 53, 213. GUTI‘MANN,M., 1976, Met. Sci. 10,337. GUTIMANN,M., 1980, Phil. Trans. Roy. Soc.A295, 169. GUTTMANN, M., and D. MCLEAN,1979, Proc. ASM Materials Science Seminar, Interfacial Segregation, eds. W.C. Johnson and J.M. Blakely (ASM, Metals Park, OH) p. 261. HARRIES,D.R., and D. MARWICK,1980, Phil. Trans. Roy. SOC.A295, 197. HASHIMOTO, M., Y. ISHIDA, R. YAMAMOTO and M. DOYAMA, 1984, Acta Metall. 32, 1. HAYES,P. and P.GRIEVESON, 1975, Met. Sci. 9, 332. HERRING, C., 1950, J. Appl. Phys. 21, 437. HIRTH,J. P. and J. R. RICE 1980, Metallurg. l h n s . 11A, 1501. HOFMANN, S., and J. ERLEWEIN, 1976, Scripta Metall. 10, 857. HOFMAWN, S., and J. ERLEWEIN, 1978, Surf. Sci. 77,591. HOFMANN, S., and P. LEJEK, 1991, Scripta Metall. Mater. 25, 2259. HOPMANN, S., P. LEJ~EKand J. AD~MEK, 1992, Surf. Interface Anal.19,601. HONDROS, E.D., 1965, Proc. Roy. Soc. A286,479. HONDROS,E. D., 1969, in: Interfaces, ed. RC. Gifkins (Butterworths, London) p. 77. HONDROS,E. D., 1970, Surface Energy Measurements, in. PhysicochemicalMeasurementsin Metals Research, Whniques in Metals Research Vol. IV,ed. R.A. Rapp Cyviley, New York) Part 2, p. 293. HONDROS,E.D., 1976, in: Grain-boundary Structure and Properties, eds. G.A. Chadwick and D.A. Smith (Academic, London) p. 265. HONDROS, E.D., 1978, Precipitation Processes in Solids, Proc. TMS-AIME Meeting 1976, Eds. K.C. Russell and H.I. Aaronson (Metallurg. Soc. of AIME, Warrendale, PA) p. 1. HONDROS,E.D., 1980%Scripta Metall. 14, 345. HONDROS,E.D., 1980b, Phil. Trans. Roy. Soc.A295, 9. HONDROS, E.D., 1986, in: Grain-boundary Structure and Related Phenomena (hoc. 4th JIM Ins. Symp.). Trans. J I M Suppl. 27, 393. HONDROS,E. D.,and P. J. HENDERSON, 1983, Metallurg. Trans.14A, 521. HONDROS, E. D., and D. MCLEAN,1974, Phil. Mag. 29,771.
1286
E.D. Hondms et al.
Ch. 13, Refs.
HONDROS,E. D., and D. MCLEAN,1976, in: Grain-boundary Structure and Properties, eds. G.A. Chadwick and D.A. Smith (Academic, London) p. 353. HONDROS,E.D., and M. P. SEAH,1977a, Metallurg. Tms. A8, 1363. HONDROS,E. D., and M. P. SEAH,1977b. Int. Met. Rev. 22,262. HONDROS,E.D., and L. E. H. STUART,1968, Phil. Mag. 17, 711. HULTGREN,R., P.A. DESAI,D.T. HAWKINS,M. GLEISERand K. K. KELLY, 1973, Selected Values of the Thermodynamic Properties of Binary Alloys (ASM, Metals Park, OH). and P. P. WYNBLATT,1978, J. Vac. Sci. Technol. 15, JOHNSON, W. C., N. G. CHAVKA,R. Ku, J. L. BOMBACK 467. JOKL,M. L., V. VITEKand C. J. MCMAHON,Jr., 1980, Acta Metall. 28, 1479. JONES, R.H., S.M. BRTJEMMER, M.T. THOMASand D.R. BAER, 1981, Metallurg. Trans. 12A, 1621. J m s , M., L. PRIE~TER and M. BIWNDI, 1994, Mater. Sci. Eng. A 185, 71. KALDERON,D., 1972, Proc. Inst. Mech. Eng. 186, 341. K A R L ~ NL., , 1988, Acta Metall. 36, 25. KARLSSON, L., and H. NORD~N, 1988, Acta Metall. 36, 13. KARLSON,L,H. Nom&, and H. ODSLIUS,1988, Acta Metall. 36, 1. KAIJR, I., and W. GUST,1988, Fundamentals of Grain and Interphase Boundary Diffusion (Ziegler, Stuttgart). KITTEL, C., 1986, Introduction to Solid State Physics, 6th edition (Wiley, New York). KOZAKEVITCH, P., and G.URBAIN1961, Mem. Sci. Rev. Metallurg. 6,401. KRAAI,D. A., and S. F~OREEN,1964, Trans. AIME 230, 833. KUMAR,A., and B.L. EYRE,1980, Proc. Roy. SOC.A370,431. KUPPER,J., H. ERHARTand H. J. GRABKE, 1981, Corros. Sci. 21,227. LATANISION, R. M. and H. OPPERHAUSER, 1974, Metallurg. Trans. 5, 483. LEA, C., 1979, Met. Sci. 13, 301. LEA, C., and E.D. HONDROS,1982, Proc. Roy SOC.A377,477. LEA,C., and M.P. SEAH, 1977, Phil. Mag. 35, 213. LEA,C., M.P. SEAHand E.D.HONDROS,1980, Mater. Sci. Eng. 42,233. LWEEK, P., 1994, Anal. a i m . Acta 297, 165. L E J ~I?,, J. A D ~ and K S. H~FMANN, 1992, Surf. Sci. 264,449. LEJEEK,P., and S. HOFMANN,1991, Acta Metall. Mater. 39,2469. ~ ( S E K ,P., and S. HOFMANN,1993, Interface Sci. 1, 161. LEJEEK,P., and S. HOFMANN, 1995, Crit. Rev. Sol. State Mater. Sci. 20, 1. L I ~ E KP.,, A.V. KRAJMKOV,Yu.N. IVASHCHENKO, M. MILITZERand J. AD1993, Surf. Sci. 280,325. LIFSHITZ, L. M., and V. V. SLYOZOV,1961, Phys. Chem. Solids 19, 35. LIU, C.T., C.L. WHITEand J. A. HORTON,1985, Acta Metall. 33, 213. LUZZI,D. E., 1991, Ultramicroscopy 37, 180. LUZZI,D.E., M. YAN,M. SOB and V. V m , 1991, Phys. Rev. B 67, 1894. LYNCH,S. P., 1988, Acta Metall. 36, 2639. LYNCH,S.P., 1989, Mater. Sci. Forum 46, 1. MARWCK,A. D., 1981, Nucl. Instr. and Meth. 182/l83,827. MASSALSKI,T.B., J. L. MURRAY, L.H. BENNETTand H. BAKKER,1986, Binary Alloy Phase Diagrams (ASM, Metals Park,Ohio). MASUDA-JINDO,K., 1986, phys. stat. sol. (b) W, 545. MATSUKO, T., K. NAKMMA, Y. TERADA and M. KIKUCHI, 1991, Mater. Sci. Eng. A 146, 261. M c W , D., 1957, Grain boundaries in Metals (Oxford Univ. Press). MCLEAN,D., 1981, Metals Forum 4,45. MCMAHON,C.J., JR., 1991, Mater. Charact. 26, 269. MCMAHON,C. J., JR., 1992, in: Clean Steels Technology, ed. R. Viswanathan (ASM, Metals Park) p. 3. MCMAHON,C. J., JR., and V. VIT~K,1979, Acta Metall. 27, 507. MCMAHON,C. J., JR., A. K. CIANELLIand H. C. FENG, 1977, Metall. Trans. SA, 1055. MENYHARD,M., B. BLUM,and C. J. MC~~AHON, JR., 1989, Acta Metall. 37, 549. MENYHARD,M., and C. J. MCMAHON,JR., 1989, Acta Metall. 37,2289. MENYHARD, M., B. ROTHMAN, C. J. MCMAHON, JR, P. LFJCEK and V. PAIDAR,1991, Acta Metall. 39, 1289.
Ch. 13, Refs.
Interfacial and surface microchemistry
1287
m m ,L.Z., J. Gmm, W. H o r n and P. VARGA,1990, Surf. Sci. 234, 197. MEZEY,L.Z., and W. HOFER, 1992, Surf. Interface Anal. 19, 618. MILITZER,M., Yu.N. IVASHCHENKO, A. V. K R m o v , P. L.EJ&K, J. W m o , and S . A . F J R ~ V1992, , Surf. Sci. 261, 267. MILITZER,M., and J. WIETING,1986, Acta Metall. 34, 1229. MnrrZER, M., and J. WIETWG,1987, Acta Metall. 35,2765. MILLER,D., 1979, Doctor of Philosophy Thesis, Univ. of Strathclyde. MILLER,M. K., 1987, Int. Mater. Rev. 32, 221. MI&$ B., M. MCLEANand E.D. HONDROS,1973, Phil. Mag. 27, 361. MUKHERJEE, S., and J.L. MORAN-L~PEZ, 1987, Surf. Sci. 188, L742. MUSCHIK,T., w. GUST,S. HOFMANN and B. F'REDEL, 1989, Acta Metall. 37, 2917. NAEARRO,F. R. N., 1948, Report of Conference on Strength of Solids, (Phys. Soc., London) p. 75. NAJAFABADI, R., H. Y. WANG,D. J. SROLOVITZ and R. LESAR, 1991, Acta Metall. Mater. 39,3071. NANNI,P., C.T.H. STODDART and E.D. HONDROS1976, J. Mater. Chem. 1,297. NICHOLAS,M. G., and C. E OLD 1976, J. Mater. Sci. 14, 1. NICOLET,M. A., 1978, Thin Solid Films 52, 415. NIMUS,H., W. = A N D and E. TAGLATJER,1993, Surface Sci. Rep. 17,252. OSGERBY, S., and T. B. GIEEONS, 1992, Mat. Sci. Eng. A 157, 63. Ch-r~m~, U., S. HOFMANN, and M. R ~ L E1993, , in: High-Temperature Ordered Intermetallic Alloys V, eds. I. Baker,R. Darolia, J.D. Whittenberger and M.H. Yo0 (MRS Proc. Symp. Vol. 288; MRS, Pittsburgh) p. 183. OVERBURY, S. H., P. A. BERTRAND and G. A. SOMORJAI,1975, Chem. Rev. 75, 547. PAIDAR,V., 1987, Acta Metall. 35, 2035. PETCH, N. J., 1956, Phil. Mag. 1, 331. PICHARD, C., M. GIJTTMANN,J. RlEu and C. Goux, 1975, J. Physique 36, C4-151. RELLICK,J. R,and C. J. MCMAHON,JR., 1974, Metall. Trans. 5, 2439. R i a , J. R., and J.-S.WANG, 1989, Mater. Sci. Eng. A 107, 23. R O E E R T S - A ~ NW.C., , 1888, Phil. Trans. Soc. A179,339. ROWLANDS, G., and D.P. WOODRUFF,1979, Phil. Mag. 40,459. R~ENAUER F.G., , 1994, Anal. Chim. Acta 297, 197. SEAH, M.P., 1975, Surf. Sci. 53, 168. SEAH,M.P., 1976, Proc. Roy. SOC.A349,535. SEAH,M.P., 1977, Acta Metall. 25,345. SEAH,M.P., 1979, J. Catal. 57,450. SEAH,M. P., 1980% J. Phys. F10, 1043. SEAH,M. P., 1980b, Acta Metall. 28,955. SEAH,M.P., 1980c, Phil. Trans. Roy. Soc. A295 265. SEAH,M.P., 1980d, J. Vac. Sci. Technol. 17, 16. SEAH,M.P., 1981, Mater. Sci. Club. Bull. 64, 2. SEAH,M. P.,1990, in: Practical Surface Analysis, eds. D. Briggs and M.P. Seah (Wiley and Sons, Chichester). 2nd ed., Vol. 1, ch. 7. SEAH,M.P., andE.D. HONDROS,1973, Proc. Roy. Soc. A335 191. SEAN,M. P., and E. D. HONDROS,1983, in: Atomistics of Fracture, ed. R.M. Latanision (Plenum, New York), p. 855. SEAH, M.P., and C. LEA,1975, Phil. Mag. 31, 627. SEAH,M.P., P. J. SPENCERand E.D.HONDROS,1979, Met. Sci. 13,307. SEIDMAN, D.N., 1991, Mater. Sci. Eng. A U7, 57. SUDMAN,D. N., 1992, in: Materials Interfaces: Atomic-Level Structure and Properties, eds. D. Wolf and S. Yip (Chapman and Hall, London) p. 58. SEKI,A., D. N. SEIDMAN, Y. OH and S. M. FOILES,1991, Acta Metall. Mater. 39, 3167. SHINODA,T., and T. NAKAMURA, 1981, Acta Metall. 29, 1631 and 1637. SKELTON, R.P., 1975, Met. Sci. 9, 192. SMITH,C.S., 1948, Trans. AIME 175, 15.
1288
E.D. Hondros et al.
Ch. 13, Refs.
SPENCE,J. C. H., and J. T m $ , 1983, J. Microscopy D O , 147. SUNDARSHAN, T. S., and M. R. LOUTHAN, JR. 1987, Int. Mater. Rev. 32, 121. S ~ HS., ,G. F. Dram and R. 0. RITCHIE, 1981, Metali. Trans. 12A, 1435. SUTTON,A. €?, and R. W. BALLUFFI,1987, Acta Metall. 35,2177. SUZUKI,S., K. ABIKOand H. KITAMURA, 1981, Scripta Metall. 15, 1139. SUZUKI,S., and M. OKU, 1990, Appl. Surf. Sci. 44,151. SWIATNICJCI, W.,S. LARTIGUE,M. BISCONDI and D. BOUCHET, 1990, J. Phys. France 51, C1-341. TAUBBR,G., and H. J. GRABKE, 1978, Ber. Bunsen Gesell Phys. Chem. 82,298. THOMAS, G. B, and T. B. GIBBONS,1980, Proc. Seven Springs Conference, eds. G.K. Tien et ai. (ASM, Metals Park, OH) p. 99. TIPLER,H. R., 1980, Phil. Trans. Roy. SOC.A295, 213. TIPLER,H. R., and D. MCLEAN,1970, Met. Sci. 4, 103. TSAI,N. H.,G. M. POUNDand F.F. ABRAHAM, 1977, J. Catal. 50,200. TSONG,T. T., 1990, Atom-Probe Field-Ion Microscopy (Cambridge UNv. Press, Cambridge). TYSON,W.R., 1975, Can. Met. Quart. 14, 307. TYSON,W.R., 1978, Acta Metall. 26, 1471. VITEK,V., and G. J. WANG,1982, J. Phys. France 43, C6-147. WAGNFX,C., 1961, Z. Electrochem. 65,581. WANG,H.Y., R. NAJAPABADI, D. J. SROLOMTZ and R. LESAR,1992, Phil. Mag. A 65,625. WATANABE, T., 1984, Res Mechan. 11,47. WATANABE, T., 1993, Mater. Sci. Eng. A 166, 11. WATANABE, T., S. KITAMURA and S. KARASHIMA, 1980, Acta Metall. 28, 455. WEI, W., and H. J. GRABKE,1986, Corrosion Sci. 26, 223. WESTBROOK, J. H., 1964, Met. Rev. 9, 415. WHITE,C. L., J. R KElsw and D. N. BRASKI,1981, Metallurg. Trans. 12A, 1485. WHITE,C. L., R. A. PAWETT, C.T. Lnr and S. M. YALISOVE, 1984, Scripta Metall. 18, 1417. WHITTLE,D.P., and J. STRINGER, 1980, Phil. Trans. Soc. A295,309. WILLIAMS, EL., and D. NASON,1974, Surf. Sci. 45, 377. WILLIAMS,T.M., A.M. STONEHAMand D.R. Hnmres, 1976, Met. Sci. 10, 14. WOLF, D., 1992, in: Materials Interfaces: Atomic-Level Stmcture and Properties, eds. D. Wolf and S. Yip (Chapman and Hall, London) p. 1. WOLF, U., E ERNST, T. MUSCHIK,M. W. Fmas and H.F. F I S C H M E1992, ~ Phil. Mag. A 66,991. Wu, R., A. J. FREEMAN and G. B. OLSON,1992, I. Mater. Res. 7,2403. WYNBLATT, P., and R. C. Ku, 1979, Interfacial Segregation, eds. W.C. Johnson and J.M. Blakely (ASM, Metals Park. OH) p. 115. YAN,M., M. SOB,D.E. LUZZI,V. VITEK,G.J. ACKLANLI,M. METHPESSFZand C.O. RODRIGUEZ, 1993, Phys. Rev. B. 47,5571. Yu, J., and C. J. MCMAHON,1980, Metallurg. Trans. 11A, 277. ZHANG,S.H., X.L. HE and T. KO, 1994a, J. Mater. Sci. 29,2655. ZHANG,S.H., X. L. HE and T. KO, 1994b, J. Mater. Sci. 29, 2663. ZHOU, Y.-X.,S.-C.FLT, and C.J. MCMAHON,JR., 1981, Metall. Trans. 12A, 959.
Further reading BRIGGS, D., and M.P. SEAH,eds., 1990, Practical Surface Analysis. 2nd ed., Vol. 1, Auger and X-ray Photoelectron Spectroscopy (Wiley and Sons, Chichester). BRIGGS,D., and M. P. SBAH,eds., 1992, Practical Surface Analysis. 2nd ed., Vol. 2, Ion and Neutral Spectroscopy (Wiley and Sons, Chichester). DOWEN, P. A., and A. MILLER,eds., 1990, Surface Segregation Phenomena (CRC, Boca Raton). JOHNSON, W.C., and J. M. BLAKELY.eds., 1979, Interfacial Segregation (ASM Metals Park, Ohio). KELLY, A., D. W. PAS-, E.D. HONDROSand C. LEA, e&., 1980, Residuals, Additives and Materials Properties. Proc. Conf. (The Royal Society, London); also appears as Phil. Trans. Roy. Soc. A295 1-341.
Ch. 13, Refs.
Intetjmial and sutjace microchemistry
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McLEAN, D., 1957, Grain boundaries in Metals (Oxford Univ. Press). RIV&RE,J. C., 1990,Surface Analytical Techniques (Clarendon, Oxford). SUTTON, A. and R. W. BALLLIFFI, 1995, Interfaces in Crystalline Materials (Oxford Univ. Press, Oxford). WOLF, D., and S. YIP,eds., 1992, Materials Interfaces: Atomic-Level Structure and Properties (Chapmanand Hall, London).
CHAPTER 14
OXIDATION, HOT CORROSION AND PROTECTION OF METALLIC MATERIALS S. R. J. SAUNDERS Division of Materials Metrology, National Physical Laboratory, Queens Road, Teddington, Middlesex, TWII OLW UK
J.R. NICHOLLS School of Industrial and Manufacturing Science, Cranjield Universiv, Cranjield, Bedfordshire, MK43 OAL, UK
R. U! Cahn and P. Haasent, eds. Physical Metallurgy:fourth, revised and enhanced edition 8 Elsevier Science BY 1996
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S.R. J. Saunders and J. R. Nicholls
Ch. 14,8 1
1. Introduction 1.1. Definitions Oxidation and hot corrosion of metallic materials are oxidation reactions (in the chemical sense) that result in the conversion of metals into compounds such as oxides, sulphides or carbides when exposed to high temperatures above about 40OOC. Oxidation processes as the name implies refer to formation of oxide layers and occur primarily during exposure to air or steam. Hot corrosion is the more general term where products other than oxides can form during exposure to air containing contaminants such as sulphurous gases from the combustion of fossil fuel or specific process conditions in petrochemical plants where high levels of carbon activity are common. In some cases exposure to molten salts occurs either directly or by condensation reactions, and in that case the term hot-salt corrosion is used.
1.2. General Protection of material exposed to aggressive environments at high temperatures depends on the properties of the corrosion product which should act as a diffusion barrier to prevent further attack by the atmosphere. Thus the layer should be chemically stable in the environment, prevent diffusion of both the reacting gases and the metal species and finally, of course, in order to maintain its effectiveness as a diffusion barrier, the layer should be mechanically stable, that is it should not crack or spall. The importance of each one of these properties will become clear as various forms of attack are discussed in this chapter. Oxidation and hot corrosion are important forms of materials degradation that often limit process conditions, which can have serious implications for plant reliability and efficiency. For example, power station boilers, petrochemical plant, gas and steam turbines, waste incinerators, diesel and internal combustion engines and nuclear reactors are all prone to degradation by oxidation and hot corrosion. The current drive toward energy saving by improved efficiency while at the same time minimising emission of pollutants places a heavy demand on the materials scientist, since in most cases these requirements mean higher operating temperatures and exposure to more aggressive environments. Clearly the high temperature behaviour of materials, therefore, can be a major factor inhibiting progress, and it is for this reason that the topic has such importance for our society. Clean coal technology is a good example of the sort of developments in process technology that have important implications for the construction materials of the plant. Alternative processes to conventional combustion have been developed which involve fluidised bed combustion with additions of limestone as a sulphur capture agent, or gasification processes where materials are exposed to reducing environments which are relatively rich in sulphurous gases. In the case of fluidised bed combustion, heat exchange tubes must be resistant to the erosive action of particulates in the fluidised bed (sand, coal ash and limestone) and, since the sulphur is retained within the bed, also resistant to attack by the calcium sulphate formed by reaction of the limestone and the
Ch. 14, 92
Oxidation, hot corrosion and protection
1293
sulphurous gases released during the combustion of coal. For coal gasification, on the other hand, process conditions are reducing to promote formation of CH,, H, and CO from the coaVsteam mixture, so that the metallic components only form a protective oxide layer if highly alloyed materials are used, and, furthermore, there is a heightened risk of sulphidation because of the relatively high sulphur potential. Again the process designer wants high operating temperatures to achieve high overall efficiency.
1.3. Outline The thermochemistry of oxidation is considered and illustrated by means of Ellingham diagrams. Kinetics of oxidation of pure metals is developed and discussed in relation to the mechanisms of oxide formation. Alloy oxidation allows for the possibility of selective oxidation, and the criteria controlling the rate controlling processes are described. The importance of the strain tolerance of the protective oxide is then considered in relation to the mechanisms that lead to stress in the layer and the possibility of stress relief. Here the effects of thermal cycling and the mismatch in thermal expansion coefficient of the oxide layer and the metallic substrate are dominant. Materials are not always exposed in simple oxidising environments, so that there is the possibility of competing reactions where oxide, sulphide or carbide may form. The conditions controlling these reactions are described. Again, simple gaseous attack can be further complicated by the presence of a deposit which can be molten or solid; here the chemical stability of the first formed layer may be inadequate, so that, for example, the oxide can dissolve when in contact with a molten layer. Test and measurement methods are an important aspect of any technology, so that a brief introduction is given to this topic before considering life prediction and modelling. Finally, protection is considered where alloy design is inadequate, or where the properties of the bulk (creep strength, for example) are incompatible with the corrosion resistance. Here a range of options is available, and the different processes for the application of protective coatings are described. Coating design is considered mechanical and chemical properties must be matched with those of the substrate. Design philosophies have evolved that exploit graded coatings to optimise both surface and interface properties. For a more detailed account of oxidation and hot corrosion the reader should consult reference books on the topic: KOFSTAD[1988a]; BWKSand MEIER[1983]; KVSASCHFWSKI and HOPKINS [1966]; HAUFFE [1967], RAPP [1983] and EVANS[1960].
2. Fundamentals of oxidation 2.1. Thermodynamics Classical thermodynamics can be used to determine the temperature and gas pressure required to cause the following reaction to take place: M + + 0 , +MO
0)
The Gibbs free energy for this reaction is: References: p. 1357.
1294
S. R. . I . Saunders and . I R. .Nicholls
AG = AGO + R T ~ + a M 'a@ where: AGO = -AGM- 3A@% + AG",, a = activity AGO = standard free energy If AGO 2 0, spontaneous forward reaction will occur. At equilibrium, AG = 0 and equation (2) reduces to the Van 't Hoff isotherm: AGO = -RTln-
'MO
a,
-ag
(3)
Since the standard free energy of elements is zero, equation (2) becomes:
AGGo = -RTln-
'MO
a M .a?
(4)
Equations (3) and (4)describe the conditions at equilibrium where both the metal and oxide are stable, or, in other words, where the formation of oxide on a metal just begins. The equation can be further simplified since the activity of a material in the pure state is unity and under conditions of low pressure and high temperature, the activity of oxygen can be approximated to its partial pressure, thus:
Therefore, knowing the Gibbs standard free energy of oxide formation at a given temperature, we can readily calculate the oxygen partial pressure required for oxidation to occur. Richardson and Jeffes have compiled the standard free energies of formation of oxides as a function of temperature, and fig. 1 shows these data. These diagrams are commonly known as Ellingham diagrams and allow the oxygen pressure for oxide formation at any required temperature to be determined directly. This is done by aligning the point 0 at the upper left of the diagram with the reaction and temperature of interest, and reading off the pressure on the scale on the right of the diagram. For example, for Al,O, at 1000°C, align the point 0 with the line representing the reaction 4/3Al+ 0, = 2/3A120 to,intersect it for 1OOO"C and read off the p(0,) of atm on the p(0,) scale at the bottom of the diagram. Also included on this diagram are scales which give similar information about oxidation in two commonly encountered gas mixtures, CO/CO, and Hm,O; in these cases, however, the critical gas mixture is given by aligning the temperature and reaction of interest with the points C and H, respectively. Similar diagrams can be constructed that indicate conditions for formation of sulphides, carbides, halides or any other reaction product of interest. (See also ch. 5, 8 5). In considering the oxidation of an alloy it is useful to decide which components oxidise preferentially. This may be determined by comparing the free energies of oxide formation; the oxide with the most negative free energy of formation forms preferential-
Ch. 14, $ 2
Oxidation, hot corrosion and protection 10-8
I295
.
10-L
,
c
I
Temperature, ' C lo00 1500
500
-0
K 10:s
,
I
,
I
.10?0
, , ,lo'!
. . I
2000
.
10"
I
Fig. 1. Ellingham diagram for some oxides of importance in the high temperature oxidation of metals and alloys.
ly. Whether a separate phase forms depends upon the concentration of the components and the mutual solubility of the various oxide phases. If no new phases form, the oxidation rate may be changed by solution of the oxides, and as will be discussed later, this alters the number of defects in the oxide, thereby changing diffusion rates.
References: p . 1357.
1296
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23. Oxide structure It is necessary to have an understanding of oxide structure before proposing models for the formation and further growth of the oxide layer. An oxide growing on a metal or alloy may be amorphous or crystalline; generally the very thin films are amorphous and thicker films crystalline. As will be shown later, oxide growth involves the movement of ions and electrons and it is important therefore to understand the restrictions imposed by the structure of amorphous or crystalline oxides upon the movement of ions and electrons. 2.2.1. Amorphous oxides These oxides can be classified according to their bond strengths; i.e., glass or network formers with covalent bond strengths greater than 300 kJ mol-', intermediates with bond strengths between 300 and 200 kJ mol-' and network modifiers with ionic bond strengths [1932], proposed certain rules for glass-forming of less than 200 kJ mol-'. ZACHARIA~EN oxides, which are useful although not universally applicable because of kinetic considerations. Oxides of the form MO and M20 do not form networks; M203 can form a network if the oxygen ions form triangles around each metal atom; MO, and M,O, form networks if oxygen tetrahedra form around each metal atom; MO,, M,07 and MO, are also probable network formers. The cations are tightly bound in a network former because of the high bond strengths. A large ion moves more easily than a small ion in such an open network because the small ions are mainly incorporated in the network so that anion transport would be expected in network formers. Since network modifiers do not form extensive networks and the bond strengths are generally low, cation transport is expected in these oxides. 2.23. Crystalline oxides Crystalline oxides which form as the result of oxidation of a metal are generally non[ 19721) and thus exhibit both ionic and electronic conductivity. stoichiometric (KOPSTAD The non-stoichiometric oxide may be either metal-rich or metal-deficit. Oxides having an excess of metal ions maintain electrical neutrality by incorporating additional electrons; this type of compound is known as an n-type semiconductor. Oxides deficient in metal maintain electrical neutrality by the formation of positive holes; a positive hole occurs where the valence state of the cation in the oxide is higher than usual; these compounds are p-type semiconductors. The non-stoichiometry of these oxides can be accommodated in a number of ways. Figure 2 depicts an n-type semiconductor with the excess metal
Zn" 2n"
o'
0' Zn" tn.. e- o'
Zn"
Z*..
Zn" Zn"
o'
Fig. 2. Zinc oxide as an example of a metal excess (n-type) semiconductor.
2n"
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Oxidation, hot corrosion and protection
1297
located at interstitial positions in the lattice. Alternatively, the oxides may have oxygen ion vacancies. Figure 3 illustrates the structure of a p-type semiconducting oxide which can in principle have both oxygen interstitials and metal vacancies. Generally only metal vacancies are found, although in compounds of the CaF, type interstitial anions occur, for example Tho,, UO,, etc. The defect structure that has been described allows easy movement of ions, cations via interstitials or vacant cation lattice sites and anions via vacant anion sites or occasionally via interstitials. It can be seen, therefore, that an n-type semiconducting oxide may allow anion or cation transport, whereas p-type oxides generally permit only cation transport. Table 1 classifies some oxides according to their semiconductingnature, where amphoteric oxides may have either metal excess or metal deficit (n- or p-type).
2.3. Kinetics An essential concept in oxidation of metals and in hot corrosion generally is that the reaction product can act as a barrier between the metal and the environment thereby controlling rates of attack. Thus, the properties of the reaction product often determine subsequent rates of reaction and the mechanism of the oxidation process is reflected in the kinetic laws that are followed. By accurately determining the kinetics of the process from relatively short-term experiments it should be possible to determine long-term behaviour by extrapolation. However, this can only be done if there is no change in the process and, as will be discussed subsequently, discontinuities are frequently observed in oxidation kinetics. There is greater certainty about the reliability of such an extrapolation if the initial data represent a significant fraction of the lifetime to be estimated. The movement of ions within an oxide layer, necessary for oxide growth, may involve a thermally activated diffusion process, or assistance by an electric field may reduce the overall activation energy to zero. The two basic mechanisms of oxide growth conveniently divide the subject into (i) the thin oxide film region (Le., growth by a field assisted mechanism), and (ii) the thick film region (Le., growth by thermal activation). The transition between the two processes usually occurs at a limiting thickness of about 100 nm. In general, the kinetics of oxidation can be described by four laws, viz., logarithmic, cubic, parabolic and linear (fig. 4). All four kinetic laws are found in the thin film region. Linear for the first few monolayers of oxide growth, logarithmic up to about 50-100 atom layers of oxide and cubic or parabolic at still greater thicknesses. In the thick-film region a parabolic law is commonly encountered as well as linear growth laws. Often a transition
Fig. 3. Nickel oxide as an example of a metal deficit @-type) semiconductor. References: p . 1357.
from parabolic (protective oxidation) to linear (breakaway oxidation) is observed and this is normally associated with mechanical failure of the protective oxide. 2.3.1. Thin film region At low temperatures, electrons from the metal are able to pass through an oxide film to the oxygen adsorbed at the outer surface; electrons do this by the tunnel effect (a quantum-mechanical process by which electrons of lower maximum energy than that of a barrier possess a finite probability of penetrating that barrier). Electrons pass through the initial cheinisorbed film to the outer surface and form oxygen anions, and metal cations form at the metauoxide interface. It has been shown that the potential drop across the film is approximately 1 V so that with a film thickness of 5 nm, the field strength would be of the order of lo7 V cm-'. The strong electric field generated is able to pull ions through the film.
tFig. 4. Oxidation kinetics.
Ch. 14, $ 2
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Oxidation, hot convsion and protection
Logarithmic kinetics have been interpreted as resulting from the rate-controlling step being either anion movement or the electron tunnelling process which creates the field across the oxide layer initially (Mm [1947]; H A ~ and E ILSCHNER[1954]), and an inverse logarithmic law can result when cation transport is the slow step (HAUPFE [1966]; EVANS[ 19601; CABRERA and Mom [ 19491). Cubic or parabolic kinetics follow from the prediction that the rate of diffusion of ions is proportional to the magnitude of the electric field, which under constant potential decreases with increasing thickness; a parabolic law follows for p-type semiconductors and a cubic law for n-type semiconductors (CABRERA and MOTT[1949]; ENGELLet al. [1954]). Linear kinetics are observed in the formation of the first few monolayers of oxide where the growth process is controlled by surface diffusion and adsorption.
2.3.2. Thick film region In the thick film (thermally activated) region, parabolic kinetics are derived by assuming that film thickening is controlled by diffusion in a concentration gradient (x2=2h‘t=kJ where w is the oxide thickness, t is the time and (or 4) is the parabolic rate constant). Linear kinetics are observed when the barrier provided by the oxide layer fails, so that either the oxide fails completely and there is direct contact between the metal and the gas, or partial mechanical failure occurs so that a constant thickness of protective oxide remains. Most practically important oxidation problems relate to the thick oxide film region and thus most emphasis will be given to the consideration of oxidation processes in this technically important region:
h’
a) Wagner’s theory WAGNER[1975] derived an expression for the rate of thickening of the oxide expressed in terms of the specific conductivity of the oxide, the transport numbers of cations, anions and electrons and the free energy decrease during the oxidation reaction. This simplified derivation assumed that the conductivity is independent of the oxygen pressure and in fact this is not the case, so that a more complex expression results. There are only three metal oxide systems that accurately follow Wagner’s parabolic equation; viz. Cu/Cu,O, Co/CoO and FeFeO, and the very good agreement between the calculated and observed rate constants is evident from Table 2. Wagner’s theory of parabolic oxidation refers only to monophase scales, but in practice most oxides layers are multiphase, for example FeO, Fe,O, and Fe,O, all occur on iron when oxidised at temperatures above 600°C. It has been shown, however, that the rate of growth of each phase obeys parabolic kinetics and that their relative thickness is independent of time but temperature-dependent. b) Deviations from Wagner’s theory One of the main assumptions in Wagner’s model is that the scale is compact, but frequently scales are observed to be porous, and therefore under these conditions deviations from Wagner’s theory are to be expected. A porous scale can form during cationic or anionic growth. For example, consider the case of nickel where upon oxidation a compact outer layer of NiO and a porous inner layer, also of NiO, are References: p . 1357.
Ch. 14, $ 2
S.R J. Saunders and J. R.Nicholls
1300
Table 2 Measured and calculated rational rate constants*
2cu + 0, T) cu,o Temp. 1000°C Pressure, atm
k, calc.
k, obs.
eq cni2sec-’
eq cm-* sec-’
8.3 x 1.5 x 2.3 x 3.0 x
6.9 x 4.8 x 3.4 x 2.1 x
6.2 x 4.5 x 3.1 x 2.2 x
lo-’ lo-’ 10” lo4
lo4 lo4 10-9 10-9
lo4 10-~
lo+ 10-9
2co + 0, -+2COO
Pressure, 1 atm Temp. OC
k, calc.
k, obs.
eq cm-* sec-’
eq cm-’ sec-’
lo00 1148 1350
1.25 x lo4 5.15 x lo4 3.15 x lo4
1.05 x IO4 1.30 x 104 4.65 x lo+
2Fe + 0,-+2FeO Pressure, 1 atm Temp. OC
k, calc.
k, obs.
eq cm-’ sec-l
eq
800 897 983
2.5 x lo4 1.1 x 10” 2.8 x lo4
2.6 x lo4 1.2 x lo4 3.3 x lo+
sec-’
* The rational parabolic rate constant, k, is used when the reaction rate is expressed in terms of equivalents of M,O, per square centimetre per second. formed. A single layer of nickel oxide forms during the early stages of growth of the oxide, but as the layer thickens a duplex structure develops, which consists of an inner region of equiaxed, fine-grained crystallites and an outer region of large columnar crystals, The inner layer is generally more pronounced on less pure materials, and is believed to be due to the presence of impurities segregating to the grain-boundaries, thereby inhibiting grain growth. Tracer diffusion studies have shown that the outer layer grows by movement of nickel vacancies along the grain-boundaries, and the inner layer by molecular oxygen penetration along microcracks and fissures which are present in the outer layer due to the build-up of stress in that layer. In the case of nickel oxide, compressive stresses result because of the constraints imposed on the oxide layer by the receding metal. A dissociative mechanism has been proposed by MROWEC[1967] for the formation of scales of this type. Metal atoms enter the scale at the metal/oxide interface and diffuse through the existing scale and react with oxygen at the oxiddoxygen interface to form new oxide. There must be corresponding movement of cation vacancies in the opposite direction so that a cationic vacancy flux arrives at the metaVoxide interface and must
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Oxidation, hot corrosion and protection
1301
somehow be removed. Whilst the film is thin, plastic deformation of the oxide is sufficient to allow continued adherence to metal, but as it thickens the oxide becomes stronger and can no longer deform, so that longitudinal cracks form at the metal oxide interface and the rate of transport of metal from the substrate to the oxide decreases. However, because of the chemical potential gradient in the scale, outward diffusion of metal can still occur. This disturbs the equilibrium at the inner surface of the scale which leads to an increase in chemical potential of oxygen at that surface and eventually oxide decomposes to form molecular oxygen, cations and electrons. The growth of an inner layer due to the presence of molecular oxygen is then possible and the processes described are illustrated in fig. 5 which shows the dissociative mechanism for double layer formation in a single-phase scale. This explanation is disputed by KOFSTAD[1985] who proposed that pores form due to plastic deformation and creep caused by growth stress in the scales. For scale growth controlled by diffusion, a gradient in defects will be developed through the scale such that a lower concentration of defects will occur in the outer layers of the scale; this will have the effect of promoting faster diffusional creep in these outer layers, so that there will be greater stress relief there and will thus avoid the development of the porous structure observed in the inner layers. K ~ P P E N BENDER and SCHUTZE [1994] have demonstrated that “pseudo”-plasticity can occur by a micro-cracking and healing mechanism. Pseudo-plasticity requires compressive deformation in one or two directions and the possibility of oxide grain movement in at least one direction perpendicular to the compressive deformation axis. The other major assumption often used in applying Wagner’s theory is that bulk lattice diffusion processes occur. This is only valid at high temperatures (>-0.6 Tq),and most industrially relevant materials are used at lower temperatures. Thus, for polycryslalline oxides, grain-boundary diffusion becomes the dominant process. This is shown very clearly by referring to fig. 6 which shows the parabolic rate constant for the oxidation of nickel as a function of temperature compared with calculated lattice and grain-boundary
Fig. 5. Schematic representation of the dissociative mechanism. References: p . 1357.
Ch. 14, 92
S. R. J. Saunders and J. R. Nicholls
1302 1400 I
1000 800 700 600 I
I
500
300
400
l l I I Growth of NiO Solid points: Measured in oxidation 0 Rhines and Connell(1977) A Atkinson et al. (1982)
2
measurements
-0 -14-
-
-
lattice diffusion -16- of Ni in NiO
.-
I
6
8
m -
10 12 1041r(IC’)
14
16
Fig. 6. Comparison of the oxidation rate of nickel with lattice and grain-boundary diffusion of Ni in NiO (ATKINSON119871).
diffusion constants (ATKINSON [19871). By extrapolation of the experimental data shown in fig. 6, it can be seen that it is only at temperatures of about 120OOC and above that the activation energies for lattice diffusion and oxide formation would be similar (230-250 kJ mol-’). At the lower temperatures the rate of oxidation is increasingly greater than would be predicted assuming control by bulk diffusion in the oxide lattice, activation energies being reported in the range 155-170 kJ/mol. The effect of prior cold-work in the nickel is to increase the oxidation rate, but the observed rate law is usually less than parabolic. Both these observations suggest that the rate of oxidation is controlled by grain-boundary diffusion in the oxide; the less than parabolic rate observed in the coldworked material occurs because the initially fine-grained oxide coarsens during the oxidation process, thereby eliminating some short-circuit diffusion paths. Models have been developed (e.g., SMELTZERet al. [1961]) to describe the oxidation reaction where the rate is controlled by dual lattice and grain-boundary diffusion in which the effective diffusion coefficient is given by: D,
= D,
+ 2(D‘S)/g
(6)
where g is the grain size normal to the growth direction, S is the grain-boundary width, and D’ and D, are the diffusion coefficients of the boundary and the lattice, respectively. ATKINSON [1987] used this model to calculate parabolic rate constants for the oxidation of nickel where grain-boundary diffusion dominates, and there was good agreement between the calculated values and those obtained experimentally, as shown in fig. 6.
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2.4. Properties of oxide layers It will have become clear from the preceding discussion that the electrical and mechanical properties of an oxide layer are of paramount importance in controlling the course of the oxidation reaction. We will now consider how these properties may be modified in order to control the rate of oxidation of metals.
2.4.1. Electrical properties (diffusion) The defect structure of an oxide may be modified by a small addition of other elements and, if the oxidation process proceeds by the Wagner mechanism, one might expect changes in the oxidation rate to occur. For example, the addition of a monovalent metal to NiO, a cation-deficit p-type semiconductor, would be expected to reduce the cation vacancy concentration and hence also decrease the oxidation rate. Conversely, if ions of a higher valency are added to a p-type oxide, then the cation vacancy concentration will increase and thus also the oxidation rate. The reverse situation would be expected for n-type oxides. Similarly, if the anion sub-lattice is altered, for example, by the incorporation of N3-, this would increase the number of vacant anion sites in an n-type semi-conductor and hence the oxidation rate would be increased if anion transport were the rate controlling step. This mechanism has been proposed to explain the differences in oxidation rates of titanium in pure oxygen (where a low value is observed) and in air (a high value). For a dopant to be effective the solubility, which is determined principally by ionic size, must be sufficiently high. A further probable restriction is that for growth processes controlled by cation diffusion the dopant should be added from the gas phase since new oxide is formed at this interface, and for anion diffusion the dopant should be added from the alloy. Thus it can be seen that the addition of minor amounts of other elements can cause large differences in the oxidation rate where the Wagner mechanism is operating. It has already been noted that the Wagner mechanism has only been quantitatively established in three cases. However, there are many more examples where qualitative effects of the type just described confirm the validity of this mechanism. In the case where grain-boundary diffusion is rate-controlling, the effect of impurities can be to alter the diffusion rates per se or to influence the microstructure. In the latter case segregation effects are likely to dominate, leading to a reduction in grain size and a consequent increase in the oxidation rate. Very few data exist to indicate which is the controlling effect, but, in the case of nickel, theoretical modelling of the grain-boundary structure by DUFFYand TASKER[1986] and experimental measurements of grainboundary diffusion rates (ATKINSON and TAYLOR[1986]) indicate that impurities often decrease grain-boundary diffusion rates (fig. 7). Thus it would appear that the effects of impurities in increasing the oxidation rate of nickel most probably results from a reduction in the oxide grain size with a consequent increase in the number of shortcircuit diffusion paths.
References: p . 1357.
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1304
T (OC)
11001000900800
-8
-16
I
700
600
500
'\
Ni(lattice)
I
I
-
in the present study
\
\
\\Cr (lattice)
-20 6
I
I
I
7
8
9
I
I
10 11 1 0 ~ (K-') 1 ~
I
I
I
12
13
14
Fig. 7. Arrhenius plot summarising the diffusion coefficients for various impurity cations in the NiO lattice, grain-boundaries and dislocations (ATKINSON and TAYLOR [ 19861).
2.42. Mechanical properties (stress generation and relief) Very recently a critical review of this topic was undertaken in a Workshop on the Mechanical Properties of Protective Oxide Scales which was published in Volume 12 of the Journal Materials at High Temperatures, 1994. It was pointed out some time ago by PILLING and BEDWORTH [19231 that, depending upon the growth processes involved, an oxide layer could develop internal stresses as a result of the growth process. If, for example, cation transport is the rate-controlling process, new oxide would form at a free surface, that is, the oxide/gas interface, and thus no stress would be imposed. However, if the oxide grows by oxygen transport, new oxide would form at the metalloxide interface and the metal lattice would then have to expand or contract to accommodate the newly formed oxide. If the volume of oxide were
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Oxidaiwn, hot corrosion and protection
1305
larger than the volume of metal consumed, the oxide would be in compression, and, if vice versa, the oxide would be in tension. The ratio, oxide volume/metal volume, is called the Pilling-Bedworth ratio, and for most metals it is greater than one. Growth stresses have also been observed in oxides which form by cation transport, so that other factors need consideration. It is clear that unless some stress-relief mechanism operates, build-up of growth stresses will ultimately result in cracking or blistering of the oxide layer, and, as was discussed earlier, this would result in a change of kinetics from parabolic (protective oxidation) to linear (non-protective or breakaway oxidation). In addition to stresses arising from the oxide volume/metal volume ratio, other mechanisms for stress generation have been proposed and recently reviewed by STOTT and ATKINSON1994. These include: epitaxial stresses, recrystallisation within the scale, the composition of the metal or oxide may change during oxidation, point defect stress, oxide growth within the oxide layer, coulombic forces due to the electric field produced by differing mobilities of the ions, and finally thermal stress (i.e., stress arising from thermal cycles where a thermal expansion coefficient mismatch exists between the oxide and substrate) which is usually by far the most important cause of stress generation in oxides formed on engineering iron- and nickel-base alloys. That zirconium oxidises to produce a coherent oxide is surprising, since anion transport is favoured in the oxide and, with a large Pilling-Bedworth ratio, one would expect considerable stresses to be developed which could cause the oxide to crack or spall. It is evident, therefore, that various mechanisms which allow stress relief by plastic deformation must operate. The following processes have been proposed, dislocation glide, grainboundary sliding, mechanical twinning, viscous flow in amorphous oxides, HerringNabarro or diffusion creep, and finally, dislocation climb processes are often observed and are probably the most generally applicable mechanism for stress relief in oxides. Where plastic deformation of oxides is insufficient for benign stress relief, catastrophic stress relief will take place by spallation and scale cracking. EVANS[1994] has developed spallation maps which can be used to predict regions of oxide stability as a function of the temperature drop during thermal cycling. These maps are based on two possible modes of scale spallation in which buckling can occur when the interface is weak and the oxide relatively strong (usually when the oxide is thin) or the wedging mode of failure where the interface is strong and the oxide is relatively weak (usually for thick oxides). Figure 8 is an example of such a spallation map, calculated on the basis of measured oxide properties for alumina formed on a ferritic steel. In simplistic terms, scale failure occurs when the in-situ stress exceeds some critical value (uc).The critical stress can be evaluated using simple fracture mechanics conand NICHOLLS [1988]), siderations to predict (EVANSand CANNON[1987]; HANCOCK whether through-thickness cracking (tensile failure) or scale spallation (compressive failure) results. For the case of tensile failure, the critical stress is given by:
where K,, is the fracture toughness and B is a measure of the defect size present in the oxide. K,, is related to the energy of fracture by K,:=2 Ear, where Eo is the oxide References:p . 1357.
1306
Ch. 14, 53
S. R. J. Saunders and J. R. Nicholls
0
1000
Calculated f:
0
lo-'
100
10'
102
Oxide Thickness, mic
Fig. 8. Notional spallation map for alumina on Fecralloy steel (EVANS[1994]).
elastic modulus and yo is the energy required to form a unit area of new oxide surface. Hence fracture mechanics analysis (EVANSand CANNON[1987]; HANCQCK and NICHOLLS[1988]; NICHOLLSand HANCOCK [ 19891) and the assessment of fracture strain energies (EVANS [1988]) are alternative approaches to solving the same problem. The in-situ stress acting at a defect, within the oxide or at the oxide/metal interface, is the summation of all stresses acting on the defect. This stress will include growth stresses, thermally induced stresses and directly applied stresses which may be tensile, compressive or bending. Hence to quantify oxide fracture, the state of stress, fracture toughness and defect distributions present within the oxide or at the mewoxide interface (depending on the mode of failure) must be capable of being assessed. This topic was recently reviewed in depth by HANCOCK and NICHOLLS[1994].
3. Oxidation of alloys The same principles apply to the oxidation of alloys as have been discussed for pure metal oxidation. The problem is more complex however, because there are at least two constituents which can be oxidized. The added complexity makes prediction of alloy oxidation rates extremely difficult. 3.1. Selective oxidation Selective oxidation of a component of an alloy will occur when the free energy of formation of its oxide is more negative than that of the oxide of the other alloy constituents. Dependent upon the difference in free energies, temperature, oxygen pressure and concentration, it is possible for the oxide of the minor constituent to form to the exclusion of all other oxides. WAGNER[1956] derived an expression for determining the
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Oxidation, hot corrosion and protection
1307
minimum atomic fraction NA(,,,in)in an alloy AB for exclusive formation of a coherent layer of the oxide AO, given that the A 0 will form in preference to BO:
where d is the interdiffusion coefficient in the alloy, Z, is the valency of A in the oxide,
k, is the parabolic rate constant for growth of oxide AO, and c is the number of gram atoms of metal per unit volume. This equation should be treated as predicting only the necessary minimum concentration; it is not a sufficient condition in itself. For example, for the copper-aluminium (0.0008 wt% Al); however, in system at 800°C the calculated value N,,-)=0.004 practice even when N, = 0.11 (5 wt% Al) some copper oxides still form. Examination of the compositional changes that occur at the alloy/oxide interface as a result of selective oxidation shows that the concentration profile is determined by two opposing processes, consumption by the oxidation process and replenishment by diffusion within the alloy. Clearly, if a protective oxide is removed for any reason and a depleted alloy is exposed to the oxidising environment, the protective layer may not be able to reform; it is this situation which is usually responsible for breakaway oxidation. Exclusive formation of Al,O,, Cr203or less frequently, SiO,, is most commonly used as a means of protecting many industrially important alloys from oxidation. Generally, the oxidation rate of
I
* A I in A1203 lattice
6
7
o~rin Cr o lattice 2 3
1 o y
, , 8
i SiO, amorphous
I
,
I
,
9
10
11
12
1 0 7 ~K-'
Fig. 9. Observed parabolic rate constants for alloys forming alumina and chromia scales compared with oxide lattice diffusion rates and the rate constant for the formation of silica on silicon; the solid circles refer to the lattice diffusion rates of Cr and A1 in Cr,O, and Al,O,, respectively, at 1180°C. References: p. 1357.
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S. R. J. Saunders and J. R. Nicholls
Ch. 14, $ 3
alumina-forming alloys is lower than that of chromia-forming alloys, but the lowest rates of all result when silica scales form (fig. 9). In the latter case this is due to the fact that silica is amorphous at temperatures up to about 1200°C and thus grain-boundary diffusion is no longer a factor controlling mass transport. In cases where several different oxides form, the scale forms in such a way that the oxide with the highest dissociation pressure (most positive AGO) forms at the oxide/gas interface and that with the lowest dissociation pressure at the metaYoxide interface. In other cases selective oxidation results in an enrichment of an alloy constituent in the oxide layer relative to the alloy concentration without exclusive formation of its oxide. If the oxides are mutually soluble the major constituent of the scale will have its defect structure altered in a manner previously described by incorporation of the solute. Where the oxides do not form solid solutions, composite oxide layers result. A commonly encountered situation is the formation of complex oxides (double oxides, spinels, etc.). This may happen when oxides of the alloying components react with one another to form a more stable oxide. Ni-Cr, Ni-A1 and Fe-Cr alloys provide good examples of this situation where spinel layers result. It is often observed that diffusion rates are much lower in spinels than in normal oxides, and protective oxide scales may consist of these complex oxides. WOOD and STOP [1983] have summarised the oxide growth mechanisms for these cases and the various possibilities are illustrated schematically for the nickel-aluminium system as shown in fig. 10, where alloys rich in aluminium eventually form an inner layer of alumina, or for more dilute alloys where the alumina forms only as an internal oxide. In the first case, after simultaneous formation of both oxides, the slower growing oxide (A1,OJ forms in the inner regions by the displacement reaction Ni-AI rich in AI
Ni-AI low i n A I AI,O,
NiO
a)* Alloy AI,O, N,iO
A!,O,
Alloy N,iO
Alloy AI,O, NIO
Alloy AI203 NiO
scale
Alloy + A1203 su bscale Fig. 10. Schematic representation of transient oxidation of Ni-A1 alloys (WOOD and STOTT[1983]).
Ch. 14, $ 3
Oxiahtion, hot comswn and protection
1309
3NiO + 2Alk + A1203 + 3Ni2+,while at the interface between the A1203and the NiO layers a solid solution reaction occurs to allow formation of the complex oxide NiAl,O,. The case of internal oxidation is discussed in more detail in the next Section.
3.2. Internal oxidation When oxygen dissolves in the alloy, the more active component (most negative AGO) may form a precipitate of oxide particles within the alloy. This phenomenon is known as internal oxidation or sub-scale formation. Internal oxidation requires a higher diffusion rate of oxygen in the alloy than that of the alloying component. This means that an oxygen gradient is established and the dissolved oxygen reacts with the most reactive [1959] studied the transition from internal to constituent to form the oxide. WAGNER external oxidation and has derived a model based on the fact that the cross-section available for oxygen diffusion is reduced by the presence of internal oxide. Above a certain alloy concentration, oxygen diffusion slows down to the extent that only an external oxide forms. Lowering the oxygen partial pressure also encourages external oxidation since the solubility of oxygen in the alloy is reduced. If internal oxidation occurs, the oxidation rate is usually high because a protective oxide layer is not established. The transition to external oxidation will result in a reduction of the oxidation rate since now a complete barrier between the alloy and the environment exists.
3.3. Intermetallic alloys Intermetallic materials are increasingly considered for engineering applications because of their high temperahue strength and low density, and a comprehensive review [1994]. Generally these materials are based on has recently been published by DOYCHAK the aluminide or silicide systems where protective alumina or silica scales can form. A particular problem relating to intermetallics, especially NbAl, and MoSi,, has been the so-called “pest”, where accelerated oxidation occurs at an intermediate temperature (550-850°C) after a brief incubation period. Early work on the mechanisms of this process proposed that it was stress-induced, but more recent work (GRABKEetal, [1991]; DOYCHAK etal. [1991]) suggests that the rapid degradation of NbAl, is the result of inward diffusion of oxygen through an outer layer of Nb,O,, continued inward diffusion of oxygen along the aluminium-depleted intermetallic grain-boundaries and precipitation of internal alumina within the intermetallic in the region unaffected by oxidation, as illustrated schematically in fig. 11.
3.4. Scale adhesion As was mentioned in the introductory section, a major problem that must be addressed once a protective film has been established is the control of adhesion or mechanical integrity. It has already been pointed out that the protective oxide layer can be subjected to stresses dependant upon the operating growth mechanisms, which usually results in compressive stresses. Additionally, however, thermal cycles (cooling to mom temperature) also result in compressive stress being applied to the oxide scale since most References: p. 1357.
1310
S.R. J. Saunders and J. R. NichoNs (a)
Ch. 14, $ 3
Amorphous external scale
N bA13 matrix
NbAljmatrix
Fig. 11. Schematic of a) the accelerated oxidation mechanism on NbAl, at 727°C and b) the initiation of pest degradation (DOYCHAK and HESBUR [1991]).
oxides have a lower coefficient of thermal expansion than the metal or alloy. The additive effect of these compressive stresses is often sufficient to cause scale spalling. Small additions of “active elements” (i.e., elements with a high affinity for oxygen) and notably the rare earths, are known to be very effective in promoting the formation of an adherent oxide layer that is resistant to thermal cycles. The active element can be added in elemental form or as an oxide dispersoid. In the latter case, a novel series of alloys produced by mechanical alloying - the so-called oxide dispersion strengthened (ODS) materials which have been developed primarily for enhanced high-temperature strength, also show good oxidation resistance (MICHELS[1977]). The alloy MA6000 (Ni-15 Cr, 4.5 Al, 4 W, 2.5 Ti,2 Mo, 2 Ta, 1.1 Y,O,, by weight) is now being used in some gas turbine applications. Early versions of this type of material, e.g., thoriadispersed (TD) alloys, were evaluated some time ago and STRINGERet al. [1972] reported that TD-Ni20Cr had excellent resistance to spalling and reduced oxidation rates compared with the simple binary alloy. As well as improving adhesion of the scale, the active element addition also reduced the growth rate and the concentration of chromium or aluminium required for preferential formation of the scale, particularly for the chromia-forming alloys. WHITTLEand STRINGER [1980] reviewed the various theories that have been proposed to account for this effect, which include enhanced scale plasticity, formation of a graded seal, modification to the oxide growth process, stronger chemical bonding at the interface, elimination of voids by inert oxide particles acting as vacancy sinks and, finally, oxide protrusions into the alloy which act as “pegs” to improve adhesion. More recently, LUTHRAand BRIANT[1986] SMEGGILet al. [1984] and LESS[1987] have proposed that segregation of sulphur to the scale/alloy interface is responsible for the poor adhesion of the oxide, and
Ch. 14, $ 4
Oxidation, hot cormsion andprotection
1311
that the effect of the “active element” is to scavenge the sulphur present in the alloy and so restore the intrinsically strong bond between the oxide and the substrate. LUTHRAand BRIANT [1986] have been unable to confirm this effect experimentally, but both FUNKENBUSCH et al. [1985] and SMIALEK[1987] have reported results that indicate that for high-purity alloys with sulphur contents of less than about 10 ppm, adherent oxide scales were formed on alumina-forming nickel-base alloys without rare earth alloying and RAPP [1994], proposed that the active element blocks active additions. PIERAGGI sites, i.e., interfacial dislocations, that support diffusion growth at the scale/alloy interface, and thereby alter the growth mechanism and hence also the adhesion of the scale. All these various mechanisms were recently reviewed by STRAWBRIDGE and Hou [1994], and it is clear that no one theory can satisfactorily explain all the experimental observations.
4. Multi-component atmospheres 4.1. Phenomenology
In many industrially relevant cases, alloys are exposed to environments containing a mixture of gases so that there exists the possibility of forming phases other than oxides. Commonly sulphides, carbides and to a lesser extent chlorides are formed, and in almost every case formation of these phases results in greatly increased rates of attack. This is because diffusion rates in sulphides and carbides are higher than in the corresponding oxide, and chlorides are often volatile, thus providing no barrier to further reaction. However, in most practical cases, alloys are used that contain reactive elements such as aluminium or chromium and there is usually sufficient oxygen present in the environment to allow formation of an initially protective oxide layer. Nevertheless, ingress of sulphur, carbon or chlorine through this protective layer can occur, leading to formation of the sulphide, carbide or chloride of the oxide-forming constituent within the alloy so that it becomes depleted in that constituent. Thus, should that oxide layer be damaged by growth stresses or differential contraction induced by thermal cycles, the alloy would be unable to reform a protective oxide. Increased rates of attack may not always be entirely due to alloy depletion effects. For example, oxide doping could occur, and it has been suggested that in the case of Cr203,S- is introduced which leads to an increase in the chromium vacancy concentration and a corresponding increase in diffusion rates. In addition, the size of the impurity ion may cause a dilation of the crystal lattice and hence also encourage diffusion. Since most metallic sulphides, carbides or halides are less stable than the oxide, continued reaction results in the conversion of these second phase particles to the oxide which does not form a coherent protective layer, as shown, for example, in the following reaction:
2MS + 0,-+ 2MO + S ,
(9)
It can be seen from this reaction that sulphur is released and is then available for further attack upon the alloy. This form of cyclic sulphidatiodoxidatiodsulphidationis frequently observed and leads to catastrophic rates of attack. References: p . 1357.
Ch. 14, 94
S.R. J. Saunders and J. R. Nicholls
1312
The process by which these aggressive species penetrate the protective oxide layer is often greatly disputed in specific cases, but in principle there are two possibilities: (a) bulk diffusion or (b) short-circuit diffusion via grain-boundaries or microcracks within the oxide layer. It will be shown in a following section that the nature of the diffusion path can sometimes be determined from the reaction products.
4.2. Prediction of reaction products A powerful method of determining the nature of the product when an alloy reacts with a complex gas mixture is to construct the relevant phase stability diagram, and with a knowledge of the composition of the gas mixture the thermodynamically stable phases are indicated. For example, the phase stability diagram for the Fe-0-S system at 800°C plotted as a function of the partial pressures of oxygen and sulphur is shown in fig. 12. Thus, for a typical combustion gas mixture containing CO,, O,, N,, H,O and about 400 ppm SO, the composition can be determined in terms of p(0,) and p(S,) and is shown as “X’ in the diagram. It can be seen that Fe,O, is the stable phase. However, if p(0J is reduced and p(S,) is increased as, for example, in a coal gasification environment, the composition of this gas mixture given as ‘Y“‘ in the diagram indicates that FeS would be the stable phase. It should be noted that the axes selected for these diagrams do not relate to the major constituents of the gas mixture, but because equilibrium is assumed in the calculation, the same result would be obtained no matter what oxygen- or sulphurcontaining species were used as a variable. For simplicity, oxygen and sulphur are usually chosen when plotting diagrams of this nature. The phase stability diagram can be constructedby considering all possible equilibriain the three component system and writing equations in terms of these components, for example: 5 C
1
I
1
I
/
r
\
\
FeS
Y
-5
2 E
I
-10
co”
n
-
-15
- 20
-25 I 15
I
-20
-15
-10 log Po,(G)
I -5
X
0
5
Fig. 12. Phase stability diagram for the Fe-0-S system at 800°C (partial pressures in pascals).
Ch. 14, $4
1313
Onidation, hot corrosion and pmrection
-+ FeO
Fe+*O,
where the equilibrium constant,
K=
aFeo ‘We)
= lo9 at 800”C
.
thus, knowing the equilibrium constant, the partial pressure of oxygen at the FeFeO phase boundary can be calculated and plotted on the diagram. Similarly all other relevant equilibria can be calculated at the temperature of interest. Use of these diagrams clearly shows the benefits of addition of elements such as aluminium and chromium in hightemperature corrosion-resistant alloys. Figures 13 and 14 are phase stability diagrams of Cr-0-S and A 1 4 4 systems, respectively, for the same conditions as were plotted in fig. 12 for Fe-0-S. It can be seen that oxide formation covers a much greater range of conditions, so that for an iron-aluminium or iron-chromium alloy, protective oxides could form in cases where, in the absence of these alloying additions, iron sulphide would appear. For example, the region delineated by a rectangle in figs. 13 and 14 defines typical conditions encountered in a coal gasificationsystem, where normally iron sulphide would form (point “Y”in fig. 12). 5
0
-5 a
a. h
(3
U
m -10
a
-s? -1 5
-20
Cr = 0 -25 -25
-20
~ 1 5
-10
-5
0
5
log P 02 (G)Pa Fig. 13. Phase stability diagram for the 0-04system at 800°C (partial pressures in pascals). References: p. 1357.
1314
Ch. 14, $ 4
S. R. J. Saunders and J. R. Nicholls
5
I
,
0
-5 0
a h
c3
v
m
A1203
-10
Q. 0
0
-1 5
\
-2 0
-25 -25
-20
-1 5
-10
log
-5
0
log N A l = O 5
P 02 (G)Pa
Fig. 14. phase stability diagram for the A I M system at 800°C (partial pressures in pascals).
An even more complex problem, however, is to consider relevant phase diagrams of an alloy in a multi-component gas mixture where oxygen-, carbon- and sulphur-containing species may result in oxide, carbide and sulphide formation. Commonly all the phase stability diagrams for individual elements are superimposed so that an approximate description is obtained. Clearly, a complete representation of the equilibria could only be depicted by using a three-dimensional diagram. These are very difficult to calculate and even more difficult to construct. Approximations, therefore, are required, and it is convenient to consider the behaviour of an element in an alloy by plotting diagrams of a single element using a reduced activity, This approximation neglects the possibility of mixed or complex oxide formation, but it is now possible to carry out the necessary calculations to produce phase stability diagrams for mixed oxide systems (BARRYand DINSDALE [1994]; PELTONetal. [1979]). If the gas mixture contains more than two active components a useful approach is to maintain one component at a constant level whilst varying the other two and then systematically alter components of interest to cover the range required. Clearly these are approximations and this needs to be borne in mind when reviewing the information available from these diagrams.
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Oxidation, hot corrosion andprotection
1315
It must be remembered, however, that these diagrams will only indicate the presence of a phase by assuming equilibrium conditions, but kinetic factors are also very important. For example, in some conditions it has been observed that oxide phases only form exclusively at partial pressures of oxygen about lo4greater than the values for the thermodynamic boundary for transition from sulphide to oxide formation, giving rise to [1983]). Figure 15 shows the position of the the so called "kinetic boundary" (NATESAN kinetic barrier for an austenitic stainless steel (AIS1 310) at 650°C on the superimposed phase stability diagrams for iron, chromium and nickel, together with the region defining typical coal gasification atmospheres shown as a rectangle in the figure. Two points of interest arise from this diagram. Firstly, that the kinetic barrier for this alloy lies within the composition used in coal gasification so that in many plants this alloy would be unsuitable, and secondly, the position of the line is also very close to the FeSFeO phase boundary. This latter observation may explain the origin of the kinetic barrier with chromia-forming iron-based alloys, since if iron diffusion occurs, sulphides would then form on the surface and promote sulphur migration through the chromia layer. 4.3. Reaction path
If one assumes that an oxide layer forms in a multi-component atmosphere and aggressive species such as carbon-, sulphur- or chlorine-containinggases are also present and are able to penetrate the oxide, then the nature of the corrosion product may depend upon the diffusion path in the oxide. If only bulk diffusion of the aggressive species occurs, then, since diffusion can only take place in a concentration gradient, the partial pressure of, for example, sulphur would be lower at the metal/oxide interface than at the oxide/gas interface. If, on the other hand, the oxide contained microcracks and the gas were able to penetrate the oxide, the gas equilibrium would no longer be controlled by the bulk gas composition because the partial pressure of oxygen would be set by the and metavmetal oxide equilibrium and not by that in the gas phase (STROOSNIJDER QUADAKKERS [1986]). In most practical cases where protection is conferred by formation of alumina or chromia, the partial pressure of oxygen at the metalloxide interface is much lower than that in the gas phase; thus for a gas containing SO,, for example, this could result in a large increase in the partial pressure of sulphur at the metauoxide interface compared with that in the bulk gas mixture. The effect of reducing the partial pressure of oxygen is clearly seen by considering the following equilibrium: so, 0,+ is, where the equilibrium constant
In the case of a nickel chromium alloy exposed to a combustion gas mixture containing and p(0,) = lo3Pa. However, at the alloy/oxide Pa, and interface where Cr,O, forms, the partial pressure of oxygen would equal this in turn would cause p(S,) resulting from 400 ppm SO, at the mewoxide interface
400 ppm SO, at 8OO"C, the p(S,) =
References: p. 1357.
1316
S. R J. Saunders and J. R. Nicholls
Ch. 14, 8 4
Kinetic Barrier \
0
0 -5 a
Q,
P -10
-15 -35
- 25
-15 log p (02)-in Pa
-5
Fig. 15. Phase stability diagrams for the Cr-0-S, Fe-0-S and Ni-0-S systems at 65OoC,indicating the range of compositions found in coal gasification atmospheres and the kinetic barrier for exclusive formation of chromia observed for AIS1 310 exposed in these conditions.
to equal 1015Pa. In many ways this is a somewhat unrealistic calculation since in most cases there may not be complete access to the gas at the metauoxide interface, and only a reduced concentration of SO, may be present. It is useful to consider the minimum concentration of SO, at that interface that is necessary for sulphide formation to occur. This can be shown to be defined by:
Figure 16 is a schematic phase stability diagram in which three possible conditions are illustrated for penetration of the oxide by either the bulk diffusion or the microcrack mechanism, and the limiting sulphur dioxide pressure is marked by a dashed line. In region 1 the p(S0,) is lower than the minimum value for sulphide formation so that sulphide cannot form by either mechanism. In region 2 transport of molecular sulphur dioxide through the scale would result in sulphide formation for p(S02) greater than the minimum, but not by bulk diffusion processes, whereas in region 3 bulk diffusion of ions would result in sulphide formation. Table 3 shows values for p(SOJ(min) for various oxides as a function of temperature, where it can be seen that the presence of highly stable oxides (large negative AGO) results in very low values of p(SO,)(min).
Ch. 14, $ 5
Oxidation, hot corrosion and protection
1317
Fig. 16. Schematic thermodynamic phase stability diagram for M-0-S systems, indicating the minimum p(S02) for sulphide formation (see text for details).
It is important to realise that if the sulphidising gas does not contain oxygen (if it is H,S for example), then no mechanism exists for increasing the p(S,) at the oxide/alloy interface, so that this rather dangerous condition which promotes rapid sulphidation would not apply.
5. Hot-salt corrosion Hot-salt corrosion is a form of high-temperature attack on metals and alloys that occurs in the presence of a molten deposit; metalwork in power station boilers, waste incinerators, gas turbines and diesel engines is susceptible to this degradation process. Most work in this area has been related to the gas turbine but the underlying principles can be readily extended to other spheres. We will consider the environment, the phenomenology of the corrosion process, the mechanisms proposed to account for the corrosion behaviour and, finally, briefly consider the implications of new types of fuel for the gas turbine.
5.1. The environment Molten deposits containing largely NqSO, andor V,O, form on turbine blades and nozzle guide vanes in the hot section of the engine. The deposits form as a consequence of combustion of sulphur- and vanadium-containing fuels which may also contain alkali References: p . 1357.
1318
Ch. 14, 5 5
S.R. J. Saunders and J. R. Nicholls
Table 3 Minimum values of p(S0J in the atmosphere for sulphide formation at the metaUscale interface by molecular transport of SO, through the scale.
P(SO3 (Pa)
Temperature, O C
co
Ni
1.6 x 2.5 x 7.9 x 6.3 x
400 600
800 1000 1200
- .
10-7
lo-* 10' 103
1.0 x 6.3 x 6.3 x 7.9 x
-
Cr
Fe 10-8
lo4 lo-' 10-1
2.0 x 1.6 x 4.0x 7.9 x 4.0 x
10-l4
lo4 10-5
10-3 lo-'
1.3 x 6.3 x 3.2 x 4.0x 2.5 x
10-33
lodu lo-'' lo4
metal impurities which are usually sodium-rich, either as NaCl or sodium naphthenate. Alternatively, the alkali metal may be ingested as an impurity in the air used for combustion, and again NaCl (seawater) is generally the main impurity, particularly for the marine gas turbine. Since the air:fuel ratio is about 501, it is clear that air quality is very important in determining the quantity of sodium sulphate formed in the combustion gases. The following equation indicates the process by which sodium sulphate forms: 2NaC1 -k H,O
-k
SO,
+ 30,+- N+S04 +- 2HCl
(13)
The combustion gases containing these impurities are in the temperature range 1100-14OO0C so that these compounds are usually in the vapour phase. However, when the combustion gas reaches the turbine blades and nozzle guide vanes where metal temperatures may be about 85OoC,condensation will occur if the vapour pressure of Na,S04 is greater than the equilibrium value for that temperature, i.e., if the dewpoint is reached. It is generally accepted that there will be severe corrosion whilst the deposit is molten. If the temperature is lowered so that the deposit solidifies, the rates of attack that are observed are greatly reduced. A problem of particular importance in gas turbines is the rate at which the sodium sulphate-formingreaction proceeds. It should be realised that the gas velocity in a turbine is about 300 m s-' and the distance between the combustion chamber and the blade can be as little as 0.3 m or for larger industrial engines 2-3 m, so that the sulphate forming reaction must be complete in about 1-10 ms. In the case of NaCl particles in the gas stream, clearly the size of the particle will affect the conversion process. It has been calculated, for example. that at 1300°C and 10 atmospheres (typical flame conditions) a 10 p m NaCl particle would only be 10%converted to sulphate in 5 ms (MCCREATH [1983]). In normal turbine operation particle sizes would be 10 p m or less, and therefore when NaCl is the main contaminant some unreacted NaCl particles would be expected to hit the blades before evaporation or conversion processes are complete. Qpically then, the environment in which gas turbine blades find themselves is one in which molten N%S04 and/or V,O, can deposit from the vapour, and also, depending upon the operating conditions of the turbine, there is the possibility of impaction by NaCl (seawater) particles; in addition NaCl vapour and HC1 gas may also be present.
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Oxidation, hot corrosion and protection
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5.2. Phenomenology of hot-salt corrosion Two types of attack can be observed. The more usual form occurs in the temperature range 800-95OoC, known as Type Z hot-corrosion and is characterised by the formation of a thick porous outer oxide layer, an intermediate layer of internal oxide particles mixed with depleted alloy and an inner region containing internal sulphide particles which leads the attack; fig. 17 is a micrograph illustrating this type of attack. It is generally observed that a maximum in rate of attack occurs at about 850-900°C; at higher temperatures, deposition of molten salts is reduced since the vapour pressure is insufficient for condensation. The other form of attack is known as low-temperature corrosion or Type ZZ and is usually observed in marine gas turbines in the temperature range 650-750°C. In this type of attack large oxide and sulphide-filled pits form, but there are no internal sulphide particles, and fig. 18 is a micrograph illustrating this type of attack. Owing to the large temperature gradient that exists along the length of a turbine blade, it is possible for both types of hot-salt corrosion to occur on one component.
5.3. Mechanism of attack We will consider in detail the more usual form of Na,SO,-induced hot-salt corrosion, 'Qpe I, and the principles applied will provide a basis for interpreting the special case of low temperature hot-corrosion or 'Qpe 11. A brief description of vanadate-induced attack will also be given.
5.3.1. Na,SO,-induced attack No single mechanism can be proposed to explain all observations but there are three main theories, viz., a) sulphidation, b) fluxing theory and c) pseudo-scale formation, and we will briefly consider the relevance of each.
a
Fig. 17.Optical micrograph of Nimonic 105 blade after service in a gas turbine, showing conventional hot-salt corrosion attack, QpI hot-corrosion. References: p . 1357.
1320
S.R. J. Saunders and J. R. Nicholls
Ch. 14, $ 5
I
Fig. 18. Optical micrograph of IN738LC blade after service in a gas turbine, showing low temperature pitting attack, ’Qpe I1 hot-corrosion.
a) Sulphidation One of the first theories to explain NqSO, induced corrosion was proposed by SIMONS et al. [ 19551 who suggested that sulphidation of the alloy occurred, as indicated by the following reactions: Triggering phase: NqSO, + 3R + Na,,O + 3 R 0 + S (14)
M+S+MS (15) NqSO, + 3MS + 4s + 3 M 0 + NqO (16) 4M + 4s + 4MS (17) Enhanced oxidation phase: 2M + 2MS + 2M.MS (18) 2M. MS + 0, + 2 M 0 + 2MS (19) The “R’ in the triggering phase is a reducing agent and this is usually the alloy, but pyrolytic carbon resulting from poor combustion of the fuel may also act as a reducing agent. The sulphidation reaction proceeds in two phases, a triggering phase followed either by the autocatalytic phase or the enhanced oxidation phase dependent upon the thickness of the melt present. With thick NqSO, layers, where there is limited oxygen access, the autocatalytic reaction is observed, but with thin layers we find the enhanced oxidation phase where the eutectic M.MS phase formed is oxidised to MO and more sulphide. The enhanced oxidation phase should be self-sustaining once the initial sulphide has formed with oxidation of the sulphide releasing sulphur for further sulphide formation. Presulphidation of many alloys often leads to continuous sulphide formation by this mechanism when the material is subsequently oxidized in air or oxygen. Autocatalytic phase:
(b) Fluxing theory
The fluxing model was initially proposed by BORNSTF~IN and DECRESCENTE [1969], and developed by GOEBEL and PETTIT[1970], and can best be illustrated by considering the decomposition of NqSO, into its acidic and basic components.
Ch. 14, $ 5
Oxidation, hot corrosion and protection
1321
NqSO, + N%O (base) + SO, (acid) (20) With basic fluxing of nickel oxide the following reaction then takes place: NiO + N%O + NqNiO, (21) and for acidic fluxing: NiO -+ SO, + NiSO, (22) The processes involved in basic fluxing are schematically represented for the case of nickel in fig. 19. It can be seen that basic fluxing is favoured by processes that increase the oxide ion activity in the Na,SO, melt. Figure 20 is a phase stability diagram for the A 1 4 4 system which illustrates acidic and basic fluxing of alumina. The formation of oxide under the molten salt would result in a decrease in the p(03 at the melt oxide/ interface and a consequent increase in the p(SJ so that there is now sufficient driving force for diffusion of sulphur into the oxide, thereby decreasing the p(S0,) and promoting basic fluxing. Acidic fluxing will be favoured by low temperatures because of the increasing stability of SO,, or by reactions of alloying constituents with Na,SO, that deplete the oxide ion concentration by, for example, complex oxide formation. MOO, + NqO 4 Na,,MoO, (23) RAPP and GOTO[1981], pointed out that in order to sustain the dissolution reaction, a solubility gradient must be present in the molten salt layer. Thus oxide can dissolve at the oxide/melt interface, migrate down a concentration gradient to a site of lower solubility where precipitation occurs, and this is believed to be the origin of the external porous oxide layer found on samples exposed to this form of attack. Type I hot-corrosion is generally believed to be the result of basic fluxing or of direct attack on the substrate by the melt through a cracked oxide layer, and Type II is the result of acidic fluxing by
Dense NiO
NiO particles
Fig. 19. Schematic illustration of the basic fluxing reactions of N+S04 with pure Ni.
References: p. 1357.
1322
Ch. 14, 3.5
S. R.J. Saunders and J. R. Nicholls
I
5 Basic fluxing N
0 A
//
4
I,/ A I ~ o ~ * o ~ -
-
I
0 .
, /
J0
2A10q2
-3
10-16
1
1-
n 0
-7
I
10-10.6
'
/10-5 Al2O3- 2Al3'+ 302-
/'
p J
/'I
- J'
z y ' '1
[3] -'
I
I I
T'
I
I
A12 S3
1
a low-melting eutectic of, for example, 38% NiS04.N%S04(m.p. 681°C) which are
stabilised by a high pso, (LUTHRAand SHORES [1980]). The solubility of oxides relevant to the hot-salt corrosion of superalloys has been measured by Rapp and co-workers, and fig. 21 is taken from a review of their work, (RAPP [1987]). It is noteworthy that there is a difference of about six orders of magnitude in basicity (i.e., +%o) between the solubility minima for the most basic oxides
c 0
Fig. 21. Oxide solubility in Na,SO, at 927°C as a function of oxide ion activity (RAPP [1987]).
Ch. 14, $ 5
1323
Oxidation, hot corrosion and protection
and the protective alumina, chromia and silica which are all acidic oxides. Chromia scales are known to be more resistant than alumina to acid fluxing. It is suggested that chromia scales are able to support a buffering action and resist large changes in the basicity of the melt, and that the solubility minimum corresponds to the conditions in the gas turbine. The solubility data do not immediately suggest the cause of the superiority of chromia relative to alumina, but RAPP [1987], suggests that this might be the result of the faster growth of chromia. Another possibility is that the alumina scale is more vulnerable to mechanical damage, which would permit direct attack on the alloy. (c) Pseudo-scale In this theory (REISING and KRAUSE [1974]) the Na2S04acts merely to transport oxygen to the alloy surface and the Mi2+ ions produced by the oxidation process are mobile in the liquid medium and move away to the liquid/gas interface where NiO particles form, producing a porous scale. Rapid oxidation ceases when the porous scale has absorbed all the Na2S04 removing it from contact with the alloy surface and hence a protective oxide layer can be reestablished. Experimental observations indicate that the NiO :Na$O, ratio is usually constant and about 50 : 1, whereas the fluxing theory which has the following overall reaction: 2N%S04+ 9Ni + 2N+0 + 6NiO + Ni,S, (24) suggests that the NiO :N+SO, ratio should be 3 : 1. However, the effects of alloy chemistry and corrosion are probably better explained by the fluxing theory than by the “pseudo-scale” model. The fluxing model is now largely used in interpreting effects of hot-salt corrosion and has gained wide acceptance.
5.33. Vanadateinduced attack Residual fuels contain relatively large amounts of sodium and especially vanadium. Figure 22 shows the phase diagram for the Na-V-O system which indicates formation
900
I
I
I
I
I
I
I
I
y, \
I
-
V
-
I
V2°5
Mole percent Nag0
Fig. 22. The quasi-binary Na,0-V,05 diagram (&MY
and WILSON[1973]).
References: p . 1357.
1326
S.R. J. Saunders and J. R. Nicholls
Ch. 14,$ 6
solution in Na,,SO, may result which would be expected to increase corrosion rates. Clearly this is a complex situation and much work is required to understand the chemistry of these Na-K-Cl-S-O molten salts. In addition to the greatly increased contaminant levels in the fuel, coal-fired gas turbines will be subjected to erosion from ash particles which despite filtration may remain entrained in the combustion gas. Some erosion tests have shown that coal ash particles 2-20 p m in diameter are capable of removing large amounts of metal in very short periods. Some testing has been carried out, however, and although severe corrosion has been observed in some cases, the catastrophic rates that might have been predicted have not been observed. The blades quickly became coated with a deposit which was largely composed of coal ash and it is possible that the “pseudo-scale” theory may explain the somewhat better corrosion behaviour. The coal ash could be acting as a porous sponge, soaking up the sulphate melt and removing it from contact with the underlying alloy.
6. Test and measurement methods The performance of materials at high temperatures is dominated by their resistance to mechanical deformation and attack by the environment. Thus, in attempting to select materials for service in these conditions it is important to have reliable means of predicting their behaviour, and whilst some information can be obtained by using the experience gained in the operation of existing plant, introduction of new materials or of different operating conditions requires data to be produced from laboratory testing procedures. High temperature corrosion can involve attack in the gas phase or the combined effects of hot gases and molten or solid deposits. As might be expected, if solid deposits can form, an increase in gas stream velocities would result in erosion rather than deposition but this is beyond the scope of this chapter. The mechanical behaviour of oxide layers is an extremely important factor determining the overall performance in corrosion and erosion processes, so that the available methods of ascertaining the mechanical properties of thin layers will also be considered.
6.1. Monitoring oxidation processes The methods used to measure the rates of oxidation, i.e., processes forming only oxide layers, are considered in this section but, as will be evident, many of the techniques are applicable to the other reactions and processes discussed in this chapter. In the first place, consider the simplest process in which a test is carried out isothermally.
6.1.1. Isothermal testing The purpose of tests of this type is to determine the rate at which the specimen is converted to oxide, and the most commonly used procedure is the gravimetric method in which the change in mass of the sample is recorded. Usually a mass gain corresponding to uptake of oxygen is observed, but if oxide spalling occurs this mass gain is reduced
Ch. 14, 56
1327
Oxiaktion,hot corrosion and protection
by the amount of metallic oxide lost, and in extreme cases net mass losses can be recorded. The simplest form of the experiment is to place the sample directly into the furnace, withdrawing it periodically and measuring the change in mass. This experiment clearly imposes temperature cycles, which are discussed later, but in some cases, particularly for thin adherent oxide layers, similar results are obtained with experiments carried out continuously. Figure 24 shows schematically a typical arrangement for conducting such experiments with continuous weighing of the sample using a microbalance. Other methods involving measurement of changes in pressure or volume, or of the optical properties of the surface, such as ellipsometry are also used, but a complete description of these and other methods is unfortunately beyond the scope of the present chapter. The reader is r e f e d to KUBASCHEWSKI and HOPKINS[1967] for full details. Oxidation rates also have been determined by measurement of the natural frequency of a freely suspended specimen by BRUCEand HANCWK [1969]; HANCOCK[1970]; HUR~Tet aZ. [1975], and as will be shown later in the section on mechanical stability of oxides, cracking and spalling of the scale can also be detected. The characteristic frequency of the first mode of vibration,f, is related to the dimension and the elastic modulus, E, of the composite rod as follows: f2
=
(4'73)4
16w Z3 (m,
+ Am)
[E, c4 + E,(b4 - c4)]
where E, and E, are the elastic moduli of the metal and oxide, m, is initial mass of the rod, Am is the mass gain, c is the radius of metal and b is the radius of the composite.
Balance beam stainless
Quartz or alumina rod
I ntroduct
+L
To pumps
Fig. 24. Schematic drawing of an automatic recording thermobalance. References:p . 1357.
1328
S. R. J. Saunders and J. R. Nicholls
Ch. 14, 46
Thus, by solving the above equation, mass changes can be derived, together with the elastic properties of this composite, as follows:
f" - f," = A
M, - M,
M, + M,
where: A is defined by m=m,(l +A) f , is frequency of the unoxidised rod M,, p,, E, are the atomic weight, density and elastic modulus of the metal M,, p,, E, refer to an oxide and P = (Eo ~ r Mo)/(Em n PO Mol
6.13. Cyclic oxidation The superimposition of thermal cycles in an oxidation experiment, cyclic oxidation, is used to encourage scale failure due to spallation (LOBB[1984]; BARRETT and LOWELL [1975]; BARRETTet al. [1978]). Hence cyclic oxidation tests are used to monitor both scale adherence and the ability of the scale to successfully repair after repetitive scale failure. The performance of materials is generally monitored gravimetrically, either continuously (LOBB[1984]; BARRETTand LOW~LL [1975]) or in a discontinuous manner (BARRETTand LOWELL[1975]; BARRETTet al. [1978]) although other monitoring methods have been used. For example, measurement of the resonant frequency of the specimen during thermal cycling has been successfully used to detect (HANCOCK[19701) the onset of scale failure, before gross spallation, and hence mass loss was observed. Figure 25 illustrates typical data produced by gravimetric tests. Curve (a) represent good performance with little scale failure; curve (b) is intermediate behaviour, that is initially protective but after a limited test duration becomes non-protective, while curve (c) shows poor behaviour with scale spallation occurring from the onset of the test. Curve (b) is characteristic of materials that have a limited reserve of stable scale forming elements, such as chromium, aluminium and silicon, so that after repeated scale failure and repair the morphology and composition of the scale changes, possibly forming less protective spinels, for example. These less protective scales spa11 more readily and rapid mass losses are recorded (BARRETT et al. [1978]).
6.1.3. Mechanistic studies Various experimental procedures have been devised specifically to obtain information about the mechanism of oxidation and should be included in any discussion of test methods. When considering mechanisms of oxide growth it is important to know the transport path (lattice or grain-boundary diffusion), where oxide growth is occurring, Le., anion or cation movement, the diffusion coefficients of the more mobile ions and the equilibrium defect concentration. Tracer and marker techniques can be used to determine the nature of mobile species and transport paths and the ROSENBURG [1960] method, using interrupted kinetics experiments, gives information about diffusion coefficients and defect concentrations. The methods most commonly used are marker and tracer and these are discussed below.
Ch. 14, 56
1329
Oxidation, hot corrosion and protection
(a)
Fig. 25. Schematic diagram of typical mass changes versus time during cyclic oxidation: (a) protective, (b) protective initially, (c) spalling.
i) Marker methods If inert material, platinum or a stable oxide, is placed on the surface of unoxidised metal, its final position in the oxide scale will give information about the transport processes that have taken place, as is illustrated in fig. 26. If the scale is pore-free the final position of the marker ideally reflects the transport processes that have taken place. However, the presence of pores can result in the marker being displaced. For example, in cases where an outer dense layer forms over an inner porous region, the marker is Marker
Me t a l diffusion Metal
(a)
Metal (b)
Metal (C )
Fig. 26. Ideal location of inert markers after lattice diffusion-controlled growth of oxide scales: (a) metal diffusion predominant; (b) oxygen diffusion predominant; (c) simultaneous metal and oxygen diffusion. References: p. 1357.
1330
S. R J. Saunders and J. R. Nicholls
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frequently found at the interface between the porous and dense layers even though cation lattice diffusion is faster than anion diffusion. In this case, the most likely mechanism is that molecular oxygen has penetrated the outer layer as the result of cracking of the scale or of the development of microchannels due to build-up of stress, so that the final position of the marker reflects movement of material but not solid-state diffusion processes. It is also suggested that with large markers, the marker itself can partially obstruct the diffusion process and some plastic flow can occur around the marker. ii) Tracer methods In this method, a radioactive form of one of the reactants in the oxidation process is used. This can be done by either depositing a thin layer of radioactive metal on the surface of the specimen or by carrying out the experiments with radioactive oxidant at some stage in the experiment; “0is commonly used in oxidation experiments. ATKINSON et al. [1979] have calculated idealised profiles for various limiting cases which are reproduced in fig. 27, where it can be seen that for the case of a scale, MO, growing by vacancy diffusion the profile depends on the effective charge, a,of that vacancy. Where grain-boundary diffusion takes place it has been assumed that there is no exchange of metal between the grain-boundary and the oxide lattice and thus the cations are immobilised. In fact, some exchange does occur and this would have the effect of broadening the profile. In most practical cases, some combination of the limiting cases referred to above is usually found, so that the resultant profile will be an appropriate mixture of the simple cases.
6.2. Mechanical failure of oxide scales By careful design of a cyclic oxidation experiment the susceptibility of a material to spalling can be assessed. Hence cyclic oxidation tests provide a qualitative method of assessing scale adherence. To quantify scale adherence, and the onset of spalling, controlled cooling experiments are necessary. Continuous monitoring permits the onset of first fracture to be detected such that from a knowledge of the temperature drop the fracture energy for scale spallation can be calculated. This approach has been used by EVANSand L ~ B [B 19841 to evaluate the interfacial fracture energy of chromia-rich scales formed on a 20Cr/25Ni/Nb-stabilised stainless steel after oxidation in CO2/2%COat 930°C.
6.2.1. Internal stress measurements Two methods have been widely used to determine the internal or growth stresses present within oxide scales; these are X-ray stress analysis (e.g., LUTHRAand BRIANT [1986]) and methods based on the deflection of a thin metallic strip oxidising on one surface only. X-ray stress measurement techniques are usually evaluated at room temperature, although some facilities are available with a high temperature capability (STOUT et al. [1989]). Hence the internal stress determinations include both growth stresses and thermally induced stresses during the cooling cycle. The X-ray determination of stresses is based on the measurement of lattice strains in a set of lattice planes of known
Ch. 14, $ 6
1331
Oxidation, hot comsion andprotection
1.C 0
E
NI vac. diffusion
? V i 0
.c
E!
Y
c 0.5
8C 0, -0
aJ
.0 U)
E
I-
O
z
0 0.5 1.0 Normalised scnle thickness, x I X
oxide / 0 2 ( g ) inter face
1
-
z
-
o
1.0
Scale thickness,x IX
Metalloxide interface
b
0.5
0
1
0.5
lo
Oxide thickness, x l X
Fig. 27. (a) Metal tracer in a scale of MO growing by metal vacancy diffusion only. (b) Distribution of M and '*Otracers in a scale growing by short circuit diffusion. (c) Distribution of M and "0tracers in a scale growing by short circuit diffusion and lattice diffusion (ATKINSONera[. [1979]).
orientation and then correlating these lattice strains with the strain calculated from elastic theory (LUTHRAand BRIANT[1986]). This approach is known as the sinz+ method. The determination of internal and growth stresses by direct deflection measurements on a thin metal beam at temperature (ROYand BURGESS119701) has been widely used by Huntz and co-workers (DELAUNAY etal. [1980]; HUNTZ[1988]). Figure 28 schematically illustrates this method of measurement. The thin metallic strip under test is only References: p. 1357.
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S.R. J. Saunders and J. R. Nicholls
Ch. 14, 5 6
Thermocouple
Alumina support
To recording mechanism Fig. 28. Schematic diagram of the deflection study apparatus.
permitted to oxidise on one side by coating the second, usually with silica. Oxidation causes the strip to bend with the degree of deflection, a measure of the stress generated within the oxide as defined by: E,X;D u, = 3L2XO
where D is the beam deflection, L is the beam length, X, and X, are thickness of the metal and oxide, respectively and E, is the elastic modulus of the metal. This measured stress is only a part of the total stress due to oxidation, as a residual stress component remains after beam deflection due to the tensile elastic strains generated within the metal beam. This method does not take account of plastic deformation of the substrate by creep, for example, and recently a detailed analysis has been carried out EVANS[1995] in which an analytical solution was developed from a relationship derived from the
Ch. 14, $ 6
,
Oxidation, hot corrosion and protection
-51
1333
Fecralloy oxide stress
3 5
- 400.0
0.5
1.0
2.0
1.5
2.5
v
Locat ion in oxide, m
Fig. 29. Stress distribution in an alumina scale formed on Fecralloy at 1100°C (EVANS[1995]).
balance of forces and the balance of momentum. Figure 29 is an example of the results obtained using this method for the case of a Fecralloy steel oxidised at 1100°C.
63.2. Detection of scale failure Three methods have been used to date to detect scale cracking both during thermal cycling and under conditions of applied load. As discussed in the section on cyclic oxidation measurements, controlled cooling experiments can reveal the critical temperature drop necessary to cause through-scale cracking (increased oxidation rates are observed) or scale spallation (mass losses are recorded) (LOBB [19841; EVANS and LOBB [1984]; BAXTER etal. [1984]; ROLLSand NEMATOLLAHI [1983]). However, this method is restricted to metal/oxide systems where mass changes induced by scale cracking can be readily detected. The vibration technique (as described in section 6.1.1) can also be used to detect the onset of scale cracking. It is very sensitive to crack formation within the oxide which results in a reduction of the stiffness of the composite system and hence a discontinuity in resonant frequency is observed (see fig. 30) when the oxide scale cracks. Continued cracking would result in a continual decrease in the recorded resonant frequency. Should oxide repair occur then a frequency transition is observed with the frequency recorded after repair approaching that observed before the cracking event. More recently, acoustic emission has replaced the vibration technique as a route to detect in situ early scale failure, and its use in the determination of oxide fracture was recently reviewed by SAUNDERSet al. [1994]. This technique is not limited to particular specimen geometries, has similar sensitivities to scale cracking and can be monitored automatically. Acoustic emission has found wide usage in monitoring cracking during thermal cycling (KHANNA [1985]; CHRIS~Letal. [1987]), superimposed Stress (SCHUTZE [19853) and during isothermal oxide growth (PRICEand BENNETT [19891). Figure 3 1 shows References: p . 1357.
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S.R. J. Saunders and J. R. Nicholls
Ch. 14, 96
Thermogravi metry
Mass gain
I
I
I
1
I
1
I I
I Resonant frequency
Acoustic emission
I I
I
'
I
'
I
I I
1 ,
;
I
I
f
/
I
A.E. events
Ti me
Fig. 30. Schematic response to breakaway oxidation, monitored using three methods.
the typical coupling arrangement of an acoustic transducer to a specimen through the use of an acoustic waveguide. Figure 31 specifically relates to monitoring acoustic events during the cracking of oxides under applied tensile loads (SCH~TZE[198S]), however, similar geometries can be used to monitor scale failure under cyclic conditions or during isothermal oxidation by combining acoustic emission with thermogravimetric studies (BENNETT e t d [1989]). Acoustic microscopy in which an acoustic signal is focused onto a sample using water as a coupling medium can also be used to detect scale cracking.
6.2.3. Measurement of the macro defects (cracks, voids and pores) present in an oxide scale As discussed earlier in this section, scale failure results when the in situ stress exceeds some critical value necessary to propagate a defect. Hence from a knowledge of the
Ch. 14, $6
1335
Oxidation, hot corrosion and protection
i
1 2 3 4 5
Tensile test specimen Corrosion test specimen Specimen grips AE - Transducer AE- Waveguide
Fig. 31. Schematic of an acoustic emission test arrangement.
stress to cause fracture, the growth stress and the defect distribution in the scale, a measure of the scale fracture toughness and scale fracture energy can be determined. HANCOCKand NICHOLLS [1988] proposed a method for assessing the composite defect size in an oxide containing multiple voids and pores. The analysis follows guidelines proposed for the assessment of defects in welds, and requires that each defect is idealised and compared with its neighbours to assess if any interaction occurs. Interacting defects are considered as a single defect and the assessment procedure continues, until a single composite equivalent defect results. Using this defect size, values of fracture toughness can be evaluated. Evidently, this process requires detailed metallographic analysis. This approach, however, has been applied (NICHOLLSand HANCOCK[1989]) to a number of scales containing active elements and has clearly demonstrated that a major role of active element additions is to reduce the macro defects present within the oxide scales with consequent benefit for scale adherence and spalling resistance.
6.3. Mixed oxidant tests Many industrial applications involve exposure of high-temperature components to complex gas mixtures, and it is necessary to devise relevant laboratory test procedures. In designing the test it is important to consider whether the atmosphere being simulated References: p . 1357.
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S, R. J. S a d e r s and J. R. Nicho1.k
Ch. 14,56
is equilibrated or unequilibrated, and, also in the context of the laboratory environment, whether adequate safety precautions exist since frequently the gas mixtures to be used contain toxic or combustible components. In this section consideration will be given to factors controlling the composition of the atmosphere and the experimental arrangements required to monitor gas composition, including any modification to experimental arrangements necessary to allow safe operation and introduction of samples.
6.3.1. Control of gas composition Normally, equilibrated gas mixtures are required, and therefore, it is common practice to assume that the main oxidising (in the chemical sense) species controls reaction rates, so that the partial pressures of oxygen (po,),sulphur (p,,), halogenic gases (for example, pq) and the activity of carbon (a,) are controlled by establishing relevant gas equilibria. It is relatively straightforward to carry out calculations to determine the partial pressure in these simple binary gas mixtures, but in multi-component atmospheres calculation of the equilibrated gas composition requires the use of iterative routines such and BARRY [1989]). The composition of the gas used in the as the MTDATA (DAVIES test will, of course, control the type of scale that forms (see section 4.2). The choice of gas mixture is usually determined by a direct simulation (for industrial atmospheres or environments containing high partial pressures of reactive species) or by using a gas with an equivalent partial pressure of active species for low po , ps,, etc. KOFSTAD[19891 has questioned this latter approach, since he considers that tke specific components of a gas mixture can individually control scale defect concentrations or microstructure thereby altering rates of attack. For example, it is known that H&O or CO/CO, mixtures of equivalent p ( 0 , ) yield different rates of attack. Representative testing, therefore, should be carried out with exact simulations of the relevant gas mixture where possible. In flowing gas streams, which are usually used to avoid depletion effects of active species (often present in only small quantities), it may be necessary to preheat the gas by passing it over a heat exchange system or in a serpentine path through the hot-zone of the furnace before contacting the samples. Some gas mixtures are slow to equilibrate so that use of catalysts is necessary, and a good example of this is the S0$3O3 equilibrium and WORRELL [19761). In which is readily established over a platinum catalyst (LUTHRA
Ch. 14, 96
Oxidation, hot corrosion and protection
1337
other cases the presence of the metal or oxide on the specimen surface is an effective catalyst, for example, iron, nickel and iron oxide. However, this catalytic activity may be poisoned by components of the gas mixture such as hydrogen sulphide (GRABKEand WOLFE19871). The nature of a hot corrosion experiment requires that the gas mixture flows in a thermal gradient, so that care is required to ensure that thermal diffusion effects do not cause unmixing of the gas by differential diffusion of the components in the temperature gradient (JOST [1952]). Generally, the heavier components will tend to concentrate in the cooler regions of the furnace for high convection velocities and vice versa for low velocities.
6.3.2. Experimentalprocedures A paramount consideration is safety. Acceptable toxicity levels are given in the Registry of Toxic Effects of Chemical Substances, and the laboratory should be regularly checked to ensure that safe levels exist in the ambient atmosphere. Monitoring the gas composition is, of course, desirable and gas chromatography and mass spectrometryare commonly used to determine the composition at ambient temperature, but it is not usually possible to measure gas compositions at the test temperature by these methods. Use of solid electrolyte zirconia probes to measure p(0J of the gas mixture within the hot-zone is being used increasingly (KIUKKOLA and WAGNER[19571). This method relies upon the establishment of the following electrolytic cell: W02 (air)fzrofi (0, unknown) (28) Here the e.m.f. generated is directly proportional to the difference between the reference (for example, air) and unknown p(O& values. These probes are available commercially or can be relatively easily made in the laboratory. There have been attempts to develop probes for the measurement of sulphur and carbon activities but these are still in their [1975]; TANIGUCHI et al. [19SS]). infancy (WORRELL 6.4. Attack by molten salts The case of gas turbine operation will be considered as an example. In designing a suitable test it should be borne in mind that in the turbine the salt deposit is constantly being replenished, and that owing to the action of centrifugal forces the deposit is also being removed to the blade tips and thus is usually quite thin. When a porous oxide layer or deposits form on a blade then the molten salt can also be trapped within pores. This is a complex situation which most researchers acknowledge cannot easily be realistically simulated in laboratory tests, so that various simplified test methods have been devised to allow some information about reactions kinetics and mechanisms to be obtained. SAUNDERS and NICHOLLS [1984] reviewed the various test methods that have been used and have classified these into crucible tests, furnace tests and burner rig tests. There is general agreement that burner rig tests gives the best simulation of operating experience, and corrosion product morphologies obtained as a result of burner rig testing agree well with those found on blades taken from operating engines. However, furnace tests with careful control of gas and deposit chemistry and deposit replacement can give a close simulation of dynamic corrosion that can be expected in service, at a very References:p. 1357.
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S.R. J. Saunders and J. R Nicholls
Ch. 14,$7
modest cost. Considerable disagreement exists about the quantitative comparison of corrosion rates from different rigs. A major factor in the confusion is that in the past operators of rigs have attempted to compare results taken from rigs where the concentration of contaminants was controlled to that expected in the turbine. Since it has and NICHOLLS [1984]) that deposition rate is the primary been established (SAUNDERS factor controlling corrosion rates, only rigs operating at the same velocity as the turbine would be expected to produce comparable rates of attack. It has been pointed out (€JANCOCK [1982]; SAUNDERS et aZ. [1982]) that contaminantflux rate (CFR) is a better parameter to control deposition and allows low velocity burner rigs to operate at comparable deposition rates to high velocity rigs and gas turbines. Recently, an international round-robin was carried out which established the validity of this approach (SAUNDERS and NICHOLLS [1995]).
6.5. Attack by solid deposits In cases where solid deposits form it is not only important to ensure that the amount and composition of the deposit corresponds with that found in service, but, because of porosity effects in altering gas composition, the structure of the deposit must also be controlled. In recent years considerable interest has developed in fluidised-bed combustion of coal, where deposit-induced corrosion can occur. In this case the deposit consists of a mixture of CaSO, (derived from sulphur capture processes in the bed) and coal ash and it is believed that the active species in the corrosion process is CaSO,. Tests were carried out in which samples were buried in various CaSO,-containing mixtures (CaSO,+C, CaSO, + CaO etc.) with or without control of the gas atmosphere. While at high temperatures these tests generally reproduced corrosion found in service they did not reproduce attack observed at temperatures below about 800°C. This is believed to be due to inadequate simulation of the pore structure of the deposit. Slurries of CaS0,-containing and SPENCER [1987], material have been used by NATESAN [1982], while SAUNDERS attempted to overcome the problem by using powder compaction techniques. In ash deposition conditions typical of those found within a fluidised bed, one cannot discount the r6le of particle impact in modifying deposit morphologies and hence corrosion (STEPHENSONand NICHOLLSC19921).
7. Life prediction modelling There is considerable interest in the ability to predict when oxide scales fail and their consequence on increased corrosion rates. This section will review “lifing” models that have been developed based on oxide failure, and hot-salt corrosion mechanisms. 7.1. Oxidationmodels The development of spallation maps (see fig. 8) and their application to the prediction of scale failure was was discussed in section 2.4.2. From this work it was shown that
Ch. 14. 87
Oxiahtion, hot corrosion and protection
1339
scale thickness in conjunction with temperature drop determines when the scale fails. This concept has been extended further to predict the lives of thermal barrier coatings (TBC) on hot-path components. The spallation of the Zr02-8wt%Y,0, TBC can be modelled on the basis of the growth of the alumina at the bond coat/TBC interface, with the number of cycles to failure, Nf, determined from a knowledge of the oxide thickness (h), the thickness of the oxide to fail in one cycle (h,) through a Manson-Coffin type relationship:
where A q is the strain required to cause failure in one cycle with no oxidation damage and b is the tensile component of the in-plane elastic strain range in the oxide at the interface, b and c are constants (MEIERetaZ. [1991]; MEIERetal. [1992]). 7.2. A probabilistic model of corrosionloss
Hot corrosion processes invariably proceed in two distinct stages; an initiation stage, where breakdown of the protective surface oxide occurs, albeit by chemical or mechanical means, followed by a propagation stage where deposited contaminants have direct access to the substrate and corrosion rates can be considered catastrophic. To model localised maximum corrosion attack (e.g., Type 11, vanadic attack and other more generalised corrosion morphologies) measurement methods must be developed that not only account for the rate of growth of the pits, but also the dispersed nature of this form of attack. The basic concept behind these models was developed by one of the authors in the early 80s and has been used to predict corrosion rates within the marine [1987]) and gas turbine (NICHOLLSand HANCOCK[1983]; NICHOLLSand HANCOCK and SAUNDERS [1990]). within diesel engines (NICHOLLS and TRINER [1990]; NICHOLLS When modelling localised corrosion the common practice of measuring mean section loss and/or maximum observed corrosion penetration clearly does not apply, and hence statistical method for evaluating metal loss due to corrosion must be used (NICHOLLS and [1983]). HANCOCK The statistical evaluation of metal loss requires that many readings of section loss (a minimum of 24 readings is recommended) be taken randomly along a corroding surface and from one component section to another. These are then evaluated using statistical techniques, for example fig. 32. In this figure, remnant coating thickness was determined at 2.5 percentiles, i.e., 39 using data points per curve. The probability of exceedance is calculated by ordering the section loss data from highest loss to lowest. Then if i is the ranking of the corrosion loss reading xi, the probability of exceedence is given by i/(n+l), where n is the total number of readings. The statistical model that is most appropriate (a normal or log normal distribution for general corrosion) can be determined from the probability plot. For example in fig, 32, the region of low corrosion loss approximates to a normal distribution. However, a few localised deep pits are observed, with a morphology typified by fig. 18. A method for modelling this extreme corrosion is required. References: p. 1357.
Ch. 14, 87
S. R J. Saunders and . I R. .NicholLs
1340 EO
-
t A
lOOh data (Sample 09.2)
300h data ISample 05.21 S00h data ISample 11.21
-
I
1 Variatio;
0.1
1
5 10 20 40 60 80 90 95 Probability o f coating thickness E plotted value
99
99.9
in mean thickness specimens.
99.99
Fig. 32. Probability plot of pack aluminide coating thickness following burner rig testing at 70OOC.
7.3. Modellingextreme corrosion Here, the life-limiting rate of corrosion is associated with the growth of the deepest pits, that is to say the extreme corrosion data In fig. 32, and from similar plots of section loss, extreme corrosion can be seen to occur for the upper 10% of metal loss readings. For the data included in fig. 32 where each point represents a 2.5 percentile, four data points can be considered extreme. This should be the minimum number of data points necessary to model extreme corrosion. However, by increasing the number of sections examined it is possible to increase the number of extreme data values available. A limited number of data points taken from multiple sections would provide better estimates of the extreme corrosion if multiple sections are available for analysis. This extreme corrosion data can be modelled using "extreme value statistics"; the best and model being a Gumbel (BERRY[1975]) extreme value model of maxima (NICHOLLS HANCOCK [1983]; NICHOLLS and HANCOCK[1987]). A necessary condition for the use of this model is that "the upper tail of the original distribution decreases at least as rapidly as the exponential function". This conditions is met for data that follow either a normal or log-normal distribution, and hence a Gumbel extreme value model can be used to m&l extreme rates of hot-salt corrosion. However, this condition should be validated (through the use of probability plots) when applying extreme value statistics to other localised corrosion phenomena. The probability density function f(x), of the Gumbel extreme value model of maxima is given by:
Ch. 14, 47
1341
Oxidation, hot corrosion andprotection
f(x) = ( V g e ) exP[l- ( x - ~ e ) / c e ] * ~ x P [ - ~ x P ( - ~p e / o e ) ]
(30)
where pe and we are the distribution parameters The cumulative distribution function, F(x), is given by: ~ ( x =)
j l f ( x ) h = ~ x P ( - ~ x P-( ~
(31)
pe)/Ve)
The expected value, E(x), is given by: E(x) = j l x . f ( x & = pe + ce y
(32)
a
where y is Euler’s constant (=0.5772), and the variance of x, V(x), is given by: V ( x ) = de,2/6
(33)
by using the log-likelihood function, L(x) given by:
the maximum likelihood estimates of p e and ue can be evaluated by solving the simultaneous equations:
These equations have to be solved iteratively. Figure 33 shows an example of the relationship between the expected arithmetic mean (E(x)) and the expected standard deviation (dV(x)) and those measured experimentally (NICHOLLS [1993]). In both cases the degree of fit is very good, confirming that these extreme corrosion data may be modelled using a Gumbel extreme model of maxima.
7.4. Development of a life prediction model Since Type II corrosion is associated with the growth of pitting morphologies, only the extreme corrosion data from these burner rig tests is considered in developing the type II hot corrosion life model. The most likely “extreme” value of corrosion (xh can be calculated and its value, of xe, that has only a 5 % chance of exceedance, is given by: xe = pe = ceIn[-ln(0.95)]
(3W
References: p . 1357.
1342
S.R J. Saunders and J. R. Nicholls
Ch. 14, 17
20
15 Elxl
10
5
0 0 (a)
5 10 15 Actual mean (xJ pm
20
0
4 8 12 Actual std. dev (SJ pm
16
Fig. 33. Relationship between (a) expected (EJ and actual mean, (b) expected (JVJ and actual standard deviation of the extreme values.
and, more generally, at any probability of exceedance (a): e ‘
= Fe = a,In[- In(1- a)]
(36b)
By modelling p e and a,as a function of exposure time a corrosion life model can be developed which has the form for any probability, xe(,=) = A + B I t + Ctln[(l-a)]
(37)
where A, B and C are constants dependant on the corrosion process, temperature etc., t is the exposure time in hours, x, is the maximum expected n p e TI hot-corrosion pit depth, and a is the risk that the pit depth will exceed x, within the exposure time (t). Clearly, the constants A, B and C depend on the rate of arrival of contaminants, i.e., the contaminant flux rate. For low contaminant flux rates it can be shown that the observed corrosion rate increases with contaminant flux and follows a power law. Incorporating contaminant flux dependence into equation 37, one obtains:
=xi: [A +E ~t+ Ctln(1-a)}
xeCln)
(38)
where A, B and C are redefined as flux independent corrosion rate constants, and take the values of 15.9, 2.2 and -0.12 respectively, for the model reproduced in fig. 34. Two alternative failure criterion exist. Figure 34 plots the predicted maximum corrosion rates, calculated using equation (38), for two sodium contaminant flux rates (0.05 and 0.13 mg/cm2. h) and for two confidence limits (95 and 99%). Also included in the figure are the curves for the most likely extreme value of Type I1 corrosion (i.e., the most likely m e I1 corrosion pit depth) at these two sodium fluxes. As can be seen, these models are capable of predicting the corrosion expected under engine service trials. It is also possible to predict the point at which corrosion penetration through the coating occurs, or when the corrosion pit is sufficient to cause fatigue failure (NICHOLLS et al. [1993]).
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0
Extreme data points from burner rig t e s t s of Pack Aluminised I N 738 a t 7OOOC
Critical pit depth for fatigue failure at Au=180MN/m2 Typical overlay coating thickness range Typical diffusion coating thickness range
_ _ _ _ _ xJ0.991 -X-
x,( 0.95 1
-- Most likely ----extreme value (x,) 100
1,000
10,000
Exposure time (hl
Fig. 34. Prediction of me II corrosion at 700'C for sodium contaminant fluxes of 0.05 and 0.13 mg cm-* h-'.
8. Developments in coating technology In discussing coating properties, performance will be assessed with reference to the desirable properties of a high-temperature coating system (table 5). In general, two types of coating process can be identified. The first requires that the composition of the surface of an alloy should be altered by diffusion, while the second involves the deposition of a metallic or ceramic overlay with improved oxidation resistance. Hence, these two processing routes are known as diffusion and overlay coating, respectively. Dijgirsion coatings provide a surface enrichment of aluminium, chromium and/or silicon which results in the formation of protective surface oxides of A1,03, Cr,O, or SO,. References: p. 1357.
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Table 5 Requirements of a high-temperature system Coating property
Requirement
Location Coating Surface
Bulk
Coatinghubstrate interface
Oxidationkorrosion Low rates of scale formation. Uniform surface attack. resistance A thermodynamically stable surface oxide. Ductile surface scales. Adherent surface scales. High concentration of scale foming elements within the coating to act as reserve for scale repair.
X X
...
...
X X X
...
X
... ... ... ...
X
X
...
Low rate of diffusion across interface at operating temperatures. Limited compositional changes across interface. Absence of embrittling phase formation during service.
...
...
X
...
...
X
...
...
X
...
X
X
...
...
X
...
X
X
...
X
...
...
X
X
Interface stability
Good adhesion
Matched coating and substrate properties to minimise coating mismatch and stress generation at coatingkubstrate interface. Optimum surface condition before coating. Growth stresses during coating formation should be minimised.
Mechanical strength Coating must withstand all stress (creep, fatigue, and impact loading) that is generated at component sqface during service. Well matched thermal expansion coefficients between coating and substrate to minimise thermal stressing and thermal fatigue.
...
... ...
... ...
Metallic overlay coatings are a specifically designed corrosion resistant alloy that are deposited onto a component surface and are typified by the M-Cr-Al-X series where M can be Fe, Ni and/or Co and X is an oxygen active metal, i.e., Y,Ce, Hf, Si or a combination of these. For this class of coating, interaction with the substrate is generally negligible during deposition. In addition, thicker coatings may be produced which can give extended service lives. The reduced substrate interactions, choice of corrosion resistant
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alloy and the ability to deposit thicker coatings results in greater design flexibility. Ceramic barrier coatings are a class of overlay coating, where corrosion protection (and thermal protection) are provided by an impervious ceramic barrier. For oxidising environments the ceramic barrier should be a stable oxide, and SiO,, Al2O3and 21-0,are the prime contenders, while nitride ceramics, TiN and Si,N, have been examined for reducing conditions. In addition to their role as a corrosion barrier, ceramic overlay coatings may be engineered to provide additional thermal protection, the thermal barrier concept, where a cooled component, for example a turbine blade is coated with a layer of ceramic (typically 0.3-0.4 mm thick) having low thermal conductivity, thereby retarding the rate of heat flux into the component surface. Several types of thermal barrier system have been developed with varying degrees of success. The preferred family of ceramics used for proprietary coatings is based on stabilised zirconias, and in particular yttria partially stabilised zirconia (Y-PSZ).
8.1. Diffision-coating processes Diffusion-coating processes have been applied for many years to improve the environmental resistance of a base alloy by enriching the surface in Cr, Al, or Si. Pack and LOCKINCTON chromising has been in widespread use since the early 1950s (SAMLTEL [1951]; SULLY and BRANDS [19671) to increase the oxidatiodcorrosionresistance of low alloy steels. In the late 1960s aluminising was first used for the protection of superalloy gas turbine aerofoils (GOWARDand Boom [1971]; GRISAFFE[1972]). There was renewed interest in siliconising and silicon-modified diffusion coatings for high temperature service in the early 1970s (FELIX and E m s [1972]; BAUERet al. [1989]) when novel solutions to the low temperature corrosion problems, associated with contaminants in industrial turbine plant burning impure fuels, were required. Siliconising had previously been dismissed as a major contender because of the strong embrittling effect on hightemperature alloys and the tendency to form many low melting point metallic and oxide eutectics (FELIXand ERDOS[1972]). Diffusion coatings can be applied to hot-gas components using several techniques, including pack cementation, slurry cementation,and metallising. A comprehensivereview of the methods of deposition of diffusioncoatings was in MEVRET et al. [19861. Fluidised bed techniques can also be used to deposit diffusion coatings on a laboratory scale (DURETetal. [1982]). This latter approach would permit the coating of large components with a close tolerance on coating thickness, as a result of the uniform temperatures and, therefore, chemical activities possible within a fluidised bed. In the pack cementation process, components to be coated are buried in a "pack", contained in a sealed retort (fig. 35). The retort is heated to the desired processing temperature under either an inert gas or hydrogen atmosphere to prevent oxidation. The exact process cycle, time, and temperature are dependent on the required coating, coating thickness, and subsequent substrate heat treatment. The pack contains three elements: i) a donor alloy that releases solute material at a known rate and hence determines the pack activity, ii) a halide activator that dissociates during the process cycle and acts to
References: p. 1357.
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Ceramic ’pacers
Components for coating
Schematic diagram of a pack cementation retort
Fig. 35. Schematic diagram of pack cementation retort.
transport solute material from the pack to the component to be coated, and iii) an inert oxide diluent to prevent pack-sintering. ’Qpical pack compositions used to produce a range of metallic coatings are given in table 6. Of these the two diffusion-coating processes that are most widely used are “aluminising” and “chromising”. The pack-aluminising process will be used as an example to illustrate a typical process cycle. During the aluminising process material from the pack is transferred to the component surface through the formation of intermediate volatile aluminium monohalide gas and therefore the coating process is probably more accurately described as a chemical vapour deposition process. Interdiffusion between the depositing aluminium and the substrate alloy results in the formation of the intermetallic coating, primarily NiAl or CoAl, depending on the alloy base, but containing to a degree most of the elements present in the base alloy either in solution or as dispersed phases. The deposition rate and morphology of the coating depend on pack activity, process time and temperature. Coatings are classified as either “low activity”, when outward diffusion of the base alloy occurs, or “high activity” when inward diffusion of aluminium
Table 6 ’Qpical pack compositions and deposition temperatures for halide activated pack cementation Coating AI Cr
occurs. In the latter case a surface layer of Ni,Al, forms and a further heat treatment is required to convert this brittle phase to NiAl. This step is usually combined with the heat
treatment required to recover substrate properties. Figure 36 illustrates a typical aluminide coating deposited onto a nickel based superalloy using a high activity coating process. The coating was deposited using an aluminising pack containing 2%A1 at 900"C, and was heat-treated for 2 h at 1120"C, then 24 h at 845°C. Clearly, the properties of the aluminide coating (or for that matter any diffusion coating) depend upon the process methodology, the substrate composition and the subsequent heat treatment. Vpically, aluminide coatings contain in excess of 30 wt%Al and are deposited to thicknesses between 30-100 ,um depending on the type of aluminide formed. They offer satisfactory performance for many aviation, industrial and marine engine applications. Under severe hot corrosion conditions, or at temperatures above 11OO"C, aluminide coatings offer limited protection. Hence modified aluminide coatings have been developed that offer improved corrosion resistance. 8.2. Modified aluminide coatings
Modified aluminides have been fabricated using one of the following techniques (DURETet al. [1982]): i) ii) iii)
Co-deposition of elements from the pack or slun-y; pretreatments of the superalloy before pack aluminising, for example, chromising prior to pack aluminising; deposition of a metallic layer using electroplating or PVD (physical vapour deposition) techniques. For example, a platinum aluminide coating is formed by depositing platinum onto the superalloy prior to aluminising.
Alloying additions include Cr, Si, Ta, the rare earths and precious metals. Many of these coatings are now commercially available. Probably, the most significant advance in this area was the development of the platinum modified aluminide class of coating. Figure 37
Fig. 36. Backscattered electron micrograph of a fully processed high activity pack aluminide coating on IN 738I.C.
Fig. 37. Backscattered electron micrograph of a platinum aluminide (FX22) coating on IN 738LC.
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illustrates the microstructure of the RT22 variant, marketed by Chromalloy. The SEM micrograph is a backscattered electron image, the platinum-rich aluminide phase within the coating being shown in light contrast.
8.3. Overlay coatings processes Diffusion coatings, by the nature of their formation, imply a strong interdependence on substrate composition in determining both their corrosion resistance and mechanical properties, hence, the possibility of depositing a more “ideal” coating, with a good balance between oxidation, corrosion and ductility has stimulated much research interest. Early coatings of this type were alloys based on cobalt (Co, Cr, Al, Y) containing chromium additions in the range 20-40%, aluminium additions between 12-20% and yttrium levels around OS%, with the most successful coating being Co25Cr14A10.5Y (TALBOOM et al. [19701). Recent coatings are more complex and are based on the M-CrA1-X system, where M is Ni, Co, Fe or a combination of these and X is an active element, for example Y, Si, Ta, Hf etc. (COWARD [1983]; CLARKE [1979]; PRATER etal. [1981]; GOEBEL etal. [1981]). The composition of the M-Cr-A1-X system is selected to give a good balance between corrosion resistance and coating ductility, while the active element addition(s) can enhance oxide scale adhesion and decrease oxidation rates. Current thinking suggests that a combination of active elements is beneficial in reducing coating degradation through their synergistic interaction. Overlay coatings have been deposited using a range of techniques including plasma spraying and physical vapour deposition (PVD). Table 7 summarises some typical commercial overlay coatings and their methods of deposition. The earliest production method was electron-beam PVD (TALBOOM [19701). However, because of the high capital cost in setting up a commercial EB-PVDplant, plasma spray methods have found wide acceptance, particularly the argon-shrouded and vacuumTable I Some typical commercial overlay coatings Composition ~~~~
plasma spray processes (NICOLL [1984]; TAYLORef al. [1985]; RESTALL and WOOD [1986]). More recently, composite electroplating (KEDWARD [1969]; FORSTERef al. [1985]; HONEYet al. [1986]) and laser fusion (POWELLet al. [1985]; STEEN[1987]) methods have been used to deposit overlay coating systems. However, coatings produced by EB PVD processes are still considered the commercial standard against which other process routes are compared.
8.3.1. Physical vapor deposition Physical vapour deposition processes encompass both electron beam (EB) evaporation and sputtering, either of which may be used in conjunction with ion-plating (see below). Commercial EB PVD coatings are produced in a vacuum environment, 10-*-1O4 Pa, with deposition rates often exceeding 25 p d m i n . Components are generally preheated in vacuum to between 800-1100°C,and are rotated within the evaporant cloud during the deposition process (BOONE[1980]). Rotation attempts to ensure uniform coverage as the process is primarily “line of sight” at these pressures. Problems such as columnar grains with unbonded interfaces (known as “leaders”) due to shadowing effects can occur as the coating grows. High deposition temperatures result in increased surface diffusivity which reduces the density of these leaders and also permits some minimal interdiffusion between the coating and substrate during the coating process cycle, ensuring good adhesion. Hence coating spallation, a problem with many processing routes is not a problem provided the substrates are properly cleaned prior to coating. Post-coating processing includes glass bead peaning and heat treatment which results in complete closure of any remaining leaders. Figure 38 is a micrograph of a Co,Cr,Al,Y coating (ATDSB) produced by an EB-PVD processing route. In the sputtering process, positive gas ions (usually argon) produced in a glow discharge or plasma bombard a target of coating material dislodging groups of atoms. These then enter the vapour phase and deposit onto substrates to be coated (BUNSHAH [1980]). Deposition rates are much slower than the EB evaporation route, generally
-*.,-
Fig. 38. Secondary electron micrograph of an EB PVD Cdr-AI-Y (ATD 5B) coating on Mar MOO2 - note the fine microstructure compared with that produced by vacuum plasma spraying, cf. fig. 39.
Fig. 39. Backscattered electron micrograph of a vacuum plasma sprayed Co-Ni-Cr-AI-Y (ATDZB) coating on IN 738LC.
References: p. 1357.
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10-20 p d h for a diode system or up to 50 p t d h for a planar magnetron system. Magnetron systems (having magnetic plasma confinement adjacent to the target) usually result in more uniform deposition and can virtually eliminate substrate heating during the deposition process. Working pressures are of the order of l-lo-’ Pa depending on W plasma is used to generate ion bombardment and whether a DC glow discharge or l hence offers excellent throwing power and good overall coverage of components to be coated. Because of the low process temperature, coatings are invariably heat-treated following deposition to produce the desired properties. Both EB evaporation and sputtering may be combined with ion-plating. Ion-plating is essentially a PVD process in a soft vacuum (1-lo-’ Pa) with evaporant depositing onto substrates held at a high negative potential (often between 2-5 kV) (TEER[1980]). During ion-plating the components to be coated are initially bombarded with positive ions, formed in the discharge, which remove oxides and other contaminants from the surface. When the surfaces are sufficiently clean the vapour source is energised and metal evaporant enters the discharge and is deposited onto the sample. With EB evaporation and ion-plating, deposition rates are typically 10-20 ptdmin. An interesting combination of sputtering and ion-plating (sputter ion-plating, SIP), using a hollow cathode sputtering target and application of a bias to the substrates held within the hollow target, has been described (COADand DUGDALE [1979]) and used to deposit coatings on turbine blade materials (COADand RESTALL[1982]). The system operates at soft vacuum (1-10 Pa). The geometry of the sputter targets and components to be coated offer very good throwing power and therefore uniform coatings. The system is heated to -300°C during coating with a bias typically of 1 kV. Deposition rates are typical of diode sputtering systems, i.e., -10ptdh. Again, heat treatment of the as deposited coating is required to obtain the desired properties.
8.33. Spraying processes Plasm-spraying has the advantage of being able to deposit metals, ceramics, or a combination of these, generating homogeneous coatings with microstructures consisting of fine equiaxed grains, i.e. no columnadleader defects. High deposition rates are possible with little significant change in composition occurring from the powder feedstock through to the coating, even when the elements in the coating have widely differing vapour pressures. The most obvious limitations of this process are that the coating process is ‘line of sight’, requiring complexed robotic manipulation for complete coverage, and that the more reactive elements may well oxidise during the spraying process if conducted in air. Porosity problems previously reported for plasma spray coatings can largely be overcome using post coating thermomechanicaltreatments. Figure 39 shows a fully processed, argon-shrouded plasma-sprayed LC022 (Co-Ni-Cr-Al-Y) overlay coating. Carrying out the plasma-sprayingprocess in a closed chamber under reduced pressure (5-7x io3 Pa), known as low-pressure plasma-spraying (LPPS) or vacuum plasmaspraying (VPS), permits close control of the gaseous environment. This removes the possibility of unwanted gas-metal reactions and also permits high particle velocities
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giving improved adhesion and higher densities, thus overcoming many of the limitations of plasma-spraying but at an increased capital plant cost. In the high-velocity oxy-fuel (HVOF) spraying process fuel is burnt with oxygen at a high pressure and generates a high-velocity exhaust jet. Fuel gases may include acetylene, kerosene, propane, propylene, hydrogen, methacetylene-propadiene (MAPP) mixture etc. The ratio of the gas flow rates determines the temperature of the flame. The temperature of oxygen-acetylene reaches a maximum at 3170°C if mixed 1.5 to 1 (by volume) and oxygen-propylenereaches the maximum temperature of 2900°C at the ratio of 4: 1 (NEDERBERGER and SCHIPPER 119901). The velocity of the exhaust jet in the Jet-Kote torch is about 2000 m s-' (KREYE[1991]). The main advantage of this process is the shorter residence time in the flame (lower powder temperature) and higher kinetic energy of the particles impacting. This produces a dense coating with less degradation of this powder during spraying. The most widely sprayed powders are carbides, although there is considerable interest in using the HVOF process to deposit M-Cr-A1-Y overlay coatings. The bond strength of HVOF-sprayed coatings can be as high as 90 Mpa (KREYE[1991]). Porosity is low, less than 196, and typical coating thicknesses are in the range 100-300 pm.
8.3.3. Laser processes Lasers can be used to form coatings directly by (i) preplacing a powder onto the substrate, (ii) blowing the powder into a laser-generated melt pool, or (iii) applying the clad in wire or sheet form. Alternatively, coatings produced by other routes such as plasma-spraying or electroplating may be modified by surface treatment with the laser. By careful selection of laser conditions, dilution of the coating by substrate melting can be minimised and large areas can be surfaced by a series of overlapping tracks. Power, beam diameter, and beam velocity are the main process variables with typical values being 1.5 kW, 5 mm, and 10 mm s-l, respectively. As with plasma-spraying it is et al. necessary to apply an inert gas shroud to reduce oxidation of the coating. POWELL [1985], suggest that adhesion is improved if some melting of the substrate occurs. Problems associated with reflectivity of the surface occur particularly with use of wire and sheet, and for the preplaced powder method there are difficulties in maintaining the powder on the surface. Thus the blown powder method is favoured by STEEN[1987]. Porosity in coatings can occur from cavities between overlapping tracks, solidification cavities, or gas evolution, but problems can generally be overcome be careful choice of processing parameters. Residual stresses in a laser coating are generally tensile and occur from restrained contraction of the prior molten clad. Cracking may result, but can usually be overcome by preheating or pretensioning the substrate before coating (STEEN[1987]). 8.4. Oxidation and hot-salt corrosion resistanceof diffusion and overlay coatings
The corrosion performance of a range of diffusion and overlay coatings has been etal. [1984]. Over the temperature range reviewed by MALE et al. [ 19821 and NICHOLLS 700-950°C the platinum-modified aluminides performed exceptionally well. The other References: p. 1357.
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diffusion coatings, chromised and chrome-aluminised coatings, generally fared worse than the conventional aluminide, although chromium-rich coatings have been shown to offer improved corrosion resistance in an industrial turbine environment (MALIKet al. [1982]; LUTHRAand LEBLANC [1987]). Overlay coatings generally perform better at higher temperatures where oxidation is the dominant failure mode (above 900OC) reflecting the good adherence of the thin alumina scales which is promoted by the presence of active elements such as yttrium. However, at lower temperatures, 65O-85O0C, corrosion rates are relatively high. Methods have been investigated to improve the traditional M-Cr-AI-Y coatings by use of a platinum underlayer and overlayers (PRATER et al. [198l]), other additions such as TI, Zr, and NICHOLLS [1984]; BORNSTEINand Hf, Si and Ta, (DURETet al. [1986]; SAUNDERS SMEGGIL[1982]). Surface modification by CVD (RESTALL and HAYMAN[1984]), PVD (GOEBELet al. [1981]) or slurry cementation techniques, (GOEBEL et al. [1981]) and reet al. [1982]; BORNSTEIN and processing of the coating surface using laser beams (DURET S ~ I [1982]) L have also been considered with varying degrees of success. Surface modification results in the formation of a duplex coating structure and this can result in improved performance, for example, a pulse-aluminised Co-Ni-Cr-Al-Y coating exhibits superior corrosion resistance at 750 and 850°C compared to its plasmasprayed counterpart (NICHOLLS et al. [19841). Similarly, silicon modifications to the et al. 1981) have also been shown to improve surface of Co-Cr-AI-Y coatings (GOEBEL resistance to low-temperature hot corrosion. In addition to corrosion resistance, surface coatings can have a major influence on the erosiodcorrosionresistance of hot-end components. As mentioned earlier, environmental resistance depends on the formation of thin protective oxide scales and therefore the interaction between an erodent and the oxide scale that is formed on a component is of particular importance in determining the erosiodcorrosion behaviour of a particular coating. The effect of particle impingement on degradation rate depends on whether the impact event produces failure within the protective oxide scale. The higher the impact energy (larger particles, higher velocities) the more likely it is that scale failure will et al. f19871; occur and this will result in an increase in corrosion rate (HANCOCK STEPHENSON[19891) and reduce incubation times to the onset of catastrophic corrosion.
8.5. Thermal stability of diffusion and overlay coatings Structural stability of coatings is also an important factor if they are to maintain their protective qualities over extended periods of time at high temperatures. Coatings degrade not only by loss of scale-forming elements to the surface, but also by interdiffusion with the substrate. This can result in additional problems such as the formation of topologically close-packed (TCP) phases, e.g., sigma, below the coating, which cause embrittlement of the substrate. In aluminide diffusion coatings, degradation results in the breakdown of the outer P-NiAl phase by the formation of a 7’-Ni,Al network which after extended exposure time can completely penetrate through the P-phase. These y’ leaders behave as short-circuit corrosion paths which results in rapid failure of the coating. The thermal stability of diffu-
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sion coatings, including pack aluminide, chromium aluminide and platinum aluminide varand STRANG [1982]; LANGand BULLOCK iants were studied in the early eighties (COOPER [1982]). These studies have shown that the thermal stability of p-NiAl can be improved by the addition of platinum. The two-layered PtAI, /3 structure degrades to single-phased p with no y' leaders even after extended exposure time, e.g., 18,000 h at 850°C (HANCOCK and NICHOLLS[1984]). This is a major factor which contributes to the improved corrosion resistance of platinum-aluminide coatings compared to conventional aluminides. Few papers have been published on the metallurgical stability of overlay coatings, probably because diffusion is not a prime factor in the formation of these coatings. Nevertheless, substrate interaction effects are still observed e.g., hafnium from hafnium modified MM200 (GOWARD[1983]) has been found to improve the hot corrosion resistance of Co-Cr-AI-Y coatings. Movement of other elements, e.g., mgsten and molybdenum, from the substrate, may well be detrimental to corrosion resistance. The beneficial effect of a platinum underlayer under marine conditions (CLARKE[19791) may well be due to a diffusion barrier effect. One of the most comprehensive studies on the stability of overlays is that of MAZARS et al. [1986], which have examined in detail the diffusion degradation of Ni-Cr-AI-Y, Co-Cr-AI-Y, and Fe-Cr-A1-Y systems. From their work they conclude that diffusion stability can be considered a life limiting factor if coatings are operated at temperatures > 1000°C for prolonged periods. However, limited interdiffusion was observed in the Ni-Cr-AI-Y/Ni based, Ni-Cr-Al-Y/Fe based, and Co-Cr-AI-Y/Fe based systems and hence these systems are of particular interest for high temperature applications.
8.6. Mechanical properties of diffusionand overlay coatings Probably the most important property of a coating for service at elevated temperature is its resistance to cracking by thermally induced stresses. A summary of the ductilebrittle transition temperatures (DBTT) for a range of diffusion and overlay coatings is shown in fig. 40,in which it can been seen that the DBTT of the aluminides is higher than that observed for many of the overlay coatings. Since the peak tensile surface strains on the turbine blades are likely to occur at relatively low temperatures, (GLENNY[1974]) then from a ductility point of view these diffusion coatings will be inadequate or only marginally adequate for relatively high strain applications. Unlike the aluminides, the ductile-to-brittle transition of overlay coatings, and hence their resistance to cracking, can be modified by varying the coating composition. It is evident from fig. 40 that, for a given Cr and Al level, the ductility of Ni-Cr-Al-Y coatings is significantly better than that of Co-Cr-AI-Y coatings. The ductile to brittle transition temperature is found to increase with Cr and A1 level. This is not unexpected since coatings with a high volume fraction of Co-A1 or Ni-A1 shows ductile to brittle transition temperatures close to those for the diffusion aluminides. This transition from ductile behaviour to brittle behaviour, as the temperature drops, is of significance when considering the thermo-mechanicalcycles seen in service. Should coatings crack on cooling, there is a danger of the crack propagating to the coating1 substrate interface. It may arrest there, may propagate along the interface or may run into the substrate component, thereby decreasing the component's life. In an investigation of References: p . 1357.
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1
Ductile/ brittle transition curves for diffusion coatings.
Ductilelbrittle transition curves for overlay coatings.
Fig. 40.Ductilelbrittle transition curves for a) diffusion coatings and b) overlay coatings.
a plasma-sprayed Ni-co-Cr-AI-Y coating on a single crystal super alloy at 65OoC, a marked reduction in the fatigue life of the coated component at low strain ranges was observed (GAYDAet al. [1987]). The mechanical behaviour of coated components is therefore of major importance but is beyond the scope of this chapter. Readers are recommended to consult the papers by WELLS and SULLIVAN 119681; STRANG and LANG [1982]; GAYDAetal. [1987]; and Au et al. 1990 for further appreciation of the subject. 8.7. Future trends in overlay coating design Although coatings are available that offer adequate oxidationkorrosion resistance for many applications, improvements in coating performance can still be made. The drive to operate plant hotter, to increase efficiency, coupled with the desire for increased component
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lives, requires that new generations of coatings be developed. As operating temperatures are increased, it no longer becomes possible to achieve the desired service lives using diffusion coatings and overlay coatings therefore are the only possible development route. For metallic systems, much work is underway to develop overlay coatings containing multiple active elements. This acknowledges that although many of the active elements (Y, Hf, Ta, Si etc.) produce similar improvements in oxidation behaviour, they function by different mechanisms and hence show maximum benefit at different operating temperatures. Since high-temperature components encounter a range of temperatures, clear benefits are possible from the inclusion of multiple active elements in a coating. But even within the M-Cr-A1-X systems, diffusion of elements between the substrate and coating can have a major influence on coating performance. Therefore, to provide long-term stability it is necessary to develop diffusion barrier coatings to minimise the interdiffusion between the coating and the substrate. Some interdiffusion is of course necessary to give good adhesion, hence the diffusion barriers must be tailored to limit the movement of particular problematic elements. Figure 41 illustrates the deposition of an intermetallic diffusion barrier at the interface of a high Cr-containing overlay coating and a superalloy substrate (MarM002). The diffusion barrier was developed to limit chromium movement from the coating into the substrate (NICHOLLS et al. [1988]) and hence reduce the likelihood of forming sigma phase within the alloy below the coating. Chromium diffusion rates were reduced by one to two orders of magnitude over the temperature range 750-1 150°C. Diffusion barrier concept based on amorphous silica have also been examined to limit coating oxidation by restricting grain-boundary diffusion (BENNETT et al. [1991]; MORSSINKOF et al. [1989]). Once the concept of a diffusion barrier is accepted as a method of providing good interface stability, one is no longer constrained in the design of the best overlay coating. No longer is substrate compatibility a requirement in specifying the overlay coating composition. By removing this constraint it is possible to design overlay coatings with optimised oxidation or corrosion resistance (NICHOLLS et al. [1989]; NICHOLLS et al.
P --
l-
1
I
-m--
Fig. 41. Graded M-C-AI-Y overlay coating and diffusion barrier (located at the coating/substrate interface) system on Mar M002, deposited by EB-PVD.
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[1993]). These studies have shown that optimum performance is achieved using different base alloy compositions. Matching the diffusion barrier coating compositions to those of the optimised corrosion resistant alloys leads to the concept of functionally graded (or multilayered) coatings. The graded coating concept is not new; it was first proposed by Boone in 1979 (BOONEe?al. [1979]). However, processing technology was not capable of varying the composition of more than one element through the coating at that time. It is now possible to vary the composition of all three base elements, Ni, Cr and Al, and also add up to three active elements as necessary. Figure 41 illustrates this new generation of EB-PVD overlay coating. The coating incorporates a diffusion barrier layer, an interfacial zone low in chromium and a zone graded outward in Cr, with a high Cr, high A1 corrosion resistant surface layer (NICHOLLSe? al. [ 19931). For low-temperature hot-corrosion resistance the surface composition should ideally be Ni-33Cr-33Al with the presence of active elements, possibly in the form of additions of yttrium, silicon and hafnium in combination. Plasma-sprayed thermal barrier coatings (e.g., fig. 42) have been used since the early 1950s and have been effectively used to protect piston crowns and valve faces in diesel engines and flare head and primary zone sections of combustors against the effects of hot spots within turbines, giving considerably improved component lives. Because of this success much attention has been directed towards the use of thermal barriers on gas turbine blades and vanes (DUVALL and RUCKEL[1982]); but only within the last decade have they been used on highly stressed turbine components within commercial gas turbine engines (FAIRBANKS and HECHT[1987]; RHYS-JONES [1990]). Strain tolerant TBC are now being developed which have a columnar grain structure that are produced by EB-PVD processing, and fig. 43 is an example of an experimental coating now being produced at Cranfield. Extending these concepts further, one can envisage the "ideal" coating for hot gas paths. It would consist of a strain-tolerant thermal barrier coating, deposited onto a bond coat which exhibited good corrosion resistance but, more importantly, matched thermal
Fig. 42. Air plasma-sprayed thermal banier coating, A TBC,B bond coat and C substrate.
Ch. 14, Refs.
Oxidation, hot corrosion and protection
1357
Fig. 43. Secondary electron micrograph of a EB-PVD thermal barrier coating, deposited at Cranfield.
expansion coefficients. Within this bond coat, composition can be graded to provide the required corrosion resistance, coating mechanical properties and a diffusion barrier at the interface with the substrate. This conceptual coating is possibly only a few years away, for research is underway into many of the fundamentals necessary to produce is functionally graded high-temperatureprotection system. The studies aimed at developing strain tolerant ceramic layer have been discussed above. Work was carried out in Germany (FRITSCHER[1987]) to develop bond coats with matched thermal expansion coefficients and the possibilities for graded corrosion-resistant coatings and diffusion barrier coatings have already been mentioned.
Acknowledgement SRJS has been supported by the UK Dept. of Trade and Industry and their support is gratefully acknowledged.
References ATKINSON, A., R. I. TAYLOR and P. D. GOODE, 1979 Oxid. Metals, 13, 519. ATKINSON, A. and R.I. TAYLOR,1986 J. Phys. Chem. Solids, 47, 315. ATKINSON, A., in “Oxidation of Metals and Alloys and Associated Mass Transport”, eds. M.A. Dayananda et al., p. 29-47, Met. Soc. of AIME, Warrendale, 1987. Au, P., R. V. DAINTY and P. C. PATNAIK,Proc. Conf. “Surface modification technologies II” pp. 729-748, Eds. T.S. Sudarshan and D.G. Bhat TMS-AIME (1990). BARREIT, C. A. and C. E. LOWELL,1975 Oxid. Met., 9, 307. BARRETT, C. A., R. G. BARLICKand C. E. LOWELL, “High Temperature Cyclic Oxidation Data” NASA TM 83655, May 1984. BARREIT,C. A., J.R. JOHNSTONand W. A. SANDERS, 1978, Oxid. Met., 12, 343. BARRY,T.I. and A. DINSDALE,1994, Mat. Sci. & Tech., 10, 1090.
1358
S. R. J. Saunders and J. R. Nicholls
Ch. 14, Refs.
BAUER,R., H. W. GRUNLINGand K. SCHNEIDER,in Proc. 1st Conf. on ‘Advanced materials for alternate fuel capable directly fired heat engines’, (eds. J.W. Firbanks and J. Stringer), p. 505; Palo Alto, CA, Electric Power Research Institute (EPRI) (1989). BAXTER,D. J., R. C. HURSTand R. T. D ~ ~ I C O T1984, T , Werkstoffe u. Korros., 35, 266. BENNETT,M. J., C. E KNIGHTS, C. E A m , A. T. TVSON,J.A. DESPORT,D. S. RICKERBY,S. R. J. SAUNDERS and K. S. COLEY,Mater. Sci. & Eng. A139 (1991) 91. J. B. PRICE,C. B. SCRUBYand K. A. STACEY, Mater. Sci. & BENNETT,M. J., D. J. BUITLE,P. D. COLLEDGE, Eng., A120, (1989) 199. BERRY,K. V., “Statistical Models in Applied Science”, John Wiley, London 1975. BIRKS, N. and G: H. MEIER,“Introduction to the High Temperature.Oxidation of Metals” Edward Arnold, London 1983. BOONE,D. H., in ‘Materials coating techniques’ (Conf. Proc. LS-106), Chap. 8; Neuilly-sure-Seine, AGARD (1980). . . BOONE,D. H., S. S m and MATIALU, Proc. 4th Conf. on ‘GasTurbine Materials in a Marine Environment’ p. 739, Annapolis, USA (1979). BORNSTEIN, N. S. and J. SMEGGIL,in ‘Corrosion of Metals processed by Directed Energy Beams’, Met. Soc. AIME, p. 147-158 (1982). BORNSTEIN, N. S. and M.A. DECRESCENTE, 1969, Trans. AIME, 245, 1947. BRUCE,D. and P. HANCOCK,1969, J. Inst. Metals, 97, 140 and 148. BUNSHAH,R. F., Proc. Conf. ‘Materials coating techniques’ Neuilly-sur-Seine, AGARD, (1980). CAB= N. and N. E MOTT, 194849, Rep Prog. Phys., 12, 163. CHRISTL,W., A. RAHMEL and M. SCHUTZE,1987, Mater. Sci. Eng., 87, 289. CLARKE,R. L., 1979, Proc. 4th Conf. on Gas Turbine Materials in a Marine Environment, (Publ. Naval Sea Systems Command, USA), pp. 189-219. COAD,J.P. and J.E. RESTALL,1982, Met. Technol., 9,499. COAD,J. P. and R. A. DUGDALE,in Proc. Conf. on ‘Ion plating and allied techniques’, London, CEP Consultants, p. 186-196 (May 1979). COOPER,S.P. and A. STRANG,in ‘High temperature alloys for gas turbines’ Conf. Roc.,(eds. R. BNnetaud et al.), 249-260, Dordmht, D. Reidel Publishing Co. (1982). DAVIES,R. H. and T. I. BARRY,“MTDATA Handbook”, National Physical Laboratov, 1989. DELAUNAY, D., A.M. HUNTZand P. LACOMBE, 1980, Corros. Sci., 20, 1109. DOYCHAK,J., Intermetallic Compounds: Vol 1, Principles Eds. J.H. Westbrook and R.L. Fleischer, 1994 John Wiley, 977. DOYCHAK,J. and M.G. HESBUR,in Microscopy of Oxidation (e&. M.J. Bennett and G.W. Lorimer) The Institute of Metals, London, 1991, p. 206. D m , D. M. and P. W. TASKER, Report TP1155 Harwell Laboratory, Oxon. UK, 1986. DURET, C., A. DAVIN,G. MARRUNISSEN and R. PICHOIR,in ‘High temperature alloys for gas turbines’ (Conf. F‘roc.) (eds. R. Brunetaud et d), p. 53-87; Ddrecht, D. Reidel Publishing Co (1982). DWALL, D. S. and D. L. RUCKLE,1982, ASME paper 82-GT-327. EVANS,H. E. and R. C: LOBB, Proc. 9th Int. Cong. on Metallic Corrosion, Toronto, 1984, Vol 2, p. 46-53, National Research Council, Ottawa Canada. EVANS, H. E., 1988, Mater. Sci. Technol., 4, 415. EVANS,A. G. and R. M. CANNON,in “Oxidation of Metals and Associated Mass Transport”,(eds. M.A. Dayananda et aL), 135-160, 1987, Warrendale PA, Met. Soc. of AIME. EVANS,H.E., 1994, Mater High Temp., 12, 219. EVANS,H. E., 1995, submitted to Mat. Sci and Eng. EVANS,U.R., ”The Corrosion and Oxidation of Metals”, Matthew Arnold, London, 1960. FAIRBANKS, J. W. and R. J. HECHT,1987, Mater. Sci. Eng. 88, 321. RLK,F! and E. ERWG, 1972, Werkstoffe u. Koms., 23,626. FORSTER, J., B.P. CAMERON and J. A. CAREWS,1985, Trans. Inst. Metal Finish, 63, 115. FRITSCHER, K., 1987, Z. Werkstofftechn 18, 265. FUNKENBUSCH,A. W., J. G. SMEGGIL and N. S. BORNSTEIN, Metall. Trans. A, 16A, (1985) 1164, and S ~ ~ I L , J. G., A. W. F~J-NKENBUSCH and N. S. BORNSTEIN, Metall. Trans. A, 17A, (1986) 923.
Ch. 14, Refs.
Oxidation, hot corrosion and protection
1359
GAYDA,J., T.P. GABB, R. V. MINERand G. R. HALFORD,Proc. TMS-AIME Annual Symp. 1987, pp. 217-223. Eds. P. K.Liaw and T.Nicholas, TMS (1987). GLENNY,R. J.E., in ‘High temperature materials in gas turbines’, (eds. P.R. Sahm and M.O. Speidel), 257; Amsterdam, Elsevier (1974). &=EL, J. A., C. S. GIGGINS,M. KRA~IJand J. STRINGER, in Proc. 2nd Conf. on ‘Advanced materials for alternate fuel capable heat engines’, (ed. J. W. Firbanks and J. Stringer), Report no. 2639SR, 7/1; Palo Alto, CA, EPRI (1981). GOEBEL,J.A. and F.S. PETTIT,1970, Met. Trans., 1, 1943 and 3421. GOWARD,G.W. and D. H. BOONE,1971, Oxid. Met., 3,475. GOWARD,G. W., in ‘High temperature corrosion’ (Conf. Roc. NACE-6), (ed. R. A. Rapp), p. 553-560, Houston, TX, NACE (1983). GRABKH, H. J. and I. WOLF,1987. Mat. Sci. and Eng., 87, 23. GRISAFFE,S.J., in ‘The Superalloys’ (eds. C.T. Sims and W.C. Hagel), p. 341; New York, John Wiley (1972). HANCOCK, P., 1982, Corns. Sci., 22, 51. HANCOCK,I?, J. R NICHOLLSand D. J. STEPHENSON, 1987, Surface and Coatings Tech. 32,285. HANCOCK,I?, and J.R.NICHOLLS, 1988, Mater. Sci. Technol., 4, 398. HANCOCK,P. and J. R. NICHOLLS,in ‘Coating for heat engines’ (Workshop Proc.), (ed. R. L. Clarke ef ai.), 31-58; Washington, DC, US Department of Energy (1984). HANCOCK, P. and J.R. NICHOLLS, 1994, Materials at High Temps., 12,209. HANCOCK,P., 1970, Werkstoffe und Korros.,21, 1002. HA^, IC, “Oxidation of Metals”, Plenum, New York, 1966. HAUPPE, K. and B. ILSCHNER,1954, Z. Elektrochem., 58,382 and 467. High Temperature. Corrosion, ed. R.A. Rapp, 1983, Houston Texas, National Association of Corrosion Engineers. HONEY,E J., E.C. KEDWARDand V. WRIDE,1986, J. Vac. Sci. Tech. HUNTZ,A.M., 1988, Mater. Sci. Technol., 4, 1079. HURST,R.C., J. B. JOHNSON, M. DAVIES and P. HANCOCK, in “Deposition and corrosion in Gas Turbines”, eds. A.B. Hart and A.J.B. Cutler, 143-157, Applied Science Publishers, Barking, 1973. ILHIT, C., “Applications of Scanning Acoustic Microscopy in Materials Science”, Oxford Univ, 1984. Thesis submitted for D. Phil. JOST, W.,“Diffusion in Solids, Liquids, Gases”, 492-501, 1952, New York, Academic Press Inc. KEDWARD,E. C., 1969, MetaUurgia, 79,225. KERBY, R.C. and 3. R. WILSON,1973, Can. J. Chem., 51, 1032. KHANNA, A. S., B.B. JHO, B. RAJ, 1985, Oxid. Met., 23, 159. KRIKKOLA, K. and C. WAGNER,1957, J. Electrochem. Soc., 104, 379. KOFSTAD,P.,“Nonstoichiometry,Diffusion, and Electrical Conductivity of Binary Metal Oxides”, Wiley, New York, 1972. KOFSTAD,P., 1989, Mater. Sci. Eng., A120, 25. KOFSTAD,P., 1985, Oxid. Met., 24,265. KOFSTAD,P., “HighTemperature Corrosion”, Elsevier Applied Science, London, 1988. KREYE, H., ‘High velocity flame spraying-pmsses and coating characteristics’. Second Plasma Technik, Symposium, Lucerne, Switzerland 5-7 June (1991), p. 3947. 0. and B. E. HOPKINS,“Oxidation of Metals and Alloys”, Butterworths, London, 1967. KUBASCHEWSKI, K~PPENBENDER, I. and M. SCHOTZE,1994, Oxid. Met., 42, 109. LANG,E. and E. BULLOCK,‘European concerted action, COST 50 -Materials for gas turbines’, EUR Report 8242 EN, (1982). LEES,D. G., 1987, Oxid Met., 27, 75. ~ B B R.C., , 1984, Themochemica Acta, 82, 191. LUTHRA,K.L. and O.H. LEBLANC,1987, Mater. Sci. & Eng., 88,329. L U ~ R AK. , L. and C. L. BRIANT,1986, Oxid. of Metals, 26, 397. LUTHRA,K.L. and D.A. SHORES, 1980, J. Electrochem. SOC.,127,2202. LUTHRA,K L, and W. L. WORRELL,in “Properties of High Temperature Alloys with Emphasis on Environmental Effects”, ( 4 s . Z.A. Foroulis and F.S. Pettit), 318-330,1976, Princetown NJ, Electrochem SOC.Inc.
1360
S. R. J. Saunders and J. R. Nicholls
Ch. 14, Refs.
MALIK, M., R. MORFIIOLI and P. HUBER,in ‘High temperature alloys for gas turbines’ Conf. Proc., ( 4 s . R. Brunetaud et al.) 87-98, Dordrecht, D. Reidel Publishing Co. (1982). Mater. Sci. Technol., (1988) 4 (No. 5 Special conf. issue - OxideiMetal Interface and Adherence). Materials at High Temps., 12 (1994). MAZARS, P., D. MARESSE and C. L o r n , in ‘High temperature alloys for gas turbines-1986‘, (eds. M. Betz et al.) 1183-1192; Dordrecht, D. Reidel Publishing Co. (1986). MCCREATH, C.C., 1983, Corrosion Sci., 23, 1017. MEIER,S. M., K D. SHEPPLER,D. M. NISSLEYand T. A. CRUISE, ’Thermalb a n k coating life prediction model development’ Transactions of the ASME, Volume 114, p. 258-263, April 1992. MEIER,S.M., K D. SHEFTLER and D.M. NISSLEY,‘Thermal barrier coating life prediction model development -Phase 2 -Final Report 2’. Pratt & Whitney Engine Company, NASA Contractors Report, CR-189111, July 1991. M,R., C. Duret and R. Pichoir, 1986, Mater. Sci. Technol., 2,201. MICHELS,H.T., 1977, Met. Trans A, 8A, 273. MORSSINKOF, R. W. J., T. FRANSEN, M. D. HENSINKVELD and P. J. GELLINGS,1989, Mater. Sci. Eng. A121, p. 44945. Mom, N.F., 1947, Trans. Faraday Soc., 43,429. MROWEC, S., 1980, Werkstoffe und Konosion, 31, 371. MROWEC,S., 1967, Corros. Sci., 7, 563. NATESAN, K., in “High Temperature Corrosion”, NACE-6, ed. R.A. Rapp, p. 336-344, NACE Houston, 1983. NICHOLLS,J. R. and P. HANCOCK,in “High Temperature Corrosion”, NACE-6, p. 198-210, ed. RA. Rapp, NACE, Houston, 1983. NICHOLLS,J. R. and F? HANCOCK, in “Plant Corrosion Prediction of Materials Performance”, (eds. J.E. Strutt and J.R. Nicholls), 257-273, Chichester, UK, Ellis Horwood, 1987. NICHOLLS,I. R. and P. HANCOCK,in “The R6le of Active Elements in the Oxidation of High-Temperature metals and Alloys”, (4s. E. Lang et aL), 1989, Elsevier Applied Science, p. 195-223. NICHOLLS, J. R., K. J. LAWSON, L. N. AL-YASIRIand P. HANCOCK, 1993, Corros. Sci., 35, 1209. NICHOLLS,J. R., K. J. LAWSON,G. C H E ~ ~ E L.RH., AL YASIRIand P. HANCOCK, in European Research on Materials Substitution (Eds. I.V.Mitchell and H. Nosbuch) p. 295-307, Elsevier Applied Science, London (1988). NICHOLLS,J. R. and S. R. J. SAUNDERS, in “High Temperature Materials for Power Engineering” @Is. E. Bachelet et aJ.), Kluwer Academic Publishers, Dordrecht, (1990), p. 865-875. NICHOLLS,J. R., D. J. STEPHENSON, P. HANCOCK, M. I. WOODand J. E. RESTALL, in Proc.Workshop on ‘Gas turbine materials in a marine environment’, Bath UK Ministry of Defence, Session 2, Paper 7 (Nov. 1984). NICHOLLS, J. R. and D. A. TRINER,“Proc. Cod. Diesel Engine Combustion Chamber Materials for Heavy Fuel Operation”, Institute of Marine Engineers, London, (1990) p. 121-130. NICOLL,A. R., in ‘Coatings and surface treatment for corrosion and wear resistance’, (eds. K.N. Strafford et aL), p. 180; Chichester, Ellis Horwood (1984). NIEDERBERGER, K. and SCHIFFER, B., Eigenschaften verschidener Gase und deren Einfluss (1990). beim thermischen spritzen. Thermische Spritzkonferenz, Essen, Germany, 29-3 1 August 1990, p. 1-5. PELTON, A.D., H.SCHMALZRIED and J. STICHLX,1979, J. Phys. Chem. Solids, 40, 1103. PIERAGGI, E. and R. RAPP, 1994, Materials at High Temps., 12,229. PILLING,N.B. and R. E. BEDWORTH, 1923, J. Inst. Met., 29,529. POWELL, J., P.S. HENRY and W. M. STEEN,preprints Conf on ‘Surface. engineering with lasers’, London, May, The Institute of Metals, paper 17 (1985). PRATER, J.T., J. W. PATTEN,D. D. HAYESand R. W. Moss, in Proc. 2nd Conf. on ‘Advanced materials for alternate fuel capable heat engines’, (eds. J.W. Firbanks and J. Stringer), Report No. 2639SR, 7/29-7/43; Palo Alto, CA, EPRI (1981). RAPP,R. A., 1987, Mat. Sci. & Ehg., 87,319. RAPP,R.A. and K. S. GOTO, in “The Hot Corrosion of Metals by Molten Salts”, eds. J. Braunstein et d.,p. 81, Electrochemical SOC..Pennington NJ, 1981. REISING, R.F. and D. P. KRAUSE,1974, Corrosion, 30, 131.
Ch. 14, Refs.
Oxidation, hot corrosion and protection
1361
RESTALL, J.E. and C. HAYMAN, in ‘Coatings for heat engines’ (Workshop Proc.), (ed. R.L. Clarke etal.) 347-357; Washington, DC, US Department of Energy (1984). RESTALL, J. E. and M. I. WOOD,1986, Mater. Sci. Tech., 2, 225. RHYS-JONES, T. N., in ‘Surface Engineering Practice’ (Eds. K.N. Strafford et al.) Ellis Horwood Ltd, p. 258-269 (1990). 1983, Oxid. Met., 20, 19. ROLLS,R. and M. NEMATOLLAHI, ROSENBURG, A. J., 1960, J. Electrochem. Soc., 107,795. ROY,C. and B. BURGEES, 1970, Oxid. Met., 2,235. SAMUEL,R. L. and N. A. JBCKINGTON: Met. Treat. Drop Forging, 18 (1951) 354-359, 407-415, 440-444, 495-502 and 506. SAUNDERS, S. R. J., M. M. NAGLand M. ScHOTzE, 1994, Materials at High Temps. 12, 103. SAUNDERS,S. R J. and J. R. NICHOLLS,1984, Thin Solid Films, 119,247. SAUNDERS, S. R. J. and J. R. NICHOLLS,1989, Mat. Sci. and Techn., 5,570. SAUNDERS,S. R J. and J. R. NICHOLLS, 1995, Materiafsat High Tempemfures, in press. SAUNDERS, S.R.J. and S. J. SPENCRR,1987, Mats. Sci. and Eng., 87, 227. SAUNDERS,S. R. I., S. J. SPENCERand J. R. NICHOLLS,‘‘Proc. Conf. Diesel Engine Combustion Chamber Materials for Heavy Fuel Operation”, Institute of Marine Engineers, London, (1990) p. 111-120. S C ~ Z EM., , 1985, Oxid. Met. 24, 199. SIMONS,E.L., G. V. BROWNINGand H.A. LEIBHAFSKY, 1955, Corrosion, 11,505. S ~ G G I LL., G., A. W. PUNKENBFH and N. S. BORNSTEIN, Spring Meeting of the Electrochemical Soc., Cincinnati, May 6-11, 1984, Extended Abstracts 84, p. 27. SMELTZER,W. W., R.R. HAERINGand I. S. KIRKALDY,1961, Acta Metall. 9,880. SMIALEK, J.L.,1987, Metall. Trans. A, 18A, 164. STEEN,W.M., in Roc. Conf. on ‘Applied laser tooling’, (eds. G.D.D. Soares and M. Perez-Amor), 131-211; Dordrecht, Martinue Nijhoff (1987). STEPHENSON, D. J., 1989, Corrosion Sci. 29, 647. STOTT,F.H. and A. A n a ~ s o 1994, ~ , Mater High Temp., 12, 195. STOUT,J. W., D. A. SHORES,J. D. Gomm and M. E. ARMACANQUI, 1989, Mater. Sci. & Eng., A120, 193. STRAWBRIDGE, A. and P. HOU, 1994, Materials at High Temp., 12, 177. S T R ~ I J D EM. R J., and W. J. QIJ~AKKERS, 1986, High Temp. Tech., 4, 141. SULLY,A. H. and E. A. BRANDES, ‘Chromium’, 2edn. Chap. 7; London, Butterworths (1%7). TALBOOM, ET., R. C. ELAMand L. W. WILSON,‘Evaluation of Advanced Superalloy Protection Systems’ Report CR7813, NASA, Houston TX(1970). TANIGUICHI,M., M. WAKAHARA, T. UCHIDAand K. HIRAKAWA, 1988, J. Electrochem. Soc. 135,217. TAYLOR,T. A., M. P. OWRS,B. J. GILLand R. C. TUCKER, J. Vac. Sci. Tech., 3 (1985) 2526. TER, D.G., Proc. c o d ‘Materials coating techniques’ Neuilly-sur-Seine, AGARD (1980). WAGNER, C., 1956, J. Electrochem. Soc., 103,772. WAGNER, C., 1975, hog. Solid State Chem., 10, 3. WELLS,C.H. and C. P. SULLIVAN, 1968, Trans. ASM, 61, pp. 149-155. W ~ L ED.P. , and J. STRINGER, Phil. Trans. R Soc. Lond., 1980, A295,309-329. WOOD,G. C. and E H. S m , “High Temperature Corrosion” (ed. R.A. Rapp), 227, 1983, Houston Texas, National Association of Corrosion Engineers. WORRELL,W.L., in “Metal-Slag-Gas Reactions and Processes”, (eds. Z.A. Foroulis and W.W. Smeltzer), 822-833, 1975, Princetown NJ, Electrochemical Soc.Inc. ZACHARIASEN, W.H., 1932, J. Amer. Chem. Soc., 54,3841.
Further reading See the list of titles at the end of 5 1.3.
CHAPTER 15
DIFFUSIVE PHASE TRANSFORMATIONS IN THE SOLID STATE R.D. DOHERTY Department of Materials Engineering Drexel University Philadelphia, PA 19104, USA
R. W Cahn and R Haasen?, eds. Physical Metallurgy: fourth, revised and enhanced edition 0 Elsevier Science BY: 19%
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Ch. 15, 01
1. General considerations 1.1. Introduction Solid-state phase transformations are a central topic in physical metallurgy, since almost all industrial alloys are heat-treated after casting to improve their properties. The heat-treatment changes the microstructure of the alloy, either by deformation followed by a recovery and recrystallization pmess (ch. 25), or by some type of phase transformation. Two main types of phase transformation are found polymorphous changes and precipitation reactions. In a polymorphous change, in for example iron, cobalt or titanium, there is a change of the metal‘s crystal structure. This affects all the atoms in the alloy and has a tremendous scope for changing the microstructure of the alloy. In a precipitation reaction, which is crucial in alloys based on aluminium, copper and nickel, which have the same crystal structure up to the melting temperature, the main method of modifying the microstructure is to alloy with elements that are soluble in the base metal at high temperature but which come out of solution at lower temperatures. The particles of the new phase are called precipitates. The name comes from the same reaction in liquid solutions where the resulting solid crystals, being heavier than the mother liquid, do faU out of solution. No such movement under gravity is expected in “precipitation” from solid solutions. In a precipitation reaction there is a change from a single phase, a, structure to a two phase, a +p, structure. The growth of the p precipitate requires both a change of composition of the matrix and, in cases where the two phases have a different structure, an interface reaction in which all the atoms that are to form the p phase undergo this structural change. In a minority of precipitation reactions such as those that precipitate Guinier-Preston (GP) zones, the two phases have the same structure and orientation. In such cases, no separate interface reaction is required to grow the precipitate, merely a change in composition. In all polymorphous and most precipitation reactions, there is a migration of an interface between two crystalline phases, and there are two possible modes of interface migration. In the first of these modes, atoms make thermally activated random jumps across the interface, a “diffusive” mechanism. In the second mode, the daughter crystal grows into the parent, by a coordinated shear-type (martensitic) motion of all the atoms N for this at the interface. The description “military” suggested by C H R I ~ I A[1965] second type of transformation is rather appropriate and leads to his equivalent label of “civilian” for the uncoordinated interface motion of atoms in the first type of transforand CHRISTIAN [1990], the altermation. In recent literature, for example BHADESHIA native terms of “reconstructive” and “displacive” are used for the two types of method for transforming one crystal structure to the new one. The present author prefers the label of civilian or reconstructive to diffusive since any reaction such as the formation of bainite in steel (a mixture of low carbon, bcc, a,iron plus iron carbide) from single phase, solid solution of carbon in fcc, y, austenite requires both a change of composition, that takes place by a true diffusive motion of individual carbon atom jumps and the y to a interface reaction. The interface process could be either military or civilian. If it were military then bainite would have to be described as being diffusive and non-diffusive!
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The civilian or reconstructive types of transformation are the subject of this chapter while the military types are the subject of ch. 15. Bainitic transformations, which appear to have both a diffusive character for change of carbon concentration and a martensitic character for the change of matrix structure, are discussed in 0 2.6. The precipitation of close-packed hexagonal, silver-rich, AgzAl from an AI-Ag solid solution, which is now universally agreed to take place by a combined diffusive long-range composition change and a displacive interface reactions is discussed in detail in 0 2.2.3 and 2.2.4. The same distinction between a military/displacive and a civiliadreconstructive transformation is seen for the movement of a grain boundary in a single phase structure. Here the military transformation is that of mechanical twinning and the civilian one is the usual mechanism of thermally activated grain boundary migration seen in recrystallization and grain growth.
1.2. Driving forces - free energy changes All structural transformations are driven, at constant temperature and pressure, by a reduction in Gibbs free energy, G, from the original to the final structure. The solution thermodynamics that are the intellectual base of the analysis of phase transformations are described in detail in ch. 5. The definition of G is:
C = H - TS, (1) where T is the absolute temperature, S is the entropy and H, the enthalpy, is given by: (2) U is the internal energy, P is the pressure, V the volume of the material. In transformations between condensed phases, the second of these terms is negligable with respect to the first term.Under these circumstances the difference between G, the Gibbs, and F, the Helmholtz free-energies becomes insignificant at least as regards work done against external pressure and the symbol F will be used throughout this chapter for the function that is minimized during a structural transformation: H=U+PV,
F = U - TS,
(la>
For a polymorphous transformation from a parent a-phase to a daughter P-phase, the driving “force”, in Joules per unit volume, which has the equivalent dimensions as a pressure, in Newtons per unit area, is given in terms of the changes in internal energy, AU,,’, and entropy, ASao’, by:
A = AUZp - T AS:@
(3)
At the equilibrium transformation temperature, T,, AFao’ is of course zero. So if AU&’
and AS,,’ are effectively constant, at a finite undercooling, AT=(T,- T), the driving force is given by: AFa; = AUzo(AT/q) = AHzp(AT/T)
(4)
For small undercoolings the constancy of AU,,’ and AS,,’ are reasonable assumptions References: p. 1497.
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Ch. 15, 8 1
but for larger undercoolings, a correction should be made to eq. (4) whenever the specific heats of the two phases are different. This is readily carried out using standard thermodynamic methods, see for example SWALIN [1972]. Table 1 gives some typical values of the latent-heat changes, AHap’, from which it can be seen that the driving force for solid-state polymorphous phase changes is much smaller than for solidification at similar undercoolings. For a precipitation reaction in a binary alloy AB, such as the one described by the free energy/composition curves in fig. 1, the free-energy changes can be readily determined. The overall free-energy change, per mole of the alloy, is I+ J, where J lies on the common tangent to the two free-energy curves. However, the free energy change, per mole ofprecipitate, is K +L where K lies on the tangent that touches the a-phase at I, the alloy composition. The intercepts M and N of the tangents at the atomic fraction of B, C,, of C, =I, are the partial molar free energies, F’,, also called the chemical potentials, p,, of component B in the a-phase at atomic fractions C, and C,, respectively. From standard solution thermodynamics, we have; &(Co) = F,”
i-RTlna,(C,),
E ( C a ) = F,”
+ RT In aB(Ca),
(33)
where F$ is the free energy of pure B in its standard state, R is the gas constant and a, is the activity of B in the a-phase at the two compositions C, and C,. M + N is then given by:
MN = G ( C a ) - G ( C o ) =RTln- ‘B
(‘a
)
‘B (‘0)
I f a is a dilute solution with C, and Ca< 1, as in the phase diagram shown in fig. 2 and in most commercial alloys, then the activity coefficient, fB, of B in the a-phase is constant (Henry’s Law) so eq. (6a) becomes:
MN = RT ln(Ca/Co).
(6b)
Fig. 1. Freeenergy-composition curve of two phases a and p showing conditions for nucleation of p from supernaturated LY of composition C,.
Ch. 15, 5 1
1367
Dsfsrcsive phase tran$onnatiom
Table 1 Latent heat changes. Element
Transformation
Latent heat change (k.J/g-mol)
TAK)
Fe
liquid to solid liquid to solid austenite to ferrite bcc to hcp
-15.5 -18.9
1809 2133 1183 1155
7.3 Fe
Ti
-0.9 -3.5
since 4CO) =f,C,. If, again as in fig. 2, p is a dilute terminal solid solution of A in €3, then KL in fig. 1 is effectively MN and the driving force per mole of precipitate, AF,,'", or per unit volume of precipitate, AF,,', are given by:
AI$ = RT In( CJC,), AF;# = [RT/V"} In(CJC,). Since C, c C, in a supersaturated solution, AFu," and AF,,' are both negative. The change from free energy per mole to free energy per unit volume is achieved through the use of V" the molar volume of the P-phase. Very often the precipitating P-phase is an intermetallic compound so that C, is significantly c 1 so that a correction must then be made, giving:
For the precipitation of an intermetallic phase such as MnA1, from an aluminium-rich solid solution, the effect of the modification to eq. (6e) is to decrease the driving force very significantly, by 1/7 for MnAl,. An extreme version of this effect is found for the growth of ferrite, a, from low-carbon austenite, y, in iron-carbon alloys. Since the precipitate is solute-depletedrather than solute-enriched,the form of the equation derived as here is slightly changed. We now have:
Tt
Fig. 2. Simple phase diagram derived from fig. 1. References: p . 1497.
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where the compositions are given in atomic fractions of the solute, which in this case is carbon. The low driving force for this reaction even when the alloy is cooled well into the two-phase, a plus y, region for example when the volume fraction of a can be 0.5 or more, makes this reaction very different from most precipitation reactions. The difference arises since the driving force for nucleation of femte, AFa,, is very small while the volume fraction of new phase, { (C, - C,)/(C, - C,)} which is also the supersaturation, that drives the diffusional growth processes (52.2.2) can be very large. That is, for the formation of ferrite from austenite and in other equivalent matrix phase changes, the nucleation of the new phase has a much lower driving pressure while growth is driven much faster than in most precipitation reactions. This idea is considered in more detail in 5 2.2.6.2”. Finally it should be noted that if the matrix phase is not a dilute terminal solid solution the activity coefficient, fB, will not be constant. The correction for this effect derived by PURDY [1971] then needs to be included. This gives:
a,
Here &, is the non-ideality factor, &a = 1+ d h f B / d h C,, derived originally to DARKEN [1948] for diffusion studies. This term can be very significant, 8, f 1 if the matrix phase from which the precipitate is forming is itself an intermediate phase such as /3 brass in Cu-Zn or Cu-A1 or an ordered intermetallic phase such as a2in the Ti-A1 system.
1.3. Stable and unstable free-energy C U N ~ S The free-energy/composition curves shown in fig. 1 are both those of stable or at least metastable phases, since any alloy on a free energy composition curve where the curvature, d2F/dC2, is positive cannot spontaneously reduce its free energy, except by nucleating a distinctly different new phase. In a phase diagram such as that sketched in fig. 3, where an a-phase, below a critical temperature T,, decomposes into solute-rich, a”,and solute-poor, a’,phases, the free-energykompositioncurve is unstable below T,,
* Just above the eutectoid temperature (-1000 K) in Fe-C,
where the carbon content of austenite is 0.8 wt%C
(Cy=0.0017),C, is almost zero. So, while the supersaturation for growth, for an alloy with 0.4 wt% C (C0=0.0008), l’2 is 0.5, the driving force for nucleation, AF‘m, is only - 4 Jlg-mole. For precipitation of a B-rich phase, such as O’, CuAl,, from an aluminium solid solution at say 1 9 0 T with an equilibrium solubility C, of 0.0002 and for an alloy composition C, of 0.02, then while l’2 is only 0.06, the driving force for the
nucleation of the copper rich precipitate, AFm8”,is 1500 times larger at - 6,350J/g-mole. As a result of these differences the polymorphous reaction in iron gives coarse grain-boundarynucleated precipitates of femte while the Al-Cu reaction gives a very different microstructure - that of very small copper-rich precipitates finely dispersed in the aluminium grains, 8 2.1, giving precipitation hardening.
Ch. 15, 3 1
Difisive phase transfonnations
a’+8
CB
1369
I
-
Fig. 3. More complex phase diagram than in fig. 2, showing the development of solid-state immiscibility in the a-phase below a critical temperature T,.
figs. 3 and 4. This instability arises in the part of the curve where the curvature, d2F/dCZ,is negative. Within this “spinodd“ region an alloy of composition i can split intoj and k and so reduce its free energy (CAHN [1968]). This can occur not only if a’ and a” are the stable phases but also when as in fig. 4,below T2,there is an even more stable P-phase. The formation of the &phase requires the system to overcome a nucleation barrier. In this case, the solute enhanced region a”,although metastable with respect to P, will still form when the rate of its formation is faster than the kinetics of the nucleation and growth of the more stable P-phase. Such regions of solute-enriched a”,sharing the same crystal lattice as a’,are known as Guinier-Preston, GP, zones.
1.4. Gibbs’s two types of transformation From figs. 3 and 4, it is possible to see the distinction between the two types of transformation first described by the father of this subject, J. Willard Gibbs. In the first, type I, of these reactions, usually described as a nucleation und growth reaction, a small region of a new phase, such as P, forms from within the matrix. The new region, the nucleus, has a completely different composition and/or structure from the parent phase, a. The nucleus is separated from the parent matrix by an interface which has an
T
Tc
Fig. 4. Free-energy-composition curve for the phase diagram of fig. 3 at a temperature below the eutectoid temperature T2.The curvature, d*F/dc, is negative in the part between the dashed lines, positive elsewhere. References:p. 1497.
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interfacial energy, c,whose units are J/m2. This change is one that can be described as being large in the magnitude of the local change in structure and/or composition but one that is spatially localized to the small part of the matrix which has transformed. The majority of the matrix remains completely untransformed at least initially, though subsequent growth of the nucleus into the matrix occurs. Other names for such “nucleation and growth” transformations are “heterogeneous” or “discontinuous” transformations, “Heterogeneous” nucleation and “discontinuous” precipitation, however, have different and quite specific meanings within the area of nucleation-and-growth reactions. As a result, it is better not to use these names for the whole class of Gibbs Type I transformations. Details of heterogeneous nucleation and discontinuous precipitation processes in various types of nucleation-and-growth reactions are discussed in 0 2. The second mode of transformation is shown by a composition fluctuation in an alloy within an unstable region of its free-energy curve. The result of such a fluctuation is that a region of initially uniform composition develops a composition wave whose amplitude grows with time, fig. 5, but with, at least initially, a constant wavelength. This second type of reaction, a Gibbs Type I1 transformation, where the initial change in composition in any local region is small but where the composition fluctuation is deZocaZized in space, is usually called spimdal decomposition, at least when the wavelength is significantly longer than diameter of an atom. When the wavelength is that of an atom, leading towards an ordered solid solution the transformation is then usually called continuous ordering. Spinodal decomposition is known to occur in many precipitation reactions, for example in concentrated Al-Zn alloys (RUNDMANand HILLIARD[1967]). The term continuous transformations is now a common name for the Gibbs type II transformation in both precipitation and in ordering and this terminology is used here in 0 3.*
distance
-
Fig. 5. Variation in solute distribution with time during a continuous reaction, Gibbs type2 spinodal decomposition.
* There is again a possible confusion here since the term continuous precipitation is sometimes used for precipitation by a nucleation and gruwth reaction that does not require a migrating high angle grain boundary. Discontinuous precipitation is the nearly universal name for precipitation that does occur behind a migrating high angle grain boundary. This name discontinuous precipitation is so commonly used that it will he used here, 5 2.5, even though an alternative name such as a moving two-phase boundary (MTPB) might be less confusing. The context of the discussion will usually avoid any difficulty. However, in this chapter, we will not use the term “continuousprecipirarion” for a nucleation and growth reaction that takes place without the assistance of a moving grain boundary.
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Difisive phase transformations
1371
1.5. First order and higher order transformations In most structural transformations in physical metallurgy there is a discontinuous change in the slope of the free energy-temperature curve as the structure changes at the equilibrium temperature (fig. 6). This type of transformation is described as first order, since there is a discontinuity in the first derivative, dF/dT, and thus a discontinuous [1933]. In second change in the internal energy, giving a finite latent heat, EHRENFEST order transformations, there is a discontinuity in the second differential, d2F/dp, and therefore in the first differential of internal energy, which is the spec@ heat. CHRISTIAN 119791discussed the thermodynamic and mechanistic classifications of structural changes of interest to metallurgists and noted that some apparent second order transformations appear to show an apparent infinity in the specific heat rather than just a discontinuity.Unlike the first order transformations, these apparent second order transformations show an increase of specific heat (i.e., a latent heat) as the temperature approaches the critical temperature.
1.6. Short-range and long-range diffusion
In polymorphous changes, such as solidification and solid state structural transformations in pure (single component) materials, and also during grain-boundary migration in recrystallization and grain growth, the only atomic process is the transfer of atoms across the interface: This is the interfacial process. For nucleation-and-growth reactions where the parent and product phases have different compositions, there are then two successive processes; firstly, long-range transport by diffusion over distances of many atomic spacings, commonly described as the difl~ssioaalprocess and secondly, atomic transport across the interface, normally a thermally activated short-range diffusional process which is again the inteijkial process. The long-range diffusion only involves a fraction of the total number of atoms in the new phase, those required to change the composition of the matrix to that of the new phase. The fraction may be nearly loo%, for example in the precipitation of a B-rich phase from a supersaturated A-rich solid solution (fig. 2). In other cases, like the precipitation of bcc ferrite from dilute fcc austenite in low-carbon steels where there is only a small change of composition, the fraction of atoms taking part in the diffusional process is very much less than one. By
TFig. 6. Variation of free energy with temperature, showing a first-order reaction at an equilibrium transition temperature Te. References: p . 1497.
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contrast all the atoms that are to be part of the precipitate must take part in the interfacial reaction. Since the two reactions, long-range diffusionand the short-range interfacial step, are successive reactions, the slowest of the two processes will be rate-controlling (p 2.3). This may be contrasted with the situation where there are two alternative reactions, for example the formation of either (i) the stable phase with the lowest free energy or (ii) a metastable phase with a higher free energy. Where there & two or more alternative reactions, then the fastest total reaction will determine which reaction occurs and this will, in consequence, determine the rate of the transformation and the resulting microstructure. This situation occurs, for example, in precipitation-hardening reactions at low temperatures and high driving forces, where metastable precipitates with low barriers to nucleation form in preference to the stable precipitate when that is more difficult to nucleate. The initial metastable reaction removes much of the driving force for the formation of the stable phase, so that the metastable product may then remain for the useful lifetime of the sample.
1.7. Techniques for studying phase transformations Since phase transformations produce a change of microstructure, almost any structural characterisation technique can be used to study a transformation. EDDIGTON[1979] reviewed this topic and detailed references to the tremendous range of published work can be found there and in the numerous conference reports on the various structural characterisation techniques. The techniques that have been used include optical, scanning and transmission electron, and field-ion microscopy. The increased resolution of the last two techniques involves a severe penalty in that the regions studied are highly localized and so the structures observed in great detail can be potentially unrepresentative. Diffraction techniques are also of vital importance, both those involving general diffraction and also those using selected-area diffraction in both transmission microscopy and at even higher spatial resolution, scanning transmission electron microscopy (STEM). At lower spatial resolution, X-ray Kossel methods, electron channelling patterns and more recently Back Scattered Kikuchi Diffraction (BKD) - also called Electron Back Scattered Patterns (EBSP) - have been of great value in investigations involving bulk samples. Of major importance, particularly for quantitative investigations, are chemical analytical studies of highly localized regions. Conventional electron-probe microanalysis allows chemical analysis down to scale of 1-2 pm, analytical electron microscopy by either X-ray emission or by electron energy loss has allowed routine analysis down to a resolution of 10 nm and below. Ultimate or near-ultimate resolution has been achieved by the atomprobe field-ion microscopy technique, the determination of groups of individual atoms by time-of-flight measurements of field evaporated atomic layers (HAASEN and WAGNER [19861, MILLER [ 19871. Other successful analytical techniques include high-resolution Auger electron spectroscopy and ion probe mass spectrometry using focussed ion beam erosion. For a detailed comparison of the techniques, a conference on microanalysis at high spatial resolution provides a detailed source of information, LORIMER [1981]. The
Ch. 15, Q 1
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1373
older techniques of measuring average solute levels throughout the microstructure should not be neglected since these give additional kinetic information to that provided by the more detailed local measurements of composition. Any physical property that is modified by the solute level in the matrix can be used for general chemical analysis. For magnetic materials the ferromagnetic Curie temperature is often used, at least for studying slow reactions or ones occuring at temperatures well above the Curie temperature. Other methods include studies of lattice parameter by X-ray, electron or neutron diffraction, semi-quantitativemeasurements of minority phase amounts by Guinier X-ray diffraction in which the white X-ray background has been eliminated, thermoelectric power and some electrochemical methods. The most commonly used however is that of electrical conductivity since, for metals, the presence of solute has the largest effect on elecbical resistivity, compared to almost all other structural defects. For ionic crystals, increased solute such as divalent magnesium ions in monovalent lithium fluoride increases the vacancy concentration, so leading to increased ionic conductivity at least in a temperature range where extrinsic conductivity is found. This technique allows very accurate chemical analysis of the solid solutions in ionic compounds. Many of these modern high-resolution techniques have only recently become available and are usually expensive in equipment costs, so much less use has been made of them than of the very widely used transmission electron microscopy which revolutionized this subject after its introduction in the late 1950s. However, as EDINGTON[1979] has shown, these latter techniques, introduced in the 1970s, have made extensive contributions to the subject and will continue to do so. Further use of these techniques may modify the present situation, which is that in many areas, the theories of phase transformations are well ahead of rigorous tests of those theories, at least, as far as quantitative experimental studies of the kinetrics of transformation are concerned. It is to be noted that there are two distinct modes of investigation of microstructural change which are used. In most cases the microstructure is investigated in samples where the reaction has been halted by rapid quenching to room temperature. The study of these static microstructures allows detailed insight of the microstructure at the highest resolution, However, the dynamic development of the reaction with time, in a particular region, cannot be followed by the static technique at least in one sample. A series of different samples quenched after different times must be used and avemged structural information obtained. The alternative technique, of dynamic or in-situ observation of structural change in a hot-stage microscope or equivalent equipment allows continuous investigation of the transformations in a selected region. This advantage is however, obtained at a severe cost of poorer resolution and with severe time limitations. The structure often changes before it can be fully characterized. An additional problem for the in-situ studies is the worry about modification of the transformation by the presence of near-by surfaces and by the effects of the illumination, usually a beam of high energy electrons. The use of higher voltage electron microscopy can reduce the first of these difficulties, by allowing thicker foils to be used, due to the greater penetration of the higher energy electrons. However, point defects especially vacancies can be induced by the higher-voltage beam and these vacancies if present at higher concentrations than the vacancies in thermal equilibrium can then modify the diffusional kinetics, see for References: p . 1497.
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example (e.g., WEAVERet al. [1978] and DOHEXTY and PIJRDY [1983]). In most cases, however, a careful use of a combination oftechniques allows the potential errors to be recognized and overcome and should enable valid conclusions to be drawn. In general, the static methods provide more accurate quantitative measurements while the dynamic or in-situ methods are better suited for giving qualitative insight into the operating mechanisms. Other techniques that can give useful dynamic insights into structural changes include calorimetry, if heat is being released or absorbed at a detectable rate, dilatometry where there are significant volume changes and for most reactions neutron diffraction is attractive since large samples can be studied both statically and, most conveniently, dynamically at temperature due to the highly penetrating nature of neutrons. (see ch. 12).
2. Nucleation-and-growthtransformations 2.1. Theory of nucleation The basis of the theory of nucleation is that when a new phase forms within a parent phase, an interface is formed between the two phases. The formation of the interface creates a local increase of free energy when the first atoms assemble in the new structure. The theory of this process, which describes the interfacial energy barrier to nucleation, was originally developed for physical chemistry by VOLMERand WEBER [1928], VOLMERand FLOOD[1934] and BECKERand DORING[1935]. The model was and FISHER[1949] to solidification of metals and subsequently applied by TURNBULL [19563. Following these solid-to-solid metallic phase transformations, see TLIRNBULL studies the concept has become fundamental to the qualitative and quantitative understanding of structural changes in physical metallurgy. The theory appears to be, qualitatively at least, highly successful both for solidification (ch. 9, $3) and in successfully describing the nucleation step in most solid-state nucleation-and-growth reactions. The subject of solid-state nucleation has been reviewed on many occasions, for example by KELLYand NICHOLSON [1%3], RUSSELL [19701, NICHOLSON [19701, CHRISTIAN [19751, RUSSELL [1980] and HAASENet al. [1984], with similar conclusions to those advanced here. For almost all diffusional solid state reactions the theory is in good agreement, at least qualitatively and semi-qualitatively,with a very large amount of experimental data. There are, however, rather few fully quantitative experimental tests of the theory; these do, however, support the theory at least of homogeneous nucleation to a surprisingly high degree. The one exception where the theory has been found not to apply, even qualitatively, is for the nucleation of new grains in the recrystallization of deformed materials, see for example DOHERTY [1978]. In this case, the nuclei of low dislocation density regions of the correct structure pre-exist in the deformed state and can develop by processes that do not require a local increase in free energy. The nucleation of recrystallization is discussed in ch. 28, $3.3. For nucleation in martensitic reactions the theory, at least to date, appears to have been somewhat less useful, as is discussed in the following chapter. The model, for diffusional phase transformations, assumes that a new phase, differing from the parent phase in structure and/or composition, is built up, atom by atom, by
Ch. 15, 92
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1375
thermally activated atom transfer across the interface. There is, as a result, a local change in the free energy, which is given, in the usual continuum description, as a decrease in the volume free energy, due to transfer of atoms from a less stable to a more stable phase, and an increase in the interjkial free energy due to the increase of the area of the interface between the two phases. In the conventional model of nucleation, the macroscopic values of these parameters are used even though it is unlikely that a cluster of only a few atoms would have the macroscopic properties that the new phase will show when it has grown to a size much larger than atomic dimensions. The total change of free energy, AFt, when a new phase of volume, V,, and interface area, A,, forms, often causing an additional increase in elastic strain energy per unit volume of precipitate, AF& is given as: U,= VBUu>+ A,u
+ VBMl,
(9)
where u is the specific energy of the a+ interface and AFa; is the free energy change per unit volume of the new /3 phase, given by the macroscopic molar free energy change, AFG, divided by the molar volume, V,, of the new phase. The first and last terms in eq. (9) are both dependent on the volume of the nucleus and are treated together. Only if the first term is larger (more negative) than the third term, which is always positive, can the reaction proceed. The volume of the nucleus increases as the third power of the nucleus size ( 4 d / 3 if the nucleus is a sphere of radius r), while the area term, which is always positive and acts as the barrier to the process, increases only as the second power of the nucleus size (h? for a sphere). Figure 7 plots eq. (9) for a spherical nucleus, and shows that at small radii the nucleus shows an increase of AFt and so is unstable and tends to redissolve in the matrix. Only when the new phase has reached the critical radius, r*, and has acquired the critical free energy increase, AF*, does the nucleus become potentially stable, so that, after the addition of one further atom, r> r*, it can grow with a continuous decrease of free energy. For a spherical nucleus the values of the critical parameters are given by: f l = -2g/(ACB+ A&"),
Fig. 7. The nucleation problem; free energy of a sphere of new phase in a supersaturated matrix, as a function of the radius of the sphere.
References: p . 1497.
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The required values of AFapv for use in these equations can be obtained from Map", for example from eqs. (4),(6c), (6e) (7) or (8), by dividing the value of AFapm, by the volume of a gram-mole of the new phase, V,. For most metallic phases, the volume of a gram-mole, containing an Avogado's number of total atoms, of any metallic phase is close to 10" m3/g.mole. The problem is to find the appropriate value for AF;; this is discussed in outline 9 2.1.2,but for mathematical simplicity for the rest of 0 2.1 it will be assumed that the strain term is negligible, as it will be for example in solidification from most metallic melts, at least above the glass transition temperatures In all cases the correct values of f l and AF* can be recovered by simple addition of AF: to AF,S'. The rate of nucleation per unit volume, Za&. can be obtained from the simple application of statistical mechanics to show that if the critical formation energy of a nucleus is A P then, at equilibrium, the concentration of critical-sized nuclei, n* in a unit volume is given by: 8=N ' exp(-AF*/kT),
(11)
W is the number of atomic sites per unit volume on which the assembly of the nucleus could have started, for homogeneous nucleation this is all atom sites. The rate of nucleation is then the product of this concentration of critical nuclei and the rate of atomic addition to the nuclei to make them just supercritical: pZ:
= N'A*v exp(-AFA/kT) exp(-AF*/kT),
(12)
A* is the number of atomic sites on the surface of the critical nucleus to which an atom can join after overcoming a growth barrier A F A (9 2.2.1) and Y is the atomic vibration frequency of an atom in the matrix at the interface, usually assumed to be of the order of 10'3/s. In most cases A* is assumed equal to the total number of atomic sites on the interface but as discussed in 8 2.2.3 this may not always be valid and atoms may be able to add to a nucleus only at ledges on the interface. The simple theory above needs to be modified to take into account the expected loss of critical-sized nuclei by their growth, during nucleation, into supercritical regions. This loss reduces the value of W to less than the value expected at equilibrium, eq. (1I), by a factor, 2, which is usually about 0.05 (RUSSELL[1970]). The near constancy of this correction arises since the rate of formation of critical-sized nuclei and their growth to become supercritical both occur at a similar rate determined by atomic diffusion onto the growing cluster of atoms. Inclusion of a factor of exp(-r/t) to allow the achievement of the expected steady state rate of nucleation, yields the result given by RUSSELL[1970]:
c,
Zip = exp(-r/t) ZN'A*v exp(-AF'/kT)
exp(-AF*/kT)
= exp(--r/t) ,:z Z = (AF*/~P~T~**)I~~.
(12b)
There are many problems with the use of the theory to predict experimentally measured quantities such as the final density of the new regions of the product phase. These
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DzDsive phase transformations
1377
difficulties include the usual lack of knowledge of the interfacial energy u which plays such a vital part in determining Insv (Z&' is proportional to the exponential of u raised to the power 3!), the difficulty in calculating the strain energy term in eq. (lob), and the difficulty in calculating how the driving force for the phase change, AFuav, falls as the reaction proceeds, so that the rate of nucleation can be integrated to give a total number of growing nuclei. Finally, as discussed in $2.4.2, a significant number of nuclei will dissolve up, by a process of Ostwald ripening, during the precipitation reaction itself, These problems are discussed by SERVIand TURNBULL[ 19661, LANGER and SCHWARTZ [19SS], K A ~ M A Nand N WAGNER[1984] and HAA~EN and WAGNER[1992]. However, despite these quantitative difficulties the theory is extremely useful in describing a large volume of experimental observations. A major success of this theory is its ability to account for the observation that nucleation increases from a rate that is almost undetectable to rates that are too fast to measure over a narrow range of underrooling, AT in eq. 4 or supersaturation, CJC, in eq. 6d.This is readily shown by use of eq. (11) for a matrix phase change driven by a free-energy function of eq. (4). Under these circumstances the concentration of critical nuclei is given by:
log$ = 28 - K / A T ~
(13)
This arises since the number of atomic sites in a cubic metre is about 10'' and K is a constant given by:
If the undercooling which gives a density of nuclei of 1 (10') m-' is AT(0) and that which gives a density lo6times as high is AT(6), then substitution into eqs. (13) shows that: AT(6)/'AT(O) = (28/22)"' = 1.1.
(144
That is, a 10% increase in supercooling will increase the density of critical-sized nuclei, and therefore the rate of nucleation, by a million times. The constant K in eq. (14) can be easily evaluated if the undercooling that gives a measurable rate of nucleation is known, by substitution of a value of N* of about lo3mW3in eq. (13). This approximate value of K allows rather an an accurate estimate of the effective interfacial energy, u,to be made using eq. (14). In many cases, for example in solidification of pure metals, the determination of the interfacial energy by use of eq. (13) has given reasonable values of this energy even when the undercooling appears to have been significantly underestimated (CANTORand DOHERTY[1979]). Since eq. 12 also contains the activation energy of diffusion, exp(--hF,/kT), the initial increase of nucleation rate with fall of temperature ceases, giving a maximum rate, Below the maximum, the nucleation rate falls rapidly as the temperature falls. This variation of nucleation rate with temperature (together with a similar behaviour for growth) causes the so-called "C-curve" behaviour seen in time-temperature-transformation, TTT, diagrams, 9 2.5.1. Another major success of the theory of homogeneous nucleation is its ability to References:p . 1497.
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understand a very large range of qualitative experience in which, in many solid state precipitation reactions, the phase that forms is not the equilibrium structure. It is frequently found that a metastable phase, characterized by good atomic fit with the matrix and therefore a low value of interfacial energy, (i, is the one that forms. The theory is compatible with this phenomenon since the rate of nucleation is so sensitive to (i; the successful precipitate will be the one that nucleates fastest, and this is usually the precipitate that has the lowest interfacial energy, even though it may not yield the most stable phase. Similarly, nucleation theory accounts for the fact that when a new crystalline phase forms in a matrix of a crystalline parent phase there is almost always a particular orientation-relationshipbetween the phases, see for example BARRETT[19521. This relationship appears to be the one that minimizes the value of the crucial parameter, the interfacial energy, in eq. lob. This idea is discussed further in Q 2.1.2. Finally it should be recognized that additional qualitative support for this picture of nucleation is provided by an apparent failure of the model. The failure arises since in many cases nucleation occurs at much smaller undercoolings than expected. In these cases it is found that the new phase forms on some defect in the parent phase, for example a dislocation or a grain boundary. The second term in eq. (9) is significantly reduced by the fact that the new phase has consumed some of the existing energy of the defect, thereby reducing the barrier to nucleation. This topic of heterogeneousnucleation is discussed in Q 2.1.4.
2.1.1. Interfacial structure and energy A major difference between the interfaces produced by nucleation during solidification from a liquid or glassy phase, and by nucleation in solid-state reactions, is in the possibility of atomic matching across an interface between two crystals. For crystal-crystal interfaces the atomic matching ranges from perfect, in fully coherent interfaces, through less perfect matching in semicoherent interfaces, to random matching in fully incoherent interfaces. The atomic structure of such interfaces is discussed in ch. 9, Q 2.3.3. A simple example of a fully coherent interface is one between two phases that have the same crystal structure, lattice parameter and orientation, giving an orientation relationship often loosely described, for systems with cubic matrix and precipitate, as “cube-cube”. For fully coherent interfaces, in a binary AB alloy, the only contribution to the interfacial energy, (i, comes from the higher energy of the unlike A-€3 bonds across the interface. Since the system is one of limited miscibility between the two phases, it is expected that the unlike bond energy h(AB) will be greater than the average energies of the bonds, [k(AA) + h(BB)]/2, as described in any discussion of the so-called quasi-chemical bond model (e.g., SWALIN [19721). Such completely perfect matching is unlikely to be found and deviations from coherency can develop as the extreme conditions of perfect matching described above are relaxed. In most reactions there will be a small difference in the lattice parameters of the two phases, a, and up. This gives rise to a misfit, 8: For small misfits and very small precipitates, the difference in atomic spacing can be
Ch. 15, $ 2
1379
DiDsive phase transformations
taken up by elastic strain in the two phases; but with larger precipitates the system can lower the resulting elastic strain energy by concentrating the strain in a cross-grid of edge dislocations spaced a distance d apart (fig. 8). Over the dislocation spacing, d, there are n atomic planes in the phase with the larger parameter and n + 1 planes in the other phase. When the misfit is much less than unity, then:
n = 1/S, and
d = (a, + as)/2S.
For such a semicoherent interface the interfacial energy has two components; a “chemical” term as discussed for the coherent interface and an additional “structural” term, due to the cross-grid of edge dislocations. For small precipitates, where the diameter of the nucleus may be less than the dislocation spacing, d, the nucleus will remain coherent, though elastically strained, and consequently will have a higher solubility because the elastic energy raises the volume free energy of the precipitate phase ( C m [1968]). An incoherent phase boundary, or a phase boundary with poor atomic fit and high energy, will have an atomic structure rather like that of a high-angle grain boundary, whose misorientation is well away from any possible coincidence-site boundary relationships (ch. 9, 9 2.2.1.1). Interfaces with poor atomic fit in two phase systems are likely to arise in one of two main ways: The first way is if, after nucleation of the precipitate, the orientation of the surrounding matrix phase is changed by the passage of a high-angle grain boundary in a recrystallization or grain-growth reaction. This phenomenon has been discussed by DOHERTY [1982] with examples of the resulting change in interfacial properties and consequent microstructural changes. The other main origin of incoherent interfaces arises when the two crystals, a and p. do not share a common crystal structure. This is the usual situation in most precipitation reactions and is inevitable in all polymorphous changes. As is discussed in the next section, under these conditions part at least of the interface is almost certain to have very poor atomic
?~ s e e b e o e o o o 6 0 6 @ @ @ 0 8 @ 0 @ 8
3
@
0 8
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
;-a,-$ I
@ 8 atoms
0A
,
atoms
Fig. 8. Semicoherent interface between two phases with the same structure but different atomic spacing and containing a set of edge dislocations a distance d apart. References: p . 1497.
1380
R D.Doherty
Ch. 15, 92
matching giving an interface that can be regarded as incoherent or at least as having a poor atomic fit and a high specific interfacial energy.
2.1.2. Equilibrium shape When the nucleus and the matrix have different crystal structures, it is almost invariably found that there is a definite and reproducable orientation relationship between [19521 gave a very complete listing of the orientation relationships the phases. BARRETT for many reactions. The orientation relationship appears to allow a good atomic fit to develop between the two crystals either along a particular plane, WESTMACMTand DAHMEN[1984], or occasionally along a particular direction, DAHMEN and WESTMACOTT [1986], Luo and WEATHERLY [1987] and DAHMEN[1994]. Qpical examples of these orientation relationships are provided by fcc-hcp/matrix-precipitate pairs in systems such as A1-Ag2Al and in the Cu-Si system. These precipitate matrix pairs show a good fit between (111) in fcc and (OOO1) in hcp. These precipitates form with these planes parallel and they provide the habit plane of the flat plate-like precipitate crystals that form in these reactions (e.g., fig. 24 below). The atomic fit between the two structures is shown in fig. 9, where it can be seen that the two structures, the matrix, a,and the
Fig. 9. Atomic structure of fcc-cph matrix-precipitate structures in the AI-Ag,AI system; (a) the fcc matrix; (b) y’ intermediate precipitate, fully coherent; (c) equilibrium Ag,AI, partially coherent; (d) the fcc {ill} stacking sequence.
Ch. 15, 52
Diffrcsive phase transformations
1381
metastable precipitate, y’, have identical atomic arrangements on the habit plane, so allowing full coherence in two dimensions. The two equilibrium phases have slightly different lattice parameters but the intermediate metastable precipitate, y‘, has the same atomic spacing as the aluminium matrix, in the close-packed plane, and so shows full coherence, and consequently very low energy, for the habit plane (the structural component being zero). Precipitates are, however, three-dimensional and the interface must lie at all possible orientations around the crystal. Consequently, the non-habit plane parts of the interface, the rims of the plate-like crystals, have much poorer atomic matching and higher interfacial energy. In other words, the magnitude of u will vary significantly around the precipitate. This can be represented in a section through the u-plot as shown in fig. 10. The u-plot is a radial plot of the interfacial energy as a vector whose length varies as a function of the orientation of the interface of the precipitate. The deep cusps in the plot occur at the orientations of good fit, corresponding to the habit plane. AARONSON et al. [19681 calculated that the ratio of energy between the good fit and the rest of the interface is 1 :2.7 for the equilibrium Ag,Al phase in Al; similar [1978]). calculations for the intermediate phase give a higher ratio of 1 : 10 (FERRANTE For anisotropic cr-plots such as the one seen in fig. 11, the Gibbs-Wulff theorem (see for and DOHERTY [1976]) predicts the equilibrium example MULLJNSE19631 and MARTIN shape of the precipitate, for the orientation relationship that gives the particular anisotropic u-plot. This shape is the inner envelope of so-called Wufffplanesthat are planes drawn perpendicular to the vector from the origin at the intersection with the u-plot, fig. 11. The Wulff plane has, of course, the orientation of the interfacial plane whose value of u is OB. Figure 11 shows as an example the Wulff plane, LBM, for the u vector, OB, and also shows the equilibrium shape of the precipitate, which provides €or facet planes at the cusps, C,, C,, C, these facet planes are the good-fit interfaces. The aspect ratio,A, of plate like precipitates, defined as the length-to-thicknessratio, R ncusp t . gives a good-fit plane where there is equal to the energy ratio u i ~ / o c o h e The is good match of crystal planes as in the fcc-hcp system discussed and in many other precipitation reactions, for example A1-8’ in the aluminium-copper system which has
Fig. 10. The (r plot, the variation of interface energy with the orientation of the boundary plane for the particular orientation that allows good fit at the cusp plane, 0,. References:p. 1497.
1382
R.D. Doherty
Ch. 15, 82
1I
_____ Wulff plane LBM Wulff planes giving the equilibrium shape
Fig. 11. The Wulff construction. The Wulff plane at B is LBM, that is normal to OB at B. The equilibrium shape is the inner envelope of Wulff phases, in this case, the cusps C,, C, and C,.
habit planes on the matrix (100) planes (fig. 15, below). For systems like the fcc-bcc system in, for example, the iron-copper system, there is only a good-fit direction, which is the close-packed direction in the fcc phase and in the bcc. This form of matching of directions provides the well-developed needle morphology, with an aspect ratio of 5: 1, reported by SPEICHand ORIANI[1965] for the Fe-Cu system. For good-fit directions, the a-plot, schematically given in two dimensions by fig. 10, must be rotated about the vertical axis, a,, to produce the appropriate three-dimensional needle or rodlike form: For plate-like crystals the three-dimensional plot is obtained by rotation about the horizontal axis, a,. In other examples the interface between fcc and bcc, for example in steels, remains platelike, see for example AARONSON et al. [1970]. Since the ends of needle crystals, and the rims of plate-like crystals, have poorer atomic fit than the coherent needle axis or habit planes of the plates, it is commonly assumed that these ends and rims are fully incoherent. This may be usually the case but it need not necessarily be true since in the fcc-hcp pair Al-Ag,AI it has been seen that the r i m s of the plates are also faceted, FERRANTEand DOHERTY [1979]. This implies that the rim is also coherent, though with a poorer fit made up by the cores of Shockley partial dislocations, H o w etal. [1985]. CHRISTIAN [1994] shows that a precipitate in a matrix cannot be fully or partially coherent on one part of the interface and incoherent elsewhere so the rims of all plate-like precipitates should be considered coherent but to have a high density of dislocations, giving high energy and mobility. It is usually assumed during nucleation that the nucleus forms with the minimum interfacial energy, since any other assumption increases the free-energy barrier to nucleation and strongly reduces the rate of nucleation. The assumption of minimum interfacial energy requires the nucleus to have the equilibrium shape, this will be plate-like or needle-like if the cr plot shows the normal anisotropy for two crystals with different crystal lattices. If the plot is, however, fully isotropic, the expected shape of the nucleus will be a sphere. In two examples of “cube-cube” relative orientations between crystals with the same structure, nearly spherical equilibrium shapes have been reported, In the bcc-bcc beta-gamma brass system studied by STEPHENS and PURDY[1975] and in the all-fcc system of Ni-Ni,X (where X can denote a mixture of several atoms, usually A1 or Ti) (see RICKS et al. [1983]), equilibrium shapes that were nearly spherical were reported. In the latter case, it seems that the precipitate develops the usually reported
Ch. 15, 32
D i m i v e phase transfotmntions
1383
cuboid morphology only with growth, and in systems when there is some small elastic misfit, 6. Both the brass and the nickel systems can evolve into a dendritic morphology during growth under condition of very low supersaturation but this is a growth form not an equilibrium shape! The shape instability leading to dendritic growth is discussed in 0 2.5. For plate-like precipitates with an equilibrium value of the aspect ratio A, that is, with A = a , / a , , the critical values of the plate radius, ab)* and nucleation barrier, A P , are found (see AARONSON et al. [1978]) to be given by:
AF* = 2561m~/{27A(AF$ + AF;)}
(+,is the interfacial energy of the poor fit rim and a,that of the good fit habit plane. 2.1.3. Strain energy Strain energy plays a vital role in the nucleation of solid-state phase changes, as shown by eqs. (9)-(19). There appear to be two different types of strain that can be developed by the formation of a new phase in a matrix phase. The first type of strain occurs only when both phases are crystalline and is that caused by a misfit, due to different lattice parameters - eq. (15), between two coherent phases. The second type of strain is that caused when the new phase occupies a different volume from the region of the matrix that it has replaced. This second type of strain can arise in various ways. One of these is when the volume per atom is different in the two structures and the precipitate grows without a change in the number of atom sites. An example would be the nucleation of the less dense bcc ferrite from austenite in pure iron. A further method for the development of this volume strain is if there is a significant difference in the rates of diffusion of two components of the alloy, An example is when zinc-rich y-brass precipitates from P-brass when there is a more rapid inward flux of the faster diffusing zinc atoms than the compensating outward movement of copper atoms. This increases the number of atomic sites in the region where y-brass is growing, giving a dilational strain. This effect has been directly observed on the surface of a brass alloy, by CLARKand WAYMAN[1970] who showed that the y-phase stood above the surrounding surface. In the interior of the sample this displacement would produce an elastic strain unless relieved by plastic deformation. A similar increase in volume arises during growth of Fe,C in Fe by rapid interstitial diffusion of carbon upto the growing precipitate. A local strain around a precipitate will produce a large energy increase if the strain is taken up elastically since the elastic moduli of metals are large. For aluminium where Young’s modulus is 70 GPa, a 1% strain would give an elastic energy of 3.5 MJ/m3 (35 J/g-mole), a 5% strain would give an energy 25 times larger. In the case of a coherent interface with a finite 8, the long range elastic distortions can be reduced plasticully by the introduction of dislocations into the coherent interface, but at the cost of raising the interfacial free energy. For the second, dilational, type of strain the elastic energy can be relaxed either plastically, by dislocation motion - usually the glide of prismatic edge References: p . 1497.
1384
R. D.Doherty
a.15, 02
dislocation loops, or by the diffusional motion of vacancies. Both of these release mechanisms, especially the diffusional motion of vacancies, occur more easily at high temperatures. Two limiting conditions are of interest to consider. The first condition would be for precipitation at low temperatures where there will usually be a high free-energy driving force (eq. 6d), since the equilibrium solubility C, is usually very small at low temperatures. The second condition occurs for high-temperature reactions, that occur with much lower driving forces. With small driving forces then the elastic strain energy opposing nucleation can easily be larger than the free-energy decrease driving the reaction. For example, precipitation of AlCu, from an aluminium-copperalloy at 800 K with a typical high-temperature supersaturation ratio C d C , = 1.1, gives a AF$ of only about 200 J/gmole which would be offset by an elastic strain of only 2.5%. For high temperature precipitation it is likely that the volume strain must be accommodated by diffusion, or by plastic accommodation. Both of these processes become faster at higher temperatures. For low-temperatureprecipitation-hardeningreactions, however, the supersaturationratios are very much larger, so the elastic strain energy is more easily provided by the large free-energy change driving the precipitation. Electron microscopy has shown that such elastic distortion does indeed occur under these low-temperature, high-supersaturation conditions (see KELLY and NICHOLSON [1963]). The resulting elastic strains are a significant part of the strengthening response of such precipitation. NEMOTO[19741 found, by direct observation using in-situ high-voltage electron microscopy, that during dissolution of cementite by decarburization of an Fe-C alloy at 70O0C,with high rates of dissolution, part of the change of volume of the reaction was provided by intense localized dislocation motion. For slower rates of dissolution such dislocation movement was not observed and the strain was assumed to be accommodated by movement of iron vacancies. The analysis of the coupled diffusion of substitutional iron atoms and rapid interstitial carbon atoms to relax localized dilational strain has been provided by ORIANI[1966] for the case of precipitate coarsening in the same, ironand BIRNBAUM [19801 cementite, system. A further example was provided by MAKENAS who reported that there was very significant plastic deformation during hydride precipitation in the niobium-hydrogen system. They charged hydrogen electrolytically into previously prepared thin foils of niobium and observed precipitation by TEM as the foils were continuously studied during cooling. Precipitation of hydride occurs by rapid interstitial diffusion of hydrogen causing a large expansion (AV/V of 18%) at temperatures too low for diffusional relaxation of the metallic atoms. Around each precipitate there was an intense tangle of dislocations with additional prismatic dislocation loops punched out in the slip directions. During resolution of the hydrides, the plastic strain was not reversible and this led to a large temperature hysteresis between precipitation and resolution. Similar examples of plastic deformation during other low-temperature hydride precipitation reactions were reviewed by MAKENAS and BIRNBAUM [1980]. It would be of interest to know if the interfacial change from the metal structure of niobium to the hydride structure during the low temperature hydride precipitation were civilian or military interface reactions, p 2.6. LEEet al. [1980, 19831 reviewed the current theoretical and experimental results on
Ch. 15, 92
D i m i v e phase transformations
1385
the influence of elastic and plastic strains on precipitation in general, and nucleation in particular. For the elastic situation the detailed theoretical analysis shows that the elastic energy is minimized when the precipitates are formed with a thin disc- or plate-like shape. This is the morphology revealed by electron-microscopic studies in all cases where Guider-Preston (GP) zones form when the solute in the zone has a different atomic size from the matrix atoms that the solute replaces. Disc-like zones are found for example in aluminium-copper and copper-beryllium alloys where there are significant size differences between the atoms. However in systems like aluminium-silver, where the atoms are almost the same size, spherical zones are found (KELLYand NICHOLSON [3963]). The shapes found for the intermediate and equilibrium precipitate in the AI-Ag system are plate-like - but this is due to the fact that the later precipitates have a different, hexagonal, structure from the fcc matrix. The observations on GP zones are therefore much more relevant to understanding the influence of elastic strain on precipitate shape than are observations on shapes of metastable precipitates, where there is a complication due to differences in precipitate crystal structure. Finally it should be noted that, as expected, the habit plane of misfitting disc-like zones appears to be in the matrix direction with the lowest elastic modulus, in most cubic metals (WERT[1976]). LEEet al. [1980] also discussed the expected plustic yielding that can occur around a spherical nucleus under the assumption of isotropic elasticity. They assumed that for precipitates larger than about 1 p m the macroscopic yield stress will operate. For smaller precipitates, however, it is likely that the yield stress will be raised by the lack of dislocations in the submicron regions. By use of a model proposed by ASHBYand JOHNSON[1969], in which dislocations can be nucleated with the help of a supersaturation of point defects, LEEet ul. [1980] were able to derive an expected yield stress for flow, as a function of the precipitate radius. Their analysis gave results that agreed well with a range of experimental results, as regards the critical size of precipitate which went from coherent, elastically strained, to semi-coherent, plastically relaxed.
Heterogeneousnucleation In many examples of nucleation in solids it is found that the nucleation sites are not distributed randomly, but are concentrated at particular sites in the matrix, usually at some type of crystal defect. The defects that act as the sites for heterogeneous nudeafion include grain boundaries, dislocations, stacking faults or the interfaces of previously formed precipitates. These sites were described in some detail by NICHOLSON[1970]. CAHN[1956, 19571provided the currently accepted analysis for nucleation of incoherent precipitates on grain boundaries and on dislocations. The analysis is based on the idea that the nucleus grows on the pre-existing defect so that the energy of formation of the nucleus is reduced by that proportion of the defect energy that is consumed by the nucleus as it forms. For nucleation on the boundary between two matrix grains (fig. 12) the grain boundary that has been destroyed is shown as the dashed line within the new crystal. The critical parameter is the semi-angle 0 determined by the ratio ~ J ~ c T where a,, is the grain-boundary energy and crUD is the incoherent energy of the interface between p precipitate and the cy matrix: 2.1.4.
References: p. 1497.
~ ~ ,
R.D. Doherty
1386
Ch. 15, 92
a grain 2
a)
bi
Fig. 12. (a) The double hemispherical cap model for the expected shape of an incoherent precipitate at an LY-CY gain boundary showing the destroyed a-a grain boundary as the dashed line. (b) The Wulff construction for this situation. (After LEEand A A R O m N 119751).
When the new phase “wets” the grain boundary, which occurs when 2u,, r, the radius of the precipitate) is given by: JR = -4?rR2D(dC'/dR),
(29)
The concentration, C, must be in the units of atoms of solute per unit volume. The relationship between C and the atomic fraction of solute C is:
tT
Tr
c r
R-
r
R-
Fig. 20. Diffusion around a growing precipitate. In the matrix phase-change (a and d). solute is diffision away from the low-solute a,while in @) and (e) solute is diffusing towards the solute-rich p.
Ch. 15, 52
DifFsive phase transformations
1401
so JR = -4rR2Ll(dC/dR)RA
V, is the volume of one atom of the alloy, which for simplicity is assumed independent of composition. The volume of the precipitate is %?, and developing the model for the case of precipitation, the increase in the number of solute atoms, AnB, in the precipitate above that of the matrix from which it forms, is given by: AnB = (4 m3)(c; - G') = (4 m3/v,)(Ca - Ci)
(31)
dAr+,/dr = 4m2 (Ca - Ci)/V,
(32)
The total flux to the precipitate, dAnB/dt is then given by: dAn - -dn, d r Bdr
d r dt
4m2 - -(Ca
V,
dr - ci)dt
(33)
This flux will be that provided by diffusion [eq. (29a)], with R = 1. The supersaturation for diffusion is usefully defined as a,, eq. (34).
a,
= (C0 - C;)/(C,- Ci).
(34)
Flux-balance from eqs. (29a) and (33) then gives: 4?rR2D(dC/dR), = 4m2(Ca- Ci)(dr/dt)
(35)
The change of sign in the flux equation (29) arises since the negative flux is one in the -R direction, which is a positive flux for precipitate growth. Provided that there is a low supersaturation, e 0.1, then during the time that it takes a solute atom to diffuse across the solute-depleted region around the precipitate, the solute distribution and the instantaneous growth rate, dr/dt, can both be regarded as essentially constant. Rearranging eq. (35) for the condition that fl,C'(F). There is always a finite Gibbs-Thomson supersaturation, S,= C,(F)/C,, for precipitates of a finite mean radius. With &=Cam there is still precipitation occurring given by eq. (66b) above. Numerical simulations of combined precipitation and coarsening were published by KAMPMANN and KAHLWEIT [1970], assuming diffusion-controlled growth and dissolution, using homogeneous nucleation theory and assuming that the supersaturation builds up steadily with time. Figure 40 shows the result of one set of calculations with fig. 40a showing the change of supersaturation, S; curve 1 is the change of S that would occur in the absence of precipitation and curve 2 the actual change of supersaturation that is determined by the nucleation and growth rates. The total number of precipitates larger than the critical size is seen in fig. 40b as the number of precipitates, WN,, normalized by dividing by the final solute atomic concentration. The number of precipitates rises during the precipitation reaction to a maximum value and then starts to
Fig. 39. Competitive coarsening during precipitation. An alloy is quenched becoming supersaturated at t = rs. The first nucleation occurs at t' and the last f . As the supersaturation, S = U C , , falls, r*(t) rises and small precipitates redissolve. (After KAMPMATWand KAHLWEIT[1970].)
Ch. 15, 9 2
Dimive phase tmnsjonnarions
1445
S
8l
d‘
Fig. 40. Numerical solution to competitive coarsening: (a) the variation in supersaturation S with time 0, curve 1 with no precipitation, curve 2 with precipitation; (b) the variation in the density of the precipitates 2,scaled with the equilibrium solute content N,(r=-). (After KAMPMANN and KAHLWEIT [1970].)
fall while the supersaturation is still large, approximately 4. Figure 41 shows the computed change of i*, the critical precipitate size in terms of the number of atoms of solute, during coarsening; this is compared to the “steady-state” loss of precipitates predicted by an analysis equivalent to the LSW analysis described in $2.4.1, shown as the dashed line. KAMPMANNand KAHLWEIT[1970] suggested that the discrepancy is probably within the experimental error of the numerical analysis used. The dimensionless time parameter, 8, scales with the diffusion coefficient, so that with the diffusion coefficient expected for liquids D = lo4 m2/s,6 = 10 corresponds to only 1Od s, but with m2/s this would be 100 s. It is clear a substitutional diffusion coefficient of D = from this model, therefore, that the number of growing precipitates found in the later stages of precipitation should be considerably smaller than the original density of growing nuclei, for example in the studies of the nucleation kinetics in the experiments
Fig. 41. Competitive coarsening during precipitation: (a) the variation in the critical size i* with coarsening time, where i* is the number of atoms in the critical cluster; (b) the rate of change of the critical size with time. (After KAMPMANNand KAHLWEIT[1970].) R4erences:p. 1497.
1446
R. D.Doherty
Ch. 15, $ 2
of Servi and Turnbull discussed in 0 2.1.5. LANGERand SCHWARTZ [1980] analytically evaluated the effect of interfacial energy driven coarsening on precipitate densities. The results reported show similar trends to those of the numerical models described here but no detailed comparison between the evaluations was made. There are several experimental studies that have been made to test the idea of competive precipitate dissolution during precipitation in the Ni-Ni,Al system. These experiments used conventional transmission electron microscopy (TEM), KIRKWOOD [1970] and HIRATAand KIRKWOOD [19771, field ion microscopy/atom probe techniques, WENDT[1981] and recently high resolution TEM, XIAOand HAASEN[1991]. The last study was discussed in 9 2.4.1 above. In all the studies a fall of at least an order on magnitude in precipitate density was found while the precipitation was continuing. The early results of WENDT [1981] are shown in figs. 42 and 43. The supersaturation, AC=G-C,, fell with time and.approached the initially expected f - ’ / 3 kinetics of eq. 66b after about 50 minutes. The initial number density fell approximately linearly after this time. However this is not the correct kinetics which should be those of eq. (66c). Figure 4 4 from the recent study of XIAOand HAASEN [1991], shows that the product of ( r - t l ) and NJt) was not constant but fell linearly with (f-4)-1’3, exactly as predicted by eq. (66d), obtained from eq. (66c), N”(t)(t- t l ) = k3 - k6(t - t y 3 .
(66d)
The recent studies of XIAOand HAASEN [19911, described above, show very clearly how both the nucleation and coarsening processes can strongly overlap -especially with very high densities of precipitates. It seems likely that this will be a general result for systems showing essentially homogeneous nucleation. For the reasons discussed in Q 2.4.1, any system with a high rate of homogeneous nucleation will very quickly achieve soft impingement - that is when the spacing between precipitate, approximately (NV)-’l3, becomes smaller than the diffusion distance, (Dt)”’. So free growth with (r-)* increasing with time is unlikely to be seen. The only conditions where free growth with (r,,,=)’ increasing with time should be expected is for systems showing heterogeneous nucleation so that not only are the precipitates widely spaced, allowing (NJ1l3 B (Dt)”’, but on a microstructural level the full supersaturation is expected to persist in regions of the matrix between the efective heterogeneousnucleation sites. Homogeneous nucleation will occur there at negligable rates at supersaturations where heterogeneous nucleation and growth dominates. KAMPMANN and WAGNER[1984] and HAASENand WAGNER[1992] show this result rather clearly through the Kampmann and Wagner “numerical” model of combined nucleation, growth and coarsening at high precipitate densities. In support of this idea is the observation that experimental studies showing (am)2 increasing with time appear to be only found in studies for heterogeneously nucleated precipitates, see, for example, the results discussed in 8 2.2.3 for plate-like precipitates, where the precipitate and matrix have different structures. Homogeneously nucleated precipitates seem limited to systems where the interfacial energy is low and isotropic giving very high densities so preventing free growth from occurring at the undercoolings usually studied. If the density of precipitates is greatly reduced then the free growth of spherical precipitates with (rmm)2increasing with time is still not seen since under these circumstances the spheres
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become unstable and grow as dendrites with a constant tip growth velocity as discussed in 0 2.2.6. Examples of homogeneously nucleated dendrites are reported in the Ni,Al-Ni system (RICKS et al. [1993]) and in the similar A1,Li-A1 system (BAUMANNand WILLIAMS [1985]). An example of a system showing heterogeneously nucleated dendrites is the y-brass&?-brasssystem (MALCOLM and PURDY [1967] and DOHERTY [1982]). References: p. 1497.
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600
1LOO
Aging Time t, Iminl Fig. 43. Measured mean particle radius, F, and calculated radius of critical nuclei, R*, versus ageing time for the alloy of fig. 37 (from WENDT[1981]).
2.4.4. Coarsening of Widmanstiitten precipitates One of the early studies of precipitate coarsening was of the growth of the needle-like and ORIANI[1965]). Despite the anisotropprecipitates of fcc copper in bcc iron (SPEICH ic shape, arising from the good atonic fit along the close-packed directions fcc and bcc, the precipitates showed apparent diffusion limited coarsening with a constant precipitate aspect ratio of about 4. The authors used a modified version of the HAM[1958, 19591 analysis to model the diffusion controlled growth. They obtained P
E 6.1OB
6 0,
- 4.1029 -
1
Ni-lZat%Al
I
ul
c-
c.
i 240a
0 0
0.010
0.020
( t - ttj"'.3 tsec"'3 1 Fig. 44.Experimental values of the product of (r-f,)N, (f)as a function of (t-tJ'" for coarsening of Ni-12 at% AI at 773K,eqs. (66c and d). After XIAOand HAAS~N[1991].
Ch. 15, 52
D i f i i v e phase transformations
1449
reasonable agreement with their experimental results. FERRANTE and DOHERTY[1979], as part of an investigation of the role of ledged interfaces in the growth of plate-like precipitates in AI-Ag, studied the coarsening of the precipitates after the completion of the initial precipitation. The plate-like precipitates in this system had much larger aspect ratios than expected for the equilibrium shapes: A > 100 while A,, = 3 (AARONSON et aZ. 119681). Ferrante and Doherty showed that with A >Aq the modified Gibbs-Thomson equations for the solubility at the incoherent rim, C,, and that at the semicoherent facet plane, C , are given by:
Here, as in 8 2.2.2, 3 is the half length (equivalent to c.e radius) of a cylindrical, platelike precipitate and a,is the interfacial energy of the incoherent rim of the cylindrical precipitate. Despite the difference in solubility between the rim and the facets of the precipitate, with A >A,, the plates showed a reluctance to equilibrate towards the expected equilibrium shape, even with very extended coarsening times. RAJAB and DOHERTY [1989] reported similar behaviour in a further study of the same system. The conclusion reached by both studies was that this reluctance of plate-like precipitates in the Al-Ag,Al system to change shape was due to the thickening reaction under the supersaturation arising from Cf< C, being inhibited by a lack of growth ledges. An unexpected result from the initial coarsening study in 1979 with these out-of-shape Ag,Al precipitates was that despite inhibited thickening, the coarsening reaction, measured by the rate of Zengthening, was very fast. However, further analysis of the results by use of the Ham, Horvay and Cahn analysis, supplemented by use of eqs. (67a and 67b), showed that the intuitive expectation, that inhibited thickening would slow coarsening, had been wrong. With inhibited thickening, so that A remains %A,, there is a much higher driving force for coarsening, as the only means of ridding the system of the excess interfacial energy, eq. (67a). Under the highly approximate condition of a constant aspect ratio, the analysis by FERRANTEand DOHERTY [19791 yielded the following prediction for the rate of growth of the mean half-length of the precipitate:
Eiq. (68) gave surprisingly close agreement with the experimental results, despite the fact that while the mean aspect ratio remained constant, the individual values of A were found to show a strong correlation with precipitate size. The correlation between A and ay indicated that there was a mean solute concentration determined by the Gibbs-Thomson solubility at the mobile ends or rims of the plates, eq. (67a). In the initial study by References: p. 1497.
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Ch. 15, $ 2
Ferrante and Doherty a fine microstructurehad been produced by low temperature ageing followed by coarsening at a higher temperature. In the later study by RAJABand DOHERTY [1989] coarsening observations were carried out isothermally at both high and low temperatures by first precipitating and then coarsening Ag,A1 at the same temperature. Under these circumstances a slow decrease of the mean aspect ratio was found, but at rates very much slower than expected for diffusion-controlledtransport from the rims of the precipitates to the facets of the same precipitate. Again a lack of thickening ledges appeared to be the reason -despite the observation that precipitate/precipitate intersections were still present. These intersections had been found to be effective sources of growth ledges under the larger supersaturations of the initial precipitation. That is, the supersaturations found during growth appeared to be sufficient to support nucleation of growth ledges while the smaller supersaturations found in coarsening appeared to be insufficient to allow a sufficiently density of ledges to form. Detailed investigations of the coarsening of plate-like precipitates with A >A, were [1981] and MERLEand MERLIN[1981]. They studied provided by MERLEand FOUQUET 8' plates in Al-Cu and found, in that system, that the precipitates formed with high and [1989] in A1-Ag, the variable aspect ratios. As in the study by RAJAB and DOHERTY highest aspect ratios of the '6 plates were found for those conditions with the lowest precipitate densities. During coarsening the aspect ratio fell towards what appears to be the equilibrium value of A,= 20. The rate of fall was fastest in those microstructures which had the highest precipitate density, see fig. 45. The implication of these results is clearly that the highest precipitate densities lead to the greatest chance of precipitate [1981] and intersection and thus a source of growth ledges. MERLE and MERLIN E19821 showed that the kinetics of the change of subsequently MERLEand DOHERTY shape could be described by a ledge model with solute transport at 225OC by irzter$ace difsusion around the incoherent edge of the precipitate. However, the agreement required the experimental measurement of ledge spacing and as noted in the discussion of growth of ledged precipitates P 2.2.4. there remains no way, at present, of predicting the ledge spacings, owing to the lack of a successful model for ledge nucleation. As noted by DOHERTY and RAJAB [1989] the inhibition of thickening during coarsening was more marked in A1-Ag than in Al-Cu for reasons that were not known. It would be good to
Fig. 45. The variation of the mean aspect ratio of the plate-like 8' precipitates in Al-Cu at 225°C. T.I. is an interrupted quench from the solution-treatment temperature, R is a room-temperature quench followed by the 225OC age, and 85 is a 5% room-temperature strain prior to ageing at 225OC. 3%, 4% and 5% are the Cu contents. Note that 2 fallsfaster with the highestdensity of precipitates. (After MERLEand FOUQUET [198I I.)
Ch. 15, 32
145 1
Difisive phase transformations
have more studies of the coarsening of plate like precipitates since these have the most common precipitate shape in successful precipitation hardening systems!
2.5. Discontinuous reactions: moving two-phase boundary MTPB reactions The reactions discussed in this section are all those occurring at a moving interface, giving a two-phase, usually lamellar or rod-like, product. The moving interface may be an interphase boundary, as in eutectic and eutectoidal decomposition, or it may be a matrix grain boundary as in the precipitation reaction conventionally described as discontinuous precipitation (GUST[19791). The moving interface is usually incoherent. This name, as previously noted in 4 I .4,is an unfortunate one since the general name for the second class of transformations considered by Gibbs, those that do not involve 119831). The nucleation and growth, is continuoustransformations(SOFFAand LAUGHLIN term “discontinuous” should then apply to all Gibbs type-I, nucleation-and-growth, transformations and not just the subset of them that occur at a moving boundary giving a two phase product. (Or a two phase product reacting at a moving interface to give a single phase product as in discontinuous dissolution.) The term “discontinuous transformations” will be used here in the narrow meaning even though an alternative term,such as “moving two-phase boundary” (MTPB) reactions would be clearer. This review is not the place to make this unilateral change of notation, but care has been taken to avoid the expression “continuous precipitation” for reactions that either are nucleated entirely within a matrix grain or for those that are, nucleated at grain boundaries and then grow away from the boundary, for example Widmanstatten femte. The term continuous transformations will be restricted to reactions which do not involve nucleation and growth (0 3).
2.5.1. Eutectoidal decomposition In eutectoidaldecomposition,the general reaction involves the change y A(Y+p. The most important example is the “pearlite” reaction, the decomposition of austenite, y, in an iron-carbon alloy containing 0.8wt% C, to give ferrite, a , and cementite, Fe,C: y + a +??e$. The name peadite comes from the similarity between the optical effect produced by light diffraction from the lamellar spacing (which is comparable wavelength of visible light) of the two-phase eutectoid and the same optical effect produced by “mother of pearl”, a biological composite material. The process can be understood from the schematic microstructure of fig. 46, for the transformation occurring isothermally at an undercooling AT below the eutectoid temperature of fig. 47. The extrapolations in the phase diagram, in fig. 47, give the compositions of the y in equilibrium with (Y and p as CJa) and Cy@);the equivalent values in the (Y phase are CJy) and Caw). If the main diffusion path is along the moving boundary, as it often is metal-metal eutectoids such as Cu-AI, then the compositions in the boundary are given as kb,, C,,((Y>and Cy@) where k:y and k& are the partition coefficients for solute between the appropriate phase boundary and the decomposing austenite. Since the separate values of the different coefficients are rarely known, an average value, kb,,is often used.
GY
References:p. 1497.
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Fig. 46. Growth of an a +p lamellar structure, showing the variation in composition, C,(a),C,@) in the y-phase, and C,(y),C,@) in the a-phase, that drive the reaction.
From figs. 46 to 48 it can be seen that there is a composition difference ACy within the yphase, ACy = C,,(cr)-Cy@),and an equivalent composition differences ACa in the a-phase and AC,, in the boundary. In the initial treatment of the diffusional problem given by ZENER[ 19461, it was assumed that diffusion took place down a solute gradient between the centers of adjacent lamellae. The mean gradient, dC/dy, is approximately 2ACy/A, in the austenite, where A, is the lamellar spacing. The solute diffusion was assumed to occur within a thickness of approximately Ad2 away from the interface, where the gradient is assumed to have the mean value, with no gradient further away from the interface. By a mass balance of the diffusing solute needed for the reaction, the following relationship between the growth velocity, v, and the spacing was obtained
where f , and fs are the relative widths of the two phases, assuming that the molar [1972]). Eq. (69) fails to take into volumes are the same in both phases (HILLERT account, however, the interfacial energy, gap,of the interface between the product
CFig. 47. The equilibrium diagram showing the origin of the composition differences seen in fig. 46.
Ch. 15, 92
DifFsive phase transformations
1453
t
F
CFig. 48. The free-energy-composition diagram that gives the diagram seen in fig. 41. The composition difference AC, is reduced to hC,.(AJ by the rise in free energy per mole of LY + p , 2 u a a V , h , ,caused by the lamellar spacing.
phases.* The origin of the lamellar equivalent to the Gibbs-Thomson effect is shown in fig. 47. The interfacial energy per unit volume of the lamellar product is 2crap/A, and so the two-phase product is raised in molar free energy by 2a,B V J A , (J/mole-') with a consequent reduction of AC, to ACY(As). If the critical spacing that reduces AC, to zero, &(rev), is included, eq. (69) can be changed to give the velocity corrected for the interfacial energy effects. The description of &(rev) is used since at that spacing, both the two-phase product with this spacing and the high-temperature austenite phase have the samefree energy and so the reaction would be reversible at As=As(rev). With AC, changed to AC&), the velocity becomes:
In deriving these diffusional equations the compositions are used in units of atoms per unit volume. However since the composition terms occur as AC/AC", use of atomic fractions will introduce only very small errors arising from small changes in the atomic volumes. In the iron-carbon system, the diffusion coefficient in the a phase product is significantly faster than in the close-packed austenite, y, so diffusion in a, is an alternative pathway which can be described by appropriate modification to eq. (70). For solute transport by diffusion in the interface the analysis by TURNBULL [1955] showed
* In the simple derivation of this Zener model, it is easiest, as here, to use the a$ interface. For the point of changing the concentrations at the moving interfaces the energies of the cyy and a@ interfaces are critical. The parameters are interrelated by the triple-point angles where the three phases meet, see JACKSON and HUNT 119661 and KURZ and FISHER[I9891. References: p. 1497.
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Ch. 15, 52
that not only must the compositions and the diffusion coefficient be changed but the diffusion thickness is reduced from Ad2 to the boundary thickness, a b , so the resultant equation is:
HILLERT[1969, 1972, 19821 has discussed these equations and the more rigorous solutions to the diffusion equations in these various cases. He also considers the possibilities that there may be some significant supersaturation left in the product phases and that some free energy, or equivalently composition difference, may be required to drive the interface. That is, the reaction front may move faster than the rate required to achieved full equilibrium and that there may some element of interface control. For full analytical details and analysis of experimental results the reviews by Hillert, quoted above, should be consulted. It is apparent from eqs. (70) and (70a) that the problem is not fully specified, even in the simple treatments, until some means of determining the operating lamellarspacing can be found. Experiment shows that there is a constant spacing at an given undercooling and that this spacing decreases as the undercooling increases and consequently h,(rev) decreases. For a eutectoid reaction with a molar entropy change ASm,at an undercooling AT below the eutectoid temperature, &(rev) is given by:
A discussion of the choice of spacing is given in 82.5.4. For the usually adopted assumption of a maximum velocity, we find As=2hs(rev). This gives the usual expression for volume diffusion control of: V =
-
DyACy
1
fafp(Cj3 - c.)
A,
DYACY
1
f.f,(Ca - c.) %(rev)
There is a simple relationship between ACy and undercooling, AT:
AC? = AT
ma
+mp
mams
Here ma and mp are the slopes of the y solubility with temperature ma = dT/dC,(a)). So combining these equations together we find the often quoted form of this valid for all undercoolings:
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Difisive phase rransformations
The equivalent equation derived by JACKSON and HUNT[ 19661has the same form as this and the triple point but is expressed in terms of the interfacial energies, rayand rpy, angles, of the moving interfaces with the parent y matrix. The growth velocity, eq. 70b, initially increases with undercooling (as A F ) due to the effect of temperature on the spacing and AC,,. However there is the usual exponential fall of the diffusion coefficient with temperature, so that although the eutectoid growth velocity initially increases with undercooling, it quickly reaches a maximum possible velocity and then the growth velocity decreases rapidly in the temperature range where diffusion becomes very slow. This qualitative pattern is always found and is of great importance in the heat-treatment of steels. In outline, the carbon-containing austenite needs to be quenched to form the high hardness martensite and to do this, the diffusional decomposition to pearlite, and the other diffusional products, allotriomorphic and Widmanstatten femte and also bainite (02.6), must be avoided. To achieve this, either very rapid quenching is required or for samples too thick to be rapidly quenched, the transformation to pearlite needs to be slowed down by inhibiting either the nucleation of pearlite or its growth. The experimental diffusional kinetics are usually reported on timetemperature-transformation (TTT) diagrams showing the time for isothermal transformation; these diagrams normally show “C-curve” behaviour (fig. 49), the “nose” of the C-curve being that for the maximum overall rate of transformation which has the usual
; 0
800
Y
1400
= 0
-
700 -
X
1200 600
15
-
20
moo
24
, e00
32
Q
41
t-
300-
6oo
46
200-
400
100-
200
w
5 500b . 2w 400’
r,
SI
62
0 -
0.5 1 2
5 IO
m2
TiME
105
io4
- SECONDS
105
106
Fig. 49. Isothermal TTT diagram for a low-alloy steel, containing 0.4% C, 0.3%Mo, 0.8% Cr, 1 % Mn and 1.8% Ni. The high-temperature transformation is firstly to precipitate ferrite (F)followed by pearlite (P+C), while the lower-temperature diffusional reaction is the formation of bainite ($2.6). (From the Atlas of Isothermal Transformation and Cooling Curves, ASM, Metals Park, OH, 1977.) Reference*: p. 1497.
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‘‘s’kinetics of a JMAK type of nucleation and growth reaction (02.4.1). In “T diagrams, of course, the x-axis is not velocity but its effective reciprocal, the isothermal holding time to achieve a certain fraction transformed. The kinetics of the experimental T’TT curve depend not only on the variation of growth rate with temperature but also the equivalent variation of nucleation rate. For the same reasons, the nucleation behaviour also show C-curve variation with temperature, § 2.1. The same type of C curve behaviour, for the equivalent reasons, is found for the diffusional precipitation of single phase a iron a (ferrite “F” in fig. 49) from y austenite (“A”) and the lower reaction of austenite to “bainite” (§ 2.6). These reactions as well as the formation of pearlite are seen in fig. 49. Although the transformation can be slowed down by an increase in austenite grain size, which increases the spacing between the grain-boundary nucleation sites, this is not the best solution, for reasons related to optimizing mechanical properties. The best method of improving the “hardenability” of steels is by slowing down the pearlite growth rate. This is achieved by addition of small amounts of substitutional metallic contents (nickel, manganese, chromium, etc.) which are soluble in the austenite but which at equilibrium will partition between the two product phases, but do so by slow substitutional dij5usioa. This idea the basis of low-alloy steels. At high temperatures the solute must partition since pearlite with uniformly distributed solute will have a higher free energy than the parent austenite. At lower temperatures, and higher undercoolings, there will be sufficient free-energy difference to allow transformation with “no partitioning” (HULTGREN[1947]) but with an inevitable decrease in the free energy driving force and therefore an increase in the critical spacing, &(rev). The “no partitioning” of ternary solute in eutectoids is comparable to the diffusionless precipitation of single-phase products discussed in § 2.2.1. The reduction in growth velocity caused by these ternary solute effects increases with the amount of the low-alloy solute additions. There has been considerable experimental interest in studies of solute partitioning in steels (see, for example, WILLIAMS etal. [1979], RIDLEYand LORIM-ER [1981], SMITH et al. [1981] and RICKS[1981]). Although it is well established that at high temperatures there is strong solute partitioning in low-alloy steel pearlites, and that with a fall in temperature the partitioning becomes incomplete, experimental difficulties appear, so far, to have prevented demonstration of the full no-partitioning of solute during the pearlite transformation. The difficulty is spatial resolution of the analysis, since at higher undercoolings the experimental spacings become rather too fine for the current analytical resolution. For the bainite reaction in steels there seems little doubt that there is no partitioning of the substitutional metallic elements between the femte, a, and the carbide phases § 2.6.
-
2.5.2. Discontinuous precipitation MTPB precipitation An equation similar to eq. (70) describes the growth of a two-phase product behind a moving matrix grain boundary that relieves a supersaturated a matrix, the reaction being CY + a’+p. This process occurs in many low-temperature precipitation reactions and arises after grain-boundary nucleation of the precipitate phase p which can then be supplied with solute by grain-boundary diffusion at temperatures where the competitive processes involve solute transport by very much slower bulk diffusion. The solute can be supplied from regions adjacent to the boundary - a process that is called the “collector
Ch. 15, 3 2
Difisive phase transformations
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plate mechanism" if the boundary does not move. However, if the boundary can be induced to migrate into one of the grains, it can then collect and transport the solute from the supersaturated grain interior that it passes through, without requiring any bulk diffusion. The precipitates initially nucleated on the boundary can then grow while remaining attached to the moving boundary giving the characteristic lamellar structure. GUST[ 19791published a very detailed review of the literature on this topic, starting from the original observation of the precipitation of copper at moving grain boundaries in Ag-Cu alloys by AGEEWand SACHS[1930]. There are several points of interest in this type of transformation. They include the problem of how and why the reaction initiates, that is why the boundary starts to move, and how it is possible for different parts of the same boundary to migrate in different directions, as reported for example by WILLIAMS and EDINGTON[1476]; the question of what determines the residual supersaturation in the product, since in many cases this is reported to be appreciable (e.g., PORTERet al. [1974]); there is also the general problem of determining which reaction, the discontinuous one or growth by volume diffusion, will occur under any given circumstances. Studies by SULONEN[1964] and more recently by CHUNGet al. [1992] have shown that many of these questions may be answerable in terms of an eZastic stress mode2 for boundary migration in discontinuous precipitation. This is equivalent to the process driving the process usually called diffusion (or chemically) induced grain boundary motion - DIGM discussed in Q 2.5.5. The role of elastic stress in both discontinuous, MTPB, precipitation and DIGM is considered in Q 2.5.5. There is extensive evidence for significant residual supersaturation in the products of discontinuous precipitation. This comes from lattice-parameter measurements and from very high resolution microanalysis by electron energy loss techniques applied to Mg-A1 alloys (PORTER eta2. [1974]) and to A1-Li alloys (WILLIAMS and EDINGTON[1976]). Figure 50 gives the chemical analysis results across a matrix grain boundary in Mg-AI. This shows (i) a discontinuous change at the boundary, characteristic of a process proceeding by boundary diffusion, and (ii) that the resulting solid solution is still supersaturated in the aluminium solute.
rio
Mg-&Bat-% AI 30min at 493K
'
100 A"
ros .s 10.8
G-10.7 10.6
a
reabtion front 1
a.
-, Distance in nm
pig. 50. Microanalysis by electron energy loss, across the migrating grain boundary during discontinuous precipitation in a Mg-AI alloy (after PORTERet al. [1974]). References: p . 1497.
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The initiation of the discontinuous reaction was studied by Tu [1972] in the Pb-Sn, solder, system at room temperature. His micrographs show regular grain-boundary nucleation of a colony of Sn-rich precipitates which grow apparently with an orientation relationship with the grain that is not consumed, when the boundary moves. Once the boundary starts to move, further precipitates form to give the operating lamellar spacing. An important question concerns the conditions required for the discontinuous reaction to dominate. This will presumably require that the discontinuous process occurs faster than any alternative reactions such as grain-interior precipitation (usually of an easy-tonucleate, usually metastable phase) or the growth of a boundary nucleated phase into the grain away from the stationary boundary. Both of these processes occur by mechanisms whose rates are determined by bulk diffusion. It might be expected, then, that the discontinuous process will take over from growth of boundary precipitates as the temperature falls, since the relative advantage of boundary diffusion over bulk diffusion increases as the temperature falls. Boundary diffusion has a lower activation energy. Results reviewed by GUST[1979] confirm this description with the discontinuous process in, for example, copper-rich Cu-In (PREDELand GUST[19751) dominating the precipitation process as the temperature of the reaction fell below 30OOC. However, if the temperature falls very far, general grain-interior nucleation dominates most precipitation reactions, at least in alloys selected for precipitation hardening (KELLYand NICHOLSON [19631). For such alloys, high temperatures of precipitation frequently cause discontinuous precipitation of the stable phase in place of the general precipitation of the strengthening metastable phases. On this basis, there should be only a narrow “window” of conditions that allows the discontinuous reaction to proceed. This is just as well since discontinuously precipitated alloys do not have good mechanical properties; having neither the high strength of precipitation-hardened alloys nor the good toughness of single-phase alloys. The studies by SULONEN[1964a and b] and by CHUNGet al. [19921, discussed in 0 2.5.5, show clearly that, in addition to these requirements, there must be an elastic stress effect due to a change in the lattice parameter of the a matrix by changes in the solute content is also required. It appears as if the elastic stress in the consumed grains drives the process so it should be possible to develop a model for the kinetics of this type of discontinuous, MTPB, precipitation based on the elastic stress model. Such a model, however, does not yet appear to have been published. 2.5.3. Discontinuous (MTPB) coarsening The ordinary coarsening process, described in 82.4, involves the reduction of interfacial energy by growth of large precipitates, at the expense of smaller ones, by diffusion either through the lattice or - for a set of precipitates all linked by grain boundaries - by boundary diffusion in stationary boundaries. LIVINGSTON and CAHN [19741 discovered a discontinuous (MTPB) coarsening reaction in polycrystalline eutectoids in C o S i , Cu-In and Ni-In, when the microstructures were annealed at temperatures close to the eutectoid temperature. Figure 51 shows a typical microstructure in which the upper grain is growing into the lower grain on the right while the opposite is happening on the left. The grain boundary is only visible in the grey Co,Si phase. The driving force for this reaction certainly involves the reduction in interfacial area between
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Fig. 51. Discontinuous coarsening of eutectoid in Co-Si alloys; a polycrystalline sample annealed at 1OOO"C for 96 h. x 1875. (From LIVINGSTON and CAHN[1974].)
the two phases. The reaction kinetics were successfully analyzed by LIVINGSTON and CAHN[1974] on the basis of boundary diffusion to give a predicted boundary velocity:
where A , and A, are the spacings before and after coarsening, Cb is the solute composition in grain boundary volais the energy of the interface between the two phases and the other terms have their previous meanings. This analysis was consistent with all the observations made. Of particular interest was the clear demonstration of which way the boundary moves. As seen in fig. 51, the grain whose lamellae lie nearly parallel to the initial boundary plane is able to grow, since it automatically has a larger effective spacing along the boundary. The only difficulty that arises here is the inevitable change of interfacial plane that occurs with this process, even though the two phases have the same relative orientations; the properties, particularly the energy of the interface, v.,, may vary with boundary orientation unless the interface is fully incoherent. A further very important point, discussed by Livingston and Cahn, was the magnitude of the spacing ratio, &/A,, which was shown to be between 5 and 7 for Cu-In and Co-Si and between 10 and 20 for Ni-In. This is discussed in more detail in Q 2.5.4. At present there is no evidence if discontinuous coarsening required movement of the grain boundary is, or is not, driven by elastic stress effects. CHUANGet al. [ 19881 studied the discontinuous coarsening reaction in an Ni-7.5at% In alloy where the initial lamellar structure had formed by discontinuous precipitation. References: p . 1497.
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They showed that the initial lamellar structure retained, as it commonly does, a significant solute supersaturation in the matrix phase and that during coarsening this residual supersaturation was essentially removed. In this case, both the residual supersaturation and the interfacial energy were driving the reaction. The authors reported that the transport mechanisms, in both the discontinuous precipitation and the subsequent coarsening reaction, were by grain-boundary diffusion. Somewhat smaller increases (by 3 to 5 times) of the lamellar spacing occurred here than in the coarsening reaction studied by Livingston and Cahn whose starting structure had been produced by eutectoid decomposition. Other examples of discontinuous coarsening had been published by GUST[1979], though without discussion. One further example has already been mentioned, the work and EDJNGTON [1976] using A1-Li alloys. At low ageing temperatures close of WILLIAMS to room temperature, the discontinuousreaction apparently involved formation by precipitation of the two-phase lamellar structure from a fully supersaturated solid solution, but at higher ageing temperatures the discontinuous reaction was one of coarsening when a lamellar, Al+Al,Li, cube-cube oriented, product grew out from grain boundaries to consume the fine coherent precipitate nucleated in the adjacent grain. The reaction halted when the spherical A1,Li distribution had coarsened by the usual LSW process, and also presumably when the aluminium supersaturationhad been removed by completion of the grain-interior reaction. An interesting result reported by Williams and Edington was that grain boundaries close to coincidence-site orientations (ch. 9, $2.2) were less able to show the discontinuous reaction than general high-angle grain boundaries. DOHERTY [19821 discussed the formation of similar coarsened lamellar structures by deformation-induced boundary migration (recrystallization) in nickel alloys containing coherent Ni,Al precipitates. Here the driving force for bounday movement was clearly the stored energy of the dislocations. and GUPTA An interesting final reaction in this series is one described by NAKKALIL [19871: this was the discontinuous (MTPB) dissolution of discontinuously precipitated AI-20% Zn when annealed close to the solvus temperature. Of interest here is that the spacing, of the dissolving two phase structure is fixed. Does the reaction have time to reach equilibrium giving a homogeneous solid solution and what role, if any, does the coherency stress play?
2.5.4. Determination of lamellar spacing in discontinuous (MTB) reactions In the initial ZENER[1946] analysis it was assumed that the system adopted the spacing that allowed the interface to migrate at the maximum velocity, which for the volume-diffusion mechanism gives A, =2A,(rev) and for interface diffusion mechanism, A, = 1.5A, (rev). PULSand KIRKALDY [1972] discussed alternative optimization criteria, including the spacing that gives the maximum rate of entropy production. This yields somewhat larger optimum spacings, which are 3A,(rev) and U,(rev), for volume and interface diffusion mechanisms, respectively. There is, however, considerable difficulty in estimating the appropriate values for &(rev) since its evaluation requires knowledge of the interfacial energy as well as the free energy driving the eutectoid reaction. Freeenergy calculations for discontinuous precipitation will be easier, particularly in dilute
a.15, 92
Dimsive phase transformations
1461
alloys. However, as LMNGSTONand CAHN [1974] pointed out, the discontinuous coarsening reaction does not have either of these difficulties, since the relevant parameter is the spacing ratio which can be easily directly measured, this is not true however if the initial lamellar structure still retains some solute supersaturation. The spacing ratios in coarsening were found by Livingston and Cahn to be considerably bigger than predicted by either of the proposed optimization criteria, maximum velocity or maximum rate of entropy production. The coarsening results therefore suggest growth well away from the maximum-velocity condition, at larger spacings. The same conclusion is also suggested by various experimental studies on discontinuous precipitation (RUSSEWand GUST[ 19791 and SPEICH[1968]), results reviewed by HILLERT[1982]. It was shown by Hillert that the experimental results lay close to the line, predicted by his diffusional model, of a plot of the residual supersaturation against lamellar spacing, but at larger values ofthe spacing than was predicted for the maximum-velocity criterion. By comparison with the equivalent problem in dendritic growth, it would appear that there are experimental reasons, firstly, for expecting growth on the large side of the spacing predicted by the maximum-velocity criterion, and secondly, for the potential successful application of some type of stability criterion for lamellar spacing. The choice of spacing will also need to be analysed in the context of the insights into the role of elastic stress described below. 2.5.5. Diffusion-induced grain-boundary migration (DIGM) In discontinuous, MTPB,reactions the available free energy of the precipitation, for example IJ in fig. 1, might be applied solely to the growing &phase alone and the growth of the &phase then may pull the depleted (Y matrix along. Alternatively the energy could be applied to both phases. HILLERT[1972, 19821 in his analysis of discontinuous precipitation has used this idea of the partition of the free energy between the two phases. In order to try to study this problem more fully, a simpler experimental situation was devised by HILLERTand PURDY [1978]. Their experiment identified a process that is now called by the title of this section, difision-induced grain-boundary migration, DIGM. CIGM, for chemically induced boundary migration, is sometimes used as an alternative name for the same process. DIGM can be achieved by alloying, or &alloying, a polycrystalline sample from the surface at a temperature at which bulk diffusion has been almost ‘‘frozen out”, but where boundary diffusion still takes place at a significant rate. HILLERTand PURDY[1978] introduced zinc into thin polycrystalline iron samples from a vapour source, an Fe-11.3wt% Zn alloy, at temperatures between 545 and 600°C.They observed boundary migration which left a layer of zinc-enriched solid solution behind the migrating boundary. They treated the driving force for this reaction is the free energy of mixing per atom, AF,,,. The effective or apparent mobility, Ma, of the boundaries was then estimated from v = MJF,.
(73)
Apparent mobilities of grain boundaries in iron, at 580°C, of Ma of about m4/J s were found which were much smaller than the values found from ordinary boundary[19681 on discontinuous precipitation mobility experiments or from the work of SPEICH References: p . 1497.
1462
R. D.Doheriy
Ch. 15, 3 2
in iron-rich Fe-Zn alloys (EA= and lo-'' m4/J s respectively). In studies of discontinuous precipitation in Cu-Co by PEROVICand PURDY [1981] values of Malower by lo4 than boundary mobilities measured in recrystallization were also reported. It was not clear, at that time, whether the discrepancy arises from extraneous sources such as solute drag or if the use of eq. (73) to determine velocity is not justified. The process of DIGM had been previously seen, for example by RHINESand MONTGOMERY [1938] and by DEN BRODER[1972] and was subsequently identified in solid-liquid mixtures by YOONand HUPPMANN[1979]. The paper of HILLERTand PURDY [1978] created a great deal of interest in the process and numerous subsequent studies of the phenomenon followed as described by the reviews of HANDWERKER [1989] and YOON [1989]. CAHNet al. [1978] and BALLUFFI and CAHN[1981] reviewed the phenomenon of DIGM and noted that it occurred under a whole range of conditions, including both alloying and dealloying in systems with either a positive or a negative deviation from ideal solution behaviour, that is, with positive or negative heats of mixing. CAHNet al. [1978] draw attention to the critical theoretical problem in DIGM and equivalently in discontinuous precipitation. This is how the overall free-energy decrease can couple with the individual atomic motions requiml for boundary migration. The problem is shown in the schematic migrating-boundary structure in fig. 52. The boundary contains solute in equilibrium with the solute content of the growing grain on the left but, apart from the boundary plane, there is no solute in the grain on the right. If an atomic layer of soluterich alloy is formed on the grain on the left by atom transfer from the right, together with solute addition down the boundary, a decrease of free energy of mixing will have occurred. However, this difference in free energy is apparently not available to the atoms in the right grain, that must jump across the boundary. CAHNet al. [1978] proposed a model for the process based on difserences in the rate of boundary diffusion, the grainboundary equivalent of the Kirkendall effect. This has the result that extra lattice sites BOUnd8fy motion
Grain 1
Grain 2
Fig. 52. Diffusion-controlled grain-boundary motion; solute atoms, solid circles, have diffused down the boundary and are causing growth of grain 1, which is being solute-enriched and therefore causing a reduction in free energy by the free energy of mixing. What is the driving force causing a net transfer of atoms across the boundary?
a.15, $ 2
Diffusive phase trans$ormations
1463
must be created, or lost, at the boundary which provides, via the climb of grain-boundary edge dislocations, a means of moving the grain boundary. There is clear evidence, for example in the Fe-Zn alloy experiments of Hillert and Purdy, that there is surface tilting at the migrating grain-boundary regions, characteristic of volume changes predicted by the boundary Kirkendall effect. The topic of DIGM has been reviewed several times recently, by HANLWERKER [19891, YOON [1989] and DOHERTY [1992] and it now seems clearly established that the driving force that moves the grain boundary is the elastic coherency stress due to the change of lattice parameter in the matrix in the layer of solute enriched material immediately ahead of the moving grain boundary. An example of this is monolayer of such enrichment shown in fig. 52. If the grain boundary is moving with a velocity v and the bulk diffusion coefficient is D, then the thickness, S,, of the solute-enriched layer, in the grain being consumed, is given by:
S, = D/v when D/v > b
(74a)
8, = b when D/v < b.
(74b)
b is the atomic diameter of the matrix atoms. CHUNGet al. [1992] suggested that eq. (74a) would be expected to apply even when D / v < b , but this fails to take into account that diffusion of solute down a grain boundary does not occur in a channel of atom sites within the boundary but occurs by atoms which are part of the su#ace layer of both grains. Figure 52 sketches the situation expected of eq. (74b). The large driving force of the free energy of mixing will ensure that atoms in a grain boundary should always be part of the surface layers of both grain in a boundary and this idea is clearly confirmed by the experimental results reported by CHUNGet al. [1992]. In the situation where r), 7) = (l/a)(da/dC) f 0, then the presence of such a solute-rich layer, of composition C, different from the matrix composition C, with either 8, = b or 6, > b, will give rise to an elastic energy per atom, AF,. This occurs since the solute modified layer which unstressed will have a different lattice parameter from the unsolute modified part of the grain but is coherently stressed to that lattice parameter. This elastic coherency energy is given by:
AFfi = xY(n)S2
(754
Here V, is the volume per atom, Y(n) is the elastic modulus that varies with the surface normal n and misfit S is the strain given by: 8 = T(C, - co)
(75b)
What appears to happen during DIGM is that the solute diffuses down the initially stationary grain boundary and enters the grains on both sides of the boundary to a depth that is approximately (Dt)’.’, where D is the bulk diffusion coefficient and t the time of diffusion. There will be a steady build up of elastic strain until on one side of the boundary the strain is plastically relaxed by the formation of a cross-grid of edge dislocations. These dislocations take up the change of lattice parameter between the References: p . 1497.
1464
R. D.Doherty
Ch. 15, $ 2
solute enriched region and the original composition. Such dislocation arrays, at a position close to the original position of the grain boundary, are commonly reported, see for example PANand BALLUFH[1982]. Once the elastic energy is plastically relaxed on one side of the boundary, then there is a driving pressure to cause migration of atoms from the grain that is still elastically strained. The velocity of the migration is given by equation (73) but with the driving force that of the elastic coherency energy, AFb, not the full free energy of mixing, AFm. The free energy of mixing drives the diffusion process down the grain boundary and into the grains so giving rise to the elastic strain energy. Since AFm > AF, the low apparent value of boundary mobility in DIGM noted above arises since the incorrect larger driving force was used for its evaluation. The elastic stress energy explanation for DIGM was demonstrated by several studies from Yoon and his colleagues using liquid-solid mixtures, as reviewed by YOON [19891 For example, RHEE and Y o o [1989], ~ using liquid phase sintered Mo-Ni established a uniform Ni content in the Mo powder particles which, after sintering, produced fully and partially liquid wetted grain boundaries. Liquid wetted boundaries were ones where the grain boundary energy, crgbr was larger than twice the solid liquid energy, crsl. The partially wetted boundaries were ones with cr,&T,c 1538OC) and again at T, ~ 9 1 2 ° C . The re-occurrence of the bcc structure at high temperatures can be rationalized by dividing the difference in free energy of bcc and fcc iron, AFa,, into a magnetic and a nonmagnetic term (ZENER[1955]). The magnetic term is negligible up to the temperatures at which magnetization of cy-Fe is nearly constant. At higher temperatures (depending on T,) the bcc structure is increasingly stabilized due to the entropy of demagnetization. Without this effect, a-Fe should transform to y-Fe at about 700°C without another transformation at 1394°C. In fig. 2, AF,, =Fa -Fy is plotted as a function of temperature. The contribution due to magnetic uncoupling in the a-phase is shown as a dashed line. This term produces the curvature of AFa,(T) as determined by JOHANNSON [1937] and the re-occurrence of the bcc structure above 1394°C. The increasing value of AF,, with decreasing temperature indicates that an increasing pressure is needed to produce the closest packed iron at temperatures below 912°C. At 20°C this pressure is 11.3 GPa, determined by both static and dynamic measurements. Iron can exist in at least three crystal structures, fcc y, bcc a and hcp 8. The closest packed phase, E , exists at pressures in excess of 11.3 GPa and at temperatures from ambient to at least 2000K. A triple point at which a, y and E coexist occurs at about 775K and 10 GPa (fig. 3). There is some evidence that a fourth or even a fifth phase [1986]) but confirmation is required. may exist POEHLER There is no experimental evidence for the existence of thermal vacancies in quenched a-Fe, perhaps because of the high ratio of energy of formation, u p to energy of motion, u,: uFJuM= 4, as compared to about 1 for the fcc lattice. Vacancies probably anneal out [1960]). It is likely that these results are strongly during quenching of cy-Fe (JOHNSON influenced by interstitial elements. The Burgers vector of the minimum energy dislocation in a-Fe is (u/2). Segments of uc100> can form by the reaction (u/2)[ 1 11+ (u/2)[ 11I] + u[100]. The
Fig. 1. Melting temperature TM,and density, p, versus atomic number Z of the elements of the first transition period.
Ch. 17, 3 1
Physical metallurgy of steels
200
1559
i
Rg. 2. Free energy difference Mq resolved into magnetic and nonmagnetic components (after ZENER [1955]).
stacking-fault energy of a-Fe (as of the other bcc transition metals) is very high. Stacking faults have therefore not been observed in a-Fe even by subtle weak beam electron microscopy. Details of the core structure of dislocations in the bcc lattice and their consequences for crystal plasticity are discussed in ch. 20, 0 3.2. The probability for the Occurrence of annealing twins is low but twins generated during cooling through the
Pressure ( GPa )
Fig. 3. Iron phase relations (SAXENAet al. [1993]). References: p . 1618.
Ch. 17, 8 1
WC.Leslie and E. Hombogen
1560
y + a transformation are common. In y-Fe the Burgers vector of the complete dislocations is (u/2). The stacking-fault energy (at lWO"C), determined from the frequency of annealing twins, is 75 mJ/m2 (NUTTINGand CHARNWK[1967]). (See ch. 28, fig, 55). Stacking faults and annealing twins are frequently found in y-Fe and its alloys. Many discontinuously during the phase transformations of iron. They are often used to determine the transformation temperatures in steels. There are discontinuities of the self-diffusion coefficient of iron at the transformation temperatures. The diffusivity of iron in (Y is about lo2 times higher than in y at the same temperatures (fig. 4).This can be explained qualitatively by the closer packing of y-Fe (BIRCHENALL [1951]). Below T, there is an anomalous decrease of the diffusion coefficient of about 60% during the transition to complete magnetic order. A change in equilibrium vacancy concentration, not magnetostriction, was used as an explanation for this behavior (BIRCHENALL and BORG [1960]). Since then, much work has been done on this anomaly (reviewed by JBNSSON [1992]); it is now attributed to a change in the activation energy for atomic jumps. An amorphous structure can be obtained for pure iron neither by vapor deposition nor by splat cooling down to temperatures close to 0 K. Such a structure is exclusively found in certain alloys of iron. (See also sonochemical approach, ch. 19, 0 3.6).
*W
Birchenol 8 Mehl, (1950) Buffinton et al., (19611 Y Graham 8 Tomlin. 11963)
+ Borg 8 Birchenall.119691 X Hettich et 01..
6
(1999)
a
10 loL K/T
12
I
Fig. 4. Self-diffusion coefficient of paramagnetic (DJ and ferromagnetic (D,) cy-iron; Q in kJ. (After J~NSSON 119921).
Ch. 17, $ 2
1561
Physical metallurgy of steels
2. Alloys of iron 2.1. Interstitial alloys The elements H, C, N and 0 can occupy interstitial sites in the lattices of iron. A11 these elements fulfill the condition r, < r,, (table 3), The interstitial voids in fcc iron are larger than those in the bcc lattice (fig. 5). The largest void in the fcc lattice, 0.052 nm in radius, is at an octahedral site; the next largest is 0.028 nm in radius. The radius of a carbon atom is 0.08 nm and for nitrogen is 0.07 nm. These atoms in solution reside in the octahedral sites, thereby expanding the lattice of austenite. In bcc iron the interstitial voids are smaller than those in austenite. Although the octahedral site is only 0.019 nm in radius as compared to 0.036 nm for the tetrahedral site, C and N atoms reside in the octahedral sites, presumably because there they have only two iron atoms as nearest neighbors rather than four in the tetrahedral sites. It is this difference in size of the interstitial sites that permits the higher solubility of C in austenite than in ferrite which in turn allows the formation of a great variety of microstructures as C precipitates from supersaturated solid solution in a iron. As shown in fig. 6, diffusivities in bcc iron depend upon the size of solute atoms. The solubilities depend on the phase with which the a-iron interstitial solid solution is in equilibrium: stable equilibria would be, for example, those with nitrogen gas or with graphite. There is a high activation energy of nucleation of these two elements in iron because of large surface energy terms. Metastable phases form instead, with lower activation energy land higher solubilities in the iron-rich solid solution. Phases that form in Fe-C and Fe-N alloys are listed in table 4. In fig. 7 the solubilities of some of these phases are shown. Which phase forms depends on temperature and time of heat treatment and on the imperfection structure of the solid solution. The probability of formation of nitrogen gas or graphite is so low that these phases rarely precipitate in steels. An
Table 3 Maximum solubility of interstitials in a-Fe. ~
element
atomic radius (for
max. solubility (at%) at temperature ("C)
coord. no. 12)
in equil. with
(nm)
~~
< 0.04
1-2x
905
0.098
0.005"
912
0.091
0.10
121
0.092
0.38
585
-0.092
0.7-13x104
906
~
* Ohtani [I9881 Swisher [1967]
References: p. 1618.
1562
W C. Leslie and E. Hornbogen
0 METAL ATOMS
METALATOMS
0 TETRAHEDRAL INTERSTICES
0 OCTAHEDRAL INTERSTICES
METALATOMS
0
Ch. 17, $ 2
METAL ATOMS
0 TETRAHEDRAL INTERSTICES
OCTAHEDRAL INTERSTICES
(b) Fig. 5. Interstitial voids in iron, (a) Interstitial voids in the fcc structure, octahedral (1) and tetrahedral (2). (b) Interstitial voids in the bcc structure; octahedral (1) and tetrahedral (2).
important exception occurs when carbon or low-alloy steels are held in the range 450"65OOC for hundreds or thousands of hours. The pearlite in the steels decomposes to ferrite and graphite. Normally Fe,C (cementite,) or less stable carbides form instead of graphite. Recently, COTTRELL[19931 has published an electronic interpretation of the high stability and curious crystal structure of cementite. In fig. 8 the metastable Fe-Fe,C diagram is shown in addition to the stable Fe-graphite diagram. CHIPMAN[1972] gives the following equations for the solubility of carbon in a-iron in equilibrium with graphite, cementite or E carbide (Fez&):
Ch. 17, 92
1563
Physical metallurgy of sfeels
CC]
]
f2/s
1500
Tcmpcroture 400 200 100
250
50 time
-9
10
f 1;
Q
-24 10
-29 10
Fig. 6. Average diffusion coefficients D of substitutional (s) and interstitial elements in iron. Q,=250 f 40 kJ/mol; in a-Fe: Q,= 83 M/mol; QN= 74 !d/mol; Q, = 95 kJ/mol.
graphite: log [C],,,
5550 T
= 7.81 - -
4040 cementite: log [CIw = 6.38 - T
&-carbide: log [ClPpm= 4.06 - 1335 (approximate) T HAS~EBE et al. [19851 agree with these solubilities for cementite and graphite if the slight deviation from the log [C]and 1/T linear relation caused by the magnetically induced anomaly below the Curie temperature is ignored. There are several important composition ranges in the Fe-Fe,C diagram (fig. 8): a. The maximum solubility of carbon in y-iron (austenite) 2.11 wt% (9.12 at%) is the limiting composition separating steel and cast iron. b. Cast iron has usually a hypo-eutectic or eutectic composition: C&=4.3 wt% = 17.3 at%. Table 4 Metastable compounds that precipitate in Fe-C and Fe-N alloys Composition
crystal structure
Nucleation sites
Formation temperatures
cementite
Fe,C
orthorhombic
dislocations, grain b.
> 200OC
s-carbide
Fe, 4c
hcp
matrix, dislocations
< 25OOC
y'-nitride
Fe4N
fcc
dislocations, grain b.
> 25OOC
a'-nitride
Fe,N
fig. 14
matrix
< 300T
References: p . 1618.
1564
WC.Leslie and E. Hornbogen TEMPERATURE.
Ch. 17, 52
OC
Fig. 7. Solubilities of C and N in a-Fe in equilibrium with different phases (average values from several investigations, after LFSLIBand KEH [1965]).
+ y-Fe(C) + Fe,C. c. Steels can be subdivided in hypo-eutectoid and hypereutectoid depending on whether their carbon content is smaller or larger than 0.77 wt% (3.48 at%). The microstructure that originates from the eutectoid reaction y-Fe(C) + a-Fe(C) + Fe,C is known as peurZire, if it is formed by lamellar growth. d. The maximum solubility of carbon in a-Fe is 0.0218 wt% (0.10 at%). At lower carbon contents, carbides can only form by precipitation from ferrite. The lower the carbon content, the lower the temperature below which precipitation can start. Below 250°C the less stable .+carbide can form. The smallest carbon contents that can be obtained currently (1-2)x lo4 wt%, correspond to the solubility at about 50°C. Nitrogen has a relatively large solubility in a-Fe as compared to carbon. The solubility of oxygen is very small (SWISHER and TURKDOGAN [1967]). The difference between the solubilities of N, C and 0 in a-iron cannot be explained on the basis of atomic-size ratios alone (table 3). Solid solubilities are determined on annealed specimens of the highest obtainable purity. Interstitial solute atoms tend to segregate to lattice defects because they offer sites that are energetically more favorable than those in the undisturbed lattice. These can be point, linear and planar defects such as vacancies, some substitutional solute atoms, dislocations, sub-boundaries, grain boundaries and free surfaces. However, in annealed specimens the concentration of such defects is insufficient to cause significant variation in the measured solubility, except perhaps of B and 0 whose solubilities in a-iron is extremely low. The important effects of segregation of solutes in steels are treated in detail later. The precipitation of interstitial solutes from solution in a-iron at low temperatures is
Ch. 17, $ 2
Fe
1565
Physical metallurgy of steels
0.5
7-Q
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0 5.5 6.0 6.5 7.0 vlkright Percanbge Carbon
Fig. 8. The F e C equilibrium diagram up to 7% carbon. Solid lines indicate Fe-Fe,C diagram; dashed lines indicate Fe-graphite diagram.
strongly affected by lattice defects, especially dislocations (table 4). The presence of the defect serves to minimize the strain energy term in the activation energy for nucleation of the precipitates, so that the distributionof precipitates is determined by the distribution of defects.
References: p. 1618.
1566
NC.Leslie and E. Hornbogen
Ch. 17, 32
2.2. Substitutionalalloys The elements marked 2 (table 1) are completely or partially soluble as substitutional elements in the iron lattices. The HumeRothery rules for solubility are well fulfilled for alloys with transition elements: large solubility of the elements in the neighborhoodof Fe (Ni, Co, Mn, Cr and V), limited solubility of Ti, and practically no solubility of Ca and K. The alloying elements have different energies of solution in CY- and y-iron. The AF,,(T) curve (fig. 2 ) is shifted in different directions depending on whether AFay is increased or decreased by the alloying element (fig. 9). This leads to an increase or decrease of the temperature range in which austenite is stable. For simplicity the curves are drawn for a- and y-phase of the same composition (Le., To-temperaturesare shown for AFq =O). We can divide the alloying elements into two groups, depending on whether they close or open the y-field. Typical phase diagrams are shown in figs. 8, 10 and 11. The following rules tell how the alloying elements behave in this respect: a) the y-field is opened by all interstitial elements, because the fcc lattice provides sites of smaller strain energy than does the bcc lattice; b) the y-field is opened by elements with a fcc or hcp structure (Cu, Au, Ni, Pt,y-Mn); c) the y-field is closed by elements that form Hume-Rothery phases with Cu, Ag and Au, even if they form a fcc lattice (AI,Si, P, Zn, Ga, Ge, As etc.); d) the y-field is closed by the bcc transition metals (V, Ti, Mo, W, Cr etc.); The phase diagram of F e C r alloys shows that small additions of chromium lower the y + a transformation temperature while the loop is closed by larger Cr-additions. ZENER [1955] has pointed out that this behavior can be explained by the magnetic term of the free energy of a-Fe (fig. 2). The addition of Cr to Fe has two effects: It shifts the AFJT) curve as shown in fig. 9 and it lowers the Curie temperature T,. Consequently the term AFaYmg,(fig. 2) appears at a lower temperature and changes the shape of the curve to that shown in fig. llb. This T,-effect should affect transform-
1400
-
1300
1200 1100
'
loo0
'
900 .
800
-
-240-180-80
0
+BO
*
AF,,
Fig. 9. The effect of solute elements with larger heat of solution in y-Fe than in a-Fe (e.g. Mo) and vice versa (e.g. Ni) on the transformation temperatures (AFm7=O), assuming constant concentrations of a and y-solid solution and no change of T, in a-Pe (schematic).
Ch. 17, 02
1567
Physical metallurgy qf steels
'
OC
ALBERTSEN, NIELSEN
& BUCHWALD, 1983
900800-
Y
2
600.
5 590-
f
w c
400.
300
-
zoo-
; I
1QO- I
\ \ I \
y"r_ I
\
t
,
I
I I
Y'I
1 \
;
Rg. 10. Proposed Fe-Ni phase diagram based on electron microscopy, electron microprobe studies, and Mossbauer spectroscopy of iron meteorites (ALBERTSEN et al. [1983]).
ation temperatures of all alloys of iron. In Fe-Cr alloys it is evident because the difference in heat of solution of Cr in a-Fe and 7-Fe is relatively small. In many alloys the a- and y-field is bounded by a miscibility gap (fig. llc). The aor y-solid solutions are then in equilibrium with an intermetallic compound or a solid solution. Special attention is drawn to the ordered bcc structures Fe,Al and Fe,Si that can form as stable coherent precipitates in a-iron. In addition metastable equilibria occur
Fig. 11. Closed y-fields in Fe-Mo, Fe-Cr, and Fe-Nb phase diagrams. References: p . 1618.
1568
WC.Leslie and E. Hornbogen
Ch. 17, $ 2
frequently in substitutional alloys of Fe which form the matrix of microstructures. Such coherent phases have been found in binary bcc solid solutions with Cu, Au, Al, Mo, W, Be (HORNBOGEN et al. [1966]). They play an important role in maraging steels. The crystal structure of the solid solution of Fe at room temperatures, as it follows from the phase diagrams (figs. 10, ll), is used to classify steels. If they consist predominantly of a-Fe solid solutions they are known as ferritic steels; solid solutions of y-Fe are the base of austenitic steels. A metastable fcc structure can be obtained, for example, from the ternary alloy indicated in fig. 12. This metastable phase is the well-known austenitic stainless steel. Precipitation of stable or metastable coherent, ordered phases (7’) is very common in the fcc iron alloys. Such (y + y’) microstructures provide the base for creep resistance of precipitation-hardened stainless steels and Ni-base superalloys.
2.3. Interstitial plus substitutional alloys Even plain carbon steels contain substitutional elements, especially Mn, present either intentionally or inadvertently. Therefore interstitials are always in interaction with atoms substituted in the iron lattice. The interaction energy as compared to that with iron, Au, can be positive or negative. Au = (uF,.) - (u,). The lattice of iron with interstitial (i) and substitutional (s) atom in solution can be expected to show a distribution of the i-atoms that depends on that of the s-atoms (fig. 13). A qualitative idea of the sign of Au can be obtained from the stability of the carbides of the s-atoms but the situation is complicated by the effects of substitutional solutes on carbon activity in cementite and in E carbide. For example, aluminum decreases the activity coefficient of carbon in femte and increases solubility; silicon increases the activity coefficient of carbon in ferrite and decreases solubility (LESLIE and RAUCH [1978]). Manganese decreases the activity coefficient of carbon in ferrite but also decreases carbon activity in cementite, the result being no change in the solubility of carbon in ferrite, at least up to 1%Mn (SAITOH and USHIODA [1989]). TPC] 1500
1000 500
0
- 50 - 100
1
I
4
I
8
\
I
12
lb
w/o Ni
Fig. 12. Section of the F a r - N i diagram showing the composition and heat treatment of austenitic stainless steel. Below M, the metastable fcc alloys become unstable and transform to bcc martensite.
Ch. 17, 92
Physical rnetallrrrgy of steels
1569
Fig. 13. Schematic drawing of the structure of interstitial (x) plus substitutional (0)solid solutions, in iron (0): (a) attraction, (b) repulsion between interstitial and substitutional solute.
A very large number of carbides can form in Fe-s-C alloys in stable or metastable equilibria. In table 5 the crystal structures of carbides of the elements of the first transition series are listed. There are the following possibilities for the composition of carbides that form in Fe-s-C alloys. a) Fe,C, solute-free carbide; b) Fe(s),C, carbide with substitutional composition of the alloy; c) [Fe$z]C, carbide with Fe replaced by a definite amount of s; d) sxCy iron-free carbide. Many [FegJC,-type carbides occur in alloy steels, still more in rapidly solidified alloys of iron, in addition to those listed in tables 4 and 5. The &C T-carbide is cubic with 96 metal and 16 carbon atoms in the elementary cell. The metal atoms are iron and group V, VI transition metals. A typical representative, [Fe,WJC, forms in high-speed steels. W, Mo and Cr can be substituted in small, Mn in large amounts in Fe,C. If a higher concentration of group V or VI elements is present a mixed carbide of the type Cr&, forms (table 5). Fe, Mn, V, Nb, Mo and W can replace Cr in this carbide in a wide range of concentrations. In tungsten steels it has a composition [Fe,,W&. Another important group of mixed carbides are the X-phases. Their structure cab be derived from substitutional compounds with the p-Mn structure. They form in Cr-Mo and Cr-W steels. C dissolves in the p-Mn structure and stabilizes it. A pure X-phase has, for example, a composition Fe,8Cr,,W, which is shifted to higher W-contents by C additions. Metastable carbides that form in ferrite are characterized by greater similarity to a-iron as compared to the more stable carbides. This allows them to form semicoherent or coherent interfaces with a-Fe. Figure 14a shows the structure which fulfills the requirements for €ull coherency if the particles are small. An ordered arrangement of interstitials that only distorts the bcc lattice (a’)has been found by JACK [1951] in Fe-N alloys. This strucmre forms during ageing of Fe-N alloys at a low temperature. The ironfree carbides Tic, VC and NbC have face-centered-cubic structures with lattice parameters 20-25% greater than those of the austenite from which they precipitate (fig. 14b). They can form as ultra-fine dispersions, which raise the yield stress of micro-alloyed steel (HSLA-steel, i.e., high-strength low-alloy steel). References: p . 1618.
1570
Ch. 17, $ 3
WC.Leslie and E. Hornbogen
Table 5 Composition and melting temperatures ["C] of carbides that form in equilibrium with transition metals (after GOLDSCHMIDT [1967]).
3. Transformation reactions 3.1. Pearlite The iron-rich side of the Fe-Fe,C phase diagram (fig. 8) provides the background for a large number of solid-state reactions. In alloy steels, reactions can be controlled by substitutional or interstitial diffusion. The microstructure of a steel can only be understood if all the individual reactions and their mutual effects are known. If an alloy of eutectoid composition is cooled below the eutectoid temperature Tpthe pearlite reaction can occur. Nucleation of two new types of crystals inside the austenite crystal is unlikely because of the high surface- and strain-energy required. Instead, transformation begins at austenite grain boundaries and moves into grains with a reaction front that leaves an aggregate of lamellar particles behind. Austenite grain comers are favored nucleation sites, followed by grain edges and planar grain boundaries. In commercial steels nucleation can occur at inclusions. In hypereutectoid steels cementite forms first during transformation of austenite, then pearlite can nucleate at the austenitecementite boundaries. Similarly, in hypoeutectoid steels ferrite forms first and nucleation can occur at ferrite-austenite boundaries. Each of these so-called pearlite nodules consists of two interlocking single crystals, one of cementite, the other of ferrite (HLLLERT [1962]). Each nodule continues to grow (at velocity G ) within an austenite grain until it impinges on another in the same grain. The pearlite reaction is completed at a time tf that depends upon the austenite grain diameter, d
d G
tf = 0.5-
where d/G is the time required for one nodule to occupy one grain. It is now generally considered that the rate-controlling step in the growth of pearlite is the rate of diffusion of carbon in the pearlite-austenite boundary modified by some volume diffusion in the austenite ahead of the
Ch. 17, $ 3
1571
Physical m e t a l l u ~ of~steels
a
b
Fig. 14. (a) Structure of the metastable compound Fe,&(a'), crosses indicate N atoms. a = 5.72 A, c = 6.29 A (after JACK 119511). (b) Structure of the carbides Tic, VC, NbC and carbonitride as Ti(C,N).
moving interface to provide the variation in carbon content of austenite at the interface. Recent studies (ZHOUand SHIFLET [1991]) have shown that pearlite growth is accomplished by the lateral migration of steps across the growth front of cementite and ferrite. The addition of a third element to the Fe-C system inevitably decreases the rate of formation of pearlite. These effects occur in two ways: a) elements that segregate to boundaries can reduce the frequency of nucleation of ferrite. Boron is spectacularly effective in this regard. b) Substitutional solutes can influence both the rate of nucleation and the rate of growth of pearlite. They can do this by changing the activities of C in the three phases involved or by partitioning between ferrite and cementite at the pearlite-austenite interface. Since the latter process requires diffusion of substitutional atoms it is most effective at elevated transformation temperatures. Molybdenum is particularly effective in decreasing the rate of growth of pearlite nodules whereas cobalt substantially increases the rate. When a hypoeutectoid steel is slowly cooled from the austenite region, ferrite forms below the upper critical temperature, which is strongly dependent upon alloy content, and continues to form until the eutectoid temperature, Ael, is reached. With more rapid cooling ferrite can form well below Ael.The ferrite takes on four distinct morphologies (AARONSON [1962]). Appearing roughly in order of decreasing temperatures, these are termed grainboundary allotriomorphs, Wdmannstatten side-plates, intragrunular idiomorphs and inbragranularplates. The Kurdjumov-Sachs relationship between austenite and ferrite: { 111 17 N (110)a c11Q>y N < I l l % is observed. The WidmannstZItten ferrite is often called acicularferrite. References: p. 1618.
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Ch. 17, 5 3
The formation of cementite plates from hypereutectoid steels follows a similar pattern but with the Pitsch orientation: (1WC8,
(55417
(010)- // (ll0)y (001)- // (225)y Figure 15 shows some of the typical microstructures which are observed in Fe-C steels of various compositions.
3.2. Martensite If austenite is cooled fast enough so that no diffusion-controlled transformation reaction can begin until an extent of under-cooling can be reached at which the fcc structure becomes highly unstable, it transforms by shear to a distorted bcc lattice, supersaturated with carbon, called martensite. The minimum cooling rate required for the formation of martensite varies widely depending upon the composition of the steel, which in turn controls the temperatures and rates at which non-martensitic transformation products form, The temperature at which diffusionless transformation starts is designated M,. The M,-temperature is always lower than the temperature To at which a-and y-solid solutions of the same composition are in equilibrium (T, AF& =0, fig. 9). The M,temperature can be above or below room temperature, and steels can be classified accordingly. For austenitic steels M, cc room temperature, martensitic steels M, > room temperature; both must of course be cooled faster than the critical rate to avoid formation of other transformation products. Between To and M, metastable austenite can transform partially into martensite under external stress. During the cooling below M, the martensitic transformation continues until it stops at the martensite finish temperature, M, Frequently, some small amount of austenite is retained. Unalloyed iron or substitutional solutions of y-iron can transform on cooling to abcc structure. As noted before, interstitial atoms occupy octahedral sites in both a and y iron. Only one set of such sites, Z, is common to both lattices. Since the carbon atoms have no time to move during the shear (displacive) transformation, the number of carbon atoms at these Z sites in bcc iron is increased, causing a tetragonal distortion of the bcc lattice. The tetragonality is minimal below about 0.20% carbon because of the association of carbon atoms with lattice defects produced by the displacive transformation (fig. 16). In addition to the shear (Bain strain) that leads to the change in crystal structure, a second deformation, a lattice invariant strain, is required to allow the martensite to correspond to the original crystal. The amount of shear for the y + a transformation is about 0.12 (WAYMAN [1964]). The internal plastic deformation takes place by twinning or slip (KELLYand NUTTING[1961]. It leads to martensite crystals that contain dislocations and thin twins. The M, of a steel depends upon a) the equilibrium temperature, Toand b) the strength of the austenite. Alloying elements affect both factors, as shown by one of several empirical equations used to calculate M,: M,(OC)=539 - 423 C - 30.4 Mn - 12.1 Cr - 17.7 Ni - 7.5 Mo (ANDREWS [19651) where the chemical symbols represent wt% of the alloying elements.
% 3. 5
-7 /’
1574
BC.Leslie and E. Hombogen
Ch. 17, $ 3
Ed .302 .29B
.2% .290
.266
Fig. 16. c/a ratio and M, for body-centered tetragonal Fe-C martensite.
All elements save Co lower M,. All the others in the equation produce solid solution strengthening of austenite and lower M,, regardless of whether they raise or lower T,,. Also, austenite can be stablilized against transformation by work hardening, by grain refinement, by deformation twins, and by prior precipitation of coherent second-phase particles, in brief, by anything that increases strength. Both nucleation and growth of martensite are inhibited by increased strength (KAJIWARA[1986], LESLIEand MILLER [1964], MEYERand HORNBOGEN [1968]). The very large effect on M, of the prior precipitation of coherent particles in the austenite of an Fe-Ni-A1 alloy is shown in fig. 17. The resistance to displacive transformation is maximized with maximum precipitation hardening. With over-aging the M, rises as Al and Ni are removed from solution. Figure 18 illustrates the microstructure after partial and complete transformation to martensite. Ms Pc) +300.
Fig. 17. Variation of the M,-temperature of an Fe, 27.4 at% Ni, 12.5 at% Al, 0.06 at% C alloy by aging the austenite at different temperatures, to form coherent precipitates.
Ch. 17, $ 3
Physical metallurgy of steels
1575
* ... - _..
Fig. 18. Incomplete (40%) and complete transformation of austenite to martensite of an 0.06 at% C, 27.4 at% Ni, 12.5 at% AI steel (MS=-69'C). The transformation starts at austenite grain boundaries. Austenite grain size determines the maximum dimension of martensite crystals. Former austenite grains can be revealed even in the completely transformed alloy. Carbide precipitates are visible at the grain-boundary nodes.
In order to obtain complete transformation to martensite a steel has to be cooled considerably below M, (fig. 19). Retained austenite is found in the spaces between the martensite. This austenite becomes work-hardened by the martensite crystals as they form, and is thus stabilized against transformation. If cooling is interrupted at a temperature below M,, then resumed after a period of time, the transformation does not continue until an additional undercooling is reached (stabilization of austenite). This can be due to strain-aging of deformed austenite and the thermally activated reduction of transformation stresses that would have aided further transformation (e.g., GLOVER [1956]). The size of martensite crystals is limited by the austenite grain size. In steels, small martensite crystals are wanted because isotropic properties and small internal stresses are wanted. Heat-treatments that lead to grain growth of austenite before cooling below M,
Temperature ("(2)
Fig. 19. Martensite that has formed below M, in a steel with 1.1 at% C and 2.8 wt% Cr after austenitizing at 1040°C. M,=230°C, the transformation is almost complete at 20°C. (After COHEN [1949].) References: p . 1618.
1576
RC.Leslie and E. Hornbogen
Ch. 17, $ 3
are therefore avoided. It has become conventional to classify the morphology of martensite as being either the lath type or the plate type. The former is typical of low to medium carbon steels. When the carbon content is below about 0.20% the structure is often body-centered cubic. The lattice invariant shear is mainly accomplished by dislocations. The units of martensite are in the shape of laths containing a high density of dislocations. The laths are grouped into larger packets. Plate-type martensite is found in alloys with M, temperatures below ambient. This structure is composed of individual lenticular plates with internal very fine twins. In many instances the structure consists of mixtures of lath and plate martensite. The conventional distinction between the two types has been questioned by TANet aZ. [1992], who found twins in lath martensite in 0.20% C steel by careful transmission electron microscopy and demonstrated that it is easy to miss these.
3.3. Bainite Although bainite was detected in a study of the isothermal transformation of austenite in 1929 (DAVENPORT and BAIN[1930]) the modes of formation of the various admixtures of ferrite and carbides comprising bainite are still matters of controversy; see the proceedings of the International Conference on Bainite, 1990. The controversy is hardly surprising, for bainite forms in a wide variety of steels, over a wide range of temperatures during the transformation of austenite isothermally or during continuous cooling and the transformation products are finely divided and only resolvable by transmission electron microscopy. The disagreements center on the relative extent of transformation by shear or by diffusional mechanisms. The answer may lie in the observation by JUNG et aZ. [1994] that bainite can nucleate by diffusion and grow by a displacive process. Regardless of the outcome, the importance of bainite lies in the desirable combination of mechanical properties it imparts to commercial steels transformed isothermally or by continuous cooling. Conventionally, the morphology of bainite is divided into two types, upper bainite and lower bainite. These terms arose from structural differences noted after isothermal transformation and may have little relevance to structures developed during continuous cooling. Bainite forms isothermally at temperatures above the M, and below that required for formation of ferrite and pearlite, or at cooling rates too rapid for nucleation of these other transformation products but not rapid enough to form martensite. The structure of upper bainite is generally agreed to consist of aggregates of ferrite platelets called sheaves with Fe,C particles between the sheaves. In lower bainite the carbides, probably E, Fe.&, are contained within the ferrite platelets. In hypoeutectoid steels the ferrite platelets form first. The controversy arises over whether the iron atoms cross the ferrite-austenite interface by shear or by diffusive jumps. The structure of lower bainite is shown in figs. 20a and b. temperatures above 400°C. There is a fourth type of reaction in iron alloys with Even at extremely high cooling rates transformation occurs above M, by a mechanism in which the rapid motion of the y-a. interface produces ferrite with the same composition as the austenite. The morphology of the transformation product differs from that of
Ch. 17, $ 3
Physical metallurgy of steels
1577
Fig. 20. (a) and (b) Microstructure of lower bainite. 3.23 wt% Cr,30 min at 35OOC. Partially transformed 0.66 wt% C (SPEICH[1962]). (a) Optical micrograph. (b) Electron micrograph (replica).
martensite because thermally activated motion of the interface occurs by individualjumps across the reaction front in a direction opposite to that of its motion. The transformed structure is almost free of the defects present in martensite. This reaction is termed a massive transformation (GILBERT and OWEN[1962]) (see also chap. 15).
3.4. lhnsformation diagrams and hardenability The evolution of microstructures in steels can be understood best by reference to isothermal (IT) or to continuous cooling transformation diagrams. These diagrams indicate the beginning and end of each reaction in the transformation of austenite at constant sub-critical temperatures or the cooling rates at which various reactions begin and end. In the making of an IT diagram, small thin specimens of a selected steel are heated into the austenite range, quickly transferred to a liquid medium held at a constant subcritical temperature, held for a pre-determined period and then quenched into iced brine and examined metallographically to determine the extent of phase transformation. The process is illustrated schematically in fig. 21. The result of many such determinations is a diagram such as fig. 22 for a typical high-strength alloy steel, AIS1 4340. Strictly speaking, each diagram is representative only of reactions in one sample of steel having one austenite grain size. Even within one grade of steel variations in chemical composition and austenite grain size will result in small changes in the times for beginning and ending of transformation. Nevertheless, such diagrams are indispensable for the planning of certain heat treatments to obtain desired microstructures and properties, as shown in figs. 23 a, b, c, and d. The IT diagram of fig. 22 shows two "C-curves", the upper one for the formation of femte and pearlite and the lower for formation of bainite, with a deep "bay" between them (bainite is not labelled in the figure). Bainite in this steel forms at constant temperatures from just above 500°C down to below M, at about 280°C. At temperatures between 500" and about,425°C the reaction does not go to completion; much austenite remains. The upper and lower critical temperatures and M, are shown. In production and heat treating of steels it is often desirable to have some knowledge References: p . 1618.
1578
Ch. 17, $ 3
WC.Leslie and E. Hornbogera
50 % 75
5
Q 100 I
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1200-
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1
104
Time, sec
Fig. 21. Diagram showing how measurements of isothermal transformation are summarized by the isothermal transformation diagram.
of microstructures following continuous cooling from austenite and this information is not readily obtained from IT diagrams. For these estimations we turn to CCT diagrams. These are developed through a combination of dilatometry and metallography. In dilatometry, temperature, time and specimen length are recorded for a variety of cooling rates. The initiation and termination of transformation are marked by an increase in specimen length and the cessation of growth, respectively. Microstructures are determined after quenching. From these data a CCT diagram can be constructed, fig. 24 (ELDIS[1978]), with the linear temperature-log time scale as used in IT diagrams. Alternatively, instead of plotting cooling curves, individual bar diameters are plotted on the abscissa instead of transformationtimes. The diagrams indicate the structures that can be obtained at the centers of the bars of the appropriate size, fig. 25 (ATKINS[1977]). Thus a bar 20 mm in diameter, air cooled, will have a structure of approximately 70% bainite, 30% martensite. A bar of 50 mm diameter, water cooled, will have a desirable structure of about 100% martensite at the center. Hardenability is commonly defined as the depth to which steel of a given composition can be quenched to yield a structure of 50% martensite - 50% other transfor-
Ch. 17, $ 3
"C
1579
Physical metallurgy of steels
"F
800 1400
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I200 600 IO00
LLI
5 500 I-
a
800 400
a
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TIME
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- SECONDS
Fig. 22. IT diagram for AIS1 4340 steel. 0.42 C, 0.78 Mn, 1.79 Ni, 0.80 Cr, 0.33 Mo. Austenitized at 845°C. Grain size ASTM 7-8. A = austenite, C = cementite, F = ferrite.
mation products. A criterion of 90% martensite also may be used. Hardenability is affected by austenite grain size, carbon content and alloy content. It is measured by the Jominy end-quench test (JOMINY and BOEGEHOLD [1938]) or by a more recent test better suited for steels of lower hardenability (GRANGE [1973]). There are various methods of predicting hardenability of steels from austenite grain size, carbon content and alloy content, reviewed by DOANE[1978]. The goal of steel producers is to attain the maximum hardenability with the minimum use of expensive alloying elements. Steels can be ordered within specified limits of hardenability (hardenability bands) and the producer can use whatever composition will yield that result. 3.5. Tempering of martensite
Ferrous martensites are highly supersaturated solid solutions of C in iron, obtained by References: p . 1618.
1580
I
Ch. 17, 9 3
WC.Leslie and E. Hornbogen CONVENTIONAL
PROOUGT
ANNEALING
r
1
ISOTHERMAL
ANNEALING
FERRITE 8 PEARLITE
\ FERRITE AND
TIM€ -LOG SCALE
PEARLITE
AUSTEMPERING
I
TIME
-LOG
SCALE
CUSTOMARY WENCHING AND TEMPERING.
W
5
-5
z
W
Y.
TIME
- LOG
SCALE
TIME- LOG SCALE
Fig. 23. Four illustrations of the use of IT diagrams in planning practical heat-treatments.
rapid cooling from temperatures at which the alloy is single-phase fcc austenite. The transformation to bcc or bct iron produces a very high density of defects - dislocations, twins, former austenite grain boundaries, vacant lattice sites and any substitutional solute atoms initially present. Generally speaking, such structures impart high strength but poor toughness. To improve toughness with some reduction in strength the steel is reheated to temperatures above ambient. The result is the removal of C from solid solution and the precipitation of carbides. The operation is simple in practice but the processes occurring are highly complicated. Despite decades of study there is still no complete agreement on all the details. These vary depending upon the carbon content and alloy content, which determine the M, temperature. If this is above ambient the tempering process (autotempering) can begin during quenching. In steels with very low M, temperatures, subsequently held at temperatures below loO°C,processes termed “aging” can occur.
Ch. 17, 0 3
Physical metallurgy of steels
I '
Log time
1581
L
Fig. 24. Schematic illustration of the construction of a CCT diagram. F-femte, P-pearlite, B-bainite, M-martensite, HV10-hardness.
UWAKWEH et al. [1991] state that there are two steps of aging, first clustering of C atoms, then the ordering of these clusters into a carbide of composition Fe,C. In contrast, SPEICHand TAYLOR[1992], consider reactions during aging to be one process of spinodal decomposition. There is a continuing tendency towards lower carbon contents in steels, for elimination of quench-cracking, improved weldability and greater toughness. There is general agreement on the processes occurring when these lower-carbon steels with high M, temperatures are quenched and tempered 1. Segregation of C to dislocations, former austenite boundaries and retained austenite 2. Precipitation of 8-carbide, Fez,&, in the temperature range 50-200°C 3. Between about 200 to 3OO0C, transformation of retained austenite to ferrite and carbide 4. Solution of 8-carbide and precipitation of y, carbide andor Fe,C. 5. Between 400 and 700"C,recovery of the dislocation structure, spheroidization of cementite and recrystallization of the ferrite. 6. If carbide-forming elements are present in substantial amounts alloy carbides such as VC, Cr,C, or MozC can replace Fe,C during tempering at above about 500°C. The hardness of a steel after tempering at any given temperature for the customary time References: p. 1618.
900
0 36
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0.80 0.020 0.020
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of one hour can be calculated from a knowledge of the chemical composition (GRANGE et ai. [ 19771). Alloying elements increase resistance to tempering, the most effective being V, P, Mo and Mn. When high-strength medium-C steels are tempered between 250 and 400°C an annoying phenomenon called tempered martensite embrittlement appears. Although the hardness after tempering decreases continuously throughout this range the toughness, as measured by Charpy V-notch tests, reaches a minimum, fig. 26 (BRIANTand BANERJI [1978]). During austenitizing, S and P atoms segregate to austenite grain boundaries (see chapter 13). The segregration remains during the transformation to martensite and during tempering, reducing cohesion at the prior austenite boundaries. Cementite precipitates at these same boundaries and serves as nucleation sites for cracks during impact loading. These cracks propagate along the weakened boundaries, leading to intergranular fracture.
Ch. 17, 94
1583
Physical metallurgy of steels
I
fracture energy
0
,200
400
600
TEMPERING TEMPERATURE C"C1
Fig. 26. Schematic illustration of tempered martensite embrittlement of a quenched and tempered steel.
Tempered martensite embrittlement can be prevented in steels of very high hardenability and very high strength by the addition of 1 to 2% Si. Silicon inhibits the growth of E carbide and cementite (OWEN[19541, KEH and LESLIE[1963]), reducing their effectiveness as crack starters and allowing such steels to be tempered in the range 260-315"C, where both yield strength and notch toughness reach a maximum.
4. Defomuztion and recrystallization 4.1. Microstructure of deformed steel
Iron and ferritic steels, in common with other bcc metals, are characterized by a pronounced dependence of the yield and flow stresses upon temperature and strain rate (CHRISTIAN [1983)). As shown in fig. 27 the yield stress of polycrystalline iron increases sharply at temperatures below about 325 K, -0.18 T,, where T,,,is the melting point in kelvins. Above that temperature a long plateau appears and the yield stress does not begin to drop until a temperature of about 800 K, -0.45 T,, is reached. Although it is not clearly shown in fig. 27, a concave-downward region appears in the yield stresstemperature plot between about 180 and 280 K (MATSUIet al. [1978]). Similarly, the yield stress increases markedly increasing with strain rate, fig. 28 (LESLIEet al. [1969]). These are intrinsic characteristics of the bcc metal lattice and are intimately connected with the decreasidg toughness of ferritic steels with decreasing temperature andor increasing strain rate. The cause of these phenomena is the slight dissociation of screw dislocations in bcc metals which renders them relatively immobile (sessile) at low temperatures. The Idissociation must be constricted by thermal activation before it can glide. In contrast, edge dislocations can glide at any temperature. In bcc metals slip can occur on any plane containing a 4 1 1> slip direction, so-called References: p . 1618.
100
-
700
600
500 Ti
- GETTERED
Fe n
400
4
0"
P
z 300
200
100
0 TEMPERATURE,
K
CI
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Fig. 27. Temperature dependence of yield and flow stresses in FS-O.IS%Ti.C z 2.5 x lo4 s-', G.S.ASTM 5-6. uEis the elastic limit, uAthe anelatic limit.
c)
4 cQ1
P
Ch. 17, $4
1585
Physical metallurgy of steels
690 552 414
345
Fig. 28. The effect of strain rate on the strength of polycrystalline bcc iron at room temperature.
pencil glide. The result is wavy slip lines on a polished surface of femte deformed at ambient temperature (fig. 29a). Deformation at temperatures well below ambient leaves a uniform distribution of the sessile screw dislocations, the mobile edge components having moved out of the crystal. During deformation of ferrite at room temperature, dense tangles of dislocations form a cell structure. The spacing between cell walls decreases with increasing strain (KEH [1961]). In fcc metals (austenitic steels) slip occurs by glide of a/2 dislocations on { 111) planes, leading to a total of 12 possible slip systems. The dislocations are confined to their slip planes because of low stacking-fault energy, the result being well-defined, straight slip lines (fig. 29b). In the presence of a fine dispersion of shearable coherent particles slip steps are well defined and widely spaced (fig. 29c). Dislocations can be generated at grain boundaries and at the interfaces of inclusions and carbide particles. The density of dislocations increases with deformation (fig. 29d), KEH and WEISSMANN [1962]. The curve shown there does not apply when a dispersion of any strong second phases is present; in those instances the density of dislocations for a given strain, i.e., the rate of work hardening, is greater than indicated by fig. 29d. Cementite, whether present in pearlite or as discrete particles, resists deformation, with the important exception of wiredrawing. Beginning with a eutectoid carbon content in the form of fine pearlite and upper bainite, a true strain of 4 reduces the thickness of Fe,C lamellae to about 2 nm, or just a few unit cells. When stressed, annealed low-carbon steels usually yield discontinuously, Le., with an abrupt drop in stress, followed by yielding at constant stress and then continuous work-
Rejerences:p . 1618.
1586
WC.Leslie and E. Hornbogen
Ch. 17, 9 4
a
Fig. 29. (a) Wavy slip bands in ferrite after 3% strain at 20°C, 200 x. (b) Traces of (1111 slip planes in austenitic steel after 2% strain, replica. (c) High slip steps in austenitic steels containing 7’ precipitates after 2% strain. (d) Relation between plastic strain and dislocation density in polycrystalline a-Fe.
hardening. The strain at constant stress is called the yield point elongation or Liiders strain. For this to occur, three requirements must be met: 1. A low density of mobile dislocations present before yielding. (Most or all of the dislocations present are normally pinned by interstitial “atmospheres”, see ch. 20). 2. Rapid generation and multiplication of dislocations. 3. A low to moderate dependence of dislocation velocity on stress. Under these conditions, when dislocations are generated heterogeneously at some spot of stress concentration they multiply rapidly by a multiple cross-slip mechanism. Edge dislocations in bcc iron move about 20 times faster than screws at a given stress, but the motion of screws is required for macroscopic yielding. The flow initiated at the upper yield stress coincides with the motion, cross slip and multiplication of screw dislocations. From the point of generation of dislocations a zone of high dislocation density (localized strain) sweeps through the specimen. This is called a Liiders bund. The cross-section of
Ch. 17, 94
Physical metallurgy of steels
1587
the specimen is reduced, raising the stress and inducing other such bands until uniform work hardening begins. Deformation of ferritic steels also can proceed by twinning. This is favored by low temperatures, high strain rates, large grain size, lack of mobile dislocations and solid solution hardening, especially by Si. As compared to deformation by dislocation generation and motion it is of minor importance. Mechanical twins can be produced in austenitic [1964b]). steels but only at very high strain rates and/or low temperatures (HORNBOGEN
4.2. Recovery and recrystallization Recrystallization can be defined as crystallographic reorientation by passage of a high-angle boundary. Recovery is the name given to all other processes that reduce the energy stored by deformation (see ch. 26). Both are thermally activated and both can be driven by stored energy of deformation but recrystallization can also be driven by the reduction of surface energy, whether the surfaces are external or internal, i.e., grain boundaries. Both can be static or dynamic. We will consider static processes first, those that occur at constant temperature after cold work. There can be several measures of recovery, such as decreases in electrical resistivity, decreases in X-ray line broadening, decreases in hardness, decreases in dislocation density and rearrangement of dislocations. These all proceed rapidly initially, then reach a plateau. Unless recrystallization intervenes they will go to completion in a range of temperature from below room temperature to about 500°C, fig. 30 (MICHALAKand PAXTON [1961]). The motion of lattice defects can be hindered by factors such as solute atoms -especially interstitials-preprecipitation clusters and nucleation of precipitates on such defects, thereby retarding or stopping recovery. Recrystallization at constant temperatures after cold working can begin by migration of existing high-angle grain boundaries, migration of sub-boundaries and by sub-grain
TIME-MINUTES Fig. 30. Recovery of a-Fe after 5% strain at ooc(MICHALAK and PAXTON [1%1]). Rderences: p. 1618.
1588
Ch. 17, $ 4
RC.Leslie and E. .Hornbogen
coalescence (Hu [1962]). These processes will begin where energy density is greatest, as at grain-boundary edges, inclusions and intersections of mechanical twins. As stated previously, recovery and recrystallization are driven by the stored energy of cold work so they can be concomitant, competing processes. After slight cold working recovery can be complete before recrystallization begins, and the rate of growth of recrystallized grains will be constant until the growing grains impinge. After heavy cold work the reduction of stored energy by recovery causes the rate of growth of new, recrystallized grains to slow markedly with increasing time at constant temperature (fig. 31). This iron was highly purified by zone-refining, which greatly enhances the rate of recovery. The rates of formation and growth of recrystallized grains can be changed by several orders of magnitude by even small additions of second elements in solid solution (VENTURELLO et al. [19631, LESLIE et al. [19631, fig. 32). However, the inhibition by fine precipitates or pre-precipitation clusters is even greater. If ferritic steels are deformed in the austenite range, dynamic recrystallization can occur immediately after working, before or during cooling. Control of the process by temperature, degree of reduction and the presence of fine precipitates is important in the production of high-strength low-alloy (HSLA) steels. The fine precipitates of AlN, Nb(C,N), V(C,N) or Ti(C,N) are used to refine austenite grain size and hence femte grain size; AlN precipitated during annealing after cold work provides a desirable texture in sheets for cold forming. Copper precipitated before cold rolling and annealing can produce a similar effect (ERAet al. [1988]). MnS, MnSe, AIN or BN can be employed in grain-oriented silicon steels to provide a
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Ch. 17, 5 5
Physical metallurgy of steels
1589
Fig. 32. The effect of small Mo additions on the velocity of the recrystallization front of iron at 5% recrystallization, 60% deformation, at% Mo as indicated (After LESLEet al. [1963]).
Second-phase particles have two different effects on the kinetics of recrystallization. Particles larger than about 1 ,urn, such as inclusions or pearlite nodules promote nucleation of recrystallization. If particles are fine, < 0.1 p m and closely spaced, dislocations after cold working are uniformly distributed, without regions of high lattice curvature required for nucleation and recrystallization is inhibited (fig. 33). The effect of second-phase particles of different sizes is further treated in ch. 26.
5. Mechanical properties 5.1. Strengthof ferrite The ideal strength of iron crystals can be measured by straining iron whiskers. Depending on crystallographic orientation, the tensile strength lies between 5 and 13 GPa and the shear strength between 3.5 and 8 GPa (BRENNER[ 19561). In contrast, the critical resolved shear stress of high-purity iron single crystals at room temperature is about 10 Mpa and of polycrystals about 30 to 40 MPa. The approximate ranges of strength of iron-base materials is shown in fig, 34. Single-phase ferrite can be strengthened by 1. grain refinement; 2. cold work; 3. solid solution additions; 4. shear-type transformation; 5. radiation. The relation between the lower yield stress and the grain size in ferrite is customarily expressed by the Hall-Petch equation.
References: p. 1618.
WC.Leslie and E. Hornbogen
1590
Ch. 17, 5 5
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where d is the average intercept grain size. The value of the exponent is now generally accepted as Although empirical, this equation is highly useful; fig. 35 is a plot covering a wide range of grain sizes. After small plastic strains the dislocation density in bcc iron is related to the flow stress, as shown in fig. 36 (KEHand WEISMANN [1963]). The relation can be expressed by
4.
a = a, + ixGbfi
Ch. 17, $5
1591
Physical metallurgy of steels
I
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Fig. 34. Strength ranges of iron-base materials.
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1592
WC.Leslie and E. Hombogen
Ch. 17, $ 5
Fig. 5. Relationship between the square root of average dislocation density and flow stress of a-iron (KEH anWEISSMANN 119631).
where a! is a proportionality constant, G is the shear modulus, b is the Burgers vector, and N is the dislocation density. Dislocation density at a given strain is greater in fine-grained than in coarse-grained iron. For uniform plastic strain the true stress-true strain relation is expressed by the empirical equation
u = KE" where cr is true flow stress, E is true strain and K and n are constants (LUDWIK [1909]). K is the true stress at a strain of unity and n, the strain hardening exponent, is the slope of a log true stress-log true strain plot. The strain hardening exponent is numerically equal to the limit of uniform plastic strain and is an indication of the stretchability of low-carbon steel sheets. The work-hardening of iron at large strains depends upon the mode of deformation. Titanium-gettered iron containing essentially no interstitial solutes can be cold drawn to a true strain of at least 10, to a strength of 1650 MPa (LANGFORD etal. [1972]). In torsion, work hardening saturates at a true strain of just under 4.8 because of dynamic recovery (YOUNGet al. [1974]). High strains are more difficult to obtain by cold rolling because of less severe geometrical restraints, but at a true strain of 2.8, work hardening was still increasing (LAKE[1977]).
Ch. 17, 8 5
1593
Physical metallurgy of steels
True solid solution hardening of ferrite refers to the interaction of the strain fields of solute atoms with dislocations when the solute atoms are immobile. It is difficult to separate dynamic and static effects when interstitial solutes are involved, but tests done at 0°C and at a fairly high strain rate indicate that the change in shear stress with change in carbon content, d.r/dC is about G / 2 . When substitutional solutes are added to bcc iron, the athermal behavior of the yield stress, shown in fig. 27, is eliminated. The yield stress is always a function of temperature. Even more interesting is the phenomenon of alloy softening. In restricted regimes of temperature and composition, solutes, whether substitutionalor interstitial, can reduce the yield stress of iron, fig. 37. The magnitude of the softening, the yield stress of the alloy less the yield stress of the iron, is comparable to the maximum strengthening. Softening can be produced in unalloyed iron by knocking iron atoms into interstitial positions by high-energy radiation (SATOHet al. [1970]). Outside the composition limits in which alloy softening occurs, solutes strengthenbcc iron. The atom size misfit is a rough guide to the magnitude of strengthening but it fails when the solute atoms are smaller than iron atoms, fig. 38. Elements of limited solubility in bcc iron tend to segregate in grain boundaries, fig. 39, (HONDROSand SEAH[1977]) with significant effects on mechanical properties, as described later (see also ch. 13). Grain boundaries are sources of dislocations and the stress required to activate these sources can vary with solute type and concentration at the boundaries. As a consequence, k in the Hall-Petch equation, a,, =a,+ k d - l n , also TEMPERATURE, 'C -200 175 150
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References: p. 1618.
1594
Ch. 17, 5 5
WC.Leslie and E. Hornbogen
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varies, C, N and P producing the largest increases. These segregation effects, equilibrium and non-equilibrium, further complicate the studies of solid solution hardening. Although the process is not technologically important, high-purity bcc iron can be hardened by shear transformations induced at ambient temperatures by passage of a shock wave generating a pressure greater than about 13 GPa. The reaction involves the shear transformation of bcc iron to E at high pressure then the reverse after the shock wave passes. The defect structure remaining resembles that of lath type martensite, without the presence of carbon. The yield strength is increased by about 540 MPa (LESLIE and SOBER[19671). A martensite-like structure of high hardness can also be generated in pure iron by ultrarapid quenching from the melt. The hardness is increased more than sevenfold compared to metal quenched in the solid state (DUFLOS and CANTOR[1978]). Iron can be hardened and embrittled by bombardment with high-energy neutrons or electrons which knock atoms from their normal lattice positions into interstitial sites ( K L m[1990]).
5.2. Properties of structural steels We are concerned here with steels that are not heat treated after manufacture. The newest of these are interstitial-free (IF) steels (TAKECHI [ 19941). They are made possible by commercially practicable steel-making techniques that can reduce C, N, 0 and S to very low concentrations; C contents can be as low as 0.002%. Sufficient Ti or Ti + Nb
Ch. 17, $ 5
1595
Physical metallurgy of steels
1o6
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is added to combine with C, N and S, reducing C in solution to less than O.OOOl% after annealing. The steels are continuously cast, hot rolled, cold-rolled and continuously annealed. The impetus for the rapid growth in production of these steels came from the References: p . 1618.
1596
WC. Leslie and E. Hornbogen
Ch. 17, $ 5
realization that continuously cast Al-killed sheet steel did not have satisfactory deep drawing and non-aging properties, especially after hot-dip galvanizing which is now in great demand for automobile construction. For good forming properties, high n (in the stresdstrain relationship (+ = KE") values and a well-developed { 111} texture are required. The extent of the latter governs the r value, which is defined as the ratio of width true strain to thickness true strain in tension tests of sheet specimens. With the assumption of constant volume: r=
In( W W f ) ln(LfWf/LiY)
where
W i= initial width Wf=final width Li= initial length
Lf= final length For deepdrawing steels r should be at least 1.5; some IF steels reach values of 2 to 3. Interstitial-free steels were developed initially to have a low yield strength and high elongation in a tension test and excellent ductility, but it is now obvious that the IF technique can be expanded to higher strength steels. P and Mn can be added for solid solution strengthening, B can be added to improve grain boundary cohesion and Cr can be added to make IF ferritic stainless steel. Many current grades of cold-rolled sheet steels may be superseded by these C-free steels. Formerly, many low-C sheet steels were not deoxidized in the furnace. As a result, C and 0 in the liquid steel reacted during cooling of the ingot, leaving an outer rim of solid steel lower in carbon than the interior. After cold-rolling and annealing the steel surface tended to be free of defects. Now the great majority of low-carbon steels are continuously cast and the rimming action cannot be tolerated, so the steel is "killed" with an addition of 0.03 to 0.04% Al. Any aluminum that does not combine with oxygen is free to form AlN, thereby minimizing subsequent strain-aging. Srrain-aging is the term applied to various changes in properties of a metal that occur by the interaction of interstitial solute atoms with dislocations during or after plastic deformation. Such interactions appear in many different alloy systems. In steels the significant interstitials are C and N atoms. When property changes occur after plastic deformation the process is called static strain-aging; when they are concurrent with plastic strain it is called dynamic strain-aging. The effects of static strain-aging on the tensile properties of a lowcarbon steel are shown schematically in fig. 40. When a specimen is strained, allowed to stand for a period of time at temperatures as low as ambient then re-strained the following effects are noted: 1. an increase in yield stress, A Y 2. a return of the Luders strain, E~ 3. an increase in ultimate tensile stress, AU 4. a decrease in total elongation, As 5 . embrittlement.
Ch.17, 55
Physical merallurgy of steels
1597
ELONGATION Fig. 40.Effects of static aging on the loadelongation curve of iron containing interstitial solutes: AY=change in yield stress due to strain-aging, e,=LUders strain after strain-aging, AU=incre.ase in ultimate tensile strength due to strain-aging, A& =decrease in elongation due to strain-aging.
There is an unfortunate consequence of the return of the Liiders strain. If cold-rolled and annealed sheets are temper rolled to flatten, then allowed to age, during subsequent forming “stretcher strains” appear. These are streaks in which the thickness of the sheet is very slightly reduced, a consequence of inhomogeneous yielding, and the appearance of the sheet is spoiled. The principal cause of strain aging in C steel is N in solid solution in femte. It is more effective than C because of its greater solubility. However, if aging after plastic strain is done at temperatures of about 100°C or above, C becomes effective. The conventional method of eliminating strain aging is to use aluminum-killed steels, but this is not completely effective because some C remains in solid solution. To bring the interstitial solutes to the necessary concentration of about one part per million requires addition of elements such as titanium, as in IF steels. Figure 41 shows the effects of dynamic strain aging on the tensile properties of a common carbon steel. When the steel is strained at temperatures between 100 and 250°C serrations appear on the stress-strain curves and the rate of work-hardening is greatly increased. Dislocations generated during plastic straining are quickly pinned by mobile interstitial solutes and more dislocations must be generated before the strain can continue. The results are an increased dislocation density for a given strain and a high rate of work-hardening. Most effects of strain-aging are detrimental but in at least one instance, bakehardening, the process proves useful. As automobile body sheets have been made thinner to reduce weight they become more susceptible to denting. To counteract that, strength must be increased, but that causes difficulties in forming. An answer is found with steels References: p. 1618.
A I S 1 1020 I
Elongation, %
Fig. 41. Stress-elongationcurves of a carbon steel stTained m tension, B = 1.75 x lo4 s-’.
5Oh
1
Ch. 17, $5
1599
Physical metallurgy of steels
containing 0.005 to 0.01% C with Al plus Nb or Ti, cold-rolled, continuous-annealed at temperatures near 90O0C, then cooled rapidly, leaving about 0.001%C in solid solution, fig. 42 (SATOHet aZ. [1992]). When the sheet is formed into a fender, painted, then baked at 170 to 2OO0C, strain aging adds an increment of strength of about 50 MPa. The latest innovation in the production of low-carbon sheet steels is the continuous casting of slabs 50 mm thick. Without being cooled the slab is reheated to rolling temperature, then passed through a five-stand hot rolling mill, rolled to desired thickness and coiled. A process somewhat further removed from commercial production is the continuous casting between water-cooled copper rolls of sheet about 2 mm or less in thickness. These require only subsequent cold-rolling and annealing (SHANGand WRAY [1989]). Questions arise as to whether current techniques of obtaining desirable microstructures, textures and surface quality will be applicable to these new products. The properties of low-C steels are determined primarily by ferrite grain size and the type and distribution of carbide particles (RICKFXTand KRISTUFEK [1949]). Ferrite grain size is determined by prior austenite grain size which in turn is controlled by the deoxidation of liquid steel, or lack thereof. As discussed before, grain growth in austenite is inhibited by precipitation of AlN, or less commonly by Nb, V or ‘I? carbonitrides. Grain size is also affected by the degree of cold work and the annealing temperature and time. Grain size can be reduced or made more uniform by re-austenitizing at a low Continuous Annealing
Al-killed -0.005%
Hot-rolling
CTiGJGT-] Bake hardening Time NbC,AlN
Extra-deep drawing Extra-deep drawing n
Fig. 42. Processing of bake-hardening and extra-deep drawing sheet steel (SATOHet al. [1992]). References:p. 1618.
1600
IC.Leslie and E. Hornbogen
Ch. 17, $ 5
temperature and air cooling (normalizing). Carbides can take the form of massive particles of cementite or volumes of pearlite, depending upon cooling rate from above A, temperature. As long as ferrite is the continuous phase, the yield strength of the steel is determined by the grain size and solute content of the ferrite. Banded ferrite-pearlite structures are common in hot-rolled steels, resulting in anisotropy of mechanical properties. The banding results from the segregation of Mn to austenite grain boundaries during heating. Hot rolling spreads the Mn-rich regions into sheets. During cooling, pearlite nucleates preferentially in the Mn-rich portions, giving rise to sheets of pearlite islands. As C content increases, pearlite becomes the continuous phase and we must consider the properties of pearlite particularly when the structure is entirely pearlite, as in eutectoid carbon steels used for rails. The yield strength of pearlite is dependent upon the et al. [19421 determined a proportionality between yield interlamellar spacing; GENSAMER strength and the logarithm of interlamellar spacing. Pearlite lamellae do not have a constant true spacing; there is a distribution of spacings about a mean value, A; A mean intercept spacing I is measured by drawing random straight lines on a micrograph. The total length of line divided by the number of lamellae intersected is defined as I: It can be shown that
(RIDLEY[1984]). The yielding of pearlitic is presumed to occur by motion of dislocations in the ferrite and BERNSTEIN[1989]). The ductility and impact toughness of lamellae (ALEXANDER fully pearlitic steels are more dependent upon prior austenite grain size than upon interlamellar spacing, large grains being detrimental (LEWANDOWSKI and THOMPSON [19861). 5.2.1. High-strengthlow-alloy steels The users of steels are greatly interested in reducing costs by eliminating heat treatments and purchasing steels that have properties equal to those of heat-treated alloy steels. These newer steels should have minimal carbon content to improve toughness and weldability and minimal expensive alloy additions. To answer this demand steel producers have developed a wide variety of high-strength low-alloy (HSLA)steels. These are intended to meet certain mechanical properties rather than specified chemical compositions. They have certain features in common, one being refinement of ferrite grains. Grain refinement is the only strengthening mechanism that simultaneously increases strength and toughness. Strengthening by precipitation of very fine particles reduces toughness so it must be offset by grain refinement. Precipitation and grain refinement are managed by control of hot-rolling temperatures and deformations. Inclusions of MnS are elongated during hot rolling thereby reducing toughness in the transverse direction so elements are added to make the inclusions resistant to hot deformation. This is termed inclusion shape control. To control the structure formed during cooling from hot-forming processes, (equiaxed ferrite-pearlite, or acicular ferrite, sometimes called low-carbon bainite), alloying elements are added to
Ch. 17,$ 5
Physical metallurgy of steels
1601
lower the transformation temperature and/or the cooling rate from the hot-rolling temperature can be accelerated. The fine precipitates used to strengthen HSLA steels and to control grain size are carbo-nitrides of Ti, Nb and V. Titanium has the advantage of also serving to change elongated sulfide inclusions into discrete particles. Additions of calcium to the liquid steel before casting also serve to control inclusion shape. The carbo-nitrides noted are chosen because of their very low solubility in austenite, Ti(C,N) being the least soluble and V(C,N) the most. By precipitating as fine dispersions in austenite they inhibit austenite recrystallization and grain growth. They are much less soluble in ferrite than in austenite; in consequence, any C, Nb, or V remaining in solution in austenite tend to precipitate at successive positions of the intermittently moving y-a boundaries leaving behind in the ferrite. planar arrays of fine, closely spaced particles only resolvable by transmission electron microscopy. This is termed interphase precipitation. These microalloy additions are often combined, Ti or Nb for grain refinement, V for precipitation in ferrite. Copper occupies a special place in HSLA steels. When added in concentrations between 0.5 and 1.0% it can remain in solution even during slow cooling from the austenite range. An object can be formed when strength is low, then aged by holding in the range 500 to 600°C. Coherent bcc particles form, containing iron as well as copper. The maximum strengthening, fig. 43, is attainedjust before the particles transform in situ et al. [1966], GOODMAN etal. [1973]). to fcc copper (HORNBOGEN [1964], HORNBOGEN Copper, in combination with phosphorus, is used in the so-called weathering steels that are resistant to atmospheric corrosion. The most common types of HSLA steel have a ferrite-pearlite structure after hot rolling. They can have yield strengths up to 700 MPa, as compared to 200 MPa in plain carbon steels with the same type of microstructure. The C content of most HSLA steels is less than 0.20%. To obtain an acicular ferrite structure after hot-rolling additions of B, Mo or Mn are used to decrease the transformation temperature. For improved toughness and weldability at high-strength levels, C must be kept below about 0.08%. Not all HSLA steels are shaped in the mills. With microalloying additions, medium carbon steels can be hot forged and with controlled cooling can eliminate the need for quenching, tempering, straightening and stress relieving. Dual-phase steels are a special category of HSLA steels. Their microstructure consists of a ferrite matrix with 10 to 20% martensite (or lower bainite) islands. The virtue of this structure lies in the fact that the formation of martensite produces stresses around each island. Yielding then occurs at low applied stress and the yielding is continuous, but because of the pnxence of the martensite islands, work-hardening is rapid. These features, together with the soft, low-carbon, e 0.10%, ferrite provide good formability and strength near 600mMPa after 10% strain. Dual-phase steels generally contain about 1.5% Mn and 0.50% Si. After hot rolling or cold rolling the steels are intercritically annealed (heated into the a -I-y fegion). The amount of austenite formed varies with the initial carbon content and the annealing temperature. During rapid cooling the austenite pools transform to martkmsite, with perhaps some retained austenite. The intercritical annealing References: p. 1618.
Ch. 17, $5
WC.Leslie and E. Hornbogen
1602
500 20
*/ 16
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---
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12
Ageing period, hours
Fig. 43. Precipitation hardening of Fe-O.91% Cu alloy, quenched from 840OC and aged at temperatures indicated (HORNBOGEN et al. [1966]).
is done in continuous annealing lines, if they are designed for sufficiently high temperatures (see, for instance, OHASHI[1988]). During forming operations the ductility can be enhanced by strain-induced transformationof retained austenite. Plastic deformation takes place almost exclusively in the ferrite component of the microstructure. Additions of strength can be gained by strain aging during paint baking. The principal market is the automobile industry. 5.3. Strength of martensite
The effect of carbon content on the hardness of untempered martensite is shown in fig. 4 4 (KRAUSS 119781). Yield strength is a linear function of carbon content up to about 0.40% (fig. 45). The yield strength of shock-transformed Ti-gettered iron containing virtually no carbon in solid solution establishes the zero point. The departure from
linearity above 0.40% C shown in fig. 44 is caused by the retention of austenite. As this phase increases in amount the hardness reaches a maximum at about 0.80% C, then decreases rapidly. The simple relation shown in fig. 45 results from a combination of several factors. The relative importance of these factors remains a matter of dispute (LESLIE and SOBER [1967], OWEN[1992]). Current interest centers on low-carbon martensites because of concern with toughness and weldability. In martensite with carbon contents of about 0.15% or less the defect structure, including prior austenite grain boundaries, provides the largest contribution to strength. At or above carbon contents of 0.30% rearrangement of carbon atoms and dislocations during the quench can dominate the strength, unless the M, temperature is below about -6O"C, when true solid-solutionhardening by carbon dominates (WINCHELL and COHEN[1962]). As a further complication it should be noted that iron-carbon martensites do not have a unique strength, the yield and flow stresses are greater in [1972]). compression than in tension, by about 15% (RAUCHand LESLIE
References: p. 1618.
1604
Ch. 17, $ 5
WC.Leslie and E. Hornbogen
80
’
-
- 500
Fig. 45. The effect of carbon content on the yield strength of untempered martensites.
5.4. Strength and ductility of tempered martensite Tempering is the process of heating martensitic steels to obtain a desired combination of strength and ductility ($3.5). High strength is easily obtained; to achieve a combination of strength, sufficient ductility to resist fracture at points of stress concentration or during impact loading, retention of ductility at low temperatures, weldability and low cost is a major challenge for designers of steels. For a given carbon or low-alloy steel the hardness (strength) can be calculated by the method of GRANGE er al. [1977]. Figure 46 is a plot of the hardness of iron-carbon alloys, quenched to martensite, then tempered for one hour at the temperatures indicated. Additions of Mn, P, Si, Ni, Cr, Mo or V do not affect the hardness of as-quenched martensite but each of them retards softening during tempering. To estimate the hardness of a given steel the hardness associated with a given carbon content and tempering temperature is read from fig. 46. To this is added an increment of hardness attributed to each of the alloying elements present, taken from plots provided. The temperature range in which tempered martensite embrittlement occurs must be kept in mind. Within the range of austenite grain sizes normally present in commercial steels, Le., ASTM grain sizes 5 to 10, there is little effect of graia size on yield strength after tempering. However, if the grain size is reduced beyond that range there is a significant gain in both yield strength and toughness. GRANGE[1966] attained ultra-fine austenite grains by four cycles of brief austenitizing at low temperature, quenching, then finally tempering at 205°C. Results are shown in fig. 47 for three kinds of steel. They can be represented by a Hall-Petch type equation
Ch. 17, 5 5
Physical metallurgy of steels
0
0.2
OA 0.6 Carbon (%)
0.8
1605
1.0
Fig. 46. Hardness of tempered martensite in Fe-C alloys (GRANGEe#al. [1977]).
uys= uo+ Kd-+
(7)
where a,,is the intercept at zero on the abscissa and d is grain diameter. The increase in yield strength was not accompanied by a decrease in ductility. Later, TOMITA and OKABAYASHI [ 19851 followed this procedure but after the last austenitizing step they transformed isothermally at 320°C to form 15 to 25% bainite, then quenched to martensite and tempered at 200OC.The result was an increase in the yield strength found by GRANGEin a similar steel, with no loss in ductility. The bainite is constrained by the enveloping martensite and presents another set of boundaries to stop the growth of martensite crystals. In instances when the steel contains elements such as V, Mo, and W, the results of tempering can be quite different. When present in concentrations between 1% and 18%, together with 0.80 to 1.50% C , these elements form the basis of high-speed tool steels. These must be austenitized at a high temperature, sufficient to dissolve most of the stable carbides of V, Mo and W. The undissolved carbides provide wear resistance to the final product. After quenching, the steels are tempered between 400 and 500°C to precipitate fine carbides. As shown in fig. 48, strength is preserved at these temperatures, which can be encountered during high-speed machining of steels. References: p. 1618.
Ch. 17, $ 5
WC.Leslie and E. Hornbogen
1606
1501
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30
mm-'/2
Fig. 47. Effect of prior austenite grain size on the yield strength of martensite in three commercial steels tempered at 205°C (GRANGE [1966]).
HRC
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high speed steel (61
60
50 YI
40
ul
2
30
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200 300 400 Mo 600 703 tempering temper at ore
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Fig. 48. Strengths of a high-carbon steel and a high-speed tool steel (table 2, no. 6) after tempering, illustrating secondary hardening of the tool steel.
Ch. 17, 9 5
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Physical metallurgy of steels
5.5. Ultra-high-strength steels 5.5.1. Maraging steels According to the definition of steel in the first sentence of this chapter, maraging steels are not steels at all, for they contain no deliberately added carbon; carbon is an undesirable impurity. They are iron-nickel alloys, commonly containing 18 wt% Ni. They also contain 3 to 5 wt% Mo, 0.2 to 1.8 wt% Ti and 0.1 to 0.2 wt% Al. Some contain Co, others have none. The heat-treatments to develop strength are simple. The alloys are hotworked at above about 920°C, austenitized at about 850°C and cooled. The M, temperature is between 200 and 300°C and the alloy is completely martensitic before reaching room temperature. The structure is iron-nickel bcc lath-type martensite, which unlike Fe-C martensite is soft, ductile and machinable. However, the rate of work hardening is low since there are no interstitial atoms available to pin dislocations. The uniform elongation in tension is only about 1 to 3%. The function of the martensite is to provide a high density of nucleation sites for the precipitation of intermetallic compounds and probably to provide paths for accelerated diffusion of substitutional solutes. The alloys are aged at about 480OC for three hours. The result is the precipitation of intermetallics of the type Ni,X, where X may be Ti, Mo or V. These particles are of the order of 10 nm in size with a number density of lou m-3 (SHAetal. [1993]). Titanium is the element that precipitates most rapidly and [1963]). When present, is the one used to control strength, fig. 49 (SPEICHand FLOREEN Co raises the activity of Mo, thus hastening its precipitation as Fe,Mo6 or Fe,Mo. Overaging results in reversion to the stable austenite + ferrite, agglomeration of particles [1967]). and softening (ME= and HORNBOGEN Maraging steels can be provided with yield strengths ranging from 1400 to 2400 MPa. The most remarkable feature of these alloys is their combination of high yield stress with high fracture toughness over a wide range of temperatures, fig. 50 (BRUCH
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1
600 500
400 300 200
0
01
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m hours at 500°C
Fig. 49. Effect of Ti content on hardness of an Fe-20% Ni martensite after aging at 500°C (SPEICHand
FLOREEN119631). References: p , 1618.
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WC.Leslie and E. Hornbogen
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20°C 100
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1000 1500 2000 Yield stress, Uy MPa
Fig.50. Fracture. toughness of an 18 Ni, 8 Co, 5 Mo, 0.1C maraging steel as a function of test temperature. and yield stress (BRUCNand HORNBOGEN[1978]).
and HORNBOGEN [1978]). Fracture toughness is not greatly sensitive to the aging treatment so at maximum yield stress the alloy is still tough. In a drastic modification of maraging steels, the nickel content is reduced from 18% to 11%, molybdenum is reduced to 1.2%, 3% Cr is added and Co is increased to 13.4% and about 0.25% C is added. The presence of Co increases the activity of C and raises the M, temperature. The alloy is austenitized-at 885"C, leaving some undissolved carbides. After air cooling the alloy is refrigerated at -73°C to transform retained austenite and finally aged at 480OC. Precipitates formed during aging appear to be M,C carbides. Some martensite reverts to austenite in the form of thin films. The yield strength can be 1650 MPa and the plane strain fracture toughness 130 MPafi. Charpy V-notch energy at -100°C can be about 32 J (AYERand MACHMEIER [1993]). 5.5.2. Modified standard steels An arbitrary definition of an ultra-high strength steel is any steel having a yield strength greater than 1380 MPa. The standard against which modifications are judged is
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Physical metallurgy of steels
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AISI 4340 and equivalent medium-carbon Ni, Cr, Mo steels produced in other nations. These can be oil quenched and tempered to any desired hardness. Properties can be improved by vacuum arc or electroslag remelting to lower the H, N and 0 contents and to reduce the number of non-metallic inclusions. An improvement is obtained by increasing the Si content of 4340 from 0.15-0.30% to 1.45-1.80% to take advantage of the well-known effect of Si in inhibiting the growth of carbide particles. This steel, called 300 M, can be tempered at a higher temperature than 4340 to develop the same hardness, thereby reducing quenching stresses. The addition of Si also moves the temper embrittlement range to higher temperatures. Tempering at 315°C produces a maximum in notch impact toughness and near a maximum yield strength. The additional Si also increases hardenability and adds a component of solid solution strengthening. A small addition of vanadium may provide fine V(C,N) particles during tempering. Crucible Hy-Tuf is a modification of AISI 4130 with increased Mn and Si and an addition of Ni. In Ladish D6AC the Cr and Mo contents are increased over those in 4340. 5.5.3. Thermomechanically-treatedsteels Ultra-high-strength steels can be produced by thermomechanical treatments. For our purposes we can divide such treatments into three groups: 1. Supercritical TMT (above Ae,) 2. Intercritical TMT (between Ae, and Ae,) 3. Subcritical TMT (below Ae,) Considering the first group, if austenite is heavily deformed without recrystallizing it takes up a very sharp texture, { 124) . During transformation the martensite also takes up a sharp texture, (112)+(111) which is retained during tempering (Hu er al. [1976]). The texture provides a high modulus of elasticity in a direction normal to the surface and an increased yield strength in compression in the through-thicknessdirection, thereby improving the ballistic properties (GHOSHand PATON[1978]). Forming in the ferrite-plus-austenite range, followed by quenching before recrystallization, yields a structure of martensite fibers in ferrite, with very high toughness in a direction normal to the fibers (GRANGE [1970]). Deformation of austenite below &, is the process called ausforming. It requires a steel having a large bay between pearlite and bainite in an isothermal transformation diagram. The steel is cooled - avoiding pearlite -heavily worked, then cooled below M,. Yield strengths of about 2600 MPa can be developed. The strengthening is attributed to refinement of martensite platelets formed in flattened austenite grains, strain-induced precipitation of carbides in austenite and development of a strong texture in the martensite. In the third category of thermomechanical treatments, a steel can be deformed as it is transforming to pearlite or bainite. As an example of the strength that can be developed in thermomechanically-treated steels, consider the following: a steel containing 0.6% C and 5% Ni was ausformed, quenched, tempered, strained 3% then aged at 12OoC, a combination of ausforming and strain aging. Yield strength was 3590 MPa, tensile strength 3630 MPa (MATASet al. References:p. 1618.
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N?C. Leslie and E. Hombogen
Ch. 17, $5
[19641) This compares with attainable tensile strengths in cold-drawn eutectoid carbon steel wire of 4800 MPa. With the exception of super-critical treatment (controlled hot-rolling) these thermomechanical processes have not been applied in industry for the following reasons: 1. Large reductions in cross section must be made, so large sections cannot be treated. 2. At the temperatures involved, loads on equipment are severe. 3. The steel must be used in the simple shape in which it is formed. 4. Joining is difficult; welding is impossible.
5.6. Tool steels Tool steels are used for the shaping of other metals or for the cutting of other hard or abrasive materials. There are seven categories of tool steels but lack of space forbids discussion of more than one, the most complex and important, high-speed steels. These are of two types, containing Mo or W. A typical composition of a W type is given in table 2. The heat treatment of high-speed steels consists of austenitizing at a very high temperature, 1250 to 1300°C to dissolve M,,C, carbides but leaving about 10% of the volume consisting of MC and M,C carbides. Hardenability is very high with a deep bay in the IT diagram. Transformation is accomplished by quenching to about 500°C to relieve thermal stresses, then air cooling. The structure is composed of martensite, about 15 to 25% retained austenite and about 10 to 20% undissolved carbides. The heat treatment is distinguished by the necessity for multiple tempering to transform the retained austenite. The first temper at 550 to 595°C “conditions” the austenite @resumably austenite loses carbon to martensite in which carbides are precipitating). The second (or successive) temper at the same temperature transforms the austenite and completes the secondary hardening process in which the precipitation of carbides raises the hardness to 63 to 65 Rockwell C, equivalent to the hardness after quenching, with greater ductility. The volume of carbides in the structure can be as great as 40%. These highspeed steels can be used at temperatures of 500°C or more.
5.7. Austenitic steels Some alloying elements, principally Ni and Mn, expand the y loop in the Fe-C system and other alloying elements can retard the y-a transformation so that the fcc structure can be metastable to ambient temperatures and below. Therefore, we must consider the differences between the mechanical properties of fcc and bcc iron. These differences are inherent in the crystal structures. Diffusivities of solutes are lower in fcc than in bcc at the same temperatures; the solubility of interstitial solutes is greater in fcc than in bcc. The stacking fault energy in bcc is very high, in fcc it is much lower but varies with solute additions. In fcc iron alloys dislocations tend to dissociate readily and cross slip is difficult. Dislocations are largely restricted to { 111} slip planes, leading to planar arrays, narrow slip lines (fig. 29), a high rate of work hardening and large uniform elongations in tensile tests. There is only a small dependence of strength on temperature or strain rate. In
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Physical metallurgy of steels
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contrast, in bcc iron cross-slip is easy; slip occurs on any plane that contains the slip direction, leading to cellular arrays of dislocations.Macroscopic deformation depends upon motion of screw dislocations which are slightly dissociated and immobile and must be constricted by thermal activation. The result is a pronounced dependence of strength on temperature and strain rate and a tendency toward brittleness at low temperatures and/or high strain rates. Because of the relatively high solubility of interstitial elements in fcc iron, strain aging is infrequent. A notable exception occurs in HadfieId manganese steel, our oldest alloy steel. This contains 10 to 14 wt% Mn and 1 to 1.4 wt% C. As a result, the fcc structure is stable at ambient temperatures. The rate of work-hardening in gouging wear is exceptionally high. Early on this was attributed to a strain-induced transformation to martensite but when the carbon content is maintained at the proper level this does not occur. Later, the high work-hardening and wear resistance were attributed to profuse mechanical twinning but when twinning is eliminated by a small addition of an element that increases the stacking fault energy, the high work-hardening remains (ZUIDEMA et al. [1987]). Because of the high concentrations of Mn and C and their mutual attraction in solution, it was concluded that dynamic strain-aging brought about by the reorientation of C members of Mn-C couples in the cores of dislocations is the principal cause of the and LESLIE[1981]). rapid work-hardening (DASTUR The most important class of austenitic steels is that of the stainless steels containing 17 to 25% Cr and 8-35% Ni. The Cr content is the basis of all stainless steels, whether ferritic, martensitic, austenitic or duplex. It provides corrosion resistance. Nickel, as stated previously, expands the y loop so that the steels remain austenitic at low temperatures. They have excellent cryogenic properties as a consequence of their fcc structure. They can be hardened by cold work, by elements in solid solution, especially N, or by precipitation strengthening. MURATAet al. [ 19931have reviewed these mechanisms. The solubility of C in the standard 18-8 stainless steel declines sharply with decreasing temperature; at 60OOC it is less than 0.01%. Since the diffusion coefficient of interstitial atoms in austenite is much lower than in ferrite precipitation of carbides is delayed. However, if heated as during welding, Cr&, carbides can form at grain boundaries, thereby causing B local depletion of Cr and providing paths for corrosive attack. This problem can be dlilleviated by reducing the C content to less than 0.03%or by adding Ti or N b in sufficient quantity to combine with C and precipitate as stable carbides.
5.8. Steels for low-temperature applications Austenitic Ni-Cr steels would be ideal for low-temperature service, except that cost is prohibitive and the strength is often inadequate. Therefore, ferritic steels must be modified in composition and microstructure to counteract their inherent tendency to fracture in a brittle manner at low temperature. In brittle fracture plastic strain and energy absorbed are negligible. Cracks propagate along grain boundaries or along { 100) planes of ferrite. A steel to be used at low temperature must fulfill the requirement that its ductile-tobrittle transition must be below the temperature of use. Questions still remain regarding the relation between methods of determining this transition temperature and behavior in service. References: p . 1618.
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In designing ferritic steels for use at low temperatures certain rules must be followed; grain size must be minimized, C content must be kept below 0.2% and the lower the better, Ni must be added and strength should be obtained by a structure of acicular ferrite or low-carbon bainite. When Ni is added in amounts of 5 or 9% the notch toughness of ferritic steel at low temperatures can rival that of austenitic steels, at a much lower cost, fig. 51. It has been known for some time that the elements Pt, Ni, Ru, Rh, Ir and Re in solid solution both strengthen and toughen bcc Fe (FLOREEN and HAYDEN[1967]). All other additions decrease toughness. The toughening effect is on the bcc lattice, not on interactions with interstitials. There is no theory pertaining to this toughening effect, but we can speculate that these metals increase cross slip of screw dislocations by reducing their dissociation and regret that Pt is not more abundant. 5.9. Segregation of solutes and steel purity
Segregation of solutes to grain boundaries in steel is important because it is closely connected to decreasing ductility. Tempered martensite embrittlement is described in section 3.5. It occurs when as-quenched martensite is tempered between 250 and 400°C. In contrast, temper embrittlement occurs in steels that are quenched, tempered at a high temperature, 600 to 700OC, then placed in service at temperatures from 350 to 575°C. After slow cooling, as in the center of heavy sections, or after long exposure to those temperatures in service, the steel becomes embrittled. This is of particular concern in the generation of electricity because of embrittlement of steam turbine rotors. As indicated in fig. 39, the grain boundary segregation of a solute is inversely proportional to its bulk solubility. However, the phenomenon of co-segregation also must be considered (GUTTMANN et ai. [1982]). When two solutes in Fe are mutually attracted, as are Mn and S or Ni and Sb, they will tend to segregate together at equilibrium. A third factor is site competition at boundaries. Carbon and P, or C and S , or B and P compete for these sites. Carbon and I3 have the desirable characteristic of enhancing grain boundary cohesion. Because of co-segregation between (Mn + Si) and (P + Sn +
-
t
c
E E z
I
-100 0 TEST TEMPERATURE . " C
-
Fig. 5 1. Low-temperature toughness of three Ni steels, with varied microstructures.
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Physical merallurgy of steels
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S Ni Cr Mo V C Mn Si P Al Sn As Sb 0.25 0.02 0.03 0.001 3.65 1.75 0.42 0.13 0.002 ~ 0 . 0 0 50.003 0.003 0.0012 This super-clean steel was made by electric-arc furnace melting under oxidizing conditions, ladle furnace refining, vacuum carbon deoxidation with argon bubbling and casting with vacuum stream degassing. It is free of temper embrittlement over the entire temperature range and safe operation at 480 to 500°C is possible, thereby increasing the efficiency of power generation. Removal of P and S from austenitic stainless steels can improve their resistance to intergranular corrosion and intergranular stress corrosion cracking (BRIANTand ANDRESEN 119881). This topic is also treated in ch. 13.
6. Other physical properties 6.1. Steels for nuclear applications Steels used in the construction of nuclear reactors, for example as canning materials, must not contain elements wjth isotopes that have a high absorption cross-section for neutrons. B, Co, Ta, Nb, Ti, Al, N are such elements that are normally contained in steels. The opposite is true for steels that are used as absorber materials or for shielding. Among alloying elements B has the best ability to absorb neutrons over a wide energy range. The macroscopic neutron-absorption coefficient depends on the €3 concentration in the steel. Because B is hardly soluble in a-Fe it is either present segregated or as particles (Fe,B). Above 2 wt% B ferritic steels become so brittle that they cannot be worked (grain-boundary brittleness). Therefore austenitic steels are often used that contain up to 5 wt% B as a dispersion of (Fe,Cr),B. The absorption coefficient of a steel is structure-sensitive, because self-shielding leads to a decrease of the macroscopic absorption coefficient above a certain particle size. Irradiation with neutrons produces a variety of effects on a very fine scale. In the C-Mn-Mo low-laploy plate steels used in the containment vessels of nuclear fission reactors the flux of neutrons can produce embrittlement (GELLES[1990]). The cause is nanoscale precipitation processes that would not be expected to occur at the temperature of use. Copper is present as a residual element in low-alloy steels, normally in concentrations of about 0.15% or less. Although solubility at temperatures below 500°C is very low the diifusivity of copper at those temperatures is so slight that precipitation does not occur. However, under irradiation the diffusivity is enhanced and the irradiation kinetics at 30OoC is about equal to thermal kinetics at 500°C. As a result, E Cu-enriched References: p. 1618.
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precipitate particles about 5 nm or less in size have formed at 288°C. These particles strengthen the steel and raise the ductile-brittle transformation temperature (StMONEN [1993]). Steels for containment vessels are now required to have extra low Cu contents. Another unexpected result of radiation is Cr depletion at grain boundaries in austenitic stainless steels, not connected with precipitation of Cr carbides. The depletion amounts of about 5 % Cr over a width of about 5 nm. The depletion correlates with increased intergranular stress-corrosion cracking. A major problem with austenitic stainless steels under neutron irradiation is development of cavities and volumetric expansion, called swelling. Ferritic/martensitic steels are less susceptible and can replace the stainless steels. Steels used in fusion reactors have problems in addition to swelling. They are exposed to very high energy neutrons. The alloying elements Cu,Ni, Mo, Nb and N are converted to radioactive isotopes with long half lives, intensifying the problem of waste disposal. New alloys, called low activation materials have been developed to replace the standard stainless steels with Mn substituting for Ni. Martensitic Cr steels with W added for strengthening appear to have [1994]). promise. This issues are treated in great detail in a recent monograph (FROST
6.2. Steels for electrical applications Steels for electrical and magnetic applications are sold on the basis of core loss, which is the amount of electrical energy dissipated as heat when the steel is magnetized by an alternating current. The reduction of core loss leads to greater efficiency in the generation, transmission and use of electrical power. Silicon is the most effective element for increasing the electrical resistivity of Fe which in turn reduces eddy current losses in alternating current magnetization. There are two types of Si steels, non-oriented and oriented. Non-oriented grades are low-C alloys containing 0.8 to 3.5% Si plus Al. They are used in less critical applications such as small motors and radio or television transformers. They are hot-rolled, coldrolled, annealed and decarburized to develop large, clean ferrite grains with no particular effort made to develop a favorable texture except perhaps by the addition of a solute such as Sb which can inhibit growth of certain grains during recrystallization. The easy direction of magnetization in bcc metals is ; to take advantage of this, grain oriented steels are made with in the rolling direction of the sheet and { l l O } planes in the plane of the sheet. When transformer cores are made, the rolling direction is put in the main flux path. Grain oriented silicon steels are the most sophisticated mass products of the steel industry. They are essentially high-punty F e S i alloys with a constant Si content of about 3.25%. Higher Si contents are not used because of difficulties in cold rolling. Production is begun by reheating slabs to the unusually high temperatures of 1315-1375"C, the purposes being to dissolve various compounds (MnS, AlN, BN, n C , MnSe, etc.) that are subsequently precipitated to serve as grain growth inhibitors and also to ensure that hot rolling is completed in the ferrite region. After cold rolling the steel is annealed to recrystallize and to remove C then annealed at about 115OOC in dry H2. The purposes of the second anneal are to develop the desired { 1lo} texture by secondary recrystallization in which most primary recrystallized
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grains are pinned by the inhibitors so that the favorably oriented grains grow at their expense and to remove the S, N, C or Se in the inhibitors by reaction with H,. Finally, the magnetic flux density and core loss can be improved by coatings that provide a smooth surface and a residual tensile stress to the very thin steel sheet (INOKUTI et al. [1993]). Si steels for transformer laminations are also treated in ch. 28, 0 4.4.2 and ch. 29, $4.2.
7. SolidiJication 7.1. Rimming steel, killed steel The solubility of oxygen in liquid steel decreases rapidly with decreasing temperature (fig. 52): Fe,+Q+ FeO. The product FeO reacts with C to form CO which evolves as bubbles in the liquid. When poured into an ingot mold this evolution of gas causes the steel to rise in the mold. The boiling action can be stopped by putting a steel cap on top of the mold - thus the term cupped steel. This procedure is often used for steels containing 0.15% C or more. If the evolution of gas, the “rimming action” is allowed to continue, the formation of honeycomb blowholes in the solidifying steel is sufficient to offset the shrinkage that occurs during solidification. The amount of gas evolved must be controlled so that a skin of solid metal forms close to the mold wall. This metal that solidifies first has about the same composition as the steel being poured but as solidification continues the remaining liquid becomes enriched in solute elements, especially S , P, C, Si and Mn. Rimmed steel has certain advantages for the manufacture of sheet products; the rim zone contains few non-metallic inclusions leading to high surface quality and ingot-to-product yield is higher than for killed steel. The term killed steel is applied to steels that have been deoxidized by addition of Al or A1 c Si to the ladle of liquid steel before the ingots are cast. Because there is little or no oxygen free to combine with C to form bubbles a shrinkage cavity called “pipe” forms at top of the ingot which must be cropped before hot rolling, thus the ingot-toproduct yield is reduced. Also, there is a greater number of non-metallic inclusions present in killed steel than in rimmed steel but less segregation of solutes. In rimmed steel the turbulence caused by gas evolution promotes segregation, so alloy steels are killed to obtain greater homogeneity. With the increased dominance of continuous casting in the production of steels and especially for sheet steels, rimmed and capped steels may soon fade into history. Gas evolution cannot be tolerated in continuous casting, so vacuum carbon deoxidation or deoxidation with a minimal addition of Al are employed. Vacuum carbon deoxidation is necessary for the production of ultra-low-carbon steels and for the removal of H,. Carbon is removed as CO. The H, content of the liquid steel is proportional to the square root of the partial pressure of H, in the vacuum chamber,
%[HI = k& The removal of hydrogen prevents the formation of internal cracks, called flakes, that occur during cooling from the forging or hot rolling temperature. References: p . 1618.
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Temperature
Fig. 52. Partial Fe-0 phase diagram.
7.2. Cast irons Cast irons are alloys of Fe with more than 2 wt% and usually 2.6 to 3.6 wt% C. By varying the composition, cooling rate and minor additions to the melt, the solidification can follow either the thermodynamically stable eutectic reaction liquid +y +graphite or the metastable reaction, liquid +y + Fe,C. During cooling through the eutectoid interval the structure can be the stable one, y + graphite +cr + graphite, or the metastable y + graphite+pearlite+graphite. These yield the common gray iron,so called because of the appearance of its fracture. If the eutectoid reaction follows the metastable path y + FeJ +pearlite+Fe,C, the result is white cast iron. Microstructures are shown in fig. 53. Which of these paths is followed depends upon the graphitization potential of the melt which in turn is determined by the C and alloy content of the melt, principally Si. Gray irons usually contain 1 to 3% Si. Silicon increases the activity of C, as commonly expressed by the carbon equivalent, CE= %C + 0.3(%Si) + 0.33(%P)- 0.027(%Mn) + 0.4(%S).Gray iron is widely used because of its low cost, strength in compression and high damping capacity. White irons contain from 1.0 to 2.0% Si. They can be annealed after casting at temperatures between 900-970°C to produce irregularly shaped particles of graphite called temper carbon, resulting in malleable iron. If slowly cooled the matrix is ferrite. More rapid cooling or subsequent heat treatments yield matrix structures of pearlite or martensite. Thus, a wide range of mechanical properties is available in these malleable irons. Ductile cast iron, also called nodular cast iron, differs from raalleable iron in that the graphite particles form as spheroids during cooling from the liquid (fig. 53). The matrix in the as-cast condition can be ferrite or pearlite or can be altered by subsequent heattreatment. Ductile iron has advantages over other cast irons in that the graphite spheroids form directly from the melt and no annealing is required as in the production of malleable iron. Strength and toughness are greater than in malleable iron and can be
Ch. 17, 57 . .
C)
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Physical metallurgy of steels L--
200x
d)
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Y
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Fig. 53. Microstructures of grey, white, and nodular cast iron: (a) graphite and ferrite; (b) graphite and pearlite; (c) cementite and pearlite; (d) nodular graphite and pearlite. (Courtesy J. MOTZ).
varied over wide ranges by heat treatment. The section size limitation on malleable iron does not apply to ductile iron. References: p. I61 8.
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In the production of ductile iron, two additions are made to the melt (SKALAND et al. [1993]). The first is the spheroidizing agent, Mg, added as ferrosilicon with 3 to 12% Mg. The Mg vaporizes and rises through the liquid iron, removing S from solution. The second is the inoculant, which provides nucleation sites for graphite spheroids. The inoculant is Ca, Sr or Ba, added in ferrosilicon containing Al. These elements form hexagonal silicates whose (001) basal planes provide interfaces with graphite that are coherent or semi-coherent with low interfacial energy. There are special grades of cast iron for particular applications; 15% Si provides resistance to corrosion, 11 to 28% Cr gives oxidation resistance, Ni provides a stable austenite matrix.
References AARONSON, H. I., 1962, in Decomposition of Austenite by Diffusional Processes, Interscience, NY, 387. ALBERTSEN, J. E,H. P. NIELSENand V.E BUCHWALD, 1983, Phys. Scripta, 27,314. ALEXANDER, D.J. and I. M. BERNSTEIN, 1989, Metallurg. Trans. ZOa, 2321. ANDREWS,K. W., 1965, J. Iron Steel Inst. 203,721. ATKINS,M., 1977, Atlas of Continuous Cooling Diagrams, British Steel Corp, 107. AYER,R., and P.M. MACHMEIER, 1993, Metallurg. Trans. 24k 1943. BIRCHENALL, C.E., 1951, Atom Movements, ASM, Metals Park OH, 112. BIRCHENALL,C.E. and R. J. BORG,1960, Trans. AIME, 218,980. BOEHLER,R., 1986, Geophys. Res. Lett. 13, 1153. BRENNER,S.S., 1956, J. App. Phys. 27, 854. BRIANT,C.L. and S. K. BANEWI,1978, Int. Metallurg. Rev. 23, 164. BRIANT,C. L. and R. L. ANDRESEN,1988, Metallurg. Trans. 19A,495. BRUCH,U. and E. HORNBOGEN, 1978, Arch. Eisenhiittenw. 49,357 and 409. CHIPMAN,I., 1973, Metallurg. Trans. 3, 55. COHEN,M., 1949, Trans. ASM, 41,35. CHRISTIAN, J. W., 1983, Metallurg. Trans. 14A,1237. COTTRELL, A.H., 1993, Mat. Sci. Tech. 9,277. DASTUR,Y.N. and W. C. LESLIE,1981, Metallurg. Trans. 12A,749. DAVENPORT, E. S. and E. C. BAIN, 1930, Trans. AIME, 90,117. DOANE, D.V., 1978, Hardenability Concepts With Applications to Steels, AIME, 351. D u a o s , E and B. CANTOR, 1978, in ‘Rapidly Quenched Metals III’ (ed. B. Cantor), London: Institute of Metals, p, 110. ELDIS,G.T., 1978, Hardenability Concepts With Applications to Steels, AIME, 126. ERA, H., M. SH’IMIZU and T. HIRUTA,1986, Metallurg. Trans. 17A,621. FLOREEN, S. and H. W. HAYDEN,1967, Trans. AIME, 239, 1405. FROST,B. R. T., 1994, Nuclear Materials (Vols. 1OA and 1OB of ‘Materials Science and Engineering’). GELLES,D. S., 1990, ISIJ Intern. 30,905. GENSAMER, M.., E. B. PEARSALL, W.S. PELLINIand J. R. Low, 1942, Trans. ASM, 30,983. GHOSH,A. K. and N. E. PATON,1978, AMMRC Report 7R 78-40. GILBERT, A. and W. S. OWEN,1962, Acta Metall. 10,45. GLOVER,S.G., 1956, in Mechanisms of Phase Transformation in Metals, Inst. Metals, London. GOLDSCHMIDT, H.J., 1967, Interstitial Alloys, Butterworths, London. GOODMAN, S. R., S. S. BRENNER and J.R. Low, Jr., 1973, Metallurg. Trans. 4,2363. GRANGE, R. A., 1966, Trans. ASM, 59,26. GRANGE,R.A., 1970, Proc. 2nd ICSMA,ASM, 861. GRANGE,R.A., 1973, Metallurg. Trans. 4,2231.
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Physical metallurgy of steels
1619
GRANGE, R. A., C. R. HRIBALand L. E PORTER,1977, Metallurg. Trans. 8 4 1775. GUTTMANN,M., Ph. DUMOULIN and M. WAYMAN, 1982, Metallurg. Trans. 13A, 1693. HASEBE, M., H. OHTANS and T. NISHIZAWA, 1985, Metallurg. Trans. 16A, 913. HILLERT,M., 1962, Decomposition of Austenite by Diffusional Processes, Interscience, NY, 197. HONDROS, E.D. and M.P. SEAH,1977, Int. Met. Rev. 22, 262. HORNBOGEN, E., 1964a, Trans. ASM, 57, 120. HORNBOGEN, E., 1964b, in High Energy Rate Working of Metals, Central Inst. for Industrial Research, OSIO, 345. HORNBOGEN, E., G. LUTGERING and M. ROTH, 1966, Arch. Eisenhuttenw. 37, 523. Hu, H., 1962, Trans. AIME, 224, 75. Hu, H., G.R. S ~ C and H R.L. MILLER,1976, AMMRC Report CTR 76-22. Hu, H., 1983, Metallurg. Trans. 14A, 85. INOKUTI, Y., K. S ~ and Y.I KOBAYASHI, 1993, ISIJ Intern. 33,957. INTERNATIONAL CONFERENCE ON BAMTE, 1990, Met. Trans. 21A, 767-889, 1343-1547. JACK,K.H., 1951, Proc. Roy. Soc. A208,200. JAFFEE, RI., T. OHHASI, 0. WATANABE and P. MACHNER,1989, Trans. Iron Steel Soc.,Feb., 45. JOHANNSON, H.,1937, Arch. Eisenhitttenw. 11,241. JOHNSON,J. W.,1960, J. Less-Common Met. 2, 241. JOMINY, W.E. and A.L. BOEGEHOLD, 1938, Trans ASM, 26,574. J~NSSON,B., 1992, Z. Metallkde. 83, 349. JUNG, Y. C., K. NAKAI,H. OHTSUBO and Y.OHMORI,1994, ISIJ International, 34, 43. KAJIWARA, S., 1986, Metallbrg. Trans. 17A, 1693. KEH, A. S. and W.C. PORR,1960, Trans. ASM, 53, 81. KEH,A.S., 1961, in Imperfections in Crystals, Interscience, NY, 213. KEH, A. S. and S. WEISSMANN, 1962A, in Electron Microscopy and the Strength of Crystals, Interscience, NY, 231. KEH,A.S., 1962b, Acta. Metall. 11, 1101. KEH, A. S. and W. C. LESLIE, 1963, Materials Science Res. 1, Plenum Publishing, NY, 208. KELLY,P. M. and J. NUTTING,1961, J. Iron Steel Inst. 197, 199. KLEUH, R. L., 1990, Metals Handbook, 10th ed.,ASM Intern, 1,653. KRAUSS, G., 1978, Hdenability Concepts With Applications to Steel, AIME, 235. KRAUSS,G., 1992 and C. J. MCMAHON,Jr., 1992, Martensite, ASM Intern, 295. LAKE, J. S. H., 1977, AIM Conference, paper 4A-2. LANGFORD, G., P.K. NAGATA, R. J. SOBERand W. C. LESLIE, 1972, Metallurg. Trans. 3, 1843. LESLIE,W. C., E. HORNBOGEN and 6.E. DIETER,1962, J. Iron Steel Inst. 200, 622. LESLIE,W.C., J.T. MICHALAK and F.W. AUL,1963, in Iron and its Dilute Solid Solutions, Interscience, NY 119. LESLIE,W. C. and R.L. MILLER,1964, Trans AIM, 57, 972. LESLIE,W. C. and A. S. KEH, 1965, in Mech. Working of Steel 2, Gordon and Breach, NY, 337. LESLIE,W.C. and J.R. SOBER,1967, Trans. ASM, 60,459. LESLIE,W. C., R. J. SOBER,S. G. BABC~CK and S. J. GREEN, 1969, Trans ASM, 62, 690. LESLIE, W.C. and G.C. RAUCH,1978, Metallurg. Trans. 9.4, 343. LEWANDOWSKI,J. J. and A. W. THOMPSON, 1986, Metaliurg. Trans. 17A, 461. LUDWIK,P., 1909, Elemente der technologischen Mechanik, Springer-Verlag, 32. MATAS,S . J., M. WILL and H. P. MUNGER,1964, Mech. Working of Steel 1, Gordon and Breach, NY, 143. MATSUI,H., S. MORIYA,S. TARAKIand H. K r m , 1978, llans. Jpn. Inst. Met. 19, 163. MEYER,W., and E, HORNBOGEN, 1968, Arch. Eisenhtittenw, 39,73. MICHALAK,J.T. and H. W. PAWN, 1961, Trans AIME, 221,850. MILLER,R. L., 1972, Metallurg Trans. 3, 905. MORRISON, W.B., 1966, Trans ASM, 59, 824. MURATA,Y., S. OHASHIand Y. UEMATSU, 1993, ISIJ Intern. 33, 711. NUTTING,J. W. and W. CHARNOCK, 1967, Met. Sci. J. 1,77 and 123. OHASHI,N., 1988, in Supplementary Volume 1 of the Encyclopedia of Materials Science and Engineering, ed. R.W. Cahn (Pergamon Press, Oxford), p. 185.
1620
KC. Leslie and E. Hornbogen
Ch. 17, Refs.
OHTAN,H.,M. HASEBE,K. ISHIDAand T. NISHIZAWA,1988, Trans. Iron Steel Inst. Jpn. 28, 1043. OWEN,W.S., 1954, Trans. ASM, 46,812. OWEN,W. S., 1992, Martensite, ASM International, 227. RAUCH, G. C. and W. C. LESLIE,1972, Metallurg. Trans. 3, 373. RICKETT, R. L. and E C. KRISTUPEK,1949, Trans. ASM 41, 1113. RIDLEY,N., 1984, Metallurg. Trans. EA, 1019. SAITOH,H. and K. USHIODA,1989, ISIJ Intern. 29,960. SATOH,A., J. M m and M. MESHI, 1970, Proc. 2nd ICSMA,ASM, 2,747. SATOH,S., S. OKADA,T. Urn, 0. HASHIMOTO, T. HANAZAWA and H. TSUNEKAWA, 1992, Kawasaki Steel Tech. Rept. No. 27, 31. SAXENA,S. K, G. SHEN and P. LAZOR,1993, Science, 260, 1312. SHA,W., A. CEREZOand G.D. W. SMITH,1993, Metallurg. Trans. 24A, 1221. SHANG,LT., and P. J. WRAY,1989, Metallurg. Trans.2 0 4 1191. SIMONEN, E.P., 1993, J. Metals, Dec., 45. SKALAND, T.. 0. GRONGand T. GRONG,1993, Metallurg. Trans. 244 2321. SPEICH, G. R., 1962, in Decomposition of Austenite by Diffusional Processes, Interscience, NY,353. SPEICH, G. R. and S. FLOREEN, 1963, Trans. ASM, 57,714. SPEICH,G. R., 1990, Metals Handbook, 10th ed., ASM Intern. 1, 423. SPEICH, G. R. and K. A. TAYLOR,1992, in Martensife (ed. G.B. Olsen and W.S. Owen), ASM Intemat., 243. SWISHER, J. H. and E. T. TURKDOGAN,1967, Trans. AIME, 239,426. TAKECHI,H., 1994, ISIJ International, 34, 1. Special Issue, Physical Metallurgy of Ultra-Low Carbon IF Steels. TAN, Y. H., D. C. ZENG, K.C. DONG,Y.H.HE and S. A. Hu, 1992, Metallurg. Trans. 23A, 1413. TOMITA,Y.and K. OKABAYASHI, 1985, Metallurg. Trans. 16A, 83. UWAKWEH,O.N.C., J-M. G ~ N I N and J-E SILVAIN, 1991, Metallurg. Trans. 2 2 4 797. VENTURELLO,G.C., C. ANTONIONE and E BONACCORSO, 1963, Trans. AIME, 227,433. WAYMAN, C. M., 1964, Introduction to the Crystallography of Martensitic Transformations, McMillan, London, 138. WINCHELL,P.G.and M. Corn, 1962, Trans. ASM, 55,347. YOUNG, C. M., L.J. A m m m and 0.D. SHERBY,1974, Metallurg. Trans. 5, 519. ZENER, C., 1955, Trans. AIME, 203,619. ZHOU,D. S. and G. J. SHTFLET, 1991, Metallurg. Trans. 22A, 1349. ZUIDEMA,B. K., D.K. SUBRAMANYAM and W. C. LESLIE, 1987, Metallurg. Trans. HA, 1629.
Further reading Bhadeshia, H. K. D. H., Bainite in steels, Cambridge Univ. Press, 1992. Decker, R. E and S. Floreen, Maraging Steels; Recent Developments and Applications, TMS-AIME, Warrendale, PA, 1988. Picker& F. B. ed. Constitution and Properties of Steels, VCH, Cambridge, U. K., 1992. Honeycombe, R. W. K. and H. K. D. H. Bhadeshia, Steels, Microstructure and Properties. 2nd Edn. Edward Arnold, London, 1995. Olson, G.B. and W. S. Owen, eds, Martensite, ASM International, Materials Park, OH, 1992. Leslie, W.C. Physical Metallurgy of Steels, Tech Books, Herndon, VA. 1991. Krauss, G. Principles of Heat Treatment of Steels, ASM International, Metals Park,, OH, 1980. Kot, R A. and B. L. Bramfitt, e&., Fundamentals of Dual-Phase Steels, TMS-AIME, Warrendale, PA, 1981. Doane, D. V. and J. S. Kirkaldy, eds., Hardenability Concepts with Applications to Steel, TMS, AIME, Warrendale, PA, 1978. Cramb, A. W. New Steel Casting Processes for Thin Slabs and Strip, Iron and Steelmaker, 15, July, 1988,45. Marder, A.R. and J.I. Goldstein, eds. Phase Transformations in Ferrous Alloys, TMS-AIME, Warrendale, PA, 1984. Metals Handbook, ASM International, 10th ed. Vol. 1, 1990. Progress of the Iron and Steel Technologies in Japan in the Past Decade, Trans. ISIJ. Vol. 25, No. 8, 1985.
CHAPTER 18
POINT DEFECTS H. J. WOLLENBERGER H a h n - M e i t n e r - I t Berlin GmbH, 0-14109 Berlin, FRG
R. W Cahn and I? Haasen?, eds. Physical MetaNurgy;fourth, revised and enhanced edition 8 Elsevier Science BK 19%
1622
H.J. Wollenbeeer
Ch. 18, 0 1
I . Introduction Point defects are lattice defects of zero dimensionality, Le., they do not possess lattice structure in any dimension. Vpical point defects are impurity atoms in a pure metal, vacancies and self-interstitials. This chapter covers the properties of vacancies and selfinterstitials, their interaction with other lattice defects, the production mechanisms and their importance for radiation damage of materials. Properties of atomic solutes per se are of interest mainly with respect to thermodynamics of alloys which are treated in chs. 5 and 6. Vacancies are produced simply by heating, in concentrations sufficiently high for quantitative investigations. To obtain similar concentrations of interstitials, they must be produced by doing external work on the crystal. Such work is done on an atomic scale by energetic particle irradiation. Collisions between the projectiles and lattice atoms cause displacements of atoms from substitutionalsites to interstitial sites. Thus, vacancies and interstitials are produced in equal numbers. As one vacancy and one interstitial together form a Freakel defect, irradiation is essentially a Frenkel defect production process. This is disadvantageous with respect to experimental research on interstitial properties, because radiation induced property changes of crystals always involve vacancy contributions. Vacancy formation is treated in $2.2.2 and Frenkel defect production in 5 3.1. Plastic deformation also produces vacancies and interstitials.Although its application is much less expensive than particle irradiation, it has not become a common procedure for point defect production because it does not allow a controlled defect production independent of complex dislocation networks. The significanceof point defects in plastic deformation is treated in chs. 21 and 22, and their recovery behavior after plastic deformation is touched upon in ch. 28. Anomalously high point defect concentrations occur in some non-stoichiometric intermetallic compounds. There, vacancies and interstitials clearly play the role of additional alloying elements and are of thermodynamicsignificance in this sense. A brief insight is given here in $ 2.3. Other methods such as rapid quenching, evaporation on cold substrates or laser annealing depend on thermally activated production, which is treated in the present chapter for controlled conditions which allow successful vacancy investigation. In pure metals and in the majority of alloys, vacancies provide thermally activated atom transport and, hence, vacancy properties directly influence the atomic transport (ch. 7). Vacancy properties give information on interatomic forces by means of specific perturbations due to the vacant lattice site. The interstitial is a very interesting defect because of the large lattice perturbation it causes. This large lattice perturbation and the related dynamic and static properties play a key role in the evolution of typical damage structures. These involve not only the microstructure but also the microchemistry of alloys. The properties of vacancies and interstitials are described in $5 2 and 3, respectively. In the late 1960s, a controversy arose on the interpretation of the generally observed radiation damage recovery. The question was whether a satisfactory explanation requires
a.18, 82
Point defects
1623
the existence of two interstitial species annihilating at quite different temperatures or needs only one interstitial species. Accordingly, quite different migration energies were assumed for vacancies and interstitials,respectively, by the advocates of the two different models. The reader is referred to the report on the panel discussion held at the Julich aE.~ ~ Conference on Vacancies and Interstitials in Metals (see further reading: S E E G E R [1970]) in order to get the flavour of the atmosphere of that period. Many papers in the literature give a lively impression. In the meantime, a large number of experimental results were reported and even completely new methods were applied. They certainly narrowed the scape of the controversy considerably. It would be impossible to outline within this article the controversy in terms of all experimental results which were claimed at the time to be of relevance. Instead of this, the scientific content and today's state of the controversy is briefly outlined in a dedicated section (8 3.3.2.5). The application of alloy materials in nuclear reactors, particularly in the core region with its high neutron flux density, causes radiation damage to be a technological problem, This is, in particular, true for the fast breeder reactor and the future fusion reactor. Therefore, 44 is dedicated to the basic mechanisms of point defect production and reactions under reactor irradiation conditions. As reactor irradiations are frequently simulated by heavy ion irradiation, the characteristics of this are treated as well. The interaction of swift heavy ions with metals has been included (0 4.3) in order to complete the description of the interaction of heavy ions with metallic materials although this topic is not directly related to reactor technology. Important property changes such as swelling, radiation-induced creep and radiation-induced atom redistribution are outlined in $8 4.4, 4.5 and 4.7. Irradiations under reactor conditions also demonstrated the pattern formation of vacancy clusters. This topic is treated in P 4.6.
2. Vacancyproperties 2.1. Theoretical background
The entropy of a defect-containing crystal is larger than that of a perfect crystal. Therefore the Gibbs free energy change resulting from changing the atomic-defect concentration c by Sc is given by:
6G = (AH'- TASf + k,Tlnc)&,
(1)
where A@ is the activation enthalpy of formation, & In c is the ideal entropy of mixing and Asf is that entropy change which arises in addition to k,lnc (excess entropy of mixing), and kB is the Boltzmann constant. In the close-packed metals, formation of vacancies requires the smallest amount of energy when compared with other lattice defects. Hence, they are dominant in thermal equilibrium. The entropy change A!,? is mainly due to the change of the phonon spectrum of the crystal by the introduced defects. For the high-temperature harmonic approximation one obtains:
References: p . I7lO.
1624
H.J. Wollenberger
a.18, 9 2
where ogiand oiare the eigenfrequencies of the crystal without and with vacancies, respectively. The phonon spectrum is changed by the change of the atomic coupling for the nearest neighbors of the vacancy and by the overall lattice volume change. The entropy change A$ is positive if on average the oiare lowered and negative if they are enhanced. The simplest model assumes nearest-neighbor interaction by a “spiral spring” with force constantf and calculates the change of the Einstein frequenciesof the nearest neighbors of the vacancy by removing one of its twelve coupling springs (in the fcc lattice). For the = 4 f / M ( M =atomic mass) is changed to =3f/M. vibration towards the vacancy, Vibrations perpendicular to the removed spring are unchanged. For all other atoms all eigenfrequencies are unchanged. Hence we obtain AS: = f(12 k, In 4/3) = 1.73 k,. Refinement of the calculation must take into account the static atomic relaxations around the vacancy (DEDERICHS and ZELLER[1980]). These change the force constants for many more atoms than the nearest neighbors. Correct consideration of this effect seems to be difficult. A negative sign of the relaxation contribution to the formation et al. [19641 and a positive one by BTJRTON entropy change was obtained by SCHOTTKY etal. [1964] to be 0.49 & for Cu, 0.92 [1971]. In total, AS,! was obtained by SCHOTTKY kB for Ag and 1.22 k, for Au, whereas BURTON[1971] obtained 1.8 kB-2.0 kB for fcc and 2.2 kB - 2.4 kB for bcc metals. The sensitivity of the result to the number of atomic shells considered as contributing to the frequency redistribution has been investigated systematically for different pairinteraction potentials by HATCHERetal. [1979]. In the model crystal about 5 x lo3atoms around the vacancy were allowed to relax statically and for 102 to 5 x Id atoms in the central region dynamical displacements (vibrations) were allowed. The pair interaction was simulated by the Born-Mayer potential introduced by GIBSONet al. [1960] for computer simulation of Cu (see also 0 3.3.1) and by the Morse potential introduced by [1966]. This last potential was smoothly cut-off at r = 1.2a, COTTERILL and DOYAMA where a is the lattice-constant by HATCHERet al. [19791. It then fits the observed lattice constant, bulk modulus and enthalpy of vacancy formation for Cu. The entropy of formation was found to depend approximately linearly on the inverse of the number of atoms allowed to vibrate. The relationship was followed up to the 19th atomic shell around the vacancy. By extrapolation, ASfv = 2.3 k, was obtained for the Morse potential and 1.6 k, for the Born-Mayer potential. The influence of the static lattice relaxation on the magnitude of AS; is demonstrated by comparing the above with 2.28 k, for the Morse potential and 2.52 k, for the Born-Mayer potential; both values are for the unrelaxed atoms (perfect lattice around the vacancy). HATCHER et ai. [1979] have also found AS;. = 2.1 kBfor a potential simulating a-Fe. The cut-off Morse potential describes the formation and migration properties of mono- and di-vacancies in Cu reasonably well. The results are unsatisfactory for the Born-Mayer and the cited long-range Morse potential (COTTWILLand DOYAMA[1966]). For the hcp Mg, ASf, = 1.5 kB-2 k, was found by using an empirical potential ( M o m and SAVINO[1981]).
Ch. 18, $ 2
Point defects
1625
Discrepancies between the results from different potentials are particularly large for the relaxution volume AVE', i.e., the volume change of a crystal caused by removing one atom from the interior and withdrawing it from the crystal. The quantity is measured by the lattice-parameter change. The Born-Mayer potential for Cu yields -0.47 atomic volume, whereas the Morse potential yields -0.02 atomic volume for both the extended and the cut-off version (DEDERICHS et ul. [1978]). With AH[, =-0.41 eV, the Born-Mayer potential fails to give a reasonable enthalpy of formation. With the corresponding values +1.17 eV and +1.29 eV for the extended and the cut-off Morse potential, respectively, satisfactory agreement with experimental values is obtained. Ab-initio calculations of vacancy formation enthalpies are obviously problematic. Reviews are by FREDEL[1970], EVANS[1977], HEALD[1977], and STOTT[1978]. The problem consists of calculating the energetics of the electron system for a vacancy containing crystal. Only so-called simple metals have been treated. It had to be assumed that the core electrons are rigidly confined to the nucleus and any charge redistribution caused by the missing atom core only affects the conduction electrons. For transition metals the vacancy effect on the outermost d electrons has not been described successfully as yet. For the simple metals two entirely different approaches have been applied. The first consists in replacing the vacancy by a repulsive impurity potential acting in a free electron gas (jellium). The second approach is based on pseudo-potential theory and assumes the vacancy effect to be weak enough to follow a linear response formalism. The results of both approaches are unsatisfactory in terms of agreement with experimental data. Reasons are the improper treatment of the effect of the local charge redistribution as well as of that of the lattice relaxation around a vacancy on the electron system. More promising seems to be the combination of selfconsistent cluster calculations (quanturn-chemical approach) with lattice defect calculations (see ADAMSand FOILES[1990]). Vacancy migration has been treated by analytical methods (FLYNN[1968]) as well as by computer simulation (BEELER[1983]). The model developed by FLYNN[1968] predicts activation enthalpies for the vacancy migration AH: in remarkably good agreement with experimental data for quite a number of metals. The model relates AH"''' to the elastic moduli. According to the author, the jumping atom passes the saddle point by means of a fluctuation of the kinetic energy which was picked up near the atom's equilibrium position where the kinetic energy takes maximum values. There, however, the atomic movement can be described by the harmonic approximation.
2.2. Experimental methods and results
2.2.1. Introductoryremarks Vacancies are #ofatomic size and therefore difficult to observe directly. However, agglomerates of numerous vacancies are visible even with a naked eye. Such agglomerates are formed in suitable diffusion samples owing to the Kirkadall eflect (SMIGELSKAS and KIRKENDALL119471). Two metals A and B are brazed to each other and then thermally annealed at high temperatures. After sufficient atom transport, one observes that the two elements diffuse into each other, but with different velocities. As a conseReferences: p. I710.
1626
Ch. 18, $ 2
H. J. Wollenberger
quence, more atoms leave the fast diffuser crystal than slow diffusing atoms arrive there. The crystal responses to this migrational loss of matter by forming voids as shown in fig. 1. The quantitative evaluation of such experiments moreover prove that the migration of single vacancies is the origin of the atom diffusion (ch. 7, 0 5.3.1.1). Field ion microscopy does resolve the atomic structure of crystals and single vacancies have indeed been imaged (SEIDMAN[1973]). But these are vacancies at the surface or just beneath a surface and might not prove the existence of this defect within the bulk of a crystal. The volume change of a crystal due to vacancy formation (0 2.2.2.2) seems to be the most direct measurable change caused by single vacancies. Other properties are treated in $Q 2.2.2.3 and 2.2.2.4. 2.2.2. Enthalpy and entropy of formation 2.2.2.1. Single vacancies and di-vacancies. For thermal equilibrium one obtains from eq. (1): c: = exp(AS,'/k,) exp(-LWf/k,T).
(3)
The determination of enthalpy and entropy of formation according to this equation requires the determination of equilibrium vacancy concentrations c,". ' h o different procedures are applied: either measurement of c, at high temperatures such that c, is indeed in thermal equilibrium during the measurement, or quenching of the sample from high temperatures to low temperatures such that the defects are immobile. The latter method avoids difficult measurements at high temperatures but requires considerable efforts to avoid vacancy losses during quenching. Review articles are by HOCH [1970], BALLUFFIet al. [1970] and SIEGEL[1978]. Even the most careful measurement of c," at high temperatures cannot yield the wanted formation enthalpy AH[,and entropy AS; of the monovacancy because c," does - -
I _ _ _
- *$,Y
Fig. 1. Cross section of a diffusion couple Cu (above) against Ni (below). Thin black line represents the Cu-Ni interface before the diffusion annealing. Black dots near the interface line are cavity images.
Ch. 18, $ 2
not
1627
Point defects
necessarily
involve
E=,
only
monovacancies. Actually
it
is
given
by
0 0 = clV + 2c,,0 +... = ne,,, where c i is the equilibrium concentration of clusters consisting of n vacancies. For equilibrium concentration measurements, higher aggregates 0
c,
than di-vacancies need not be considered owing to their negligibly small concentration in the normal cubic metals. We then have in the dilute solution approximation: c," = clv 0 + 2c& = exp(ASi/k,) exp(-AH,i/k,T) + 2g%exP[
(2As,f,- Asiv)/kB]
"P[
(miv
- 2mk)/kBT]s
(4)
where A$., and A& are entropy and enthalpy of binding of the di-vacancy, respectively. For the di-vacancy formed by two vacancies at nearest-neighbor sites in the fcc structure, the ageometry factor becomes gZv=6.In bcc and hcp structures different divacancy configurations may exist simultaneously with non-degenerate values of g,,, AS; and AH;. Up to now, there is no reliable method of measurement which separatesci 0 and c,, .Hence, the formation enthalpy as derived from the temperature dependence of c," is an effective one: d In c,"(T) AH,f = - d(l/k,T)
where G denotes the Gibbs free energy. For bcc and hcp metals, eq. (5) becomes much more complicated due to the possible Occurrence of different di-vacancy configurations (SEEGER[1973a]). Ekpation (5) introduces a temperature dependence of AH! which must be considered whenever A& is determined over a large temperature range. Binding enthalpies of the di-vacancy have been calculated for Cu,Ni, Au, and Pt to amount to 0.08, 0.07, 0.08, and 0.11 eV, respectively (KLEMRADT et al. [1991], 0 2.2.5). Such values would not lead to measurable deviations from the Arrhenius behavior. 2.2.2.2. Dfierential dilatometry, The classical method of direct c," determination is difSerential dilatometry (WAGNERand BEYER[1936], SIMMONS and BALLUFFI[1960a, b, 119561, which relates the microscopic 1962, 19631). It is based on a theory by ESHELBY volume change A V r of randomly distributed dilatation centers (point defects) to the total volume change of the defectcontaining crystal as compared to the perfect crystal. The total crystal volume change consists of the Eshelby contribution plus that caused by adding to the total crystal volume the; atomic volumes of those atoms which are taken out of the interior of the crystal in order to produce vacancies. The crystal volume change can be measured by the length change, Al/E, of the sample for cubic crystals and the average microscopic (lattice cell) change, A d a , by the lattice parameter change of the same sample. One obtains: C,
= 3(Al/Z
- Au/u),
(6)
Rderences:p . 1710,
1628
Ch. 18, $ 2
H.J. WoUenbever
which is correct for cubic crystals and the small vacancy concentrations, usual in such metals. Precise measurements of this type are extremely difficult. For illustration of the [196Oa] are orders of magnitude to be measured, the A1 data by SIMMONSand BALLUFW shown in fig. 2. From these data the authors derived AH: = 0.76 eV. Since then AH: has frequently been determined also by means of the methods of measurement described below with the result of values ranging from 0.60 eV to 0.77 eV with 0.67 f 0.03 eV as [1978]. Recommended values for the other metals are given recommended by BALLUFFI [1991] and SCHULTZ in table 1. A complete collection of data was given by EHTWART [1991al. The relaxation volume is given by AVv?/fl =3Aa/c: (where R is the atomic volume). The diffuse X-ray scattering cross section (DXS, $3.3.2.2.) is given by the product of c, and the square of the scattering amplitude caused per vacancy (DEDERICHS [1973]) for low vacancy concentrations and random spatial distribution. The scattering amplitude is essentially given by the elastic moduli and the dipole force tensor which describes the atomic displacements resulting from the insertion of the vacancy. ASAV,? is determined by the same quantities and can be derived from A d a measurements as shown above, the combination of DXS and Aa/a measurements allows the evaluation of both, c, and AVvm' (EHRHART etal. [1979]). The DXS - Aa/a measurements yielded AV:' =-0.15 il for gold when measured at 4 K. The DD method yielded -0.5 R instead (SIBADIONS and B A L L U P[1962]). ~ EHRHART etal. [1979] trace this difference back to a potential temperature dependence of AVvml.For other metals, AV:.' data are given in table 1. (See also ch. 12, 03.3). The third method determines the volume of vacancy formation AV;'= AVfl+CI by measuring the quenched resistivity increment (as a measure for c," at the quenching temperature Tq) as a function of temperature and hydrostatic pressure. The order of magnitude of the pressure dependence is such that for gold, a 6 kbar pressure increase corresponds to a temperature decrease of about 30 K around 900 K for constant c,". The way of determining AVvmffrom (Aa/a), and Ap, measurements on quenched
h.
L
q
'6
; m
14
e4 U
'2
lo
400
450
500
550
600
650
T 1°C Fig. 2. Differential dilatometry in vacancy equilibrium conditions for aluminium.I W L is the relative length change of a sample, Aa/a is the relative lattice-constant change. (AfterSIMMONS and BALLIJFFI [196Oa].)
Ch. 18, $ 2
1629
Point defects Table 1 RoDerties of vacancies in uure metal.
Glossary: Property symbol
Definition and units
mrn
vacancy concentration at the melting temperature (x )'01 Activation entropy of formation (units of Ig), refemng either to single vacancy values or to effective values depending on data evaluation Activation enthalpy of formation (in eV) refemng to single-vacancy or effective values, depending on data evaluation Relaxation volume of the vacancy (in atomic-volume units) Electric resistivity contribution per unit concentration of vacancies (in units of IO+ a m ) Activation enthalpy of migration (in eV)
Measurement symbol
Method of measurement for data evaluation
DD
Differential dilatometry Electrical resistivity of quenched,irradiated and cold worked samples respectively Qansmission electron microscopy Positron-annihilation spectroscopy Diffuse X-ray scattering Nuclear magnetic resonance Muon spin rotation experiments Thermal helium desorption spectroscopy
SIEGEL[1982a] DOYAMA and KOEHLER[I9621 BALLUFFI [1978]
Ag (mer,fee)
C"(Td A$
I\Hf
1.11 f 0.05
P"
1.3
Aft
0.66 f 0.05
AI (aluminium,fee) c&)
a @"
Av? P Y
AH?
9.4 0.7 0.67 zk 0.03
DD, APp4 DD, 4op4
-0.05 1.o
DXS
7.2 0.72 0.93f 0.04 -0.15
DD
1.70
%
0.61 f 0.03
SIEGEL[1982b] SIEGEL[1978] BALLUF~ [ 19781 SEGEL[1978] S~ILGFER etal. [1987a] EHRHART et d. [1986] SIEG!X [1978] SIEGSL [1982b]
An (goM,fee) dTd
as: I\Hf
AV?
P.
A e
0.71 f 0.05
Dpg, HDS, Aula, valid at 4 K
SIMMONSand BALLUF~ [1%2] SIEGEL [1978] SIEGEL[1982a] EHRHARTet aZ. [1979] EHRHART et d. [1979]
SONNENBERGand DEDEK[1982]
References:p. 1710.
1630
propero,
Ch. 18, $ 2
H. J. Wollenberger
Value
Measurement
Reference
DD DD DD PAS, APpirra
MATSUNO [1977] Mnrmo 119771 M n m o [1977] CORBEL et al. [I9871
DD DD PAS PAS, pressure dependence
JANOT and GEORGE [1975] JANOT and GEORGE [1975] EHRHART[1991] DICKMANet al. [1978] EHRHART [1991] EHRHART
Be (beryllium, hcp) 0.8
w
Bi (bismuth, rhombohedral) COM) 6.17 1Ls: 0.3 .Iwt 0.35 AH? 0.5 Cd (cadmium, hcp) 4.5 AS: 0.5 @" 0.46 f 0.05 A V -0.54 0.4 f 0.05 cV(TM)
w
Co (cobalt, hcp)
Aq'
-0.05
DXS,valid at 4 K
AH?
0.72
APiu
Cr (chromium, bcc) M" 2.0 f 0.2 -0.95
w
Cu (copper, fcc) C"(Td 2.1 f 0.1
AS: Aft Avt" P"
A e
2.8 1.6 1.28 f 0.05 -0.25 0.62 0.70 f 0.02
and SCHONFELD [1982] DANDERand SCHAEPER119771
PAS
LOPER etal. [1985] SCHULTZ [1991]
DD
DIPPERT et al. [1987] BERGER et aL [1979] DIFFERTet al. [I9871 SCHAEFER etal. [1987b] EHRHART er al. [1986]
% DD
DXS, low tempratwe value 4%' DD
BERGER et al. [1979] BALLUFFI [1978]
Fe (iron, bcc)
AT'
-0.05
mv(fern)
1.79 f 0.1 1.85 f 0.1 1.59 - 1.73; 0.55 1.11 f 0.2
Lwt (Pa) AH? (ferro)
In (indium, tetragonal) A$ 5 0.52 f 0.04 A T -0.6 (0.31)
a
w
K (potassium,bcc) 0.34
AH? 0.038 Li (lithium, boe) @"
AH:
0.48 0.038
A d a , AT DXS
EHRHART [1985] DE SCHEPPBRetal. [1983] S m w et al. [1989] pSR, ferromagnetic state FURDERER et al. [I9871 TAKAKI et al. [1983] e"-* S ~et al. [19891 I PAS PAS
PAS PAS Q,""-H,'
QY-W b i r r
Q:"-w
WEILER and SCHAEFER[1985] EHRHART [1991] DICKMAN et al. [1977] WEILER and SCHAEFER [1985] WALLNER ef al. [1986] WALLNER et al. [1986] SCHULTZ [1991] WALLNER [I9831
and NOWICK [1967] LEADBETTER et al. [1966] FED= and NOWICK[1967] ITO et al. [1982], HAM and NANAO[1982] KNODLEand K O ~ E[19781 R SCHROEDER and SCHILLING [I9761
Pt (platinum, fcc) CdTM) 9.4 ut 0.4 M" 1.35 f 0.05 A F -0.28 P" 5.75 AH? 1.43 f 0.05
DD 4099
FEDER
DD
%
4%
4%
FIM,Apq, upper limit
Re (rhenium, hcp)
AH:
2.2 0.5-0.6
VANDENBOWet al. [1968] VANLXENBORRE et al. [1972] References: p . 1710.
1632 ~operty
Value
Measurement
Reference
e""-H",
DE D m ef al. [I9871 EHRHART[1991] DE DIEGO et al. 119871
Sb (antimon, rhombohedral) CLT, 10 @"
w
Ch. 18, Q 2
H. J. Wollenberger
1.4 -0.3
PAS
Sn (tin; P-Sn (for T 2 13'C): tetragonal, bct (As)) A$ 1.1 DD, PAS cVVd S 0.3 DD @" 0.51 PAS 0.3-0.4
BALZER and SIGVALDASON [1979] BALZERand SIGVALDASON [1979] SHAH and CAT2 [1984] MCILWAINet al. [1975]
Ta (tantalum, bee) 3.1 0.7
Q?"-W
SCHULTZ [I9911 FABERet al. [19741
PAS
KIMef al. [1984] KIM ef d. [1984]
w
a LsHEl
Th (thorium, fcc)
Aft Aff?
1.28 f 0.2 2.04
V (vanadium, bce) @" 2.2 f 0.4 0.7 0.5
w
e""-# PAS, life time
Q;"-w, PAS
JANOTet a!. [1982] ,%KJLTZ [1991] SCHULTZ 119911
valid for low T
PARK et d. [1983] PARK ef U l . [1983] PARKet al. [1983] PARK et Ul. [1983] MUNDYetal. [1987]
W(twWh P" C,(TM)
U"
Aft
w
7 3 3.2 3.6 f 0.2 1.70 f 0.1
Zn (zinc, hcp)
c,(TM)
G A@
AF' Aff:
5 f 0.3 If1 0.54 f 0.03 -0.6 0.42 f 0.02
DD DD
DXS
Zr (zirconium, hcp) W" > 1.5
PAS
AF'
-0.05
DXS,valid at 4 K
0.54-0.62 0.6-0.7
PAS,
w
bi,
BALZER and SIGVALDASON [1979] BALZERand SIGVALDASON [1979] EHRHART [1991] EHRHART and SCHONFELD 119791 EHRHART [1991] HOODri986i EHRHA~T and SCHONFELD [1982], EHRHART et a[. [1986] NEELY119701 HOODet al. [1984], HOODand SCHULTZ[1988]
Ch. 18, $ 2
Point defects
1633
samples must be based upon well known pv values. As these are obtained by use of absolute c," data from DD measurements this method goes back essentially to DD data but does involve all quenching problems. For the bcc metals AVvrervalues do not exist because of difficulties in determining absolute c," values (SCHULTZ[1991]). For hcp metals, similar values have been reported as for the above-quoted fcc metals. 2.2.2.3. Positron-annihilation spectroscopy. The second common method of measuring vacancy concentrations in thermal equilibrium is positron-annihilation spectroscopy (PAS). Review articles on PAS application for studies of vacancy properties [1973b], DOYAMA and HASIGUTI[1973], TRIFTSHAUSER [1975], SIEGEL are by SEEGER [1978], M m W m [1979], WEST [1979] and HATJTOJARVI[1987]. High-energy positrons injected itlto metal crystals are rapidly thermalized by electron-hole excitations and interactions with phonons. The thermalized positron diffuses through the lattice and ends its life by annihilation with an electron. The lifetime depends on the total electron density occurring along the diffusional path of the positron. Vacancies obviously trap positrons in a bound state, and because of the missing core electrons at the vacant lattice site, the local electron density is significantly reduced. This condition causes the lifetime of trapped positrons to be enhanced by 20430%as compared to that of free positrons in the perfect lattice. Consequently, positrons in a vacancy-containing crystal end their lifes by annihilation either as free positrons or as trapped positrons. Critical discussion on [1991]. The lifetimes for non-thermal trapping was given by mum and HEHENKAMP both fates are different and the probability of trapping is proportional to the vacancy concentration. Lifetime measurements are possible as y-quanta are emitted at the birth of a positron as well lils at its decay. Fortunately, thermalization happens within about one picosecond whereas the average lifetime in the metal crystal is in the order of 200 ps. With PAS, infopnation is extracted not only from the positron lifetimes but also from the angular correlation between the directions of the emitted annihilation y-rays and from the Doppler broadening of these y-rays. These quantities yield information on the net momenta of the annihilating electron-positron pairs. They allow a distinction to be made between annihilations with the higher-momentum core electrons and the lower-momentum valence or cohduction electrons. Vacancy conceqtrations are determined by means of lifetime spectrum measurements as well as rnomeqtum techniques (angular correlation and Doppler broadening). The former avoids additional assumptions on positron-annihilation parameters but requires high-resolution m&suring techniques and expanded data deconvolution. Although the deconvolution of momentum-distribution data requires critical assumptions on the temperature dependence of positron-annihilation parameters, these methods have become more popular for AH: determinations. In fig. 3, vacancy concentrations derived from angular correlatioq measurements are shown for Cu and Au. The range of measurement by differential dil#ometry is indicated for comparison. The PAS data extend to about two orders of magnitude lower vacancy concentration than the differential dilatometry data. As monovacmcies certainly predominate in this concentration range, PAS studies are of great importance as a complement to differential dilatometry on the one side, and to resistivity measbrements in quenched samples on the other side. I
References: p. I710.
1634
Ch. 18, $ 2
H.J. Wollenberger T 1°C
I
lo00
"
600
800 I
I
I
500 I
400
-3
-5
l@/T / K-' Fig. 3. Arrhenius plot of the vacancy concentration as derived from positron annihilation spectroscopy for gold (full symbols) and copper (open symbols) according to TRIFTSHKUSER and MCGERVEY[1975]. The left hand side of the arrows represents the range covered by the differential dilatometry.
23.2.4. Resistivity measurements after quenching. All problems of measurements at temperatures near the melting point can be avoided if equilibrium vacancy concentrations can quantitatively be quenched from such temperatures Tq to temperatures which immobilize the vacancies. Properties which are proportional to the vacancy concentration could then be measured on quenched and unquenched samples for direct comparison. This idea appeared so challenging that large efforts were put into the development of quenching methods following the pioneering work by KAUFFMANand KOEHLER [1952]. Methods and problems are reviewed by BALLUFFI et al. [1970] and SIEGEL [1978]. There are indeed problems inherent to the quenching process. During quenching, the vacancies are still highly mobile in a significant part of the total temperature interval being passed through. The migrating vacancies are able to react with other defects or with one another, with the following consequences: (i) Vacancy losses to sinks such as dislocations, grain boundaries and surfaces. The quenched concentration c, will therefore be smaller than c,"(Tq).The inhomogeneous distributions of sinks will cause locally varying concentrations. (ii) Vacancy clustering, which causes repartitioning of the cluster size distribution existing at Tq.High-order clusters are favored in comparison to the equilibrium distribution at T4' By modelling the vacancy reaction scheme for the conditions of quenching, vacancy ef al. loss and repartitioning of cluster sizes were studied in great detail (BALLUFFI [ 19701). With the aid of such calculations, quenching results obtained for systematically varied quenching rates could be corrected and extrapolated to infinite quenching rates in the case of Au (FLYNNet al. [1965]). Systematic studies of this kind were performed also on A1 (BASS[1967] and BERGER et al. [1978]). An entirely different approach was successfully followed by LENGELER [1976] and
Ch. 18, $ 2
1635
Point akfects
LENGELER and BOURASSA [1976]. They were able to rapidly quench single crystals of Au and Cu with dislocation densities so low that vacancy losses are negligible even for high Tq values. The effect is demonstrated in fig. 4. Quenching data were reviewed by BALLUFFI[1978].
2.2.3. Activation enthalpy of migration 2.2.3.1. Problems of methods of determination. The common way of investigating the migration properties of vacancies consists of quenching a sample from high temperatures and subsequent annealing at increasing temperatures in order to induce the diffusion-controlled annihilation of the excess vacancies at the annealing temperature (further reading: Seeger et aE. [1970]). The residual resistivity has mostly been taken as a measure of the vacancy concentration in the lattice. This property is certainly preferable to all others in view of its convenient application and high sensitivity. The isochronal recovery behavior of a pure metal containing point defects is sketched in fig. 5. The resistivity increase recovers in distinct steps which are labelled according to VAN BUEREN[1955]. Stage I is observed in irradiated samples only. Stage II occurs in irradiated and plastically deformed samples. Quenched samples show resistivity recovery at temperatures varying from stage III to stage IV, depending on the individual metal. Stage Y is caused by recrystallization (ch. 28) and occurs in all samples which contain larger defect agglomerates. As the first recovery stage of quenched samples does not
I
8
9
IO
I II
I 12
10 4/r I K-’
Fig. 4. Arrhenius plot of the quenched-in residual resistivity of gold. The influence of the dislocation density Ndis shown by the curve for quenched polycrystals. (After LENGELER[1976].) References: p . 1710.
1636
ch. 18, 82
H. J. Wollenberger
100
90 80 70 60
50
40 30
20
0 20
40
60 00
200
400 600
Fig. 5. Isochronal recovery curve of the electrical resistivity of Cu electron irradiated at 4 K.
generally coincide with stage III of irradiated samples, a straightforward classification of this stage was not possible. This situation created the stage ZZZ controversy (0 3.3.2.5). The resistivity method is not appropriate for deducing information on details of the underlying defect reactions. Its inability to indicate secondary vacancy reactions like clustering and trapping at impurities beside annihilation is one of the origins of the stage I11 controversy. It was not before the mid seventies that other methods such as PAS or pertarbed y-y angular correlation (PAC) came up, which allowed a quantitative separation of different vacancy reactions. A breakthrough was achieved by applying PAS in order to determine the temperature at which vacancies become mobile. Another significant step was taken when the ability of nuclear probe atom methods such as Mossbauer spectroscopy and PAC to distinguish between interstitials and vacancies was discovered. As many as 21 experimental results are listed by BALLUPPI [1978] which favor stage III interpretation by vacancy migration. Since then methods such as the perturbed y-y angular correlation measurements have been applied fruitfully. Their results also favor [1982]). the vacancy interpretation (see, for example, PLEITERand HOHENEMSER 2.2.3.2. Two selected pieces of evidence for vacancy migration in stage 111. Among the earlier results, the positron-annihilation behavior in stage III seems especially conclusive with respect to vacancy migration id that stage. The important quantity is the - Z,”)l The energy of the positronso-called lineshape parameter R=l(Z: - Z,f)/(Z,‘ annihilation y quanta is Doppler-broadened because of the non-vanishing momenta of the
Ch. 18, $ 2
1637
Point defects
annihilating electron-positron pairs. The center of the curve (intensity versus energy) is caused by low-momentum pairs, i.e., involves conduction and valence (v) electrons, whereas the branches arise from high-momentum pairs, i.e., involve core (c) electrons. The quantity I is simply a suitably broad section of the Doppler line integral, one below the maximum (I,)and the other below the branches (13. The upper subscripts t andf refer to trapped and free positrons, respectively, as obtained from the irradiated and unirradiated state. The lineshape parameter R was found to be considerably larger for positrons trapped by voids than for those trapped by single vacancies (MANTLand TRIprs~iiusER[1978]). For trapping by dislocation loops its magnitude is close to that for single vacancy trapping. The resuIt from measurements in electron-irradiated Cu through stage III recovery is shown in fig. 6. The corresponding resistivity-decrease between 200 K and 300 K amounted to about 85% of its value at 200 K. The increase of R indicates a further fractional decrease of core electron density as compared to the presence of merely single vacancies below 200 K. This can only be visualized by vacancy agglomeration in three-dimensional arrangements. The effect of interstitial-type dislocation loops which could be formed in stage III according to the two-interstitial model was found to produce R = 0.62 f0.03 in neutron-irradiated aluminium (GAUSTER et al. [ 19751). Interstitial agglomeration can therefore be excluded as an origin of the observed R enhancement. [ 19781 assume the three-dimensional agglomerates to be the MANTLand TRIFTSWAUSER nuclei of the vacancy loops observed at higher temperatures by electron microscopy. The same behavior of R as in Cu has been observed in the bcc Mo whereas in A1 the and TRIFTSobserved increase in R did not significantly exceed the error limit (WNTL K~~USER [1978]). Fortunately, the PAC technique indicated the arrival of a vacancy type defect in stage III at the probe atom Cd in A1 (RINNEBERGetal. [1978], RINNEBERG and HAAS [1978]). The PAC technique essentially measures the hyperfine interaction of the probe-atom nucleus with the electric crystal field. Interstitials or vacancies situated at the nearestneighbor sites of such probe atoms obviously cause sufficiently large field gradients to produce measurable signals. Review articles on this type of application of PAC are by [1982], WICHERT[1982] and NIESEN [1981]. The PAC signals PLEITER and HOHENEMSER are determined by the electric field gradient at the site of the probe-atom nucleus and are I
0.5I 30
I
'
I
100 200 300 500 Annealfng temperature I K
50
Fig. 6. Lineshape parameter R of low-temperature electron-irradiated Cu upon isochronal annealing (after MANTLand TRIFTSHAUSER 119781). References: p. 1710.
1638
H.J. Wollenberger
Ch. 18, 92
in that sense specific for the type of defect which is placed at a nearest-neighbor site. If it happens that the probe atom in a given host metal traps the vacancy, a quenched sample containing probe atoms will exhibit the PAC signal caused by the vacancy. Similarly one obtains the signal caused by the interstitial when one irradiates the sample at low temperatures and anneals through stage I or irradiates in stage II. In this way one obtains the “fingerprints” of vacancy and interstitial. Clear evidence for the arrival of vacancies at the probe atoms in stage I11 has been obtained for Al, Ag (BUTT et al. [1979], DEICHER et al. [1981a]), Au (DEICHER et al. [1981b]), Cu (WICHERT et al. [1978]), Pt (MULLER [1979]), Cd and Zn (SEEBOECK et al. [1982]). The PAC signal also gives information on the deviation of the electric field-gradient tensor from axial symmetry. When measurements are performed in single crystals, the tensor orientation with respect to the crystal orientation can be obtained. Knowledge of orientation and deviation from axial symmetry provides valuable criteria for answering the question whether mono-, di-, or tri-vacancies decorate the probe atom and cause the observed signal. Indeed, the PAC technique has resolved a number of different configuret al. [1983]). The variety of ations for the metals quoted above (review by RECKNAGEL defect arrangements trapped at In in Au after different treatments is illustrated in fig. 7. From the field-gradient tensor properties, it has been concluded that defect 3 is a monovacancy at a nearest-neighbor site, defect 4 is a planar vacancy loop and defects 1 and 2 are multiple vacancies in different arrangements (DEICHERet al. [1981b]). It should be kept in mind that the spectrum of visible vacancy arrangements does depend on the nature of the probe atom because of the required trap property. Nevertheless, the variety of arrangements to be seen in fig. 7 gives a good impression of the complexity of the stage 111 processes which, in the earlier controversy, were often identified with either monovacancy or monointerstitial reactions.
Annealing temperature I K
Fig. 7. Fraction of In atoms in gold which emit perturbed y-y angular correlation signals upon isochronal annealing after quenching and low-temperature irradiation with the particles indicated. The different types of lines indicate different types of signals and hence, different defects trapped by the probe. atoms (after DEICHER et al. [1981b]).
ch. 18, 52
1639
Point defects
2.233.Experimental determinationof AHvm.The primary problem of experiments involving quenching and subsequent annealing consists in the large variety of defect reactions which are likely to occur simultaneously. In addition, the boundary conditions for these diffusion-controlled reactions are significantly influenced by the quenching and annealing procedure. The following conditions are important: (i) locally inhomogeneous distribution of vacancies, due to losses at sinks or cluster formation (8 2.2.4), (ii) timedependent sink concentration and population of different configurations during the annealing process, e.g., by nucleation and growth of vacancy clusters, (iii) sink effciencies which depend on both, vacancy concentration and temperature, (iv) vacancyimpurity interactions, (v) drift diffusion effects caused by elastic interaction of vacancies and sinks. Computer simulation by means of the rate equation approach shows that the main defect parameters can hardly be inferred from a standard analysis of isothermal and isochronal data if, for example, the actual clustering kinetics are unknown (JOHNSON [19683). The experimentally observed influence of clustering is illustrated in fig. 8 which shows for Au and A1 the observed temperatures T,of annealing stages as function of the quenching temperature Tp' Within the temperature range shown for T ,vacancies either annihilate at fixed sinks or form immobile clusters which dissociate at considerably higher temperatures than 450 K. The behavior of T,for Al is illustrated in more detail in fig. 9 which shows the temperature-differentiatedisochronal recovery curve for the T;s indicated. Peak B was quantitatively interpreted by a single diffusioncontrolled annihilation process with an activation enthalpy of 0.65 eV, which was ascribed to monovacancy migration. Peak A could not be interpreted by a single process and is characterized by an effective activation enthalpy of 0.44-0.5 eV. Obviously, at low vacancy concentrations as quenched from low temperatures, monovacancies are the dominant defects. For higher vacancy concentrations as quenched from temperatures above 50O0C, multiple vacancies are formed which migrate faster than monovacancies and accelerate the annihilation process. This feature describes, in general, the annihilation behavior of quenched-in vacancies in most metals, although the separation between the processes involving
350
. Y
200
300
400
500
600
Tq I "C
Fig. 8. Dependence of the temperature T ,characterizing the temperatures of vacancy-annihilation stages in isochronal annealing experiments on quenching temperature Tqfor Au and Al (after BALLUFPI 119783). References: p. 1710.
1640
Ch. 18, 82
H.J. Wollenberger
jl ,pf, Tq = 5OO0C
*
B
‘Y\*
1
A
6
\
8
Tq = 400°C
A
Tq
-100 -0
0
300’C
1 IO0
T 1°C
Fig. 9. Temperature-differentiated isochronal resistivity recovery curve of Al for different quenching temperatures Tq (after LEVYet al. [1973]).
monovacancies and those involving mobile multivacancies is not so obvious as in Al. In order to obtain reliable data, the boundary conditions must carefully be controlled in the experiments (LENGELER and BOURASSA [1976], fig. 4) and/or systematically varied solutions of clustering models must be fitted to the data for evaluation (SAHU et al. [1978]). The results of such fits are not single sets of enthalpy and entropy changes for migration of single vacancies but “fields of existence” for possible parameter sets which include also the vacancy-binding parameters up to higher clusters. Enthalpies of monovacancy migration are listed in table 1. The stage 111 controversy arose when the recovery stage III of irradiated samples always occurred at lower temperatures than the main recovery stage in quenched samples. A systematic comparison is by BALLUPPI [1978]. Asan example, the temperatures of recovery T,measured by more than 10 groups of authors in Pt are compared in fig. 10. The widespread data-scatter for both types of treatment as well as the systematic difference between irradiation and quenching is obvious. It is quite suggestive to take the discrepancy of 200 K between the most often observed recovery temperatures in irradiated and quenched samples as overwhelming evidence for the migration of two entirely different defects, as was done by the advocates of the two-interstitial model. According to this model, stage 111 in irradiated metals is caused by the migration and subsequent recombination of the so-called stable interstitial which arises from the metastable one by conversion (SEEGER[1958, 19751). Its enthalpy of migration is supposed to equal the effective enthalpy of migration found for stage HI,whereas the
Ch. 18, $ 2
1641
Point defects
Quenched
0 0
E
OCO
.Q c L
00 0 000 0 000 0000
k
a
I
*.
Irradiated
L.
d
E
9
0 00
0 0 0 0 000 0 0 0 0 400
500
600
700
800
vacancy migration enthalpy follows from the respective quantity found in quenched samples. The behavior of prequenched and irradiated samples has been studied for A1 and SCHILLING [1969], (BAUER [1969]), Au (LEEand KOEHLER[1968]) and Pt (DUESING et al. [1972]). Typical curves for Pt are shown in fig. 11. The splitting of S~NNENBERG the single stage III in the unquenched state into two stages in the prequenched state has et al. [19707 in terms of a mono-di-vacancy model which been interpreted by SCHILLING takes into account the different concentrations and local distributions of sinks in the quenched and unquenched samples. Very careful measurements and data evaluation revealed a monotonic variation of AH; when proceeding through the stage 111-IV region. The data were quantitatively interpreted by a mono-divacancy model with AH: = 1.45 eV, AH; = 1.00 eV and AH..=0.15 eV (SONNENBERG et al. [1972]). BALLUPP~ (1978) compared the reaction processes in stage 111for the irradiated and quenched state by plotting AHe; against Ta, the T, values averaged over all reported measurements, as shown in fig. 12. The data for the different metals lie well on straight lines through the origin, but quenched and irradiated states are characterized by different lines. The equality AHG/kB.Tafor the different metals indicates the close similarity of the parameters determining rmgration and annihilation, such as frequency factor, sink geometry and density, and initial defect concentration. The different slopes for quenched and irradiated states indicate deviation of at least one of these parameters. The difference can be explained by a number of jumps-per-vacancy until annihilation which is larger by a factor of lo3in the quenched state than in the irradiated state. This order of magnitude had already been derived earlier (SONNENBERG etal. [1972], JOHNSON [1970], SCHILLING etal. [1970], LEEand KOEHLER [1968], L w etal. [1968]).
References: p. 1710.
1642
Ch. 18, $ 2
H.J. Wollenberger
150
200
300
400
600
800
Temperature I K
Fig. 11. Temperaturedifferentiated isochronal recovery curves of Pt after low-temperature. electron-irradiation. Shown is the influence of pre-quenching and that of initial defect concentration (the ratio of quenched-in resistivity to irradiation-induced resistivity is large for the upper curves, small for the lower curves). The curves are normalized by different values of Ap,,, (see ordinate). Merely for that reason a deviation between dashed et al. [1970].) and solid curves occurs below 350 K.(after SCHILLING
2.2.4. Agglomeration Vacancy agglomerates which are large enough to be observable in the electron microscope have been studied in great detail (ch. 11). The topology of configurations involves dislocation loops, stacking-fault tetrahedra and voids. Review articles are by EYREet aE. [1977], KIRITANI [1982], and SHIMOMURA etaE. [1982]. Since the resolution of common electron microscopy is limited to 1-2 nm, the observable agglomerates certainly contain more than ten vacancies (see also $4.2.2). About the same resolution limit holds for diffuse X-ray scattering (EJBHARTet al. [1982] and LARSONand YOUNG[1982]). Field ion microscopy does allow imaging of agglomerates consisting of less than ten vacancies (WAGNER[1982]). It has, however, not been applied yet to questions like shape and size distribution of small vacancy agglomerates as formed by the encounter of migrating vacancies.
Ch. 18, 92
1643
Point defects
2.0
-
1.6 1.8
. 2
E*
1.4
-
1.2
-
5
0
auemh
+ had 100
2 0 0 300 4 0 0 5 0 0 6 0 0 7 0 0 -
800 9 0 0 1 0 0 0
Ta IK
Fig. 12. Effective enthalpies of defect migration versus T, (definition see fig. 7) averaged over the results by different authors for quenched and inadiated samples (after BALLUFFI [1978]).
While experimental information on configuration and stability of di-, tri- and tetravacancies is lacking, computer simulations give some information by using a special nearest-neighbor interaction potential constructed to similate the bcc a-Fe (JOHNSON [1964]). This potential was also used for simulating a fcc model crystal which behaves similar to Ni with respect to the elastic properties. This model yielded for the single vacancy: AH; = 1.49 eV, AH: = 1.32 eV and AVE' =-0.15 atomic volume. The stable di-vacancy consisting of the vacancies at nearest-neighbor sites is characterized by AH; =0.25 eV and AHc =0.9 eV. The most stable tri-vacancy, shown in fig. 13a, yielded AH; =0.75 eV and AH: = 1.02 eV. The migration includes an intermediate dissociation step. Reorientation of the trivacancy with a lower activation enthalpy than AH: occurs by jump of the atom which forms a tetrahedron together with the tri-vacancy (fig. 13b). The stable tetravacancy is that shown in fig. 13d with AH; = 1.51 eV. The stability of the three-dimensional tetrahedron as compared to the two-dimensional rhombic configuration (fig. 13c) might be a direct consequence of the nearest-neighbor interaction potential (DEDERICIIS et ckl. [19781). For the a-Fe mbdel crystal AH: = 0.68 eV was found for the nearest-neighborjump. The most stable di-vacancy configuration places the vacancies at second-neighbor distances AH; =Ck2 eV. Migration happens either via the metastable nearest-neighbor References: p . 1710.
1644
H.J. Wollenberger
Ch. 18, $ 2
Fig. 13. Tri- and tetra-vacancy configurations in the fcc skcture (see text).
configuration or via a fourth-neighbor configuration, both with an activation enthalpy being about the same as for the mono-vacancy migration. The most stable tri-vacancy has two nearest-neighbor and one second-neighbor spacings with =0.49 eV (JOHNSON and BEELER [19771). The first immobile aggregate was found to be the tetravacancy. The local vibration densities of states of the neigbouring atoms of vacancy clusters were calculated for Cu, a-Fe and a-Ti by YAMAMOTO [1982]. Formation and migration enthalpies of di-vacancies have frequently been derived from experiments as described in 532.2.2 and 2.2.3 (EHRHART[1991]). But the derived numbers result from fitting models to measured data by assuming that eq. (4) holds. However, direct evidence for divacancies causing the observed deviation of c:(l/T) from linearity has not been reported. Another reason would be a temperature dependence of the activation enthalpies of vacancy formation (AUDIT [1982], GANNEand Q u M [1982], VAROTSOSand ALEXOPOULOS [19821).
23.5. Interaction with solutes The interaction of vacancies with solutes controls solute diffusion, also known as impurity diffusion, and the self-diffusion of the solvent atoms. This topic is treated in ch. 7. Here, we briefly look at the interaction energies and their influence on c,". In theory, a number of different approaches have been followed in the past in order to obtain quantitative information about the vacancy solute interaction (review by DOYAMA [1978]). Nowadays ab initio calculations are performed by means of the KKR Green's function method based upon the density functional theory in the local spin density approximation (KLEMRADTet al. [1991]). The method allows one to calculate the nearest neighbor interaction of vacancies with 3 d and 4 sp solute atoms in Cu and Ni as well as with 4 d and 5 sp solute atoms in Ag and Pd. As a result the data shown in fig. 14 were obtained. A positive sign of the interaction energy means repulsion and a negative sign, attraction between vacancy and solute. For the sp solute atoms in Ag and Pd, the attraction is approximately proportional to the valence difference. Such proportionality was discussed for a long time in the literature for experimental data and was often
Ch. 18, 52
1645
Point defects
1 I
(a)
I
I
I
I
1
I
In
Zr Nb Mo Tc Ru Rh Pd Ag Cd
0
.
1
(b)
I
I
1
1
1
1
,'>
Sn
Sb
2
1
1
1
1
1
1
2
1
1
1
1
1
1
Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Go Ge As Se
Fig. 14. Calculated interaction energies of a vacancy with a solute atom of the element given by the abscissa for the solvents silver (-) and palladium (--) in (a) and copper (-) and nickel (---) in @) according to KLEMRADT e t d . [1991].
compared with a proportionality to the lattice parameter according to a different approach of modelling (review by BENHDEK [1978]). The same behavior is seen for the sp solute atoms in Cu and Ni. The significantly different behavior of the 3 d solute atoms in Cu and Ni as compared to that of the 4 d solute atoms in Ag and Pd is due to their magnetic moment. The magnetic exchange energy reduces the repulsive energy to very small values. KLEMRADT et al. [1991] find remarkably good agreement of their data with experimental ones as obtained from solute diffusion measurements. The authors also report on first results for binding energies of a second solute atom with a vacancy solute pair. This configuration and high order clusters are of great interest for the understanding of the dependence of solvent and solute diffusion on solute concentration. The direct experimental method of determining AH:= for vacancy solute atom pairs is the comparison of DD measurements for dilute alloys and the pure solvent metal. In the dilute alIoy ce is enhanced by a term being proportiorial to c,,exp[-(AHC-~,)/k,al. Further methods are equilibrium PAC measurements and resistivity measurements after quenching, both comparing dilute alloys and pure solvent metals (DOYAMA [1978]). A careful comparison of DD measurements and PAS measurements in Cu and dilute Cu-Ge alloys performed by K L m and HEHENKAMP [I9911 yields good agreement for the References: p. 1710.
1646
H. J. Wollenberger
a.18, 9 2
results from the two methods after a reassessment of the PAS models applied for the evaluation of c,".
2.3. Vacancies in ordered alloys Long-range atomic order is a common phenomenon in intermetallic phases, (ch. 4). Examples are Cu,Au and CoGa. For many intermetallic compounds the critical temperatures for the order-disorder transformation exceed the melting temperature, which indicates high ordering energies. For such alloys the point defects show additional features to those known from pure metals or disordered solid solutions. Let us inspect the already quoted CoGa which belongs to the group of &brass electron compounds and crystallizes in the CsCl structure. This structure is made up of two cubic primitive sublattices a and P occupied by A (Co) and B (Ga) atoms, respectively. The latter atoms are positioned at the body-centered sites of the first lattice. Vacant lattice sites in the two sublattices are not likely to be energetically equivalent, owing to their different nearest neighbor atomic shells. The vacancy fractions in the two sublattices may be quite different, which can be inferred from the following naive picture. The vacancy in the a-sublattice is surrounded by only B atoms. This arrangement might suggest for the enthalpies of formation AH: = AH: where the right hand side describes the pure B metal case. This assumption is indeed confirmed for a number of group VIII-group IIIA compounds. For Ga we have AH: = 0.5 eV leading to vacancy concentration in the Co sublattice of about 10% at 900°C. The more detailed treatment by MIEDEMA[1979] yields AH," = 0.48 eV while equilibrium measurements yielded 0.23k 0.06 eV. The same reasoning leads to AH! = 1.4 eV and, hence, vacancy fractions in the p-sublattice being many orders of magnitudes smaller than that in the a-sublattice. Certainly, such large total amounts of vacancies will significantly determine macroscopic properties of the material. How are the excess vacancies in the a-sublattices being formed? An excess vacancy (excess with respect to the vacancies in the P-sublattices) can be formed only by transferring an A atom into the P-sublattice. Such an atom forms an anti-site defect (also called antistructure atom), Le., a point defect specific for ordered alloys. Because of the requirement of equal numbers of sites per sublattice the one anti-site defect must be accompanied by two vacancies in the a-sublattice. We have a triple defect. The creation of anti-site defects introduces the ordering energy as a controlling parameter for concentration and mobility of the vacancies in the a-sublattice. The usual nearest neighbor jump distance ( 4 11>/2) must be replaced by the next nearest neighbor distance (). In particular, the vacancies are retained from annihilation during quenching or even slow cooling. By this means, vacancy concentrations of the 1% order of magnitude can easily be observed at room temperature for a number of intermetallics. On the other hand, the degree of order does not only depend on the ordering energy but also on the formation enthalpy of the a-vacancies. This condition leads to a curved Arrhenius plot for the degree of order as being observed. For some of the intermetallics significant repulsive interaction of the vacancies has been observed and explained by the electrical charge which may be as high as one
Ch. 18, $ 3
Point defects
1647
electron per vacancy (for FeAl, KOCHand KOENIG[1986]). Repulsive interaction leads to dispersed vacancy distribution as observed for FeAl whereas formation of large voids (50-100 nm in diameter) is observed for NiAl. For NiAl and for CoAl the vacancy concentration is significantly smaller than for FeAl and CoGa. The latter compounds show a higher degree of intrinsic disorder than the former ones. Long-range ordering of vacancies has been observed for a number of more complex intermetallics (LIUPINGand DUNLOP[19SS]). An important feature of many intermetallic compounds is their stability with respect to deviation from stoichiometry. The compound CoGa is found to be stable from 45 to 65 at.% Co for slowly cooled specimens. The vacancy concentration in the Co sublattice amounts to about 10% for 45 at.% Co and falls below 0.1%at 65 at.% Co. At the stoichiometric composition it amounts to about 2.5%. At first sight, these vacancies might be taken as structural ones, i.e., formed to assure the lattice stability for the respective composition. On the other hand, the above-mentioned sluggish approach towards the thermal equilibrium concentration of vacancies does substantially aggravate the discrimination between thermal and structural vacancies. For CsCl structure compounds the existence of structural vacancies is still discussed controversially in the literature (KIM [1986], WEVER[ 19921). ]For the intermetallic compounds of more complex structures the above-treated questions are studied to even lesser extent by systematic means which is mainly due to the great difficulties with the preparation of reliable sample material. Nevertheless, for a few cases, most interesting properties have been studied in great detail. One example is the Zintl phase P-LiAI which is of great interest as an anode material for lithium sulfur batteries. At room temperature this material contains 0.4% and 7% Li vacancies for 47 at.% AI and 52 at.% AI, respectively. The activation energy for Li self-diffusion is of the order of 0.1 eV. The Li vacancies are long-range ordered at 80 K (KIM [1986]). The intermetallic compounds Nb,Sn, V,Ga, and Nb,Ge (A15 structure) are of great interest because of their excellent superconducting properties. These properties are closely related to a specific feature of the A15 structure which contains linear chains of the transition metal atoms. The transition temperature for the superconducting to the normal state was found to depend strongly on the quenching temperature for V,Ga. This dependence has successfully been interpreted in terms of the anti-site defects (Ga atoms within the V atom chains) produced by thermal disordering at the quenching temperatures (BAKKER [1987]). Diffusion properties of intermetallic compounds are reviewed by WE= [1992]. (See also ch. 7, 05.2). A good recent review is by DENOWON[1995].
3. Selj-interstitials 3.1. Production of interstitial atoms
3.1.1. Introduction The interstitial formation energy of a few eV is easily provided by irradiating a crystal with energetic particles. For example, an electron of 400 keV energy transfers References: p . 1710.
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Ch. 18, $ 3
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19 eV recoil energy to a Cu nucleus via head-on collision. The maximum energy transferred to Cu by fission neutrons of 2 MeV amounts to 125 keV. The fundamental problems of radiation damage in materials intensively stimulated research regarding particle-lattice-atom interaction and the Frenkel defect production which results. In the present section we deal with this matter with regard to the atomic defect concentration c d produced by a givenjuence @ (time-integrated flux density 0) of particles penetrating a crystal. Among the numerous questions on the Frenkel defect production this point is particularly important for the evaluation of specific (microscopic) defect properties from radiation-induced changes of macroscopic materials properties.
3.1.2. A t o d c displacementcross-sectionfor electron irradiationand the production of stable Frenkel defects The atomic fraction of collisions between projectiles and target atoms for a fluence Q, is given by crQ,. The term collision ought to be specified in order to render the crosssection u a welldefined quantity. Electrons penetrating a crystal interact with the electrons as well as with the nuclei of the target atoms. In metals, the electron-electron interaction produces heat and, hence, requires cooling of irradiated samples. At the same time it causes an angular spread of the electron beam around the direction of incidence. This spread is of relevance for the angular resolution of threshold-energy determinations (see below) and for the flux-density determination in any kind of electron-irradiation experiment except for very thin samples. The electron-nucleus interaction causes the displacement of atoms from regular lattice sites to interstitial positions. The necessary specification of this collision is given by the minimum recoil energy required for permanent displacement of an atom. Only collisions with recoil energies T larger than the displucement threshold Td are of relevance for the Frenkel defect production. The total cross-section can be written as the recoil-energy integral of the differential cross-section da,
taken from the displacement threshold energy to the maximum transferred energy (headon collision), E is the electron energy. The differential cross-section for the scattering of a relativistic (Dirac) electron by a point nucleus was calculated by MOTT [1932]. Relativistic electron scattering favors mean recoil energies and reduces high ones when cos4 8 holds, where 8 compared to Rutherford scattering, for which da/dT = T 2= is the starting angle of the recoiling atom with respect to the incident electron-beam direction (further reading: CORBETT [19661). The lattice structure causes T, to depend on the recoil impact direction with respect to the lattice orientation. This dependence is shown in fig. 15 for Cu (method of measurement see further below). Irradiation of textureless polycrystalline samples means randomness of the recoil impact directions with respect to the lattice orientations. If for Tdin eq. (7) the absolute minimum Td,minof the threshold-energy surface were taken, u would be calculated much larger than the experimental total displacement cross-section. All those collisions with
TZ
Ch. 18, 9 3
Point defects
1649
Fig. 15. Displacement threshold energy Td as a function of the atomic recoil impact direction within the fundamental orientation triangle in fi (after KING et al. [1981]).
T>Td,,, but with impact directions for which T
O and interstitial-interstitialinteraction, i.e., AV?‘ = AV,”’ > O we have an attractive interktion along and repulsive ones along and . For d=O the dipole interaction vanishes but the multipole interaction does not. The influence of higher-order and POLLMANN [1972]. A terms of the perturbation expansion was studied by DEDERICHS numerical solution free of approximations was given by MASAMURA and SINES[1970]. Higher multipole interactions decrease with r-5 or even faster (SIEMS[1968], HARDY and BULLOUGH [1967]). Of special interest is the induced interaction based upon the [1968]). For the interaction of polarizability of the defects. It depends on r as r4 (SIEMS two interstitials in Cu or A1 with their high polarizabilities, the induced-interaction [1975]). energy is comparable to the dipole interaction only for r S 3 nnd (TRINKAUS The above described relationships are certainly not applicable to interstitialsapproaching one another as close as is necessary to form multiple interstitials. This range of interaction has been studied by computer simulation (INGLEet al. [1981]). Quite detailed information on static and dynamic properties of di- and tri-interstitials for different Cu potentials were obtained. The stability of configurations was studied up to clusters consisting of 37 interstitials. The activation enthalpies of binding were found to be larger than the activation enthalpies of migration for all cases. Dissociation of multiple interstitials must therefore not be expected according to this fcc model. The relaxation volume change per clustering interstitial amounts to -5% to -15% of AV,Tel. The formation enthalpy per interstitial decreases by about 30% when going from two to ten interstitials. The stable di-interstitialconsists of parallel dumbbells at nearest-neighbor sites, tilted
Ch. 18, $ 3
Point defects
1663
by a small angle (< 10') in the plane. Stable tri-interstitials are formed by mutually orthogonal dumbbells at nearest neighbor sites. Adding one atom in the octahedral lattice position to the tri-interstitial configuration yields the stable fourinterstitial. Larger clusters arise from the four-interstitial cluster by adding further dumbbells such that each is equidistant from the central octahedral interstitial and is aligned orthogonally to its nearest-neighbor dumbbell. Such three-dimensional structures are more stable than two-dimensional ones for fewer than nine interstitials and are less stable than these for more than 13 interstitials. The stable two-dimensional clusters are platelets of octahedral interstitials on planes. The activation enthalpy of di-interstitial migration was found to lie below or above that of mono-interstitial migration, depending on the details of the potential. For the higher-order clusters, the migration enthalpies increase rapidly with the number of interstitials in the cluster. The dynamic behavior of the di-interstitial is essentially similar to that for the monointerstitial. A variety of resonant and localized modes occur. According to the resonant modes the thermal displacements of the dumbbell atom are nearly as large as for the single dumbbell. The change of the,elastic constant C , per interstitial is even larger than that of the single dumbbell. Although the quantitative results depend significantly on the details of the pair-interaction potential used for the simulation, the general dynamic behavior makes at least di-interstitials a very interesting object for experimental studies. For information on interstitial loop formation from MDC studies see Q 4.2.2.
3.33. Experimental methods and results 333.1. Relaxation volume. Introduction of an additional atom into a perfect finite crystal enhances the volume of this crystal by the relaxation volume of the self-interstitial, AF" . Because of possible relaxation anisotropies, interstitials of concentration ci must be introduced randomly with respect to the anisotropy axes in order to produce the correct total volume change ciAK."'. The method of determining AYE' is given by the [1978]) or the diffuse appropriate evaluation of the Huang scattering (EHFWART scattering between the Bragg peaks (HAUBOLD and MARTINSEN[1978], 503.2 and 3.3.2.2, also ch. 12). As the relaxation volumes of interstitial and vacancy enter quadratically into the Huang scattering intensity and AV:' is considerably smaller than AV?', the former needs to be known only approximately. The experimental results are collected in table 4. The deviation of Ab',.' from one atomic volume can be imaged as that additional volume which must be spent because of the perturbation of the regular packing of spheres by the inserted interstitial atom. Perturbation of the fcc structure obviously requires more additional volume than the less dense bcc structure. 3.3.2.2. Configuration. Experimental information on interstitial configurationscomes from diffuse X-ray scattering and mechanical (5 3.3.2.6) or magnetic relaxation experiments. The open question of the potential existence of two interstitial configurations in one and the same metal as postulated by the two-interstitial model generated a strong impact on the development of both the theory of the point-defect-induced diffuse X-ray [1973]) and the measuring technique (EHRHART et aE. [19741). scattering (DEDEEUCHS References: p . 1710.
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Ch. 18, $ 3
H.J. Wbllenberger
As the result of these efforts, the configuration of the interstitial in Al,Cu, Ni, Mo, Fe and Zn has been determined. For the fcc metals the split interstitial, for the bcc the 1
co
cr
Fe
~0.15
0.36
, , and . The excitation energies per atom were chosen around 1 eV/atom and the radial impacts were given simultaneously to about 600 atoms forming the cylinder around the assumed projectile trajectory. The excitation energy was dissipated at the crystal boundaries by properly arranged stochastic walls. As a result, the formation of one to two Frenkel defects was observed per event. The results indicate that Frenkel defect production might be possible by electronic excitation via a Coulomb explosion. Experimental work proved a significant influence of the electronic excitation on the defect production above a certain threshold of (dE/d;lc),. Figure 46 shows the thermal recovery of the radiation-induced resistivity increase after irradiation with the given ions and energies. Attention is focussed to the stage I annealing. While the 0.5 MeV proton irradiation produces the wellknown stage I annealing, the amount of recovery becomes smaller when the electronic stopping power of the projectiles is increased. The effect is References:p. 1710.
1692
Ch. 18, $ 4
H. .I Wollenberger .
0
120MeV
'"1
126 MeV
A
0
0 5 4 MeV
'H
20 30 40 Temperature I K
10
B
m 3-a
20
50
60
"'1
c
100 MeV
0
120 MeV
"Si
1 1 5 MeV
"F
-
A
:.8MeV
'OAr
0
84MeV
'*C
c
2
1 7
20
40
80 Temperature I K
60
100
0
120
Fig. 46. Electrical resistivity recovery upon isochronal annealing and recovery rates for heavy-ion irradiated Cu and Ni (after IWATAand IWASE[19901).
Ch. 18, $4
Point defects
1693
stronger for Ni than for Cu. The recovery reduction occurs at the low temperature side of stage I indicating that the fractional population of close pairs and correlatedly recombining freely migrating interstitials (0 3.3.2.4) is reduced. By increasing mass and incident energy of the projectiles not only (Wdx)),is enhanced but also the recoil spectrum due to elastic nuclear collisions is changed. In order to separate potential effects arising from both of the mechanisms it is reasonable to plot the stage I recovery one time et aE. vs. the weighted mean elastic recoil energy T,/zfrom nuclear collisions (AVISRBACK [1978] and the other time vs. (dE/dx),. Increasing T,,, means essentially fractional decrease of single Frenkel defect production in favor of defect cluster production. Both are done in fig. 47 for Ni. The two branches (open and closed symbols) in fig. 47a tell us that equal values of lead to completely different stage I populations, depending on the incident ion energy. Fig. 47b shows all datapoints describing just one curve. Obviously, the electronic stopping power is responsible for the stage I reduction. For Cu the datapoints describe just one curve already in the T,,,-plot indicating negligible influence of (dE/dx), on the stage I population. The metals A1 and Pt were found to behave like Ni while Ag behaves like Cu. The two groups of metals deviate by their electron-phonon coupling (0 4.2.1 and $4.2.2) with a strong coupling for Ni and Pt and a weak one for Cu and Ag. The authors explain these observations by Frenkel defect production via elastic nuclear collisions and the reduction of the recovery in stage I by subthreshold annihilation (0 3.1.2) of formerly produced Frenkel defects as a result of the electronic excitation. The behavior of the defect production rate as function of the defect concentration confirms this interpretation. The rate is constant initially and begins to decrease with increasing defect concentdon when the accumulated defect concentration allows for a measurable annihilation rate (IWATA and IWASE[ 19901). For the metals Ti, Co, and Cr, DUNLOPet aE. [1993] observed positive electrical resistivity damage rates which increase linearly with (dE/dx),. The effect is largest for Ti. For Ti irradiated by GeV F% ions electron microscopy resolved alignments of small defects along the incident ion beam direction. The defects appear by the same contrast as known for dislocation loops. The defect size appears to stay below 10 nm. By irradiating with 4 GeV U ions at 20 K the phase transformation from the hcp equilibrium a-phase to the hexagonal o-phase is induced. Without irradiation, this phase transformation is initiated under hydrostatic pressure of the order of 10 kbar. For Fe the authors find the Ni type behavior for irradiation with ions up to (dWdx), = 40 keV/nm. Beyond this value Fe behaves like Ti, Co, and Cr. DUNLOPet al. [1993] point out that each of the metals Ti, Fe, Co, and Cr does form various allotropic phases and exhibits soft modes in the phonon spectra, which are associated with the displacive phase transformation. This correlation might suggest that the damage detected by means of the electrical resistivity increase consists of second phase precipitates also for Fe, CQ, and Cr. Up to now there is no experimental evidence for direct Frenkel defect production by electronic excitation in pure metal crystals. But, the deposited energy is obviously sufficient to induce close Frenkel pair recombination(subthresholdcollisions) with strong electron-phonon coupling. The specific collective and radial impact generation emanating from the swift ion path may well trigger displacive phase transformations. This has been demonstrated, in particular, for the crystal-amorphous transition in intermetallic comReferences: p . 1710.
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H.J. Wollenberger
Ch. 18, 44
PKA median energy TIn I eV
Electronic stopping power / MeV /(mg Icm') Fig. 47. Fraction of stage I recovery for heavy-ion-irradiatedNi (a) vs. weighted average recoil energy of the irradiating ions indicated, (b) vs. electronic stopping power of the irradiating ions indicated (after IWATAand IWASE[1990]).
pounds. In compounds like NiZr, and NiTi which can easily be amorphized by other kinds of irradiation (elastic collisions) or plastic deformation, ion tracks of amorphous material of about 8 nm diameter are produced for (a/&), > 50 keV/nm (800 MeV Pb or U ions). For Cu,Au no phase transformation by electronic excitation has been observed. In amorphous materials, fast-ion irradiation causes strong plastic flow perpendicular to the ion beam direction (KLAU"NZER[1992]). This phenomenon is again caused by
1696
Ch. 18, $ 4
H.J. Wollenberger
(composition) of the materials and irradiation temperature are of considerable influence on the swelling rate. The general feature of the swelling behavior of relevant structural materials is sketched in fig. 49. We note a number of important characteristics: (i) significant swelling starts after accumulation of an incubation fluence, (ii) saturation of swelling appears not to exist within tolerable swelling levels (except a few void lattice forming materials), (iii) ferritic steels show considerably less swelling than austenitic ones, and, (iv) cold-working enhances the incubation fluence. For comparison, the behavior of Mo is shown: this develops a void lattice (0 4.6). In the present section we focus our attention on the basic processes controlling void nucleation and growth. This view will enable us to understand the above characteristics in terms of general defect reaction schemes. Although voids have been known for a long time from the Kirkendall effect (5 2.2.1) (see also ch. 7, 5.3), they were not expected in irradiated materials as all earlier investigations showed only planar aggregates of vacancies (dislocation loops, stacking fault tetrahedra). Moreover, such aggregates were found to dissolve at temperatures at which voids are still being formed. Research of the nucleation problem was always impeded by the fact that impurities, in particular, gaseous ones will certainly stabilize three-dimensional vacancy agglomerates. In addition, Fast Reactor irradiations not only produce Frenkel defects but also relatively large amounts of He resulting from the large cross-sections for ( n p ) transmutations of frequently present metal isotopes. On the other hand, it proved to be difficult to prepare sufficient gas-free or generally impurity-free samples to study homogeneous void nucleation under etaE. charged particle irradiation. MDC studies for Cu and H as impurity (SHIMOMURA [1993]) revealed that the tri-vacancy relaxation towards the stacking fault tetrahedric structure may be suppressed by involving H atoms. However, the trapping energy for H is small for vacancy agglomerates up to 5 vacancies. From 6 vacancies onwards, trapping
Solution
Standard cold-worked austenitic steels
$? 20 \
-
Commenial ferritic steels
6,
3 (0
Molybdenum
(void lattice)
10 cold-worked
*O
50
100
150
Fluence l d p a Fig. 49. Swelling vs. fluence for different materials and conditions (schematic) after EYREand MAITHEWS 119931.
Ch. 18, 54
Point defects
1697
of H stabilizes the void structure. An H atom which is trapped in the hexa-vacancy pushes out four atoms which are positioned in the tetradedral 10-vacancy of 6-vacancy stacking fault tetrahedra. This stabilization is equivalent to the suppression of the relaxation of a 3-vacancy to a stacking fault tetrahedra. The void structure becomes more and more stable between 10 and 20 vacancies, but needs an increasing number of H atoms for the stabilization. The study also shows that vacancy agglomerates with less than 15 vacancies become thermally unstable at the peak swelEing temperature (see M e r below). Gaseous impurities are now generally accepted as necessary prerequisites for the stabilization of the void structure. The fission product He has received most attention et al. [1981], STOLLERand O D with respect to its role in void nucleation (GLASGOW [1985], PARKERand RUSSELL[1982]). Void formation by electron irradiation demonstrates the nucleation for pure single vacancy production. Here, void densities are often and SINGH[1988]). enhanced at and near grain boundaries and dislocations (HORSEWELL This inhomogeneity corresponds to the well-known impurity segregation. For cascade-producing irradiation, intra-cascade clustering and the considerable fluctuation of the defect concentration of freely migrating defects (which escape from cascades) on void nucleation, has been studied in some detail (KITAJIMAet al. [1985], TRINKAUS [19901). I1 turns out that these effects should have only minor influence on the effective void nucleation rates, mainly because of the small probability for the direct agglomeration of large enough numbers of vacancies. Gas-containingthree-dimensionalvacancy agglomerates(cavities) are termed bubbles, the size of which is controlled by the gas pressure, i.e., the ratio of the numbers of gas atoms to vacancies and the metal surface energy (TRINKAUS [1983], MANSUR[1993]). The transition to apparently unlimited void growth as seen in fig. 49 is correlated with the critical bubble radius, which is given by the deviation of the effective gas conditions from those of the ideal gas, the surface energy, and the excess vacancy flux to the cavity. Hence, extension of the incubation fluence requires materials design such that the gas and/or vacancy fluxes towards bubble embryos are reduced. The excess vacancy flux feeding the void growth was generally assumed to arise and from biased vacancy and interstitial annihilation at sinks (reviews: BRAILSFORD BULLOUGH[1978], BULLOUGHand WOOD[1980], MANSUR [1993] and further reading: Wiffen et al. [1979], Nygren et al. [1981]). The common explanation of such biased vacancy and interstitial annihilation at sinks considers the elastic interaction between the migrating point defects and dislocations as causing a drift diffusion towards the dislocations. The interaction energy for a dilatation centre A with the volume change AV; in the strain field of an edge dislocation is proportional to AV,"/r, where r is the distance of A from the dislocation core. The proportionality to AV; makes the interaction strength quite different for vacancy and interstitial (Cu, for example: AV: =-0.2, AVY = 1.6). In a given material, the topologies and concentrations of saturable sinks (e.g., sessile dislocations), unsaturable sinks and saturable traps composes a complex scheme of spatially varying defect reactions. The swelling theories were based upon the chemical rate equation approach which assumes spatially hOmogeneous defect reactions. Spatially inhomogeneousreactions like References: p. 1710.
~
1698
H.J. Wollenberger
Ch. 18, 54
point-defect annihilation at an extended localized inexhaustable sink (e.g., sessile edge dislocation) were incorporated into the rate equations by appropriate divergence terms for the defect concentrations. For a long time, freely migrating single interstitials and vacancies were assumed to be the only reactants for traps and sinks and in addition the displacement rates according to eq. (10) were taken as production rates for these freely migrating defects. Intra-cascade recombination and clustering (5 4.2.2) were widely neglected. The wealth of experimental data on the microstructural evolution under all kinds of irradiation (further reading: Klueh et al. [1992], Ishino et al. [1991]) as well as the progress in modelling suggest that satisfying agreement between the early theories and experiments whenever achieved may be attributed to the large number of less wellknown parameters which were involved in the theories and adjustable within relatively wide ranges. However, taking the various defect reactions into account allowed a satisfying qualitative explanation of the swelling response to the various materials variables and irradiation conditions. For a brief illustration, we look at the influence of cold-working and alloy composition on the irradiation temperature dependence of the swelling (fig. 50). The cold work controls the vacancy supersaturation via dislocation sink strength and bias. The temperature controls the defect mobility and, by this, the stationary defect concentration which in turn determines the relative influence of the recombination. At low temperatures the recombinationrate is large because of the slowly migrating vacancies. The biased sink annihilation is suppressed. At higher temperatures the latter is favored at the expense of recombination. At the same time, vacancy loops start decaying which enhances the vacancy supersaturation.At even higher temperatures, the stationary vacancy concentration approaches thermal equilibrium values and, hence, does not provide the supersaturation required for void formation. Slowing down of one of the migrating defect species can be achieved by adding solutes which temporarily trap interstitials ($ 3.3.2.8). In this way swelling is reduced (see, for example, POTTERet al. [1977]) as shown in fig. 50b. The peak swelling temperature depends on the displacement rate because of the increasing dominance of recombination with increasing displacement rate (steady-state defect concentration). Quantitative interpretation of the vacancy supersaturation by the annihilation bias alone became more and more difficult when new experimental work revealed swelling rates up to 5%/dpa in neutron-irradiated Ni (KIRITANIet al. [1990]) and in Cu containet al. 119921). Moreover, detailed investigation of local variations of ing 0 (YAMAKAWA the swelling rates showed maxima up to 7% 1dpa for Al in the neighborhood of grain boundaries and in subgrains surrounded by dislocation walls (HORSEWELL and SINGH C19881). The width of these maximum swelling regions is of the order of lod m. A swelling rate of S%/dpa requires an excess vacancy flux to the voids by the same fraction of the total flux of the freely migrating single defects. Hence, for the less probable case that all interstitials annihilate at sinks different from the voids, the fractional freely migrating defect escaping from the cascades must be 5% or significantly larger for the probable case of partial interstitial annihilation at voids. Such values do not fit theoretical and experimental findings ($4.2.2). In addition, the width of loa m is considerably larger than the average diffusion distance of single vacancies and interstitials for the sink strength observed for respective irradiation conditions. These
Ch. 18, $ 4
Point wects
24
1699
a
Temperature I "C
I
I
b
I
1
I
r : t"\ Fe-15Cr-20 Ni
40
0.3 Si
v) 30 20
0
575
625
675
Temperature ii "C
725
Fig. 50. Influence of pretreatment and solute additions on void-swelling: (a) 316 type stainless steel after 14.1x IOn neutrons/cm*,o - annealed, A 10%cold worked (cw), 0 - 20%cw, v - 30% cw (from F L m and KBNFIELD [1976]); (b) Fe-15 at%Cr-20 at% Ni with quoted Si additions after a fluence corresponding to 140 displacements per atom (from JOHNSTON etul. [1976]).
-
References: p . 1710.
1700
H.J. Wollenberger
Ch. 18, 44
difficulties can be overcome by taking into account the production of glissile interstitial loops during the cascade event (TIUNKAUS et al. [19921). Such loops are known to move by climb and glide. Small interstitial loops are believed to be faulted. However, MDC studies for Cu have shown that small interstitial loops do transform spontaneously from the faulted to the highly glissile unfaulted configuration (FORENAN et al. [1991]). Such loops would perform a one-dimensional glide with an estimated activation enthalpy of about 0.1 eV. For small loops of four interstitials, the direction of the Burgers vector may change to equivalent lattice orientations with an activation enthalpy of about 0.5 eV. Such loops would migrate by describing a zig-zag path until they are trapped or annihilated by proper components of the microstructure. The significantresult of the onedimensional glide motion is the much larger free travel distance than the diffusion distance of the three-dimensionally migrating defect for the same microstructure. Under these circumstances, spontaneously produced small interstitial loops may be trapped or annihilated having a very small chance to get involved in recombination or to encounter voids. Numerical calculations showed that the free travel distance may indeed be of the order of lo4 m and thus explain the required vacancy supersaturation in the above mentioned width of maximum swelling regions near grain boundaries. By this explanation, the annihilation bias is supplemented by a production bias in the sense that the numbers of freely migrating vacancies and interstitials escaping the cascade are biased by the gliding interstitial loops. More specific experimental work is necessary to confirm this defect reaction path. The remarkable swelling resistance of the ferritic steels (fig. 49), and of other bcc metals and alloys, was earlier ascribed to the small elastic interstitial dislocation interaction due to the small relaxation volume of the interstitial in these structures (table 4). The small elastic interaction would result in a small drift diffusion effect towards the dislocations and, hence, a small annihilation bias (SNIEGOWSKI and WOLFER[19841). MDC studies of the interstitial dislocation interaction, however, showed that interstitials are likely to be trapped near dislocations (KURAMOTO [19921) and thus kept available for recombination. More detailed experimental information leads to the conclusion that impurity trapping is the leading effect among a number of other defect reactions which are of greater influence on the vacancy supersaturation than corresponding reactions are in fcc metals and alloys (LITTLE119931). The interstitials C and N and the substitutional Si are particularly seen as being effective. The C-vacancy binding enthalpy seems to be as high as 0.85 eV and would therefore act as an effective recombination catalyzer even at high temperatures. Also, Cottrell atmospheres around dislocations may screen the sinks and reduce climbing rates. 4.5. Radiation-induced creep
The thermally activated creep of materials is treated in chs. 20-25. An important feature of this phenomenon is the exponential temperature dependence of the creep rate, L., for a fixed stress, cr. If the creep experiment is done under irradiation, L. becomes nearly independent of temperature below about TM/2. This behavior originates from the high radiation-induced vacancy- and interstitial-concentrationwhich allows sufficient disloca-
Ch. 18, $ 4
1701
Point defects
tion climb and deformation, despite the small diffusivities of these defects (reviews: LUCAS [1993], MANSUR[1992] and further reading: Carpenter et al. [1980], Gittus [1978], Woo and McElroy [1988]). For not too large applied stresses a,the creep rate b is found to be proportional to u and to the defect production rate. One of the successful models uses the stress-induced preferential absorption (SPA) of defects at dislocations capable of climbing (HEAm and SPEIGHT[1974]), as illustrated in fig. 51. If again the elastic-energy-controlled drift diffusion is assumed to determine the defect current towards dislocation sinks, the orientation of the dislocations with respect to the external stress will control the rate constants for defect annihilation. According to the model calculation, dislocations of type I (fig. 51) absorb more interstitials than vacancies, and vice versa for type 11 dislocations. The resulting climb along the directions indicated cause the volume-conserving elongation of the sample parallel to u.Within the voidswelling regime, the excess interstitial flux to the dislocations determines the creep rate, which then obtains a contribution proportional to the swelling rate. The SPA mechanism is certainly of fundamental importance for the radiationinduced creep but the really observed behavior is more complex due to the complex microstructure of the structural materials (dislocation interactions, grain-boundary diffusion, chemical heterogeneities). At high stress levels the creep rate becomes proportional to 2 which has been explained by a climb-enabled dislocation glide mechanism (MANSUR[19791). 4.6. Self-organizationof point defect agglomerates
Many metals and alloys develop three-dimensionally ordered arrangements of voids (void lattices) when irradiated with heavy ions or neutrons at temperatures around 0.45 T,,,. Figure 52 shows the void lattice in Nb, for example. The phenomenon is observed for fcc, bcc and hcp structures. In the cubic metals, the void lattices are isomorphous to the atomic lattice. For the hcp metals, only planar ordering parallel to the basal planes is observed. The void radii amount to about 5 nm while the void lattice parameters are about ten times larger. This ratio varies between 5 and 15 for the various metals with the exception of Al, for which it amounts to about 40. The ratio is nearly independent of the irradiation temperature. The minimum fluence required for the lattice formation ranges
r I
I 01 D l
I
L
1
I I IQ ' D
I
1
Fig. 5 1. Dislocation climb directions by preferential vacancy and interstitial absorption under applied stress u.
References: p , I710.
1702
H.J. Wollenberger
Ch. 18, $ 4
Fig. 52. Void lattice in Nb irradiated at 1073 K with 7.5 MeV Ta ions to fluences of (a) 140 dpa and (b) 300 dpa.
from 5 dpa to 400 dpa, depending on the metal species. This fluence also depends on the irradiation temperature and on the gas and impurity level of the material. In pure metals void lattices are not observed for electron irradiation. Electron irradiation of disordered void arrangements as produced by ion irradiation did not induce the formation of the lattice. As the total swelling caused by the void lattice generally amounts to a few percent, the total number of displacements is by two to four orders of magnitude larger than that of the vacancies accumulated in the void lattice. Correspondingly, the energy temporarily stored in the total number of defects produced (given by the total fluence level) is large compared to the energy stored in the void lattice. The ordered structure is the result of the highly energy dissipative process.
Ch. 18, $ 4
Point defects
1703
Sufficiently long irradiation with ions or neutrons at temperatures around 0.3 T, leads to periodic arrangements of dislocation walls (see fig. 53, for example). The existence ranges for disordered and ordered vacancy agglomerates are well distinguished in the defect production rate vs. temperature diagram as shown in fig. 54. At still lower temperatures, simple cubic lattices of stacking fault tetrahedra are observed in electron irradiated fcc metals and alloys (ZAISERet al. [1992]). Earlier theoretical treatments of void lattice formation were based upon equilibrium thermodynamics (minimization of the total energy) and the elastic interaction of the microstructural components (WILLIS[19753). These attempts failed as they can neither explain the observed dependence on temperature and irradiating particle species nor the lattice formation in isotropic metals like W.The now generally accepted explanation considers the void lattice as the result of a self-organizing open system far from thermodynamic equilibrium (MARTIN[19751, KRISHAN[19821). The various theoretical approaches reported since follow this concept but deviate by the various defect reactions taken into account and considered to be dominant (ABROMEITand TRINKAUS [1992], HAHNERand FRANK [1992], review: ABROMEIT [1989]). The simplest defect reaction system describing a heavy ion or neutron irradiated crystal already fulfils the conditions for self-organization (HAKEN[19771). The irradiation produces vacancy loops and freely migrating vacancies and interstitials by constant rates. The loops grow by vacancy
Fig. 53. Periodic arrays of planar {Ool) walls of dislocations and stacking fault tetrahedra in copper irradiated with 3.4 MeV protons around 100' C to 2 dpa fluence. Imaged in projection (after JKGER et al. [1990]). References: p. 1710.
1704
H.J. Wollenberger
Ch. 18, 4 4
annihilation and shrink by interstitial annihilation. The annihilation rates are biased as described in Q 4.4.For simplicity we neglect recombination, which is allowed for small defect production rates and/or high temperatures. The important characteristics for the self-organization are (i) the diffusion length of the reacting vacancy and interstitial commensurates with the characteristic length of the evolving ordered structure, (ii) the reaction rates controlling the loop arrangement are non-linear as they are given by the concentrations of the loops times the concentrations of the migrating defects. Mathematically) the system is described by a proper chemical rate equation system which is analysed by means of the linear stability analysis with respect to the stability of spatially homogeneous defect distributions. Our diffusion-controlled reaction system does not contain any information about lattice orientation. Hence, it will not give predictions on the relationship between the ordered structureand the lattice orientation. Nevertheless, we stay with this model in order to understand the physical origin of the self-organization. The stability analysis considers the behs;vior of concentration fluctuations superposed to the spatial averages. The fluctuations are described by a Fourier series. One component is shown in fig. 55. The index s stands for sink and c, is proportional to the vacancy loop concentration. Because of the annihilation bias the fluctuation amplitudes of the interstitial concentration ci and of the vacancy concentration c, are different in the way as shown in the figure. This difference causes amplification of the fluctuation of the loop concentration Sc,/cc,> as the vacancy concentration is larger than the interstitial concentration at the maximum of the loop concentration and higher vacancy arrival rates than interstitial arrival rates induce loop growth. The growing fluctuations induce vacancy and interstitial fluxes from the regions of positive fluctuation amplitudes to those of negative ones (fig. 55). These fluxes are initially larger for the interstitials than for the vacancies due to the difference in the fluctuation amplitude. However, they reduce this difference until total equilibration. The linear stability analysis shows that this amplification of 6c./ vanishes when the fluctuation wavelength A goes to zero and to infinity. The fluctuation of the loop concentration must therefore have a maximum at a certain
-1 . ;
I-
m - 3 t
.
Fig. 54. Map of periodic cluster arrays for Cu irradiated with neutrons, heavy ions, and protons, v - random, c3 - periodic cluster array, 0 - periodic voids and dislocations (after JAGER and T ~ ~ U U [1993]). S
a.18, p4
1705
Point a'ejfects
0
1
0.5
xlh
Fig. 55. Sketch of the spatial fluctuation of the concentrations of sinks (s). vacancies (v) and interstitiak (0.
finite value of A. It becomes obvious that the driving force for the self-organization is the annihilation bias for the two diffusing reactants and the fact that one reactant increases the sink strength of the loops (by growing) and the other (interstitial) decreases it. When the bias had the opposite sign the system would not be self-organizing. Adding recombination or secondary sinks (grain boundaries) or traps to the reaction system would reduce the driving force for self-organization to the measure at which the loops contribute to the total sink and trapping strength. Introducing a production bias (0 4.4) shifts the annihilation bias threshold accordingly: the difference of the sink strength for interstitials and vacancies must be larger than the respective difference of the production rates. At low temperatwres where the recombination dominates the sink annihilation, loop ordering will be impossible in accordance with fig. 54. At high temperatures the vacancy loops decay but void formation becomes possible if nucleation is favored and self-organization may be possible when they dominate as sinks and the respective annihilation rates are biased. The linear stability analysis yields the initial conditions for which homogeneous cluster distributions may become unstable. It does not give information on the shape of the evolving structure. Such information is obtained by analysis of the stationary References: p . 1710.
1706
Ch. 18, 54
H. J. Wollenkrger
solutions in the instability regime. As a result one obtains the concentration fluctuations for vacancy loops as shown in fig. 56. The regions of high-loop concentrations are much smaller than those of low concentrations and with proceeding fluence the separation between the empty and the highly populated regions becomes even sharper and the wavelength slightly increases. The stability analysis for defect reaction systems controlled by isotropic diffusion cannot give information about the symmetry relationship between the ordered defect arrangement and the crystal structure. Anisotropic diffusion controlled by the crystal lattice symmetry must influence the process of pattern formation. Two different attempts have been made to introduce anisotropic diffusion: (i) the diffusion of the vacancies and interstitials is assumed to be asymmetric in the sense that the vacancy migrate threedimensionally while the interstitials migrate one-dimensionally (dynamic crowdion, gliding interstitial loops) or two-dimensionally (EVANS[1987], HAHNERand FRANK [1992], DUBINKO [1991], JAGER and TRINKAUS [1993]), (ii) the elastic interaction between (predominantly) the interstitials and the ordering vacancy cluster introduces the crystal lattice symmetry, leads to respective anisotropic drift diffusion and thus controls the shrinkage of the clusters (BORODMet al. [1993]). The effect of one-dimensionally migrating interstitials can easily be seen in fig. 57 which shows how a mispositioned void is shifted. Arriving interstitials whose path is not blocked by neighboring voids induce shrinkage at the one side and three-dimensionally migrating excess vacancies arriving within the screened regions induce growth on the other side. In a three-dimensional crystal lattice the evolution of the void lattice would therefore start with linear ordering along the migration paths of the interstitials or with planar ordering within one of the planes containing the maximum number of the migration lines. The fundamental question with this explanation of the pattern formation concerns the structure of the one-dimensionally migrating interstitial. Stability of the dynamic crowdion has not been confirmed by the MDC studies for fcc and bcc metals
3
%
e b‘
4
* 1
-1.0
0.0
-0.5
X
0.5
1.o
/A,
Fig. 56. Spatial variation of the loop concentration for two different maximum minimum ratios (afterABROMEIT and TRINKAUS 119921).
Ch. 18, 54
1707
Point defects
Fig. 57. Rearrangement of voids by one-dimensionally migrating interstitials (after FOREMAN [1972]).
up to now (see 8 3.3). In addition, the difference of the formation enthalpies for the dumbbell interstitial and the crowdion were found to only measure a few tens of an eV. Such values are at variance to the values required for the explanation of void lattices observed at the upper temperature limit and the observed maximum void lattice parameters. The mean free path of the crowdion between its creation and conversion to a stable dumbbell interstitial must be at least of the order of the void lattice constant. The opposite tendency of the mean free path to decrease and the void lattice distance to increase with increasing temperature defines an upper temperature limit for the void lattice formation. The observed maximum void lattice distances and the upper temperature limit requires a difference between the enthalpies for interstitial conversion and migration of the crowdion of values between 0.5 and 1 eV in contrast to the findings from MDC studies. Of special interest in this regard is the behavior of Au for which the absence of a pronounced annealing stage I has been interpreted by the conversion of the crowdion to the stable dumbbell configuration at very low temperatures (FRANKand SEEGER[ 19831). Therefore, the observed bubble lattice (JOHNSON et al. [ 19831 cannot be explained by a one-dimensionally migrating interstitial. Similar arguments hold for Nb. The gliding interstitial loops discussed in 0 4.4 might explain the symmetry relationship in a consistent manner. For hcp metals two-dimensionally migrating interstitials are more likely to be stable and would explain the observed void lattice geometries. The various attempts to explain the symmetry relationship by anisotropic drift diffusion due to the elastic interaction have been successful in rationalizing the symmetry relationship (MARTIN [19771) but quantitative explanation seems to be successful for sinks with significant elastic asymmetry (as dislocation loops) only.
References: p . 1710.
1708
Ch. 18, $4
H.J. Wollenberger
4.7. Radiation-inducedsolute segregation Irradiation of a crystal causes the supersaturation of vacancies and interstitials in the lattice. At irradiation temperatures such that both defect species are mobile the defects will annihilate by recombination and annihilation at sinks. For very low sink densities the annihilation rate will be controlled by the recombination reaction. As the defects migrate randomly recombinations will occur homogeneously in space. For descriptions in terms of the rate equation approach the defect annihilation is considered by the “lossy medium”. This treatment neglects an important sink effect. Unsaturable sinks like surfaces or non-sessile dislocations cause persistent defect fluxes under stationary circumstances. The situation in front of such a sink is sketched in fig. 58 for a solid and D; of solution consisting of A and B atoms. If the partial diffusion coefficients 0,’ the constituents A and B of a binary alloy due to vacancy migration are different, the vacancy flux J, to the sink will cause different atom fluxes JAand JB with the result of
’:t I
CV
I
I Jv
;
V 8 1
I
X-
X-
X-
Fig. 58. Inverse Kirkendall effect produced by vacancy (a, b) and interstitial (c. d) fluxes to sinks.
Ch. 18, 54
Point defects
1709
a depletion of the faster transported constituent near the sink as shown in fig. 58b. The process is the inverse Kirkendall effect (SMIGELSKAS and KIRKENDALL [1947]). If the interstitial flux Ji provides atom fluxes as shown in fig. 58c constituent A will be enriched as shown in fig. 58d. For steady state conditions and rvs=risthe concentration gradient VcA was derived by WIEDERSICHet al. [1979] to be proportional to [(D,’/D,’) - ( D ~ / D ~ ) ] V c , . Atomic redistribution of this kind also occurs if one of the defect species forms a mobile tightly bound complex with one of the constituent atoms and annihilates at the sink. In this case, the complex forming constituent will be enriched at the sink. One realizes that such atomic redistributions may lead to the formation of new phases. The most simple case is the irradiation of an undersaturated solid solution of B in A. When the redistribution amplitude is large enough, the solubility limit will be exceeded near or at the sink and an intermediate phase will be formed whenever nucleation conditions allow it. At sufficiently high irradiation temperatures the process may be reversed by the action of thermal vacancies after stopping the irradiation. Theoretical description of the radiation-induced segregation has been reviewed by WIEDERSICH and LAM[19831. For studies of precipitation near internal sinks high voltage electron microscopy has proved to be a very valuable tool. Electron energies from about 0.5 MeV upwards are sufficient to produce Frenkel defects during the imaging. The beam intensity is extremely high such that the number of atomic displacements produced by irradiation in nuclear reactors can be exceeded by orders of magnitude within minutes. Radiation-induced formation of ordered phases like Ni,Si can thus be imaged easily by means of the dark field technique. For Cu-Be the formation of radiation-induced Guinier-Preston zones (WOLLENBERGER and KELL[1989]) has been studied by means of field ion microscopy and that of the long range ordered y-phase CuBe by transmission electron microscopy (KOCH et aE. [1981]). The experimental material on radiation-induced segregation was reviewed by REHN and OKAMOTO [1983]. Radiation-driven atomic redistribution is not necessarily bound to sink annihilation of the defects. If the rate of Frenkel pair recombination depends on the alloy composition, which might be the case for a strong influence of the chemical potential, the recombination rate would become inhomogeneous in space. This again would create persistent defect fluxes which could amplify existing concentration fluctuations (MARTIN [19801). The field of phase transformations in irradiated materials has been reviewed by RUSSELL[1993].
Acknowledgement The author is deeply indebted to Drs. C. Abromeit and V. Naundorf for many enlightening discussions and for critical reading of the manuscript.
References: p . 1710.
1710
H.J. Wollenberger
Ch. 18, Refs.
References A B R o ~ C., , 1983, J. Phys. F Met. Phys. 13,L169. ABROMEIT,C., 1989, Int. J. Modem Phys. B 3, 1301. &ROMEIT, C., 1994, J. Nucl. Mater. 216, 78. ABROMEIT, C., and H. TRINKAUS, 1992, Solid State Phenomena 23 & 24, 181. ABROMEIT, C., and H. WOLLENBERGER, 1983, Phil. Mag. A6, 951. ADAMS, J. B., and S. M. FOILES, 1990, Phys. Rev. B 41, 3316. ADLHART, W., G. FRITSCH and E. L ~ ~ H E1975, R , J. Phys. Chem. Solids 36, 1405. APPLEBY,W. K., D. W. SANDUSKY and U.E. WOLFF, 1972, Fluence and Temperature Dependence of Void Formation in Highly Irradiated Stainless Steels, in: Radiation-Induced Voids in Metals, eds. J. W. Corbett and L.C. Ianniello (Conf-710601, Natl. Techn. Inf. Service, Springfield, VA 22161) p. 156. AUDIT, P., 1982, Vacancy-Thennal Expansion in Aluminium, in: Point Defects and Defect Interactions in Metals, eds. J. T h u r a , M. Doyama and M. Kiritani (vniv. of Tokyo Press, Tokyo) p. 291. AVERBACK,R. S., R. BENEDEK and K. L. MERKLE,1978, Phys. Rev. B 18,4156. BAKKER,H., 1987, Mater. Sci. Fonun 15-18,1155. BALLUFH, R. W., 1978, J. Nucl. Mater. 69/70, 240. BALLWFI,R W., K. H. LIE, D. N. SEIDMANand R. W. SIEGEL,1970, Determination of Concentrations and Formation Energies and Entropies of Vacancy Defects from Quenching Experiments, in: Vacancies and Interstitials in Metals, eds. A. Seeger, D. Schumacher, W. Schilling and J. Diehl (NorthHolland, Amsterdam) p. 125. BALZER,R., and H. SIGVALDASON, 1979, Phys. Stat. Sol. (b) 92, 143. BASS, J., 1967, Phil. Mag. 15,717. BAUER,W., 1969, Radiation Effects 1, 23. BECKER, D. E., E DWORSCHAK and H. WOLLENBERGER, 1973, Radiation Effects 17, 25. BEELER,J.R., Jr., 1983, in: Radiation Effects Computer Experiments (North-Holland Publ. Co.,Amsterdam). BENDER,O., and P. EHRHART,1983, J. Phys. F.13, 911. BENEDEK, R., 1978, J. Phys. F Met. Phys. 17, 569. BENNEWWN,K.H., and L. TEWORDT, 1960, Z. Naturf. 1% 772. BERGER,A. S., D. N. SEIDMANand R. W. BmLum, 1973, Acta Metall. 21, 123. BERGER,A. S., S. T. OCKERSand R. W. SJEGEL, 1979, J. Phys. F 9, 1023. BERGER,A. S., S. T. OCKERS,M. K. CHASONand R W. SIEGEL, 1978, J. Nucl. Mater. 69/70,734. BIGET,M., E MAURY,P. VAJDA,A. LUCAWNand P. LUCASSON, 1979, Phys. Rev. B 19, 820. BILGER,H., V. HIVERT,J. VERDONE,J. L. LEVEQUEand J. C. SOULIE,1968, Point Defects in Iron, in: International Conference on Vacancies and Interstitials in Metals, Julich (JLil-Conf-2, p. 75 l), unpublished. BIRTCHER, R C., W.HERTZ, G. FRITSCH and J. F. WATSON,1975, Very Low Temperature Electron-Irradiation and Annealing of Cold and Lead, in: hoc. Int. Conf. on Fundamental Aspects of Radiation Damage in Metals, Gatlinburg, 1975, eds. M. T. Robinson and F. W. Young, Jr. (CONF-751006, Natl. Techn. Inf. Service, Springfield, VA 22161) p. 405. BLEWITT, T. H., 1962, in: Radiation Damage in Solids, ed. D. S. Billington (Academic, New York) p. 630. BORODIN,V. A., A.I. RYAZANOV, and C. ABROMEIT, 1993, J. Nucl. Mater. 207,242. BRAILSFORD, A.D., and R. BULLOUGH,1978, J. Nucl. Mater. 69170, 434. BRANDT,W., 1974, Appl. Phys. 4, 1. BRANDT,W., and H. F. WAUNG, 1968, Phys. Lett.27A, 100. BULLOUGH,R., and J. R. WILLIS,1975, Phil.Mag. 31, 855. BULLOUGH, R., and M. H. WOOD, 1980, J. Nucl. Mater. 90, 1. BURTON,J. J., 1971, Phys. Rev. B 5, 2948. BUTT,R., R. KEITELand G. VOGL, 1979, HMI-Report, unpublished. CANNON,C. P., and A. SOSIN,1975, Radiation Effects 25,253. CARO,A. and M. VICTORIA,1989, Phys. Rev. A40,2287. CAWTHORNE,C., and E. J. FULTON,1967, Nature 216,515.
Ch. 18, Refs.
Point dt$ects
1711
CHAMBRON, W., J. VWONE and P. M o m , 1975, Determination of Point Defects Symmetry in a Cubic Lattice by Magnetic Relaxation, in: Proc. Int. Conf. on Fundamental Aspects of Radiation Damage in Metals, eds. M. T. Robinson and E W. Young, Jr. (CONF-751006, Natl. Techn. Inf. Service, Springfield, VA 22161) p. 261. COLTMAN,R R , C.E. KLABUND6, J. K. REDMAN and A.L. SOUTHERN,1971, Radiation Effects 7,235. CORBEL, C., P. BOIS,P. MOSERand I. LEMAHIEU, 1987, Mater. Sci. Forum 15-18, 721. CORBETT,J. W., R.B. SMITHand R.M. WALKER,1959, Phys. Rev. 114, 1442 and 1460. COTTERILL, R. M. J. and M. DOYAMA, 1966, Energies and Atomic Configurations of Line Defects and Plane Defects in fcc Metals, in: Lattice Defects and Their Interaction, ed. R. R. Hasiguti (Gordon and Breach, New York) p. 1. DANDER,W., and H.E. SCHAEPER, 1977, Phys. Stat. Sol. (b) 80, 173. DAOU,J. N., P. V A I D A A., LUCASSON,P. LUCASSONand J. P. BURGER,1985, Phys. Lett. 107A, 142. DAUSINGER, E,1978, Phil. Mag. A37,819. DE DIEGO,N., C. HIDALGOand P. M o m , 1987, Mater. Sci. Forum 15-18, 193. DE SCHEPPER, L., D. SEGERS,L. DORIKENS-VANPRAFI;'M.D o ~ m sG. , KNUYT, L. M. SrALS and P. Mosw, 1983, Phys. Rev. B 27,5257. D E D E R I ~P., H., 1973, J. Phys. F. 3, 471. DEDERICHS,P.H., and J. POLLMANN 1972, Z. Phys. 255,315. DEDERICHS, P. H., and R. ZELLER1980. Dynamical Properties of Point Defects in Metals, in: Springer Tracts in Modem Physics, Vol. 87, eds. G. H6hler and E.A. Niekisch (Springer. Berlin) ch. 6.4. DEDERICHS, P. H., C. LEHMANN and A. SCHOLZ,1973, Phys. Rev. Lett.31, 1130. DEDERICHS, P.H., C. LEHMANN, H.R. SCHOBER,A. SCHOLZand R. ZELLER,1978, J. Nucl. Mater. 69/70, 176. DEICHER,M., E. R E C ~ A G Eand L 9%. WICHERT,1981%Radiation Effects 54, 155. DEICHER,M., 0. Ecm, E. RECKNAGELand TH. WICHERT,1981b, Geometrical Structure of Lattice DefectImpurity Configurations Determined by TDPAC,in: Nuclear and Electron Resonance Spectroscopies Applied to Materials Science, 4 s . E.N. Kaufmann and G. K. Shenoy (Elsevier-North-Holland,New York) p. 435. DENOWON,C., 1995, in: Intermetallic Compounds - Principles and Practice, eds. J. H. WESTBROOK and R. L. FLZISCHER wiley, Chichester), Vol. 1, p. 559. DIAZ DE LA RUBIA,T.and M. W. GUINAN,1992, Mater. Sci. Fonun %99,23. D ~ E R TH., J., K. SONNENBI~RG,W. SCHILLINGand U. DEDEK, 1972, Radiation Effects 15, 115. DICKMAN,J.E., R. N. JEFFERY and D. R GVS~AFSON, 1977, Phys. Rev. B 16, 3334. DICKMAN, J.E., R. N. JEFFERY and D. R. GUSTAFSON, 1978, J. Nucl. Mater. 69/70, 604. DIFFERT,K., A. SEEGER and W. 'Ikon, 1987, Mater. Sci. Forum 15-18, 99. DIMITROV, O., and C. DIMITROV, 1985, Radiation Effects 84, 117. DOYAMA, M., 1978, J. NucL Mater. 69fl0.350. DOYAMA, M., and J.S. KOEHLBR,1962, Phys. Rev. 127,21. DOYAMA, M., and R. M. J. COTTERILL,1967, Energies and Atomic Configurations of Point Defects in fcc Metals, in: Lattice Defects and Their Interactions, ed. R. R. Hasiguti (Gordon and Breach, New York) p. 79. DOYAMA, M., and R. R. HASIGUTI,1973, Cryst. Lattice Defects 4, 139. DRITTLER, K., H. J. LAHANNand H. WOLLENBERGER, 1969, Radiation Effects 2, 51. DUBINKO,V.L, 1991, J. Nucl. Mater. 178, 108. DUBSING,G., and W. SCHILLING,1969, Radiation Effects 1,65. DUNLOP,A., D. L ~ U and R A. BARBU,1993, J. Nucl. Mater. 205, 426. DWORSCHAK, F., 6. HOLFELDER and H. WOLLENBERGER, 1981, Radiation Effects 59,35. DWORSCIIAK, F., H. WAGNERand P. WOMBACHER, 1972, Phys. Stat. Sol. (b) 52, 103. DWORSCHAK, F., R. LENNARTZ, J. SELKEand H. WOLLENBERGER, 1978, J. Nucl. Mater. 69/70,748. DWORSC~K, E,TH.MONSAUand H. WOLLENBERGER, 1976, J. Phys. F 6,2207. E~KER, K.H., 1982, krhandl. DPG (VI) 17,892. EKRHART,€?, 1978, J. Nucl. Mater. 69/70, 200. EHRHART, P., 1985, Mater. Res. SOC.Symp. 41, 13.
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H.J. Wollenberger
Ch. 18, Refs.
EHRHART, P., 1991, in: Atomic Defects in Metals, ed. H. Ullmaier, Landoldt-Barnstein, ed.0. Madelung, Group III, Vol. 25 (Springer Berlin) p. 88. EHRHART, P., and B. SCH~NFELD, 1979, Phys. Rev. B 19, 3896. EHRHART, P., and B. SCH~NFELD, 1982, Self-lnmtitial Atoms and their Agglomerates in hcp Metals, in: Point Defects and Defect Interactions in Metals, eds. J. Takamura, M. Doyama and M. Kiritani (Univ. of Tokyo Press, Tokyo) p. 47. P.,and E. SEGURA,1975, X-ray Investigation of Interstitials and Interstitial Clusters after Low EHRHART, Temperature Electron-irradiation and Thermal Annealing of Gold, in: Roc. Int. Conf. on Fundamental Aspects of Radiation Damage in Metals, Gatlinburg, 1975, eds. M. T. Robinson and F. W. Young, Jr. (CONF-751006, Natl. Techn. Inf. Service, Springfield, VA 22161) p, 295. EHRHART, P., and W. SCHILLING,1974, Phys. Rev. B8.2604. 1982, Agglomerates of Intersptial Atoms and Vacancies in EHRHART,P., B. SCHBNPBLD and K. SONNENBERG, Electron-Irradiated Copper, in: Point Defects and Defect Interactions in Metals, eds. J. Takamura, M. Doyama and M. Kiritani (Univ. of Tokyo Press, Tokyo) p. 687. EHRHART,P.,H.D. CARSTANEN,A.M. FATTAH and J. B. ROBERTO, 1979, Phil. Mag. A 40, 843. and W. SCHILLING, 1974, Investigation of Point Defects and Their Agglomerates EHRHART,P., H. G. HAUBOLD in Irradiated Metals by Diffuse X-ray Scattering, in: Festkorperprobleme XlV/ Advances in Solid State F'hysia, ed. H. I. Queisser (Weweg. Braunschweig) p. 87. EHRHART,P., K.-H. ROBROCK and H. R. SCHOBER, 1986, in: Physics of Radiation Effects in Crystals, eds. R. A. Johnson and A.N. Orlov, Elsevier Science Publishers, p. 3. EMRICK,R.M., 1982, J. Phys. F 12, 1327. ENGLISH, C. A., A. J.E. FOREMAN, W.J. PHYTHIAN, D. J. BACUNand M.L. JENKINS,1992, Mater. Sci. Fomm 97-99, 1. ERGINSOY, C., G.H. VINEYARD and A. ENGLERT,1964, Phys. Rev. 133, A595. J. D., 1956, The Continuum Theory of Lattice Defects, in: Solid State Physics, 3, eds. F. Seitz and ESHELBY, D. Tumbull (Academic, New Yo&) p. 79. EVANS,I. H., 1987, Materials Science Fonun 15-18, 869. EVANS,R., 1977, Calculation of Point Defect Formation Energies in Metals, in: Vacancies '76, eds. R.E. Smallman and J. E. Harris (The Metals Society, London) p. 30. 1993, I. Nucl. Matex 205, 1. EYRE,B.L., and J. R. MA=& EYRE,B. L., M. H. h m o and R. E. SMALLMAN, 1977, Electron Microscopy Studies of Point Defect Clusters in Metals, in: Vacancies '76, eds. R. E. Smallman and J. E. Hams (The Metals Society, London) p. 63. FABER, K., and H.SCHULTZ,1977, Radiation Effects 31, 157. and H. SCHULTZ,1974, Scripta Metall. 8,713. FABER,K., J. SCHWEIKHARDT &DER,R., and A. S. NOWICK,1967, Phil. Mag. 15,805. FEE=, K., D. HOFFMANNand H. WOLLENBERGER, 1970, Cryst. Lattice Defects 1,245. FINNIS,M., P. AGNEWand A. J.E. FOREMAN,1991, Phys. Rev. B 44, 567. FLINN,J. E., and T. A. KENFIELD, 1976, Neutron Swelling Observations on Austenitic Stainless Steels Ircadiiated in EBR-II, in: Correlation of Neutron and Charged Particle Damage, Proc. of the workshop in Oak Ridge, ed.J. 0. Stiegler (CONF-760673, Natl. Techn. Inf. Service, Springfield, VA 22161) p. 253. FLYNN,C.P., 1968, Phys. Rev. 171,682. FLYNN,C. P., 1975, Thin Solid Films 25, 37. PLYNN,C.P., I. BASSand D. LAZARUS,1965, Phil. Mag. 11,521. POLES,S., M. 1. BASICES and M. S. DAW,1986, Phys. Rev. B33,7983. FOLWEILER, RC., and F.R BROTZEN,1959, Acta Metall. 7,716. FOREMAN, A. J. E., 1972, Harwell Report, AERE-R 7135. FOREMAN, A. J. E., C. A. ENGLISH and W. J. PHYTHIAN, 1991, AEA Technology Harwell Report, AEA-TRS2028 and 2031. FRANK, W., A. SEEGERand R. SCHINDLER, 1979, Radiation Effects 40, 239. 1983, Radiation Effects 71, 157. FRANK,W., and A. SEEGER, -EL., I., 1970, Theory of Point Defects in Metals, in: Vacancies and Interstitials in Metals, eds. A. Seeger, D. Schumacher, W. Schilling and J. Diehl (North-Holland, Amsterdam) p. 787.
Ch. 18, Refs.
Point defects
1713
FORDERFX,K., K . 2 DORING,M. GLADISCH,N. HAAS, D. HERLACH, J. MAJOR, H.-J. MUNDINGER, J.
ROSENKRANZ, W. SCHAFER, L.SCHIMMELG, M. SCHMOLZ,W. SCHWARZ and A. SEEGER, 1987, Mater. Sci. Forum 15-18, 125. GABRIEL, T. A., J. D. AMBURGEY and N. M. GREW, 1976, Nucl. Sci. and Eng. 61,21. GANNE,J. P., and Y. QuM, 1982, Intrinsic Thermal Expansion of Point Defects in Metals, in: Point Defects and Defect Interactions in Metals, eds. J. Takamura, M. Doyama and M. Kiritani (Univ. of Tokyo Press, 1982) p. 232. GAUSTER,W.B., S. MANTL,T. SCHOBERand W. TR~.SH.XUSER, 1975, Annealing of Dislocation Loops in Neutron-Irradiated Copper Investigated by Positron Annihilation, in: Proc. Int. Conf. on Fundamental Aspects of Radiation Damage in Metals, Gatlinburg, 1975, eds. M. T. Robinson and E W. Young, Jr. (CONF-751006, Natl. Tech. Inf. Service, Springfield, VA 22161) p. 1143. GEHLEN, P. C., I. R BEELERand R L. JAFFEB, eds., 1972, Interatomic Potentials and Simulation of Lattice Defects (plenum, New York). M. MILGRAMand G.H. VINEYARD, 1960, Phys. Rev. 120, 1229. GIBSON,J.B., A.N. GOLAND, GLASGOW,B.B., A. Sr-Amm, W.G.WOLFERand F. A. GARNER,1981, J. Nucl. Mater. 103/104,981. GRANATO,A.V., and T.G. NILAN,1965, Phys. Rev. 137, A1250. D. L. JOHNSON,G.G. SETSER,P. WALLACE and A. WONG,1982, GRANATO,A.V., I. HOLDER,K.L. HULTMAN, in: Point Defects and Defect Interaction in Metals, eds. J. Takamura, M. Doyama and M. Kiritani (University of Tokyo Press, Tokyo) p. 360. GRANATO, A. V., 1993, J. Non-Crystalline Solids 156-158, 402. GUINAN,M.W., C. L. SNEADand A. N. GOLAND, 1973, Radiation Effects 20, 33. H~~HNER,P., and W. FRANK,1992, Solid State Phenomena 23 8124,203. HAKEN,H., 1977, Synergetics (Springer Verlag, Berlin). HARA, Y., and S. NANAO,1982, in: Point Defects and Interactions in Metals, Proc. of Yamada Conf. V, eds. J. Takamura, M. Doyama, M. Kititani (University of Tokyo Press, Tokyo) p. 595. HARDY, J.R., and R. BULLOUGH,1967, Phil. Mag. 15,237. HATCHER, R., D.R. ZELLERand P.H. D~ERICHS.1979, Phys. Rev. B 19,5083. HAUBOLD,H.-G., 1975, Study of Irradiation-Induced Point Defects by Diffuse Scattering, in: Proc. Int. Conf. on Fundamental Aspects of Radiation Damage in Metals, Gatlinburg, 1975, eds. M.T. Robinson and E W. Young, Jr. (CONF-751006, Natl. Techn. Inf. Service, Springfield, VA 22161) p. 268. HAUBOLD,H.-G., 1976, Rev. Physique Appl. 11,73. HAUBOLD, H.-G., and D. MARTINSEN, 1978, J. Nucl. Mater. 69/70,644. HAUTOJARVI,P., 1987, h4ater. Sci. Forum 15-18, 81. HEALD,P. T., 1977, Discrete Lattice Models of Point Defects, in: Vacancies '76, eds. R. E. Smallman and I. E. Harris (The Metals Society, London) p. 11. HEALD,ET., and M.V. SPBIGHT,1974, Phil. Mag. 29, 1075. HEINISCH, H., 1981, J. Nucl. Mater. 103/10$, 1325. HEINISCH,H., 1990, Radiation Effects and Defects in Solids 1l3,53. HERTZ,W., and H. PEBL, 1975, J. Phys. F 5, 2241. HETTICH,G., H. MEHRSR and K.MAIER, 1977, Scripta Metall. 11,795. HOCH,M., 1970, Equilibrium Measurements in High-Melting-Point Materials, in: Vacancies and Interstitials in Metals, eds. A. Seeger, D. Schumacher, W. Schilling and J. Diehl (North-Holland, Amsterdam) p. 81. HODGES,C.H., 1970, Phys. Rev. Lett. 25,285. HOLDER,J., A. V. GRANATOand L. E. REHN, 1974, Phys. Rev. B10, 363. HOOD, G.M., 1986, J. Nucl. Mater. 139, 179. HOOD, G. M.,and R. I. SCHULTZ,1988, J. Nucl. Mater. 151, 172. 1984, J. Nucl. Mater. 126, 79. HOOD,G. M., R. J. SCHULTZand J.A. JACKMAN, HORSEWELL, A. and B,N. SINQH, 1988, ASTMSTP 955, 220. HOW, M., and M. L. SWANSON, 1982, Ion Channeling Investigations of the Interactions between Irradiationproduced defects and solute atoms in metals, in: Point Defects and Defect Interactions in Metals, eds. J. Takamura, M. Doyama and M. Kiritani (Univ. of Tokyo Press, Tokyo) p. 53. HULTMAN, K.L., J. HQLDBRand A. V. GRANATO,1981, J. Physique, Colloque C5-42, Suppl. 10, 753. HUNTINGTON, H.B., 1942, Phys. Rev. 61, 325.
1714
H. J. Wollenberger
Ch. 18, Refs.
HUNTINGTON, H. B., 1953, Phys. Rev. 91, 1092. IMAFUKU, M.,R YAMAMOTO and M. DOYAMA, 1982, Computer Studies of Self-Interstitids in Magnesium, in: Point Defects and Defect Interactions in Metals, eds. J. Takamura, M. Doyama and M. Kiritani (Univ. of Tokyo Press, Tokyo) p. 145. INGLE, K.W., R.C. PERRIN and H. R. SCHOBER,1981, J. Phys. F 11, 1161. ISHINO,S., 1993, J. Nucl. Mater. 206, 139. ISHINO, S., S. IWATA, Y.MATSUTANI and S. TANAKA, 1977, Computer Simulation and Neutron, Heavy-Ion and Electron Irradiation Correlation, in: Radiation Effects in Breeder Reactor Structural Materials, Scottsdale, eds. M. L. Bleiberg and J. W. Bennett (Met. SOC.AIME, Warrendale, PA) p. 879. ITO,K., Y. Omu and S. TANIOAWA, 1982, in: Positron Annihilation, Proc. 6th Int. Conf. on Positron Ann., eds. P.G. Coleman, S. C. Sharma, L. M. Diana (Noah-Holland h b l . Co., Amsterdam) p. 514. IWATA, T., and A. IWASE,1990, Radiation Effects and Defects in Solids 113, 135. JACKSON,J. J., 1979, Phys. Rev. B 20, 534. JXGER,W., 1981, J. Microsc. Spectrosc. Electron. 6,437. JAGER,W., and H. TRINKAUS, 1993, J. Nucl. Mater. 205,394. JXGER, W., T. P. EHRHARTand W. SCHILLING,1990, Radiation Effects and Defects in Solids ll3, 201. JANOT,C., and B. GEORGE,1975, Phys. Rev. B 12,2212. JANOT,C., B. GEORGE and P. DELCROIX, 1982, J. Phys. F 12,47. JANOT,C., D. MALLEJAC and B. GEORGE,1970, Phys. Rev. B 2,3088. JENKINS, M.L., and M. WILKENS,1976, Phil. Mag. 34,1155. 1993, J. Nucl. Mater. 205, 16. JENKINS,M. L., M. A. KIRK and W. J. PHYTHIAN, JOHNSON,P.B., D. J. MAZEI,and J. H. EVANS,1983, Radiation Effects 78, 147. JOHNSON, R.A., 1964, Phys. Rev. 134, A1329. JOHNSON,R. A., 1965, J. Phys. Chem. Solids 26,75. JOHNSON,R. A., 1966, Phys. Rev. 145,423. JOHNSON, R.A., 1968, Phys. Rev. 174,684. JOHNSON,R.A., 1970, Phys. Rev. B 1,3956. JOHNSON,R. A., 1973, J. Phys. F 3, 295. JOHNSON,R.A., and E. BROWN,1962, Phys. Rev. 127,446. JOHNSON, R. A., and J. R. B m m , 1977, Phys. Rev. 156, 677. W.G., T. LAURITZEN,J. W. ROSOLOWSKI and A.M. TURKALO, 1976, J. Metals 28, 19. JOHNSTON, JUNG,P., 1981a, Phys. Rev. B23, 664. JUNG,P., 1981b, Radiation Effects 59, 103. JUNG, P., 1991, in: Atomic Defects in Metals, ed. H. Ullmaier, Landoldt-Barnstein, ed. 0. Madelung, Group 111, Vol. 25 (Springer Verlag, Berlin) p. 1. JUNG,P., R. L. CHAPLIN,H. J. FENZL,K. REICHELTand P. WOMBACHER, 1973, Phys. Rev. B 8,553. KANZAKI,H. J., 1957, J. Phys. Chem. Solids 2,24. UPINOS, V. G., Y. U. N. OSETSKII and P. A. PLATANOV, 1990, J. Nucl. Mater. 173,229. KAUPFMAN, J. W., and I. S. KOEHLER,1952, Phys. Rev. 88, 149. KHELLAP,A., R. M. EMRICKand J. J. VUILLEMIN, 1988, Phys. Rev. B 37, 6717. KIENLE,W., W. FRANK and A. SEEGER, 1983, Radiation Effects 71, 163. KIM, S.M., 1986, Phys. Rev. B33, 1509. KIM,S.M., and A.T. STEWART, 1975, Phys. Rev. B 11,4012. KIM, S.M., J. A. JACKMAN, W. J. L. BUYERSand D.T.F~ERSON,1984, J. Phys. F 14,2323. KINCHIN, G. H., and R. S. PEASE, 1955, Rep. Prog. Phys. 18, 1. KING,W.E., and R. BENEDEK, 1981, Phys. Rev. B 23, 6335. and M. MESHII, 1981, Phys. Rev. B 23,6319. KING,W.E., K. L. MERKLE KJRITANI,M., 1982, Nature of Point Defects and Their Interactions Revealed by Electron Microscope Observation of Their Clusters, in: Point Defects and Defect Interactions in Metals, eds. J. Takamura, M. Doyama and M. Kiritani (Univ. of Tokyo Press, Tokyo) p. 59. KIRITANI, M., T.YOSHIIE,S. KOJIMA,Y. SATOHand K. HAMADA,1990, J. Nucl. Mater. 174, 327. KIRITANI, M., 1993, J. Nucl. Mater. 206, 156. KIRK, M. A., T. H. BLEWITC and T. L. SCOTT, 1978, J. Nucl. Mater. 69/70, 780.
Ch. 18, Refs.
Point defects
1715
K~TATIMA,K., Y. AKASHI and E. KURAMOTO, 1985, J. Nucl. Mater. 1338~134,486. KLAUMUNZER, S., 1992, Mater. Sci. Forum 97-99, 623. KLEMRADT,U., B. DRITTLEX, T.HOSHINO,R. ZELLER,P.H. DEDERICHSand N. STEFANOU, 1991, Phys. Rev. B 43,9487. KLUIN,J.-E., and T. HEELENKAMP, 1991, Phys. Rev. B 44, 11597. KNODLE, W.C., and J. S. KOEHLER,1978, J. Nucl. Mater. 69/70,620. KNULL, H., U. DEDEKand W. SCHILLING, 1974, J. Phys. F4, 1095. KOCH,J. M., and C. KOENIG, 1986, Phil. Mag. B 54, 177. KOCH,R., R.P. WAHIand H. WOLLENBERGER, 1981, J. Nucl. Mater. 103/104, 1211. KOLLERS, G., H. JACQWS,L.E. REHN and K. H. ROBROCK, 1981, J. Physique, Colloque C5-42, Suppl. 10,729. KONIG,D., I. V U L and ~ W. SCHILLING,1964, Phys. Stat. Sol. 7,591. KUSTER,W., and H. P. ~ E R 1965, , Z. Metallkunde 56, 760. KRISHAN,K., 1982, Radiation Effects 66, 121. QO-L~, H., 1970, Studies of Point Defects in Metals by Means of Mechanical and Magnetic Relaxation, in: Vacancies and Interstitiah in Metals, eds. A. Seeger, D. Schumacher, W. Schilling and J. Diehl (NorthHolland, Amsterdam) p. 667. KUGLER,H., I. A. SCHWIRTLICH, S. TAKAKI,K. YAMAKAWA, U.ZIEBART,J. PETZOLDand H. SCHULTZ,1982, Stage III Recovery in Electron-Irradiated bcc Transition Metals, in: Point Defects and Defect Interactions in Metals,eds. J. Takamura, M. Doyama and M.Kiritani (Univ. of Tokyo Press, Tokyo) p. 520. KURAMOTO, E., 1992, J. Nucl. Mater. 191-194, 1297. KUSMISS,J.H., and A.T. S T J ~ A R T 1967, , Adv. Phys. 16,471. LAM,N. Q.. N. V. DOANand Y. ADDA,1980, J. Phys. F 10,2359. LAM,N. Q.,N. V. DOAN,L. DAGENSand Y. ADDA,1981, J. Phys. F 11, 2231. LARSON, B. C., and F. W. Y O m G , Jr., 1982, Vacancy and Interstitial h p s in Irradiated Metals, in: Point Defects and Defect Interactions in Metals, eds. J. Takamura, M. Doyama and M. Kiritani (Univ. of Tokyo Press, Tokyo) p. 679. L A u P ~ I ~ A., R , W. FRANK,M. R t k r . ~ A. , S e m m and M. WILKENS,1981, Radiation Effects Lett. 67,95. LEADBETTER, A. J., D. M. T. NEWSHAM and N. H. PICTON,1966, Phil. Mag. 13, 371. LEE,C., and J.S. KOEHLER,1968, Phys. Rev. 176, 813. LEGRAND, l?, J. MORILLOand V. P~NTIKIS, 1993, Radiation Effects and Defects in Solids 126, 151. LEHMANN,C., 1977, Interaction of Radiation with Solids, in: Defects in Crystalline Solids, vol. 10, eds. S. Amehckx, R. Gevers and J. Nihoul (North-Holland, Amsterdam). LEHMANN, C., A. SCHOLZ and W. SCHILLING,1981, Computer Simulation of Radiation-Induced Defects, film (16 mm, colour, 32 min) produced and distributed by E m .Huschert Filmstudio (Weststr. 34-36, D-40597, DUsseldorf-Benrath, Germany). LEIBFRIED,G., 1965, Bestrahlungseffektein Festkiirpern (Teubner, Stuttgart). LEIB~IED,G., and N. BREUER,1978, Point Defects in Metals 1, in: Springer Tracts in Modern Physics, vol. 81, eds. G. H6hler and E.A. Niekisch (Springer, Berlin). LENGELZR,B., 1976, Phil. Mag. 34,259. LENGELER, B., and R.R. BOURASSA, 1976, J. Phys. F6, 1405. LENNARTZ,R., F. DWORSCHAK and H. WOLLENBERGER, 1977, J. Phys. F 7,201 1. W,V., J. M.LANOREand J. HILLAIRET,1973, Phil. Mag. 28, 373. LIITLE, E. A., 1993, J. Nucl. Mater., 206, 324. LIU PING and G. L. DUNLOF,1988, J. Mater. Sci. 23, 1419. LOPER, G. D., L. C. S ~ S K I M R M.,K. CWN and R. W. SIEGEL,1985, in: Positron Annihilation, eds. P. C. Jain, R.M. Singru, K.P. Gopinathan (World Scientific Publ. Co., Singapore) p. 461. LOSEHAND,R., E RAu and H. WGNZL,1969, Radiation Effects 2,69. LUCAS,G.E., 1993, J. Nucl. Mater. 206, 287. LUCASSON, P, G., 1975, in: Fundamental Aspects of Radiation Damage in Metals, Proc. Int. Conf. Gatlinburg, Tenn., USA, eds. M.T. Robinson and F. W. Young (USERDA Conf. 751006PlP2) p. 42. LUCASSON,P. G., and R.M. WALKER,1962, Phys. Rev. 127,485. Lwm, Y.N., M. DOYAMA and J. S. KOEHLm, 1968, Phys. Rev. 165,787. IMACKENZE, J. K., T.W.CRAIGand B. T. A. MCKEE,1971, Phys. Lett. 33A, 227.
1716
H.J. Wollenberger
Ch. 18, Refs.
MAIER,K, H. MEHRERand G. REIN, 1979~.2.Metallkunde 70,271. MAIER,K., H. METZ, D. HERLACH,H.E. ScHAEFER and A. SEECER,1978, J. Nucl. Mater. 69170,589. MANNINEN, M., R. NIEMINEN, P. HAUTOJ~VI and J. ARPONEN,1975, Phys. Rev. B 12,4012. MANSEL,W., J. MARANQOS and D. WAHL, 1982, J. Nucl. Mater. lOt?/loS, 137. MANSUR,L. K., 1979, Phil. Mag. A 39, 497. MANSUR,L. K, 1992, Materials Science Forum 9749,489. MANSUR,L. K., 1993, J. Nucl. Mater. 206, 306. V ! ,S. and W. TRIFTSHAUSER, 1978, Phys. Rev. B 17, 1645. MARADUDIN, A. A., E. W. MONTROLL,G. H. WEISSand I. P. IPATOVA, 1971, Theory of Lattice Dynamics in the Harmonic Approximation, Solid State Phys., Suppl. 111, eds. H. Ehrenreich, F. Seitz and D. Turnbull (Academic, New York). MARTIN,G., 1975, Phil. Mag. 32,615. MAFCTIN, G., 1977, J. de Physique, Colloque C-7, suppl. 4, 38, 419. MARTIN,G., 1980, Phys. Rev. B21, 2122. MASAMURA, RA., and G. SINES,1970, J. Appl. Phys. 41,3930. MATSUNO,N., 1977, J. Phys. SOC. Jpn. 42, 1675. MAURY,E, A. LUCASSON,P. LUCASSON,J. LE HERICY,P. VAJDA,C. Dlhnmov and 0. DIMITROV,1980, Radiation Effects 51, 57. and P. LUCASSON, 1978, Radiation Effects 38, 53. MAURY,E, M. BIGET,P. VAJDA,A. LUCASSON MCILWAIN, J., R. GARDINER,A. SOSINand S. MYHRA,1975, Radiation Effects 24, 19. MEECHAN, C.J., and A. SOSIN,1959, Phys. Rev. 113,422. METZNER,H., R. SIELEMANN, S : K L A U M ~ Z and E R E. HUNGER,1987, Mater. Sci. Forum 15-18, 1063. MIEDEMA,A.R., 1979, Z. Metallkunde 70, 345. MIJNARENDS, P. E., 1979, Electron Momentum Densities in Metals and Alloys, in: Topics in Current Physics 12, ed. P. Hautojtkvi (Springer, Berlin) ch. 2. MISEK,K., 1979, Czech. J. Phys. B29, 1243. MONSAU,TH.,and H. WOLLENBERGER, 1980, unpublished results. M o m , A.M., and E. J. SAVINO, 1981, Phys. Rev. B 23, 6494. MOSER,P., 1966, MBm. Sci. Rev. MBtallurg. 63, 431. Mom, N.F., 1932, Proc. Roy. SOC. A 135,429. MULLER,H.-G., 1979, thesis, UNv. of Bonn, unpublished. MGLLER,M., 1986, Reports of the Hahn-Meitner-InstitutBerlin, B 433. MUNDY,J.N., S. J. ROTHMAN, N. Q.LAM,H. A. HOW and L. J. NOWICKI,1978, Phys. Rev. B 18, 6566. MUNDY,J. N., S. T. OCKERSand L. C. SMEDSKYAER, 1987, Mater. Sci. Forum 15-18, 199. MYHRA,S., and R. B. GARDINER, 1973, Radiation Effects 18, 39. MYHRA,S., and R. B. GARDINER,1975, Radiation Effects 27, 35. NEELY,H. H., 1970, Radiation Effects 3, 189. NICKLOW,R.M., W. P. CRUMMETTand J.M. WILLIAMS,1979, Phys. Rev. B20,5034. NIESEN,L., 1981, Hyperfine Interactions 10,619. NORGETT,M. J., M.T. ROBINSONand LM. TORRENS,1974, Nucl. Eng. Design 33,50. NOWICK,A. S., and B. S. BERRY,1972, Anelastic Relaxation in Crystalline Solids, in: Mater. Science Series, eds. A.M. Alper, J.L. Margrave and A. S. Nowick (Academic, New York). O’NEAL, T.N., and R. C. CHAPLIN,1972, Phys. Rev. B 5,3810. OKUDA, S., 1975, Experimental Studies on Self-Interstitids in bcc Metals, in: Proc. Int. Cod. on Fundamental Aspects of Radiation Damage in Metals, Gatlinburg, 1975, eds. M. T. Robinson and F. W. Young, Jr. (COW-751006, Natl. Techn. Inf. Service, Springfield, VA 22161) p. 361. PARK,J. Y., H.-C. HUANG,R. W. SIEGELand R. W. BALLUFFI,1983, Philos. Mag. A48, 397. PARKER, C. A,, and K. C. RUSSELL,1982, in: Effects of Radiation on Materials, eds. H. R. Brager and J. S. Pemn (ASTEVI Publ. 04-78 2000-35, Philadelphia, PA.) p. 1042. WRY, W., G. VOGLand W. MANSEL,1982,Z. Phys. B46, 319. PICHON,R:, E. BISOGNI and P. MOSER,1973, Radiation Effects 20, 159. P L E I m , E, and C. HOHENEMSER, 1982, Phys. Rev. B 25, 106.
Ch. 18, Refs.
Point Meets
1717
POTER, D. I., L.E. REHN, P. R. OKAMOTO and H. WEDERSICH,1977, Void Swelling and Segregation in Dilute Nickel Alloys, in: Radiation Effects In Breeder Structural Materials (AIME, New York) p. 377. 1976, Phys. Stat. Sol. @) 71,515. PROFANT,M., and H. WOLLENBERGER, RECKNAGEL,E., G. SCHATZ and TH. WIECHERT, 1983, in: Hyperfine Interactionsof Radioactive Nuclei, Topics
in Current Physics 31, ed. J. Christiansen (Springer Berlin) p. 133.
REHN,L. E. and P. R. OKAMOTO,1983, in: Phase Transformations during Irradiation, ed. F. V. Nolfi, Jr. (Appl. Science Publ., London) p. 247.
REHN, L.E. and R. C. BIRTCHER,1993, J. Nucl. Mater. 205, 31. REHN,L. E., and K. H. ROBROCK,1977, J. Phys. F 7, 1107. REHN,L. E., K.H. ROBROCKand H. JACQUES,1978, J. Phys. F 8, 1835. RINNEBERG,H., and H. HAAS,1978, Hyperline Interactions 4, 678. RINNEBERG, H., W. SEMMLER and G. ANTFSBERGEX, 1978, Phys. Lett. A66,57. RIZK,R,, P.VAJDA,E MAWRY,A. LUCASSON and F? LUCASSON, 1977, J. Appl. Phys. 48, 481. RIZK,R., P. VAJDA,F. MAURY,A. LUCASSON, P. LUCASSON,C. DIMITROVand 0. DIMITROV,1976, J. Appl. Phys. 47,4740. ROBERTS, C. G., W.P. RICKEY and P.E. SHEARIN, 1966, J. Appl. Phys. 37,4517. ROBROCK,K. H., 1982, Study of Self-Interstitial-Atom-Solute-Atom-Complexes by Mechanical Relaxation, in: Point Defects and Defect Interactions in Metals, eds. J. Takamura, M. Doyama and M. Kiritani (Univ. of Tokyo Press, Tokyo) p. 353. ROBROCK, K. H., 1983, The Interaction of Self-Interstitial Atoms with Solute Atoms in Metals, in: Phase Stability and Solute Redistribution, eds. E Nolfi and J. Gittus (Appl. Sci. Publ., London) p. 115. ROBROCK, K.H., and B.R. SCHOBER,1981, J. Physique, Colloque C5-42, Suppl. 10, 735. ROBROCK, K. H., and W. SCHILLINQ,1976, I. Phys. F6, 303. ROTH, G., H. WOLLENBERGER, C. ZECKAUand K. LUCKE,1975, Radiation Effects 26, 141. RUSSELL,K.C., 1993, J. Nucl. Mater. 206, 129. SAHU,R €?,K. C. JAINand R. W. SIEGEL,1978, J. Nucl. Mater. 49/70, 264. SCHAEFER,H.-E., 1987, Phys. Stat. Sol. (a) 102,47. SCHAEFER,H.-E., D. BW'ITEWEG and W. DANDER,1975, Defects in High Purity Iron after 27 K Electron Irradiation, in: Proc. Int. Conk on FundamerltalAspects of Radiation Damage in Metals, Gatlinburg, 1975, eds. M.T. Robinson and E W. Young, Jr. (CONF-751006, Natl. Techn. Inf. Service, Springfield, VA, 22161) p. 463. SCHAEFER, H.E., 1982, in: Positron Annihilation, Proc. 6th Int. Conf. on Positron Ann., 4 s . P.G. Coleman, S. C. Sharma, L. M. Diana (North-Holland Publ. Co., Amsterdam) p. 369. SCHAEFER,H.E., R. GUGEL~~EIER, R. SCHMOLZ and A. SEEGER, 1987a, Mater. Sci. Forum 15-18, 111. SCHAEFER, H. E., W. STUCK,F. BANHART and W. BAUER,1987b, Mater. Sci. Forum 15-18, 117. SCHILLING,W., 1978, J. Nucl. Mater. 69/70,465. SCHILLING, W., and P. TISCHER,1967, Z. Angew. Phys. 22, 56. SCHILLING,W., G. BURGER,K. ISEBECK and H. WENZL,1970, Annealing Stages in the Electrical Resistivity of Irradiated fcc Metals, in. Vacancies and Interstitials in Metals, eds. A. Seeger, D. Schumacher, W. Schilling and J. Diehl (North-Holland, Amsterdam) p. 255. SCHINDLER,R., W. FRANK, M. RUHLE,A. SEEGERand M. WILKENS,1978, J. Nucl. Mater. 49/70,331. SCHOBER,H.R., V. K. TEWARY and P. H. DEDERICHS,1975.2. Phys. B 21, 255. SCHOBER, H.R., 1989, in. Phonon '89, eds. S. Hunklinger, W. Ludwig and G. Weiss, Vol. 1 (World Scientific, Singapore) p. 444. SCHOLZ,A., and C. LEHMANN, 1972, Phys. Rev. B6,813. SCHOITKY,G., A. SEEGERand G. SCHMID,1964, Phys. Stat. Sol. 4, 439. SCHROEDER, H., and B. STRITZKER,1977, Radiation Effects 33, 125. SCHROEDER, H., and W. SCHILLING, 1976, Radiation Effects 30, 243. SCHROEDER, H., R. hmmTz and U. DEDEK,1975, Recovery of Pure Lead after Electron Irradiation at 4.7 K and below 3 JC, in: Proc. Int. Conf. on Fundamental Aspects of Radiation Damage in Metals, Gatlinburg, 1975, 4 s . M. T. Robinson and F. W. Young, Jr. (CONF-751006, Natl. Techn. Inf. Service, Springfield, VA, 22161) p. 411.
1718
H.J. Wollenberger
Ch. 18, Refs.
SCHROEDER, K.,1980,Theory of Diffusion-Controlled Reactions of Point Defects in Metals, in: SpringerTracts in Modem Physics, vol. 87, eds. G. H6hler and E.A. Niekisch (Springer, Berlin) p. 171. SCHULTZ,H., 1987, Mater. Sei. Forum 15-18, 727. SCHULTZ, H., 1991, Mater. Sci. Eng. A 14, 149. SCHULTZ, H., 1991a, in: Atomic Defects in Metals, ed. H. Ullmaier, Landoldt-B6rnstein, ed. 0. Madelung, Group In, Vol. 25 (Springer Berlin) p. 115. SCHUMACHER, D., 1970, in: Vacancies and Interstitials in Metals, eds. J. Diehl, W. Schilling, D. Schumacher, A. Seeger (North-Holland Publ. Co., Amsterdam) p. 889. SCHWIRTLICH, I. A., J. PETZ~LD and H. SCHULTZ,1987, Phys. Stat. Sol. (a) 104,297. SEEBOECK, R., W. ENGEL,S. HOTH, R. KEITEL and W. WITTHUHN,1982, Vacancies in Zinc and Cadmium Produced by Proton- and Electron-Irradiation, in: Point Defects and Defect Interactions in Metals, eds. J. Takamura,M. Doyama and M. Kiritani (Univ. of Tokyo Press, Tokyo) p. 271. SEEGER, A. and E. MANN, 1960, J. Phys. Chem. Solids 12, 326. SEEGER,A., 1958, On the Theory of Radiation Damage and Radiation Hardening, in: Proc. 2nd U. N. Int Conf. on the Peaceful Uses of Atomic Energy, vol. 6 (United Nations, Geneva) p. 250. SEEGER, A., 1970, Radiation Effects 2, 165. SEEGER,A., 1973a, Cryst. Lattice Defects 4, 221. SEEGER, A., 1973b. I. Phys. F3,248. SEECIER, A,, 1975, The Interpretation of Radiation Damage in Metals, in: hoc. Int. Conf. on Fundamental Aspects of Radiation Damage in Metals, Gatlinburg, 1975, eds. M. T. Robinson and F. W. Young, Jr. (CONF-751006, Natl. Techn. Inf. Service, Springfield, VA 22161) p. 493. SEEGER, A., 1982, Phys. Lett.89A, 241. SEEGER, A., and H. KRONMULLER, 1987, Mater. Sci. Forum 15-18,65. SEEGER, A,, and W. FRANK,1983, Radiation Effects 71, 157. SEEGER,A., E. MANNand R. V. JAN, 1962, J. Phys. Chem. Solids 23,639. SEGURA, E., and P. EHRHART, 1979, Radiation Effects 42, 233. SEIDMAN,D.N., 1973, I. Phys. E Met. Phys. 3, 393. SEIDMAN, D.N., 1978, Surf. Sci. 70,532. Smz, E, and J. S. KOEHLER,1956, Displacement of Atoms during Irradiation, in: Solid State Physics. vol. 2, eds. E Seitz and D. 'hmbull (Academic, New York) p. 305. SHAH,N., and A.L. CATZ,1984, Phys. Rev. B 30,2498. SHIMOMURA, Y., K.YAMAHAWA, K. KITAGAWAand H. ODA, 1982, Studies of Point Defect Clusters with an Electron Microscope, in: Point Defects and Defect Interactions in Metals, eds. J. Takamura, M. Doyama and M. Kiritani (vniv. of Tokyo Press, Tokyo) p. 712. SHIMOMURA, Y., M. W. GUINANand T. DIAZDE LA RUBIA,1993, J. Nucl. Mater. 205,374. S ~ IY., , H.-E. SCand A. SEEGER,1989, in: Positron Annihilation, Proc. 8th Int. Conf. on Positron Annihilation, Gent, Belgium, 1988, eds. L. Dorikens-Vanpraet, M. Dorikens, D. Segm (World Scientific Publ. Co., Singapore) p.. (?) SHIRLEY, C. G., and R.L. CHAPLIN,1972, Phys. Rev. B 5,2027. SIEGEL, R.W., 1978, J. Nucl. Mater. 69/70, 117. S ~ E LR.,W., 198% in: Positron Annihilation, Proc. 6th Int. Conf. on Positron Ann., eds. P. G. Coleman, S. C. Shatma, L. M. Diana (North-Holland Publ. Co.,Amsterdam) p. 351. SIEGEL, R. W., 1982b, in: Point Defects and Interactionsin Metals, Proc. of Yamada Conf. V,eds. J. Takamura, M. Doyama, M. Kiritani, University of Tokyo Press, Tokyo, p. 533. SEW, R., 1968, Phys. Stat. Sol. 30, 645. SIGMUND, P., 1972, Rev. Roum. Phys. 17,969. SIMMONS,R. O., and R. W.BALLUPPI, 1960a, Phys. Rev. 117, 52. SIMMONS, R.O., and R W. BALLUPPI,1960b, Phys. Rev. 119,600. SIMMONS, R. O., and R. W. BALLUFFI,1962, Phys. Rev. 125, 862. SIMMONS, R. O., and R. W. BALLUPFI, 1963, Phys. Rev. 129, 1533. SINGH,B. N. and S. J. ZINKLE,1993, J. Nucl. Mater. 206, 212. SMIGELSKAS, A.D. and E.O. KIRKENDALL,1947, Trans. AIME 171, 130.
..
Ch. 18, Refs.
Point defects
17f9
SNIEG~WSKI ,J .J. and W. G. WOLF@&1984, Proc. Topical Conf. on Ferritic Alloys for Use in Nuclear Energy Technology, ATME,1984,579. SONNENBERG, K., and U. DEDEK,1982, Radiation Effects 61, 175. SONNENBERG,IC., W. SCHILLING, H. J. DIBBERT,K. MIKAand K. SCHRO~ER, 1972b, Radiation Effects 15, 129. SONNENBERG, K., W. SCHILLING, K. MIKAand K. DETTMANN, 1972a, Radiation Effects 16, 65. SOSIN,A., and W. BALIER,1968, Atomic Displacement Mechanism in Metals and Semi-Conductors, in: Studies in Radiation Effects in Solids, ed. G. J. Dienes, vol. 3 (Gordon and Breach, New York). SPIRIC, V., L.E. m, K.H. ROBROCK and W. SCHILLING,1977, Phys. Rev. B 15,672. STOR.E., and G.R ODETTE,1985, J. Nucl. Mater. 131, 118. STOTT,M. J., 1978, J. Nucl. Mater. 69/70, 157. SULPICE, G., C. MINIER,P. M o m and H. BILGER,1968, J. Physique 29,253. TAKAKI,S., J. Fuss, H. KuGL!%, U.DEDEKand H. SCHULTZ, 1983, Radiation Effects 79,87. TENENBAUM, A., and N.V. DOAN,1977, Phil. Mag. 35, 379. TEWORDT, L., 1958, Phys. Rev. 109.61. THEIS,U. and H. WOLLENBERGER, 1980, J. Nucl. Mater. 88, 121. TRIFTSHAUSER,W., 1975, Festkoprprobleme XWAdv. in Solid State Phys., ed. H. J. Queisser (Weweg, Braunschweig) p. 381. TRIFTSHAUSER, W., and J.D. MCGERVEY,1975, Appl. Phys. 6, 177. TRINRAUS,H., 1975, Theory of Polarization-Induced Elastic Interaction of Point Defects, in: Proc. Int. Conf. on Fundamental Aspects of Radiation Damage in Metals, Gatlinburg, 1975, eds. M.T. Robinson and F.W. Young, Jr. (CONF-751005. Natl. Techn. Inf. Service, Springfield, VA 22161) p. 254. TRINKAUS, H., 1983, Radiation Effects 78, 189. TRINKAUS, H., 1990, J. Nucl. Mater. 174, 178. -US, H., B.N. SINGHand A. J.E. FORFMAN,1992, J. Nucl. Mater. 199, 1. TRINKAUS, H., B. N. SINGHand A. J. E. FOREMAN,1993, J. Nucl. Mater. 206,200. TRINKAUS, H., V. NAUNDORF,B.N. SINGHand W.H. Woo, 1994, J. Nucl. Mater. 210,244. TZANETAKIS,P., J. HILL^ and G. REVEL,1976, Phys. Stat. Sol. (b) 75,433. URBANK., and N. Y ~ D A 1981, , Phil. Mag. A 44, 1193. URBAN,K., B. SAILB, N. YOSHIDAand W. ZAG,1982, in: Point Defects and Defect Interaction in Metals, eds. J. Takamura, M. Doyama and M. Kiritani (UNv. of Tokyo Press, Tokyo) p. 783. VAIDA,P., 1977,Rev. Mod. Phys. 49,481. VANBWREN, H.G., 1955, Z. Metallk 46,272. VAN DEN BEUKBL, A., 1970, in: Vacancies and Interstitials in Metals, eds. J. Diehl, W. Schilling, D. Schumacher, A. Seeger (North-Holland Publ. Co., Amsterdam) p. 427. VANDENBORRE, H., L. STALSand J. NIHOWL,1968, Int. Conf. Vacancies and Interstitials in Metals (JiilichConf. 2) p. 802. VANDENBORRE, H., L. STALS, K. SONNENBERG and J. NIHOUL,1972, in: Defects in Refractory Metals, eds. R. De Batist, J. Nihoul, L. Stals (Centre d'Etudes de I'Energie NuclBaire, Mol, Belgien) p. 49. 1982, Phys. Stat. Sol. (b) 110.9. VAROTWS,P., and K. ALEXOPOULO~, VOOL,G., and W. MANSEL,1975, Mossbauer Studies of Interstitials in fcc Metals, in: Proc. Int. Conf. on Fundamental Aspects of Radiation Damage in Metals, Gatlinburg, 1975, eds. M. T. Robinson and F. W. Young, Jr, (CONF-751006, Natl. Techn. Inf. Service, Springfield, VA 22161) p. 349. VOGL, G., W. MANSELand P. H. DEDERICHS,1476, Phys. Rev. Lett. 36, 1497. WAGNER, C., and J. B m , 1936, Z. Phys. Chem. B32, 113. WAGNER,R., 1982, Field Ion Microscopy, in: Crystals, Growth, Properties and Applications, ed. H.C. Freyhardf (Springer, Berlin). WAITE, T. R., 1957, Phys. Rev. 107,463 and 471. WALLNER,G., 1983, qissertation, Techn. LJniversitXt Miinchen. G., K. B ~ N ~ N andGU. DEDEK,1986, J. Phys. F 16, 257. WALLNER, WEILER,W., and H.E. SCHAEFER,1985, J. Phys. F 15, 1651. WENZL,H.,1970, Physical Properties of Point Defects in Cubic Metals, in: Vacancies and Interstitials in Metals, eds. A. Sebger, D. Schumacher, W. Schilling and J. Diehl (North-Holland, Amsterdam) p. 363.
1720
H.J. Wollenberger
Ch. 18, Refs.
WEN=, H., E KERSCHER,V. FISCHER,K. EHRENSPERGERand K. PAPATHANASSOPOULOS, 1971,Z. Naturf. 26, 489. WEST,R.N., 1973, Adv. Phys. 22,263. WEST, R. N., 1979, Positron Studies of Lattice Defects in Metals, in: Topics in Current Physics 12, ed. P. Hautojhi (Springer, Berlin) ch. 3. W m , H., 1992, in: Diffusion in Solids - Unsolved Problems, ed. G.E. Murch (J’rans. Tech. Publ., Aedermannsdorf, Switzerland) p. 55. WICHERT,TH., 1982, PAC Study of Point Defects in Metals, in: Point Defects and Defect Interactions in Metals, eds. J. Takamura, M. Doyama and M. Kiritani (Univ. of Tokyo Press, Tokyo) p. 19. WICHERT,TH.,M. DEICHER, 0. ECHT and E. RECKNAGEL,1978, Phys. Rev. Lett. 41, 1659. WIEDERSICH, H., and N. Q. LAM,1983, in: Phase Transformations during Irradiation, ed. F. V. Nolfi, Jr. (Appl. Science Publ., London) p. 1. WIEDERSICH, H., P. R. OKAMOTU and N. Q. LAM,1979, J. Nucl. Mater. 83.98. Wrurs, J. R., 1975, J. Mech. Phys. Sol. 23, 129. WOLLENBERGER, H., 1965, Fruhjahrstagung DPG Freudenstadt, unpublished . WOLLENBERGER, H., 1970, Production of Frenkel Defects during Low-Temperature. Irradiations, in: Vacancies and Interstitials in Metals, eds. A. Seeger, D. Schumacher, W. Schilling and J. Diehl (North-Holland, Amsterdam) p. 215. WOLLENBERGER, H., 1975, Defect interactions above stage I, in: Proc. Int. Conf. on Fundamental Aspects of Radiation Damage in Metals, Gatlinburg, 1975, eds. M. T. Robinson and E W. Young, Jr. (CONF-751006, Natl. Tech. Inf. Service, Springfield, VA 22161) p. 582. WOLLENBERGER, H., 1978, J. Nucl. Mater. 69/70, 362. WOLLENBERGER, H., 1983, in: Physical Metallurgy, Vol. 2, eds. R. W. Cahn, P. Haasen (Elsevier Science Publishers, Amsterdam) p. 1140. WOLLENBERGER, H., and B. KUL, 1989, J. Nucl. Mater. 169, 126. WOUENBERGER,H., and J. WURM,1%5, Phys. Stat. Sol. 9, 601. Woo, C.H., A. A. SEMENOVand B.N. SINGH,1993, J. Nucl. Mater. 206, 170. WOOD, R. E, and M. MOSTOLLER, 1975. Phys. Rev. Lett. 35,45. YAMAKAWA, K., I. MUKOUDA and Y. S ~ O M U R1992, A , J. NucL Mater. 191-194,396. YAMAMOTO, R., 1982, Lattice Vibrations around a Vacancy and Vacancy Clusters in Metals, in: Point Defects and Defect Interactions in Metals, eds. J. Takamura, M. Doyama and M. Kiritani (vniv. of Tokyo Press, Tokyo) p. 120. YOUNG, E W., Jr., 1978, J. Nucl. Mater. 69/70, 310. ZAISER,M., W. FRANK and A. SEEGER,1992, Solid State Phenomena 23 8z 24,221. ZELLER,R., and P. H. DEDERICHS,1976, Z. Phys. B 25, 139.
Further reading Bleiberg, M. L., and J. W. Bennett, eds., 1977, Radiation Effects in Breeder Reactor Structural Materials, Roc. Int. Conf. Scottsdale 1977 (AIME, New York). Carpenter, G.J. C., C. E. Coleman and S.R. MacEwen, eds., 1980, Fundamental Mechanisms of RadiationInduced Creep and Growth, F’roc. Int. Conf. Chalk River 1979, J. Nucl. Mater. 90. Corbett, J. W., 1966, Electron Radiation Damage in Semiconductors and Metals, in: Solid State Physics Suppl. 7, eds. E Seitz and D. Tumbull (Academic, New York). Corbett, J. W., and L. C. Ianniello, eds., 1972, Radiation-InducedVoids in Metals, hoc.Int Conf. Albany 1971 (CONF-710601, Natl. Techn. Inf. Service, Springfield, VA 22161). Flynn, C. P., 1972, Point Defects and Diffusion (Clarendon Press, Oxford). Gavillet, D., M. Victoria, B. N. Singh and A. Horsewell, eds., 1993, Proc. Workshop on Time Dependence of Radiation Damage Accumulation and its Impact on Materials Properties, J. Nucl. Mater. 206. Gehlen, P. C., J. R. Beeler, Jr., and R. I. Jaffee, eds., 1971, Interatomic Potentials and Simulation of Lattice Defects, Battelle Inst. Materials Science Colloquia, Seattle and Harrison, Hotsprings (Plenum, New York). Gittus, I., 1978, Irradiation Effects in Crystalline Solids (Appl. Science Publ., London).
Ch. 18, Refs.
Point defects
1721
Green, W.V., M. Victoria,T. Leffers and B.N. Singh,eds., 1990, Radiation Effects and Defects in Solids 113, 1. Hautojhi, P., ed., 1979, Positrons in Solids, Topics in Current Physics, vol. 12 (Springer, Berlin). Hlihler, G., and E. A. Niekisch, eds., 1978, Point Defects in Metals I, Springer Tracts in Modem Physics, vol. 81 (Springer, Berlin). Hlihler, G., and E. A. Niekisch, eds., 1980, Point Defects in Metals II, Springer Tracts in Modem Physics, v01.87 (Springer, Berlin). Holland, J. R., L. K. Mansur and D.I. Potter, eds., 1981, Proc. Symp. On Phase Stability during Irradiation (Met. SOC. AIME, Warrendale, PA). Ishmo, S., M. Kintani, T.Kondo and J.L.Scott, eds., 1991, Fusion Reactor Materials, J. Nucl. Mater. 179-181. Klueh, R. L., R. E. Stoller and D. S. Gelles, eds., 1992, Fusion Reactor Materials, J. Nucl. Mater. 191-194. Krippnec, M., ed., 1969, Radiation Damage in Reactor Materials, Proc. Symp. Vienna 1969 (IAEA, Vienna). Nelson, RS, 1968, The Observation of Atomic Collisions in Crystalline Solids, in: Defects in Crystalline Solids, vol. 1, eds. S. Amelinckx, R Gevers and I. Nihoul (North-Holland, Amsterdam). Nygren, RE., R.E. Gold and R. H. Jones, eds., 1981, Fusion Reactor Materials, Proc. 2nd Topical Meeting on Fusion reactor Materials, Seattle, J. Nucl. Mater. 1031104. Peterson, N. L., and R. W. Siegel, eds., 1976, Properties of Atomic Defects in Metals, Proc. Int. Conf. Argonne, J. Nucl. Mater. 69/70. Proc. Workshop ‘Time Dependence of Radiation Damage Accumulation and its Impact on Materials Properties”, 1993, J. Nucl. Mater. 206. Robinson, M.T.,and F. W. Young, Jr., eds., 1975, Proc. Int. Conf. on Fundamental Aspects of Radiation Damage in Metals, Gatlinburg 1975 (CONF-751006, Natl. Techn. Inf. Service, Springfield, VA 22161). Schilling, W. and H. Ullmaier, 1994, Physics of Radiation Damage in Metals, in: Nuclear Materials, Part 11, 179, ed. B. R.T. Frost, (Volume 1OB of Materials Science and Technology, ed. R W. Cahn, P. Haasen and E. J. Kramer, VCH, Weinheim) Seegcr, A., D. Schumacher, W. Schilling and J. Diehl, eds., 1970, Vacancies and Interstitials in Metals (NorthHolland, Amsterdam). Singh, B. N., A. Horsewell, M. Victoria and W. V. Green, 1990, e&., J. Nucl. Mater. 174, 125. Stiegler, J. O., ed., 1976, Correlation of Neutron and Charged Particle Damage, Proc. Workshop Oak Ridge 1976 (CONF-760673, Natl. Techn. Inf. Service, Springfield, VA 22161). Stiegler, J. O., ed., 1979, Workshop on Solute Segregation and Phase Stability During Irradiation, Gatlinburg 1978, J. Nucl. Mater. 83. Wiffen, F. W., I.H. DeVan and J. 0. Stiegler, eds., 1979, Fusion Reactor Materials, Proc. 1st Topical Meeting Miami, J. Nucl. Mater. 85/86. Woo, C. H., and RJ. McElroy, eds., 1988, J. Nucl. Mater. 159, 1.
CHAPTER 19
METASTABLE STATES OF ALLOYS R. W. CAHN and A. L. GREER Department of Materials Science and Metallurgy University of Cambridge Cambridge CB2 3QZ, England
R. W Cahn and I? Haasenf’, eds. Physical Metallurgy; fourth, revised and enhanced edition Q Elsevier Science BY 19%
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R. W Cuhn andA. L. Greer
Ch. 19, 0 1
1. Introduction 1.1. General features Much of the development of practical metallic materials has involved dealing with metastable states, Just what is meant by “metastable” we will consider more carefully in § 1.3; for the moment we use it to describe any state with a Gibbs free energy higher than the lowest value corresponding to stable equilibrium. Conventionally the access to metastable states is achieved by solid-state quenching. Examples are the quenching of steels from the austenite phase field to form martensite (ch. 17, $3.2) and the quenching of solution-treated Al-Cu alloys to obtain a supersaturated solid solution suitable for agehardening ($5.1, below), These examples illustrate the point of obtaining metastable states: the relaxation of the constraint of being in equilibrium gives access to a wider range of structures, some of which can be expected to show superior properties. They also illustrate that the metastable state may be of interest as the final product or as an intermediate on the way to the final product. At least since the 1920s there has been interest in developing new materials by increasing the departure from equilibrium, mainly by rapid quenching of the liquid. This grew into a large research effort following the work of Duwez and colleagues in 1959-60 (reviewed by D u w ~ [I967]), z in which gun splat-quenching was used to demonstrate complete, metastable, solid solubility in Ag-Cu and metallic glass formation in Au-Si and Au-Ge. Rapid liquid quenching has since been used in a very wide range of fundamental studies as well as being adopted as a significant commercial production process, particularly for magnetic materials. In the early 1980s interest turned also to solid-state processing as a way to obtain alloys in states far from equilibrium. This built on a great deal of earlier work, including the development in the 1960s of mechanical alloying to produce high-temperature alloys (BENJAMIN [1970]). The microstructural manifestations of the departures from equilibrium achieved by novel processing routes can be classified under five headings: (i) Increased defect concentrations. These include increased concentrations of vacancies (and to a lesser extent interstitials), dislocations, stacking faults, twin boundaries and grain boundaries. Another characteristic to be put in this category is an increased level of chemical disorder (that is, reduced order parameter) in ordered solid solutions and compounds. (ii) Microstructural rejinement. This involves finer scale distributions of different phases and of solute. Examples include finer dendrite arm spacings, eutectic spacings and precipitate diameters. (iii) Extended solid solubility. A stable crystalline phase may be found with solute levels beyond the solubility limit at the ambient temperature, or indeed beyond the equilibrium limit at any temperature (see account of solute trapping in ch. 8, $5 5.2, 8.1.7). (iv) Metastable phases. A metastable phase may form during processing. It may be one that is not found in equilibrium under any conditions in the system of interest, or it may be so found but appear metastably under different conditions of composition, temperature or pressure. This heading includes crystalline and quasicrystalline intermetallic compounds.
Ch. 19, § 1
Metastable states of alloys
1725
(v) Metallic glasses. These are a special case of a metastable phase. When the processing is by cooling a liquid they form not by a phase transformation, but by a continuous congealing of the liquid. Some of the kinetic aspects of rapid solidification have been dealt with in ch. 8. The present chapter is concerned more generally with the wide variety of methods for producing alloys in states which are far from equilibrium, that is, with novel methods which lead to larger departures from equilibrium than can be achieved by conventional methods. It is also concerned with the metastable products themselves, and in particular with the metallic glassy state, which is the most distinctive product of the novel methods. Few surveys have sought to cover metastable states from such a broad perspective; another recent example is a book chapter (MARTIN,CANTORand DOHERTY [1996]).
1.2. Methods for achieving metastability Here, an outline is given of some of the very diverse methods which have been applied, and links between them are discussed. Details of the methods are presented in 3 3. Most of the effort on developing new production methods has focused on cooling rate. This depends on the dimensions of the sample, the heat conduction in the material, and the rate of heat transfer to the surrounding quenching medium. If the sample to be quenched is initially solid, its dimensions are fixed and often not ideally suited to rapid heat extraction. In addition, efficient heat transfer may be difficult to achieve. On the other hand if the sample is initially liquid, it can be flattened into a thin sheet in intimate contact with a good solid heat sink. This is the basis of the Duwez method, subsequently developed into diverse continuous production techniques described in 3 3.1. In addition to solid heat sinks, the heat from the liquid alloy can be extracted into another liquid or a gas. The most common rapid liquid quenching methods give products in droplet, wire, ribbon or sheet form, with at least one dimension as small as 10 to 100 pm. The cooling rate is commonly 10'- lo6 Ws.Rapid liquid quenching is also readily achieved when the surface of a bulk sample is melted; the bulk itself then acts as the heat sink (8 3.5). The melting can be by a scanned continuous heating beam or by a stationary pulsed beam. For laser beams, pulse durations can be ultra short - in the nanosecond or even picosecond range - giving very thin melted layers and quenching rates as high as 1014 K/s; this is the highest liquid quenching rate which can be achieved &IN and SPAEPEN [1984]). The objective of rapid heat extraction is to obtain rapid movement of the solid-liquid interface. The latent heat of freezing released at the interface is extracted through the growing solid into the quenching medium. However, the latent heat can alternatively be extracted into the liquid ahead of the interface if that liquid is sufficiently undercooled. The methods for obtaining highly undercooled liquids are described in 5 3.2; they have attracted much interest because of their suitability for fundamental studies of the mechanisms of metastable phase formation. A number of other techniques are associated with the concept of rapid quenching, though less obvi6usly than with the liquid case. When a solid is grown from atoms in the vapor phase (generated by evaporation or sputtering), the effective quench rate in the References: p . 1818.
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R. U! Cahn and A. L Greer
Ch. 19, Q 1
process of condensation is as high as 10l2K/s (83.3). Other methods in which the mobility of the atoms joining the solid is rapidly reduced are those based on chemical or electrochemical precipitation or deposition from solution (5 3.6). Irradiation of a solid leads to locally disturbed configurations which regain thermal equilibrium very quickly; again the effective local quench rate is of the order of 10l2K/s. Amorphization by irradiation is considered in $3.4. Chemical reactions can proceed towards equilibrium in stages, and the intermediate stages can yield a metastable phase. Just this is found in the solid-state amorphization reaction (8 3.7) in which an amorphous alloy can be produced by the reaction of two solid metallic elements. Severe mechanical deformation can lead to metastable states (8 3.8). The deformation forces the production of disturbed configurations, and in this way can be considered akin to techniques such as irradiation. On the other hand it can be used to bring different phases into intimate contact, promoting solid-state reactions which can yield metastable phases. Finally, high pressure can induce metastability.
1.3. The nature of metastability A classification of thermodynamic states is illustrated in fig. 1. In stable equilibrium the free energy (the Gibbs free energy for a system under constant pressure) is at the lowest possible minimum. The minimum shown schematically in the figure may represent the stable state of a single phase or of a phase mixture. A metastable state is one in internal equilibrium, that is, within the range of configurations to which there is access by continuous change, the system has the lowest possible free energy. However, if there were a large fluctuation - the nucleation of a more stable phase - transformation to the new phase would occur. A good example of a thermodynamically metastable state is an undercooled liquid; this has well defined equilibrium properties at each
Fig. 1. A schematic illustration of possible thermodynamic states of alloys: (a) unstable equilibrium, (b) metastable equilibrium, (c) stable equilibrium, and (d) unstable non-equilibriumstate. The vertical axis shows Gibbs free energy, the horizontal, a parameter representing changes in stn~cture.
Ch. 19, § 1
Metastable states of alloys
1727
temperature, but contact with a nucleant will result in rapid transformation to the stable crystalline phase. A state of unstable equilibrium is destroyed by any fluctuation, however small; an example would be a precipitate dispersion (subject to Ostwald ripening) in which all the precipitates have exactly the same size. And there are unstable states, not in equilibrium at all. The term “metastable” was invented by the German physical chemist, Wilhelm Ostwald, in 1893. In his textbook (OSTWALD [1893]) he defined the newly named concept as follows: “Of a physical system, persisting (in its existing state) when undisturbed or subject to disturbances smaller than some small or infinitesimal amount, but passing to a more stable state when subject to greater disturbances.” As elucidated by TURNBULL [1981], all solids of practical interest are configurationally frozen, that is, under the conditions of observation, the atomic configuration does not undergo significantchanges. In this way microstructures which are thermodynamically unstable are readily found, and are indeed much more common than microstructures in true equilibrium, whether stable, metastable or unstable. Examples are a supersaturated solid solution in the spinodal range, a dispersion of precipitates of varied size, and a glass; all of these can relax toward lower free energy states continuously, without any nucleation barrier. That these states persist at all is of course because of kinetic barriers to atomic motion. As suggested by Turnbull, the best practice may be to describe a microstructure in terms of its thermodynamic state before configurational freezing set in. In this way, a glass would be considered metastable and a supersaturated solid solution in the spinodal range would be considered unstable. A common practice, however, is to term all such configurationally frozen states “metastable”. Although such a usage is approximate at best, we shall adopt it in this chapter. By considering the local potential wells between which atoms must make diffusionaljumps, even states which are thennodynamically unstable may be thought of as kinetically metastable. Such kinetic metastability can exist only where thermal activation of atomic jumps is required. Transformations which do not require thermal activation cannot be suppressed; martensitic transformations come in this category, and Turnbull suggests that crystallization of amorphous pure metals does also. Taking the definition of a metastable state to be any state which has a free energy in excess of that of stable equilibrium, it is then natural to quantify the degree of metastability by the value of the excess. The novel production techniques and microstructures associated with large excess free energies are the focus of this chapter. By novel methods such as those outlined in 0 1.2, excess free energies as high as RT, (where T, represents the average melting point of the elements in the system) can be retained in configurationally frozen states. Such values are up to twice the highest achieved by conventional processing. In the various microstructural effects of novel processing methods (0 l.l), the excess free energy can be seen to arise from three basic kinds of metastability (TURNBULL[19811): morphological, compositional and structural. In morphoZogicaZ metastability, the excess free energy comes from defects of all kinds including interphase boundaries; it is associated with (i) increased defect concentrations and (ii) microstructural refinement. Compositionalmetastability is found in (iii) extended solid solubility. SrmcturaZ metastability is associated with (iv) metastable crystalline References: p. 1818.
1728
R. U! Cahn and A. L.. Greer
Ch. 19, $ 2
phases and (v) metallic glasses. Typical excess free energies associated with these types of metastability are: morphological I0.1 RT, structural 1 0 . 5 RT, compositional I1.0 RT, As described by Turnbull, such excess values can be considered to be attained in general by a process of energizing the system followed by quenching to effect configurationalfreezing. Considering the processing routes outlined in 0 1.2, examples of energization can be seen to include melting, generation of gaseous species by evaporation or sputtering, dissolution, irradiation or mechanical deformation. In the solid-state amorphization reaction, the energization is the creation (by deposition or mechanical working) of interphase interfaces. The quenching, typically characterized by a cooling rate, is needed to reach a configurationally frozen state while retaining a large excess energy. It is useful to consider what constitutes configurational freezing. This does not occur at a single well defined temperature, but reflects the range of types of atomic motion. Roughly, however, it can be considered that diffusive motion of atoms within bulk phases (that is, lattice diffusion in a crystal, or diffusion in a glass) is frozen out below a reduced temperature (T/T,) of = 0.4. For grain-boundary and free-surface diffusion, the corresponding reduced temperatures are = 0.1 and = 0.2. Though a highly metastable material may have a lower temperature for effective configurational freezing because of for example a larger grain-boundary area, the fundamental temperatures for freezing the various atomic processes stay the same as in materials near equilibrium. Thus the retention of large excess free energies in practical materials is not necessarily thwarted under normal service conditions by any increased susceptibility to structural change. Configurationally frozen normal materials, even when processed by novel methods, could be in thermodynamically stable states, and such states by definition have the largest driving forces for their formation. It is reasonable to ask, therefore, why metastable materials are so commonly obtained. They must be favored kinetically. The nucleation of one phase in another is easier with lower interfacial energy between the phases, and that energy tends to scale with the entropy difference between the phases. Growth of one phase in another is favored if substantial solute partitioning is not required. Thus a transformation is in general more likely if the new phase is more similar to the old, either in entropy or composition. This has been described as a “step rule” (TURNBULL [19Sl]), and it will often lead to metastable products.
2. Formation of metallic glasses (amorphous alloys) 2.1. Formation and thermodynamics The metastable alloy structure most associated with novel processing methods and with large departures from equilibrium is the metallic glass. Strictly, a glass is formed when a liquid which is cooled fails to crystallize and instead continuously and uniformly congeals into a solid. The glass has essentially the structure of the liquid, combined with
Ch. 19, $ 2
Metastable stufes of alloys
1729
the resistance to shear deformation of a solid. Non-crystalline, amorphous, alloys can be obtained by any of the methods outlined in 0 1.2, and not just those involving quenching of a liquid. In this section we show that amorphous alloys made by liquid quenching can be considered as true glasses and that those prepared by other methods are not significantly different. The thermodynamics of the glassy/amorphous state are considered in relation to the formation of amorphous alloys. We start by pointing out that amorphous and glassy states are not constrained to be the same, and illustrate this with the case of silicon. Liquid silicon has a high density and is metallic. Crystalline silicon has tetrahedral covalent bonding. Amorphous silicon, which can be made by vapor deposition as well as by solidification from the liquid, also has local tetrahedral coordination with covalent bonding (POATEet al. [1987]). In its bonding type, solid amorphous silicon is therefore quite distinct from the liquid and cannot be regarded as a glass, which if formed would be dense and metallic. For example, this has the result that crystallization of amorphous silicon is a quite different transformation from crystallization of liquid silicon, as illustrated by the different temperature dependences of the two processes (STIFPLER,EVANS and GREER[1992]). Figure 2 shows schematically the changes in properties which occur when a liquid is cooled into the crystalline or glassy states. At the higher temperatures, range A, the melt is in equilibrium. It is extremely difficult (though possible under certain circumstances, PEMPEZKO and GLJ~ITER [19831) to superheat a crystal above the thermodynamic melting point, so that metastable states are very rare in range A. In defining the boundary between ranges A and B, we take Tf(the equilibrium freezing temperature); this is not necessarily the same as T,, the melting temperature, for alloys with a solidus-liquidus gap, but we ignore the complexities arising from an interposed solid + liquid range. In effect, the diagram is for a system in which all the transformations are without solute partitioning. In range B, the melt is thermodynamically undercooled and will crystallize rapidly if a critical nucleus is provided. At least in the higher temperatures in this range the liquid has a low viscosity, and crystallization can be avoided only because fast cooling gives no time for nucleus formation. At the lowest temperatures in this range the liquid viscosity increases rapidly with falling temperature, and a glass can still be formed even in the presence of nuclei because the low atomic mobility stifles crystal growth. Devitrification studies (0 4.3) provide evidence for copious nucleation occurring in the later stages of a quench without significant overall transformation of the liquid. In range B, the viscosity varies over some fifteen orders of magnitude, and though most of the range is not accessible to direct measurement (because crystallization would intervene during the measurement), the curve can be estimated by interpolation between experimentally accessible points in the neighborhood of P and Q. At all temperatures in range B the atomic mobility is adequate for the liquid to remain in configurational equilibrium (that is, in an internal equilibrium in which it has the lowest free energy of any state which it can reach by cofitinuous change). Crystallization, if it occurs, involves a discontinuous change in properties (as shown, for example, by volume in fig. 2). Crystallization can occur only in range B, and if it is avoided on cooling, range C is reached. Range C is defined by the glass fransition References: p . 1818.
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R. W Cahn and A. L Greer
Ch. 19, 5 2
-
TEMPERATURE
Fig. 2. The glass transition: variation of several properties with temperature. See text.
temperature T,, below which the liquid configuration is congealed in a pattern which corresponds to equilibrium at Tg.Experimentally, Tgis close to the temperature at which the viscosity reaches 10" Pa.s. The nature of the transition at Tg can be seen by considering the temperature dependence of liquid/glass properties, for example volume and viscosity as shown in fig. 2. Above Tg the properties show a strong temperature dependence, reflecting the configurational changes in the liquid. Below Tgr however, atomic motions in the melt are so slow that during a quench (as presupposed in drawing fig. 2) there is no longer time for configurational changes; the mobile melt has become a rigid glass. Corresponding to the lack of configurational change in the glass, its properties have a rather weak temperature dependence, roughly matching the behavior of the crystal. It is important to recognize that Tgis not a thermodynamically defined temperature; its location is determined wholly by kinetic considerations. If the quenching rate is reduced (but not so much as to permit crystallization in range B), then the liquid
Ch. 19, 82
Metastable slates of alloys
1731
configurations can stay in equilibrium to a lower temperature, that is (TJ1is lowered to, say (TJP This slower quench entails a smaller volume (higher density) and higher viscosity for the glass. Thus different glassy states can exist; $4.2 deals with the structural relaxation by which glassy states can change. On simple annealing glasses densify and evolve towards an ideal glassy state represented by the extrapolation of the equilibrium properties, shown on fig. 2 for the viscosity. How dense could an ideal glass be? This intriguing question was first addressed by KAUZMANN[1948]. He pointed out that extrapolation of liquid properties to lower temperatures would soon (that is, at temperatures not far below experimental T,s) lead to what he termed an apparent paradox in which the characteristic difference between liquid and crystal properties would be inverted. For example, the liquid would become denser than the crystal and it would have lower entropy. While not thermodynamically forbidden, structurally such an inversion seems very unlikely. Kauzmann proposed that some transformation, either an ideal (non-kinetic) glass transition, or some type of spontaneous crystallization, must intervene to prevent the paradox being reached. He suggested that the isentropic point (at which the extrapolated liquid entropy matches that of the crystal) could be taken to represent the ideal glass transition temperature and that the observed glass transition (found at finite rather than infinitely slow cooling rates) occurs somewhat before the ideal transition. There has been a steady flow of papers attempting to make sense of Kauzmann’s entropy catastrophes (as they are termed nowadays), not only in Kauzmann’s original version at low temperatures, but also other catastrophes in the neighborhood of the IiquiUvapor transition, which the most recent study (&SHORE and SHOBHA [1994]) has sought to relate to van der Waals’s well-known critical temperature. The Kauzmann paradox remains very much on the research agenda. In connection with fig. 2 we have described the classical features of glass formation, well known for oxides and polymers. Metallic systems are not so well known as glassformers, yet experiments have consistently shown that they fit into the same picture. Around Tgrmetallic glass-forming systems show all the characteristics (volumetric, rheological, thermal) of the conventional glass transition illustrated in the figure. On annealing, the properties of metallic glasses change in a way characteristic of the structural relaxation of eonventional glasses. Most importantly, some easy glass-forming compositions are sufficiently stable that measurements are possible in range B without crystallization. In this way the link between the glass and liquid is directly established (KUI and TURNBULL [ 19851). Measurements of properties and structures (and, as we will see later in this section, thermodynamic measurements) suggest that amorphous metallic phases produced by other methods are (within the range of structure expected) identical to glasses produced by liquid quenching. Of central importance in considering glass formation and the properties of glassy or amorphous phases is the atomic mobility. Low mobilities permit the glass to be formed and a glassy or amorphous phase to persist. Much of the discussion of atomic mobility in dense non-crystalline systems has centred on the concept of free volume, which we now examine. As seen in fig, 2, a faster quenched glass (1) has a lower density than a more slowly quenched one (2) or, in other terms, “1” has a largerfree vohme than “2”. Free volume was originally introduced and analysed by polymer physicists (FOXand References:p. 1818.
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R. a! Cahn and A. L Greer
Ch. 19, 52
FLORY [1951]); F~MACHANDRARAO et al. [1977a, b] give a full discussion of the concept. In the simplest terms, free volume is the unfilled space between the atoms or molecules (which at once poses the problem of defining the contours of these entities); the central concern is with the space in excess of that which would exist even in ideal closepacking. The free volume can be defined in terms of thermal expansion starting from an ideal glassy state, or in terms of the hole theory of liquids in which free volume is postulated to consist of identifiable holes with a rather narrow size distribution. The crucial point is that, the smaller the free volume, the more sluggish is any structural rearrangement. The viscosity, 77, varies with free volume approximately according to the Doolittle equation, originally derived for polymers: q =A exp(B/f,), wherefT is the relativefree volume defined by (VT-V,)/V,, where V , V, are the volumes of material at temperatures T and 0 K respectively. The equation is valid throughout range B in fig. 2, but cannot be applied in the glassy state itself. The contentious question whether the free volume is more or less uniformly distributed, or else concentrated in holes which are probably at least the size of the smallest atoms in the glass, is still open, as reviewed by CAHN [1979]. Glass stability against annealing-induced crystallization (which will be considered further in 54.3) is directly related to atomic mobility. Figure 3 shows that the crystallization temperature has a good correlation with hole formation enthalpy; a stabler glass has a higher formation energy for holes, therefore fewer holes form, the total free volume is smaller and thus crystallization more sluggish. In range B (fig. 2) the relative free volume remains in thermal equilibrium. On cooling it decreases particularly rapidly as Tgis approached, leading to a very sharp rise in viscosity. Below T, free volume is frozen-in. In the glassy state, free volume is a very useful parameter for describing and analysing the property changes which occur on annealing; this is exemplified in the work of SPAEPENand TAUB[19831 on viscosity and of CHA~ON and MIZOGUCHI [1987] on atomic diffusion (see 0 4.2).
i
1000
0
0
0
/'
u 1-x
m,,
Tx Fex
200
100
AH,,(kJlmdl
Fig. 3. Dependence of the measured crystallization temperature, Tx,of various binary metallic glasses on the formation enthalpy of a hole the size of the smaller of the constituent atoms: the hole formation enthalpy is calculated from thermochemical data. (After BUSCHOW[19821).
Ch. 19, 92
1733
Metastable states of alloys
Different types of glass-forming system, network formers like SO2, ionic liquids and those showing van der Waals bonding, do show different types of behavior, reflecting the different sensitivities of the liquid structure to temperature changes. The so-called strong liquids like SiO, have a near-Arrhenius dependence of atomic mobility (characterized by the viscosity). On the other hand an organic liquid like toluene is afiagile liquid with a strongly non-Arrhenius temperature dependence following the Doolittle equation. The distinction is fully discussed by ANGELL[1995]. Metallic liquids fall between these extremes, nearer to the fragile limit. It is characteristic of such systems that there is a large step in specific heat at Tg.As shown in fig. 2c, the specific heat of a glass-forming liquid alloy is significantly greater (perhaps as much as two times greater just above T,) than that of the corresponding crystalline solid. Integration of liquid and solid specific heats over the temperature interval between Tf and Tgshows that the heat of crystallization of the glass at Tgshould be significantly less than the latent heat of melting at T,; this is confirmed by direct measurement and provides a way of estimating the liquid specific heat in the undercooled region when values are not otherwise available (GARRONE and B A ~ Z Z A T[1985]). I The large specific heats of glass-forming liquids relate directly to the loss of entropy on cooling. Glass-forming alloys typically show a strongly negative heat of mixing of their components. This heat is greater at lower temperatures, again reflecting the increased order in lower-temperature liquids. Figure 4 shows the entropy difference between liquid and crystalline states in an easy glass-forming system for which reliable specific heat data are available. The ideal glass transition temperature (zero entropy difference) is just a little below the measured glass transition (determined as in fig. 35, below). The proximity means that thermodynamic parameters can be used to and GREER [1987]. estimate glass transition temperatures, as shown by BATTEZZATI
f.
b q Temperature (ti)
Fig. 4. Excess entropy of liquid Pd,NiJ'$,
as a function of temperature, calculated from specific heat data of
EVANSet aZ. [1987]. The excess entropy has the value of the entropy of fusion, ASf, at the eutectic temperature, Te, and extrapolates to zero at the ideal glass transition temperature. This is seen to be very close to both the observed kinetic glass transition, Tg,and the ideal transition temperature, To,calculated from the hole theory of Dmw and RAMAcHAMlRARAo [19841. References: p. 1818.
1734
Ch. 19, 8 2
R. W Cahn and A. L Creer
The thermodynamics of the glassy state itself can be estimated using specific heat measurements as above. In addition, electrochemical (E.M.F.) measurements on the glass can be used to determine its thermodynamic parameters directly; the procedures have been discussed by BORMANN and Z~LTZER [1992]. CALPHAD (CALculation of PHAse Diagrams) methods involve the determination of a consistent set of thermodynamic parameters for an alloy system by fitting the phase diagram. When this takes account of specific heat and other data (such as E.M.F. measurements) reliable estimates are possible, even in the undercooled IiquiUglass regime. SAUNDERS and MIODOWNIK [1986] first successfully applied this approach to interpret amorphous phase formation by et al. [1988]). solid-state reactions, and it has since been developed further (BORMANN A striking example of the link between the liquid and an amorphous alloy formed in the solid state, and of the use of thermodynamics in interpreting experimental results, is provided by inverse melting. This phenomenon occurs in the metastable body-centred cubic solid solution in the Cr-Ti system. When a solid solution of Cr55Ti45is annealed below -750°C it transforms polymorphically to an amorphous phase of the same composition. Yet the bcc phase itself forms from the liquid. Thus if the amorphous phase is thermodynamically continuous with the liquid, its Gibbs free energy must equal that of the bcc at two temperatures (fig. 5). The upper temperature is the normal melting/ freezing point. At the lower temperature there is an equilibrium between the amorphous and crystalline phases, but the amorphous phase is the more stable phase below the temperature; this is an inverse melting point, because in equilibrium at this point a crystal would melt on cooling. The Cr-Ti system has been subjected to a CALPHAD analysis and this predicts behavior of the form shown in fig. 5. The corresponding metastable phase diagram is shown in fig. 6; in this diagram only three phases are included - the
CI
F
a
\
7
5
8
-
-
-4-
-6 -
-a -
-
Temperature (“C) Fig. 5. Free energy curve of the liquid/amorphous phase with respect to the bcc solid solution for Ti-55 at% Cr. Hcp titanium and bcc chromium are taken as reference states at each temperature. T,,, and Tb denote the melting and the inverse melting temperatures, respectively (After BORMANN[1994]).
Ch. 19, $ 2
1735
Metastable states of alloys
20001 n
0
e
1500
Q,
L
J c
f E" e
I000
'
"
"
1
'
"
'
"
'
'
1
"
1
'
I
1 1 3 t
0
j
bcc
75 Cr Concentration (at.%) 25
50
100
Fig. 6. Metastable phase diagram of the T i 4 system as calculated by the CALPHAD method, considering only equilibria between the hcp and bcc solid solutions and the amorphous or the liquid phase (After BORMANN
119941).
hcp and bcc solid solutions and the liquid. Intermetallic compounds are excluded. The metastable diagram is relevant to experiments on solid-state amorphization because at the temperatures used the compounds have difficulty in nucleating. As shown in fig. 6, the amorphous phase appears at low temperatures. The thermodynamic analysis shows that this is really the same phase as the liquid. Consideration of the temperature dependence of the free energies in fig. 5 shows that the liquid or amorphous phase at the inverse melting point must have a lower entropy than the bcc solid solution. This remarkable result is possible because the amorphous phase is strongly chemically ordered, whereas the solid solution is disordered. The greater chemical order in the amorphous phase outweighs its greater positional entropy. Given the chemical order in the amorphous phase, it is possible to regard it as a type of intermetallic compound, as its phase field in fig. 6 would imply. The analysis of inverse melting was concerned with a transformation occumng near thermodynamic equilibrium (albeit a constrained equilibrium in which intermetallic compounds do not appear). Other types of solid-state amorphization have prompted consideration of what the ultimate limit to crystalline stability might be - if a liquid or amorphous phase did not nucleate, how far could the crystal be superheated? FECHTand JOHNSON [1988] have considered this, and show that the proliferation of vacancies in the crystal would lead to its entropy becoming greater than that of the liquid at a finite temperature (roughly twice the thermodynamic melting temperature T J . In a further application of Kauzmann's concept of a structural paradox, they suggest that this point would represent the absolute stability limit of the crystal. More generally, how far could the crystal exist beyond its equilibrium phase field? Its thermodynamic stability could be reduced not only by heating, but also for example by changing its composition or by References: p , 1818.
1736
R. W Cahn and A. L. Greer
Ch. 19, $ 2
injecting defects (by irradiation or deformation). These points have been considered by Em,DES&AND JOHNSON [1989], who suggest that melting or amorphization could become a continuous second-order transition at high enough solute contents. It is as yet unclear whether any observed amorphization fits this pattern, or occurs closer to thermodynamic equilibrium.
2.2. Compositions of amorphous alloys Amorphous alloys are now known for many compositions. Within a given alloy system, the range of compositions which can be made amorphous depends on the production method and conditions. For example, with rapid liquid quenching, the range of glass formation is wider for higher cooling rates. However, even at extreme cooling rates (for example in pulsed laser quenching, Q 3.5), the glass-forming composition ranges remain limited. Table 1gives some examples of compositions which can be made amorphous by ‘standard’ techniques such as melt-spinning (0 3.1). More complete listings [1976], SURYANARAYANA [1980] and JOHNSON [1986]. may be found in TAKAYAMA The main categories are: (i) Late transition metai-metalloid. This is probably the most studied category of amorphous alloys, as it includes the Fe-, Co- and Ni-based alloys which are commercially exploited because of their soft magnetic properties ( S m [1993]). The alloys consist of one, or a combination of, the late transition metals and one, or a combination, of the metalloids. The late transition metals are here considered to be in groups VIII and IB. The metalloids are in groups IIIB, IVB or VB, including AI and Sb. Amorphous alloys are mostly formed by rapid liquid quenching, and the glass-forming compositions are centered around deep eutectics, typically in the range 15 to 25 at.% metalloid. Binary amorphous alloys in this category are common, but the glass-forming ability is generally found to be higher when there are more components, probably because the eutectics are deeper. (ii) Early transition metal - metalloid. Early transition metals from groups IVA, VA and VIA also form amorphous alloys in combination with metalloids, with compositions again centered on deep eutectics in the range around 20 at% metalloid. Partial substitution of an early transition metal for the late transition metal in category (i) can improve the glass-forming ability. When the alloys contain a refractory metal, the crystallization temperatures can be particularly high, and this has led to interest in early transition metal-based glasses as diffusion barriers (D~RNER et al. [1991]). (iii) Early transition metal - late transition metal. These alloys of group VIII or IB elements with group WA, VA or VIA elements form the second most-studied category. When they are made by rapid liquid quenching, the glass-forming composition ranges are typically much wider than in the metalloid-containing alloys (categories i and ii), covering several eutectics and intermetallic compounds. Although the glass-forming ability is often greatest around the centre of the composition range, Fe, Ni and Co alloys with as little as 9 at% Zr have been made amorphous. These alloys have attracted much attention because they are the main category which can be made amorphous by solidstate reaction (§ 3.7).
Ch. 19, 02
1737
Metastable states of alloys
Table 1 Examples of glass-forming alloys", classified into the categories discussed in the text. Type
Alloy
Glass-forming range
Refmnces
x=12-25 x= 14-22 3141 ~~17-18.5, ~=34-36.5
HASEGAWA and RAY119781 L w s and DAVIES[1976] DONALDand DAVIES[1980] SRIVASTAVA et al. [1972] D u w ~ z[1%7] LUBORSKY [1977] POLK and CHEN[I9741 MAKINOet al. [1990] MAITREPIERRE [1970]
WOUEand MASJMOTO[1980] FISCHER et al. [1978] DBRNERet al. [1991] FISCHER et al. [1978] RUHLetal. [1967] RAY etal. [1968] CLEMENS et al. [1984], GREGAN etal. [I9811 HORVATHet a/. [1988] GIESENet al. [ 19761 FISCHER e t d . 119781 INOUEet al. [19901 INOUE et al.
[1988b]
DUBOISet al. [I9921 HOL~ER and KELTON[1991] DUBOISet al. [I9921 INOUEet al. [1988a] INOUE et al. [1988a] INOUE et al. [I9901
x=18-34 X= 17-22 x=l8-26 ~=32-50
AGYEMAN et al. [1979] AGYEMAN et al. [I9791 JOHNSON et al. [1975] BUSCHOW[I9791 INOUE e l d.[I9891
(viD Actinide-based
U,CO.
~=25-32 x=12.5-47..5
CALKAet al. [19771 SOMMER e t d . [1978] KIM er al. [I9901 HASEGAWA and TANNER [1977]
x=24--10
RAY and Muss0 [1976]
a All compositions are quoted in at.%. The amorphous alloys have been formed by a variety of techniques, mostly rapid liquid quenching. The compositions given are those of representative alloys and do not imply that amorphous phase formation at other compositions in the same system would not be possible. For binaiy alloys, however, some indication is given of glass-forming ranges.
References:p. I818.
1738
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R. W Cahn and A. L.Greer
(iv) Aluminum-based alloys. Aluminum-based amorphous alloys have attracted attention only quite recently, when ductile compositions with >75 at.% Al were discovered; these are of interest because of their high specific strength (9 4.4.1). The alloys most studied have 5 to 15 at% of a lanthanide and 5 to 15 at% of a late transition metal. Amorphous phase formation is also found in the binary aluminum-lanthanide systems, at approximately 10 and 50-80 at% lanthanide. Other amorphous aluminum alloys are based on additions of an early transition metal and a late transition metal, or of a late transition metal and metalloid. (v) Lanthanide-based. Alloys of a lanthanide with 10 to 35 at.% metalloid can form amorphous alloys. Also, binary alloys of the lanthanides, or Y, with late transition metals (groups Vm and IB) have deep eutectics at lanthanide-rich compositions, and some glass-forming ranges have been found around these. An alloy of this type (La-Au) was the first to demonstrate solid-state amorphization by interfacial reaction (SCHWARZ and JOHNSON [1983]). Of particular interest are alloys from the same Al-lanthanide-latetransition metal systems as yield ductile Al-rich amorphous alloys. When lanthanidebased, e.g., La5,Al,,Ni,o (INOUEet al. [1989], these alloys exhibit unusually high glassforming abilities. The critical cooling rate for glass formation is exceptionally low e 10 Ws)and the amorphous phases are remarkably resistant to crystallization, crystallizing only at temperatures 100 K above Tg.(Most amorphous alloys crystallize rapidly at or slightly above Tg)The very good glass-forming ability has the consequence that bulk amorphous samples (for example, rods of diameter up to 1 cm) can be cast (INOUEet al. [1993]), in marked contrast with the thin sections associated with the rapid quenching required for more noma1 glass-forming alloys. (vi) Alkaline-earth bused. The group IIA metals can have quite wide composition ranges for amorphous alloy formation when alloyed with a metal from groups IB, IIB, IIIB,IVA, VIIA or VITI. A ternary addition of a lanthanide can greatly improve the glass-forming ability (KIMet al. [1990]). For example, compositions such as Mg,Cu,Y,, show glass-forming ability similar to that of the lanthanide-aluminum-transition metal alloys discussed in (v), and can similarly be cast in bulk (LI et al. [1992], INOUE et al. [1993a1). (vii) Actinide-based. A number of amorphous binary alloys based on actinides are known, which do not fit into a clear pattern. The examples in table 1 illustrate the main types of amorphous alloys, but there are many others. The table emphasizes binary compositions, but many of the compositions which form amorphous phases most readily have three or more components, and some of these may be difficult to categorize. As amorphous alloys are optimized for various applications, multicomponentcompositions are developed. An example is the composition Fe,3,5Cu,Nb3Sil,.,B9,designed to devitrify to a fine-grained structure with good soft magnetic properties (8 4.3). Like most complex compositions, however, this is based on a simple amorphous alloy type, in this case type (i). Other compositions, such as ~ ~ l . z ~ l ~ , ~ C u l z . ~ Nhave i l o been , ~ e zdeveloped z,~, to obtain exceptionally good glassforming ability (PEKER and JOHNSON [1993]. These recent findings have been put in their [1993]. Subsequently, INOUE [1995] has found a whole context in a reportage by GREER series of easy-forming glasses, quaternary and quinary compositions of the type
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Zr,,A1,,(Co,Ni,Cu),.,, e.g., Zr~A175Ni~,,Cu17~5 which, like Johnson's alloy, can be prepared as an amorphous cylinder as much as 16 mm in diameter. In an alloy with many components crystallization may be strongly inhibited; this is sometimes referred to as the confusionprinciple. The confusion is increased if the atoms are of differing sizes. In the first example given, zirconium and titanium atoms are large, copper and nickel of intermediate size, and beryllium has the smallest atoms of any metallic element. Alloys of this type also have an exceptionally wide temperature gap between the glass transition and the temperature at which crystallization begins. Other than these broad generalizations, other criteria for selecting compositions to form amorphous phases readily are considered in the next section. We show here (fig. 7) seven phase diagrams, one from each of the categories in table 1, each with the glass-forming ranges (for cooling rates such as are achieved in meltspinning). It is very clear that glasses form preferentially near deep eutectics, where viscosity is high before the thermodynamic freezing temperature is reached on cooling, although the glass-formingranges are usually not symmetricalabout the eutectic composition. These phase diagrams link with several of the criteria discussed in the next section.
2.3. Criteria for formation of amorphous phases In this section, we concentrate initially on glasses made by rapid solidification;those made by irradiation, mechanical milling or interdiffusion will be discussed afterwards. The generic term most commonly used to represent what we are trying to interpret here is gluss-forming ubilizy, or GFA. This is usually measured by the critical cooling rate, R,, which is necessary to turn a melt into a glass, avoiding crystallization: when this is in the experimentally accessible range for mainline rapid solidification processes such as meltspinning, approximately up to lo7Ws,then the alloy can be vitrified by such a process. If theory tells us that R, is higher than this, then at best tiny amounts of glass can be made by some much faster quenching process such as in-situ quenching with nanosecond or picosecond laser pulses (0 3.5). MOTORIN[19831 calculated from first principles the expected homogeneous nucleation rates of crystals in supercooled pure metal melts, using known physical parameters as input. For silver, copper nickel and lead, minimum cooling rates of lQ'2-1013 W s were derived, which demonstrates clearly enough why no pure metals have been amorphized by melt-quenching. Not even picosecond laser-quenching has been successful in this respect. Thus practically interesting GFAs refer only to alloys - as mentioned in 0 1.3, TURNBULL[I9811 did propose that crystallization of pure amorphous metals would come into the category of transformations which do not require thermal activation and thus cannot be suppressed. The range of ideas and approaches which have been proposed to attain an understanding of why some alloy systems can be vitrified and others not, and of what determines the composition ranges over which metallic glasses can be made in practice, is disconcertingly broad, There are almost as many criteria for glass formation as there are for the good lifa, and almost as little agreement among the protagonists as to which is preferable. References: p . 1818.
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2000
800 I 500
1
e Composition (at. %)
Fig. 7. Quilibrium phase diagrams for examples of each of the categories (i) to (vii) listed in table 1. Glassforming ranges are indicated.
An early attempt to come to grips with this difficult issue was by P ~ L and K GIESSEN [1978]. The best comprehensive overview of the development of these criteria is by RAMACHANDRARAO [1984]. CAHN[1986a], HAFNER[1986] and MASSALSKI [1986] have published more specialised treatments. Here, only an abbreviated review is feasible. In spite of a tendency by some theorists to regard this claim as culpably naive, there can be no doubt that atomic size is the single factor that plays the major role in
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determining GFA. The recognition that the constituent atoms in a melt must have sufficiently different Goldschmidt radii to permit glass formation goes back to very early experiments on vapor-quenching, in a classic study by MADm et aZ. (1967). A critical radius mismatch of - 15% fitted the results, and this value has been accepted ever since. This criterion might be termed an anti-Hume Rothery criterion, since Hume-Rothery’s celebrated law laid it down that a radius mismatch not exceeding 15% was necessary for extensive solid solution formation in terminal alloy phases. - Mader himself had simulated glass formation by jiggling populations of wooden balls of different sizes on a tray, and found that the radii had to differ sufficiently for “crystallization” to be prevented; subsequently, SIMPSONand HODKINSON (1972) performed the first of several simulations of this kind with rafts of soap bubbles of 1 mm diameter which better simulate real interatomic forces, and again found a critical radius mismatch of - 15%. The next stage was a series of attempts to create GFA maps, in which radius mismatch was plotted along one axis of a graph, and some measure of bond strength such as the negative heat of mixing or heat of evaporation along the other - the idea being that strong interatomic bonds render crystallization more difficult. One of several such attempts in this period was by GIESSEN(1981). Such graphs suggest that the bond strength does indeed play a part, but it is a “weak” variable compared to the size mismatch. It should be noted, however, that strong bonding can itself modify an effective atomic size: this was demonstrated by BECHETand REGAZZONI [1991] who showed that the effective size of late-transition-metal atoms in amorphous aluminum-rich alloys is significantly reduced by the strong bonding between the elements. A very important development of the radius mismatch approach is due to EGAMIand WASEDA[1984], followed by a further development of the same ideas by EGAMIand A m [1987]. These authors were interested in calculating the microscopic stress levels at the scale of individual atom groups in a glass or a crystalline solid solution. Briefly, they concluded that in a glass, neither the local stress fluctuations nor the total strain energy vary much with solute concentration when these quantities are normalized with respect to the elastic moduli, whereas in a crystalline phase the strain energy rises steadily with solute content. Thus, beyond a critical solute content, glass becomes favored over crystal formation, and in this way, not only GFA but also the glass-forming composition range can be calculated. The conclusion is that cB- (vB-vA)= 0.1, where cBmin is the minimum solute concentration of B in A required for a glass to form and the v’s are atomic volumes. The agreement with observation is good for many systems, though melt-quenching by normal melt-spinning is not necessarily fast enough to give agreement with Egami and Waseda’s criterion. Figure 8, for the Ni-Nb system, shows.at the bottom the results of applying Egami and Waseda’s criterion, and also three different experimentally determined glassforming ranges for different quenching methods. The “faster” methods agree very well with the theoretical criterion. Several othek theories, of a thermodynamic nature, based on atomic volumes have appeared. One dersion is due to RAMACHANDRARAO[1980]: he estimated the departure from ideality of melts, in terms of a kind of deviation from a liquid Vegard’s Law,and concluded, both’theoretically and by comparison with experiment, that melts with a large
-
References: p . 1818.
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Nb
Atom fraction
Nl
sputtering
1
[
ns laser quenching
1
I melt-spinning
................... 3 m . x
I
J
prediction
1 Fig. 8. The Ni-Nb phase diagram, with experimentally determined glass-forming ranges for three methods of progressively diminishing effective quenching speed, together with the range predicted by Egami and Waseda’s theory.
deviation, in the sense of having an anomalously small atomic volume, were most likely to form glasses. This finding was interpreted in terms of the enhanced viscosity associated with a small atomic volume (which implies little free volume available to support atomic motion in the melt). - YAVARI et al. [I9831 independently developed a closely related idea, suggesting that a zero (or negative) change of specific volume on melting of a crystalline species favors glass formation on subsequent rapid solidification. In effect, their idea is that if a crystal is denser than the melt from which it grows, then in growing it rejects free volume into the melt and thereby reduces its viscosity; thus crystal growth becomes self-catalytic. Contrariwise, a less dense crystal raises viscosity and favors glass formation. Theoretical physicists have developed a number of models to rationalize GFA. The best known is that due to NACELand TAUC[1977]. They proposed that a glass is most likely to form if its electronic energy lies in a local metastable minimum with respect to composition change. They showed that if the structure factor corresponding to the first, strong peak of the diffuse X-ray scattering curve, Kp, satisfies the relationship K, = 2 kF, where kF, is the wave vector at the Fermi energy, then the electronic energy does indeed occupy a local minimum. A number of familiar glasses, in the Au-Si, Au-Ge and Co-P systems for example, accurately obey this criterion, but there are also exceptions, e.g., some obscure glasses in the Cs-0 and Rb-O systems. This kind of approach, using modem approaches such as pseudopotential theory, has been taken much further in recent years, but a discussion would take us too far here. For further details, the reader is referred to reviews by HAFNER[1981, 19861. However, as we shall see below, the
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ability, demonstrated by Hafner, to make theoretical estimates of the glass transition temperatures of unknown glasses offers a novel way to estimate GFA. Theories based more explicitly on the need to prevent nucleation of a crystalline phase take two main forms. First, there are models based on an explicit calculation of homogeneous nucleation rates (this begs the question whether heterogeneous nucleation plays a role). The standard approach here is due to DAVES(1976). He adapts a theory of isothermal crystallization kinetics due to Uhlmann and calculates the cooling rate necessary to bypass the “nose” of the time-temperature-transformation plot thus calculated. (The nose is the minimum time required to initiate homogeneous nucleation, at some temperature well below the thermodynamic freezing temperature). The principal difficulty here is to estimate the viscosity of a supercooled melt, but there are a number of empirical relationships which allow a rough estimate to be made. (There is now available a critical survey of known viscosities of molten metals and alloys, by BATTEZZATI and GRFXR [1989].) Figure 9 shows some calculated critical quenching rates obtained in this way, and fig. 10 shows how the calculated value of R, varies with composition across a phase diagram. It will be seen from this second figure that, as a number of people have pointed out and as follows from Davies’ theory, the ratio TdT, (where Tgis the glass transition temperature, Tfis the thermodynamic freezing temperature) is a crucial figure of ment in determining GFA, for purely kinetic reasons associated with the need to avoid crystallization. The lower this ratio, the more viscous the melt becomes before it is ever undercooled and the more difficult crystallization becomes, thus enhancing GFA and reducing R,. A somewhat related, more thermodynamically biased approach derives from a paper
I
t 0
, 0.2
,
,
0.L
0.6
10.8 : -
Tg ’ 6 Fig. 9. Calculated critical quenching rates, R,, for glass formation, and corresponding estimated limiting ribbon thickness for wholly glassy ribbons. Key to alloys: (1) Fe≪ (2) Au7,Ge,,Si,; (3) Fe,B,,; (4) Fe.,l,5Ni415Bl,; (5) Co,,Si,,B,,; (6)F ~ S i I & B l(7) I ; Fed13C7;(8)Pd,,Si,,; (9) NkNb,; (10) Pd7,,5CkSilas;(11) PddiJ‘d,. (After DAVIS [1978]). (An updated version of this plot has just appeared, DAVIES[1995].) ~
References:p. 1.818.
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Experimental c
v1
Y
.-C 0 -m
- 2
5
10
15 20 Atomic % Si
25
Fig. 10. Equilibrium freezing temperature, T, glass transition temperature, Tk figure of merit, TdT, and calculated critical quenching rate, R,, for a range of Pd-Si solid solutions.
by J. W. CAHN(1980) on the thermodynamics of metastable equilibria. From this, the role of the relative values of Tg and To becomes clear; here, To is the temperature at which liquid and solid of the S Q composition ~ have the same free energy. A glass forms readily if T, > To;what this means is that the melt rigidifies before reaching the temperature at which freezing without composition change (and thus without long-range diffusion), also known as solute-trapping, becomes possible. The implications of this are clearly set out in a review by MASSALSKI (1986) and applied to a detailed analysis of GFA across the Cu-Ti system in a paper published about the same time (MASSALSKI and WOYCHIK[1985]). The problem is that, before a glass has actually been made, its T, is not known. Here, HAFNER’S [1983, 19861 ability to calculate glass transition temperatures comes into its own. In fig. 11, for the Ca-Mg system, the calculated To values are shown, and also two estimates for Tg,an upper limit based on the “entropy-crisis” or Kauzmann model, and a lower limit based on a free-volume model. (For a fuller explanation, see HAPNER (1983)) Theoretical composition ranges for glass formation are shown for two cooling rates; here the criterion T,>T, is the central consideration; agreement with experiment is quite good. Another detailed thermodynamic analysis of metallic glass formation near eutectic troughs was published by HIGHMOREand GREER[1989]. Another set of theories is based on the postulate that metallic melts are not homo-
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Fig. 11. Calculated phase diagram, Tovalues and upper and lower limits (T&) for T, for the Ca-Mg system, with theoretical and experimental glass-forming ranges. (After HAFNER[1983, 19861).
geneous in composition but contain compositional clusters. Such clusters are held to aid crystal nucleation, and their absence or weak development to aid glass formation. Contrariwise, short-range order (the converse of clustering), which is believed (though there is a singular absence of diffraction evidence on this point) to increase with falling temperature of a glass, just as it does in a crystalline solid solution, should enhance GFA. (See, for instance, WAGNER’S [1986] survey of SRO in metallic glasses; at least, it has been established for Cu-Ti that a glass has higher SRO than the melt from which it is quenched (SAKATAe?al. [198 l]).) Indeed, there is some evidence that some Cu-Ti alloys show clustering in the melt whereas the glass has SRO. This field of research, which has a large literature, has been somewhat beset by controversy: for further details, the reader is referred to RAMACHANDRARAO [1980, 19841. Finally, we return to a less sophisticated “figure-of-merit” approach which has been References:p . 1818.
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quite successful in rationalizing GFA. DONALDand DAW [1978] long ago recognized the awkwardness of theories which related GFA to quantities (such as TJTJ which involve T,, when normally this is unknown until a glass has been made and examined. They proposed that a good test of GFA is the extent to which the equilibrium freezing temperature of an alloy melt is depressed below the ideal value, which they calculated simply by linear interpolation between the freezing points of the constituent metals. This simple method was then developed by WHANG[1983], who took into account the modification required to allow for possibly extensive solid solution of one metal in the other, in the solid state. This is necessary because solid solubility reduces the slope of the To versus composition curves (like those shown in fig. 11). Whang generated figureof-merit maps in which one axis gave TLR, defined as T L R = AT/T,”, where AT is the difference between the ideal freezing temperature for an alloy (T,”), defined as above, while along the other axis is C,, , a simple measure of the amount of the solubility of the minor constituent in the major at the eutectic temperature -a large value of TLRimplies a severely depressed liquidus, while a small C,, implies a steeply sloping To versus composition curve ... both factors favouring easy glass formation. In fact, the maps so generated show a clear boundary between glass-forming and non-glass-forming alloys DUBEYand RAMACHANDRARAO [1990] have developed Whang’s model to show that most eutectic phase diagrams can be expected to show an asymmetry of GFA, in the sense that glass formation is easier just to one side of the eutectic composition than on the other; the melt-spinning range indicated in fig. 8 shows an example of this, and several of the diagrams in fig. 7 exemplify the same feature. Whang’s theory was then adapted by TENDLER[I9861 to show, for a series of Zrbased alloys, that alloys which according to Whang’s criterion should be good glassformers are also those in which there is anomalouslyfast difusion, in the special sense introduced by WARBURTON and TURNBULL [1975] (see ch. 7, 3 4.3.2).Figure 12 shows one of Tendler’s figures for a series of Zr-M alloys. All the alloys showing fast diffusion (Zr with Cr, Mn, Fe, Co, Ni, Cu and Be) are also glass-formers. For fast diffusion, the solute atom must be much smaller than the solvent atom (for details see Tendler’s paper) and this clearly also favors GFA. In fact, some years ago TURNBULL[1976] had predicted just such an association between GFA and fast diffusion. Care must of course be taken in setting up such diagrams as fig. 12 because, as shown by Bechet and Regazzoni, large radius ratio disparities (leading to strong bonding and to fast diffusion) can modify effective atomic sizes. Fast diffusion is also linked to glass-forming mechanisms through the fact that in the solid-state amorphization reaction (SSAR), one atomic species diffuses very much faster than the other. This by no means exhaustive overview of the models and theories that have been advanced to make sense of glass-forming systems and ranges might well seem discouraging, because at first sight they are mutually exclusive. In fact, hidden cross-connections undoubtedly exist: the linkage between Whang’s thermodynamic approach (related to terminal solid solubilities) and Tendler’s association between fast diffusion and GFA clearly comes from a correlation of both solid solubilities and fast diffusion with atomic size ratios. Perhaps in due course even the electronic criteria studied by Nagel and Tauc and by Hafner may prove to be linked with some of the other ideas, e.g., the free volume
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o Ga
AI
5
I h
Fig. 12. A number of Zr-M alloys plotted on a Whang graph, separating glass-formers (GFA) from non-glass[1986]). formers. (After TF.NDLER
approach due to Ramachandrarao and Yavari. Our own view is that simple geometry atomic sizes - will prove to be the main criterion that in various subtle ways incorporates the others.
2.3.1. Criteria for amorphization by irradiation, and mechanically aided and induced amorphization Some alloys can be amorphized either by irradiation with electrons or heavier particles (see 93.4) or by heavy deformation, if need be followed by annealing (see P 3.8). In 9 3.4, below, a brief account is offered of an unresolved controversy regarding the relative roles of two amorphization criteria - the need to destroy the (chemical) long-range order in an irradiated compound to a sufficient degree versus the need to attain a critical concentration of point defects. The criteria for amorphizability in the case of mechanically aided or induced amorphization. (see for instance SCHULTZand HELLSTERN[1987]) are generally assumed to be similar to those applying to irradiation, but no critical comparison of the two families of techniques appears to have been made. There is an important new approach, however, due to BEKEetal. [1991]. They estimated the additional elastic (mismatch) energy which is stored in an ordered solid solution when its long-range order is destroyed, whether mechanically or by irradiation; this depends, of course, inter alia on the atomic sizes. This energy is then compared with the ordering energy, and it was concluded that amorphization is possible if (a) the (virtual) critical disordering temperature is above the melting temperature, and (b) the ratio of the References: p. I818.
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Ch. 1 9 , § 3
elastic mismatch energy to the ordering energy exceeds a critical amount. A comprehensive test against experimental facts gave widespread though by no means universal agreement.
3. Practical methods of creating metastable phases and microstructures In what follows, various techniques are outlined, both those involving solidification of a melt and those which operate entirely in the solid domain. In exemplifying what can be achieved, emphasis is placed upon amorphization, but most of the methods have of course also been used to make metastable crystalline phases. 3.1. Rapid quenching from the melt
All casting methods involve a chill zone close to the mould surface; the metal in the chill zone cools more rapidly than the interior of the casting, and its grain morphology and composition are accordingly different. Such a zone will normally cool at a rate of up to some hundreds of Ws. The modern techniques of Rapid Solidzjkation Processing (RSP), with which this chapter is principally concerned, however, involve cooling rates in the range 104-107 Vs,and require an approach quite distinct from the traditional casting methods. Atomization of liquid metal to form small droplets -one of the methods widely used today - goes back to the last century, and as recounted by JONES [1981] and more recently by C m [1993] in historical surveys, several investigators in the period 1925-1955 developed variants of chill-casting which gave estimated cooling rates up to lo5U s , while several studied extension of solid solubilities resulting from such cooling rates. However. the remarkable present-day developments in RSP were initiated by Duwez in 1959-60. Duwez, in California, set out to establish whether a continuous metastable series of solid solutions could be created in the Cu-Ag system, to bring it into line with the Cu-Au and Au-Ag systems which both show continuous solid solubility, as required according to the familiar Hume-Rothery rules. For this purpose, he argued, the melt would have to be frozen fast enough to inhibit nucleation of two distinct fcc phases. Having unsuccessfully tried a number of other approaches, he designed the Duwezgun, a device in which a gaseous shock wave atomizes a drop, = 10 mg, of molten alloy and projects the microdroplets into contact with a copper substrate, or chill block, to produce small foils, or splats. The technique was by some given the onomatopoeic nickname splat-quenching; this term, disapproved of by Duwez, fell out of use together with the original gun,as improved methods were developed. In 1960, Duwez and his collaborators were successful in making a continuous metastable series of Cu-Ag alloys without any two-phase region, and in the same year also discovered the first metallic glasses made by rapid quenching, in the Au-Si and Au-Ge systems (DUWEZ et al. [19601). The essential condition for rapid quenching from the melt, as realized in the Duwez gun, was that small particles of liquid metal should rapidly be flattened into a thin sheet in intimate contact with a good heat sink. More generally - and this was the real importance of the innovation wrought by Duwez -
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only by starting from the molten metal was it possible to achieve really high rates of cooling in the solid. The importance of RSP arises both from the changes brought about in the freezing process and from the more rapid cooling of the solid so formed -but it should be noted that not all meltquenching methods in fact create fast cooling in the solid stage. The inadequacy of melt-spinning in this respect emerges from some striking observations concerning age-hardening kinetics (see 0 5.1). The Duwez gun was used for a decade to investigate the crystallographic and calorimetric characteristics of numerous metastable splat-quenched crystalline phases and of a few glasses, notably P$,,Si,, which was for a number of years a standard composition for those interested in the properties of metallic glasses. Duwez [1967] has described these pioneering days, and also the background to his original experiments. The splat-quenching principle was developed to provide faster cooling rates still, in the form of the rotating-wing quencher; attempted by various early investigators and designed into an effective form by SOMMER et al. [19871: here a disc fitted with vertical wings rotates very rapidly about a vertical axis, in high vacuum, and a drop of melt is allowed to fall into the rotating wings. The instrument has recently been used to quench out disordered Ti,A1 (BRAUNet al. [1994]), a good test of efficacy in fast cooling. The next stage came when the first methods of making continuous rapidly quenched ribbons were invented. All these depend on contact between a thin liquid ribbon and a moving chill block. The old method of single-roller melt-spinning was re-invented by Pond in 1958 and later published (PONDand MADDIN[1969]); twin-roller melt-spinning is due to CHENand MILLER[1970] and was improved by MURTYand ADLERE19821 but is not very efficient and is little used now; melt-extraction was developed by MARINGER and MOBLEY[1974]. The fourth major technique, much used in basic research, is the drop-smashel; also known as piston-and-anvil quenchel; in which an alloy drop is levitation-melted inside a conical induction coil, released and quenched between two moving copper surfaces to form a disc (HARBuR et al. [ 19691, BEGHIet al. 119691 and CAHNet al. [1976]). The principles of these four methods are indicated in fig. 13. Of these methods, melt-spinning and its variant pZanur$ow-casting (fig. 14), which allows wide sheets to be made, approaching a meter in width, are in practice the most important; the kinematic and heat-flow aspects of these processes are analysed by SHINGUand ISHIHARA [1993]. Their analysis produces an analytic relationship between ribbon thickness and substrate velocity: ribbon thickness varies as the - 0.8 power of substrate velocity, ranging from 70 p m for 15 m / s to 20 p m for 50 d s . (See also ch. 29, 0 5.2). Plasma-spraying is a method which in a sense represents a return to Duwez’s microdroplet approach an electric discharge between two electrodes in argon provides energy to melt alloy globules which are then projected by an argon jet on to a substrate to build up a deposit. The starting matenal is pre-alloyed powder, 50-100 mm in diameter. The hot gas jet needs to be blown aside to ensure rapid cooling of the impinging droplets. GAGNI~ and ROY[ 19821 have succeeded in plasma-depositing layers of a Ni-Cr-Si-B glass on to a metallic substrate. A full account of the use of this technique was published by SAPAIand HERMAN[1981], and some of the metastable phases made in this way are discussed by HERMANand BHAT[1980]. The method can also be used to solidify fine droplets in flight for subsequent compaction, though it has References: p . 1818.
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Induction coil
w <E7 e
(d)
(C)
Fig. 13. Principal methods of rapid quenching from the melt: (a) drop-smasher, using levitation-melting by induction; the pistons are pneumatically or electro-magnetically accelerated and come into action when the. falling drop breaks a light beam; (b) melt-spinning; (c) pendant-drop melt-extraction (there is a distinct version with the wheel dipping into a melt surface); (d) twin-roller quenching device.
been little used in this mode. A simple flame-heated metal-spraying gun has also been used to deposit glassy layers of Fe,Ni,P,,B, (SHINGU[1979]). A number of different methods have been used for the production of RSP wires. One is free jet melt-spinning of a molten alloy, usually a steel, through a nozzle into a gaseous quenching medium (MOTTERNand PRIVOTT [1978]). The problem here is to avoid Rayleigh instabilities which tend to break up the free jet: this is best achieved by alloying the steel with aluminum, quenching into carbon monoxide and relying on the surface oxide layer formed on the freshly formed jet to resist its break-up. Other alloys can be correspondinglytreated. An alternative approach (OHNAKAand FUKUSAKO [ 19781, I N O ~ et al. [ 19821, OHNAKA [1988]) relies on a variant of melt-spinning: the alloy jet impinges on the inner surface of a rotating vessel containing water which is held in an annular shape by centrifugal force. The method is known as the in-rotating-water splat-processing (INROWASP) (fig. 15). This, like the free-jet method, produces wires 100-200 p m in diameter. Rayleigh instabilities are avoided by carefully matching the speed of the liquid metal jet and the rotational speed of the water annulus. There is evidence from comparative relaxation studies (INOUEet al. [ 19831) that a glass in the form of wire is solidified at about the same rate as the same glass in the form of a melt-spun ribbon several times thinner, but (because of continuing close thermal contact) cools faster after solidification (an important aspect of rapid solidification that is too often ignored). In contradiction to this, a very recent study of the microstructure of Ni,Al converted into wire by the inrotating-water method (CHIBAand HANADA[ 19961) provides some (though not conclusive) microstructural evidence that wires made in this way have solidified more slowly than melt-spun ribbons of equivalent thickness. -Another method of rapid solidification which has recently enjoyed a revival is the Tuylor wire method: here a thick wire is encapsulated in a glass which is heated and drawn down, together with the molten wire
within; the change in cross-section combined with heat withdrawal by the encapsulating glass provides effective rapid cooling. Apart from the rapid-freezing aspect, this is a good method of making ultrafine wires (see review by DONALD[1987]). LEBOand GRANT[ 19741 published a key paper on production of RSP aluminum alloy powder by atomization and also of splats against a cold metal surface, followed by consolidation of the powder at a temperature and for a time designed to prevent deterioration of the desirable properties induced by RSP. Another key paper, describing the compaction and properties of splat-quenched Al-Fe alloys, was by THURS~ELD and STOWELL [1974]. All subsequent research and development of industrial uses of RSP as applied to crystalline alloys has been based on this approach. Atomization methods in use [19911) include ultrasonic atomization centrifugal (reviewed by SURYANARAYANA atomization in helium and quenching of droplets between rollers to make flakes (SANKARANand GRANT[1980]). Consolidation is either by warm extrusion or by hot isostatic pressing. Even cold explosive compaction can be used (MORRIS[1980]). An alternative approach which is increasingly finding favour is to pulverize a brittle crystalline melt-spun ribbon. A smallscale variant, capable of producing extremely fine, submicron powders, is electrostatic extraction of particles from a liquid-alloy tip drawn out into a point by a strong electric field (PERELet aZ. [198l]). This apparatus has been used for a detailed metallographic study of the microstructures of RSP aluminum alloy globules (LEVI and MEHRABIAN [19821; the globules were fine enough for TEM examination without further thinning). Yet another method which is used from time to time is spark-erosion, using a standard workshop machine for the purpose. Electrodes and a vibrating charge of blocks, all of the same alloy, are spark-eroded in a dielectric, which can be a hydrocarbon, water or liquid argon. Small drops of the alloy are heated to an extreme degree and turn into a plasma which is then quenched by the surrounding dielectric fluid. The technique is described by BERKOWITZ and WALTER[19871, and CAHNet al. [19881have exemplified the drawbacks arising from carburisation or oxidation of the alloy by the dielectric fluid; this is why liquid argon is to be preferred (BUCKLOW and DRAIN[1964]). All the foregoing methods have been systematically reviewed and compared in a [19911; he also includes recent, advanced major overview by SURYANARAYANA atomization methods, some employing lasers, which are now used to make tonnage quantities of atomised alloy powders. The historical background to the early methods of rapid solidification is reviewed by CAHN [1993]. 3.1.1. Cooling rates in rapid solidification processing The rate of cooling of the melt before, and of the solid after solidification can be accessed by direct measurement. Alternatively, the cooling rate at the freezing stage can only be deduced by measurement of a microstructural feature of the resulting crystalline alloy, most commonly the secondary dendrite arm spacing (ch. 8, $7.1) or, where apposite, eutectic lamellar spacing; but such a microstructural approach is not available when the product is a glass. Dendrite arm spacings are determined by both the temperature gradient in the solidifying melt, GL,and the velocity of the melVsolid interface, V. The cooling rate,
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dT/df,=G,.V. The spacing of primary dendrite arms is not suitable for cooling rate measurement because its relationship with GLand with Vfollows distinct power laws, whereas for secondary dendrite arms, both variables are related to the arm spacing, I, by an inverse 1/3 power law. This is demonstrated in fig. 16, (JONES [1982]), based on a range of measurements for Al-Cu and A1-Si alloys. For the fastest cooling rates, a microthermocouple projecting into the melt in a piston-and-anvil quenching unit is used (ARMSTRONG and JONES [1979]). The dendrite method is not quite so reliable for nickeland iron-based alloys; the eutectic-lamella approach, used for melt-spun aluminum alloy ribbons and relying on extrapolation from slower, measured cooling rates, gave an estimate of 3 x lo6 K/s in the neighborhood of the freezing temperature (CHATTOPADHYAYand RAMACHANDRARAO [1980]). The limitations of such indirect methods are: (1) they are not applicable to glass formation; (2) they can give an estimate only of the effective cooling rate (related to the solidification time and specimen thickness) in the temperature range in which solidification takes place, and not of the subsequent cooling rate in the solid state; (3) each family of alloys requires separate calibration; (4)the dendrites may coarsen after solidification and so yield misleading results. For all these reasons, direct measurement is now regarded as preferable, difficult though it is. The early direct measurements of liquidquenching rates which have been published have referred to the piston-and-anvil technique. Thus, HARBURetal. [1969] made use of the metals (suitably insulated) of two dissimilar pistons as the legs of an intrinsic thermocouple, whereas DUFLOSand CANTOR[1982, 19871 used a minute intrinsic thermocouple, electrically shorted by the melt itself. Pyrometry of the droplet has also been employed (e.g., KATTAMIS et al. [1973]). Melt-spinning poses a more difficult et al. [19821 used calibrated black-andmeasurement problem altogether: WARRINGTON
0.071
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Fig. 16. Dendrite spacing, A, as a function of cooling rate, E , near the melting temperature, for A 1 4 to 5 wt% Cu (open circles) and A1-7 to 11 wt% Si (solid circles). The plot refers to secondary dendrite arms. The line shown represents leIB=const.=SO mm(K/s)’”. (After JONES [1982]). References: p . 1818.
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white photography, while later techniques have focused on the use of calibrated photometry of color photographs of the surface being investigated (GILLENand CANTOR and CANTOR[1986]) and the use of infrared imaging devices, which [1985], BEWLAY last have proved particularly useful for spray-deposition (GRANTet aE. [1989], GRANT and CANTOR[1990], BEWLAYand CANTOR[1990]). Figure 17 shows some results by Duflos and Cantor for a drop-smasher, while fig. 18 shows photometric measurements relating to melt-spinning; the sharp decrease in cooling rate when the ribbon loses contact with the copper wheel should be noted. It is quite clear that the cooling rate varies substantially between the freezing point and ambient temperature. Thus at 15OO0C, near the freezing-point of a steel, the cooling rate for piston-quenched steel or iron samples 25-50 ,um in half-thickness (the proper measure) is 106-107 Ws;the higher values apply to the very rapid-acting piston-and-anvil apparatus designed by CAHNet al. [1976]. The cooling rate at 1000°C is = 10 times slower than at 1500"C, and at 50O0C, 2&30 times slower again. It is evident that the
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Fig. 17. Data from many cooling curves obtained by embedded microthermocouple and oscilloscope traces during piston-and-anvil drop-smashing of iron, replotted as cooling rate, p, versus temperature, T.The voltages cited are electromagnetic accelerating voltages; higher voltages imply faster impact. The weights refer to specimen sizes; larger specimens, heated for longer periods, had greater superheat. (After Dwms and CANTOR [19821).
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1600 U 1
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\ wheel velocity = 24 m/s
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custom of characterizing the cooling rate by a single figure is unacceptable, unless the temperature range to which such a figure applies is specified. Single-roller melt-spinning turns out to be slower than piston-and-anvil quenching (HAYZELDEN [1983]): thus, for a 40 mm-thick steel ribbon, the measured cooling rate is 5 x Id Ws at 1500°C and only lo4 W Sat 1000°C (because the ribbon has by then left the wheel). It is generally accepted that dual-roller quenching is even slower than single-roller melt-spinning overall, in spite of the presence of double chill block, because the duration of contact between the ribbon and the chill block is so brief. However, the initial cooling rate appears to be very high (DAVIESet al. [1978]). In the case of rapid quenching by atomization, cooling rates have been estimated from dendrite spacings: thus, for a gasatomized superalloy, cooling rates ranging from 103-106 Ws have been estimated as References: p . 1818.
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particle diameters range from 400 to 10 p m (DUFLOS and STOHR [1982]). The preferred approach at present., however, is by means of heat-transfer calculations, as exemplified for conventional casting in ch. 8., 0 2. The crucial parameter here is the Nusselt (or Biot) number, N, of both specimen and “chill block” (the latter, in the case of a droplet solidifying in flight, is gaseous). N=M/k, where h is the heat-transfer coeflca‘ent,d, is the effective specimen or chill-block thickness and k is the thermal conductivity. As h increases, cooling changes from Newtonian, via intermediate to “ideal” cooling, with progressively larger temperature gradients in the specimen. h can be estimated from [1982]), but measured cooling rates in the case of piston-and-anvil quenching (CANTOR for powder atomization it is necessary to compute h from first principles for specified boundary conditions (h is very low for pure radiative transfer and several orders of magnitude larger for favorable convective cooling). The theory has become extremely [1982], MEHRAcomplex: full treatments have been published by LEVIand MEHRABIAN BUN [I9821 and CLYNE[1984];reference should be made to these papers. The cooling rate varies of course as a function of droplet diameter and depth within the droplet, and is perturbed by the release of latent heat in a way which depends on the undercooling that preceded the start of freezing. A particularly important regime is with heat-transfer rapid enough to permit hypercooling, that is, suppression of nucleation till the sample has undercooled so far that the release of the entire latent heat is insufficient to reheat the sample to the equilibrium freezing point: freezing can then go to completion without any further heat extraction. The analysis by CLYNE[1984] concludes that there are two distinct approaches to generating very rapid crystallization from the melt: one can either hypercool, via very small droplets, or else accept a lesser degree of undercooling and arrest the subsequent recalescence by means of highly efficient heat extraction. In the theory of atomization, it turns out to be more useful to compute solid/liquid interface velocities rather than cooling rates, since the former relate more closely to interface stability. Mehrabian cites values for 50 p m aluminum droplets of 4 m / s at the start of freezing, dropping to 0.04-0.13 m / s in the intermediate stages of freezing.
3.2. Solidification of highly undercooled liquids When a liquid alloy is cooled, solidification normally starts at a small undercooling below the liquidus temperature, triggered by nucleation on heterogeneities. However, liquid alloys are typically rather resistant to homogeneous nucleation. The most thorough kinetic analysis has been by TURNBULL [1952] of liquid mercury in which homogeneous nucleation appears to occur only at undercoolings as high as 35 to 40% of the absolute melting temperature. Thus there is a possibility of achieving large undercoolings if heterogeneous nucleation can be suppressed or at least inhibited. This is achieved by having a clean melt and ensuring that it does not make contact with a solid container; various methods are illustrated in fig. 19. In the emulsion method (exploited by TURNBULL [1952] and more recently by PEREPEZKO[1984]) the liquid alloy is dispersed in an inert carrier liquid. When the number of alloy droplets exceeds the number of potent heterogeneous nucleants originally in the alloy, some of the droplets must exhibit substantial undercooling. However, for the
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GLASS
Fig. 19. A schematic illustration of the most common methods for inhibiting heterogeneous crystal nucleation and thereby obtaining highly undercooled liquid alloys: (a) droplet emulsion, (b)substrate, (c) entrained droplet (mushy state quenching), (d) fluxing, and (e) drop-tube or levitation. (After PEREPEZKO[1984]).
emulsion to be stable the droplets must be coated and this coating may itself act as a nucleant. The coating problem is not present in the fluxing technique in which a single alloy drop is surrounded by viscous flux or slag, which, in addition to isolating the drop from contact With solid container, may have a cleaning action (KuI, GREERand TURNBULL [ 19841). Truly containerless solidification can be achieved for droplets in free-fall in evacuated drop-tubes (GREER[ 19941) and levitated, for example electromagnetically (SCHLEIPet al. [1988]). A few examples of maximum undercoolings which have been achieved for various metals are shown in table 2. When a melt is highly undercooled, solidification when it does occur is rapid and shows all of the microstructural effects outlined in 8 1.1. Crystal nucleation can even be avoided completely, yielding a glass (F'EREPEZKO and SMITH [1981]). The techniques for obtaining undercooled melts axe of interest both for production of metastable states and for fundamental studies. The emulsion and fluxing methods, for example, can readily be used to measure not only undercooling but. also the specific heat of the undercooled liquid, an important thermodynamic parameter. Of particular interest is that electromagnetic levitation of a droplet can be used for quantitative studies of solidificationvelocities at selected undercoolings. Studies of undercooled alloy melts have been comprehensively reviewed by HERLACH et al. [1993]. Among the phenomena investigated are grain refinement (GREFB[1991]) solute trapping and the nucleation and growth of metastable phases. These topics are discussed in some detail in several sections of ch. 8, P4.2. References: p , 1818.
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Table 2 Some examples of maximum undercoolings achieved for pure metals, using various techniques. Much more extensive surveys will be found ‘in KELTON [1991], HERLACH er al. [1993] and GREER119941, from which these data are taken. Metal
3.3. Deposition by evaporation or sputtering The rapid-quenching methods described above have been used to make not only crystalline alloys but also metallic glasses. Several glass-making methods which do not depend on melt-quenching will be outlined here. These methods may collectively be termed pseudo-RSI? One group of such methods depends on the deposition of vapor on cold substrates (often held at liquid-nitrogen or even liquid-helium temperature). These methods are in fact of older ancestry than true RSP methods and were used to make various amorphous and supersaturated crystalline materials in the 1930s (Kramer [1934, 19371) and 1950s (BUCKELand HILSCH[1956]), generally in the form of thin films. (For more details, see CAHN’S 1993 review). Sometimes thermal evaporation is used to produce the vapor (MADER[1965] and SINHAet al. [1976]), but cathodic sputtering is more usual; if a powerful high-rate sputtering apparatus is used, glass samples several p m thick can be [1978]). These techniques are made, though at very high cost (see review by DAHLGREN now waning, in spite of the fact that more alloy compositions and wider composition B N R U T( ranges can be made glassy by vapor methods than by liquid-quenchingLLU [1981]). Perhaps a reduction of interest in these methods is a recognition that in metallurgy, as opposed to microcircuitry, thin films have only a modest role. 3.4. Amorphization by irradiation A further approach is heavy irradiation (to a dose of at least one displacement per atom) of a crystalline alloy (generally an ordered intermetallic compound), using either neutrons or ions (RECHTINet al. [1978], ELLIOTTand Koss [1981]). As judged by calorimetric tests, such “artificial” glasses are structurally very similar to the same alloys vitrified by RSP. A related “solid” approach to making a glass is by implantation into a pure metal of ions of a different element, most usually metalloid ions. Phosphorus has been most effectively used. The bombarding ion both destroys crystalline order and, by
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occupying strategic locations in the glass structure, stabilizes it. Self-ion bombardment (e.g., Ni’ into nickel) never generates a stable glassy structure in a metal (although apparently it can do so in silicon). The technique has been reviewed by BORDERS[1979], and there is later research by RAUSCHENBACH and HOHMUTH[1982], and by RUSSELL [1985]. Very recently, a complete and up-to-date review by How [1995] has been published. Here there is space only for a thumbnail sketch: - A striking group of observations is on electron-induced amorphization. MORI and FUJITA[19821 have established the temperature dependence of the critical dose for amorphizing NiTi with 2 MeV electrons, while MORIet al. [1983] have observed the surprising phenomenon of initial amorphization preferentially at dislocations, in the same alloy. The theoretical background, including a discussion of critical irradiation doses, is presented by CARTER and GRANT[1982], It is important to recognize that glassy layers made in this way are less than one micron thick. It is now well established that the intermetallic compound must be disordezd by the irradiation below a critical level of long-range order; it then et al. [1984, 1985, 19861 ). Mom et al. becomes unstable towards amorphization (LUZZI E19841 nevertheless incline to an alternative (or additional?) criterion, requiring a large critical vacancy concentration. CAHN and JOHNSON [ 19861 have critically discussed the relative merits of these models. A more recent discussion of these rival models is by MOTTAand LEMAIGNAN[1992]: they assemble evidence that a critical degree of destruction of long-range order is generally a necessary but not a sufficient criterion and that a critical concentration of one type of point defect (either vacancies or interstitials) is also necessary, and they answer earlier criticisms suggesting that the required point defect concentrations are not in practice achievable. MENDOZA-ZELIS et al. [1982] describe the amorphization of nickel films by phosphorus ion implantation. Electrical resistivity indicates that the glass thus prepared is structurally identical to glasses of the same compositions made by evaporation or electrodeposition. (The last two approaches to glass-making, together with the related technique of electroless plating, with a partial but extensive list of compositions prepared in these ways, are outlined by DIETZ[ 19771.) Ion implantation is also used to amorphize crystalline silicon; some of the consequential research is outlined in the next section. A technique closely related to ion implantation is ion-beam mixing, in which two discrete layers at a surface are mixed on an atomic scale by bombardment with “neutral” ions such as A+ or Xe’. Thus, Au-Si glasses have been made by ion-mixing of a gold-coated silicon substrate; this same glassy alloy has been made by RSP of the melt, evaporation on to a cold substrate, laser-treatment and ion-mixing. Ion-beam mixing is reviewed by FOLL~TAEDT [1991]. The methods mentioned in the last two paragraphs effectively involve treatments at or close to metallic surfaces, which is the subject of the following section.
3.5 Rapid solidification processing of surfaces A benefit of the various rapid solidification methods that operate at surfaces is that only a very thin layer of the solid needs to be melted and the underlying solid then can act as a built-in heat sink. The usual instrument employed for this purpose is a laser, References: p. 1818.
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either continuous-acting (CO, type), used in scanning mode, or pulsed (ruby type). The most common way of rapid quenching at surfaces involves no local change of composition, but it is possible to coat the surface with a thin layer of another substance and metastably alloy it into the melted surface; in this way, expensive substances need only be used in small quantities. RSP at surfaces has in practice been used for two very different purposes: (1) There is a large body of research (which can only be briefly exemplified here) on the lasertreatment of semiconductor surfaces, silicon in particular, and (2) laser treatment has been used on a large scale for the hardening of surfaces of steel and some other industrial alloys. Figure 20 shows the variables involved in the treatment of a surface by a continuous wave laser. Heat-flow calculations, for adequately large heated spots, can be simplified by neglecting sideways heat flux and treating the problem in terms of one-dimensional [1982]. He heat flow: calculations of this kind have been published by MEHRABIAN assumes that the heat which can be usefully absorbed is limited by the onset of vaporization; on this basis, increasing the laser power (which entails faster scanning over the surface) yields a shallower melt, faster refreezing and a steeper thermal gradient. Typical figures, for aluminum, are 650 ,urn or 6.5 ,um melt depth for 5 x lo8or 10'' W/m2 heat flux,respectively. The estimated solid-liquid interface velocities are 0.035 and 3.5 m / s , respectively. KEAR et al. [1981] estimated that, again taking vaporization as a limiting factor, quench rates up to I d K/s can be achieved for a melted layer = 10 pm thick. More recently, LINand SPAEPFN [1984] showed that a pulsed laser can provide cooling rates as high as 1014K/s.
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Fig. 20. Schematic illustration of laser beam-substrate geometry during rapid surface melting and refreezingby laser treatment. (After MEHRABIAN [1982]).
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The use of Q-switched ruby lasers so designed as to give extremely short pulses in the nanosecond (or even picosecond) range has been highly fruitful in research on silicon. A good example of research on metastable structures in silicon is the body of work on the melting of amorphous silicon. A layer of amorphous silicon can be created at the surface of a silicon crystal slice by ion-implanting either silicon itself or another species such as indium. It was first suggested in 1978, independently by Spaepen and Turnbdl and by Bagley and Chen, that such an a-silicon layer might undergo a firstorder melting transition at a temperature several hundred degrees lower than the melting temperature of crystalline silicon (although a-silicon cannot have an exact themodynamically defined melting temperature). The melting of a covalently bonded amorphous solid to give a metallically bonded liquid here is bound to be discontinuous. This was duly confirmed by laser-heating in situ of the a-silicon layer, with a = 3 ns pulse: what is done here is to make transient measurements of properties, in particular, electrical conductance and optical reflectance, which vary with the state of the silicon, with a time resolution of a few nanoseconds. Indeed, the investigators found that the molten layer can then explosively crystallize: the release of latent heat makes the process self-sustaining. Fig. 21, from a review by POATEet al. [1987], shows transient conductance and reflectance measurements on such a specimen, and also indicates the evolution with time of the molten and subsequent crystallized layer at the expense of a-silicon. (Because of the large latent heat released, crystallization under these circumstances is “explosiveyy).The transient measurement approach can even be used to assess whether the crystallizing layer is nucleated at the surface or in the interior of the melt (PEERCYet al. [1990]); the conclusion was that nucleation is internal. Figure 22 shows the time-dependent depth of
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the molten layer deduced from reflectance measurements (because reflectance changes as melt thickness changes) for different laser intensities. It can be seen that the dimensions involved are very different from those calculated for typical experiments with metals, as indicated in the preceding paragraph. At the other extreme, laser treatment of steels, titanium and aluminum alloys has been widely practised, with and without powder injection into the laser-melted surface pool to change local composition. The field has recently been reviewed by MORDIKE[1986, and BERGMANN [1993]. Thus, aluminum-silicon alloys can be 19911 and by SCHUBERT effectively “modified” to refine the scale of the eutectic and enhance mechanical properties thereby, without the use of the usual modifying dopants. Similar effects can be achieved with cast iron. The nature of nucleation of crystalline phases in a surface melt-pool created by laser irradiation can be deduced by examination of the resultant microstructure, and the conclusion was drawn that nucleation at the free surface, at the melt/substrate interface, and homogeneously within the melt, can all be generated by varying the conditions of irradiation. For further details, Mordike’s review should be consulted. A further approach is to make layered alloys and use laser pulses to convert these into a homogeneous metallic glass. Thus, LINand SPAEPEN[1982] made an iron-boron glass with as little as 5 at% B (much less than is required when less extreme methods are used) by picosecond laser treatment of a finely layered Fe/Fe,B sandwich.
3.6. Electrochemical, electrolessand sonochemical deposition of amorphous phases It was discovered long ago by Brenner at the NBS that Ni-P alloys containing more than = 10 at% P can be made by electrodeposition, to give an ultrahard surface coating.
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This industrially useful method has been steadily improved over the years (BRENNER 119631). The most recent development has been an examination of the influence of pulsed as opposed to steady cumnt (LASHMORE and WEINROTH [1982]), and it appears that Ni-P glass deposits made under these two regimes are structurally distinct (LASHMORE et al. [19821). Little has been published about other compositions that can be macle in amorphous form by electrodeposition, but there is evidence that some research has been done under conditions of industrial secrecy: apparently Co-W-B is another alloy family that can be made amorphous in this manner. Also, bright amorphous chromium (presumably stabilised by solutes) can be electrodeposited from chromic acid solution etal. [1986]; TSAIand Wu [1990]). with additives (HOSHINO Electroless plating has already been mentioned in passing in Q 3.4, in connection with DIETZ’S1977 review. Much research has been done on this technique, which is based on electrochemical principles though no external potential difference is applied. It has very largely been applied to the deposition of Ni-P coatings applied from solutions containing hypophosphite ions, which can be amorphous or nanocrystalline and are very hard. This extensive research, much of it concerned with the control of deposition rates, is reviewed by ROOSet al. [1991]. A recently discovered, distinctly exotic way of making an amorphous metal is via sonochemistry. This term relates to the technique of injecting energy into a chemically reacting system by means of ultrasound. The key investigators here have been the chemist K. Suslick and his group. Ultrasound is conveyed to a liquid mixture of chemicals in the simple apparatus shown in fig. 23. Suslick has explained how high temperatures and consequential ultrarapid cooling arise in such an apparatus (SUSLICK [1993], [1995]). Vapor-filled bubbles are formed in the liquid through resonant absorption of sound energy and oscillate in size at, typically, 20 kHz. When a bubble has grown so large that it is no longer resonantly coupled to the ultrasound field, it collapses at extreme speed (“cavitation”) under the influence of surface tension and in the process, the vapor is heated adiabatically to a high temperature; immediately after this stage, the surrounding liquid cools the vapor as fast as it was heated. The temperatures reached and the timescale involved have been measured, among other techniques, by means of sonoluminescence, the emission of light flashes during bubble collapse, and temperatures near 5000 K have been estimated (FLINT and SUSLICK[1991]). (The very existence of sonoluminescence has been doubted until, recently, BARBERand PUTTERMANE19921 measured the time-dependent radii of sonoluminescent bubbles at the same time as the sonoluminescent output. MADDOX[ 19931 has discussed the implications of these striking observations). The duration of the high temperature is estimated to be of the order of 100 ns and the cooling rate thereafter is around 10” K/s, much faster than any of the other techniques discussed hitherto. “Nearly pure” (>96% Fe) amorphous iron powder, of 1-2 nm particle size, has been made by the sonochemical decomposition of iron pentacarbonyl, a liquid (SUSLICKet al. [1991]). The powder crystallizes at the surprisingly low temperature of = 308°C. This approach does not appear to have been applied yet to nickel carbonyl.
References: p . 1818.
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Fig. 23. A typical sonochemical apparatus with directly immersed ultrasonic horn. (After SUSLICK [1993]).
3.7. Solid-state amorphization reactions (SSAR) Since the amorphous phase in alloys can be a well defined metastable state, it can be produced by reaction, enabled by thermal annealing. The initial "energized" state (see 5 1.3) is simply an aggregate of unreacted elements. The first metallurgical demonstration of a solid-state amorphization reaction was by SCHWARZ and JOHNSON[1983], who showed that thin polycrystalline gold and lanthanum films in a multilayered stack would react at 50-100°C to give an amorphous alloy. (This research followed a slightly earlier discovery of a related phenomenon: the compound Zr,Rh, in a metastable crystalline form, could be amorphized by reacting it with hydrogen to form a metastable amorphous hydride (YEHer al. [ 19831; this finding has been broadened more recently by AOKIer al. [ 19911, who discovered hydrogen-induced amorphization in numerous binary intermetallic compounds in which one constituent is a hydride former). A significant number of elemental pairs have since been found to undergo similar reactions similar to that between gold and lanthanum; these include Ni-Zr, Cu-Zr, Fe-Zr, Ni-Ti, Ni-Nb, Cu-Ti among others, as reviewed by JOHNSON [1986] and also in the first conference devoted to this phenomenon (SCHWARZ and JOHNSON [ 19881). The amorphization reaction is associated with two characteristics of the elemental pairs, one thermodynamic and one kinetic: a strongly negative heat of mixing in the amorphous state, and
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a marked asymmetry in diffusivity, that is, one element diffusing much faster than the other. Since diffusional asymmetry is most likely associated with a size difference, it can be seen that the characteristics needed for the amorphization reaction have much in common with those favoring glass formation generally ($2.3). The negative heat of mixing provides the driving force for the reaction, but it is associated with a tendency for the formation of highly cohesive intermetallic compounds. However, at the low annealing temperature of the reaction (=250°C, for example, in the case of Ni-Zr) these compounds have difficulty in nucleating, and then the metastable extrapolated liquidus curves shown in bold in fig. 24 become relevant. The amorphization reaction is then interpretable as eutectic melting occurring below Tgand therefore yielding an amorphous solid rather than a liquid. In the case of Ni-Zr, nickel is a fast interstitial diffuser in crystalline zirconium, whereas zirconium is a slow substitutional diffuser in crystalline nickel. This difference in behavior, in which the Ni can diffuse several orders of magnitude faster than the Zr at the same temperature (GREEXet al. [1990]), applies also in the amorphous phase. Diffusional asymmetry of this kind appears to apply in the other systems showing the solid-state amorphization reaction, the high mobility of one species enabling the reaction to occur, the low mobility of the other species inhibiting nucleation of the equilibrium intermetallic compounds. Systematic experiments on Ni-Zr have established that the amorphous phase is et al. nucleated at grain boundaries in the polycrystalline zirconium films (SCHR~DER [I9851 and VREDENBURG et al. [1986] have confirmed this by a beautiful experiment in which they demonstrated the impossibility of launching an amorphization reaction in a bilayer consisting of polycrystalline nickel deposited on single-crystal zirconium. It is mostly found that the amorphization reaction proceeds on a planar front at a diffusion-controlled rate which diminishes as the amorphous layer thickens. Beyond a critical thickness of reacted layer, that is, below a critical front velocity, a crystalline
2000
0 Ni
20
40
60
80
100 Zr
Zr (at. 7 ' )
Fig. 24. Ni-Zr equilibrium diagram, a system exhibiting solid-state amorphization. The bold lines show the and GREER[1989]). terminal liquidus curves and their metastable extrapolations. (After HIGHMORE References:p. I818.
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R.W Cahn &A. L Greer
a.19, 4 3
compound appears. Its appearance has been quantitatively interpreted by HIGHMORE [1990] in terms of transient nucleation theory. The intermetallic nuclei thus formed can grow at the expense of the previously formed amorphous layer (MENGet al. [1988]). Thus, if a fully amorphous product is to be obtained by the reaction, initial thicknesses of the elements in a multilayer stack must be small, typically below 100 nm. Multilayers with such thin layers can be made by thin-film deposition, but then the final product is likely restricted to a thin-film form, and this could be obtained more readily by initial codeposition of the elements. S S A R in multilayers has been illuminatingly examined by JANKOWSKI et al. [1995]. The more general issue of the relation between “nucleation of disorder” in SSAR, melting and the destruction of atomic long-range order in a solid solution has been discussed by C m and JOHNSON [1986] and brought up to date in a stimulating thermodynamic approach by WOLF etal. [1990].
3.8. Amorphwationby mechanical processing When the diffusion-induced amorphization reaction was first discovered, it was soon recognized that deposition by evaporation was not the only way of making the initial multilayers: it was also possible to roll or wire-draw composites of foils or wires of the constituent metals (to reduce the scale of the microstructure and to weld the constituents) and then to anneal the product. In fact, the production of ultrafine filamentary composites by swaging, wire-drawing or rolling of mixtures of two metals, notably for use as superconductors, was pioneered in the 1970s long before the SSAR had been discovered, and reviewed by BEVK[1983]. Soon after the discovery of SSAR, ATZMONet al. [1985] rolled stacks of alternating 20-200 mm foils of Ni-Zr, Ni-Er and Cu-Er to very heavy reductions and then annealed them: partial conversion to amorphous phase was observed. SCHULTZ [1987] worked with similar Ni-Zr stacks: his amorphous phase had a composition different from that of the average of the stack, implying that the zirconium had become exhausted before the nickel. It appears that Schultz may have been the first to observe this kind of mechanically aided amorphization (SCHULTZ [1989]). Bordeaux and Yavari conducted an extensive programme of experiments on coldrolled multilayer stacks of systems such as Ni-Zr, Ni-Nb, Cu-Zr, Cu-Ti and Fe-Ti, in the late 1980s (see review by BORDEAUX and YAVARI[1990a]). X-ray diffraction established that a measure of amorphization took place during the rolling process itself, but annealing was required to complete it. They were able to amorphize AI-Pt stacks completely during heavy rolling (BORDEAUX and YAVARI[ 199Obl); this kind of process is termed mechanically induced amorphization, to distinguish it from mechanically aided amorphization They made careful kinetic measurements of amorphization rates, using XRD and DSC, and were able to show (in spite of difficulties occasioned by the fact that the as-rolled layer thicknesses had a broad distribution) that amorphization was much faster than in an interdiffusion experiment at the same temperature, in the absence of cold work. The comparison was also complicated by the fact that deformation causes local heating at the foil interfaces, which may transiently approach +lo0 K as measured by thin-foil thermocouples (MARTELLI etal. [1990]). From their kinetic studies of the rate
Ch. 19, $ 3
Metastable states of alloys
1767
of thickness increase of the amorphous layers, Bordeaux and Yavari deduced that the activation energies for nickel diffusion through the layers were much lower than those previously measured for undeformed multilayer stacks. This suggested that “holes” localised low-density regions in a metallic glass generated by deformation - had accelerated diffusion (see ch. 7, 0 9.1.2). It is possible that the extreme defect concentrations generated by very heavy deformation altered the free energy balance between the various phases sufficiently to accelerate the kinetics significantly. In parallel to these studies, ball mills were introduced as an alternative way of imparting heavy deformation and thus bringing about amorphization without the need for annealing. This was originated by YERMAKOV[I9811 in Russia, and soon after taken up by SCHWARZand KOCH[1983] in America, who prepared Ni-Ti and Ni-Nb amorphous phases by ball-milling. In this process, grinding balls, usually of specially hardened steel, are combined with a small amount of powder, either mixed metallic elements or powder of a prealloyed intermetallic compound, in a container which can be rotated or vibrated or subjected to a combined form of motion. The end-results are found to be sensitive to the particular form of grinding motion employed. Both mixed elements and prealloyed powders often behave in the same way with respect to amorphization. Details of the extensive body of research on this process of mechanically induced amorphizationwhich has been accumulated within a few years can be found in a book chapter by KOCH [199 11. The accumulated evidence suggests that mechanically aided amorphization and mechanically induced amorphization are variants of a single process. The criteria which govern what systems can be amorphized in this way are discussed at the end of 0 2.3.
3.9. Pressure effects In considering phase transformations, we have mostly been concerned with changes in temperature. Yet pressure is an equally important thermodynamic variable. As we have seen, metastable phases or microstructures can be achieved as a result of rapid temperature changes. Of particular interest, then, is that pressure changes (because they do not involve a conductive process) can, especially for samples of large cross-section, be much more rapid than temperature changes. There are several ways in which pressure and pressure changes can be used to attain large departures from thermodynamic equilibrium. When pressure is varied, new phases can appear and pressure is then directly of interest in providing access to a wider range of structures. Some of these structures can, in fact, be obtained without application of pressure. For example, a metastable phase of pure bismuth obtained at high undercooling in a droplet emulsion experiment is a highpressure phase; thermodynamically it can form because the liquid is cooled below its (metastable) melting point, and it must be kinetically preferred to the stable phase (PEREPEZKO and ANDERSON [1980]). In most cases, however, application of pressure is necessary to cause a transformation (usually in the solid state) to a high-pressure phase. The question is whether this phase will be retained when the pressure is again reduced to ambient, Temperature is commonly used to effect retention under ambient conditions. The general procedure is as follows: Pressure is applied, the temperature is raised to References:p . 1818.
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R W Cahn and A. L Greer
Ch. 19, 9 3
permit the transformation to the high-pressure phase to take place, the temperature is reduced to lower the atomic mobility, and the pressure is then reduced with the atomic mobility sufficiently low to prevent decomposition of the high-pressure phase. In this way metastable high-pressure phases are obtained. The classic example of such a phase is diamond. Research in this area has concentrated on compounds, such as InSb, GeP, Nb,Ge, Nb,Te. Before the advent of oxide-based high critical temperature superconductors, high-pressure phases such as Nb,Ge were of considerable interest for their superconducting properties. JAYARAMAN [19801 has reviewed metastable phase synthesis by this route. The metastable compounds produced can in some cases decay [1990]). into amorphous solids (BATTEZZATI A special case of a transformation induced by pressure is melting. This of course occurs only in those systems which show a contraction in volume on melting. Of these, by far the most widely studied is ice. Ice held at 77 K and subjected to a pressure of 10 kbar undergoes amorphization (WHALLEY[19881). Thermodynamically this result is straightforward the high pressure takes ice to its melting point despite the low temperature. Although there is some superheating, the ice does melt, albeit to an amorphous phase rather than liquid water, because 77 K is below Tg for water. A similar technique could find application in some semimetal systems showing similar volume changes on melting. Just the converse effect has been exploited in a wide range of alloys showing the more usual behavior of a volume expansion on melting. For these materials application of pressure increases the melting temperature. Thus rapid pressure application to the liquid alloy can induce rapid solidification. After the solidification is complete, the temperature is reduced to ambient, not necessarily rapidly. The crystalline products can show all of the features of materials rapidly solidified by temperature reduction, such as refined microstructure, extended solid solubility and metastable crystalline phases. An and RAJASEKHARAN [1986], and the field example of work in this area is that by SEKHAR has been comprehensively reviewed by REDDY and SEKHAR[1989]. The rapid pressure application technique does not lend itself to forming amorphous phases. The application of pressure increases the thermodynamic driving force for crystallization of the liquid without as large a reduction in atomic mobility as would be brought about by the corresponding temperature change. Nevertheless, application of high pressure should decrease atomic mobility and this has been successfully exploited. For example, C u S n alloys held under pressure can be heated by electric resistance, and then quenched by turning off the electrical current. When this is done, Cu-(12-17 at.% Sn) glassy phases can be made, which is not possible under ambient atmospheric pressure (BATTEZZATI [1990]). PONYATOVSKY and BARKALOV [19911have developed this process, which they call thermobaric quenching, and have made glasses in the Cd-Sb, Zn-Sb and A1-Ge systems. Apart from the pressure effects outlined above, there is a further series of recent observations referring to silicon. Silicon undergoes a transformation to a metastable amorphous structure when subjected to an indentation with a diamond pyramid (e.g., PHARR et al. [1992]), a phenomenon linked with the metunon-metal transition treated by Mott many years ago. The amorphous phase remains metastable after the indenter is
Ch. 19, 94
Metastable states of alloys
1769
removed. The complicated pedigree of this observation has been explained by C m [19921. Distinct from this stress-induced crystal-to-amorphous transition which is observed at ambient temperature is another, discovered by F’mouz et aZ. [1990] who found that indentation of silicon in the temperature range 400-650°C leads to a metastable, martensitic transformation from diamond-cubic to a hexagonal crystal structure; germanium behaves similarly.
4. Metallic glasses: structure and properties 4.1. Structure!
In a single-phase solid alloy the distinctive feature of the glassy state is that it has, in essence, no microstructure. There is no crystal lattice in which to have defects such as grain boundaries or dislocations. But the glass does have a structure in the way its atoms are arranged, and this structure is important in determining its properties. Relaxation (84.2) of the glassy structure on annealing can, for example, lead to increases in viscosity by a factor of up to lo5. Structure and properties can also show a natural or induced anisotropy. This section reviews the great deal of work which has been done to determine and describe the structures of metallic glasses. Although it is not possible to obtain full information as for a crystal structure, the main features of glass structure have now been established. The problem of determining the structure of a crystal consists merely of identifying the coordinates of all the atoms in the unit cell; though the task is very difficult for large and complex unit cells, yet in principle (if the “phase problem” can be solved) the structure can be determined precisely. Conversely, for a glass, the structure can described only on a statistical basis: there are no unit cells and the environments of different, chemically identical atoms will necessarily vary. Herein lies the difficulty of determining and describing the structure of any glass (or liquid), and the intellectual challenge no doubt accounts for the attention devoted to the problem. Almost all the analysis of glass structure has been based on pair distribution functions (PDFs), because they can be determined from scattering experiments. PDFs give a measure of the probability of finding an atom centre at a distance r from an average central atom; they are often given in a reduced form G(r)=4mb(r)-p0], where p(r) is the number of atoms per unit volume at a distance r and po is the number of atoms per unit volume in the sample as a whole. The PDF gives statistical information about distances only, and this is insufficient to specify the structure. The angular distribution of interatomic vectors can readily be extracted from structural models, but cannot be determined from conventional scattering experiments. The determination of glassy structures is based on the comparison of measured PDFs with those calculated from trial structural models. PDFs can be determined with good precision (especially with the advent of synchrotron-based high-intensity sources of Xrays or neutrons (EGAMI[1994])) and can discriminate between models. When an acceptable match is obtained, there is still the problem that the solution is not unique; this is a more severe problem for glasses than for the analogous determination of crystal References: p . 1818.
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R.W Cahn and A. L Greer
Ch. 19, 4 4
structum, because of the angular information missing from PDFs. The discussion which follows is based on PDFs determinable in diffraction experiments, but at the end of the section less conventional methods for structural analysis will be introduced; some of these methods are capable of yielding angular information missing from PDFs. The PDF of a glass is determined by Fourier inversion of scattering data, for example from normalized X-ray scattering curves of the kind shown in fig. 25. For an alloy glass one set of data is sufficient to determine a radial distributionfunction (RDF)in which the different types of atoms are not distinguished. Alternatively, a partialpair distribution function (PPDF) can be determined for each type of atom pair. For a binary alloy there are three types of pairs, and to determine the three PPDFs it is necessary to perform at least three scamring experiments in which the two types of atom have differing relative scattering powers. An example of the type of analysis which can be achieved is given in fig. 26 showing the Ni-Ni, B-B and Ni-B PPDFs in a Ni,,B,, glass. For a ternary alloy there would be six PPDFs, and an analysis of this, or greater, complexity has not yet been attempted. There are various ways of achieving the differing relative scattering powers for the determination of PPDFs. These are: (a) Different radiations: X-rays, neutrons and electrons are all possible, but the poor precision available with electrons is a problem. (b) Anomalous dispersion: For X-rays the effective atomic scattering factors can be altered by using a wavelength close to an absorption edge. 600 -
500 -
I
m
+ .C
3
f 400
e e0
-
+ .-
.-C
- 300 -
.-
u)
C m
c
c ._
2 200-
.I
a,
c
0 0
cn
10.0 -
28/O
Fig. 25. Scattered x-ray intensity versus scattering angle, 0, for amorphous Ni8,Blp (After CARGILL, private communication, 1979).
Ch. 19, 5 4
1771
Metasfablestares of alloys
n
I
0.2
0.4
0.6 I-
0.8
1.O
1.2
(nm)
Fig. 26. Paaial pair distribution functions for amorphous Ni,,B,,, as measured (full curves) by LMARTER ef a!. [1982] and as calculated (broken curves) by DUBOISetal. [I9851 from the ‘chemical twinning’ model.
(c) Polarized neutrons: The scattering from ferromagnetic samples is different for different planes of polarization. (d) Isotopic substitution: Isotopic substitution leaves the structure of the glass unaffected, but may strongly alter neutron scattering lengths. The PPDFs in fig. 26 were determined in this way. Three Ni,,B,, samples were prepared, one with natural nickel, one with 62Niand one with a mixture of 60Niand 62Nito give an effective scattering length of zero. In this last case there was no scattering by nickel, and Fourier inversion of the neutron scattering data yielded directly the B-B PPDF. References: p . 1818.
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R. K Cahn and A. L Greer
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(e) Isomorphous substitution: It may be possible to substitute one element for another of similar type so that the structure is likely the same. X-ray scattering can then be. used with the different atomic scattering factor. The experimental and computational procedures for determination of RDFs and PPDFs have been described by WAGNER[1983], EGAMI[1981] and SuzUKI[1983]. Assessment of the accuracy of pair distribution functions (obtaining a figure of merit analogous to the R (Rietveld) values currently used to assess the quality of a crystal and GASKELL [1982]. structure determination) has been considered by LIVESEY The basis for structure determination is the comparison between measured and calculated PDFs. Yet even without such comparison the measured PDFs reveal basic features of the glassy structure. The first peak in the radial distribution function indicates the nearest neighbor separation and the area under the peak gives the average coordination number. The separation is as expected for a condensed phase, and the coordination numbers are typically in the range 11.5 to 14.5, indicating that metallic glasses are densely packed, as would be indicated by an extrapolation of liquid properties. When PPDFs are examined it is found that the distribution of the different elements in a glass [1976] who can be far from random. This was first established by SADOC and DIXMIER found that in a Co,,P,, glass the relevant PPDF showed that P-P nearest neighbor pairs were completely avoided. This avoidance of metalloid neighbors seems to be a general feature of transition-metal-metalloid alloys (types (i) and (ii) in $2.2) in their normal composition ranges. When the metalloid content is increased, as for example in Ni,B,, et aZ. [1984], metalloid contact becomes inevitable, but is still much studied by COWLAM less prevalent than in a random alloy. Table 3 shows evidence for such chemical ordering in a variety of glasses. The derivation of a suitable order parameter has been considered and CARGILL[1985]. by SPAEPEN For the metal-metalloid glasses showing strong chemical ordering, more complete data from measured PPDFs are given in table 4. The coordination numbers, of metal atoms around a central metalloid, 8 to 9, and around a central metal atom, 10 to 12, are both well defined. Together with the rather small spread in interatomic distances, a limited variability of local atomic environments is suggested. Further progress on elucidating glass structure requires comparison with structural models. The modelling must use as much incidental information as can be gleaned about the sizes, bonding and coordination of the constituent atoms: observations of density, atomic Table 3 Observed and hypothetical coordination numbers in some metallic glasses.
Alloy, A-B
A neighbors of B (actual %)
A neighbors of B if there were no ordering (a)
100 100 88.8 56.2
80 81
70.1 70
64 57 65 57
Ch. 19, $4
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Metastable states of alloys
Table 4 Atomic coordination in metal-metalloid metallic glasses. Alloy
The mean interatomic spacing, i, the standard deviation, u,and the coordination number, from GASKELL [1985].
radii in crystalline forms, interaction potentials (Lennard-Jones, Morse, etc.), interbond angles where appropriate are all taken into account in constructing a model. The actual construction was initially done physically, with balls and springs, or just steel balls, but now is always done by computer simulation. Models for covalently bonded amorphous solids such as silicon involve distinct problems from metallic glass models (only the former incorporate directed atomic bonds) and we discuss only the latter here. There are many difficulties in comparing measured and computed RDFs. The experimental determinations are limited by the range of scattering vector lengths which is accessible, and the modelling is limited by the finite size of the model; the problems and best techniques are comprehensively discussed by FINNEY[19831. Probably the first structural model suggested for metallic glasses was that based on microcrystallites, i.e., the suggestion that the “glass” is simply a polycrystalline solid in which the grain size is very small. Early work by CARGILL[1970] showed clearly that the microcrystallite model does not provide good quantitative fits to measured RDFs of glasses such as NiT6PM. Recently the interest in nanocrystalline materials has led to a reexamination of this question. There are examples of apparently glassy alloys for which the structure can be successfully modelled assuming nm-sized grains of an icosahedral phase. A key observation, however, is that such materials do not transform in the way expected for true glasses; on annealing they transform ta a polycrystalline aggregate of larger grains by a process of continuous grain growth rather than by nucleation and growth (CHENand SPAEPEN [1991]). For true metallic glasses, which form the great majority of cases, structural modelling has mostly been based on either the dense random packing @RP) model of BERNAL[1964], or the local-coordinationmodel of GASKELL [1979]. Bernal’s empirical model originally referred to the DRP of hard spheres all of the same diameter and was based on physical measurements with steel balls kneaded in a rubber bag. The packing is statistically reproducible, and the structure can be considered to be made up of only five simple polyhedra (canonical holes), shown in fig. 27, the References: p . 1818.
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R u! Cahn and A. L. Greer
Ch. 19, $ 4
Fig. 27. Polyhedra formed by dense random packing of hard spheres, according to Bernal.
smaller holes (tetrahedra and octahedra) being in the great majority. The topology of DRP is related to that of the continuous random network first proposed for tetrahedrally bonded oxide glasses by ZACHARIASEN [1932] (CONNELL[1975], CHAUDHARIet al. [1976]). The average coordination number in the DRP of hard spheres is well defined and is 12.3 (SCOTTand MADER[1964]). Models based on the DRP of hard spheres give fair agreement between computed and measured RDFs. Two deficiencies in the model can be identified: the spheres in the modelling are infinitely hard, and only one type of atom is included. With realistic interatomic potentials, sequential computer relaxation of soft spheres can be used to increase the packing density of the model in better agreement with measured densities. In this way the fit between computed and measured RDFs becomes rather good in some cases, notably for vapor-deposited nominally pure metals (VONHEI~ENDAHL [1975]). The extension of this type of work to two-component glasses was undertaken by BOUDREAUX and GREGOR[1977]. For glasses of the metal-metalloid type they constructed a DRP with a mixture of metal and metalloid atoms and imposed a rule that there should be no nearest-neighbor metalloid pairs. While a model of this type gives a much better fit to experimental RDFs, there are still significant problems, particularly in matching PPDFs. The modelling of metal-metalloid glasses is now based on the local coordination model of GASKELL[1979]. The coordination of the metalloid typically by nine nearestneighbor metal atoms in glasses of this type (table 3) is the same as in many crystalline metal-metalloid compounds. In these compounds the basic coordination polyhedron is a trigonal prism (giving six nearest neighbors, with three more slightly further away). The prisms are linked in one of two ways (fig. 28). GASKELL[I9791 proposes that the glass structure is a non-periodic array of such prisms. Computer models assembled in this way and then relaxed under appropriate atomic potentials give rather good fits to measured distribution functions. The second nearest neighbor metal-metalloid distance in measured PPDFs can correspond distinctly to one or the other of the distances expected from the types of linkage in fig. 28. Thus there is evidence for medium-range order in the glasses in which the packing of the prismatic units resembles that in particular crystd structures. More evidence for this is provided by the structure in the B-B PPDF in fig, 26 out to quite large distances; this structure cannot be reproduced by a model based on randomly packed trigonal prisms. The prisms themselves appear to arise not from any directed bonding, but from a compromise between the tendency to the closest possible packing and the need for
Ch. 19, 44
Metastable states of alloys
1775
Fig. 28. The packing of trigonal prisms as found in (a) Fe,C and (b) F%P.Each prism has a metalloid atom at its center. The long-dashed l i e indicates the second-nearest metalloidmetal distance in each case. (After GASKELL [1983]).
space to insert metalloid atoms between the preponderant metal atoms (GASKELL [1985]). The local coordination model can be adapted for cases other than metal-metalloid glasses around the 20 at% metalloid composition. MEHRAet al. [ 19831 have applied it to ( R u ~ ~ Z T , ~ glasses ) ~ - ~ Bwith ~ 0.40 < x < 0.53 - Le., an unusually high metalloid content. They show that no DRP model can give even moderate agreement with the scattering data from such glasses, while a form of irregular prismatic packing based on the Ru,B crystal structure gives good, though not perfect agreement of RDFs. The local coordination model has been developed into the chemical twinning model by DUBOIS et al. [ 1985a1, who point out that trigonal prismatic coordination is common not only in metal-metalloid compounds but also in many metal-metal compounds in which the minor component has a radius 65 to 85% that of the major component. The model then can be applied to a wide range of glass types. In crystalline compounds trigonal prisms can be considered to result from chemical twinning; for example the cementite (Fe,C) structure is obtained in this way from the hcp lattice. The smaller atoms of the minor component are on the twinning planes, and the frequency of these planes is adjusted to match the composition. In modelling glassy structures, regular twinning is permitted only over a short correlation length of the order of 1 to 2 nm. The type of twinning is chosen to match the counterpart crystalline compound. As shown by the dotted curves in fig. 26, the fits to PPDFS obtained in this way can be quite good. The chemical twinning model has the distinction of being perhaps the only structural model for metallic glasses which has led to the discovery of new glass-forming compositions. According to the model, glass-forming ability should be greatest when there are competing types of twinning in the alloy. This idea has been used in developing glassforming compositions in aluminum-based alloys (DUBOIS et QZ. [1985bI). As exemplified in fig. 7, most glass-forming alloy systems are able to form glasses over a range of compositions. In many such systems, physical properties vary substantially with composition. Structural models of the kind discussed above give some reason to expect that there could be clear discontinuities in some properties and fig. 29 gives some measured examples. It has been emphasized that pair distribution functions are limited in the information References: p. 1818.
2.60 (r
2
I
I I 15 20 25 30 X (ATOMIC PERCENT METALLOID)
5 'V
b
600
400
1200
q
I
*
1
0
0.05
0.10
.
1000
y 0.1
"."
C
z4
zo1I0
'
12
'
ii
'
16
'
f8
20
'
22
24
'
2%
I
'28
X - at. % metalloid
Fig.. 29. Instances of discontinuities in the variation of properties with composition in metallic glasses: (a) electrical resistivity and nearest-neighbor distance in (MO,,~RU~,~)& glasses (after JOHNSON 119821); @). density, d, and Vickers microhardness,H,, in F%,8,,SjB,,, glasses (after AL BUAT etal. [1982]; (c) densities of F e B glasses (after Wyeral. [1977]).
Ch. 19, $ 4
Metastable states of alloys
1777
which they provide. For this reason there has been much interest in more local probes of atomic environments. One such technique is EXAFS in which the fine structure of an Xray absorption edge is analysed to obtain information on the local environment of absorbing atoms. Unfortunately a structural model is still required and the details of the model are important (so that the influence of atom positions and atom identities is hard to separate). Nonetheless, useful results have been obtained; generally they confirm the conclusion from PPDF studies that the nearest neighbor environment in a glass is well [1985]). defined and similar to that in the corresponding crystal (WONG[1981], GURMAN We turn now to techniques which can provide some information on the angular distribution of nearest neighbors. Nuclear magnetic resonance (NMR) and Mossbauer spectroscopy, through the interaction of the nuclear quadrupole moment of the probe atom with the local field gradient, can provide information on the symmetry of the local environment. The extraction of further information is unfortunately very model-dependent. N M R experiments in general show local symmetry similar to that found in the corresponding etal. E19831 have shown that crystalline phases. In Ni-B glasses, for example, PANNISOD the variation of the environments around "B nuclei is modelled well by a combination of the environments in crystalline Ni,B and Ni,B,. Each of these phases has trigonal prismatic coordination of the boron. Specifically the environment of the boron in the glasses is not similar to that in crystalline Ni2B (which does not have trigonal prisms). Miissbauer spectroscopy is a less discriminating test of structural models, but again there is evidence for crystal-like structural units (WHITTLEet al. [19851) and for local chemical et al. [1985]). order (SCHOLTE Field-ion microscopy (FIM) permits the imaging of individual atoms on the surface of a needle tip, and in combination with an atom probe the identity of the atoms can be established. By field evaporation, atomic layers can be stripped from the tip, and in this way a catalogue of atomic dimensions in three dimensions can be built up. Even though this has not been fully achieved for a metallic glass, there have been significant results. For example, PILLERand HAASEN [1982] have detected that a Fe4$Ji40B20glass is inhomogeneous on a scale of 2 to 4 nm. One phase is boron-poor, the other contains -25% €3, and Piller and Haasen interpret the structure as arising from a competition between DRP and defined local coordination. DE HOSSON[1983] has determined planar RDFs from F W images of the palladium atoms in Pd7,Cu5Si,,glass, and has shown that they are consistent with RDFs obtained from scattering experiments. Of greater importance is that he has determined the angular distribution of nearest-neighbor interatomic vectors (unobtainable from scattering experiments). This distribution has peaks at 60°, 90°, 120" and 150'. The strong peak at 60" is evidence for tetrahedral packing, but the other peaks suggest that other polyhedra may be present. High-resolution transmission electron microscopy (HREM) is not a useful tool for studying the structure of amorphous materials at the nearest-neighbor level; the resolution is barely sufficient and there is a severe problem with superposition even in thin foils. On the other hand, HREM can be a useful probe of medium-range structure. GASKELL and SMITH[1980] have shown that in a Pd-Si glass there are domains about 2 nm across in which atomic positions appear to be correlated. This is consistent with the correlations observed in PDFs and may lend support to the chemical twinning model discussed above. References: p. 1818.
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R. W Cahn and A. L.Greer
Ch. 19, 54
4.2. Relaxation
When a metallic glass is annealed in the vicinity of its glass transition temperature, many physical and mechanical properties change, in pursuit of metastable equilibrium.. (The equilibrium must be metastable because the equilibrium state in this temperature range is crystalline). All such changes entail atomic dz@sion, which has been extensively studied in metallic glasses. The phenomenology and mechanism of diffusion in metallic glasses is fully treated in ch. 7, 99, and will not be repeated here. However, that treatment omits reference to one striking characteristic of diffusion in metallic glasses, i.e., the diminution in,diffusivity when a metallic glass is annealed in effect, the process of diffusion tends to choke off that process itself. This phenomenon is analogous to the gradual drop in diffusivity when a crystallime metal has been quenched from a high temperature so that excess vacancies are retained, and these vacancies gradually anneal out (5 5.1, below). Figure 30 shows an instance of an anneal-induced fall in diffusivity, in this instance, of boron in a nickel-niobium glass (Urn et aZ. [1986]). Very precise measurements of x-ray absorption (CHASONand MIZOGUCM[1987]) have shown (fig. 31) that the fall in diffusivity in such a glass is correlated with an increase in density. i.e., with a reduction in free volume, which is the “carrier” of diffusion. The drop in diffusivity saturates after
1000/T in Fig. 30. Diffusivity of “B in Ni,Nb,
K-’
glass. 0 unrelaxed; * relaxed 420 s at 878 K. (After K u m et al. [1986]).
Ch. 29, 54
1779
Metastable states of alloys
I
10-22
I 0.4
0.2
-+-8 8 ton 8
1
0.6
0.8
-AV (%I v
! 0 (b)
0
1
I
I
I
120 240 Time in min
I
3 0
Fig. 31. Diffusion in a metallic glass as affected by a relaxation anneal. (a) Interdiffusivity, d, in a F e T i amorphous multilayer at 160°C, plotted against change in atomic volume, AV/V . The solid line is a fit to the free-volume theory of diffusion. (After CHASONand MIZOGUCHI [1987). (b) Relative change in the density of a P&Si,, glass as a function of annealing time at 260°C, determined from changes in x-ray absorption. (After C I y r s o et ~ al. [19851).
a time if the annealing temperature is not too low: this indicates that the free volume tends towards a metastable equilibrium level characteristic of the annealing temperature. This aspect of diffusion is further discussed by CAHN[1990]. The change in properties that accompanies an anneal of a metallic glass in the neighborhood of Tg is generically called relaxation. This is by no means specific to metallic glasses: for instance, the refractive index of an oxide glass, of the kind used for lenses, changes steadily as it is annealed close to its glass transition. Properties which have been pressed into service to examine the consequences of structural relaxation include, in particular, density or length changes, elastic modulus, Rgeerences: p . 1818.
Ch. 19, 54
R.U! Cahn and A. L Greer
1780
Table 5 Structural relaxation in amorphous alloys: properties affected and types of relaxation. Property affected by relaxation
Density Length changes Thermal expansion coefficient Enthalpy Young's modulus
Type of relaxation"
Selected references
(i) (ii) (iii) (iv)
(v)
I D D D I
GERLINGetal. [1988], TOULOUI etal. I19851 HUIZERand VAN DER BEUKEL[19871 KOMATSUet d.[1983] Y WOLDT[1988], I N O etal. ~ [1983] Y KOEBRUCCEand VANDERBEUKEL [1988], C m et d.[1983] Y B E S O N e r d [1982], DENG and ARGON [1986] Y VOLKERTand SPAEPEN[1989] Y WILLIAMSand EGAMI[1976] Y AKHTARand MISRA119861 and LEAKE[19791, %AMI [1978a] Y GR~ER KELTONand SPAEPEN[1984], BALANZATe t d . [1985] Y K E L ~ Nand SPAEPEN[1984], BALANZATet d. [1985] POSGAYetal. [1985] y. INOUS et at! [1983], MULDFXeta1 [1983] Y NISHIet al. [19861 Y Guo et d.[ 19861 GUOet at! [1986] Y CACIU~~O et al. [I9891 BAXTER[1986]
Hardnesdyield stress Viscosity Relaxation time for stress relief Dfisivity Curie temperahm Electrical resistivity
I I Y I Y D LD Y
Tempemture coefficient of resistivity Internal friction Strainatfracture Saturation magnetization Magnetic after-effect Induced magnetic anisotropy Structure factor (X-ray, neutron) Extended X-ray absorption fine structure Positron lifetime Superconducting transition temperature MSssbauer spectra Corrosion resistance Dcnsity of states at Fermi level Debye temperature Magnetostriction coefficient
D, I
1, D
D D D
D
Y Y D D Y Y I I, D I
r
DE V R et al. ~ [19881 JOHNSON[1981]
Y Y Y Y
Y
STR6M-oLSEN etat! [1988] MASLIMOTO et al. [1986] ZOUGMORf? e t d . [1988] ~OVOMORE? et al. [I9881 CUNATet al. [1988], TARN~CZI et al. [1988]
a The types of relaxation are: (i)monotonic relaxation; (ii) reversible relaxation near the glass transition temperature Ts;(iii) reversible relaxation well below TE;(iv) memory effects; and (v) effects of production conditions. I and D indicate an increase or decrease, respectively, of the proper value in monotonic relaxation; Y indicates occurrence of that type of relaxation.
viscosity, diffusivity (as we have seen), Curie temperature (for ferromagnetic glasses), resistivity, and a number of others as shown in table 5 . One can make a distinction between monotonic relaxation and wholly or partially reversible relaxation. Fig. 32 shows some examples of observed relaxation kinetics. In this figure, (a), showing change of density deduced from a reduction in length at 623 K, is a good instance of monotonic relaxation: if the annealing temperature is raised above 623 K, the length does not again increase. The elastic (Young's) modulus, however,
‘72 623
0
b
1000
time Is]
1551
I
I
1
1
2
3
L
IC
5
0
log t (SI
0
2
4
6
2o t Lain)
40
6G
-
8 1 0 1 2 1 4 tJhl
Fig. 32. Manifestations of relaxation of metallic glasses: (a) Irreversible length change of a F e J V i i , glass; @) short-term reversible changes in Young’s modulus of Co,Fe,Ni,,,B,,Si,, glass cycled between 623 and 723 K, (c) irreversible changes in Young’s modulus in the same glass at various temperatures, starting from the asquenched state; (d) changes in the Curie temperatm of F@, glass as a function of holding time at various temperatures above the Curie temperature ... substantially reversible; (e) reversible changes of coercive field of the abovementioned Co-rich glass, a composition of low magnetostriction, during cyclic annealing. References: p . 1818.
d
1782
R. W Cahn and A. L. Greer
Ch. 19, 94
shows both monotonic behavior (graph (c)) and reversible behavior once the monotonic process has gone to completion (graph (b)). The ferromagnetic Curie temperature shows this same double behavior, though in fig. 32 there is only space to show the monotonic form, These two main types of relaxation are further subdivided in the table, according in particular to the temperature range in question: the meaning of the columns is indicated at the bottom of the table. It is striking that (column (iii)) even though the kinetics of structural rearrangement well below Tgare so slow that monotonic relaxation cannot go to completion (i.e., to metastable equilibrium), yet reversible cycles of the type seen in fig. 32b are possible in this low temperature range. EGAMI[1978] first observed this with respect to the Curie temperature; he found cycles just like those in fig. 32b, cycling between 250" and 300°C. Egami proposed that there are two distinct types of atomic process during structural relaxation: topological short-range ordering (TSRO) involves a rearrangement of atomic sites and a reduction of free volume (increase of density); it is this process that is associated with normal relaxation near T8'The other process, compositional (or chemical) short-range ordering (CSRO), is a statistical rearrangement of chemically distinct species on fixed sites, and can occur at lower temperatures. This classification has been widely accepted as providing the best interpretation of the many observations of relaxation in metallic glasses. )&AMI [1986] published a critical examination of his hypothesis some years after he first put it forward. - It would provide useful confirmation of Egami's central hypothesis, that reversible property changes are associated with changes in CSRO, to compare the magnitude of such changes in those glasses which are known to have substantial amounts of CSRO in equilibrium, like Cu-Ti for instance, and others, such as Fe-Ti, which have been found to have little or no equilibrium CSRO. No such comparisons have yet been performed. However, there is mounting evidence, when the magnitude of reversible property relaxation is studied as a function of composition for a series of glasses in a ternary system, that the magnitude is greatest for compositions which permit the maximum exchange of species, and this supports the identification of reversible relaxation with changes in CSRO (GREER[1993b], STR~M-OLSEN et aZ. [1988]). Direct structural studies by XRD have provided only limited evidence. Changes in TSRO are so small that any changes in x-ray scattering measurements, like those shown in fig. 25, are so slight that they often vanish in the experimental scatter. However, EGAMI[1978b] did succeed in demonstrating significant changes in the x-ray diffraction pattern of a metallic glass, and showed that the densification on annealing involves a change in coordination number and not a reduction in the mean nearest-neighbor distance. The same features have been found for relaxation of amorphous silicon, where changes in the x-ray pattern are more substantial (ROORDA[19901). -Reversible CSRO changes have not been detected by XRD,but rather by Mossbauer measurements, which are highly sensitive to nearest-neighbor geometry ( ( S T R ~ M ~ L S etEal.N [ 19881). X-ray diffraction has, however, been successfully exploited in relation to another form of annealing - stress-induced directional ordering. (HILZINGER[1982)). This is closely related to field-induced directional ordering, in which a magnetic field is applied while the metallic (ferromagnetic) glass is heated the glass so treated possesses an induced uniaxial magnetic anisotropy, and it has been well established since the 1950s
Ch. 19, $ 4
1783
Metastable states of alloys
(GRAHAM[1959]), for crystalline solid solutions which behave similarly, that this effect is caused by directional order, that is, a very slightly different proportion of chemically unlike nearest neighbor pairs measured parallel and perpendicular to the originally applied field. In fact, the rate at which the anisotropy is established provides a rough [1980]). It appears that the stressmeasure of diffusion rates in glasses (LUBORSKY induced effect is more pronounced: SUZUKIet a!. [19871, in a striking experimental tour de force, were able to show directly, by energy-dispersive x-ray diffraction from a stressannealed Fe&&lo,Si,,B, glass, done successively with two orthogonal diffraction vectors, that directional short-range order (called "bond-orientational anisotropy" by them) was induced by the stress-annealing and decayed gradually on subsequent stressfree annealing. These investigators presumed that what they had observed was really a change in TSRO, with changes in the absolute numbers of bonds in the two directions; A EGAMI[1991]) recognized however, a later study from the same laboratory ( T o ~ and that chemical as well as topological anisotropy must be involved in the phenomenon. These exact studies show how difficult it is to secure definitive information about TSRO and CSRO. There is no space here to discuss in any detail the effects of irradiation on TSRO and CSRO: The effects of neutron irradiation on density and associated changes in ductility (see first entry in table 5 ) are linked with changes in TSRO, while the studies of Hillairet etal. [1985]), using electrical resistivity as a marker, have and his group (e.g., HILLALRET generated good though indirect evidence of substantial changes in CSRO within atomic displacement cascades in glasses such as Ti50Be&r,o. The kinetics of relaxation became a disputed problem when the crossover efSect was discovered, in respect of the Curie temperature and elastic modulus (GREERand S P - ~ P E N [1981]; SCOTT and KURSUMOVIC[1982], respectively). Another name for what was discovered is memory efect. What this involves can conveniently be illustrated by an older study of changes in refractive index of an oxide glass (MACEDOand NAFOLITANO [1967]). Figure 33 and its caption explain how such an experiment is done. If there were only a single relaxation process, the plot would simply show a horizontal line when the
6
z (1 W 2 t51450 -
-z .c
0
x 1.51110 0
-c
-0
Equilibrium line
L F v,
1.51130 0"
100 111
II
I I
II
200 300 400 Time in minutes
Fig. 33. Refractive index vs time in a '%rossover"experiment on a borosilicate glass. The glass was held at temperature TIuntil the index reached a value corresponding to equilibrium at a second, higher, temperature T,. The temperature was then increased abruptly to T,:instead of remaining constant, the index followed the curve shown, retuming eventually to the T, equilibrium value. (After MACEDOand NAPOLITANO i19671).
References: p . 1818.
1784
R. U? Cahn and A. L. Gwer
a.19, 54
temperature was raised from TI to T,. Experiments like this show that there is not a oneto-one relationship between a property and the internal state of the glass, in the sense that the same value of the property can be associated with distinct structural states generated by different heat-treatment programs. There must be at least two distinct processes, but el al. [1983], in fact with respect to metallic glasses, detailed analysis has shown (GIBBS LEAKEet al. [19881) that an entire spectrum of processes with distinct activation energies must be involved. This kind of experiment, again, shows the near-impossibility of associating different aspects of relaxation with precisely defined structural changes. Nevertheless, a number of investigators have attempted to interpret relaxation kinetics in terms of the reduction of free volume, and this kind of theory (e.g., VAN DEN BEUKEL and SIETSMA [1990]) is linked with theories of the glass transition, which are also concerned with ideas of critical free volume. Some remarks about how structural relaxation influences the creep behavior of metallic glasses will be found in $ 4.4, below. 4.3. Crystallization
As emphasized in $2.1, a metallic glass can be considered to be an undercooled liquid, and as such it is susceptible to crystallization. While there may be alloy compositions for which a metallic glass is more stable than any single crystalline phase, the glass always has a higher free energy than a combination of crystalline phases. Crystallization of metallic glasses has been widely studied, and a number of reviews are available (e.g., KOSTERand HEROLD[1981], KOSTER [1984], KOSTERand Sc[1993]). There are several reasons for interest. It is important to characterize the stability of metallic glasses in order that desirable properties may be preserved under service conditions. Crystallization of a glass (also known as devitrijication) is very much slower than conventional solidification of a liquid, and it can readily be interrupted by lowering the temperature. This facilitates fundamental studies of the processes of nucleation and growth, in particular of nucleation kinetics difficult to study quantitatively in the liquid state. A glass is formed only if crystal nucleation and growth are substantially avoided during liquid quenching; thus crystallization studies are of relevance for understanding glass formation. Finally, it is increasingly recognised, and is now a matter of commercial practice, that useful partially or fully crystalline microstructures can be obtained by crystallizing metallic glasses: this processing route is analogous to that used for glassceramics. We compare first the basic kinetics of crystallization from the glassy state and solidification from the liquid state. Figure 34 is a schematic illustration, for a glassforming alloy, of the variation of the homogeneous crystal nucleation frequency and the crystal growth rate over the temperature range from the liquidus down to below the glass-transition temperature Tg.As the undercooling is increased, the thermodynamic driving force for transformation to the crystalline state increases and the atomic mobility decreases. The forms of the temperature dependences of homogeneous nucleation and growth are readily derived from standard expressions; both show maxima at intermediate temperatures because of the lack of driving force at high temperature and the lack of
Ch. 19, 54
Metastable states of alloys
1785
r
1 H
m 0
4
high rl
Temperature
-
r,
Fig. 34. Schematic representation of the homogeneous nucleation frequency and the growth rate of crystals in an undercooled melt or glass. The melting temperature, Tm,and the glass transition temperature, T,,are shown, together with (shaded) the regions for conventional solidification and for nystallization of amorphous alloys, The temperature ranges in which homogeneous or heterogeneous nucleation dominate are marked, indicating that either can dominate for crystallization of amorphous alloys.
mobility at low temperature. The maximum rate occurs at a lower temperature for nucleation than for growth, because of the need for the driving force to overcome the solid/liquid interfacial energy. As seen in fig. 34, conventional solidification occurs at rather low undercooling with moderate growth rates on heterogeneous nuclei. Crystallization of a glass occurs near T, at very low growth rates on nuclei which could be heterogeneous (predominant at lower temperature) or homogeneous. In conventional solidification the rate of latent heat production is matched by the rate of external heat extraction, but there can be significant temperature gradients. As liquids are undercooled further, nucleation may occur at temperatures for which the atomic mobility is still large; crystal growth is then rapid and the release of latent heat is more rapid than the heat extraction, leading not only to temperature gradients but to a rapid overall temperature References: p . 1818.
1786
Ck 19, 84
R. W Cahn and A. L . Gwer
rise (recalescence). For crystallization of the glass, however, the crystal growth is so slow that the transformation can be isothermal, with negligible temperature gradients. The solidification processes considered in chapter 8 mostly occur close to the liquidus, at temperatures above the range in which recalescence is significant, The crystallization processes of concern in this chapter mostly occur near the glass transition temperature at temperatures below the recalescence range. While solidification and crystallization can be viewed within one framework as indicated in fig. 34, significant differences can be expected. Taking dendritic growth as an example, the rate-limiting processes in solidification are heat and solute redistribution at the growing tip, whereas in crystallization the rate is limited by the mobility of the crystaVglass interface itself and by solute redistribution. Nevertheless, the common aspects of the processes in the two regimes have for example enabled heterogeneous nucleation in a metallic glass to be used to study particle inoculation (grain refinement)processes in conventional solidificationprocessing. In such and GREER[1994] have embedded inoculant TiB, particles in an a study, SCHUMACHER amorphous aluminum alloy. Crystallization of the alloy to a-A1occurs on the particle surfaces, and novel microscopical studies of the nucleation mechanism are possible because the crystallization proceeds so much more slowly than conventional solidification. Figure 35 shows a typical differential scanning calorimeter (DSC) trace obtained on heating a metallic glass. In the DSC the power necessary to heat the sample through any Cl
i 270
310
350
390
430
470
t 0
Temperature (“c) Fig. 35. Differential scanning calorimeter record during the heating, at 20K/min, of Pd,&u,Si,, glass. 6 mg sample. The ordinate represents power released (exotherm, positive displacements) or absorbed (endothems, negative displacements). TBis clearly differentiated from the first crystallization peak, Txl.
Ch. 19, 54
Metastable states of alloys
1787
small temperature interval is automatically compared with the power necessary to heat a reference sample (undergoing no transformations) through the same temperature interval. As plotted in fig. 35, an upward deflection corresponds to heat release from the sample. In this example of a comparatively easy glass-former, the glass transition is evident and shows the increase in specific heat when the glass changes to liquid. Before the glass transition is reached, temperature cycling experiments can reveal irreversible heat releases corresponding to structural relaxation in the glassy state (0 4.2). The glass transition is followed by crystallization. As shown in this case, there are often separate peaks in the DSC trace revealing distinct stages in the crystallization of the sample, the integrated area under each peak giving the associated enthalpy change. The different stages may correspond to successive crystallization of different parts of the sample (perhaps arising from composition changes induced by the crystallization processes themselves), or to transformations from metastable crystalline states towards the final equilibrium state. As the thermodynamic driving force for crystallization of the glass is very high and the atomic mobility is low, it is likely that there will be metastable phases (or combinations of phases, for example a metastable eutectic) before equilibrium is reached. In thermodynamic terms, three types of initial crystallization can be distinguished. In polymorphic crystallization the glass transforms to a single crystalline phase of the same composition. The crystalline phase is not normally a solid solution of the base element because glass formation would be difficult if such a crystallization were possible and because any crystallization which did occur would likely involve solute partitioning. The phase formed in polymorphic crystallization is normally an intermetallic compound. In primary crystallization a single crystalline phase is formed which has a composition different from that of the glass. Solute partitioning is involved and the crystal growth is often dendritic. Complete transformation to the initial phase may not be possible and the remaining glassy matrix of changed composition may undergo a distinct crystallization. The third type of' crystallization is eutectic, in which two crystalline phases grow cooperatively. There is no overall composition difference between the glass and a eutectic colony. Within the colony straightforward lamellar structures are only rarely observed, and the microstructure is on a fine scale which does not vary strongly with crystallization conditions. Microstructures arising from the three types of crystallization are shown in fig. 36. Ternary eutectic crystallization is also known, in which three phases form cooperatively, The growth rate of the eutectic colonies is particularly slow in such cases, which are usually associated with very easy glass formation (GARCIA-ESCORIAL and GREER[1987]). Another form of decomposition of a glassy phases is separation into two glassy phases (TANNER and WY[19801). These phases can then crystallize separately. The driving forces for the various forms of decomposition are illustrated schematically in fig. 37. Experimental studies of crystallization can be based on isothermal anneals or on continuous heating of the kind used for fig. 35. In either case, DSC (which in effect measures enthalpy changes in the sample) or measurement of other property changes (notably electrical resistivity) can be used to derive the volume fraction x transformed as a function of time f. Isothermal x vs t measurements are often analysed according to References:p . 1818.
1788
Ch. 19, 4 4
R. W Cahn andA. L Greer
Johnson-Mehl-Avrami kinetics, x(t) = 1 - exp[-(Kt)”], where K is a temperature-dependent rate constant and n is an exponent which can relate to the modes of nucleation and growth (see, e.g., ch. 28, $3.2). It has been frequently pointed out, however, that such analyses are subject to errors of interpretation and must be supplemented by microscopical studies. In particular, if the transformation in a sample differs from place to place, quite erroneous conclusions may be reached, as has been elucidated by HOLZERand KELTON [1991]. X-ray diffraction and related techniques are useful for identifying the crystalline products, but microstructural techniques are essential for understanding transformation kinetics. The scale of crystalline volumes developing on annealing a metallic glass can vary enormously, from a few nanometers up to as much as a millimeter in a very good glass-forming alloy. Depending on the scale, transmission electron microscopy (TEiM), scanning electron microscopy of polished and etched surfaces or optical metallography may be appropriate. Recently, intense radiation sources, such as Xrays from a synchrotron, have permitted structural studies with very fine time resolution. Time-resolved studies during crystallization can reveal transient phases and other important aspects of the process (KOSTERet al. [1991]). An example of a kinetic analysis is given in fig. 38 from the work of GREER[1982] on F e d , glass which crystallizes eutectically to a mixture of a-Fe and a metastable boride, / -
A-
’r
i
Ch. 19, $ 4
Metastable states of alloys
1789
(C) Fig. 36. Transmission electron micrographs showing: (a) Polymorphic crystallization: Fe,,B, glass annealed at 400°C for 10 min (reproduced with the kind permission of U. KBster). (b) Primary crystallization: FG,,5B,.,,Si, glass annealed at 445°C for 20 min, showing a-iron dendrites. (c) Eutectic colony in Feai&,, glass, annealed 1 h at 390°C.showing y-(Fe, Ni) + (Fe, Ni),B. Marker= 10pm.
I Composition,
Fig. 37. Schematic free-energy curves for a binary alloy system, showing how the glass or amorphous phase can lower its free energy by undergoing: (a) separation into two amorphous phases; (b) polymorphic crystallization into the supersaturated solid solution a; (c) polymorphic crystallization to the metastable compound fi; (d) polymorphic crystallization to the equilibrium phases y ; (e) primary crystallization of a;(0 eutectic crystallization to a and metastable fi; and (9) equilibrium eutectic crystallization to a and y. References: p . 1818.
Ch. 19, $ 4
R. W Cahn and A. L Greer
1790
Q6
-.-
c
'c
E
0.4
c
experimental
\ V
x
calculotcd
V
-
--- -
0.2
0.0 660
680
700 TEMPERATURE [K)
720
740
Fig. 38. Crystallization kinetics, dx/dt, deduced from continuous DSC heating data, for Fe&, glass 32 mm thick (solid curves), compared with fitted curves derived from a model. The lower peak corresponds to a preannealed sample. (After GREER[1982]).
tetragonal F%B. The traces in fig. 38 are derived from DSC data and show the rate of change of transformed fraction during continuous heating. The data (main peaks) can be well matched by a numerical model which assumes eutectic growth from a fixed number of quenchedin nuclei. The population of these nuclei can be determined from the numerical fitting and was found to be of the order of 10"/m3, in agreement with direct "EM observations. It was suggested that these nuclei most likely arose by homogeneous nucleation in the melt during quenching. The higher-temperature small peaks in fig. 38 arise from a thin layer on one side of the melt-spun ribbons which contains no quenched-in nuclei. Nucleation kinetics in metallic glasses can be determined by direct counting of the numbers of crystals, by inference from overall transformation kinetics, or by measuring the crystal size distribution in a partially transformed sample. Examples of the latter are shown in fig. 39. The types of nucleation behavior which have been observed are illustrated schematically in fig. 40.Nucleation can occur during an anneal and can be homogeneous or heterogeneous, showing site saturation (see ch. 15, Q 2.1). In either case transient effects may be evident. Alternatively, in some cases annealing merely leads to crystal growth on pre-existing nuclei. Homogeneous nucleation is perhaps most clearly shown in Fe&iJ',,B, glass, in which nucleation kinetics of type (b) (fig. 40)are found. Even for this well studied glass, however, a fully quantitative fitting of the kinetics to nucleation theory has remained elusive. The steady-state homogeneous nucleation frequency Is (m-3 s-') should be of the form (KELTON [1991]):
Ch. 19, $ 4
1791
Metastable states of alloys
0 observed
.~__._. : calculated
05
10
1.5
20
smoli 0x1s of prolate crystals
[
,urn 1
Fig. 39. Typical histogram of crystal diameter distribution in a partially crystallized metallic glass: polymor[1993]). phous crystallization in Fe,,Ni,&. (After KOSTER and SCI-I~NEMANN
.-W U
3
z Lc
0 L
W
n
E
3
z
Time Fig. 40. Schematic variation in the number of nuclei with time in an isothermal anneal for the following nucleation types: (a) steady-state homogeneous; (b) transient homogeneous; (c) steady-state heterogeneous; (d) [1988]). transient heterogeneous; (e) quenched-in active nuclei. (After GREER References: p. 1818.
Ch. 19, $ 4
R W? Cahn and A. L Greer
1792
where AGv is the free energy driving force (J/m3) for crystallization, cr is the crystal/ glass (or liquid) interfacial energy (J/m2), q is the glass or liquid viscosity (Pa s), kB is Boltzmann's constant, and A is a prefactor. Of the three main materials parameters in this expression, q can be measured, AG, can be estimated by various methods, but CT is not known. As reviewed by KELTON[1991], all reasonable assumptions about the form of the temperature dependence of q, AG,, and cr, when substituted in the above equation, fail to fit the full set of measured steady-state nucleation frequencies, shown in fig. 41. If (+ is independent of temperature, then a plot of Fq vs l/TAG, should be straight, but as can be seen in the figure this is so only over the central portion of the data. The value of A derived from fitting the central portion is reasonable for homogeneous nucleation. It is possible to fit the entire data set by assuming a temperature-dependent CT,but then the fitted value of A is much reduced and characteristic of heterogeneous nucleation (GRANLSYet al. [19941). Thus, even for this well studied glass the steady-state nucleation kinetics cannot be interpreted with certainty. While the crystallization of metallic glasses offers great potential for a quantitative test of nucleation theory in metallic systems, the potential remains largely unrealised, particularly because of problems with determining the materials Parameters. Some aspects are already clear, however. The non-linear variation of free energy of crystallization with undercooling (arising from the excess specific heat of the liquid, as discussed in 0 2.1) must be taken into account to explain quite basic experimental findings, for example that the activation energy for nucleation [1994]). is typically greater than that for crystal growth (GREER In addition to FeaiJ",,B, there are other metallic glasses for which the nucleation kinetics suggest homogeneous nucleation. For some glasses (for example the aluminum-
29
c
26
E ol 0
23
20
6
7 ioai
8
9
T( A G ~
Fig. 41. Crystal nucleation in amorphous Fe@iaPI4B6; the graph points are from two experimental studies. The plot of logpv, vs l/TAGvz (where p is the homogeneous nucleation frequency and 7 ) is the viscosity) should be linear for a constant (i.e., temperature-independent) solidfliquid interfacial energy.
Ch. 19, $ 4
1793
Metastable states of al6oys
rare-earth-transition metal glasses discussed later in this section) the population density of nuclei, up to lOZ/m3, is so high that homogeneous nucleation seems the only as shown in fig. 38) plausible hypothesis. In yet other cases (for example Fe,$, crystallization is by growth on a large population of quenched-in nuclei. The large population of such nuclei and its strong dependence on quench rate suggest transient homogeneous nucleation during the rapid quenching of the alloy. Analysis of the transient effects suggests that in some cases, inhibition of nucleation is essential if glass and GREER[1986]). This is illustrated for the case formation is to be possible (KELTON of Au,,Si,, in fig. 42. The dashed line in the lower figure indicates the low level of volume fraction transformed (lod) which is usually taken as the condition for successful formation of a (metallic) glass; it can be seen that this level is achieved with transient nucleation for a quench rate of le U s , whereas for steady state nucleation the unrealistic rate of 10, JUSwould be needed. Thus, crystallization studies and their analysis reveal that for the formation of some metallic glasses from the liquid-high quench rates are absolutely necessary. In marked contrast, there are other metallic glasses for which crystallization is clearly dominated by heterogeneous nucleation, the homogeneous nucleation frequency being and GREER very small. Examples are the good glass-formers Pd4Ni&Pm (DREHMAN
- 10m
. i 0
'-
.-
53 1oQ -
Au,,Si,
-7Steady= state nucleation
-
; 10' L
z'
1
-
=I-
g o lo-a
Transient nucleation
(Y
a5 10-m
>
10'
io5
106 10' Quench rate in K/s
10'
Fig. 42. (a) The number of nuclei and (b) the transformed crystal fraction calculated for quenching molten Au81Si19 at various rates, assuming either steady-state or transient homogeneous nucleation. (After =TON md GREER[1986]).
References: p . 1818.
1794
Ch. 19, $ 4
R. B Cahn and A. L Greer
[19841) and Pd,7~,cu6si,6,5 (KIMINAM and SAHM [19861. That the predominant IIUCleatiOn is heterogeneous can be exploited, as processing under very clean conditions can permit bulk glass formation at very low cooling rates (as low as 1 K/s for Pd,Ni,P, (DREHMAN and GREER[19841). Surfaces are often preferred sites for crystallization of a metallic glass. The main reason is modification of surface composition, for example by oxidation. There are clear possibilities to manipulate surface composition to enhance or impede nucleation, or to select crystallizing phases different from those which would be formed in the bulk. [1984]. Surface crystallization can Comprehensive work in this area has been by K~STER be used to improve bulk properties of a metallic glass (particularly magnetic properties) by inducing a compressive in-plane stress in a thin ribbon (OK and MORRISH[198l]). WEI and CANTOR[1989] made a detailed study of surface crystallization of Feai,B, and found that it is enhanced if the alloy is first phase-separated by relaxation and then abraded at the surface, or else the surface is enriched in nickel by electroplating and the glass is then annealed. Sometimes, surface oxidation can have the opposite effect of inhibiting crystallization at the surface: thus, with Pd40Ni,Pzo, a thin NiO layer protects the glass from local loss of phosphorus; such loss is here the prime cause of preferential surface nucleation. Removal of the original surface can drastically modify crystallization behavior: thus, NhB,, glass (which normally crystallizes from the surface with a strong accompanying [loo] fiber texture), loses the quenched-in nuclei responsible for this if the and SC-MANN [1993]); their chapter can be consulted surface is etched off (K~STER for more detailed information concerning crystallization at surfaces. Finally in this section we consider crystallization of metallic glasses as a route to desirable microstructures. This concept is well known for glass-ceramics. For metallic glasses it has been comparatively neglected, but is receiving increasing attention. It may seem strange to adopt an extreme processing route (such as melt-spinning) to avoid crystallization and form a glass, only then to crystallize it. The important point is that the crystalline product made from the glass can have a very fine and uniform microstructure, h e because the high effective undercooling favors nucleation, and uniform because of the uniformity of the glass itself and the lack of macrosegregation during crystallization. Devitrification offers a particularly attractive way of making nanocrystalline material (with grain size less than 100 nm) cheaply, controllably, and in comparative bulk without the need for consolidation. The first serious work on the exploitation of devitrification was by RAY [1981] on alloys of Fe, Ni, Al, Cr, Mo, Co and W in multiple combinations, with 5 to 12 at.% of B or other metalloids as aids to glass formation. Similar alloy developments have been etal. [1985]. Ni-Mo-B and Ni-Al-Ti-X-B made by DAS etal. [1985] and VINEBERG and later other Ni-Mo-B alloys with added Cr were made by melt-quenching, comminution and consolidation by extrusion or HPping. During the processing, ordered phases including Ni4Mo, Ni,Mo, Ni,Mo and Ni,(Al,Ti) are precipitated from the crystallized matrix, together with stable boride precipitates. This family of alloys is now manufactured commercially under the trade name Devitrium@(VINEBERG et al. [1985]), and the best examples have impressive high-temperature properties, exceeding high-grade tool steels. Work in this area is continuing; for example, ARNBERGet al. [1991] have
-
Ch. 19,54
Metastable states of alloys
1795
developed a range of tool steels by crystallizing Fe-Cr-MA-B or Fe-Cr-Md-V glasses. The most important industrial application of metallic glass crystallization has been in the production of the modern family of permanent magnets based on the high-coercivity phase Nd2Fe,,B. These materials can be made either by sintering an alloy powder, by melt-spinning the alloy to form a glass and then devitrifying, or by direct quenching to a nanocrystalline structure. The melt-quenching method was introduced by CROATet al. [1984], and has been fully commercialized. The process has been reviewed by B u s c ~ o w [1986]. Figure 43 provides an overview of the production possibilities by direct quenching or by heating a glass. The hard magnetic phase is desired, the alternative soft magnetic phase must be completely avoided, and a fine grain size is desirable for high coercivity. In the materials of most interest, grain sizes range from 14 nm to 50 nm. Interest in nanocrystalline magnetic materials now extends to the opposite extreme: very soft magnets. Glassy alloys based on Fe-BSi are well known as soft magnetic materials. If the composition is modified by addition of copper (apparently to enhance crystal nucleation) and niobium (to impede crystal growth), crystallization of such alloys leads to an isotropic microstructure with a very fine grain size of 10 to 20 nm (YOSHIZAWA et al. [1988]). At this scale the local magnetocrystalline anisotropies are averaged out, and soft magnetic properties result. The materials, now marketed under the trade name Finemet@,are of interest in comparison to glassy soft magnetic alloys because they can have very low magnetostriction; the magnetostrictions of a-Fe crystallites and of the residual amorphous phase in the microstructure are of opposite sign and substantially cancel (HERzER [1991]). Nanocrystalline materials made by devitrification of metallic glasses may also be of interest for their mechanical properties. Aluminum-based alloys with rare earth and late transition metal additions up to a total of 10 to 15 at. % can either by direct quenching
---
continuous cooling quenching and annealing
t magnetic phase
Time
-
Fig. 43. Schematic time-temperatumtransfomtion diagram to rationalize optimum conditions for producing M-Fe-B permanent magnets. (After WARLIMONT [1985]). References: p. 1818.
Ch. 19, 54
R. W! Cahn and A. L Greer
1796
or by devitrification yield microstructures - nanometer scale - with exceptional strength, discussed further in P 4.4.1. Consolidation of glassy or partially devitrified material may lead to a fully crystalline product, but one in which a fine-scale microstructure with desirable properties is retained. A particular property of interest is superplasticity. An alloy of the aluminum - rare earth - transition metal type has shown superplastic elongations of more than 500% at strain rates as high as l/s (Higashi etal. [19921). Even after consolidation and deformation, the mechanical properties (for example, a tensile strength -900 MPa) are superior to those of conventional highstrength aluminum alloys. 4.4. Plastic deformation and fracture
The principal plastic characteristics of metallic glasses are, first, their great strength (the flow stress is typically as much as 1/50 of the shear modulus); second, their fracture toughness, which in the normal state is very much greater than that of comparably strong fibrous materials such as graphite; and third, the mechanisms of flow and fracture which are entirely distinct from those found in crystalline materials. Table 6 compares the flow stresses of a number of ductile metallic glasses, both on an absolute basis and normalized for density, with those of some other ultra-high-strength wires or fibres. Metallic glasses can be seen to be among the strongest practical materials known (the adjective is to distinguish them from single-crystal whiskers). Metallic glasses deform plastically by two entirely different regimes. Under tension, at high stresses near their fracture limit,they deform extremely inhomogeneously: shear Table 6 Tensile properties of various continuous filaments and ribbons. Material
S-glass (Si0,-A1,03-MgO) C fibre (high-yield PAN tYPeY B filament (on W core)” SIC microcrystalline filamenta Kevlar fibre (organic polymer)b High-C steel wire Fe&,, met. glassb Ti&e&rlo met. glass T i a G S i , met. glass” Cu&rm met. glassb a
Yield stress cry/GPa
Relative density d
uyd’/GPa
5.0
2.5
2.0
3.2 2.5-4.5
2.6 2.6
3.5 2.8
4.2 3.6 2.3 2.5 1.8
Indicates heat-resistant materials. Indicates materials with some ductility.
Young’s modulus E/GPa
E d’/GPa
85
34
1.7 1.0-1.7
490 380
153 146
2.6
1.4
200
77
1.5 7.9 7.4
1.9 0.55 0.5
135 210 170
90 27 23
4.1 3.9
0.55 0.65 0.25
105 110 85
26 28 12
7.3
Ch. 19,94
Metastable states of alloys
1797
bands form on planes close to the orientation of maximum resolved shear stress. These are very thin (typically 10-20 nm thick according to DONOVAN and STOBBS[1981] and high-speed cinematography and acoustic emission indicate that the bands form extremely rapidly. NEUHAUSER[1978] found a formation time for individual bands in Pd,,Si, < 2 ms.) At low stresses, there are no shear bands and deformation is by homogeneous creep. A schematic deformation map separating the two deformation regimes is shown in fig. 44.The ductility in inhomogeneous deformation is usually only about 1% in tension, whereas many glasses can be rolled to reductions exceeding 50% if sufficiently hard and stiff miniature rolls are used. Creep rates in most creep experiments have been so low that total creep strains in reasonable times are less than 1%, but it is possible at temperatures close to T,to hot-press a metallic glass foil such as Fe$iaPI4B6 to strains
A
Fig. 44. Deformation map for an idealized metallic glass. T8 is the glass transition, T, the crystallization temperature, T, the liquidus temperature, T is the shear stress, 1.1 the shear modulus, p is shear rate. (After SPAEPEN [197'7l).
References: p . I818.
1798
R. W? Cahn and A. L Greer
Ch. 19, 84
of the order of unity (PAT~ERSON et al. 119781). This approach has been used in Japan (MASUMOTO [1994]) to shape minute but complex objects such as a micro-gearwheel from a metallic glass. The mode of ductile fracture of a metallic glass is unique. Fracture following inhomogeneous flow always takes place by separation along a previously active shear band - i.e., it is strictly a shear failure - and the fracture surface shows a “river pattern” of fine ridges which can be exactly mimicked by simultaneously shearing and pulling apart a pair of glass slides coated with vaseline. Accordingly, it has been proposed that the material in a shear band which is about to fail behaves in a liquid-like manner. Shear along one of the shear bands clearly generates additional free volume and this lowers the viscosity locally and can eventually lead to separation along the shear band. The creation of free volume can be deduced from the measured decrease (= 0.15%) in density after extensive inhomogeneous shear (CAHNet al. [1984]). The complex [1983]. fracture mechanics of metallic glasses are reviewed by KIMURAand MASUMOTO There is no record of the appearance of a fracture in creep; apparently glasses subject to slow homogeneous flow undergo perfect viscous flow with no structural deterioration that could lead to fracture. When a metallic glass is stressed at a high temperature, as remarked, it undergoes homogenous flow by a purely viscous process. As creep proceeds, free volume is progressively annealed out and since the viscosity is directly linked to free volume, the creep rate under constant stress progressively diminishes. The relation between viscosity, q, stress, u,and strain rate, L., is: q = (113) a/&.Instead of a constant-stress creep test, one can also measure stress relaxation at constant strain or perform a bend-stress relaxation experiment (a ribbon is elastically bent and held at a fixed radius while it is annealed, and then released to see how far it springs back). A brilliant series of experiments to study these processes, mostly with PhSi,, glass, was performed by Spaepen and Taub and reviewed by them (SPAEPENand TAUB[1983]). According to the Stokes-Einstein relation, q = kT/Db, where D is a self-diffusivity and b is a characteristic distance of the same order as the interatomic distance (NACHTRIEB [1976]). The rise in viscosity on relaxation due to reduced free volume should thus be associated with a fall in diffusivity. However, there have been increasing doubts as to the applicability of the Stokes-Einstein equation to metallic glasses, and a number of experiments claimed to show that the equation breaks down have recently [1994]. Van den Beukel has been critically discussed by VANDENBEUKEL and SIETSMA recently proposed that, for metallic glasses, qD2 is constant and has advanced reasons for preferring this relationship (DUINEet al. [1992, 19931). The free volume model of flow and viscosity, originated by Cohen and Tumbull, can be outlined as follows: Each atom is confined in a cage defined by its nearest neighbors; that cage can vary in volume, and the free volume is that part of a cage in which the atom can move without an energy change. Normally the atom can only oscillate, but if there is a density fluctuation there is a computable probability (dependent of course on temperature) that the atom will jump to a neighboring cage if the fluctuation generates an instantaneous local free volume exceeding a critical amount. Such jumps define both viscosity and creep rate under stress. Relaxation diminishes the average free volume and
Ch. 19, $4
Metastable states of alloys
1799
IO”,
I
Fig. 45. Rate of change of viscosity of Fe40NLP,4B6 glass with time as a function of temperature, as measured by three different experimental techniques. (After SPAEPENand TAUB[1983]).
thus increases the viscosity, linearly with time as it turns out. Figure 45 shows how the (linear) rate of increase of viscosity varies with temperature for a metal-metalloid glass. Attempts to interpret such relaxation-induced changes quantitatively are still beset by some difficulties, notably the need to assume that the concentration of flow defects (unlike free volume collectively) to some extent changes reversibly with temperature. A ”flow defect” here is a particular configuration of free volume of a kind which favors atomic jumps, as sketched above. (The notion that the reversible element in creep behavior is linked, as suggested for other aspects of relaxation, with changes in CSRO, does not seem to have been seriously analysed.) For details of the theoretical considerations involved, the reader is referred to Spaepen and Taub’s review, and to other detailed reviews by CHATJDHARI et al. [1983] and SPAEPEN et al. [1986]. There is a surprising measure of reversibility in the viscosity (a value characteristic of a given temperature TI can be attained directly by annealing a quenched sample at that temperature or by annealing at TI a sample previously equilibrated at a lower temperature (VOLKERT and SPAEPEN[1989])). It is undoubtedly difficult to define in an acceptable operational manner what is meant by a defect in an amorphous structure. Very recently, SPAEPEN[1994] has proposed regularities governing the evolution of a population of defects (density fluctuations) in a metallic glass of different sizes (singlets ... i.e., effectively a single missing atom .. doublets, triplets and quadruplets ), with a range of shifts from one to another among these categories. The idea is that singlets govern
.
References: p . 1818,
1800
R. W Cahn and A. L Greer
a.19,§4
diffusion, doublets govern plastic flow, triplets can annihilate partially and quadruplets are large enough to collapse and disappear entirely. This model is consistent with the apparent inapplicability of the Stokes-Einstein relationship and the fact that different regularities describe diffusion and flow. The figures cited in table 6 indicate that there is scope for exploiting the very high strength of metallic glass ribbons or wires in composites (especially in view of their usually excellent corrosion resistance), but surprisingly not much has been done to pursue this. HAGIWARAet al. [1985] shown that iron-based INROWASPed glass wires, strengthenedby additions of niobium, tantalum, chromium or molybdenum, combine high yield strengths with particularly good fatigue strengths. At about the same time, OHNAKA [1985], the main originator of the INROWASP process, suggested the use of wires made by this process as motor tire reinforcement, and subsequently an American company showed that Fe-Si-B, with an appropriate coating, fulfil the requirements of tire reinforcement, which are: a high tensile strength, good adhesion to rubber, and excellent resistance to fatigue and corrosion (OGINO[1986]). A recent study (VAIDYAand SUSRAMANIAN [1990]) has shown the potential of metallic glass ribbons made by meltspinning to reinforce glass-ceramics (i.e., devitrified oxide glasses). The melt-spun ribbons are assembled with the oxide glass precursor in powdered form; the assembly is sintered at 400-450°C, to full density, and sintering is followed by devitrification of the powder without devitrification of the metallic glass. Large increases of strength and toughness were achieved with as little as 1 vol% of reinforcing ribbons. Apart from the good strength of uncrystallized (undevitrified) metallic glasses, recent researches on aluminum-base glasses have revealed the remarkable properties that can be attained by partid devitrification. This topic is covered in the next section. 4.4.1. Partially devitrified (nanostructured) metallic glasses As emphasized in the previous section, metallic glasses can have very high yield strengths, approaching the ideal strength (that is, the strength shown by a dislocation-free material). In steels, because of the microstructural changes associated with the ferriteaustenite transition, conventional materials can also reach a significant fraction of the ideal strength. However, for light alloys based on A1 or Mg such a transition does not exist, and the hardening based on precipitates or dispersoids tends not to be so effective. For light alloys, therefore, the very high specific strengths (that is, strengtwdensity) of amorphous alloys are of particular interest. As outlined in $2.2, there are amorphous alloys based on aluminum with rare earth and transition metal additions and these alloys are ductile. Their flow strengths can be as high as 1250 MPa (BLANK-BEU~ERSIXWIT [1991]), about twice as high as the best precipitation-hardened conventional dloy, and immensely harder than the commercial purity element. Partial or complete devitrification usually embrittles a metallic glass. In some cases, however, the ductility can even be improved (HILLENBRAND et aZ. [19821). In the case of the Al-based glasses, partial devitrificationleads to increased tensile strength -as much as a 50% increase an the corresponding amorphous alloy - while ductility is retained. The partially devitrified materials, introduced in 04.3, have a fine dispersion of 3- to 10-nm A1 crystallites in an amorphous matrix. They are obtained by quenching at below
Ch. 19, $ 4
1801
Metastable states of alloys
the critical rate for full glass formation or by annealing an initially fully glassy material. The devitrification can also be induced by mechanical working. These materials, developed in Japan (KIM et al. [1990]) and the United States (CHENet al. [1991])show optimum tensile strength for a crystalline volume fraction of 25%, and the values are quite exceptional - as high as 1560 MPa for Al,,N~Ce,Fe, (INOUE et al. f19921). The strengths of the devitrified materials are compared with other aluminum alloys, including amorphous alloys, in fig. 46.This shows that as the microstructural scale of aluminum alloys is decreased the strength increases. The reasons why a nanometer-scale dispersion of crystallites should give such a pronounced hardening effect are not yet clear. However, the aluminum crystals appear to be too small to contain any dislocations.
-
4.4.2. Thermal embrittlement Many metallic glasses are capable of a moderate degree of deformation at room temperature: the process is heterogeneous and involves the formation of shear bands
--0.1 I
l
nrn I
IO
l
I
0.1
Prn I
IO
0.1
I
I
I
I
I
I
mm
4 1500 A
v
-r-
f ~1000
amorphous alloy
L
-L
I
v, QI
5 500 c CU
t-
I I
-
0 Microstructural Scale (m) Fig. 46. The tensile strengths of aluminum alloys compared as a function of microstructural scale: (a) commercial purity aluminum, (b) the strongest conventional precipitation-hardened aluminium alloys, (c) amorphous, or partly amorphous, aluminum-rare earth-transition metal alloys, consolidated and crystallized (HIGASHIet al. [1992], OH"m.4 ef al. [1992]), and (d) amorphous aluminum-rare earth-transition metal alloys partially devitrified (either during quenching or by annealing) to give nanometer-scale crystallites ((haef al. [1991], IWUE et al. [1992], KIM ef al. [1991]). The diameter of an aluminum atom, and the wide range of measured tensile strengths for fulZy amorphous AI-TM-Ln alloys are shown for comparison (TM= transition metal, Ln=lanthanide). References:p. 1818.
1802
R.W Cahn and A. L Greer
Ch. 19, 94
approximately along a plane of resolved shear stress. A ductile metal ribbon can generally be bent sharply back on itself without fracturing. Some glasses, mostly those based on iron (one might call these “glassy steels”) show (1) a ductibbrittle transition temperature, and (2) the phenomenon of t h e m 1 embrittlement ,which is marked by an increase in the ductile-brittle transition temperature on annealing well below the crystallization temperature. Figure 47 exemplifies these phenomena. The mechanism of this embrittlement has given rise to much investigation and argument. SPAEPEN [1977] first put forward the idea that shear band formation (morphologically similar to slip bands in crystalline metals) is to be attributed to a multiplication of free volume at an incipient band, which reduces the flow stress locally so that the shear band develops preferentially. As mentioned above, CAHNet al. [19841 showed that the overall density reduction in a Pd-Cu-Si glass due to a 40% rolling reduction was 0.14%, which implied a much larger density reduction within the n m w shear bands themselves. This notion was further analyzed by ARGON et al. [1985], and they concluded, from measured overall density changes in heavily deformed glasses (rolled or wire-drawn) and the observed shear-band geometry, that within a shear band itself the local dilatation due to free volume can be. as large as 50%. CAHN et al. [1984] further showed that the excess free volume in the shear bands was dissipated during a relaxation anneal, while KRISHNANANDand CAHN (1975) had earlier shown that after such a relaxation anneal and renewed plastic deformation, the new population of shear bands was not in the same sites as the old, Le., the old shear bands had been locally “healed” by the relaxation anneal. It can thus be taken as well established that free volume plays a central role in both homogeneous (creep) and heterogeneous plastic flow in metallic glasses. The theories to explain thermal embrittlement fall into two categories: (1) the hypothesis that annealing removes excess free volume and (2) the hypothesis that the homogeneous glass structure separates into two distinct amorphous phases. The basis of [1986] have the first hypothesis is clear from the foregoing: further, Wu and SPAEPEN assembled detailed evidence on thermal embrittlement consistent with this model. The et al. [1976], who second hypothesis leans on observations such as that due to WALTER by Auger electron spectroscopy established an enrichment of phosphorus at fracture surfaces of a Fe&JiaP14B6glass, and PILLERand HAASEN [1982] who used field ion microscopy to demonstrate the formation, during relaxation annealing of Fe&&,, of minute zones enriched in boron to 25 at.%. However, tests by X-ray small-angle scattering of some other glasses have failed to demonstrate phase separation of this kind, whereas other experiments have given contrary results. The literature is quite extensive. A possible marriage between the two types of hypothesis has been proposed in an interesting paper by YAVARI[1986]. By analysing the volume per metal atom in F e B glasses of different compositions, he shows that a local enrichment of B will need to “suck” free volume from the adjacent matrix, reducing the free volume concentration there. He suggests, therefore, that the phase separation hypothesis resolves itself, indirectly, into the free volume hypothesis. According to Yavari, the fact that some additives (e.g., Ce) delay embrittlement to higher temperatures, whereas others such as Sb promote it, can also be interpreted on the basis of his model. -Another observation
ch.19, § 4
Metastable states of alloys
1803
350OC-2hr Te,=97OC
i-+--i
4
.i100 2
c
n ! m
.-WC
E
n
-s
10"
u
z
LL
10-2
I
I
I
I
I
I
50 70 90 113 130 150 1 0 Testing temperature in O C
(4
180
1LO 100
Y
60
.C 0
-20
-6C -1oc
-180-
(b)
2L0
280
320 TI in OC
'
'
360
'
'
LOO
Fig. 47. Thermal embrittlement: (a) Strain at fracture in a bending test as a function of temperature for a Fe,,Be,,Si.,oC, glass, annealed for 2 h at 350°C after melt-spinning. TBDis the ductile-brittle transition temperature. (b) Change of TBDwith isochronal anneals at progressively inrreasing temperatures. (After SPASPEN et al. [ 19861).
References: p . 1818.
1804
R. W Cahn and A. L. Greer
Ch. 19, 34
(YAMASAKI et aZ. [1985]) to the effect that Fe-based glasses whose compositions fall close to an equilibrium eutectic composition embrittle much more sluggishly, if at all, is attributed to a reduced tendency to phase separation for such compositions. The clearest demonstration to date of the central role of free volume in determining the incidence of thermal embrittlement comes from an important series of publications by GERLINGet al. [1985, 1988, 1989, 19901 relating to Fe,Nia, glass. Briefly, they have established in circumstantial detail how this glass is embrittled by the loss of free volume on annealing: for instance, the embrittlement behavior varies with ribbon thickness, because the quenched-in free volume varies also. They have further shown that a glass which has been thermally embrittled can be reductilized by neutron irradiation, which creates fresh free volume. The restored ductility can in turn be removed by a second relaxation anneal. 4.5. Other properties of metallic glasses
Apart from their mechanical properties, some metallic glasses are beginning to be exploited for their magnetic, chemical and diffusion properties, and as precursors for crystallization (devitrification). Ferromagnetic (mostly iron-based) glasses, as used nowadays in transformers, are treated in chapter 29 and will not be discussed here. Precursors for devitrification are treated in 004.3 and 4.4.1, and this includes a brief account of permanent magnets made by this route. There is space here only for a summary account of the other properties. Many metallic glasses have exceptionally good corrosion resistance, even in quite strong acids, and this is clearly due to their structural homogeneity, which obviates the creation of the electrolytic microcells that enhance corrosion of crystalline metals. For some uses, metallic glasses can be used “as is”: thus, Fe-Cr-P-C ribbons were used for the active element in electromagnetic filters to remove rust from water (KAWASHIMAet al. [19851); since the field is strongest at the edges of ribbons, narrow ribbons were used. Their high corrosion resistance ensured a long life for the filter elements. For other purposes, it is necessary to put a glassy coating on a crystalline metallic substrate; this can be done by either laser-glazing or sputter-deposition. DIEGLEand HASHIMOTO [1988] and HASHIMOTO [I9931 have shown that deposits made by sputtering can provide quite remarkable corrosion protection: thus, a-Cu-Ta (anywhere from 20 to 80 at% Ta) passivates spontaneously even in 12N HCI, and behaves much better than pure crystalline tantalum, itself notably resistant to acids. HASHIMOTO [1993] has also surveyed the use of metallic glass coatings in electrocatalytically active electrodes, for instance, for use in chlorine manufacture by electrolysis. However, for lack of space the reader must be referred to Hashimoto’s excellent review. There has also been extensive research, published in the chemical literature, on the use of metallic glasses (in the devitrified condition) as catalysts for gas-phase reactions such as the hydrogenation of carbon monoxide: here again, the reader is referred to an excellent review by MOLNARet al. [1989], as well as to Diegle and Hashimoto’s conference proceedings. Finally, metallic glasses have found an exotic use as diffusion barriers in microelec-
Ch. 19, $ 5
1805
Metastable states of alloys
10-u700 600
10m
a5 10"
-
c .-
500
'.'.
100 Polycrystal
a Au in NiNb
1.0
1.2 1.1 1000/T in K
1.6
Fig. 48. DiffiLsivity of gold in amorphous and in polycrystalline Ni,,Nb,,. The righthand ordinate shows the diffusion distance in a period of one year. (Aftyer D o ~ ~ ~ e [1982]). tal.
tronic circuits, especially for those to be used at temperatures of several hundreds of degrees Celsius, for which they are superior to most crystalline alloys (CAHN[1986], DE REUS [1995]). This works because in the temperature range of interest, atomic transport in polycrystalline metals is predominantly by grain-boundary diffusion, which is absent in glasses. Figure 48 shows this: the diffusion of gold in a-Ni55Nb45is seen to be seven orders of magnitude slower than in the same alloy after it has been devitrified (which however requires higher temperatures to do). Ta-Ir glass, which crystallizes at over 900°C, has been effectively used to make diffusion barriers between silicon substrates and Y-Ba-Gu-0 ceramic superconductors layers (DE REUS et al. [1988]).
5. Rapid-solidification-processed(RSP) and other metastable crystalline alloys 5.1. Age-hardeningof light alloys Before we treat the properties of RSP alloys, it is appropriate here to remind the reader that metastable crystalline states are by no means restricted to alloys which have undergone extreme cooling rates and often indeed, solid-state quenching is all that is required. The two such states that have been known and studied for longest are martensite in steels and the age-hardened state in light alloys, Al-Cu in particular: both these are products of solid-state quenching. Both were cited as archetypes in the first paragraph of this chapter. Martensite is discussed in chapters 16 and 17; here we devote some paragraphs to the key phenomenon of age-hardening, since the structural changes (as distinct from the hardening mechanism) are not treated elsewhere in Physical Metallurgy. Age-hardening was discovered accidentally in 1906 by a German engineer, Wilm, References: p . 1818.
1806
R. W Cuhn and A. L Greer
Ch. 19, 55
who examined an aluminum alloy containing 3.5 wt% copper, 0.5 wt% magnesium and 0.5 wt% manganese, and his findings were eventually published in 1911. His alloy, quenched from 500°C at which temperature it had been homogenised, hardened substantially over a period of days at room temperature. This process came to be called agehardening. The original paper is not cited here, though it contains much of interest; it is too difficult to locate. Wilm’s paper (in translation) and extracts from many other of the early key papers have been reproduced in a most useful book about age-hardening by MARTIN[1968], which also includes an extended commentary about the meaning of the early discoveries. Wilm did not understand why his alloy hardened and this was left to three American metallurgists, Merica, Waltenburg and Scott, who in 1919 established the shape of the solvus curve for copper in aluminum, establishing that the solubility increased sharply with temperature, and they were the first to suggest that the precipitation of the equilibrium phase, CuAl,, was the basis of the hardening, although it could not be seen in the microstructure. (In fact, the age-hardening of Wilm’s alloy, which he called ‘cduraluminyy, was later found to be due not only to the copper but also to the precipitation of a form of the phase Mg,Si, the silicon being an unintended impurity in the alloy). Understanding did not develop further until in 1938 two independent notes appeared side by side in the journal Nature, by GUINIER[1938] and PRESTON [1938], soon followed by more substantial presentations by these same authors. They showed by h u e or oscillating-crystal x-ray diffraction photographs that age-hardening of Al-Cu alloys was associated with a pre-precipitation process: thin “zones” parallel to { 100) became enriched in copper, without change as yet in crystal structure, and the lattice spacing was concurrently locally reduced because copper atoms are smaller than aluminum atoms. They also indicated that these zones (which later came to be called Guinier-Preston, or G-P, zones) developed into better defined intermediate crystal structures at a later stage of ageing. These later, distinct, phases in Al-Cu came to be called Off, 8’ and B (all tetragonal), and all coherent with the matrix on { 100). 8 is in fact the equilibrium CuAl, phase; 8’ and B are formed only when ageing is done above room temperature. The association of the successive stages of hardening with the development in the preprecipitation structures was soon demonstrated, first by Gayler in 1940. Research on both the structural and the hardening stages proceeded intensively in the two following decades: an important synthesis was by SILCOCK et al. [1953-541. Soon after this, the age-hardening stages began to be examined by transmission electron micrography, and such matters as forms of heterogeneous nucleation of zones and transition phases were examined. A definitive review of what had been discovered about these stages came a few years later (KELLY and NICHOLSON [1963]). This review covered not only Al-Cu alloys but also others, such as Al-Ag in which the initial age-hardened zones are spherical and Al-Mg-Si alloys in which they are rod-shaped. After that, the fast pace of research in this field slackened, but x-ray studies, especially by small-angle scattering (see ch. 12, 9 5.2) have continued. An up-to-date outline of age-hardening in light alloys (aluminum, magnesium and titanium base) has been published by POLMEAR[19951. Figure 49 shows the metastable solubility curves for the various intermediate phases as deduced from the researches in the 1950s and 1960s. GP zones and B” nucleate
Ch. 19, 8.5
1807
Metastable stares of alloys
Cu,at.% 0
1
2
3 i
Fig. 49. Solubilities of copper in aluminum in the presence of various, mostly metastable phases as a function of temperature.
homogeneously, 9‘ forms preferentially at dislocations and the equilibrium phase, 0, at grain boundaries. The succession of stages follows Ostwald’s (and Turnbull’s) “rule of stages”: the zones or particles of each stage undergo reversion (Le., re-solution) before the next phase is formed; the initial presumption that each stage formed by atomic rearrangement of the earlier form proved to be mistaken. Reversion was studied in detail by experiments in which an age-hardening treatment at one temperature was interrupted by a shift to a higher or lower temperature (e.g., KOSTFJRand KAM[1938]); the sudden shifts of hardness, later supplemented by structural studies, showed that structural reversion took place. The kinetics of aging were studied not only by x-ray diffraction and hardness measurements but also by resistivity measurements, conducted by Turnbull and his associates in the late 1950s on Al-Cu alloys. The. resistivity is closely linked to the process of formation of GP zones. These systematic studies were of great importance because they proved unambiguously that the rate of zone formation (or clustering, as they called it) at mom temperature was orders of magnitude faster than would have been predicted by extrapolating measured diffusivities of copper in aluminum (DESORBO et al. [1958]), and the same thing was found for the age-hardening kinetics of A1-Ag (TTJRNBULL and TREAFTIS [1957]) and later of Al-Zn. The activation energy of the process clearly is that for vacancy migration, showing that there was a metastable excess of quenched-in vacancies which accounted for the rapid aging. The striking effect of these References: p. 1818.
1808
a.19, 5 5
R. W Cahn and A. L. Greer
2l 2.4
?.I\
I. f
0
E
d I
I
0
2
4
6
8
10
12
14
16
18
20
Time, min
Fig. 50. Fractional change in resistivity vs aging time at 0°C for an A1-2at% Cu alloy after the indicated quenching sequences. (Holding times were 1 h at O T , 2 min at 200°C). (After TU~NBLJLL et al. [1960]).
excess vacancies emerges from fig. 50, taken from a later study by TURNBULLet al. [1960]: here it is seen that a reversion treatment at 200"C, interspersed with aging at OOC, greatly slows down aging kinetics because such reversion quickly gets rid of a large fraction of the excess vacancies (the authors believed that this happens because the reversion treatment creates dislocation loops which act as efficient vacancy sinks). Another feature of these studies was a comparison of aging after quenching into different fluids: the faster the quench, the more rapid the ultimate aging kinetics. This series of studies constitutes one of the great classics of physical metallurgy. (See also ch. 7,0 8.1). Turnbull's experiments stimulated, after a long interval, some very interesting investigations of concentrated AI-Mg alloys, by VAN R o o m et ul. [1988]. Using scanning differential calorimetry, these authors compared, using both melt-quenched and solid-quenched specimens, the kinetics of both the formation of GP zones at room temperature, and their subsequent redissolution during heating. (The melt-quenching took place by melt-spinning, producing 30 p m thick ribbons, the solid quenching was carried out by reusing these same ribbons). It was shown that in the solid-quenched specimens, both GP zone formation and GP zone redissolution were much faster than in the meltquenched specimens. This was attributed to the greater efficacy of the solid-state quench in "freezing" in excess vacancies. That, in turn, is no doubt due to the greater effectiveness of a quench in water in maintaining a fast cooling rate down to quite modest temperatures, whereas melt-spinning leads to very fast solidification but slow cooling in contact with air after solidification is complete. This reinforces the message, conveyed in 5 3.1, that a quenching method must take into account the measured cooling rate in the temperature range of interest, and is in harmony with the observation cited there, that inrotating-water wire quenching is more effective in maintaining fast cooling in the solid state than is melt-spinning. In recent years, research has been focused on the Al-Zn alloys, mostly by means of the small-angle scattering of x-rays or neutrons (see also ch. 12, 05.2). These alloys
Ch. 19, $ 5
Metastable states ofalloys
1809
decompose by a spinodal mechanism (see ch. 15, 3 3.1) below a well defined spinodal temperature, though the similarity of aging kinetics above and below this temperature (HENNIONet al. [19821; GUYOTand SIMON[19821) indicate that the spinodal singularity does not play any substantial part in the decomposition kinetics. Many other relevant references, as well as a detailed explanation of spinodal decomposition as observed in Al-Zn alloys, can be found in a book chapter by BINDER[1991].
5.2. General characteristicsof rapid-didfication-pfication-processed (RSP) crystallime alloys The principal changes which can be brought about in crystalline alloys consequent on RSP have already been outlined in 3 1.1. They include: (1) Extension of solid-state solubility; (2) refinement of grain size, with possible modification of grain shapes and textures; (3) reduction or elimination of microsegregation; (4) formation of metastable phases; ( 5 ) achievement of high point-defect concentrations. [1978], and All of these have been surveyed in a general review by HONEYCOMBE various aspects, grain refinement and metastable phase formation in particular, are treated elsewhere in this chapter. We have already seen, in $ 2.1, how thermodynamic and kinetic considerations, the variation of Towith composition in particular, affect solute trapping -that is, metastable extension of solid solubility. Aluminum alloys have been far more studied than any others, especially those with transition-metal solutes, and fig. 51, taken from MIDSONand JONES [1982], summarizes available results. It is plain that the solubility enhantxment factor is in general much larger for solutes with a small equilibrium solubility. (This is reminiscent of the inverse correlation between intensity of segregation at grain boundaries and equilibrium solubility, see ch. 13, $4.2.) Midson and Jones analyze these results in terms of the morphological criterion and Jones has also sought to interpret the lattice parameters of some greatly supersaturated solid solutions in terms of rival physical models (JONES[1983]). An early but substantial collection of experimental data on extended solubility in many different solvents was assembled by ANANTHARAMAN and SURYANARAYANA [1971]. WOODand HONEYCOMBE [1981] review the available information on extension of solubility in iron-base alloys. Their figures together with some additional ones assembled by CANTOR[I9821 are collected in table 7. In addition, C solubility in martensites is enhanced up to 2.0 wt% and in austenites up to 3.5 wt% before cementite is detected; but results are very variable and dependent on the measurement technique used. Boron, which is almost insoluble in equilibrium in pure a-iron (=50 ppm) was forcibly dissolved by RSP up to 1 wt %; as with aluminum alloys, it appears that the least soluble elements have the highest enhancement factor. Extended metastable solid solubility of course leads to greater volume fractions of pre-precipitation structures (GP zones) and consequently enhanced hardening. Much of the work on this has been purely empirical and there is rather a dearth of basic studies of pre-precipitation and precipitation mechanisms in RSP alloys. The rejection of solute References: p . 1818.
1810
R
Ch. 19, 8 5
Cahn and.4. L Greer
I
121314 21
*32 40 ATOMIC
-50 58 74-79
NUMBER
Fig. 51. Solubility extensions of various solutes in aluminum. The black bars indicate the increase beyond and JONES equilibriumroom-temperature solubilities achieved by rapid solidificationprocessing. (After WSON [19821).
from the matrix and the identity and density of zones and phases need to be studied and compared with those for solid-quenched alloys of the same type.The rejection of solute and from supersaturated RSP A1-12 at% Cu alloy has been studied by KRISHNANAND CAHN [1976] and the decomposition products of RSP A1-17 at% Cu by SCOTTand LEAKE[1975]. The behavior of supersaturated RSP Al-8 wt % Fe alloys was examined [1982], while a by JACOBS et aZ[1974], Al-Fe-Cr alloys by YEARIMand SHECHTMAN range of supersaturated Al-Zr alloys were examined in detail by SAHINand JONES [19781 Table 7 Extensions of solubility for substitutional solutes in rapidly solidified binary iron-base alloys. Solute
and Horn et aZ. [1982]. It is important to emphasize that to obtain the full benefits of improved mechanical properties from RSP of alloys, it is necessary to develop special solute-rich compositions and not to restrict oneself to standard compositions used for normal casting. One of the most common consequences of RSP is refinement of microstructure, including such features as dendrite arm spacings, eutectic lamellar spacings, particle dispersions and grain size. When there is solute partitioning, rapid solidification leads to shorter diffusion distances and thereby to a refined structure. Even without partitioning, however, for example in pure metals, there is refinement of grain size. Sizes in the range 1 to 5 ,urn are quite common, and in splat-quenched foils can be as small as 0.01 pm. The refinement of grain size by RSP may have a number of distinct origins (quite apart from refinement brought about by the addition of inoculant particles). The possible [1991]. As shown in fig. 34, solidification at high origins have been reviewed by GREER and undercooling should favor homogeneous nucleation and yield finer grains. BOSWE~LL CHADWICK119771 attempted thus to rationalize micrograin sizes in aluminum. In general, however, recalescence precludes grain refinement occurring in this way, except in glass-forming alloys near Tg.After rapid quenching, for example by melt-spinning, the grains are not necessarily equiaxed. Melt-spun ribbons have a have a fine equiaxed chill zone at the surface next to the quenching wheel, but this is followed by columnar growth, with column width increasing through the thickness of the ribbon because of competitive growth of differently oriented grains. The initial grain size at the wheel-side of the ribbon is governed by the heterogeneous nucleation frequency and the crystal growth rate. A simple analysis based on the temperature dependence of these quantities suggests that the grain size should be inversely proportional to cooling rate, in good agreement with experiment (GREER119911). Rapid solidification also occurs in highly undercooled bulk melts, and it is known that grain-refined structures are obtained for undercoolings beyond a critical value (as seen, for example, in the work of KATTAMIS and FLEMINGS[1970]) (see also c k 8, 5 11.2). There has been much investigation of this phenomenon, and it appears that it is not due to copious nucleation in the melt, but rather to a break-up of the initial dendrites (Schwarz et al. C1994)). Consequencesof a fine grain size include substantial “Petch-strengthening”at ambient [1978]) as well as an enhanced liability to supertemperature @UFI,OS and CANTOR plasticity in RSP alloys (e.g., CHEESEand CANTOR[1980]), in the sense that superplasticity persists to higher strain rates. Quite generally, fine grains enhance hotformability and thus RSP alloys are easier to shape as well as often having better mechanical properties after forming and heat-treatment, than slowly solidified analogues. Fine grains also have consequences for martensitic transformations in steels and for creep in superalloys; these are discussed below. Few studies have been reported of preferred orientations in as-cast structures produced by RSP. BLAKEet al. [1982] found remarkably perfect alignment of (0001) planes of Zn and Cd parallel to the surface of melt-spun ribbons; several other metals formed weaker textures of various types. The texture of high-purity (99.999%) zinc was found to deteriorate with time, and this was tentatively attributed to selective grain growth at ambient temperature. KAN et al. 119821 have melt-spun ribbons of highly References:p . 1818.
1812
R. W Cahn and A. I, Greer
Ch. 19, 5 5
alloyed F e S i (6.5 wt% Si, unlike the usual 3 wt%) and examined the preferred orientation. Grains with (100) [Om orientations grew preferentially to give a sharp planar texture, leading to much reduced alternating current losses. The texture-sharpening is no doubt an instance of surface-controlled grain growth (ch. 28, 0 4.4.2). The general technological implications of the combination of extended solubility, microcrystallinity and reduced segregation have been lucidly reviewed by GRANT[19781. KOSTER and SCKUNEMANN [1993] provide a broad-based overview of phase transformations in rapidly solidified alloys. 5.3. RSP light alloys
A particularly important practical objective is the long-standing requirement of substantially increasing the elastic stiffness of aluminum alloys for airframe applications. Elastic moduli are not structure-sensitive and so this objective is best approached by introducing large amounts of stiff intermetallic compounds; initially, those based on etal. [1947-48]), manganese were preferred. When this was first attempted (DUDZINSKI the segregation associated with the slow freezing used led to coarse, uneven particle distributions and the alloys were hopelessly brittle. Once RSP methods had become established, numerous investigators recognized that this approach was the answer to the original difficulties. It should be noted here that while there is evidence that meltquenching traps fewer excess vacancies than does solid-state quenching ( $ 5 . l), yet meltquenching is more effective in permitting large supersaturations of solute; this is helped by fast cooling through the freezing temperature. One company began using a form of planar flow-casting to produce alloy ribbons which were then chopped into small fragments and compacted by hot extrusion or by one of several rapidly acting high-temperature processes, such as impact forging or explosive consolidation, which restrict or eliminate structural coarsening during compaction (ADAM and LEWIS[1985]). These authors, for instance, give elastic properties of hot-extruded A1-(8-12)Fe2V alloys: the Young’s modulus for pure aluminum is 70 GPa, and the alloying enhanced this value by up to 37% without substantial embrittlement. The enhancement is due to a combination of enhanced solubility and fine dispersion of intermetallic compounds. Since the 1985 account, development of this kind of alloy has continued and further improvements of properties has been achieved @AS and FROIS [1993]). Thus, silicon has been added to make AI-FeV-Si alloys, in which the chief dispersoid is Al,,(Fe, V),Si. Figure 52 shows the high-temperature strength of different members of this family, with different contents of this dispersoid. The top plot refers to an alloy with a stiffness almost 40% up from that of pure aluminum, though its fracture toughness is only moderate; the bottom curve refers to a somewhat weaker alloy which is highly formable; it has been drawn into wire. Enhanced stiffness is a special requirement of modern airframe aluminum alloys, but so is strength, notably at elevated temperatures, because of aerodynamic heating in flight). Figure 53, from Adam and Lewis’s 1985 review, shows the success which had already been achieved 10 years ago in attaining this objective. All these advanced alloys contain iron: ever since Thursfield and Stowell’s 1974 study, this has been the primary
Ch. 19, 8.5
Metastable states of alloys
r
la,
I813
Tensile Strength
v)
0 4 0
.-
27% silicide phase
VI
C
16% silicide phase
M 3wc
Yield Sfrength 'O0
i
v)
jl
20
0
' . '
la,
0
Mo
- ' 300 w '
3mc
'
503
- 600 ' *. 700'
Tempaa~urzO F Elongation
"i
& b FfS061 1
NS0812
IO
NS1212
3wc
Fig. 52. Tensile pmperties of RSP and consolidated AILFe-V-Si alloys as a function of temperature. (After DAS and FRO= 119931).
solute of choice (iroh has a small equilibrium solubility and a large enhancement of metastable solubility through RSP). In the figure, 70775-T6 and 2219-TS51 are non-RSP high-grade alloys, and the "AFWALprogram goal" was a military objective of that time. It can be seen that, even then, the new alloys came close to matching the state-of-the-art References: p . 1818.
1814
Ch. 19, 8 5
R. ICahn andA. L Greer
K 100
300
4CO
sco
600
700 600
80 500
40
2 300
40
a
200
20 100
'0
100
200
300
400 500 Temperature
r9
600
700
800
900
Fig. 53. Elevated-temperature strengths of advanced aluminum alloys. (After ADAM and LEWIS[1985]).
aeronautic titanium alloy, Ti-6A1-4V. A notable feature of such alloys, arising from their microstructural homogeneity, is their good corrosion resistance. For the range of alloys based on A1-Li, which has received intense attention in the past decade because of the prospect of much enhanced specific stiffness (lithium reduces density and enhances stiffness), RSP has been extensively explored since the early work of SANKARANand GRANT[1980], and at present there is still much dispute as to the relative benefits of standard ingot processing and RSP. It has been shown that RSP allows larger lithium contents than does ingot metallurgy (5 wt% as opposed to 2.5) without causing embrittling coarse dispersoids (MESCHTER et al. [1988]). Also, RSP alloys show less anisotropy than do those made by ingot metallurgy. However, the economics of the process may rule it out. A recent overview of Al-Li-based alloys (GRIMES [1990]) asserts that early womes about the adequacy of the ingot route and claims that powder metallurgy was essential, had largely vanished by 1990. Das and Froes' overview of 1993 also covers the striking progress that has been made with RSP magnesium-base alloys: these alloys are always hovering on the verge of large-scale application and then do not make progress, largely because of corrosion problems. RSP alloys have improved corrosion properties as well as improved strength, and their prospects may now improve. The overview also covers the considerable work done with RSP titanium-base alloys. The review must be consulted for further details. 5.4. Steels
The quite extensive early work on microstructure of RSP steels has been fully reviewed by WOODand HONEYCXMBE[1981]. Extension of solid solubilities in iron was outlined in 55.2. The findings concerning the precipitation morphology of various
a.19, $ 5
Metastable states of alloys
1815
carbides, both in simplified “model” steels and in commercial compositions, do not lend themselves to summary and the review should be consulted. The principal driving force for research in this field has been the desire to improve the characteristics of tool (“high speed”) steels. Much of the research has centred on simplified model alloys such as Fe-W-C. The most striking finding (see for instance S m and HONEYCOMBE [1978]and RAYMENTand CANTOR[1981])is the fact that for the fastest quenches and for solute-rich steels, S-iron is retained; it is only at intermediate quench rates that austenite is retained in such steels, and at conventional cooling rates, martensite is the major phase. The kinetics of hardening during heat-treatment are not much affected though the pattern of carbide precipitation is refined and, in particular, carbide segregation, a serious problem, is minimised (KASAKand DULIS[1978]). The peak hardness is however raised for optimum treatments (RAYMENTand CANTOR[1978]). Practical developments based on RSP of tool steels are surveyed by JONES [1982]. In scientific terms, one of the most striking sets of findings concerns the modification of the martensitic transformation by RSP. INOKUTI and CANTORE19761 found a variation of microhardness values for a series of RSP Fe-Ni alloys, as shown in fig. 54a; values for the solid-quenched alloys are shown for comparison. This was traced to a very pronounced change in martensite start (MJ temperatures on cooling (fig. 54b) This curve, and also the a -+ y start temperatures on reheating, were obtained calorimetrically: the martensitic alloy was heated slowly in a DSC until it had all transformed back to y and then slowly cooled again; M, (y+a) was marked by a pronounced exothermic peak, LY + y by an endotherm (INOKUTI and CANTOR [1982]).The striking feature was that the depressed M, caused by the original rapid quench from the melt does not disappear on repeated cycling between CY and y; only a prolonged y anneal restores the alloy to normal behavior (labelled “solid quench” in fig. 54b). It is therefore not the rapid cooling per se which reduces the Ms, so it can only be the austenite structure resulting from that quench. In fact, various earlier studies not involving RSP have shown a relationship between austenite grain size, d, and M, (e.g., for Fe-32% Ni, AM,=40-100 K for d reduced from 200 to 10 pm). In the RSP alloys, d is only 4-5 p m and this leads to a drop in M, of up to 200K for Fe-17% Ni. Thermal cycling, via the reverse shear transformation, restores the fine austenite grains, and only a long y anneal gives the y grains a chance to grow substantially. Such austenite grain growth seems often to be particularly sluggish in RSP steels (SUGAet al. [1980]); a micrographic study suggested that stable, extremely fine sulphide dispersions at austenite grain boundaries in RSP steels are responsible for such retardation of grain growth. (KELLYand VANDERSANDE [19$0]). A slight reduction in the cooling rate (substituting melt-spinning for piston-andanvil quenching) generated massive transformation instead of martensite (HAYZELDEN and CANTOR[1984-85]). A study of Fe-Ni-C alloys, with lower martensite start temperatures (HAYZELDEN and CANTOR [19821,using a scanning calorimetry approach, similarly showed M, depressed by a very fine grain size generated by RSP. Figure 29a shows that pure iron can also form martensite. The structure, formation conditions and crystallography of this martensite were studied by DUFLOSand CANTOR [1978];hardness goes up with martensite/fernite ratio, and for 100% martensite, the microhardness of pure iron is 600-700! A Hall-Petch plot (ch. 17, 0 5.1) for pure iron is References: p . 1818.
1816
R. K Cahn andA. L Greer
Ch. 19, 55
800r
k
Splat quench
I
0
1
10
I
I
I
20
I
30
I
I
I
40
Ni content (wt %)
r
Ni content (wt%)
Fig. 54. (a) Microhardness values for liquidquenched and solid-quenched Fe-Ni alloys. (b) y +LY (Mb and a 4 y transformation temperatures for a range of Fe-Ni alloys. (Both after INOKUTIand CANTOR119821).
Ch. 19, $ 5
1817
Metastable states of alloys
Liq.quenched martensite
Liq.quenched ferrite
/ I
Ssquenched ferrite
,
10 30 50 GRAiN SiZE-V2( mm-V2)
Fig. 55. Microhardness vs (grain size)-’” for pure iron. (After D m o s and CANTOR[1978]).
shown in fig. 55, taking the thickness of a martensite lath as “grain size”. It can be seen that the remarkably high hardness of martensitic iron can be largely, or even wholly, attributed to “grain-size strengthening”. Recently, the limit of applicability of the Hall-Petch straight-line plot for very small grain sizes, because of insufficient space for dislocation pileups, has been discussed by Smith et al. [1995]; clearly, a grain size of about 0.35 ,urn is still within that limit. A study by MAWELLA et al. [19821 of a melt-spun ultra-strong steel indicated that the considerable strength increase generated by melt-spinning was primarily due to refinement of the martensite structure. It has also been established that high-carbon cast iron (2.4-3 wt% C) following RSP by centrifugal atomization consists largely of retained austenite. Subsequent annealing produces a microcrystalline(a! + carbide) structure. Even at 650°C, this microcrystalline structure is so stable that the material behaves superplastically (EISELSTEINet al. [1983]). The conversion of brittle cast iron into a superplastic material is a particularly striking indicator of the changes in metallurgical characteristics obtainable by RSP.
5.5. Superalloys
Apart from tool steels and aluminum alloys, the other principal category of alloys to which RSP has been extensively applied is that of the superalloys, both those used for turbine discs and those destined for blades. The two Reston Conferences, in 1977 and 1979, are the principal source of information up to now on these studies; some other publications are by TRACEYand CUTLERE19811 and by SMUGERESKY [1982]. JONES [1982] also provides a lucid summary. Most investigators up to now have used atomized powders, especially those made by centrifugal atomization in helium. However, Dunos and STOHR[1982] have demonReferences:p. 1818.
1818
R. U! Cahn andA. L Greer
Ch. 19, Refs.
strated that melt-spinning can provide higher quench rates and thus comminuted meltspun ribbon provides an alternative material for subsequent hot consolidation, most commonly done by hot isostatic pressing. The benefits of RSP include the possibility of forcing more solute into the y and y’ phases, producing alloys fine-grained and homogeneous enough to permit easy hot-working, even superplastic fo~mingin extreme cases, and uniformity of structure throughout a large piece. The microstructural implications of a change from gas atomization to melt-spinning of superalloys have been examined in some detail by LIEBERMANN et al, [1983]. SHOHOJI et al. [1982] have also examined the structure of melt-spun superalloys and estimated cooling rates. INOUE et al. [1983b] melt-quenched a number of superalloys consisting of y’ only. They found that these RSP alloys, instead of being almost perfectly ordered with large antiphase domains, were imperfectly ordered with APDs well below 0.1 mm in diameter, and were free of grain-boundary segregation. These structural modifications lead, in several alloys, to greatly enhanced ductility and some increase in strength. However, this much-cited research has recently been called into question, because the authors used melt-spun ribbons for microstructural examination and INROWASP-wires for mechanical testing, presuming that both forms had similar microstructures. CHIBAand HANADA [1996] have now demonstrated that the wires had quite a different microstructure from the ribbons, apparently because of slower cooling through the freezing-temperature. There is, however, a major problem. The fine grain sizes produced by RSP, while they facilitate forming, leave the alloy very liable to creep at high temperatures but low stresses. This can be overcome by a process of directional recrysfallizufionof the RSP alloy after it has been consolidated and formed the process consists of a slow passage through a steep thermal gradient, to form large elongated grains rather like those generated by directional solidification. However, as SMUGERESKY [19821 indicates, the prolonged period spent at high temperatures during consolidation, forming and directional recrystallization may damage the desirable phase structure generated by RSP, and this problem is attracting further attention. The technique is currently applied primarily to oxide-dispersion-strengthened superalloys like MA 6000 (which contains 1.1 wt% Y,03). Such alloys cannot be turned into coarse or single crystals by directional solidification, and thus directional recrystallization is the solution of choice (ROBERTS [1993]). The mechanism of this process has been examined in detail by BALOCHand BHADESHIA [1990], and computer simulation has recently cast further light on it (GODPREYand MARTIN[19951).
References ADAM,C. M. and R. E. LEWIS, 1985, in: Rapidly Solidified Crystalline Alloys, eds. S. K.Das, B. H. Kear and C . M. Adam (The Metallurgical Society, Warrendale, PA), p. 157. AGYEMAN,K., R. MULLERand C. C. Tsm, 1979, Phys. Rev. B19, 193. AKHTAR,D. and R.D. K.MISRA,1986, Scripta Metall. 20, 627. AL BIJAT, S., R. IRALDI,J. M. DUBOIS,G. LE CAER and C. %TE, 1982, in: Proc. 4th Int. Conf. on Rapidly Quenched Metals. Eds. T.Masumoto and K. Suzuki (Japan Institute of Metals, Sendai) p. 375. ANANTHARAMAN, T.R. and C. SURYANARAYANA, 1971, J. Mater. Sci. 6, 1111. ANGELL,C. A., 1995, Science 267, 1924.
Ch. 19, Refs.
Metastable states of alloys
1819
AOKI K., X.G. LI and T.MASUMOTO,1991, Mater. Sci. Fng. Al33.565. (7th Conf. RQW. ARMSTRONG, G.R. and H. JONES, 1979, in: Solidificationand Casting of Metals (The Metals Society, London; now Institute of Materials), p. 456. ATZMON,M., K.M. UNRUHand W.L. JOHNSON, 1985, J. Appl. Phys. 58,3865. BALANZAT, E., J.T. STANLEY, C. MARY and J. HIUAIRET, 1985, Acta Mktall. 33,785. BALOCH,M.M. and H. K.D. H. BHADESHIA, 1990, Mat. Sci. Twhn. 6, 1236. BARBER,B. P. and S. J. PUTTERMAN, 1992, Phys. Rev. Lett. 69, 3839. BATTEZZATI, L., 1990, Phil. Mag. B61, 511. BATTEZZATI, L. and A. L. GREER, 1987, Int. J. Rapid Solidification 3,23. BATTEZZATI,L. and A. L. GREER,1989, Acta Metall. 37, 1791. BAXTER,D. V., 1986, J. Non-Cryst. Solids 79.41. BECHEI,D. and G. REOAZZONI, 1991, Mater, Sci. Eng. A134, 1120. BEGHI,G., R. MATERAand G. F’IATTI, 1969, J. Nucl. Mater. 31,259. BEKE,D. L., P.I. L~EFF and H. BAKKER,1991, Acta Metall. et Mater. 39, 125, 1267. BENJAW, I. S., 1970, Metall. Trans. 1, 2943. BERKOWTZ,A.E. and J.L. WALTER, 1987, J. Mater. Res. 2,275. BERNAL, J.D., 1964, Proc. R. SOC. (Lond.) A284,299. BEVK,J., 1983, Annu. Rev. Mater. Sci. 13, 319. BEWLAY, B.P. and B. CANTOR,1986, Int. J. Rapid Solidification2, 107. BEWLAY, B. P. and B. CANTOR,1990, Metall. Trans. 21B, 899. BINDER,K., 1991, in: Phase Transformations in Materials, ed. P. Haasen, Vol. 5 of Materials Science and Technology, eds. R. W. Cahn, P. Haasen and E. J. Kramer (VCH, Weinheim) p. 405. BLAKE,N. W., E A. HAMES and R. W. SMITH,1982, in: Rapidly Solidified Amorphous and Crystalline Alloys, Proc. Mater. Research SOC. Annual Meeting, Boston, 1981, eds. B. H. Kear, B. C. Giessen and M. Cohen (North-Holland, Amsterdam) p. 363. BLANK-BEWERSDORFT, M., 1991, J. Mater. Sci. Lett. 10, 1225. BORDEAUX, E and A.R. YAVARI,1990%J. Phys. France 51, C-249. BORDEAUX, E and A.R YAVARI,1990b. I. Appl. Phys. 67,2385. BORDERS, J.A., 1979, Annu. Rev. Mater. Sci. 9, 313. BORMANN, R., 1994, Mater. Sci. Eng. A179/A180, 31. (proc. 8th Int. Conf. on Rapidly Quenched and Metastable Materials, Sendai, 1993). BORMANN, R. and K. ~ L T Z E R 1992, , Phys. Stat. Sol. (a) 131,691. BORMANN, R., E GXRTNERand K. Z~LTZER, 1988, J. Less-Comm Metals 145,283. BOSWELL,P.G. and G. A. CHADWICK, 1977, Scripta Metall. 11,459. BOUDREAIJX, D. S., and J.M. GREGOR,1977, J. Appl. Phys. 48, 152 and 5057. BRAUN,J., M. ELLNERand B. PREDEL,1994, Z. Metallkde. 85,855. BRENNER,A.. 1963, Electrodeposition of Alloys: Principles and Practice, 2nd edition (Academic Press, New York). BRESSON,L., J. P. CHEVALIERand M. FAYARD, 1982, Scripta Metall. 16,499. BUCKEL,W. and R. HILSCH, 1956, Z, Phys. 146,27. BUCKLOW,I.A. and L.E. DRAIN, 1964, J. Sci. Instr. 41,614. BUSCHOW,K. H. J., 1979, J. Less-Common Metals 66, 89. BUXHOW, K. H. J., 1982, Solid State Commun. 43, 171. BUSCHOW,K.H. J., 1986, Mater. Sci. Reports 1, 1. CACIUBO, R., M. STEFANON, W. S. HOWELU, A. K. SOPER,P. ALLU, F. VINAI, S. MELONE and F. RUS~CHELLI, 1989, Physica B E 6 & 157,220. CAHN,R. W., 1979, in: Alliiges et Maeriaux Amorphes. 21* Colloque de M6tallurgie, Saclay, 1978 (Centre d’Etudes Nucl&res de Sacley. Gif-sur-Yvette) p. 137. CAHN, RW., 1986, J. Vac. Sci. Technol. A4, 3071. CAHN,R. W., 1990, in: Basic Features of the Glassy State, eds. J. Colmenero and A. Alegria (World Scientific, Singapore), p. 282. CAHN,R. W., 1992, Nature 357,645. CAHN,R. W., 1993, in: Rapidly Solidified Alloys, ed. H. H. Liebermann (Dekkw, New York), p. 1.
1820
R. U? Cahn and A. L Greer
Ch. 19, Refs.
C m , R. W. and W.L. JOHNSON,1986, J. Mater. Res. 1,724. CAHN,R. W., J.L. WALTERand D. W. MARSH,1988, Mater. Sci. Eng. 98,33. CAHN,R. W., K.D. KRISHNANAND,M. LARIDJANI, M. GREENHOLZand R. HILL, 1976, Mater. Sci. Eng. 23, 83. CAHN,R. W., N.A. PRATTEN, M.G. SCOTT,H.R. SINNINGand L. LEONARDSSON, 1984, in: Mater. Res. Soc. Symp. Proc. 28, 241. CALKA,A,, M. MADHAVA, D. E. POLK,B. C. GIESSEN, H. MATYJAand J.B. VANDER SANDE,1977, Scripta Metall. 11,65. CANTOR,B., 1982, Rapidly Solidified Amorphous and Crystalline Alloys, Proc. Mater. Research SOC.Annual Meeting, Boston, 1981, eds. B.H. Kear, B.C. Giessen and M. Cohen (North-Holland, Amsterdam), p. 317. CARGILL,G. S., 1970, J. Appl. Phys. 41, 12. CARTER,G. and W.A.. GRANT,1982, Nucl. Instr. and Meth. 199, 17. CHASON,E. H., A.L. GREBR,K.E KELTON,P. S. PERSIAN,L. SORENSEN, E SPAEPEN and A. H. WEBS, 1985, Phys. Rev. B32, 3399. CHASON, E.H. andT. MIZOGUCHI, 1987, Mat. Res. Soc. Symp. Proc. 80,61. CHATTOPADHYAY, K. and P. RAMACHANDRARAO, 1980, J. Mater. Sci. 15,685. CHAUDHARI, P., J.F. GRACZYK, 0.HENDERSON AND P. STEINHARDT, 1976, Phil. Mag. 31,7272. CHAUDHARI,P., E SPAEPENand P. J. STEINHARDT, 1983, in: Glassy Metals 11, eds. H. Beck and H.4. Giintheccdt (Springer, Berlin) p. 127. CHEESE,R. and B. CANTOR,1980, Mater. Sci. Eng. 45, 83. CHEN,H.S. and C.E. MILLER, 1970, Rev. Sci. Instr. 41, 1237. CI-IEN, H. S., I. T. KRAU~E, A. I N OAnd ~ T.MASUMOTO, 1983, Scripta Metall. 17, 1413. CHEN,H., Y HE, G. J. S m u n and S. J. POON,1991, Scripta Metall. Mater. 25, 1421. CHEN, L.C. and E SPAEPEN,1991, J. Appl. Phys. 69,679. CHIBA,A. and S. HANADA,1996, Intermetallics 4 (in press). CLEMENS,B. M.,W.L. JOHNSON and R. B. SCHWARZ,1984, J. Non-Cryst. Solids 61&62, 817. CLYNE,T. W., 1984, Metall. Trans. 15B, 369. CONNELL,G. A. N., 1975, Solid State Commun.16, 109. COWLAM, N., W. GUOAN,P.P. GARDNER and H. A. DAVIES,1984, J. Non-Cryst. Solids 61-62,337. CROAT,J. J., R. W. HERBST,R. W. LEEand E E.PINKERTON, 1984, J. Appl. Phys. 55,2078. CUNAT,Ch., H.R. HILZINGER and G. HERZER, 1988, Mater. Sci. Eng. 97,497. DAHLGREN, S.D., 1978, in: Proc. 3rd Int. Conf. on Rapidly Quenched Metals, Brighton, ed. B. Cantor (Metals Society, London), vol 2, p. 36. DAS, S . K. and E H. FROES,1993, in: Rapidly Solidified Alloys, ed. H. H. Liebermann (Marcel Dekker, New York), p. 339. DAS, S. K, K. OKAZAKIand C. M. ADAM, 1985, in: High Temperature Alloys - Theory and Design, ed. J. 0. Stiegler (The Metallurgical Society, Warrendale, PA) p. 45 1. DAVIES,H. A., 1976, Phys. and Chem. of Glasses 17, 159. DAVIFS, H.A., 1978, in: Proc. 3rd Int. Conf. on Rapidly Quenched Metals, Brighton, 1978, ed. B. Cantor (Institute of Metals, London), vol. 1, p. 1. DAVE%H. A., 1995, in: Nanostructured and Non-Crystalline Materials, eds. M. V w u e z and A. Hernando (World Scientific, Singapore), p. 3. DAVIES, H. A., B. G. L,EWIS and I. W. DONALD,1978, in: Rapid Solidification Processing, principles and Technologies. Proc. Conf. on Rapid Solidification Processin, Reston, 1977, eds. R. Mehrabian et aL (Claitor, Baton Rouge, LA), p. 78. DE HOSSON,J.Th. M., 1983, in: Amorphous Materials: Modeling of Structure and Properties, ed. V. Vitek, (Met. Soc. AIME, Warrendale PA) p. 147. DENG,D. and A .S . ARGON,1986, Acta Metall. 34,2011. DE REUS,R., 1995, in: Intermetallic Compounds, 4 s . J. H. Westbrook and R. L. Fleischer (Wiley, Chichester), Vol. 2, p. 603. DE REUS, R., E W. SARISand T.S. BALLER,1988, J. Less-Common Metals 145, 621. DESORBO,W., H.N. TREAFTIS and D. T U R N B U L L , Acta I ~ ~ ~Metall. , 6,401. DE VRES, J., G. W. KOEBRUGGE and A. VANDENBEUKEL, 1988, Scripta Metall. 22,637.
Ch. 19, Refs.
Metastable states of alloys
1821
DIEGLE,R.B. and K. HASHIMOTO (eds.), 1988, Proc. Symposium on Corrosion, Electrochemistry and Catalysis of Metallic Glasses (The Electrochemical Society, Pennington). DIETZ,G., 1977, J. Magn. Magn. Mater. 6, 47. DONALD,I. W., 1987, .I. Mater. Sci. 22, 2661. DONALD,I. W. and A. H. DAVIES,1978,J. Non-Cqst. Solids 30,77. DONALD,I. W. and A.H. DAVIES,1980, J. Mater. Sci. 15,2754. DONG, X. Z., W. FEXNENGEL and H. KRONMULLER, 1982, Appl. Phys. A28, 103. DONOVAN, P.E. and W.M. STOBBS, 1981, Acta Metall. 29,1419. WRNER, W., H. M~HRER, P. J . POKELA, E. KOWALAand M.-A. NICOLET,1991, Mater. Sci. Eng. B10, 165. D o n e , B.L., P.S. PEEXCY,J. D. WILEY,J. H. F’EKEPEZKO and J.E. NORDMAN, 1982, J. Appl. Phys. 5,6168. DREHMAN, A. J., and A. L. GREER,1984, Acta Metall. 32, 323. DUBEY,K. S. and P. RAMACHANDRARAO,1984, Int. J. Rapid Solidif. 1, 1. Dvso~s,J. M., 1992, in: Trends in Non-Crystalline Solids, eds. A. Conde, C. E Conde and M. Mil& (World Science, Singapore) p. 343. DWIS, J. M., P. H. GASKELLand G. Le CAER, 1985a, in: Rapidly Quenched Metals 0,eds. S. Steeb and H. Warliimont (North-Holland, Amsterdam) p. 567. DUBOIS, J. M., G. LE CAR\ and K. DEHGHAN,1985b, in: Rapidly Quenched Metals 0,eds. S. Steeb and H. Warlimont (North-Holland, Amsterdam) p. 197. DUDZINSKI,N., J. R. MURRAY, B. W. MOTTand B. C J U L ~ R S1947-48, , J. Inst. Metals 74, 291. DUFLW,B. and B. CANTOR,1978, in: Proc. 3rd Int. Conf. on Rapidly Quenched Metals, Brighton, 1978, ed. B. Cantor (Institute of Metals, London), vol. 1, p. 110. DUFLOS,B. and B. CANTOR,1982, Acta Metall. 30,323. Dvnos, B. and B. CANTOR, 1987, J. Mater. Sci. 22, 119. DUFLOS,B. and J. E STOHR, 1982, J. Mater. Sci. 17, 3641. D u m , P. A., J. SETSMAand A. VANDEN BEUKEL,1992, Acta Metall. Mater. 40,743. DUINE,P. A., J. SETSMAand A. VANDENB m e ~ 1993, , Phys. Rev. B 48,6957. D u w ~ zP. , , 1967, Trans.Am. SOC. Metals 60,607. Duwsz, P., RH.WILLENSand W. KLEMENT,1960, Nature (London) 181, 869; J. Appl. Phys. 31, 1136 and 1500. FGAMI,T.,1978, Mater. Res. Bull. 13, 557. EGAMI,T., 1978b, J. Mater. Sci. W, 2587. EGAMI,T., 1981, in: Glassy Metals I: Ionic Structure, Electronic RTRANSFQRT and Crystallization (Springer, Berlin) p. 25. EOAMI, T., 1986, in: Amorphous Metals and Semiconductors, eds. P. Haasen and R.I. Jaffee (Pergamon, Oxford) p. 222. &AMI, T, 1994, Mater. Sci. Eng. A179/A180, 17. ELLIOTT, R. 0. and D. A. Koss, 1981, J. Nucl. Mater. 97,339. EISELSTI~IN,C.E., 0.A. RWANO and O.D. SHERBY,1983, J. Mater. Sci. 18,483. EVANS, P. V., A. GARCIA-ESCQRIAL,P. E. DONOVAN AM) A. L. GREER,1987, Mater. Res. SOC. 57,239. FFCHT,H. and W.L. JOHNSON,1988, Nature 334,50. FECHT, H., P. J. DESL~and W. L. JOHNSON,1989, Phil. Mag. B 59, 577. FINNEY,J. L., 1983, in: Amorphous Metallic Alloys, ed. E E. Luborsky (.Butterworth, London) p. 42. FISCHER, M., D.E. POLKand B. C. GIESSEN,1978, in: Proc. 1st Int. Conf. on Rapid Solidification Processing (eds. R. Mehrabian, B. H. Kear and M. Cohen (Claitor, Baton Rouge), p. 140. FLINT,E. B. and K. S. SUSLICK,1991, Science 253, 1397. FOLLSTAEDT,D.M., 1991, in: Processing of Metals and Alloys, ed. R.W. Cahn, Vol. 15 of Materials Science and Technology, ais. R. W. Cahn, P. Haasen and E. J. Kramer (VCH, Weinheim) p. 247. GAG&, A. and C. ROY, 1982, High Temp. Technology, 1. 93. GARCIA-EWORIAL, A. and A L . G m , 1987, J. Mater. Sci. 22,4388. GARRONR, E. and L. BATIEZZATI,1985, Phil. Mag. B52, 1033. GASKELL,P. H., 1979, J. Non-Cryst. Solids 32, 207. GASKELL, P.H., 1983, in: Amorphous Materials: Modeling of Stmcture and Properties, ed. V. Vitek (The Metallurgical Society, Warrendale, PA) p. 47.
1822
R. E? Cahn and A. L.Greer
Ch. 19. Refs.
GASKRLL,P.H., 1985, in: Glass, Current Issues, eds. A. E Wright and J. Dupuy (Martinus Nijhoff, Dordrecht) p. 54. GASKELL,P. H. and D. J. SMITH,1980, J. Microscopy ll9,63. GERLING,R, EP. SCHIWIANSKY and R. WAGNER,1985, in: Rapidly Quenched Metals (V), eds. S. Steeb andH. Warlimont (North-Holland, Amsterdam) p. 1377; 1988, Acta Metall. 36, 575; 1989, Acta Metall. 36, 575; 1990, Int. J. Rapid Solidification 5, 137. GIBBS,M. R. J., J.E. E m s and J. A. LEAKE,1983, J. Mater. Sci. 18, 278. GIESSEN,B. C., M. MADHAVA, D. E. POLKand J. B. VANDERSANDE,1976, Mater. Sci. Eng. 23, 145. G-N, B.C., 1981, Proc. 4th Int. Conf. on Rapidly Quenched Metals. Eds. T. Masumoto and K Sumki (Japan Institute of Metals, Sendai) p. 213. GILLBN,A.G. and B. CANTOR,1985, Acta Metall. 33, 1813. GODFREY,A. W., and J. W. MARTIN,1995, Phil. Mag. A72,737. GRAHAM,Jr., C. D., 1959, in: Magnetic Properties of Metals and Alloys. Seminar volume. (Amer. SOC.Metals, Cleveland) p. 288. 1994, Scripta Metall. Mater. 31,601. G ~ h hL.,, I. EGRY, L. RATKEand D.M. HERLACH, GRANT,N. J., 1978, in: Rapid Solidification Processing, Principles and Technologies. Proc. Conf. on Rapid Solidification Processing, Reston, eds. R. Mehrabian et al. (Claitor, Baton Rouge, LA), p. 230. GRANT,P. S. and B. CANTOR,1990, Powder Metallurgy 33, 144. GRANT,P. S., W.T. KIM, E.P.BEWLAY and B. CANTOR, 1989, Scripta Metall. 23, 1651. GREER,A.L., 1982, Acta Metall. 30, 171. GREER, A.L., 1988, J. Less-Common Metals 140,327. GREER,A.L., 1991, Mater. Sci. Eng. A133, 16. GREER, A.L., 1993a, Nature 366,303. GREER, A. L., 1993b, in: Rapidly Solidified Alloys, ed. H. H. Liebennann (Dekker, New York), p. 269. GREER, A. L., 1994, Mater. Sci. Eng. A178, 113. GREER,A. L. and J. A. LEAKE, 1979, J. Non-Cryst. Solids 33, 291. GREER, A.L. and E SPAEPEN, 1981, in: Structure and Mobility in Molecular and Atomic Glasses, Annals N. Y. Acad. Sci. 371, 218. L.-U. AAEN-AND-, R. E. SOMEKH,J. B @ ~ G Eand R J. JANTING,1990, Mat. GREER,A. L., K. DYRBYB, Res. Soc. Symp. Proc. 371, 3. GREGAN,G. P. J., Y. D. DONGand M. G. SCOTT,1981, in: Proc. Int. Conf. on Metallic Glasses, Vol. 2 (Kultura, Budapest, Hungary) p. 197. GRIMES,R., 1990, in: Second Supplementary Volume to the Encyclopedia of Materials Science. and Engineering, ed. R. W. Cahn (Pergamon, Oxford), p. 667. GUNER,A., 1938, Nature 142, 569. G m , H. Q., H. KRONM~LLER, N. MO~ER and A. HOFMANN,1986, Scripta Metall. 20, 185. GURMAN,S. J., 1985, in: Proc. 5th Int. Conf. on Rapidly Quenched Metals, eds. S. Steeb and H. Warlimont (North-Holland, Amsterdam) p. 427. GUYOT, R and J. P. SIMON,1982, in: Dynamics of Ordering Processes in Condensed Matter, eds. S. Komura and H. Furukawa (Plenum, New York), p. 211. HAFNER,J., 1981, in: Glassy Metals I, ed. H.4. Giintherodt. (Springer, Berlin) p. 93. HAFNER,J., 1983, Phys. Rev. B 28, 1734. m, J., 1986, in: Amorphous Metals and Semiconductors. Eds. P. Haasen and R.I. Jaffee (Pergamon, Oxford) p. 151. HAGIWARA, M., A. INOUE and T. MASUMOTO, 1985, in: Rapidly Quenched Metals (V), eds. S. Steeb andH. Warlimont (North-Holland, Amsterdam) p. 1779. HARBUR,D.R., J. W. ANDERSON and W. J. MARAMAN, 1969, Trans. Met. Soc. A m 245, 1055. HASBGAWA, S . and R. RAY,1978, J. Appl. Phys. 49,4174. HASEGAWA, D. R. and L. E. TANNER,1977, J. Appl. Phys. 49, 3211. HASmhacurO, K.,1993, in: Rapidly Solidified Alloys, ed. H.H. Liebermann (Dekker, New York), p. 591. HAYZELDEN, C., 1983, D. Phil. Thesis, Sussex University.
Ch. 19, Refs.
Metastable states of alloys
1823
HAYZELDBN, C., and B. CANTOR, 1982, in: Proc. Int. Conf. on Solid+Solid Phase Transformations,Pittsburgh, 1981, eds. H.I. Aaronson, D. E. Laughlim, R.F. Sekerka and C. M. Wayman (The Metallurgical Society, Warrendale, PA), p. 1397. HAYZELDEN, C. and B. CANTOR,1984-85, I ~ J. L of Rapid Solidification 1,237. H~NNION, M., D. RONZAUD and P. GUYOT, 1982, Acta Metall. 30, 599. HERLACH.D.M., R.F. COCHRANE, I. EORY, H. J. FXHT and A.L. GREER, 1993, Int. Mater. Rev. 38, 273. HERMAN,H. and H. BHAT,1980, in: Synthesis and Properties of Metastable Phases, eds. E. S. Machlin and T. J. Rowland (The Metallurgical Society, Warrendale, PA) p. 115. G., 1991, Mater. Sci. Eng. A133, 1. HERZ~R, HIGASHI,K., T. MUKAI,S. TANIMURA,A. INOUE,T. MASUMOTO, K. KITA,K. OHTERAand J. NAGAHORA, 1992, Scripta Metall. Mater. 26, 191. HIGHMORE, R. J., 1990, J. de Physique France (Colloques) 51,C-37. HIGHMORE, R. J. and A. L. GREER,1989, Nature 339, 363. HILLAIRET, J., C. MARY,E. BALANZAT, J. HAMANT and M. MASSON[1983], Radiation Eff. 70, 131. HILLENBRAND, H.-G., E. HORNBOGEN and U. K ~ R 1982, , in: hoc. 4th Int. Conf. on Rapidly Quenched Metals, eds. T. Masumoto and K. Suzuki (Japan Institute of Metals, Sendai) p. 1369. HILZMGER, H.R., 1982, in: Proc. 4th Int. Conf. on Rapidly Quenched Metals, eds. T.Masumoto and K. Suzuki (Japan Inst. of Metals, Sendai), vol. 2, p. 791. HOLZER,J. C. and K. F. KELTON,1991, Acta Metall. Mater. 39, 1833. HONEYWMBE, R. W. K., 1978, in: Proc. 3rd Int Conf. on Rapidly Quenched Metals, Brighton, 1978, ed. B. Cantor (Institute of Metals, London), vol. 1, p. 73. HORI,S., S. SAJIand A. TAKEHARA, 1982, in: Proc. 4th Int. Cod. on Rapidly Quenched Metals, Sendai 1981, eds. T. Masumoto and K. Suzuki (Japan Inst. MetAls, Sendai), p. 1545. I., J. Om,K. PFAHLFXand W. UIILFERT, 1988, Mater. Sci. Eng. 97,409. HORV~TH, HOSHINO,S., H. A. LAITINENand G.B. HOFLUM),1986, J. Electrochem. Soc. l33,681. HOWE, L. M., 1995, in: Intermetallic Compounds: Principles and Practice, eds. J. H. Westbrook and R. L. Fleischer (Wiley, Chichester), Vol. 1, p. 791. H ~ R E. , and A. VAN DENBEUKEL,I987, Acta Metall. 35,2843. INOKUTI, Y. and B. CANTOR,1976, Scripta Metall. 10, 655. INOKUTI, Y.and B. CANTOR,1982, Acta Metall. 30,343. INOUE,A., 1995, in: lproc. Int. Symp. on Metastable, Mechanically Alloyed and Nanocrystalline Materials (1994), Materials Science Forum, 179/180,691. INOUE, A., and T. MASUMOTO,1980, Sci. Rep. Res. Inst. Tohoku Univ., 284 165. INOUE,A., M. HAGIWARA and T. MASUMOTO, 1982, J. Mater. Sci. 17,580. INOUE,A, M. HAGIWARA, T. MASUM~TO and H. S. CHEN,1983, Scripta Metall. 17,1205. INOUE,A., H.TOMIOKA and T. MASUMOTO, 1983b, Metall. Trans. 14A,1367. INOUE, A., K. OHTERA, A. P. TSAIand T. MASUMOTO, 1988a, Japan J. Appl. Phys. 27,L280. INOUE,A., K. O~'rm.4.and 'HI MASUMOTO, 1988b, Japan J. Appl. Phys. 27,L736. INOUE,A., 'F. ZHANGand T. MASUMOTO,1989, Mater. Trans. Japan Inst. Met. 30, 965. INOUE, A., N. MATSUMOTO and T. MASUMOTO, 1990, Mater. Trans. Japan Inst. Met. 31,493. INOUE, A., Y. HORIO,Y.H.KIM and T. MASUMOTO, 1992, Mater. Trans. JIM 33,669. INOUE, A., T. NAKAMURA, N. NISHIYAMA, T. SUGITAand T. MASUMOTO,1993%Key. Eng. Mater. 81-83,147. JACOBS,M.H., A.G. D m m and M. J. STOWELL,1974, J. Mater. Sci. 9,1631. JANKOWSKI, A. E, P. SANDOVAL and J. P. HAYES, 1995, Nonstruct. Mat. 5,497. JAYARAMAN, A., 1980, in: Synthesis and propexties of metastable phases, eds. E. S. Machlin and T. J. Rowland (Met. SOC.AIME,Warrendale, PA) p. 75. JOHNSON, W.L., 1981, in: Glassy Metals I, eds. H.-J. GUNTHERODTand H.BECK(Springer, Berlin) p. 191. JOHNSON, W.L., 1982, in: Metastable Materials Formation by Ion Implantation, Proc. Mater. Res. Soc. Annual Meeting, Boston, 1981, eds. S.T. Picraux and W. J. Choyke (North-Holland, Amsterdam) p. 183. JOHNSON, W. L., 1986, h g r . Mater. Sci. 30, 81. JOHNSON, W.L., S. J. POONand P. D w z , 1975, Phys. Rev. B 11, 150. JONES,H., 1981, in: Ultrarapid Quenching of Liquid Alloys, Treatise on Materials Science and Technology, vol. 20, ed. H. Herman, (Academic, New York), p. 1.
1824
R. l+! Cahn and A. L.. Greer
Ch. 19, Refs.
JONES,H., 1982, Rapid Solidification of Metals and Alloys (Institution of Metallurgists, London; now Institute of Materials). JONES, H., 1983,Scripta Metall. 17,97. KAN, T., Y. ITOand H. SEIIMANAKA,1982,J. Magn. Magn Mater. 26,127. KASAK,A. and E. J. DULIS, 1978, Powder Metallurgy 21, 114. KATTAMIS, T. Z.and M. C. FLEMINGS, 1970,Metall. Trans. 1, 1449. KATTAMIS, T.Z.,W. E. BROWERand R. MEHRABIAN, 1973,J. Cryst. Growth 19,229. KAWASHIMA, A., K.ASAMI,T. SATOand K. HASHIMOTO, 1985,in: Rapidly Quenched Metals 0,eds. S.Steeb and H. Warliimont (North-Holland, Amsterdam) p- 1671. KEAR,B. H., J. N. MAYER,J. M. POATEand P. R. STRUTT,1981,in: Metallurgical Treatises, eds. J. K. Tien and J.F. Elliott (TMs-AIM& Warrendale, PA), p. 321. KELLY,A. and R.B. NICHOLSON, 1963, Progr. Mater. Sei. 10, 149. KELLY, T.F. and J.B. VANDERSANDE,1980,Rapid Solidification Processing: Principles and Technologies 11. Proc.(2nd) Conf. on Rapid SolidificationProcessing, Reston, 1979,eds. R. Mehrabian etal. (Claitor, Baton Rouge, LA, p. 100. KELTON,K. F., rgy, Solid State Phys. 45,75. KELTON, K.F. and F. SPAEPEN,1984,Phys. Rev. B30,5516. KELTON,K. F. and A. L. GREER,1986,J. Non-Cryst. Solids 79,295. KLIEK,M. M., D. W. PALMERand B. CANTOR,1986,Acta Metall. 34, 1455. KIM,S.G., INOUE,A. and MASUMOTO, T., 1990,Mater. Trans. Japan Inst. Met. 31,929. KIM, Y.H., A. INOUEand T. MASUMOTO, 1990,Mater. Trans. Japan Inst. Met. 31,747. KIM, Y.H., A. INOUE and T. MASUMOTO, 1991,Mater. Trans. Japan Inst. Met. 32, 334. &AMI, C.S. andP.R. SAHM,1986,ActaMetall. 34,2129. KIMURA,H. and T. MASIJMOTO, 1983, in: Amorphous Metallic Alloys, ed. F.E. Luborsky (Butterworth, London) p. 187. KISHORE,K. and H. K. SHOBHA,1994,J. Chem. Phys. 101,7037. KOCH,C. C., 1991,in: Processing of Metals and Alloys, ed. R. W.Cahn, Vol. 15 of Materials Science and Technology, eds. R. W. Cahn, P. Haasen and E. J. Kramer (VCH, Weinheim) p. 193. KOEBRUGGE, G. W. and A. VANDENBEUKEL,1988,Scripta Metall. 22,589. KOMATSU, T., K. MATIJSITAand R. YOKOTA,1983,J. Non-Cryst. Sol. 72,279. KOsm, U., 1984,in: Amorphous Metals and Non-Equilibrium Processing, ed. M. von Allmen (Editions de physique, Paris) p. 175;Z. Metallkde. 75,691. K~STER, W. and K. KAM, 1938,Z. Metallkde. 30,320. KOSTER,U. and U. HEXOLD,1981,in: Glassy Metals I, eds. H.-J. Giintherodt and H. Beck (Springer, Berlin) p. 225. U. and SCH~~EMANN, 1993,in: Rapidly Solidified Alloys, ed. H. H. Liekrmann (Dekker, New York), K~STER, p. 303. KOSTER, u., U. SCHONEMANN, M. BLANK-BEWERSDORFF, S. BRAUER, M. SurrON and G.B. STEPHENSON, 1991,Mater. Sci. Eng. A133,611. KRAMER,J., 1934,Ann. Phys. 19,37. KRAMER,J., 1937,Z.Phys. 106,675. KRISHNANAND,K.D. and R. W. CAHN,1975,Scripta Metall. 9, 1259. KRISHNANANDand CAHN, 1976,in: Proc. 2nd Int. Conf. on Rapidly Quenched Metals, Cambridge, MA, 1975, eds. N. J. Grant and VB.C. Giessen, part 1 (MlT Press, Cambridge, MA) p. 67. Ktn, H.-W. and D. TURNBULL, 1985,Appl. Phys. Lett. 47,796. KUI, H.-W., A.L. GRFERand D. TURNBULL, 1984,Appl. Phys. Lett. 45,615. KURSUMOVI~, A.,M. G. SCOTT,E. GIRTand R. W. CAHN,1980,Scripta Metall. 14 1303. LAMPARTER, P.,W.SPERL,S. STEEBand J. B L ~ Y1982, , Z. Naturforsch. 37a, 1223. LASHMORE, D. S. and J. F. WBINROTH. 1982,Plating and Surface Finishing (August) p. 72. LASHMORE, D. S., L. H. BENNETT, H.E. SCHONE, P. GUSTAFSON and R.E. WATSON,1982,Phys. Rev. Lett. 48,
1760. LEAKE,J. A., E. WOLDTand J.E. E v m , 1988,Mater. Sci. Eng. 97,469. LEBO,M. and N. J. GRANT,1974,Metall. Trans. 5, 1547.
Ch. 19, Refs.
Metastable states of alloys
1825
LEVI, C.G. and R. MEHRABIAN,1982, Metall. Trans. 13A, 13. LEWIS,B.G. and H.A. DAVIET,1976, Mater. Sci. Eng. 23, 179. LI,Y., H. J O and~H.A. DAVIES,1992, Scripta Metal. et Mater. 26,1371. LIEBERMANN, H.H., R.E. MAXWELL,R. W. SMASHEY and J.L. WALTER,1983, Metall. Trans. 14A, 1817. LIN, C.-J. and E SPAEPEN,1982, Appl. Phys. Lett. 41,721. LIN, C.-J. and E SPAFPEN, 1984, Mat. Res. SOC. Symp. Proc. 28, 75. LIVESFY,A.K. and P.H.GASKEU, 1982, in: Proc. 4th Int. Conf. on Rapidly Quenched Metals, Sendai, 1981, eds. T. Masumoto and K. Suzuki (Japan Inst. of Metals, Sendai) p. 335. LUBORSKY,EE., 1977, in: Amorphous Metallic Alloys, ed. E E. Luborsky (Butterworth, London) p. 360. LUBORSKY, F. E., 1980, in: Ferromagnetic Materials, ed. E. P. Wohlfahrt (Butterworth, London), vol. 1, p. 451. LUZZI, D.E,, H. MOR&H. FUJITAand M. MESHII,1984, Scripta Metall. 18,957. LUZZI, D.E., H. MOR&H. FUJITAand M. MESHII, 1985, Scripta Metall. 19,897. LUZZI,D. E., H. MORI,H. FUJITA and M. MESHII,1986, Acta Metall. 34, 629. MACEDO,P.B. and A. NAPOLITANO,1967, J. Res. Nat.Bur. Standards 71A, 231. MADDOX,J., 1993, Nmre 361, 397. MADER,S., 1965, J. Vac. Sci. Technol. 2, 35. MADER,S., A. S. NOWICK and H. WIDMER,1967, Acta Metall. 15, 203, 215. MAITREPJERRE, P. L.,1970, J. Appl. Phys. 41, 498. MAKINO,A., INOIJE,A. and T. MASUMOTO, 1990, Mater. Trans. Japan Inst. Met. 31, 884. MARINGER, R. E. and C. E. MOBLEY,1974, J. Vac. Sci. Technol. 11, 1067. MARTELLI, S., G. MAZZONE, A. MONTONE and M. VITTORIANTISARI, 1990, J. Phys. France, 51, 241. MARTIN,J. W., 1968, Precipitation Hardening (Pergamon, Oxford). MARTIN,J. W., B. CANTORand R. D. DOHERTY,1996, chapter 3 in: Stability of Microstructure in Metallic Systems (2nd edition) (Cambridge University Press). MASSALSKI, T.B., 1986, in: Amorphous Metals and Semiconductors.Eds. P. Haasen and R I. Jaffee (Pergamon, Oxford) p. 178. T.,1994, Mater. Sci. Eng. A179/A180, 8. (Proceedings of 8th Int. Conf. on Rapidly Quenched and MASUMOTO, Metastable Materials, Sendai, 1993). hlASmoTO, Y., A. INOUE, A. KAWASIMA,K. HASIMOTO, A.P. TSAIand T. MASUMOTO, 1986, J. Non-Cryst. Solids 86, 121. MAWELLA,K. J. A., R. W. K. HONEYCOMBE and P. R. HOWELL,1982, J. Mater. Sci. 17, 2850. MEFIRABIAN, R., 1982. Int. Metall. Rev. 27, 185. MENDOZA-ZELIS, L., L. THO&, L. BROSSARD, J. CHAUMONT, Q. BOLAS and H. BERNAS,1982, Phys. Rev. B26, 1306. MENG,W.J.,C.W. NIEH,E. MA, B. FULTZand W.L. JOHNSON,1988, Mater. Sci. Eng. 97, 87. MESCHTER,P. J., R. J. LEDERICK and J. E. O'NEAL 1988, in: Proc. Int. Conf. on PM Aerospace Materials, A Metal Powder Report Conf., Luzern, 1987 (MPIF, Amer. Powder Met. Inst., Princeton, NJ) p. 27.1. MIDSON,S. P. and H. JONES,1982, in: Proc. 4th Int. Conf. on Rapidly Quenched Metals, Sendai, 1981, eds. T. Masumoto and K. Suzuki (Japan Inst. of Metals, Sendai) p. 1539. MOLN~R,A., C. V. SMITH,and M. BARTOK,1989, Advances in Catalysis 36, 329. MORDKE,B. L., 1986, ed., Laser Treatment of Materials (DGM Infonnationsgesellschaft Verlag, Oberursel, -Y). MORD~KE, B.L., 1991, in: Processing of Metals and Alloys, ed. R. W. Cahn, Vol. 15 of Materials Science and Technology, eds. R W.Cahn, P. Haasen and E. J. Kramer (VCH, Weinheim) p. 111. MORI, H. and H. FIJJITA,1983, Jap. J. Appl. Physics 21, L484. MOM, H., FUJITA. H. and M. FUJITA,1983, Jap. J. Appl. Phys. 22, L94. Mom, H., H. FUJITA, M. TENDO and M. FUJITA,1984, Scripta Metall. 18,783. MORRIS, D.G., 1980, in: Rapid SolidificationProcessing: Principles and Technologies 11.Proc. (2nd) Conf. on Rapid SolidificationProcessing, Reston, 1979, eds. R. Mehrabian et al. (Claitor's, Baton Rouge, LA, p. 372. MOTORIN,V.I., 1983, phys. stat. sol. (a) 80, 447. MOTTA,A. T. and C. LEMAIGNAN, 1992, in: Ordering and Disordering in Alloys, ed. A. R. Yavari (Elsevier Applied Science, London) p. 255.
1826
RN!CahnandA.LGreer
Ch. 19, Refs.
MOTTERN, J. W. and W. J. P R I V ~eds., , Spinning Wire from Molten Metal (Amer. Inst. Chem. Engrs., New York), A.1.Ch.E. Symp. Series 74, no. 180. MULDER,A.L., S. VANDERZWAAGand A. VANDENBEUKEL,1983, Scripta Metall. 17, 1399. MURTY,Y. V. and R. P. I. ADLER, 1982, J. Mater. Sci. 17, 1945. NACHTRIEB, N.H., 1976, Ber. Bunsengesellschafi Phys. Chem. 80,678. NAGEL,S. R. and J. TAW, 1977, Solid State Comm. 22, 129. NEUHAUSER, H, 1978, in: Proc. 3rd Int. Conf. on Rapidly Quenched Metals, Brighton, 1978, ed. B. Cantor (Institute of Metals [now Materials], London), vol. 2, p. 431. Nmn, Y., T.KAI, M. TACHI, T.JSHIDAIRA and E. YAJIMA,1986, Scripta Metall. 20, 1099. O ~ I N OT., , 1986, French patent 2 569 143. OHNAKA, I., 1985, Int. J. Rapid Solidification 1,219. OHNAKA, I., 1988, in: Supplementary Volume 1 to the Encyclopedia of Materials Science and Engineering, ed. R. W.Cahn (Pergamon, Oxford), p. 584. OHNAKA,L and T. FTJKUSAKO, 1978, J. Japan Inst. Metals 42,415. OHTERA, K., A INOUE, T. TERABAYASHI, H. NAGAHAMA and T. MASUMOTO, 1992, Mater. Trans. Japn Insty. Met. 0 33,775. OK, H.N. and A.H. MORRISH,1981, Phys. Rev. B23,1835. OSTWALD,W., 1893, Lehrbuch der allgemeinen Chemie, 2 (part l), p. 517. PANNWD, P., I. BAKONYIand R. HASEGAWA,1983, Phys. Rev. B 28,2374. PATTERSON, J., A. L. GREER,J. A. LEAKEand D. R. W. Jom, 1978, in: Proc. 3rd Int. Conf. on Rapidly Quenched Metals, Brighton, 1978, ed. B. Cantor (Institute of Metals [now Materials], London), vol. 2, p. 293. PEERCY,P. S., J. Y. TSAOand M.O. THOMPSON, 1990, I. Mater. Res. 5, 1463. PEKER,A, and W.L. JOHNSON, 1993, Appl. Phys. Lett. 63, 2342. PEREL,J., J. E MAHONEY, B. E. KALENSHER and P. D w z , (1981), in: Advances in Metal Processing, eds. J. J. Burke, R Mehrabian and V. Weiss (Plenum, New York), p. 79. F%REPFZKO, J. H., 1984, Mater. Sci. Eng. 65, 125. PEREPEZKO, J. H. and I. E. ANDERSON,in: Synthesis and Properties of Metastable Phases, eds. E. S. Machlin and T. J. Rowland (Met. Soc. AIME,Warrendale, PA) p. 31. PEREPEZKO, J.H. and J. S. SMITH 1981, J. Non-Cryst. Solids 44,65. PHARR,G. M., W. C. OLIVER,R E COOK, P. D. KIRCHNER, M. C. KROLL.T. R. DINGERand D. R. CLARKE, 1992, J. Mater. Res. 7, 961. FTLLER, J., and P. HAASEN,1982, Acta Metall. 30, 1. F’IROUZ, P., R. CHAM, U. DAHMEN and K. H. WESTMACOTT,1990, Acta Metall. et Mater. 38, 313, 323, 329. POATE,J.M., P. S. PEERCY and M. 0.THOMPSON,1987, Mat. Res. Soc. Symp. Proc. 57, 465. POLK,D.E. and H.S. CHEN,1974, J. Non-Cryst. Solids, 15, 165. POLK, D.E. and B.C. GIESSEN,1978, in: Metallic Glasses. Eds. J. J. Gilman and H. J. Leamy (Amer. Soc. Metals, Metals Park, Ohio) p. 1. POLMFAR,I. J., 1995, Light Alloys, 3rd edition (Edward Arnold, London) POND,R. and R. MADDIN,1969, Mater. Sci. Eng. 23,87. P~NYATOVSKY, E. G. and 0.I. BARKALOV,1 9 1 , Mater. Sci. Eng. A133,726. POSGAY,G., E J. KED-, B. ALBERT,S. KrSS and I.Z. HARANGUZQ,1985, in: b. 5th Int. Conf. on Rapidly Quenched Metals, eds. S. Steeb and H. Warlimont, (North-Holland, Amsterdam)) p. 723. -ON, G.D., 1938, Natnre 142,570. RAMACHAXDRARAO, 1980, Z. Metallkde. 71, 172. RAMACHANDRARAO,P., 1984, in: Metallic Glasses - Production, Properties and Applications. Ed. T. R. Anantharaman (Trans Tech, Aedennannsdorf, Switzerland) p. 31. RAUSCHEN&ACH, B. and K.HOHMUTH,1982, phys. stat. sol. (a) 72, 667. RAY,R., 1981, J. Mater. Sci. 16, 1924,2927. RAY, R., B. C. GIESSEN and N. J. GRANT, 1968, Scripta Metall. 2, 359. RAY, R. and E.Musso, 1976, US Patent 3 981 722. RAY, R., R. HASEGAWA,C.P. CHOUand L. A. DAVIS,1977, Scripta Metall. 11,973. RAYMENT, J. J., and B. CANTOR1978, Metat Sci. 12, 156.
Ch. 19, Refs.
Metastable states of alloys
1827
RAYMENT,J.J., and B. CANTOR,1981, Metall. Trans. l2A, 1557. RECHTIN, M.D., J.B. Vander Sande and P.M. BALDO,1978, Scripta Metall. 12,639. REDDY,G. S. and I.A. SEKHAR,1989, in: Principles of Solidification and Materials Processing, eds. R. Trivedi, J. A. Sekhar and J. Mazumdar, (Oxford & IBH, New Delhi) Vol. 1, p. 83. ROBBRTS,W.T., 1993, in: Third Supplementary Volume to the Encyclopedia of Materials Science and Engineering, ed. R. W. Cahn (Pergamon, Oxford), p. 1616. ROORDA,S., 1990, Doctoral thesis, University of Utrecht, Netherlands. ROOS,J. R., J.-P. CELISand M. de BONTE,1991, in: Processing of Metals and Alloys, ed. R. W. Cahn, Vd. 15 of Materials Science and Technology, eds. R. W. Cahn, P. Haasen and E. J. Kramer (VCH,Weinheim), p. 481. RUHL,R C., B. C.GIESSEN,M. COW and N. J. GRANT,1967, Acta Metall. 15, 1693. RUSSELL,K.C., 1985, in: Proc. Int. Seminar on Solute-Defect Interaction, Kingston, Ontario, August 1985, eds. G. Saimote, G. Kidson and G. Purdy (Pergamon, Oxford) p. 317. SADOC, J.F. and J. DIXMIER, 1976, Mater. Sci. Eng. 23, 187. SAPAI,S. and H. HERMAN, 1981, in: Ultrarapid Quenching of Liquid Alloys, Treatise on Materials Science and Technology, vol. 20, ed. H. Herman, (Academic, New York), p. 183. SAHIN, E. and H. JONES, 1978, in: Proc. 3rd Int. Conf. on Rapidly Quenched Metals, Brighton, 1978, ed. B. Cantor (Institute of Metals, London), vol. 1, p. 138. SAKATA, M., N. COWLAM and H.A. DAVIES,1981, J. Phys. F 11, L157. SANKARAN, K. K. and N. J. GRANT, 1980, Mater. Sci. Eng. 44, 213. SARE, I. R. and R. W. K.. HONEYCOMBE, 1978, J. Mater. Sci. 13, 1991. SAWNDERS, N. and A. €? MIODOWNIK, 1986, J. Mater. Res. 1, 38. SCHLEIP, E., R. WILLNECKBR, D.M. HERLACHand G.P. G~RLER, 1988, Mater. Sci. Eng. 98,39. SCHOLTE,P.M.L. O., M. TEGGE,F. VAN DER WOUDE,K.H. I. BUSCHOW AM) I. VINCZE, 1985, in: Proc. 5th Int. Conf. on Rapidly Quenched Metals, eds. S. Steeb and H. Warlimont (North-Holland, Amsterdam) p. 541. SCHR~DER, H., K.SAMWER and U. K~STER, 1985, Phys. Rev. Lett. 54, 19. SCHUBBRT,E. and H. W. BERGMANN, 1993, in: Rapidly Solidified Alloys, ed. H. H. Liebermann (Marcel Dekker, New York),p. 195. SCHULTZ,L., 1987, Mat. Res. SOC.Symp. Proc. 80, 97. SCHULTZ, L., 1989, in: New Materials by Mechanical Alloying Techniques, eds. E. Ant and L.Schultz, (DGM Informationsgeselhhaft, Oberursel, Germany) p. 53. SCHULTZ, L. and E. HELLSTERN,1987, Mat. Res. Soc. Symp. Proc. 80, 3. SCHUMACHER, P. and A. L. GREER, 1994, Mater. Sci. Eng. A1811A182, 1335. SWARZ, M., A. KARMA, K. ECKLERand D.M. HERLACH, 1994, Phys. Rev. Lett. 73, 1380 (correction, ibid. 2940). SCHWARZ, R.B. and W.L. JOHNSON, 1983, Phys. Rev. Lett. 51,415. SCHWARZ, R.B. and C.C. KOCH,1983, Appl. Phys. Lett. 49, 146. SCHWARZ, R.B. and W.L. JOHNSON (eds.), 1988, Proc. Conf. on Solid-state Amorph~ngReactions, J. LessCommon Metals 140. SCOTT, G.D. and D.L. MADER, 1964, Nature 201,382. SCOTT,M.G.and J. A. LEAKE,1975, Acta Metall. 23, 503. SCOTT,M. G. and A. Ku~SvMoVrt,1982, Acta Metall. 30, 853. SEKHAR,J.A. and T. RAJASEKHARAN, 1986, Nature 320, 153. SHINGU,P.H. and K. N. ,-sI 1993, in: Rapidly Solidified Alloys, ed. H. H. Liebermann (Marcel Dekker, New York), p. 103. SHOHOJI, N., H. A. DAW, H. JONESand D. H. WARRTNGTON, 1982, in: Proc. 4th Int. Cod. on Rapidly Quenched Metals, eds. T. Masumoto and K. Suzuki (Japan Inst. of Metals, Sendai) p. 1529. SILCOCK,J. M., T. J. HEALand H. K. HARDY,1953-54, J. Inst. Metals 82, 239. SIMPSON, A. W. and P. A. HODKINSON, 1972, Nature 237, 320. SINHA,A. K.,€3. C. GIESEN and D. E. POLK,1976, in: Treatise on Solid-state Chemistry, ed. N. B. Hannay (Plenum, New York) vol. 3, p. 1. SINNING, H.-R., L. LEONARDSSON and R. W. CAHN,1985, Int. J. of Rapid Solidification 1, 175.
1828
R. U?Cahn and A. L. Gwer
Ch. 19, Refs.
SMITH,C. H., 1993, in: Rapidly Solidified Alloys, ed. H. H. Liebermann (Marcel Dekker, New York), p. 617. S m , T.R., R. W. ARMSTRONG,P. M. HAZZLEDINE,R. A. m u m and C. S. PANDE,1995, in: Mat. Res. SOC.Syp. Proc. 362,31. SMUGERESKY, J. E., 1982, Metall. Trans. 13A, 1535. So-, E,G.DUDDEKand B. -EL, 1978, Z. Metallkde. 69, 587. SOMMER, E,T. LANGand B. PREDEL,1987, Z. Metallkde. 78,596. SPAEPBN, E,1977, Acta Metall. 25,407. SPAEPEN, E, 1994, Mater. Sci. Eng. A179/A180, 81. SPAEPEN, E and G.S. CARGILL,1985, in: Proc. 5th Int. Conf. on Rapidly Quenched Metals, eds. S. Steeb and H. Warlimont (NorthHolland, Amsterdam) p. 581. E and A. I. TAUB,1983, in: Amorphous Metallic Alloys, ed. E E. Luborsky (Butterworth, London) SPAEPEN, p. 231. SPAEPEN,E,S. S. TSAOand T. W. WU, 1986, in: Amorphous Metals and Semiconductors, eds. P. Haasen and R. I. Jaffee (Pergamon, Oxford) p. 365. SRIVASTAVA, P. K., B. C. GIESSENand N. J. GRANT,1972, Metall. Trans. 3,977. STIFF'L~, S.R, P.V. EVANSand A.L. GREER,1992, Acta Metall. 40, 1617. STR~M-OLSEN, J. O., R. BR~NING, Z ALTOUNIAN and D.H. RYAN,1988, J. Less-Common Metals 145,327. SUGA,M., J. L. Goss, G.R. OLSENand J. B. VAN DER SANDE,1980, in: Rapid Solidification Processing: Principles and Technologies II. Proc. (2nd) Conf. on Rapid Solidification Processing, Reston, 1979, eds. R. Mehrabmn et al. (Claitor, Baton Rouge, LA, p. 364. SURYANARAYANA, C., 1980, Rapidly Quenched Metals - A Bibliography (Plenum, New York). SURYANARAYANA, C., 1991, in: Processing of Metals and Alloys, ed. R.W. Cahn, Vol. 15 of Materials Science and Technology, eds. R. W. Cahn, P.Haasen and E. J. Kramer (VCH, Weinheim), p. 57. SUSLICK, K. S., 1993, in: Supplementary Volume 3 of the Encyclopedia of Materials Science and Engineering, ed. R. W. Cahn (Pergamon, Oxford), p. 2094. SUSLICK,K. S., 1995, Mater. Research Bull. 20 (April), 29. SUSLICK,K.S., S. B. CHOE,A. A. CICHOWLAS and M. W. G ~ S T A F 1991, F , Nature 353,414. SUZUKI, K., 1983, in: Amorphous Metallic Alloys, ed. E E. Luborsky (Butterworths, London), p. 74. SUZUKI,Y., J. HAIMOVICH and T. EOAMI, 1987, Phys. Rev. B35,2162. TAKAYAMA, S., 1976, J. Mater. Sci. 11, 164. TANNER,L. and R. RAY, 1980, Scripta Metall. 28, 633. TARN~CZI, T., A. LOVASand C. KOPASZ,1988, Mater. Sci. Eng. 97,509. TENDLER, R.H. de, 1986, J. Mater. Sci. 21, 64. THLRSFIELD, G. and M. J. STOWELL, 1974, J. Mater. Sci. 9, 1644. TOLOUI,B., A. KURSUMOVIC and R. W. CAHN,1985, Scripta Metall. 19, 947. TOMIDA,T. and T. EGAMI,1991, Mater. Sci. Eng. A133, 931. TRACEY, V. A. and C. P. CUTLER, 1981, Powder Metallurgy 24, 37. TSAI,R.-Y. and S.-T.Wu, 1990, J. Electrochem. Soc. 137,2803. TURNBULL, D., 1952, J. Chem. Phys. 20,411. TURNBULL, D., 1976, J. de Physique 37, C4-1. TURNBULL, D., 1981, Metall. Trans. lZA, 695. TURNBULL, D. and H. N. TREAFTIS, 1957, Acta Metall. Mater. 5, 534. TURNBULL,D., H. S. ROSENBAUM and H. N. T R E A ~ I S1960, , Acta Metall. 8, 277. VAN DENBEUKEL,A. and J. SIFISMA, 1990, Acta Metall. et Mater. 38, 383. VANDENBEUKEL,A. and J. SIET~MA, 1994, Mater. Sci. Eng. A179/A180, 86. (Proc.8th Int. Conf. on Rapidly Quenched and Metastable Materials, Sendai, 1993). VANROOYEN, M., J. A. SINTEMAARTENSDIJK, and E. J. M ~ M E I J E R1988, , Metall. Trans. 19A, 2433. VINEBERG, E. J., E. K. OHRINER,E. P. WHELANand G. E. STAPLEFORD, 1985, in: Rapidly Solidified Crystalline Alloys, eds. S. K. Das, B. H. K e a and C.M. Adam (The Metallurgical Society, Warrendale, PA) p. 306. VOLKERT,C.A. and E SPAEPEN, 1989, Acta Metall. 37, 1355. VON HEIMENDAHL, L., 1975, J. Phys. F 5, L141. VREDENBURG, A.M., J. E M. WBSTENDORP,E W. SARIS,N.M. van der PERSand Th. D. KEIJSER, 1986, J. Mater. Res. 1. 775.
Ch. 19, Refs.
Metastable states of alloys
1829
WAGNER,C. N. J., 1983, in: Amorphous Metallic Alloys, ed. E E. Luborsky (Butterworth, London) p. 58. WAGNER,C. N. J., 1986, in: Amorphous Metals and Semiconductors. Eds. P. Haasen and R. I. Jaffee (Pergamon, Oxford) p. 54. W a r n , J.L., F. BACONand E E. LUBORSKY, 1976, Mater. Sci. Eng. 24, 239. WARBURTON, W. K. and D. -VU, 1975, in : Diffusion in Solids: Recent Developments, eds. A. S. Nowick and J. J. Burton (Academic Press, New York) p. 171. WARLIMONT, H., 1985, in: Rapidly Quenched Metals (Proc. 5th Conf.), S. Steeb, H. Warlimont, eds. (NorthHolland, Amsterdam), p. 1599. 1989, Acta Metall. 37,3409. WEI, Gao and B. CANTOR, WHANG,S.H.,1983, Mater. Sci. Eng. 57, 87. WHALLEY, E., 1988, J. Less-Common Metals, 140, 361. WHITTLE, G.L., P. WELLS,S.J. CAMPBELL, A. CALKAAND A.M. STEWART, 1985, in: Proc. 5th Int. Conf. on Rapidly Quenched Metals, eds. S. Steeb and H. Warlimont (North-Holland, Amsterdam) p. 545. WILLIAXIS, RS.and T. EOAMI, 1976, IEEE Trans. Mag. MAG12,927. WOLDT,E., 1988, J. Mater. Sci. 23, 4383. WOLF, D., P.R. O w o m , S. YIP, J. E LUTSKOand M. KLUGE,1990, J. Mater. Res. 5, 286. WONG,J., 1981, in: Glassy Metals I, eds. H. J. Giintherodt and H. Beck, (Springer, Berlin). WOOD, J. V. and R. W. K. HONEYCOMBE,1981, in: Ultrarapid Quenching of Liquid Alloys. Treatise on Materials Science and Technology, vol. 20,ed. H.Herman (Academic, New York) p. 117. WU, T. W. and E SPAEPEN, 1986, in: Mechanical Behavior of Rapidly Solidified Material, eds. S. M. L.Sastry and B. A. Macdonald (TMS, Warrendale, PA) p. 293. YA~USAKI, T, M. TAKAHASHI and Y. OGINO,1985, in: Rapidly Quenched Metals (V), eds. S. Steeb and H. Warlimont (North-Ilolland, Amsterdam) p. 1381. YAVARI, A.R., 1986, J. Mater. Res. 1, 746. YAVARI, A.R., P. HICTERand P. DSsRfi, 1983, J. Chim. Physique 79,572. YEARIM, R. and D. SHECHTI~AN, 1982, Metall. Trans. 13A,1891. Ym, X. L., K. S A P ~ W E and R W. L. JOHNSON,1983, Appl. Phys. Lett.42, 242. YERMAKOV, A. Ye., V. A. BARINOVand Ye. Ye. YURCHIKOV,1981, Phys. Met. Metall. (translation of Fiz. Met. Met.) 54, 935. YOSHIZAWA, Y., S. O G w and K. YAMAUCHI, 1988, J. Appl. Phys. 64,6044. ZACHARIASEN, W. H., 1932, J. Am. Ceram. Soc.54,3841. ZOUGMOR~~, E, J. C. LASAUNIASand 0.B~THOUX, 1988, J. Less-Common Metals 145, 367.
Further reading Numerous international conferences have been devoted to the field of rapidly solidified alloys since 1970, recently at 3-yearly intervals; initially the conference title referred to “rapidly quenched metals”, latterly to “rapidly quenched and metastable materials”. The proceedings of most of these are cited in the Bibliography, in relation to specific papers, as are other important proceedings such as those of the Reston Conferences on Rapid Solidification Processing and other ‘one-off‘ meetings. Most of these will not be listed separately hem, but it should be recognized that these proceedings are prime sources in this broad field. A few selected overviews are listed below. Others are recommended in the body of the chapter. T. R. Anantharaman and C. Suryanarayana, Review: A decade of quenching from the melt, J. Mater. Sci. 6 (1971) 1111. R. W. Cahn, Metallic Glasses, Contemp. Phys. 21 (1980) 43. F.E. Luborsky, in: Ferromagnetic Materials, vol. 1, ed. P.E. Wohlfahrt (North-Holland, Amsterdam. 1980). ch. 6, p. 451. (Detailed quantitative data on magnetic properties, densities, etc., of many metallic glasses are included). H. Herman, ed., Ultrarapid Quenching of Liquid Alloys, Treatise on Materials Science and Technology, vol. 20 (Academic Press, New York, 1981). (The chapter by Li, on mechanical properties, tabulates T8, T,, densities, elastic moduli, etc., of many metallic glasses).
1830
R W Cahn and A. L Greer
Ch. 19, Refs.
H. Jones, Rapid Solidification of Metals and Alloys (Monograph No. 8, The Institution of Metallurgists, London; now the Institute of Materials; 1982). S.R. Nagel, Metallic Glasses, Adv. Chem. Phys. 51 (1982) 227. R. Mehrabian, Rapid Solidification, Int. Metals Review 27 (1982) 185. E E. Luborsky, ed., Amorphous Metallic Alloys (Butterworth, London, 1983). The most complete survey of metallic glasses. E Spaepen and D. Turnbull, Metallic Glasses, Ann. Rev. Phys. Chem. 35 (1984) 241. S. K. Das, B.H. Kear and C.M. Adam (eds.), Rapidly Solidified Crystalline Alloys (The Metallurgical Society, Warrendale, PA, 1985). S. M. L. Sastry and B. A. MacDonald (eds.), Mechanical Behavior of Rapidly Solidified Materials (The Metallurgical Society, Warrendale, PA, 1985). T. R. Anantharaman and C. Suryanarayana, Rapidly Solidified Metals (Trans Tech. Publications, Switzerland, 1987). R. W. Cahn, Metallic Glasses,in: Glasses and Amorphous Materials, ed. I. zanycki, Vol. 9 of Materials Science & Technology (VCH,Weinheim, Germany, 1991), p. 493. C. Suryanarayana, Rapid Solidification, in: Processing of Metals and Alloys, ed. R W. Cahn (Vol. 15 of Materials Science and Technology, ed. R. W. Cahn, P. Haasen and E. J. Kramer (VCH, Weinheim, G e m n y ) p. 57. H. H. Liebermann (ed.), Rapidly Solidified Alloys (Wiley, New York, 1993). (Including, in particular, chapters by R. W. Cahn on history, A. L. Greer on relaxation and diffusion, T.Egami on metallic glass structure, and U. Kijster and U. Schiinernann on phase transformations in RSP alloys.) A. L. Greer, Amorphous phases, in: Intermetallic Compounds, ed. J. H. Westhrook and R. L. Fleischer (Wiley, New York, 1995), p. 731. Also: W. L. Johnson, Metastable Phases, ibid., p. 687. A. L. Greer, Metallic Glasses, Science 267, (1995) 1947. A.L. Oreer, Crystallization of Amorphous Alloys, Metall. Trans. 26A (1995) (in press). H. Bakker, G.F. Zhou and H. Yang, Mechanically Driven Disorder and Phase Transformations in Alloys, Progr. Mat. Sci. 39 (1995) 159. J. W. Martin, B. Cantor and R. D. Doherty, Stability of Microstructure in Metallic Systems (2nd edition) (Cambridge University Press, 1996). Classified bibliographies of papers concerned with rapid solidification(covering both crystallineand amorphous materials): H. Jones and C. Suryanarayana, Annotated Bibliography 1958-72. J. Mater. Sci. 8 (1973) 705. C. Suryanarayana, Rapidly Quenched Metals: A Bibliography, 1973-79 (Plenum, New York, 1981). C. Suryanarayana, Bibliography on Mechanical Alloying and Milling (Cambridge International Science Publishing, 1995).
- SUBJECT INDEX 1st Volume: Pages 1-942 2nd Volume: Pages 943-1830 3rd Volume: Pages 1831-2740 “page number ff” signifies that treatment of the topic continues on the following page(s).
Growth of solid from liquid, see “Solidification” Growth steps, see “Growth of precipitates, (with) ledges” Guinier approximation, 1163ff Guinier-Preston zones, see “Pre-precipitation”
- -vacancy interaction, 1651ff --: close pairs, 1653 --: spontaneous recombination, 1651ff - X-ray scattering from, 1136 Interstitial positions -: body-centred cubic structure, 1562 Invar alloys, 2540 ff Inverse melting, 1734ff Ion-beam mixing, 637 Ionic bond, 61 Ionicity, degree of, 61, 137 Ion microprobe analysis, see “Secondary-ion microscopy” Ion-probe techniques, 989 ff Ion-scattering spectroscopy, 1214ff Iron, 1556ff -allotropy, 20ff, 30ff, 1416 --: effect of substitutional solutes, 1566ff --: property changes at phase change, 1560 --: thermodynamics, 1558 --: role of entropy of demagnetization, 1558ff - carbides, 1563ff -- precipitate microstructure, 2053 - -carbon phase diagram, 771, 1565 - -carbon solid solution, 1561ff. 2035 --: precipitation hardening, 2052ff --: discontinuous yield, 2052ff - -chromiumxobalt permanent magnet alloys, 2517 ff -, cleavability, 2217 -: diffusion rates of interstitial and substitutional solutes compared, 1563 - dislocation density in deformed iron, in relation to flow stress, 1590ff -: fatigue behavior, 2334ff, 2382 -: interstitial alloys, 1561ff, --: fatigue, 2334ff --: flow stress, 2035,2037 -: interstitial plus substitutional alloys, 1568ff - nitrides, 1563ff, 1571 - -nitrogen solid solution, 1561ff, 2037 -: octahedral and tetrahedral voids, 1562 -, phase transition, 1416 (see also “allotropy”) -, phosphorus in, 1208, 1214, 1215, 1237, 1272, 1582,2271 -: properties of pure element, 1557ff -: solubility of elements in, 1556, 1563 -: strength of ferrite, 1589ff -: substitutional alloys, 1566ff --: effect on form of gamma-field, 1566ff
-, sulphur in, 1224, 1582 -: vacancies in a-iron, 1558 -: yield stress, in dependence on temperature and grain size, 1583ff Iron aluminides, 2078 - as soft magnetic materials, 2533 ff Iron-chromium alloys, 2035 Iron-oxygen-sulphur system, 1312ff, 1316 Iron-silicon steels, see “Silicon steels” Irradiation (effects) -: amorphization, 1758ff - (in) amorphous alloys, 1804 -: atom redistribution, 640 ff, 1708 -: biased point-defect annihilation, 1697 -: cavities, electron microscopy of, 1066 -: defect clusters, 1689ff -: dislocation wall lattice, 1703 - effects, miscellaneous, 1682ff --: collision (displacement) cascade, 1684ff --: displacement spike, 1684 --: intracascade defect reactions, 1688ff -: electron, see “Electron irradiation” -, fast heavy-ion, 1690ff - -enhanced diffusion, 635 ff - -induced creep, 1700ff - -induced Guinier-Preston zones, 1709 - -induced phase transformation, 643, 1709 - -induced precipitation, 640ff - -induced segregation, 64Off, 1708 -: loss of.order, 1687 -: swelling, 1695 ff --, reduction, 1698ff -: void formation, 1695ff (see also “cavities”) -: void rearrangement, 1706ff -: void lattice, 1701ff
Janecke coordiites, 518 Jellium, 861 Jogs, 1853ff, 1904 Johnson-Mehl-Ami-[Kolmogorovl (JMA[K]) kinetic equation, 1435ff, 1788,2421, 2674 -: relation to soft impingement, 1435ff -: necessity for a spatially uniform driving force, 1436 Jominy test, 1579 Jones theory of solid solubility, 151ff K a g o m e net, 234,246 Kauzmann paradox, 1731 Kerr effect, 957
&a1 (electron) probability density, 56 ff Radiation effects, see “Irradiation effects” Radius of gyration, 1164 Rafting, see “Superalloys” Random walk motion - in a crystal, 546 ff - in a glass, 649ff Raoult’s law, 436 Rapid quenching from the melt, see “Solidification, rapid” Rapid solidification processing (RSP), see “Solidification, rapid” and “RSP crystalline alloys” -: pseudo-RSP, 1758 Rare. earth metals, see “Lanthanides” Rayleigh instability, 880 Reaction equilibrium in solutions, 447ff Read-Shockley equation, 1870,2412 Reciprocal lattice, 68, 1043, 1097ff
Sdject index
Recovery from deformation, 2401 ff -: aluminum, 2403 ff -: annealing textures, effect of prior recovery on, 2418 -: cell formation, 1978, 2412ff -: cell evolution, 1980, 2418 -: cell size in relation to Row stress, 1981 - (in) copper.. .does it exist?, 2402 ff, 2407 -: (role in) creep, 1973ff -: dislocation density reduction, 1978 -, dynamic, 1924, 1929, 2003, 2030ff, 2127, 2408 --: dynamic secondary recrystallization, 2486 -: fatiguestrain enhanced, 2408 -: impurity influence, 2403 -: iron alloys, complete recovery, 2405 ff -: kinetics, 2405 ff --, theories of kinetics, 2417ff -, (of) mechanical properties, 2405 ff -, meta-, 2408 ff -, ortho-, 2410 -: polygonization, 2410 ff - (in) steels, 1587 -: stored infernal energy and its recovery, 2401 ff -, stress-enhanced, 2406 ff Recovery of electrical resistivity after irradiation, 1692ff Recovery of electrical resistivity after quenching , 1634ff -: resistivity per interstitial, 1654ff -: resistivity per vacancy, 1629ff -: stage I, 1667ff -: stage II, 1674ff -: stage III controversy, 1622ff, 1636ff, 1640, 1670ff -, use to determine volume of vacancy formation, 1628 - (and) vacancy concentrations, 1634 Recrystalliiation, 2419f -: annealing textum, see “Annealing textures” -: classification of phenomena, 2400 - diagram, 242 1ff -, directional, 1818, 2205 -, dynamic, 1999ff, 2453ff -: grain-boundary migration, see “Grain boundary” -: grain growth, 870ff, 2474 ff --, epitaxial, 2491 ff --: grain-size distribution, 2476 --: impurity influence, 2475
S29
-- kinetics, 2476ff --: mechanism, 2474ff
-- (in) nanocrystalline materials, 2479ff
-- (and) pores, 2642ff --: second phase influence, 887ff, 2476
-- (and) sintering, 2492ff, 2642ff --: stagnation in thin fibs, 2490ff --: texture inhibition, 2477 --: thickness inhibition, 2476ff -- (in) thin films, 2489ff - kinetics, 1588,2421ff --: effect of minor solutes on precipitates in steels, 1588ff -: laws of recrystallization, 2419 -, metadynamic, 2164, 2454 -: neutron radiation influence, 245 1 -: nucleation, 2425ff --: models, 2427 --, oriented, 2427ff --: role of inhomogeneity of orientation after deformation, 2428ff --: strain-induced grain-boundary migratiolL 2435ff -- nucleation, stimulated, 216, --: subgrain coalescence, 2435 ff --: techniques of investigation, 2425 --, twin-based, 2438ff - (of) ordered alloys, 2471ff --: antiphase domain creation during, 247 1ff --: retardation of grain-boundary migration, 2471 ff -, primary, --: annealing textures, 2205 --: critical strain, 2420 --: growth of grains, see “Grain-boundary, migration” --: hot working, see “dynamic” --: impurity influence, 2423 ff --: kinetics, 1588, 2421 ff --: Kronberg-Wilson rotation, 2440 --: microgrowth selection, 2435 --: nucleation of grains, see “nucleation” -: recrystallization-controlled rolling, 2455 -: retardation due to recovery , 2424 -, secondary, 2482ff --: driving force, 2483 --, dynamic, 2486 --: role of disperse phase, 2485, 2487 -- (and) sintering, 2492 --, surface-controlled, 2487 ff -- texture, 2486ff, 2488
x - r a y absorption and scattering -: absorption coefficients, 1120ff -: absorption edge, 1121 -: angle of total reflection, 1131
Subject index -: Bragg peak broadening, 1132ff
-: Bragg peak intensity (and changes), 1118 , 1132ff -: Bragg peak shifts, 1130ff -, Compton scattering, 112 -detectors, 1130, 1139 -: diffuse scattering, 1118ff, -- between Bragg peaks, 1139ff (see also “monotonic Laue scattering”) -- components, 1148ff --near Bragg peaks, 1134ff -- (due to) point defects, 1664ff -: extended X-ray absorption fine (FXAFS), 1183, 1777 -: fluorescence, 1121 -: Honl corrections, 1121 -: Huang scattering, 1135ff, 1147, 1665, 1673ff - inelastic scattering, 1126ff --, coherent, 1126 --, incoherent, 1127ff -: isomorphous and isotopic substitution, 1771ff -: line broadening due to plastic deformation, 1924 -: pair distribution function, 1769ff -: phonon role in inelastic scattering, 1126 - (from) point-defect clusters, 1136ff -: peak shifts due to plastic deformation, 1924 -: radial distribution function, 1770 -: scattering, 1116ff -: singleparticle scattering function, -: size-effect scattering, 1132ff -: small-angle scattering, 1161ff (see also “Small-angle scattering...”) -: spurious radiation, -: surface sensitivity, 1131 ff -: thermal diffuse scattering, see “diffuse scattering” -: X-ray photoelectron spectroscopy, 1213ff -: X-ray sources, 1121, 1128ff -: X-ray topography, 988 -: Zwischenreflex scattering, 1139
Y i e l d anomaly, see “Anomalous flow behavior in L1, phases” Yield, discontinuous (yield phenomenon) - in fcc solid solutions, 2028 ff - in lithium fluoride, 1938ff - in non-ferrous metals, 1939, 1941 - in polymers, 2695 ff
s39
- in semiconductors, 1938 - in steels, 1585ff, 1596ff, 1869, 1938, 2053 ff
- (due to) strain softening, 1939ff Yield stress - (in terms of) continuum mechanics (yield criteria), 1946ff --: Mohr diagram, 2129ff --: (for) polymers, 2698 ff, 2709 --: von Mises condition, 1946, 2590, 2698 --: Tresca criterion, 2698 -: critical resolved shear stress for glide, 1885ff -, dependence on cell (subgrain) size, 1930ff, 1981, 1984,2416 -, dependence on dislocation density, 1925 -, dependence on grain size, 2168 ff (see also “Hall-Petch relationship”) -, dependence on mesh length, 1923 -: dependence on order, 2059 ff -, Fleischer-Friedel theory, 1903 -: forest dislocation cutting, 1903 ff -, Labusch, 2020 -, mechanisms determining, 1894ff --,extrinsic, 1896ff --, intrinsic, 1894ff -, metal-matrix composites, 2584 ff -: particle resistance, 1897ff - of polymers, 2693 -: solute resistance, 1896 -: superposition of different resistances to plastic deformation, 1905ff -: threshold stress, 2185 ff Young’s modulus, 1880
z e n e r relationship, 1009,2159,2467 ff. 2642 Zener relaxation, 567 ff Zero creep technique, 1210,2630 Zinc, recovery from deformation, 2401 ff Zinc group metals - crystal structures, 21 ff Zintl phases, 225 ff Zirconium -, purification by electromigration, 619 -: fast diffusion, 595 -: allotropy, 20, 24 Zone-hardened AI-Cu alloys, 205 1 Zone-melting (zone-refining), 719ff Zone-refined iron, 1588
I The late ProJ Peter Haasen
Colour picture onffont cover: Simulation of an alloy dendrite growing into a supercooled liquid using the phase-field method. The colours show variation of composition (atomic fraction Cu) in the liquid and solid for parameters approximating a Ni-Cu alloy with 0.41atomic fraction Cu. See ch. 8, par. 7.5 (Courtesy -r’Nilliam J. Boettinger and James A. Warren).
ISBN
o 444 89875 I
Phqs e talICl -
Robert W. Cahn and Peter Haasen
FOURTH,REVISED A N D
(-I),editors
E N H A N C E D EDITION A
PHYSICAL METALLURGY VOLUME I11
LIST OF CONTRIBUTORS
A.S. Argon E. A n t H. K. D. H. Bhadeshia H. Biloni J. L Bocquet W. J. Boettinger G. Brebec R.W. Cahn G.Y. Chin? T. W. Clyne R.D. Doherty H.E. Exner R. Ferro D.R. Gaskell H. Gleiter A.L. Greer P. Haasent J. P. Hirth S. Hofmann E.D. Hondros E. Hornbogen G. Kostorz
C. Laird P. LejEek W.C. Leslie Y. Limoge J. D. Livingston E E. Luborsky T.B. Massalski J. R. Nicholls AD. Pelton D.G. Pettifor D.P. Pope M. Riihle A. Saccone S. R.J. Saunders M.P. Seah W. Steurer J.-L. Strudel R.M.Thomson C.M. Wayman M. Wilkens A.H. Windle H. J. Wollenberger
PHYSICAL METALLURGY Fourth, revised and enhanced edition
Edited by
Robert W. CAHN
Peter HAASEN?
University of Cambridge
University of Giittiplgen
VOLUME m
1996 NORTH-HOW AMSTERDAM-LAUSANNE-NEW
YORK-OXFORD-SHANNON-TOKYO
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Printed in The Netherlands
SYNOPSIS OF CONTENTS Volume 1 1. 2. 3. 4.
5. 6. 7. 8.
9.
Crystal structure of the metallic elements Electron theory of metals Structure and stability of alloys Structure of intermetallic compoundsand phases Appendix: Quasicrystals Metallurgical thermodynamics Phasediagrams Diffusion in metals and alloys Solidification Microstructure
Volume 2 10. Surface microscopy, qualitative and quantitative 11. Transmission electron microscopy 12. X-ray and neutron scattering 13. Interfacial and surface microchemistry 14. Oxidation, hot corrosion and protection of metallic materials
h e r Riihle, Wlkes Kostorz Hondros, Seah, Hofmann, LejFek Saunders, Nicholls
15. Diffusive phase transformations in the solid state
16. Nondiffusive phase transfomtions 17. Physical metallurgy of steels 18. Point defects 19. Metastable states of alloys
Volume 3 20. Dislocations 21. Mechanical properties of single-phase crystalline media: deformation at low temperatures 22. Mechanical properties of single-phase crystalline media: deformation in the presence of diffusion 23. Mechanical properties of solid solutions 24. Mechanical properties of intermetallic compounds 25. Mechanical properties of rnultiphase alloys 26. Fracture 27. Fatigue 28. Recovery and recrystallization 29. Magnetic properties of metals and alloys 30. Metallic composite materials 3 1. Sintering processes 32. A metallurgist’s guide to polymers V
DISLOCATIONS J. P. HIRTH Mechanical and Materials Engineering Department WashingtonState University Pullman, WA 999164-2920, USA
R. W! Cahn and F! Haasent, eds. Physical Metallurgy; fourth, revised and enhanced edition 6 Elsevier Science BY 1996
1832
J. l? Hirrh
Ch. 20, 8 1
I . Elementary geometrical properties The concept of crystal dislocations was introduced by POLANYI [19341, OROWAN[19341 and TAYLOR [1934], although the elastic properties of dislocations in isotropic continua had been known since 1905 (TIMPE, also VOLTERRA [1907]). Dislocations are defects whose motion produces plastic deformation of crystals at stresses well below the theoretical shear strength of a perfect crystal. In fig. la, b and c the glide motion of an edge dislocation is shown to cause plastic shear strain. One can imagine a virtual process of cutting the crystal on a glide plane, shearing the cut surface by a shear displacement vector b, the Burgers vector, and gluing the cut surfaces together, creating the edge dislocation in fig. lb. The dislocation bounds a slipped area and is a line defect. It is characterized by the Burgers vector b and by a unit vector 5 tangent to the dislocation line at a point in question. The same dislocation could be formed by opening a cut under normal tractions, fig. Id, and inserting a plane of matter. In order that the deformation not produce a high energy fault on the cut surface, b is usually a perfect lattice vector as illustrated for an edge dislocation in fig. 2. The choice of the 2 sense of E is arbitrary, but once chosen, the f. sense of b is fixed by the following convention: imagine a perfect reference crystal, select a vector 5 in it, and construct a closed circuit in it, right-handed relative to Then construct the same Burgers circuit in the real crystal, as shown for example in fig. 2. The vector SF connecting the start of the circuit to the finish is the Burgers vector of a dislocation if it is contained within the circuit. In this operation, the circuit must not pass through the nonlinear core region within an atomic spacing or two of the dislocation line. For the edge dislocation, b is seen to be perpendicular to E. In fig. 3 a screw dislocation can be imagined to have been created by a cut and displacement operation, or simply by shearing the slipped area by motion of the dislocation in from the surface of a perfect crystal. The screw dislocation has b parallel to 5. It is right-handed if b points in the same direction as 6 as in fig. 3; left-handed otherwise. In general, fig. 4, the slipped or displaced surface can be arbitrary, the dislocation line can be arbitrarily
Fig. 1. (a-c) shear of a crystal under a shear stress w by an amount b by passage of an edge dislocation; (d) creation of same dislocation under normal stress w once added material is placed in the opened cut.
ch.20, I1
1833
Dislocations
Fig. 2. An edge dislocation in a simple cubic crystal. A Burgers circuit is also shown, projected toward the viewer for clarity so that it passes through the center of atoms not shown. The sense vector 5 points out of the page-
Fig. 3. A screw dislocation in a simple cubic crystal.
+b I I
.
.
Fig. 4. A mixed dislocation. References: p . 1872.
1834
J. F! Hirth
Ch.20, $ 2
Fig. 5. Three dislocations meeting at a node. Arrows on lines indicate sense of 5. Equivalent Burgers circuits also shown.
curved, and the dislocation is called mixed when the angle p between b and 5 is neither 0 nor an integer multiple of ~ / 2 . Some other properties follow directly from the above definitions. If 5 is reversed, the sense of b is also reversed as seen from fig. 2 since the circuit is also reversed when 5 is reversed. Since the dislocation line bounds a displaced area, the line cannot end within otherwise perfect crystal but can only end at a free surface, a grain boundary, a secondphase interface, or a dislocation node. A node is a point where two or more dislocation lines join. Translation of a Burgers circuit along 5 without the circuit cutting through a dislocation core does not change the vector SF or thus the total b (imagine such an operation for fig. 2); such circuits are called equivalent Burgers circuits. Thus, if a dislocation denoted by its Burgers vector, b,, splits into two dislocations b, and b,, enclosed by equivalent circuits, fig. 5, an analog of Kirchhoff‘s law applies and b, =bz+b3.If the 5 for dislocations 2 and 3 are reversed, then the signs of b, and b, change by the earlier axiom and: S,bi=O for dislocations meeting at a node if all sense vectors 5 are selected to point toward the node.
2. ElasticJields of dislocations 2.1. Displacements and stresses
Consider the right-handed screw dislocation with the geometry as indicated in fig. 6. Imagine that the cylindrical region is a portion of an infinite continuum. In cylindrical coordinates, the displacements can be deduced by inspection: u, = u, = 0,y = be/27r.
(1)
Differentiation and the use of Hooke’s Law gives the strain and stress fields:
a , = 2 p , = pb/2m,
(3)
with p the shear modulus. The edge dislocation, fig. 7, has displacements u, and u,#O but u,=O. Thus, the solution for this plane-strain problem proceeds by deducing the Airy stress function q. Guided by the screw dislocation result that (T l/r and observing in fig. 2 that (a,+aee) --sin 0 to give compression above, and tension below the glide plane 8 = 0, one finds:
-
Ch. 20, 62
Dislocations
1835
Fig. 6 . A right-handed screw dislocation along the axis of a cylindrical region of radius R and length L.
with u =Poisson’s ratio. Differentiation of eq. (4) gives: u r n
1 a* 1 - =dZ* = -r- ar + -rz 88’
pb sin 6 2741 - v)r’
uee= a 2 =urn, ~ us*
h2
urz= y(a, +urn).
This field has the physical features required by fig. 2; compression above the glide plane, tension below it, and maximum shear stresses on the glide plane 8 =O. The strains are given by Hooke’s Law and their integrals give the displacements, see HIRTHand LOTHE [1982]. For the screw dislocation, the strain energy density is w = uiz/2p.The total strain energy per unit length in the cylindrical region in fig. 6 is:
W = laRwrdrd8 = -1n-. pbz
R
4n-
ro
Fig, 7. An edge dislocation.
References: p , 1872.
1836
J. I? Hirth
Ch. 20. $ 2
Here a core cutoff radius r,, is introduced to avoid divergence. This is necessary because the use of linear elasticity gives an artificial divergence in eel,eq. (2), as r +ro =b. For the edge dislocation, an alternate procedure is simpler. From thermodynamics, the strain energy is equal to the isothermal reversible work done on the system in creating the strain. For fig. 7, imagine creating the dislocation by making a cut on the surface 8 =0, r>0, and displacing it by b, applying stresses ule(8 = 0) to the cut to accomplish the displacement. Since u, = b finally, the average stress in the process is h, and the (8 = O)b dA. The total strain energy per unit length surface work on an element dA is hle is then given by the integral of this surface work over A:
W =
ln-.R 5,: 3a,8(0 = 0)bd r = 4 4r*bz 1 -v) ro
(7)
The displacement and stress field for a mixed dislocation is given by decomposition of b into edge and screw components be and b, and use of the above expressions with be= b sinP and b, = b cos /3 substituted for b. In general, the strain energy, being quadratic in u,cannot be superposed, but the screw and edge components of the mixed dislocation have no common stress components, so W/L can also be added,
where E @ ) is called the energyfuctor. 2.2. Peach-Koehler force
As just discussed, the strain energy is the isothermal reversible work done on the system, which, for a system under constant external pressure, is the Gibbs free energy G of the system. For an isothermal, constant pressure system, AG for a hypothetical reversible displacement of the system in some reaction coordinate c, is a criterion for whether the displacement in the actual case tends to occur spontaneously (forward if AGO, not if AG=O). In a general sense, we define a virtual thermodynamic force as F = d G / & , . Then if F>O, the system tends to change spontaneously in the direction of the reaction coordinate. We are interested in whether a dislocation will tend to move spontaneously in the presence of a stress field, so the appropriate reaction coordinate is the position r of the dislocation. Imagine the process, fig. 8, of cutting an area dA = n dA = dl x Sr = dZ(5 x Sr), displacing the dislocation by Sr, removing or adding matter if necessary, and gluing the surfaces together. If surface forces are needed to prevent the cut from relaxing, work can be extracted (AG < 0) as the displacement is accomplished in the hypothetical process. The self-stresses of the dislocation are symmetric and hence do no net work in the process. However, any other stresses u acting in the core region can extract work. The stresses produce surface forces u -dA and in the displacement b extract an amount of work 4 G =SW,given by
SW = b*mdA = b * ~ * x( g&)SI = & . ( b * x~ 4)Sl.
Ch. 20, $ 2
1837
Dislocations
Fig. 8. Displacement of a segment dl of dislocation.
-
But by the definition of the virtual thermodynamic force F,SW=F Sr. Thus, the virtual [1950]): force is (PEACHand KOEHLER
where ciir is the permutation operator (ce= 1 or -1 for or cjZlor their cyclic permutations, 0 otherwise). Consider the example of fig. 7, with only 1 nonzero, and with only b, = b nonzero. According to eq. (9), a shear stress u2,produces a glide force F,/L=a,,b,;a normal stress u,,produces a climb force FJL=-u,,b,; other stresses produce no force. For the screw of fig. 6, with = 1, b3= b a stress a,,produces a glide force F I / L = u 2 , , b , while a stress aI3 produces a (cross-slip) glide force F2/L=-ul,b3. Thus, the Peach-Koehler equation (9) reveals the direction of prospective spontaneous motion of a dislocation in the presence of stresses u.Theforce is a virtual thermodynamic one and must not be confused with a mechanicalforce.
e3=
2.3. Dislocation interactions interaction forces and energies also can be developed from eq. (9). Consider first the interaction between parallel dislocations A and B in fig. 9, with t3= 1 and b, = b. Equation (9) then gives for the glide force on B:
FB
B
A
B
1 = ~ 3 u , , b=l
L
ccb2
2741 - v)x '
since ut,=@&(e= 0, x =X ) =pb/27r( 1 - v)X. The interaction force is repulsive between like-sign edge dislocations; attractive between opposite-sign dislocations. For more general parallel dislocations, such as A and C in fig. 9 the use of eq. (9) is more complicated, involving more terms. However, in cylindrical coordinates with r = R at dislocation C, the radial interaction force has the simple form
References: p. 1872.
1838
J. f? Hirth
Ch. 20, 5 2
Y
Fig. 9. Parallel dislocations;6: points out of the page.
Equation (11) is seen to reproduce eq. (10) for the edge case and to give F/L=pb2/2?rR for parallel like-sign screw dislocations. There is a weak oscillating F J L component as well in the general case (NABARRO [1952]). The interaction energy as a function of R can be determined by integration. For example, for the above screw case
y,,, =
RF,
-dr 4 L
pb2
R
2n
4
= --In-,
where Ra is a reference position and can be chosen to equal r,,, for example. A procedure for more general interactions is exemplified by the perpendicular screw case of fig. 10. Coordinates are selected for dislocation A so that the stress field can be conveniently and simply expressed by eq. (3), ~~~=pb/2?rt‘. The stress tensor is then transformed to coordinates x, y, z fixed on dislocation B to make the use of eq. (9) easy, giving uYz =-pb cos28’/2?rh. Equation (9) then readily gives (F:IL) =-pb2 cos2B’/2?rh for the local value of the interaction force, the locus of which is plotted in fig. 10. This procedure works for defects other than dislocations. Defect A could be a solute atom, a precipitate, or the like, and, provided one can express its stress field, d,the interaction force on B can be determined as above. The equal and opposite nature of the net forces in fig. 10 is fortuitous; in general, Newton’s law does not apply for these virtual thermodynamic forces; for example, there can be interaction torques between dislocations. All of the interaction forces have the characteristic feature that they vary inversely with the spacing of dislocations. This result has important implications for work hardening and macroscopic descriptions of plastic deformation on the basis of dislocation models. The macroscopic flow stress scales inversely with forest dislocation density, or,
Ch. 20, $ 2
Dislocations
1839
2'
Fig. 10. lbo perpendicular right-handed screw dislocations separated by a distance h; light arrows represent the interaction force.
equivalently, directly with the square root of the dislocation density (EMBURY[1971]; LAVRENTEV [1980]), Le., the total length of dislocation line per unit volume. The exact reasons for such scaling laws remain elusive, but both long-range elastic stresses (the dislocation interaction) and short-range network and junction formation (with Orowan bowing as discussed subsequently) can both produce this behavior. Indeed, both appear to contribute (BASINSKIand BASINSKI [1979]; NABARRO [1986]). Also, the flow stress scales inversely with dislocation cell size (KUHLMANN-WILSDORF [19661). Recent ideas on the latter relation are critically reviewed by KUBIN [1993] and include the concepts of dislocation mesh lengths and (fractal-lie) scaling (KUHLMANN-WILSDORF [ 19891; HANSENand KUHLMANN-WILSDORF [1986]); fractal behavior of slip lines (NEUHAUSER [ 19901); and several reaction-diffusion continuum models. The examples presented here are all for infinite straight dislocations. Stress, force, and energy expressions are available for generally curved dislocations in the form of line integrals, but, except for special cases such as circles, the integrals can only be solved numerically. Analytical results are available in both the isotropic and anisotropic elastic cases for straight-line segments, angular dislocations and infinitesimal loops. The results are very lengthy, beyond the scope of the present work, and can be found in NABARRO [1967] and HIRTH and LOTHE [1982]. These results are directly applicable to some arrays, such as stacking-fault tetrahedra, and can be used to obtain fairly accurate approximate results for curved arrays such as extended dislocation nodes. The segment approximation is being used currently in the forefront topic of computer simulation of large arrays of interacting dislocations (KUBIN, ESTRINand CANOVA[ 19901; DEVINCRE and CONDAT[ 19921). 2.4. Surface effects
At a free surface, equilibrium requires that no forces act on a surface element. If the area element A is defined by a local normal unit vector R, the condition is a - A =Aa 'n =O. For References: p . 1872.
1840
J. F! Hirth
Ch. 20, 8 2
dislocations in finite media, this surface condition modifies the elastic field. As an example, suppose that the surfaces in fig. 6 are free surfaces bounding the cylinder. The stresses a,,from eq. (3) would act on the end surface A, and produce a net torque on the end. These extraneous stresses are removed by superposing equal and opposite tractions on the surface, solving the elastic problem for a cylinder under such tractions, and adding the resultant strain and displacement field to the infinite medium result. For the case of fig. 6, a net twist appears in the cylinder, the added stresses are ae,=pbr/?rR2, and the [1953]). free energy of the system is decreased by AW/L=-pb2/h (ESHELBY Similarly, for the edge dislocation in a finite cylinder, fig. 7,stresses a, and a, must be imposed on the cylinder surface r=R. In this case the free energy decreases by AW/L=pb2/8?r( 1- v). Surface stresses thus characteristically produce energy changes -pbz/lO. These changes are of the order of uncertainties in W/L because of uncertainties in core energy and in the use of isotropic elasticity, so they can consistently be neglected. When a dislocation is parallel to a free surface, the superposed field often can be determined by an “image” construction. Consider the screw dislocation in fig. 11. Rotation of coordinates shows that eq. (3) gives stresses un= -pby/2.rrR2 acting on the free surface where r =R. A solution which gives opposite stresses which exactly cancel these extraneous stresses is that of an opposite-sign dislocation at the mirror-image position relative to the surface. Thus, the field in the body is that of the real dislocation ue,=pb/2?rr and that of the image dislocation a,, =-pb/2?rf. At the site of the real =?r, f = U ) = p b / h h . Thus, by eq. (9), the dislocation, the image field is a,,,=-o,,~((e free surface exerts an image force
attracting the dislocation to the free surface. For the edge dislocation, the simple image construction is insufficient and an added stress field is needed to satisfy the free surface boundary condition (HIRTHand LOTHE [1982]). However, the force associated with the free surface tractions is still that of the simple image dislocation. This result is general, Lothe’s theorem [1976]; for single dislocations inclined to the surface, the simple image construction gives the correct force
Fig. 11. A right-handed screw dislocation. parallel to and a distance A from a free surface.
Ch. 20, 8 2
Dislocations
1841
produced by the surface tractions. This theorem has been verified for a number of nearand HAZZLEDINE [1980]. For two or surface dislocation configurations, e.g., SHAIBANI more dislocations near the surface, the simple image construction is insufficient. The added stress field of one dislocation, discussed above for the edge, would produce a force on the other dislocation. Also the theorem is valid only in the linear elastic approximation and breaks down very near the surface where core effects become important (FILLIPOV, GAIDUKOV and MAKSIMOV [1985]). There are other methods for calculating the fields of dislocations near free surfaces. For example, MUSKHELISHVILI [19751 presents a method of solution of Cauchy integrals. Which method one uses is a matter of taste, but for a single dislocation near a planar surface, the simple image method is straightforward.
2.5. Line tension For arbitrary dislocations in a deformed crystal, the determination of W/L is a most formidable task. However, a rough approximation can be made. For a random array of dislocations, WILKINS[1969] showed that the outer cutoff R in the self energy exceeds the dislocation spacing. However, the dislocations tend to rearrange to polarize their elastic fields and reduce the energy, with R approaching the interdislocation spacing. On the average, since like-sign dislocations repel while opposite-sign ones attract, a dislocation tends to be surrounded by opposite-sign dislocations at an average spacing A. In accord with St. Venant's principle, a pair of dislocations of opposite sign are sources of equal and opposite stress fields which should cancel one another over distances greater than -A from the pair. Thus, the strain field of the dislocation is limited to a radius R = A. Hence, since as already discussed, core and surface terms can be neglected, eq. (8) applies roughly for the arbitrary case with R = A. Moreover, A where p is the dislocation density (m/m3). Qpical values of p range from -10" m/m3 for annealed crystals to m/m3 for cold worked crystals, corresponding to lo4 m < h < m. With ro=bn3x lo-'' m and uzO.3, the range of W/L in eq. (8)is from 0.28 to 1.18 pb2. Since p and b are constant for a given material, W/L is a constant with the above approximations. A stretched linear-elastic string has a constant energy per unit length W/L numerically equal to the line-tension force S restraining the string, since aW/aL= W/L= 5. In the above approximation the dislocation can be thought of similarly as a stretched line constrained by line tension forces acting parallel to the line:
Here g is a numerical factor varying from 0.3 to 1.2 with an average value of roughly
0.7,or in a cruder approximation, 1/2. The simple line tension analogy is qualitatively useful. For example, in this approximation a bowed-out dislocation or zig-zag dislocation is unstable with respect to a straight dislocation in the absence of stress. Also, the equilibrium angles at a node are determined by the line tensions. References: p . 1872.
1842
J. I? Hirth
ch.20. $ 2
In a more accurate representation, one must account for the variation of W/Lwith /3 in eq. (8). E@) is a minimum for the screw orientation /3 =O or m, so a toque exists tending to twist the dislocation into screw orientation. Including this factor, but retaining the concept that S is the change in W with change in the total line length of a dislocation, DEWIT and KOEHLER[1959] found:
z 82F) [: d2E(B)l
S=-+
=-
I+-E@)
8/32
In the simple approximation that S = W/L,the value of S for an edge exceeds that for a screw, eq. (8), by a factor 1/(1- Y)2+ 1.4.With the inclusion of the factor a2(W/L)/3fl2, which tends to stabilize screws and destabilize edges, the situation is reversed and S for a screw exceeds that for an edge by a factor (1 + v)/( 1 - Y)(1 -2v) = 4.6. Even more accurately, rather than use the rough St. Venant approximation, one should compute the self and interaction energy. This can be done by the straight line segment equations mentioned previously. For the configurationsin fig. 12, Hmm et al. [1%6] find:
with g a numerical factor equal to 6.41, 1.11, and 0.18, respectively, for cases a, b, and c. The result is understandable qualitatively. In case a, segments A and B are equal and opposite and so work must be done against their attractive interaction force in forming the loop. In case b, segment B is missing, so less work is needed and S is smaller. In case c, there is even less attractive segment length and S is lowest. Interestingly, as first [1948], the line tension of a bowed segment depends on noted by MOTTand NABARRO the bowout length L and not on the outer cutoff R. This result is important in the theory of bowout in solution-hardened or dispersion-hardened systems. Advanced solutions of the bowout problem include a many-straight-line-segment, and BACON[19751 and a cubic spline fit anisotropic elastic calculation by SCATTERGOOD et al. [19921. Using an anisotropic elastic version of line integral solution by DUESBERY the straight segment approach, SCHMID and KIRCHNER [1988] found that equilibrium shape of complete dislocation loops was rounder for small loops than for large loops which developed regions of low curvature. The small loop configuration reflects the dominant role of the segment-segment interaction energy, while the large loop shape is controlled by the /3 dependence in eq. (15).
Fig. 12. Bowed-out configurations of an edge dislocation; b is normal to the page.
a.20,§3
Dislocations
1843
3. Crystal lattice efects 3.1. Peierls barrier PEIERLS[1940], in a model elaborated by NABARRO [1947], modeled the periodic potential in the glide plane experienced by a straight dislocation lying in a low-index direction. In the model, the bonds across the glide plane were considered to interact by an atomic potential, while the remainder of the lattice was elastic. The model is useful in giving an analytical result that removes the artificial core divergencies of the Volterra dislocation, eqs. (2), (3) and (5). However, as discussed by HrrrTH and LOTHE[1982] and by BULLOUGH and TEWARY [1979], the model is physically unrealistic (compared to a cylindrical atomic model centered on the dislocation) and predicts that symmetry positions are positions of energy maxima instead of minima. Nevertheless, atomic calculations indicate that the potential is roughly equivalent to the sinusoidal form of the Peierls-Nabarro result. Also, the prediction of the Peierls-Nabarro result that the magnitude of the energy maxima are greater for screw dislocations than for edge dislocations is supported by the results of nonlinear, anisotropic elastic estimates (PRINZ et al. [ 19831) and by atomistic calculations (SONDHIet al. [1992]). Thus, it is useful to retain their result as an empirical representation of the barrier. For a dislocation displaced by a distance x from a symmetry position (such as in fig. 2), the potential is:
where a is the period, usually equal to b, and Wp is the Peierls energy. The maximum slope of this oscillating potential gives the Peierls stress, up,needed to overcome the barrier at 0 K,but the barrier is usually overcome instead by kink formation as discussed below. In the original model the explicit form for Wp for an edge dislocation was
wp= Pb2 (,,, L 2 4 1 - v) -y -
[
exp -b(?v))’
where 6 is the width of the spreading of the core in the glide plane and d is the glide plane spacing. An equivalent form without the factor (1 -v) holds for a screw dislocation. This form is not verified in atomic calculations, but the qualitative trends of a marked decrease in upor Wp with an increase in core spreading or d are retained and are useful in predicting crystal slip systems. Because d and b appear in the exponent in eq. (18), the relation suggests that the smallest b and largest d should correspond to the observed slip system, in good agreement with experiment when dislocation extension is not an important phenomenon. As an indication of the change of Peierls energy with slip system, the ratio of Wp for the closest packed plane slip system, (1lo)-( 111}, to that of the next closest packed plane system, (110)-{ loo}, is 0.36 in fcc crystals. While the empirical form of eq. (18) is useful, it has several deficiencies in that it does not include expected variations of Wp with temperature or with stress components et al. other than the shear stress resolved on the appropriate glide system. DOBROMYSLOV References: p. 1872.
1844
J. R Hirth
Ch. 20. 9 3
[19841 have obtained experimental evidence that the Peierls stress in Mo varies linearly with the normal stress acting on the glide plane, in accord with the latter expectation. At present, there are insufficient results from atomistic simulations to suggest empirical modifications to eq. (18) for these added effects. A dislocation line containing kinks is shown in fig. 13. The configuration of a kink such as B is determined by a balance between line tension forces, which tend to make w in fig. 13 large and hence straighten the line, and the forces associated with the Peierls barrier, which tend to make the kink abrupt. In the simple line-tension model of eq. (14), this force balance gives an equilibrium kink width w and kink formation energy W,given in terms of the total elastic energy W/Lby
w = a(W/2Wp)?
w, = (2a/74(2WWP)?
Qpically, W increases successively for bcc metals, ionic crystals and covalent crystals and lO-*W, W,- 0.01, 0.05 and 0.1 p d and w= 70,20 and with values Wp- lo4, 7a, respectively, see DUE~BERY [1983] and SUZUKI and SEEGER [1992]. For fcc and hcp metals Wp is much smaller, so that Peierls barrier effects are only important well below room temperature. Kinks are of three types: geometric kinks such as B in fig. 13 are present by geometric necessity since the segment AE spans one Peierls hill. At thermal equilibrium, a dislocation line contains a concentration of thermal kinks of - ( l / b ) exp(-WJkT). Finally, since the motion of a dislocation produces an offset of b in the entire lattice, as illustrated in fig. 1, intersection kinks can be produced on each dislocation if two nonparallel dislocations intersect.
3.2. Core structure and energy With the advent of fast computers, atomic calculations for the nonlinear core region have become prevalent. They often have the drawback of using atomic pair potentials which do not accurately represent real metals. However, they do approximate real metal behavior fairly well and they are most useful in delineating classes of core behavior for different crystals. Such calculations have been performed using either lattice statics or lattice dynamics as reviewed by PULS [1981]. With the development of flexible boundary methods, in which a cylindrical atomic region concentric with the dislocation core is matched compatibly with a surrounding elastic continuum, the use of lattice dynamics in real space is very efficient according to Puls. The reliability of calculations of this type in
Fig. 13. Double kink CD and single kink B in a dislocation segment AE. Dashed and solid lines are maxima and minima in the varying lattice potential, respectively.
a.20, 53
Dislocations
1845
mimicking real materials has improved with the development of more accurate semiempirical potentials beyond pair potentials (BASKES e?al. [19921; CARLSSON [19901). One important feature of these results is related to the volume of a dislocation. In the linear elastic approximation, the net volume change produced by a dislocation in a crystal is zero. The atomic calculations, both for metals and for ionic crystals, reveal that both edge and screw dislocations produce a volume expansion of about an atomic volume per close-packed atomic plane cut by a dislocation, i.e., the volume-equivalent of a row of interstitial atoms. As also discussed by Puls, calculations of the core energy, while fewer, are of interest. In the reversible process of creating the edge dislocation, $2.1, in addition to the elastic energy of eq. (8) stored between r, and R, elastic energy flows into the core and is stored there as core energy. The core energy can only be estimated by atomic calculations. Rather than adding a core term to all energies, it is convenient to retain the results of eqs. (6) to (8) and to incorporate the core energy by adjusting the value of r,. In terms of a parameter a defined by r,=b/rw, estimates summarized in HIRTHand LOTHE [19821 indicate that a = 3-5 for ionic crystals with screw values larger than those for edges, while a 0.5-2 for both partial and perfect dislocations in metals. Results for covalent crystals vary from about a = 1 to 4, with the lower values for reconstructed cores and the higher values for unreconstructed cores (DUESBERY et al. [ 19911; NANDEDKAR and NARAYAN [1990]). One of the successes of atomic calculations has been in the understanding of the properties of screw dislocation cores in bcc metals, reviewed by V ~ [1974] K and by SEEGER and WOTHRICH[1976]. Figure 14 illustrates the result which indicates splitting of a [; 1113 screw dislocation into a dislocation q[1113 at the origin and three dislocations (=q)[lll] at B, C, and D. Because the splitting is only over atomic dimensions and because of the variable nature of q, these are calledfractional dislocations. For the glide sequence in fig. 14, the value of vpis large since the fractional dislocations must contract before motion can occur. Also, because of nascent extensions on { 112) planes, also indicated in fig. 14. up is asymmetric, being smaller in the so-called easy twinning direction than in the opposite direction, in agreement with experiments. The configuration of fig. 14a can exist in an alternate configuration rotated by T about the vertical in the page. The two configurations have opposite handedness, or polarity, and hence break the symmetry of the (110) direction along which the dislocation lies. A new localized type of defect, a flip, occurs where the configurations change from one to the other (SEEGER and WUTHRICH[ 19761). Because the fractional dislocations must at least partially constrict when the polarity changes, the flip can also be thought of as a type of constriction. Since kinks and flips can superpose, there is a wide variety of such defects, particularly in bcc metals where there are several low-index directions that are potential Peierls valleys in each glide plane. Computer simulations of metals indicate that the energies of flips are of the same order of magnitude as those of kinks (WUTHRICH [19771; DUESBERY [19831). In diamond cubic crystals, simulations of partial dislocations show that core reconstruction can break symmetry, again leading to the possibility of flips (DUESBERY et al. [ 19911). Concerning dislocation cores in intermetallic compounds see ch. 24, $2. References: p. 1872.
1846
J. I! Hirih
Ch. 20, 5 3
Fig. 14. (a) Core.dissociation of screw dislocation;(b-e.) Motion of screw under stress producing glide on the average on the dashed (112) plane shown in e.
33. Stacking faults and partial dislocations In close-packed fcc and hcp crystals, stacking faults on { 111) planes have relatively low energies and, consequently, perfect dislocations extend to form parrial dislocations bounding a stacking fault. As an example, we consider the fcc case and the commonly observed intrinsic stacking fault, formed when a { 111} plane is removed from the normal stacking sequence ABCABCABC and the crystal is collapsed together normally to form the fault ABCA:CABC (see discussion of fig. 17e). The notation of THOMPSON[1953] is useful. If one connects the atom centers of the four atoms near a corner of a unit cell, one forms the Thompson tetrahedron, fig. 15. The faces are the { 111} glide planes and the edges are the *(110) Burgers vectors (glide directions). Then AB, etc., denote Burgers vectors and (d) denotes glide plane. Now consider the view of the glide plane in fig. 16. A perfect dislocation AB would shear the crystal and move an atom from 1 to 2. Clearly, in a central-force atom model, the path along the potential “valleys” (1)-(3)-(2) would be more favorable energetically. Yet, at the midpoint (3) the configuration is that of an intrinsic stackingfault. Hence, motion occurs first by the partial SB, creating the fault, followed by the partial A6, annihilating the fault. The actual configuration is shown in fig. 17a. The equilibrium spacing is achieved when the repulsive forces between the partials, eq. (1l), just balance the attractive force y of the fault. Here y is the stacking fault energy (J/m*),numerically
ch. 20, § 3
Dislocations
1847
Fig. 15. Thompson tetrahedron.
equal to the surface tension (N/m) (see ch. 4, 89.2). The partial dislocations A& etc., of the glide type, are also given on the Thompson tetrahedron, fig. 15, and are called Shockleypartials. The correct sequence to form an intrinsic fault is established if: viewing the fault along positive 6 and viewing the plane as if one were outside the tetrahedron [as for (d) in fig. 171, place the Greek letter outside the fault. An example is given in fig. 17b for the dislocation AB bending from plane (d) to (c). Use of the continuity axiom of 0 1.1 reveals that a stair-rodpartial y8 must exist at the bend. Stair-rods are sessile, since their glide would produce a very high energy fault. Some other important arrays are also shown in fig. 17. Figure 17c shows a Lomer[1964]) formed by the interaction of BA(d) and DB(a), the Cottrell barrier (TEUTONICO barrier strength being associated with the attractive interaction of B6 and n B to form as. The symmetric array in fig. 17c is the original result. More detailed elastic calculations show that this is actually a rather flat saddle point with the true equilibrium configuration being asymmetric with relative extensions of the two arms in a ratio in the range 3 to 4 depending on the values of the elastic constants (KORNERet al. [1979]). The asymmetrical barrier has been found in computer simulations (GAO and BACON[1992]). However, the symmetrical configuration is often seen in experimental observations (e.g., ELKAJBAJI et al. [1988]), possibly because of the influence of nonlinear core-core repulsions. Other types of analogous barriers (Hmm [1961]) are observed in the asymmetrical dissociated configuration (ELKAJBAJI et al. [1988]). Figure 17d is an
Fig. 16. Partial dislocation Burgers vectors in fcc crystals. References: p . 1872.
1848
J.P. Hirth
a.20, $ 3
Fig. 17. Extended arrays in fcc. View is parallel to the (111) planes in a, b, e, and f, and perpendicular to it in d. Faults are intrinsic except for those labeled E which are extrinsic.
extended node, and 17e is an extended dipole. Figure 17f shows the removal of a plane of atoms by vacancy condensation to form a fault bounded by Frankpartials of the type Ds, the vector magnitude equalling the glide plane spacing. Frank partials are also sessile, but, as indicated by their formation process, they can climb by vacancy absorption or emission. Figure 17g is a stacking fault tetrahedron, formed from a Frank loop such as that in fig. 17f. with all faces corresponding to faults. Figure 17h is a stacking fault lozenge (KUMARand HIRTH[1992]). Finally, fig. 17i represents the GALLAGHER [I9701 fault. The parallel arrays of figs. 17a, c, e, h and i, for which interaction forces can be easily calculated in the anisotropic elastic case, are particularly suited for estimates of stacking fault energy on the basis of transmission electron microscopy observations. For example, a recent compilation for the case of GaAs is given by GERTHSENand CAR= t19931. Exuiqsic faults, equivalent to insertion of a plane of atoms into the normal sequence, ABCABbCABC have energies about equal to those of intrinsic faults (GALLAGHER [1970]) but are rarely observed. Extension to form this fault involves pairs of partials
Ch. 20, Q 3
1849
Dislocations
bounding each side. The more complex movements to form these pairs probably hinder this type of extension, accounting for its rare appearance. Analogous faults and partials form in hcp crystals, diamond cubic crystals, and in layer structure crystals such as mica and graphite (AMELINCKX [ 19791). The fault energy is too large for meaningful extension to form a stacking fault in pure bcc metals, so extension into fractional dislocations occurs instead, as already discussed. Among these other structures, there is great interest currently in diamond cubic and structurally related compounds for optoelectronic applications, so this structure serves as an example for more complex structures. A projection of the GaAs sphalerite structure is shown in fig. 18. Dislocations in principle can glide between the triply bonded { 111} planes, the glide set, or the singly bonded planes, the shuffle set (HIRTHand LOTHE [1982]). In conformity with the now standard notation (AMELINCKX [1979]), the dislocations are identified by the atoms at the end of the inserted plane in edge orientation and the appropriate plane, glide (g) or shuffle (s). Hence, the four dislocations are Ga(s), As&), A s ( s ) and Ga(g). Since one often cannot determine between (s) or (g), the dislocations are also identified by the direction in which they move under stress, with a including Ga(s) and As(g) and fl including As(g) and Ga(s). Stacking faults analogous to those in fcc crystals occur in the glide set plane, and dissociation can occur directly for glide set dislocations. Shuffle set dislocations can dissociate into a glide set partial (and stacking fault) and a complex partial, zonal, because it extends over two planes. Glide in such structures is difficult to analyze because of the more complex core structure, the possibility of core reconstruction and the influence of electronic core states and electronic charge effects (HIRSCH[1981]). Recent discussions of such effects for the case of GaAs are given by YONENAGA and SUMINO [1992], MARKLUND and YONG-LIANG [1991] and ALEXANDER and TEICHLER[1991].
(52,
(111)
I a-type
P - type
Fig. 18. GaAs lattice projected normal to [lTO]. a represents atoms in the plane of the paper and /3 represents atoms in the plane below. [lll] is perpendicular to the plane of the paper and appears as a horizontal trace. Planes with ABC contain Ga atoms, those with abc contain As atoms. Positions where inserted planes would terminate on dislocations if appropriate cuts were made. are designated. References: p . 1872.
1850
J.?? Hirth
Ch. 20. 5 3
Conrrary to earlier assumptions, recent work indicates that dislocations slip on the glide [19891) although motion on both sets is suggested by some set (LOUCHET and R~IBAULT observations (CHIANGet al. [1980]). Because of the many types of dislocations in these structures, there is also a wide variety of partial dislocations as discussed for 111-V compounds, for example, by DE COOMAN and CARTER[ 19891. 3.4. Ordered alloys
As an example of ordered alloys, we consider the L1, structure, with Ni,Al as an illustration. Antiphase boundaries (APBs) occur where atom pairs across the interface have the correct atomic positions for the structure but the wrong chemical nature. Dislocations of the type AB (in the Thompson notation of fig. 15) that would be perfect dislocations in the disordered crystal become imperfect in the ordered crystal and must bound APBs. Perfect superdislocations are of the [110] type (2AB) and [lo01 type (2ABXD). In the ordered lattice, the dissociation of AB into partials AS and SB creates a complex stacking fault, i.e., a fault structurally equivalent to an intrinsic stacking fault in fcc but with a superposed APB. Dissociation of 2AB into 2AS and 2SB creates a superlattice intrinsic stacking fault (SISF). A key to the understanding of deformation behavior and associated dislocation configurations in this structure is that the APB energy varies with orientation, being less for { 100) than for { 111) planes. Early regular solution models (FLYNN[1960] as modified by HIRTHand LOTHE [1982]) indicated a large difference in these energies. Values determined by correlation with transmission electron microscopy observations support the trend that the energy for { 100) is less but show that the difference is much smaller, by about 20 to 60 percent (DIMIDUKet al. [1993]). Some of the important dislocation arrays in the L1, structure are presented in fig. 19. A superdislocation dissociated in a planar manner into four partials is shown in fig. 19a. This configuration is convenient for the determination of fault energies (e.g., DIMIDUK et al. [1993]; BALUCe? al. [1991]). Figures 19b and c show variations of the KearWilsdorf lock, the important feature being the extension normal to the { 111} planes. A superdislocation dissociated into superpartials and an SISF is shown in fig. 19d, while a further dissociation of one of these superpartials is depicted in fig. 19e (SUN ef al. [1991a]). An APB tube, which can be formed by cross-slip, is presented in fig. 19f (VIDOZand BROWN[1972]). Figure 19g shows a climb-dissociated loop (in perspective, unlike the other figures) which can be envisioned as a cross-slipped configuration for the screw portions of the loop but a climb-dissociated configuration for the edge and mixed and YOSHIDA[1993]). Figure 19h is the PAIDARet al. (PPV) [1984] portions (TAKASUGI configuration, distinguished from the Kear-Wilsdorf lock by forming a cross-slipped segment only locally over a limited length of the dislocation. The climb dissociation in fig. 19i has been observed by DOWNet al. [1988]. In addition to these defects, barriers and other reacted arrays analogous to those in fig. 17 can form. Of course, additional barriers, such as the double Lomer-Cottrell barrier (SUN et al. [1991b]), can exist because of the more numerous partials in dissociated superdislocations.
Together, the concepts of Peierls energy and stacking faults are sufficient to explain most observed slip systems, with core structure and image effects having an influence in a few special cases. Equation (18), or its equivalent in a more exact model, predicts that the Peierls stress upis least for the slip system with the smallest b and largest interplanar spacing d, that is, b along a low-index direction and d normal to a closed-packed plane. The value of upincreases rapidly with an increase in b or decrease in d, so the Peierls energy criterion would predict only a few low-index slip systems for a given crystal. A low stacking-fault energy is another factor favoring a given glide system, both because the extension of dislocations tends to restrict glide to the fault plane and because upis less for Shockley-type partials. The above ideas are borne out for observed slip systems as reviewed by HIRTH and L o r n [1982]. (110)-{ 111) is the predominant slip system in fcc crystals, fitting both criteria, while (110)-{ 100) slip, on the second closest packed plane, appears occasionally. In hcp crystals with large c/a ratios, giving a relatively large d for the basal plane, which is also a low-fault energy plane, the (1 120)-(0002) basal slip system is favored and is observed. With low c / a ratios, pyramidal slip or prismatic slip become favored and are observed. In bcc crystals, the low-index (111) direction is invariably the slip direction. This non-close-packed lattice has a smaller difference between the largest { 110) and next largest { 112) d spacing, and both slip planes are observed. As discussed in $3.2, the pertinent glide plane for screw dislocations is determined by the behavior of fractional dislocations in the core. The situation for edge dislocations is unresolved, but there is some indication of fractional extension on { 112) planes at low temperatures. Slip that is called anomalous because it does not occur on the slip system of a given type that experiences the maximum resolved shear stress is sometimes observed. For (1 11) { 112) slip in bcc crystals,the core arrangement for motion in, say, a [1113 direction --differs from that in the [l 1 13 direction, so one expects the Peierls stress to differ as well, giving slip asymmetry. The fractional dislocations in fig. 14 reflect this expectation. F~ANCIOSI[1983] has shown that in addition to effects of the Peierls stress, hardening associated with intersections with specific secondary forest dislocations influences this asymmetry. For Ta, anomalous (111) { 110) slip is observed to be profuse at low strains at 77 K (WASSERBACH [1985]). The observation was explained by the mechanism and KUBIN [1975]: because of both elastic interaction and the proposed by L~UCHET formation of intersection jogs (see below) that pin the dislocations, screw/screw forest intersections should be particularly potent in providing hardening. If forest hardening is dominant, one then expects that the slip system with the lowest screw forest dislocation density to operate, and this criterion does explain the observations. In fcc crystals, latent hardening of secondary slip systems varies for different systems, primarily at strains of 2 percent or less and most prominantly for crystals with low stacking fault energy (FRANCIOSI [1985]). This effect is also a form of slip asymmetry. The latent hardening is related to the intersection of systems that form intersectionjogs, extended barriers and junctions.
Ch. 20, 8 3
1853
Dislocations
Some ordered alloys, such as the widely studied case of Ni,Al, exhibit anomalous behavior of slip in that the yield stress increases with increasing temperature (see the review by DMDUK [1991]). The effect had been associated with pins of the type in fig. 19h, which were postulated to form more easily with increasing temperature (PAIDARet al. [1984]; GREENBERG etal. [1978]), although a role of the Peierls stress has also been suggested (CAILLARD et al. [1988]). However, a new view has appeared, stemming from observations of large kinks in screw dislocations that otherwise were locked in the Kear-Wilsdorf configuration in fig. 19b and c (SUNand HAZZLEDINE [1988]). This has led to further observations (COURET ef al. [1993]), simulations (MILLS and CHRZAN [1992]) and other theories (DIMIDUK[1991]; HIRSCH [1992]; VEYSSIERRE [1989]) supporting the view of dislocation glide by a kink formation and lateral propagation mechanism. (See also ch. 23, 24.) In thin crystals, where surface slip nucleation is important, unusual slip systems are observed in the sense that they do not have the maximum resolved shear stress (VESELY [I972]; L o r n [1974]). These results are associated with image effects, either favoring glide by enhancing kink nucleation for screw dislocations oblique to the surface, or suppressing glide by pulling a dislocation more normal to the surface and creating jogs on it. The presence of surface steps that provide favorable sites for dislocation nucleation for dislocations with components of their Burgers vectors normal to the surface (DREGIA and HIRTH[1991]) can give rise to unusual slip. Nonlinear core effects also may have some influence on this phenomenon.
3.6. Jogs An offset in a dislocation line which has a component normal to the glide plane i s called a jog. A jog in a screw dislocation is simply a kink in the cross-slip plane and has properties as discussed in $3.1. A jog in an edge dislocation is shown in fig. 20a. Rearrangement of the jog to a kink-like configuration would involve a very large misfit energy, so the jog remains sharp. Therefore, jogs have energies Wjof the order of the dislocation line energy multiplied by the jog height or pb3.These energies are 10-100 times larger than the formation energies for kinks, W,.Consequently, the concentrations of thermal jogs are much less than those of thermal kinks at a given temperature. Intersection jogs and geometric jogs also exist, analogous to the equivalent kinks. Climb occurs at jogs by vacancy formation or annihilation at jog sites. Self-interstitids could cause climb also, but their formation energy is so large in metals that they are
-
+b Fig. 20. Jogs in (a) a perfect edge dislocation and (b) an extended edge dislocation in fcc.
References: p. 1872.
1854
J. I! Hirth
ch.20,54
only important in the case of radiation damage. For a perfect dislocation, the only constraint to double jog nucleation or jog motion, the processes which produce climb in an analogous way to glide by kink processes as shown in fig. 13, is the formation of the pertinent defect, the double jog or vacancy. An extendedjog in a fcc crystal is depicted in fig. 20b. For a superjog, where the jog height is a multiple of the interplanar spacing, the extended jog is composed of partial dislocations and stacking faults, as shown. For a unit jog with height equal to a single interplanar spacing, the stair-roddipole $ 4exists ~ only over core dimensions. Rather than stair-rods and stacking faults, the jog is then better viewed as a jog-line, equivalent to a row of 1/3 vacancies for the present example (HIRSCH[1962]). Clarifying earlier puzzlement, work stemming from that of CARTER119791 has shown that jogs of all heights, while constricted to some extent, exist in extended configurations: a high-resolution transmission electron micrograph is presented by SAKAet al. [19851 for the fcc case. The extension of the jog evidently provides a constraint to both jog nucleation and motion. In early work, it was thought that the jog would have to constrict to a perfect dislocation to climb. However, as shown by BALL UP^ and GRANATO[1979] a vacancy itself can dissociate into the jog line and cause climb without constriction. Observations by CHERNSet al. [1980] show that extended dislocations can also nucleate double jogs without constriction. Thus, the constraint to climb caused by extension, while present, is less than once expected. Direct climb by this mechanism has been observed, for example, for GaAs (CHERNSand F'EUILLET[1985]) and Si (THIBAULTDESSEAUX et al. [1989]).
4. Dislocation behavior at low homologous temperatures 4.1. Kinkmotion
In bcc metals at low temperatures, glide occurs by kink motion. For low stresses, kinks move and annihilate but nucleation easily replenishes them. The dislocation velocity v is the area swept per unit length per unit time or, fig. 13, the product of the equilibrium number of kinks of both geometric signs 2nk,the kink velocity v,., and the distance between Peierls valleys b. vk is given by the kink mobility DJkT, where Dk is the kink diffusivity, and the driving force from eq. (9), ub2,
-
Uk = -
kT
For larger stresses, kinks are swept to the end of a segment of length L before they can be replenished, and double-kink nucleation is controlling. The formation energy 2wk of a double-kink then enters the velocity expression, which becomes:
and LOTHE[1982] and Further developments and detailed derivations are given by HIRTH
Ch. 20, $ 4
1855
Dislocations
by SEEGER [1981], who also notes that the phenomenon of soliton motion can be described by the same form of kinetics in the latter case. The above relations were derived by LOTHE and HIRTH [19591 in the assumption that w k is the formation energy of an isolated kink and this was revised by SEEGER and SCHILLER[1962] to include the stress dependence in w k arising from kink-kink interaction energies. When eq. (20) holds, kink spacings are typically large, especially in ionic and covalent crystals (e.g., see PICHAUDet al. [1986]) and the stress dependent emendation is negligible. When eq. (21) holds, and especially for metals, the kink-kink interaction term is important (QUENETet al. [1987]). Dislocation motion by this mechanism has received much recent attention in the case of semiconductors and compound semiconductors, where dislocation motion in nearly dislocation-free thin films can be observed directly by transmission electron microscopy. Results by HULLeral. [1991] for the Si& system agree well with the above theoretical equations. In general, the above approach, when used to interpret both dislocation motion or plastic flow at low temperatures and low-temperature internal friction, see 8 4.3, give self-consistent estimates of Wk,with values in rough agreement with the estimates in 8 3.1 (recent examples include SUZUKIand SsIiGER 119921 and Bmom etal. [1987]). In some cases, the inclusion of the stress dependence arising from the kink-kink interaction energy is necessary (QUENET et al. [1987]). All of the above discussion pertains to the case of kinks moving by diffusion under the control of phonodelectron damping. At very large stresses at low temperatures, underdamped, high-velocity kink motion is possible. In this case, as demonstrated by EYKHOLTet a1. [1989], new phenomena are observed, such as the interpenetration of opposite-sign kinks instead of their mutual annihilation. 4.2. Point forces and bowout
For fcc and hcp metals or for bcc transition metals above about mom temperature,
kT>,w k and the effects of the Peierls barrier are removed by thermal fluctuation. In prin-
ciple, dislocations under a driving force could then accelerate until their motion was controlled by damping produced by phonon scattering or radiation. However, except in shockloading, dislocation motion becomes hindered by obstacles and breakaway from the obstacles becomes rate-controlling. A review of this subject is given by KOCKSet al. [19751. Bowout at obstacles is illustrated in fig. 21. For small bowout, the force on a dislocation segment ubh is transmitted to pinning points via line tension forces acting on
F
F
fy F
x
Fig. 21. Bowout from pinning points.
References: p . 1872.
1856
Ch. 20, 44
1.I! Hirth
the pinning points. In the simple line tension case, eq. (14), the resultant force in the direction of bowout acting at the pinning point is pb2 cos(+/2). The opposite stabilizing force F is the interaction force of the dislocation and a pinning point, often so localized that F can be considered a point force. At equilibrium, these forces are all equal and for a linear array of obstacles as in fig. 21.
For stresses u F,; breakaway of dislocations from the pinning
+
points occurs, relevant to solid-solution or precipitation-hardening and to dislocationdislocation intersection. For obstacles so strong that F 1r/3, see KOCKSetal. [1975] and ALTINTASand MORRIS[1986]. Since h e 1 for this case, the resisting force is less than that for a regular array. This can be rationalized in terms of easy breaking points with large Ai or large pi (fig. 25b) within the statistical distribution. Whether the form of eq. (29) holds for real arrays and other than in the approximation of eq. (25) remains an open question. For any version of breakaway from point obstacles, where a constant line tension is assumed, the dependence c = A-’ of eqs. (22) and (29) is retained. For discrete obstacles, other effects appear as shown in fig. 2%. If the particles are harder than the matrix, the usual case, image repulsion leads to a repulsion distance x, which, together with the finite size of the obstacles D, reduces the effective spacing from A to A=. In a refined line-tension model, the dimension A replaces L in eq. (16) and this expression replaces that of eq. (15) in eq. (29). Further, when the interactions between adjacent loops are included, b would be lowered because of the attraction between opposite sign portions 1 and 2 in fig. 25, an effect that could be modeled by further References:p . 1872.
1862
J. I! Hirth
Ch. 20. $4
reducing the outer cutoff in eq. (16) to the dimension D (Asmu [1964]). On the other hand, the harder particle would screen the interaction between segments 3 and 4 which would tend to partially offset the previous effect (HIRTH[1976]). Including all of these effects, one finds, instead of eq. (29):
u =@ A cos(
[.( F)+
f)
A].
Here, h=h(&,Ai,Pi) as above, D is the nominal outer cutoff parameter, the factor c> 1 accounts for the elastic inhomogeneity between the particle and matrix, and A arises from long-range loop-loop interactions. The factor A introduces a (weak) dependence on A. With all of these modifications, the simple u oc A-' OrowadFriedel relation is no longer valid. A few experiments support the form of eq. (30). MESSERSCHMIDT et al. [1985], in a study of loop bowout in MgO, found that bowing is easier than predicted by either eq. (30) or for a single bowed loop as calculated by the anisotropic, straight-line segment case (SCATIXRCOOD and BACON [1975]), implying an effect of the factor A. Also, numerical calculations including looploop interactions (DUESBERY and SADANANDA [1991]) verify that the added h dependence of u arises from the factor A. Still, there are insufficient results of this type to predict a generally applicable constitutive relation of the type of eq. (30) so that the approximate form of eq. (29) is probably sufficient with the present state of understanding. When the critical stress is exceeded for the case of fig. 25c, either the particles are cut or the loops bypass the particles, recombine, and leave a closed loop around each particle, in the Orowan process. Subsequent loops can form, giving rise to circular pile-ups (or their blunted versions) and attendant work-hardening in dispersion-hardened systems. 4.7. Work-hardening
In the absence of strong pinning points, work-hardening is dominated by dislocationdislocation interactions. Early theories for work-hardening emphasized one single mechanism, but it now appears that a number of dislocation mechanisms can contribute. The loop length scaling of eq. (22) still applies for free segments, with dislocation interactions providing the pinning (see KUHLMANN-WILSDORF [19661). Here, we briefly enumerate the hardening interactions. For forest interactions of a dislocation with other dislocations threading its glide plane, the initial interaction is the elastic one of fig. 10, as verified by APPEL[1989]. In the process of intersection, energy must be supplied to form intersection jogs and kinks. The latter process is more important for extended dislocations, but the relative importance of the two contributions is unresolved for perfect dislocations. Continued pinning is provided by sessile jogs on screw dislocations. Forest dislocations can be provided by glide on secondary slip systems or by several special mechanisms. One of these is dipole formation, which can be analyzed using fig. 23. If after cross-slip, h is less than the critical value to operate the new segment as a
Ch. 20, 5.5
1863
Diskxalions
source, the original dislocation and the new segment interact to form dislocation dipoles, pure edge in character. A dipole loop could also form by the condensation of vacancies into a disc which collapses. In materials of low stacking fault energy the latter process leads to the formation of stacking fault tetrahedra, fig. 17. Another possibility in the process of intersection is the formation of attractive junctions, illustrated in fig. 26, which can form for either dissociated or undissociated dislocations (SAADA [1960]). The reacted part of fig. 26b is the Lomer-Cottrell barrier of fig. 17. Because Wi, is negative between the reacting dislocations, energy must be supplied to remove the attractive junctions to complete the intersection process, and therefore their formation contributes to hardening. Grain boundaries or Lomer-Cottrell barriers extensive in length are sufficiently strong barriers to sustain dislocation pile-ups (Hmm [1972]; THOMPSON [1977]). These not only generate back stresses to stop the source producing the pile-up dislocations, but also generate long-range internal stresses, both effects leading to hardening. The ideas discussed here provide the dislocation mechanisms to be considered in work-hardening. The integration of these models to provide a macroscopic workhardening relation is discussed by ARGON [1995].
5. Dislocation behavior at high homologous temperatures 5.1. Osmotic climb forces
At temperatures greater than about O.4Tm,vacancy contributionsbecome sufficient for dislocation climb to be important in deformation and recovery. In the absence of stress, the equilibrium vacancy atomfraction is: c," = exp(-G,/kT),
(31)
where Gf is the formation energy of a vacancy. In the presence of external pressure P, the formation of a vacancy as in fig. 27a requires extra work PSV. Here S V is the volume change on vacancy formation, equal to the atomic volume v, minus the relaxation volume 6v+6v' arising from the partial collapse of neighboring atoms into the vacant site. The term Sv is the local volume change and Sv' is an image contraction of the
Fig. 27. Creation of a vacancy at B by depositing the atom from that site to (a) site A at a step on a free surface or @) site A at a jog on an edge dislocation.
crystal associated with satisfying the free surface boundary condition. In the presence of an internal pressure p at site b there is extra work p(6v + Sv'). Thus, the equilibrium atom fraction becomes
{-[ Gf + Pv, - ( P + p ) (SV + SV')] / k T ) = c," exp {-[ Pv, - ( P + p ) (8u + SV')] /kT}
c = exp
= coexp (-Pv,/kT). It is convenient to include the image terms in co and use this value as a reference concentration for dislocation problems so that first-order effects of stress can be separated from weak effects arising from internal pressure gradients (HRTH and LoTm [1982]). For the creation of a vacancy by the process of fig. 27b, the net work is
6W = -u,,b,ah + cv= -crllva + kTIn(c/cO).
(33)
Here a,,b, is the Peach-Koehler force on the edge dislocation, eq. (9); a is the length and h the climb distance of an atomic unit of dislocation line, so that blah= v,, Gvis the vacancy chemical potential referred to a reference state co, eq. (32), and c is the actual vacancy concentration. The net force in the x,climb direction per unit length is
Here F,,and F, are the elastic and osmotic climb force on the dislocation, respectively.
a.20,§5
Dislocations
1865
More generally, Fel would be the total Peach-Koehler force in the x, direction and could include forces on screw components of mixed dislocations, see 8 2.2. Let us consider some local equilibrium examples. At local equilibrium the total force F/L=O and eq. (34)gives: c = co exp(rl,,u,/kT).
(35)
For a dislocation under uniform hydrostatic pressure P, rI,=-P, eqs. (32) and (35) give the same local equilibrium value of c at the dislocation and at a free surface, and no vacancy diffusion or climb occurs. A dislocation under stress ullwith local concentration c near a stress free surface with concentration co experiences a vacancy concentration gradient and hence a tendency to climb. Often uIlvackTat high temperatures and the gradient can be linearized, c - coz cOa,,vJkT. If local equilibrium does not obtain, either Fd or F, can give a net climb force. For example, a quenched-in concentration c of vacancies producing a supersaturationrelative to co at the final temperature gives an osmotic climb force given by eq. (34). For quenches from near T, to T,,,/2 or less, the osmotic forces are large, corresponding to stresses of -p/10 or greater, of the order of theoretical strengths of perfect crystals. Thus, there is little constraint to climb or to nucleation of vacancy discs which collapse to dislocation loops after quenches from near T,. A special case is the development of a helical dislocation from a straight screw dislocation in which local equilibrium is established by a balance of the line-tension forces, tending to return the helix to a straight line, by the osmotic force on the edge component of the helical dislocation. 5.2. Jog drag
A climb-type process that can contribute to an overall glide process is the drag of jogs by screw dislocations. A jog on a screw is sessile, as seen in fig. 28, since the jog itself has edge character. Yet, whether it is a unit jog or a superjog, it can move along with the screw dislocation by vacancy absorption or emission. A steady-state configuration could be that of fig. 21 with the osmotic forces at the jogs balancing linetension forces from the loops bowed out by the applied glide force. With opposite-sign jogs, the local equilibrium concentrations at the jogs would be given by eq. (35) but with opposite signs of the exponent. Thus, concentration gradients would exist and the rate of
Fig. 28. Unstressed screw dislocation (dashed line) containing a jog, and glide of the dislocation under stress with pinning at the sessile jog.
References: p . 1872.
1866
Ch. 20, 55
J. €! Hirth
climb would be controlled by vacancy diffusion from one sign jog to the other. Relevant to ch. 22, the concentration gradient in this case would involve a difference of exponential~,eq. ( 3 3 , together giving a stress dependence -2 sinh(cr,,vJkT). For bowout forces giving point forces at the jog pinning points F > F: =G&,/v,, breakaway will occur in the sense that the dislocation will move rapidly in glide, athennally generating a string of vacancies (or interstitials) in the wake of the jogs. This can happen because the energy released on glide by a unit atomic distance exceeds the formation energy G, of the appropriate point defect. The forces required for this process are modest when jog spacings are large and this process undoubtedly contributes to the large excess supersaturations of vacancies and interstitials produced in work hardening and detected in recovery measurements (see WOLLENBERGER [19951).
53. Climb In the absence of constraints, local equilibrium vacancy concentrationsare maintained at jogs and the determination of the climb velocity of a dislocation becomes a diffusion boundary-value problem with the jogs as sources and sinks. Often the jog spacing is so small or the pipe diffusivity of vacancies along the core is so large that a dislocation acts as a perfect line source and sink and dislocation lines can be considered as the sources and sinks (BALLUFFIand GRANATO [1979]). The boundary-value solutions are beyond the scope of this work, but consideration of constraint is of interest. In the presence of strong pinning, dislocations must climb to critical configurations as in fig. 22 to activate Bardeen-Hemng sources. In the critical configuration, the sum of the elastic and osmotic forces, eq. (34), must balance the line tension forces @b/A). For straight perfect dislocations, there are no indications of constraints preventing the attainment of local equilibrium vacancy concentrations. For extended dislocations, jog nucleation and jog motion are complex and could provide constraint (53.6). However, experiments on the relatively low stacking fault energy metal gold show that the climb efficiency is greater than or equal to 0.1 (SEIDMANand BALLUFFI[1966]). Here the climb efficiency was defined as the climb rate of a dislocation, under moderate to small osmotic forces produced by quenching, divided by the maximum climb rate under pure diffusion control. In general, constraint would be negligible when the relaxation time for diffusion from source to sink over a distance L,7 L 2 / D , with D, the diffusivity of vacancies, is large compared to the time to nucleate and move jogs over a typical segment length A. In most cases T is expected to be large compared to the nucleation time in the gold quenching experiment. Thus, generally, one expects constraint to be absent so that it is a good approximation to treat climb as a pure diffusional boundaryvalue problem.
-
5.4. Solute drag
In analogy with the vacancy case already discussed, there is an interaction energy pSv between a dislocation with internal pressure field p and a solute atom with a characteristic local volume change 6v, equivalent to a ball inserted into a hole of different size in an elastic medium. A straight screw dislocation has no p field and thus no interaction.
Ch. 20, 5 5
1867
Dislocations
For the edge dislocation of fig. 7, p = - ( u m + u , +u,)/3. interaction energy is:
Hence, from eq. (5), the
yw= &sv(l+ v) sin e 313(1- v)r
*
The solute concentration in local equilibrium in the presence of p , again analogous to 85.1, is: C =
1-c
A exp(-Wb, / k T ) = A exp( 1-c, 1 - c,
).
(37)
where c, is the bulk equilibrium concentration and terms other than r and sine in eq. (36) are included in p. Here it is important to use the Fermi-Dirac form, since for positive exponents c + 1 as r + 0, and the simpler Boltzmann form would artificially diverge. With 6v positive, corresponding to a solute atom larger than the solvent atom, solute is depleted from the compressed region above the glide plane where sin8 is positive, and attracted to the extended region (see fig. 2), as is physically reasonable. The solute of eq. (37) comprises the Corrreffatmosphere. In addition, solute atoms are attracted to either screw or edge dislocation cores with a binding energy EB for the appropriate case, forming a “core atmosphere”. Also, interstitial atoms in bcc metals can form “Snoek atmospheres”. These interstitials produce an asymmetric tetragonal strain in the octahedral or tetrahedral sites with the maximum principal strain aligned along (100) directions. A stress which produces extension along a [loo] direction would then favor preferred occupation of that type of site with a corresponding negative interaction energy. All three atmospherescan produce dislocation drag and breakaway phenomena. As an example, the Cottrell atmosphere is considered here. The formation of the atmosphere is again a pure diffusion problem. For a discussion of the diffusion solutions, see BAIRD[1971]. (See also COTTRELL [1953]). The atmosphere-drag problem involves a diffusion solution, in a moving reference frame, that is quite lengthy and complex. However, quite good approximate results can be found by use of a simpler dissipation theorem (HIRTHand LOTHE [1968]; HILLERT and SUNDMAN [1976]). The diffusion flux J in a volume element 6V is related to the chemical potential gradient V E by the Einstein relation:
J = - - Dc VC. kT The rate of dissipation of free energy in the element is:
with the use of eq. (38). However, for a problem where the driving force for diffusion is the Peach-Koehler force F, the total work done on the system per unit time is F - v where v is the dislocation velocity. Thus, the total force is given by:
Rqferences: p . 1872.
1868
J. k? Hirth
Ch. 20. 5 5
1
kT -dV. J-J F-v = D c Diffusion can occur in response to a concentration gradient Vc or to a potential gradient VW. At steady state in the moving reference frame, the total flux is zero, -DVC
- (D C / ~ Tvw ) - U(C
- c,,) = 0,
so the static frame flux J can be written
J = -DVC - (Dc/kT)VW = U(C
- c,).
(41)
The key to the use of the dissipation theorem is that, except for velocities so large that drag would be meaningless anyway, eq. (41) remains approximately valid with the static rest values substituted for J . Let us consider a dislocation-type problem with a driving force ub in the x direction opposed by a drag force and with a dislocation velocity dx/dt. Equation (40)then reduces to
As the dislocation moves through the solution, diffusion occurs to tend to establish the equilibrium concentration of eq. (37), producing the drag force of eq. (42). At low velocities, c is given by eq. (37) and the substitution of eq. (41) into (42) gives F kTv ---I L D -
(C-c0)2
c
d V.
(43)
In this regime F / L increases with increasing v. At high velocities c = c, V c= 0 and one must use the form J = (-Dc/kT) V W giving:
E = -jc,(VW)*dV. D L
kTv
In this regime F / L decreases with increasing v. The form of the stress-velocity curve is thus that shown in fig. 29. The maximum value ucb= 17 c& occurs at a critical velocity v,n 4DkT/P, according to numerical
vc Fig. 29. Drag force-velocity relation for a dislocation with a Cottrell atmosphere of solute atoms.
Ch. 20, $ 6
Dislocations
1869
solutions of Several types (TmucHI and ARGON [1979]; FUENTES-SAMANIEOO et al. [1984]). In the region of positive slope in fig. 28, stable flow occurs. Large stresses can push the dislocation past the peak to velocities in the negative-slope region where breakaway occurs, with the dislocation accelerating until some other mechanism such as phonon damping controls its motion. At about v,, breakaway can occur, but if the dislocation should be slowed down, repinning can occur, and the behavior can oscillate from pinned to unpinned motion. There are two possible mechanisms for repinning: bulk diffusion to a temporarily slowed or arrested dislocation (VAN DEN BEUKEL[1975]) or pipe diffusion from a forest dislocation in contact with an arrested dislocation (MULFORD and KOCKS [1979]). As discussed by SCHLIPP[1993], the means of distinguishing between these models are subtle and both remain possible mechanisms on the basis of current analysis. These phenomena relate to macroscopic flow behavior in tension through the Orowan relation: B = mpbv,
(45)
where 6 is the strain rate, m is a strain resolution factor and p is the average mobile dislocation density. At temperatures Tw T,,D is large and the slope dab/dv is small, eq. (43). Thus, for a given B , a large v can be attained with a small c7 to satisfy eq. (45) and p will tend to be small. For T e T,,D is small, the slope is large, and even the maximum stress a, is insufficient to produce a sufficiently large v to satisfy eq. (45). Then breakaway to another high velocity region will occur and again p will tend to be small. For T= T,,a =a,,v = v, will be small compared to the other cases, p will tend to be large, and oscillating breakaway-repinning behavior will occur. The breakaway and repinning correspond to macroscopic yield point and serrated flow phenomena. These effects are associated with low ductility which would correlate with the expected larger values of p needed to satisfy eq. (45). Actually, eq. (45) is a very rough approximation and should be replaced by a tensor and LOTHE[1982], ch. 22) with sums of expressions like eq. relation (e.g., see HIRTH (45) over segments of dislocations and the inclusion of segment-segment interactionenergies. While simulations of such many-segment arrays are underway (KUBIN [1993]), there is not as yet sufficient data to give an explicit tensor form to replace the approximate eq. (45).
6. Grain boundaries Grain boundaries represent a broad topic CHADWICK WICK and SMITH[1976], POND [1989], and GLEITF~R [1995]), far beyond the scope of the present treatment. There are two aspects,however, that provide a connection between dislocations and other topics in this text. These are grain-boundary energy and grain-boundary dislocations. (See also ch. 9, $2.2.5). Consider the simple edge-dislocation tilt boundary in fig. 30. The geometry of fig. 30 shows that the number of incomplete planes terminating to form the length h of tilt boundary is n = (2h/b) sin(8/2). Hence the mean spacing between dislocations in the boundary is: References: p. 1872.
1870
J. R Hirth
Ch. 20, $ 6
Fig. 30. A simple tilt boundary (a) and its representation in terms of planes ending at the boundary (b).
with the approximate form holding for small angles 8. The tilt array of fig. 30a is a minimum-energy array because the tension and compression fields of adjacent dislocations overlap and tend to cancel out over distances of the order of D according to St. Venant’s principle. The energy per dislocation is thus given by eq. (8) with R = D = b/O. Since there are 1/D=8/b dislocations per unit area of boundary, the strain-energy [ 19501) is: contribution to the boundary free energy per unit area (READand SHOCKLEY
The term A contains the core energy, and while there are linear elastic estimates of A in terms of the core cutoff r,, they are not expected to be very accurate for reasons discussed previously. Indeed, the measurement of A for small 8 is a good way to estimate the core energy. An equation with a similar &dependence holds for screw or mixed dislocation [ 19601 show that eq. (47) is valid up to 8 = 5’ for boundaries. Experiments by GJOSTEIN copper. Fits of the form of eq. (47) are possible for much larger angles, but the work of Gjostein shows that these do not properly match low-angle experimental data. Interestingly, WOLF [1990] has performed atomistic calculations of grain boundaries for a variety of materials and found that a version of eq. (47) with 8 replaced by sin8 empirically extends to values of 8 much larger than expected (Le., where a hyperbolic functional dependence is expected, HIRTHand LOTHE [ 19821). Many high-angle grain boundaries can be regarded formally as superposed arrays of dislocations spaced atomic distances apart. This description has little advantage over a purely atomic disregistry view as reflected, for example, by the polyhedral structural unit (ASHBYet al. [1978]; PONDetaf. [1978]). However, for high-angle grain boundaries near twin boundaries, the coincidence-lattice, DSC lattice model of BOLLMANN [ 19701 is appropriate. These are illustrated for a first-order { 111]-(112) twin in fcc in fig. 31a. The DSC is essentially the coarsest lattice containing all lattice sites from both constituent crystals of the twin. If a dislocation passes through the twin boundary, continuity of Burgers vector ($1) requires that a grain-boundary dislocation (GBD) is left in the
Ch.20,96
1871
Dislocations
,Coincidence lattice
5 6
s2
L
M
(b) Fig. 31. View parallel to { 11 1) of a firstder { 111)-(112) twin in fcc showing the coincidence lattice and the DSC lattice (a). This view provides a reference lattice for the Burgers circuit in (b) which also shows the GBD in the. DSC lattice. For faulted crystals, the Burgers circuit must start and end on the fault plane.
boundary. As illustrated in fig. 3lb, these GBDs have Burgers vectors correspondingto lattice vectors of the DSC lattice, small compared to' those of perfect lattice dislocations. While the DSC model is useful in some cases, in others, such as hcp twins, there is no applicable DSC lattice. In such cases, the symmetry theory of POND[1989], describing the two lattices that meet at the boundary as superposed to form a dichromatic complex, predicts and describes possible interface defects, including GBDs, in terms of perfect vectors of the dichromatic complex. Even for irrational boundaries or interfaces, a series of defects belonging to nearby orientations can be used to describe the boundary (PONDand HIRTH[19941). References: p. 1872.
1872
J. I! Hirth
Ch. 20, Refs.
' b i n boundaries of relatively low order are expected also to correspond to low-energy configurations or cusps on a grain-boundary energy-orientation plot (CHADWICK and SMITH[19761). Near these cusp orientations small deviations are accomplished by the superposition of small-angle boundaries composed of GBDs. Thus, the percentage of boundaries with dislocation structure is greatly increased if one includes the GBDs. Extensive verificationsby means of observationsof such GBDs in grain boundaries have been made with the use of electron transmission microscopy, e.g., see LN and BALLUFFI [ 19851.
References ALEXANDER,H. and H. TEICHLER, 1991,in: Materials Science and Technology. vol.4. Bectronic Structure and Propemes of Semiconductors, ed. W. S c h t e r (VCH. Weinheim) p. 248. v.I. and v. L. INDENWIM, 1975. SOV. PhyS. USp. 18, 1. ALTINTAS,S.and J. W. MORRIS, Jr., 1986. Acta Metall. 34.810, 809. AMELINCKX, S., 1979,in: Dislocations in Solids, ed.F.R.N. Nabam, (North-Holland, Amsterdam)vol. 2,p. 67. APPEL,F., 1989,Phys. Stat. Sol. 116a, 153. ARGON,A., 1995,ch. 21. this volume. ASHBY. M.F., 1964,Z.Metalk 55. 5. ASHBY. M.F.. E SPAEPEN and S. WILLIAMS,1978,Acta Metall. 26, 1647. BAIRD,J.D.. 1971,Met Rev. 16, 1. BALLUFFI,R.W. and A.V. GRANATO. 1979, in: Dislocations in Solids, ed. F.R.N. Nabam, (Noah-Holland, Amsterdam) vol. 4, p. 102. BALIJC, N.. H.P.K~N-ELUERand M.J. M w , 1991,Phil. Mag. 64A. 137. BASINSKI.S.J. and Z.S. BASINSKI,1979. in: Dislocations in Solids, ed. F.R.N. Nabam, (North H ~ l l ~ d , Amsterdam) vol. 4. p. 261. and A. NEEDL.MMAN, 1992. Mater. Sci. Eng. A159, 1. BASKES,M.I., R G . HOAGLAND BENOIT,W., M. BUJARDand G. GREMAUD,1987,phys. Stat. Sol. 1427. VAN DEN BEUKEL,A.. 1975,Phys. Stat. Sol. 3Oa 197. BOLLMANN,W.,1970,Crystal Defects and Crystalline Jnterfaces (Springer, Berlin). BULUIUGH,R. and V.K. TEWARY, 1979, in: Dislocations in Solids, ed. F.R.N. Nabarm (North-Holland, Amsterdam) vol. 2,p- 1. CAILLARD, D., N. CLEMENT, A. CovRer. P. LOURS and A. COIJJ~U,1988, Phil. Mag. Lett. 58,263. CARLSSON,A. E., 1990,Solid State Phys. 43,1. CARTER, C.B., 1979,Phys. Stat. Sol. 54a, 395. CHADWICK, G.A. and D. A. SMITH,1976. Grain Boundary Structure and Properties (Academic, New York). CHERNs. D.. P.B. HIFKSCH and H. SAKA. 1980,Roe. Roy. Soc. (London) 371A. 213. CHERNs, D. and 0. FEUILLET, 1985, Phil. Mag. S A , 661. CHIANG, S.W.,C. B. CARTERand D. L. KOHLSI'EDT,1980,Scripta Metall. 14,803. CHOU,Y.T. and J.C.M. LI, 1969, in: Mathematical Theory of Dislocations, 4. T. Mura (Am. Soc. Mech. Eng.. New York)p. 116. DE CoobmN, B.C. and C.B. CARER, 1989. Acta Metall. 37.2765. COTTRELL,A. H.. 1953,Dislocations and Plastic Flow in Crystals (Clarendon Press, Oxford), p. 133. COURET, A.. Y.Q. SUNand P.B. HIRSCH.1993. Phil. Mag.67A, 29. D~VINCIE, B. and M. CONDAT,1992, Acta Metall. Mater. 40,2629. D m m , D.M., 1991, J. Phys. m,1,1025. DIMIDUK. D.M.. A. W. THOMPSON and J. W. WILLUMS,1993. Phil. Mag. 67A, 675. DOBROMYSLOV, A.V., G.V. DOL.GUCH and I. A. PWETURINA, 1984,Phys. Stat. Sol. ?Ha,559. DOWN, P., P. BEAUCHAMP and P. VEYSSIERE. 1988.Phil. Mag. 58A. 923. D ~ I AS.A. , and J. P. HIRTH, 1991,J. Appl. Phys. 69,2169. ALSHITS,
Ch. 20. Refs.
Dislocations
1873
DUESBERY, M. S., 1983,Acta Metall. 31, 1747, 1759. DIJFSBERY,M. S., B. Joos and D. J. MICHEL,1991,Phys. Rev. B43,5143. DIJFSBERY,M. S. and K. SADANANDA, 1991, Phil. Mag. 63A, 535. DWBERY, M. S., N. P. LOUATand K. SADANANDA,1992,Phil. Mag. 65A,311. FLKAJBAJI, M.. H. 0.K. KIRCHNER and J. THIBAULT-DESSEAUX, 1988,Phil. Mag. 57A, 631. EMBURY,J.D., 1971, in: Strengthening Methods in Crystals, eds. A. Kelly and R.B. Nicholson, (Applied Science Publishers, London) p. 331. ESHELBY,J. D., 1953,J. Appl. Phys. 24, 176. EYKHOLT,R., S.A. TRUGMAN and D.J. SROLQVITZ, 1989,J. Appl. Phys. 65,4198. FANTOZW,G., C. ESNOUP, W. BENOITand I. G. RITCHIE, 1982,Prog. Mater. Sci. 27.31 1. FIUIPOV,A. P., G. N. GAIDIJKOV and S.K. MAKSIMOV, 1985,Phys. Stat. Sol.YOa, 215. FLEEHER, R. L., 1964,in: Strengtheningof Metals, ed. D. Peckner @einhold, New York) p. 93. FLINN,P., 1960, Trans. TMS-AIME 218, 145. FRANCIOSI. P., 1983,Acta Metall. 31, 1331. FMNCIOSI. P., 1985,Acta Metall. 33, 1601. FRIEDEL, J., 1956,Les Dislocations (Gauthier-Villars, Paris) p. 123. FUENTEESAMANIEGO, R., R. GASCA-NERIand J. P. HIRTH, 1984,Phil. Mag. 49A,31. GALLAGHER, P.C. J., 1970,Metallurg. Trans. 1,2429. GAO,F. and D. J. BACON,1992. Phil. Mag. 6 6 4839. GERTHSEN.D.and C.B. CARTER, 1993, Phys. Stat. Sol. 136a,29. GJOSTEIN, N. A., 1960, Acta Metall. 8,263. GLEITER,H., 1995,ch. 9,this volume. GRANATO,A.V. and K. LOCKE,1956.3. Appl. Phys. 27,583. GREENBERG,B. A., M. A. IVANOV, Yu.N. GORNOS~YREV and L. E. KARKINA, 1978,Phys. Stat. Sol. 49a, 517. HAASEN, P., 1995,ch. 23,this volume. HANSEN,N. and D. KUHLMANN-WILSDORF, 1986,Mater. Sci. Eng. 81, 141. HILLERT,M. and B. SUNDMAN,1976,Acta Metall. 24,731. HIRSCH,P.B., 1962,Phil. Mag. 7,67. HIRSCH,P.B., 1992,Phil. Mag. 65A,569. HIRSCH,P. B., 1981,in: Defects in Semiconductors, eds. J. Narayan and T.Y. Tan (North-Holland, Amsterdam) p. 257. HIRSCH, P.B. and S. G. ROBERTS,1991. Phil. Mag. 64A,55. HIRTH,J. P., 1961,J. Appl. Phys. 31,700. HIRTH,J. P., 1972,Metallurg. Trans. 3,3047. HIRTH,J.P., 1976,Scripta Metall. 10,755. HIRTH, J.P. and J. LOTHE, 1982, TheoIy of Dislocations, 2nd Ed. (Wiley, New York); 1992, Krieger (Melbourne, Florida). HIRTH,J. P., T.J-NG and J. LOTHE, 1966,J. Appl. Phys. 37, 110. HOAGLAND, R. G., 1994,J. Mater. Res., 9, 1805. HULL,R., J. C. BEAN,D. BAHNCK,L.J. PETICOLAS,Jr., K. T. SHORTand F.C. UNTERWALD, 1991,J. Appl. Phys. 70,2052. INDENBOM,V. L., B. V. PETUKHOV and J. LOTHE, 1992,in: Elastic Strain Fields and Dislocation Mobility, eds. V.L. Indenborn and J. Lothe (North-Holland, Amterdam) p. 489. IIJNIN, Yu.L., V. I. NIKITENKO, V. I. ORLOV and B. Ya. FARBER,. 1991,Sov. Phys. JETP 73, 1079. KAMAT,S.V. and J. P. HIRTH,1990,J. Appl. Phys. 67,6844. KOCKS, U.F., A.S. ARGONand M.F. ASHBY,1975,Prog. Mater. Sci. 19, 1. KORNER,A., H. SCHMIDand E PRINZ,1979,Phys. Stat. Sol. 51% 613. KUBW, L. P., 1993,in: Materials Science and Technology, eds. R.W.Cahn,P. Haasen and EJ. Knuner (VCH, Weinheim) p. 137. KUBIN, L.P., Y. E~TRINand G. CAWVA, 1990, in: Patterns, Defects and Material Instabilities, eds. D. Walgraef and N. M. Ghoniem (Kluwer,Dordrecht) p. 277. KUHLMANN-WILSDORF, D., 1%6, in: Work Hardening, eds. J.P. Hirth and J. Weertman (Gordon and Breach, New York) p. 97.
1874
. I F?.Hirth
Ch. 20. Refs.
KUHLMANN-WILSDORF, D., 1989, Mater. Sci. Eng. AlW. 1. KUMAR,A.M. and J. P. HIRTH,1992, Phil. Mag. 65A, 841. LABUSCH,R., 1972, Acta Metall. 20,917. LAVRENTEV, E E, 1980. Mater, Sci. f i g . 46,191. LN,J.S. and R.W. BALL^, Phil. Mag. 5 2 6 713. LOHNE, 0..1974, Phys. Stat. Sol. (a) 25, 209. LOTHE, J., 1967, Phys. Norvegica 2, 153. LOTHE, J. and J.P. HIRTH,1959, Phys. Rev. 115,543. L~UCHET, E and L. P. KUBIN,1975, Acta Metall. 23, 17; Scripta Metall. 9,911. Loucm, E and J. THIBAULT, 1989, Struct. Prop. Dislocations in Semiconductors(Inst. Phys., London) p.47. MARKLIJNDS. and W. YONG-LIANG. 1992, Solid State Comm. 82, 137. MESSERSCHMIDT, U.,J. SHU and H. S c m m , 1985, Phil. Mag. A51.781. MILLS.M. J. and D. C. CHRZAN. 1992, Acta Metall. Mater, 40,3051. MOTT, N. E and E R. N. NABARRO, 1948, in: Report on Strength of Solids (Physical Society. London) p. 1. MULFORD,R. A. and U. E KOCKS. 1979, Acta Metall. 27, 1125. MUSKHELISHVILI, N. I., 1975, Mathematical Theory of Elasticity, ch. 19 (Nordhoff, Leyden). E R. N., 1947, Roc. Phys. Soc. (London) S9,256. NABARRO, NABARRO, F.R. N., 1952, Adv. Phys. 1,269. NABARRO. F. R. N., 1967, Theory of Crystal Dislocations (Oxford Univ. Press, Oxford). NABARRO, F.R.N., 1986, in Strength of Metals and Alloys 7, eds. H. J. McQueen et al. (Pergamon, Oxford) p. 1667. NANDEDKAR, A. S. and J. NARAYAN, 1990, Phil. Mag. A61, 873. NEUHAUSER, H., 1990, in. Patterns, Defects and Material Instabilities, eds. D. Walgraef and N.M. Ghoniem (Kluwer, Dordrecht) p. 241. OROWAN,E., 1934,Z phys. 89,605 and 634. PAIDAR, V., D. P. POPEand V. VITEK,1984, Acta Metall. 32,435. PEACH,M. 0. and J. S. KOEHLER, 1950, Phys. Rev. 80,436. PEWLS, RE., 1940, Proc. Phys. SOC.(London) 52.23. PICHAUD,B., P. JEANand E MINORI, 1986. Phil. Mag. MA, 54. POLANYI, M., 1934, Z. Phys. 89, 660. POND. R. C., 1989, in: Dislocations in Solids, ed. F.R.N. N a b m (Elsevier, Amsterdam) vol. 8, p. 1. POND, R.C. and J. P. HIRTH,1994, Solid State Phys. 47,287. POND, R.C., D. A. SMITH and V. VITEK.1978, Scripta Metall. 12,699. PRINZ,E,A. KORNER and H. 0.K. KIRCHNER, 1983, Phil. Mag. 47A, 441. PULS. M.P., 1981, in: Dislocation Modeling of Physical Systems, eds. M.F. Ashby et al. (Pergamon, Oxford) p. 249. QuENa, B., 0. GRENAUD,M. BUJAUDand W. BENOIT,1987. J. Phys., Colloq. C8 48, 125. READ, W.T. and W. SHOCKLEY, 1950, Phys. Rev. 78,275. RICE,J. R., 1992, J. Mech. Phys. Sol. 40, 239. RICE,J. R. and R. THOMSON, 1974, Phil Mag. 29,73. SMDA,G., 1960, Acta Metall. 8, 841. T. IMURA and J. WATANABE, 1985, Phil. Mag. 52A. L7. SAKA.H., S. SUGAWARA, SCATIXRWD, R.O. and D. J. BACON,1975. Phil. Mag. 31, 179. SCHWPF, J., 1993, Scripta Metall. Mater. 29, 287. SCHMID. H. and H.O. K. ~ R C H N E R , 1988, Phil. Mag. S A ,905. SCHOFCK, G.and W. PUSCHL,1991, Phil. Mag. @A, 931. SEFGER. A., 1981.2. Metall. 72.369. SEEGER, A. and P. SCHILLER,1%2. Acta Met. 10,348. SEEGER,A. and C. WO~HRICH,1976. Nuovo Cim. 33B. 38. SEIDMAN. D. N. and R. W. BALLUPPI, 1966. Phys. Stat. Sol. 17, 531. SHAIBANI, S. J. and P. M. HAZZLEDINE, 1981, Phil. Mag. 44A, 657. SMITH.E., 1968, h. Roy. Soc. (London) A305.387. SONDHI,S., R.G. HOAGLAND and J.P. HIRTH,1992, Mater. Sci. Eng. A152, 103.
Ch. 20, Refs.
Dislocations
1875
SUN, Y.Q.and P.M. HAZZLEDINE,1988,Phil. Mag. S A ,683. SUN, Y. Q.,M. A. CRIMP and P. M. HAZZLEDINE,1991a,Phil. Mag. MA,223. SUN,Y.Q.,M. A. Crimp, P. M. HAZZLEDINEand A. COURET,1991b,Phil. Mag. MA, 31 1. SUZUKI,S. and A. SEECER, 1992,Phys. Stat. Sol. 1314 727. TAKASUGI, T. and M. YOSHIDA,1993,Phil. Mag. 67A,447. TAKEUCHI, S. and A. S. ARGON,1979,Phil. Mag. 40,65. TAYLOR,G.I., 1934,Roc. Roy. Soc. (London) A145.362. TEUTONICO, L. J., 1964,Phil. Mag. 10,401. THIBAULT-DESSEAUX, J., H.0.K. KIRCHNER and J.L. PLJTAVX, 1989, Phil. Mag. @A, 385. THOMPSON, A. W., 1977, Work-Hardening in Tension and Fatigue (Met. Soc. AIME, Warrendale, PA). THOMPSON, N., 1953,Proc. Phys. Soc. (London) 66B,481. THOMSON, R., S.J. ZHOU,A. E. CARLSONand V. K. TEWARY, 1993,Phys. Rev. B47,7710. TIMPE, A., 1905,2. Math. Phys. 52, 348. VESELY,D., 1972,Scripta Metall. 6,753. VEYSSIERE, P., 1989,in: High Temperature. Ordered Intermetallic Alloys 111, eds. C.C. Koch et al. (Materials Research Society Symposium Proceedings, Pittsburgh, PA) 133,p. 175. VIDOZ, A. E. and L. M. BROWN, 1972,Phil. Mag. 7, 1167. VITEK,V., 1974, Cryst. Latt. Defects 5, 1. VOLTERRA, V., 1907,Ann. Ecole Norm. Super. 24,400. WASSERBACH, W., 1985, Phil. Mag. 51A,619. WILKENS,M., 1969,Acta Metall. 17, 1155. DEWIT,G. and J. S. KOEHLER,1959,Phys. Rev. 116, 1113. WOLF, D., 1990,Acta Metall. 38,781, 791. WOLLENBERCER, H. J., 1995,this volume, ch. 18. WUTHRICH,C., 1977,Phil. Mag. 35,337. YONENAGA, I. and K. SUMINO,1992,I. Appl. Phys. 71,4249; 1993.73,1681.
Further reading Amelinckx, S., 1964,Solid State Physics, Suppl. 6. Friedel, J., 1964,Dislocations (Addison-Wesley, Reading, MA). Hirth, J. P.. and J. Lothe, 1992,Theoty of Dislocations (Krieger, Melbourne, FL). Nabarro, F. R. N., ed., 1979,Dislocations in Solids, vols. 1-8 (North-Holland, Amsterdam). Rabier, I., A. George, Y. Brechet and L. Kubin, 1993,Dislocations 93, vol. 35-36, Solid State Phenomena (Scitec, Zug, Switserland). Read, W. T., Jr., 1960.Dislocations in Crystals (McGraw-Hill, NY).
CHAPTER 21
MECHANICAL PROPERTIES OF SINGLE-PHASE CRYSTALLINE MEDIA: DEFORMATION AT LOW TEMPERATURES A.S. ARGON Massachusetts Institute of Technology, Department of Mechanical Engineering, Cambridge, MA, U.S.A.
R. W Cahn and I! Haasent, eds. Physical Metallurgy:fourtk revised and enhanced edition 0 Elsevier Science BY 19%
1878
A. S. Argon
Ch. 21, 5 1
1. Overview In this chapter we discuss the deformation of single phase crystalline solids both from the point of view of actual mechanisms and to some extent from the point of view as continua. In the discussions of the precipitate particle resistance we will by necessity consider solids with a light concentration of heterogeneities. Outside of this natural excursion, however, we will not consider the broad subject of mechanical properties of composites. Moreover, our coverage will concentrate on mechanistic explanations of phenomena and will not develop any operational aspects in the form of deformation boundary value problems. References to such developments, however, will be given throughout the chapter wherever they are considered to be particularly relevant. The subject of primary concern in this chapter and the closely allied chapter 22 is the modes of response of a crystalline solid to applied stresses. In the present chapter we develop the responses that occur in the absence of diffusion. The complementary discussion of deformation processes in the presence of diffusion where that controls either the recovery processes which govern the deformation rate, or enters directly in the production of strain by material transport is given in chapter 22. Deformation of amorphous solids will be considered only very briefly in section 11 where their mechanisms are contrasted with the mechanisms of strain production in crystalline solids. We will use widely accepted terminology for phenomena of the different constituitive connections between strain, strain rate, temperature and stress. We call the time independent, completely reversible, linear connection between stress and strain components elasticity. All other forms of time and deformation rate dependent connections are collectively called inelastic. Among inelastic, however, we distinguish a number of different patterns of behavior. We call anelastic a linear but temperature and time dependent, completely reversible connection between stress and strain, arising from topologically isolated thermally activated shear relaxations in an elastic background structure. The formal theory of linear viscoelasticity, which we will not discuss, is a direct outcome of this connection. We call visco-plastic the temperature and moderately deformation-rate dependent, non-linear, and usually irreversible, connection between stress and strain. It is more meaningful to consider this as a constitutive connection giving the dependence of the deformation rate on temperature, stress and the defect microstructure. We will consider two limiting forms of this behavior: first the high temperature and low stress limit of linear, liquid-like, viscous flow, and second, the low temperature, high stress and nearly deformation rate independent form of plastic flow. The latter will be of primary interest. At elevated temperatures where the deformation-rate controlling defect microstructure can decompose by diffusion assisted processes, the visco-plastic connection becomes much more strongly deformation-rate dependent. This behavior we will refer to as creep and delay its discussion for chapter 22. In the discussion of these phenomena we emphasize primarily the mechanistic aspects of the processes at a moderately advanced level but also show the operational forms in which these constitutive connections become useful in the solution of actual macro-scale technological problems.
Ch. 21, 92
Mechanical properties of single phase crystalline media
1879
2. Kinematics of deformation 2.1. Elasticity as an affine transformation Hookean linear elasticity can be most generally characterized as a nearly instantaneous connection between any element of the symmetrical strain tensor and all elements of the symmetrical stress tensor, or vice-versa. In the first case of the strain-stress connection, the connecting tensor is known as the compliance tensor, while in the second case the connecting tensor is called the elastic constant tensor. These tensor connections are discussed in many excellent references (NYE[ 19571; HWTHand LOTHE [19821). For our purpose it is important to note that the source of the elastic response is at the atomic level and is a direct measure of the interatomic resistance to separation or shear. In a homogeneous crystal subjected to a homogeneous increment of stress, all atoms experience the same relative displacements, resulting in a homogenous increment of strain in the entire crystal, down to the scale of atoms. This can be viewed as an afine transformation, and the “material point” where the connection is considered can be taken down to the atomic scale where the material ceases to be a continuum. This property of elastic response is fundamentally different fiom all inelastic deformations which are inhomogenous over much coarser scales. We note that the elastic response of a solid manifests itself in two physically distinct forms: dilatation and shear. In an isotropic idealization of the elastic connections the dilatation 8 is proportional to the mean normal stress (negative pressure), a,,through the bulk modulus K,and the shear strain y is proportional to the shear stress u through the shear modulus p, i.e.,
where
u = applied shear stress
3
]
Tr(a - oml) 2 4.,
e = E,, + E22 + E33 = T+); y = tangential shear strain
(4)
(5)
[2Tr(e- 4O I r ] t ,
(6)
where the * indicates the 3-dimensional interpretation (see also sect. 9.1). In the above expressions cr, e, and Z are the stress tensor, the strain tensor, and the identity tensor respectively; ut,etc., and E , , etc., are the normal stress and strain components, and Tr stands for “trace”. The RHS expressions in Eqns. (4) and (6) define the two invariants References: p. 1951.
1880
A.S. Argon
ch. 21. 8 2
of the stress and strain tensors respectively, which provide the basis of the physically distinct nature of the dilatational and shear responses of a solid. The well-known Young's modulus E, and Poisson's ratio, v, giving the uniaxial tensile stresdstrain connection and the fractional lateral contraction, measure a combination of bulk (dilation) and shear responses. Therefore, they are less desirable forms of representation. For an isotropic solid they are related to the bulk modulus and shear modulus by the expressions:
.
E = - 9K 3K" 1+P
1-- 2P v=----- 3K 2P 2+-
*
3K
The elastic connections introduced above are defined always at zero strain. As strains increase in magnitude, the connections become progressively non-linear and ultimately lead to volumetric decohesion or ideal shear. The self similar bulk decohesion obeys a remarkably simple scaling relationship arising from a Universal Binding Energy Relation that has been demonstrated by ROSEet al. [1983] which can be given for uniaxial tension simply as: a11
= EOSI exP(-%)
(8)
where E, is the uniaxial, zero strain, Young's modulus (or is (lu)Ed(l+v)(l-2~)for the case of only uniaxial strain deformation), and a is the reciprocal of the ideal uniaxial decohesion strain. From here it is clear that the strain dependent decrease of the Young's modulus can be obtained readily as: E(€) =
$(I
- cu~ll)exp(--Q~ll),
(9)
The corresponding ideal shear response at large shear strains, is less well understood because of shear induced break-down of symmetry. It is a subject under active study. The fundamental atomic level bases of the elastic connections and temperature dependence of the elastic constants have been widely discussed. While first principles connections are still rare, these are becoming better understood (see, e.g., NASTERand WILLAIME [1995]). Reliable experimental information on many pure solids and binary solid solutions for both single crystals and polycrystal averages are available in the literature (see, e.g., SIMMONS and WANG[1971]). While dilatational and shear responses of an elastic solid are mechanistically distinct, they are not uncoupled. The presence of a pressure results in the stiffening of the shear response and the presence of a shear stress alters crystal symmetry, and, therefore, affects the bulk modulus. (For an operationally useful treatment see RICEet al. [1992]).
2.2. Kinematics of inelastic deformation 2.2.1. Plasticity resulting from transformations Inelastic deformation cannot be exhibited by a perfect crystal. It requires for its local development a defective solid state which permits the various forms of local strain producing configurations to take shape. In the broadest sense, all inelastic deformation
ch.21, $ 2
Mechanical properties of single phase crystalline media
1881
can be viewed as arising from a set of shear transformations, occurring in small volume elements, stimulated by the prevailing local stress state and emanating from the variety of crystal or structural defects. Thus, inelastic deformation is always inhomogenous on the mechanism scale and must be defined as an average over a volume element that must contain enough elementary transformations to result in an acceptably smooth process. The smallest such volume element above which the inelastic response can be considered smooth will be labeled as a representative volume eZement (RVE).It represents the scale of the so-called “material point” in any continuum field description of the process. We consider first the kinematic aspects of inelastic deformation, separate from their energetic aspects, or the driving forces that cause them to develop. While such separation is instructive in clarifying the kinematics, it is not realistic in understanding the overall nature of the development of shear relaxations. Therefore, caution is necessary. Consider, as shown in fig. la, a shear transformation in a small volume element sl,, occurring inside the large volume V. We define by .eTthe unconstrained transformation strain (tensor) as it would deveIop freely outside the body. When constrained by the elastic surroundings the transformation strain of the interior is only E‘ where all distant stresses are relaxed. If the volume element flf is an ellipsoid and the exterior volume V is very large (infinite) the inner field is homogenous (ESHELBY [1957]). When the outer body is of finite size the effect of the transformed volume on the outside is to set up a compensating effective homogenous “image strain”, in the same sense as the transformation strain inside fir (ESHELBY[1954]). Thus, if a total volume fraction c of material has undergone the same transformation then the total volume average strain in the large [1940]),* body as a whole becomes (NABARRO E
= CET.
(10)
If there are m different types of transformations, as depicted in fig. 1, each producing separately unconstrained transformation strains E: and having volume fractions c, then collectively they produce an overall volume average strain, E
=
C,E,T, rn
provided that c, remains relatively small so that mutual interactions between volume elements f&,, rn can be neglected, and all transformed volume elements have the same mechanical properties.
23.2. Plasticity resulting from dislocation glide When the inelastic deformation is produced by a uniform shear Sy in lenticular volume elements with principal planes, normal to a unit normal vector n, and shear
* In this, and subsequent sections of this chapter the strains that are being referred to are all inelastic strains, or plastic strains. When they are experienced they occur in an elastically strained background. Thus, they should be conceived of as the residual strains that remain after the deformed body is unstressed. References: p, 1951.
A. S. Argon
1882
Ch.21, 4 2
Fig. 1. Inelastic strain production: a) shear transformation producing a free standing transformation shear strain e’ = y in a volume SZ, of a large sample of volume V; b) average strain e produced in a volume V as a result of three sets of transformations e’“ (m = 1,2,3).
direction parallel to d, the average inelastic strain increment in the sample is still the volume average of all inelastic shears and is given by =
(T)jA + (dinj+ dini)& +
= cay (dinj + dini)
where h is the average thickness of the “lentils”, diand ni are the components of the unit vectors d and n in the direction of the macro axes i and j, the area integral is over the principal plane areas of the entire set of sheared lentils and c is the total volume fraction of sheared lentils. Clearly, if the shear is produced by a set of m separate transformations having distinct plane normals n”’,shear directions d“ and shear increments Sy,,,, the overall increment of plastic strain in the macro sample is
a&,;
= ~c,ay,cY;
(124
m
where at;=
+ (qn; + di”n,”)
Wb)
is the so-called Schmid strain resolution tensor. Finally, when the shears are due to individual translations of dislocation loops, or sweeping of dislocation segments of
Ch. 21, 5 2
Mechanical properties of single phase crystalline media
1883
portions of slip planes having unit normal vectors n with Burgers vectors in the direction d and having magnitudes b, the local shears degenerate into discontinuous translations and the strain increments, hereafter to be called plastic, of a set of m separate slip systems become, as a special case of eqs. (12a) and (12b),
Here a! is of the same form as given in eq. (12b) and the swept-out area Sa, is for the entire length of dislocation line of the specific set m under consideration. Shear operations of this type, depicted in fig. 2, and resulting from relative translation of parts of the crystal over each other in the direction of the Burgers vector, are called slip." Often, single-slip systems are of interest in mechanistic considerations, for which eq. (13) simplifies to
a&;
(--)."
= b6a
(14)
where 6a is again the entire area increment swept by mobile segments of dislocations on the slip planes oriented in relation to the external axes by aiias depicted in fig. 2. When this motion can be well characterized as an average displacement Sx of a total segment length A of dislocations,
where the final form pM (= 1)has been introduced as the mobile dislocation (segment) Y density, that is, the mobile line length per unit volume.
Fig. 2. A shear strain increment 6y in a volume V resulting from a collection of translations (slip) over a total area Sa with unit normal vectors parallel to n and in the direction d lying in the planes.
* We will use dip to refer to an accomplished translation and glide to refer to the conservative motion of dislocations. References: p. 1951.
1884
A. S. Argon
Ch. 21, 8 2
Finally, whenever time enters into the motion of dislocations to establish a kinetic basis of the process, eq. (15) readily transforms into a rate expression as
Equation (16) relates the glide rate, based on the dislocation velocities ito the external strain rate, and constitutes a kinematic expression based on dislocation mobilities. In certain instances the kinematics of the strain rate needs to be viewed differently where a characteristic incubation (activation) time fa must elapse to form an embryo. This then produces an increment of volume fraction, Sc, of the transformation 2,or results in an increment of area Sa of translation of dislocation loops, within a time much shorter that r,,. This constitutes a nucleation controlled deformation process and gives rise to a plastic strain rate of
sc
CP = -ET la
due to transformations, or CP IJ
Sal = b -- a.., fa
v
IJ
due to slip. It has applications to phenomena suc as twinning, ma^ rnsitic transformations, or in the case of dislocation glide to problems of thermally overcoming specific slip plane obstacles.
23. Lattice rotations accompanying slip Slip results in lattice rotations that are instrumental in the development of deformation textures which we will discuss briefly in section 8. The concepts are illustrated schematically in fig. 3. Figure 3a illustrates a reference
Fig. 3. Kinematics of the various components of crystallographic slip and rotation: a) initial undeformed lattice, b) plastically sheared lattice, c) plastically sheared and rigidly rotated lattice.
Ch. 21, 9 3
Mechanical properties of single phase crystalline media
1885
volume element along with the unit vectors n"' and 8' representing the slip plane normal and the slip direction of the system m. Without loss of generality, the instantaneous configuration can be taken as the external shape reference. At fixed lattice orientation an increment of crystallographic shear Sy" on the system m generates an increment of plastic strain 84 and an increment of lattice rotation provided by the mth system, is given by:
wj,
88; = 8y"(Y,7
sw;
=8y"g
(1% b)
where
p,; = 4 (dyny - d p q )
(19)
is the skew symmetric counterpart to the Schmid strain resolution tensor. The resulting intermediate deformed configuration of the body is depicted in fig. 3b. It must now be noted that the total crystallographic strain increment and plastic-strain induced rotation must be given by sums over all possible slip systems k as:
m=l
nt=l
Homogenous (or homogenized on the macroscale) slip processes alone, in general, alter lengths and orientations of fiducial lines and angles drawn on the element, but leave the lattice unchanged. Lastly, the intermediate configuration could be subject to a strain-free rigid body rotation increment SWg which carries both material line elements and lattice vectors to their final orientations. This is shown as fig. 3c. This final stage of rigid body rotation is important in the operational analysis of deformation processing problems, but of no great interest to us here. While all the kinematic relations discussed above have been derived on the basis of shears and translations of planes parallel to each other, they are applicable without change to dilatations as well. Thus, for example, if the areas in eq. (13) or eq. (14) were to be separated (or contracted) by a magnitude b, parallel to the unit normal vector n, this would be reflected in the definition of the resolution factors a#;in this case of thickening or thinning of the area, nid. The type of dislocation motion which accomplishes this is called climb.
3. The mechanical threshold of deformation 3.1. The critical resolved shear stress for glide It is intuitively clear that large strain distortions can only be accomplished by shear and that, as a result, the driving forces for it must involve shear stresses, or at least, a critical deviation from a state of pure mean normal stress (section 9). In that sense it is possible to conceive of a fundamental thought experiment carried out at 0 K, in which a nearly perfect crystal containing the necessary complement of microstructural or lattice defects is subjected to increasing levels of shear stress, while the shear strain rate, over References: p, 1951.
1886
A.S. Argon
ch. 21. $ 3
Fig. 4. Idealized threshold behavior: at T = 0 the deformation threshold is sharp. at T > 0 it is softened by thermally assisted processes, and ?(T) is less than .7(0).
and above the ubiquitous rate-independent elastic flexure, is monitored and plotted against the shear stress. Figure 4 shows the likely outcome, of no detectable shear strain rate, for shear stresses u less than a definite value .i(O). When u =.i(O) the shear strain rate should undergo an abrupt and, perhaps, only inertially limited increase through an apparent initiation of inelastic deformation by a specific mechanism. Clearly, in real materials with the usual internal variability of defect structure, the transition from no deformation to a high rate of deformation should be somewhat less abrupt but still exhibit a threshold behavior. We take this material property +(O) as the mechanical threshoM to initiate deformation by a specific mechanism such as, e.g., an intrinsic lattice resistance to dislocation glide or resistance to such motion of a field of solute atoms, precipitate particles, and the like, associated with a specific level of defect structure. Thus, under this idealization we expect that
p=0
for
p .> 0
(and large) for
u < +(O)
(1 9 4
u = ‘i(0)
(19b)
and that u >+(O) should be unattainable under quasi-static conditions with a given initial defect structure. At T>O the above behavior should be ameliorated in two important ways. Since the basic process governing .iwill usually be an elastic interaction on the atomic level, an increase in temperature should decrease +(T) ( AG') however, the net strain rate is adequately given by
Ch. 21, 5 4
Mechanical properties of singre phase c y t a l l i n e media
1891
(31) the familiar Arrhenian expression. 43. Principal activation parameters for plasticity
Before considering specific processes of crystal plasticity, we develop a number of generally useful and instructive thermal activation parameters that characterize systems deforming close to their mechanical threshold. In this range (see fig. 6c)
AG' = VIyu(7(y) - a)dy = VrAyO(7)d7= rAyd(7)flrd7 Y.
where Aya is the system activation strain, and Ayd the cluster activation strain respectively, the two being related by eq. (20). For many, but not all, of the thermally activated local deformation processes, the energy barrier is governed by an elastic interaction mechanism which scales by some set of elastic constants, but usually by the shear modulus, p, which has a known temperature dependence resulting from anharmonic effects. Thus, we can write
AG' = AGOP(T) Po
(33)
where poand AGOrefer to a reference state which as a first approximation can be taken at 0 K. Moreover, we recognize that
AG' = AH'
- TAS'
(34)
and that
so that
Two important and readily measurable properties of the activated process that convey important information about its nature are: (a) the shear activation volume Avi and (b) the pressure activation volume Aui or the activation dilation. These we define as
References: p . 1951.
1892
A. S. Argon
ch.21, 84
The determination of these two quantities and other activation parameters, by experimental means, is of great interest in probing the mechanism of the activated process. These additional quantities include M,A f , AG'. The kinetic expression of the deformation experiment, eq. (31), is the principal vehicle for probing, through a number of so-calledjump experiments, which in increasing complexity involve: the imposed strain rate (at constant temperature and pressure; T, p); the temperature (at constant stress and pressure; u,p); and finally the pressure (at constant strain rate and temperature, y , 7). From such jump experiments we obtain
where reflects the important strength differential (SD)effect. To obtain the final form of the expressions we have noted that yo is given by eq. (27) in which the pressure and temperature dependence is negligible, and the stress dependence, usually coming through the stress dependence of mobilizable configurations c@), is very much lower than the principal term when a < +(see section 7.3.8). Through the principal temperature dependence of eq. (33). we arrive at (see also S m et al. [1973])
Since for the usually expected regular behavior of materials d p / d T O,the dislocation line can advance in thermally activated steps by initiating double kinks into the neighboring lattice valley, followed by the propagation of the kinks along the dislocation line. An
Ch. 21, 94
Mechanicalproperties of single phase crystalline media
1895
approximate analysis (KOCKSet al. [19751) of this activated configuration based on line notion of dislocations gives the following useful scaling relations for the activation free energy AG’ and the activation area Aa* of the saddle point configuration,
-4 4) 2
AG* = Aa
?T&
2
==b-;).
1-
(43)
?Tb2
While the above elementary consideration suffices for our overall view, it must be noted that the lattice resistance mechanism harbors unusual complexities since it may affect more strongly the edge or the screw dislocation and is sensitive to the actual atomic structure of the core, which cannot be treated by dislocation line notions. More advanced analyses of the thermally activated advance of dislocations against the lattice resistance can be found elsewhere, ranging from more elegant line analyses to fully atomistic forms ( ~ R and N RAJNAK[1964]: GUYOTand DORN[1967]; KOIZUMIet al. [1993]; PULS [19811). 4.3.1.2 The dislocation resistance. The interactions of dislocations with each other have different manifestations. We divide these interactions or resistances into two distinct parts: (a) the interplane resistance and (b) the intra-plane resistance (WIRSCH and HUMPHREYS [19701). The first relates to the interaction of mobile dislocations on parallel slip planes with each other which is a characteristic of the “flow state” and is relatively unaffected by strain hardening. The second incorporates dislocation forest cutting resistances (cutting of glide dislocations through other less mobile dislocations penetrating through the glide plane), and longer range stress fields resulting from dislocation groups and deformation induced misfit. Of these, we will discuss the forest-cutting resistance together with the particle resistance, in section 4.3.2 as a pseudo-particle, and the discussion of the longer-range resistances we will delay to section 7 on strain hardening. Here we consider only the interplane resistance which governs the slip line spacing that we will consider in section 5 below. During plastic deformation, dislocation fluxes (see also section 7.3.5) of positive and negative type have to flow through each other in opposite directions. This results in encounters between opposite edge, and opposite screw dislocations depicted in fig. 8. For planes of given spacing H the maximum interplane resistance 7 for edge and screw dislocations are:
If dislocation encounters occur at stress levels less than these resistances, they should trap each other in dipolar and multipolar arrangements if they are of edge nature. This is a primary source of strain hardening in Stage I of FCC and HCP crystals. If the encountering dislocations are of screw type they would, most likely, annihilate by cross-
References:p . 1951.
1896
A.S. Argon
Ch. 21, 54
Fig. 8. Passing of two opposite types of edge dislocationson planes a distance H apart.
glide. The energetics of this mechanism of pairwise encounters are too unfavorable for the resistances to be subject to reduction by thermal activation.
433. Extrinsic mechanisms 4.3.2.1 The solute resistance. The resistance to dislocation motion by misfitting substitutional or interstitial solute atoms is one of the more powerful mechanisms of strengthening a pure metal, albeit mechanistically one of the most complex. The process has been considered first by Mom and NABARRO[1940], followed by many refined considerations reaching the present day (for an earlier summary see KOCKSet al. [1975]). Much of the complexity arises from the collective and relatively weak interactions of misfitting solute atoms with dislocations which may tug the dislocation line in all directions but polarize against the direction of motion. Detailed considerations discussed by KOCKSet al. [1975] result in a threshold resistance ?sol given by:
where k/26 is the peak resistive force of an individual solute atom against a dislocation, normalized by twice the line tension, (for reasons that will become apparent in the discussion of the precipitate particle resistance below), c is the volume fraction of the solute. This form of the resistance has been experimentally verified by TROJANOVA and HAASEN [1975] in dilute silver alloys, as fig. 9 shows. The temperature and strain rate dependence of the resistance shows unusual complexity, some of these are discussed by HAASEN[1976]. A major complexity, however, arises because in many instances the solute atoms can be affected by and can migrate with the dislocation in the form of a solute atmosphere which forms dynamically polarizable clouds around the dislocation and from which the dislocation can be tom free with an attendant reduction of resistance, until the dislocation is again clogged up with solute, etc., resulting in serrated flow. Such behavior has been studied by a number of investigators.For a discussion of these see MULFORDand KOCKS [1979a]. For further discussions of the temperature dependence of solid solution strengthening see chapter 23 and also KOCKS[1985].
Ch. 21, $ 4
Mechanical properties of single phase crystalline medla
1897
6.0 x
kl2t
Fig. 9. The dependence of the solute resistance on the normalized peak resistive force (varying with solute concentration) in dilute alloys of silver (re-scaled from TROJANOVA and HAASEN[1975]).
4.3.2.2 Particle resistances. The resistance resulting from second-phase particles to dislocation motion is called precipitate resistance if the particles are potentially shearable, by the dislocation. If they are not shearable, the process is referred to as dispersedparticle strengthening. Here we will discuss both processes. There are a variety of mechanisms contributing to the precipitate particle resistance that have been identified. These can be divided into two groups: (a) contact interactions, and (b) interactions at a distance. The prominent mechanisms of the first category are depicted in fig. 10 and include: formation of an interface step by shear, generation of internal faults such as antiphase boundaries in ordered coherent particles, interface loops around particles for which substantial differences between lattice parameters of the particles and the matrix exists, lattice resistance inside particles, and finally differences in core energy of dislocations lying inside vs. outside of the particles. The longer-range interactions between particles and dislocations are depicted in fig. 11, and include the size misfit interaction, and the modulus misfit interaction, for particles that respectively have a size difference between themselves and the matrix cavity they fill, or have important differences in elastic properties. In all instances it is assumed that the particle lattice is coherent with that of the matrix so that there are no kinematic restrictions to the entry of dislocations. In all these cases the dislocation evokes a response within the particle along an effective interaction distance. Thus, the fundamental property of interest is the specific force-distance curve K = KQ) for these various interactions. Each of these separate forcedistance interactions exhibits a peak value k that is of interest in determining the plastic resistance of the crystal at 0 K. Establishment of the specific force-distance curves for any component mechanism of a particle requires considerable sophistication in analysis and cannot be discussed here. It must suffice to state that such interaction contours can be determined in principle for each component mechanism and that these must be combined to determine the overall interaction curve with its overall maximum resistance:
References: p. 1951.
1898
Ch. 21, 8 4
A.S. Argon
’
interface step
-c
(a)
0’ internal fault
(3)
(c)
,interfaqe loop
0’ lattice friction .-.-....
(dl
core energy difference
Fig. 10. Short-range, “contact”, mechanisms resulting in resistive forces on dislocations: (a) interface step; (b) internal fault production; (c) interface loop production to preserve continuity of Burgers vector between inside and outside, (d) large internal lattice resistance; (e) difference of core energy between particle and matrix.
R=CK,
(47)
i
which determines the plastic resistance of the plane. When a dislocation touches or otherwise interacts with a particle to test its strength, it does so by partly wrapping itself around the particle to exert increasingly larger components of the line tension force on the particle in the direction of advance as
(a)
@)
Fig. 11. Long-range mechanisms resulting in resistance to dislocations: (a) particle misfitting in either size or shape; (b) particle with different shear or bulk modulus.
Ch. 21, 5 4
Mechanical properties of single phase crystalline media
1899
depicted in fig. 7. Thus, the evoked particle resistance K in any instance is balanced by the components of line tension force of the cusped dislocation. The maximum force a dislocation can exert on a particle in this way is equal to twice the line tension 28, which makes it useful to consider the resistance of particles in units of 28. Hence, any particle with strength 12/28>1 is impenetrable by dislocations. A dislocation can overcome such particles onIy by going around them. When Z?/28< 1, the dislocation overcomes the particle by cutting it. Of these two limiting forms that for impenetrable obstacles results in dispersed-particle strengthening, and that for penetrable particles results in precipitation strengthening. When impenetrable particles with 12/28> 1 are distributed randomly in a plane with mean center-center spacing I and having an overall volume fraction c, the flow stress is governed by bowing of dislocations through the gaps of average spacing 1 in semicircular segment configurations. This gives rise to a threshold plastic resistance of
where r0 (< 1) is the actual particle radius and p (- 0.85) is a statistical factor accounting for the variation in particle spacings in the plane, away from the mean, which the dislocation tests. Clearly, for a given particle size the maximum plastic resistance is
orowanparlunctcr
-heb) 1.13b
Fig. 12. Dependence of the dispersed particle resistance on geometrical particle field parameters (ASHBY[ 19691, Courtesy of MIT Press). References: p. 1951.
1900
A. S. Argon
Ch. 21, 94
proportional to cn, while for a given particle volume fraction, Z depends on l/r, over the range in which I?/26> 1. The form of eq. (48) with some second-order corrections has been experimentally well verified as fig. 12 shows (ASHBY[1969]). In this mechanism when dislocations circumvent the impenetrable particles they leave closed so-called Orowan loops (see OROWAN[1948]). Figure 13 shows a typical case of Orowan loops surrounding impenetrable particles (HIRSCHand HUMPHREYS [19691). The consequence of such loops in producing a long range back stress (BROWNand STOBBS [1975]; ASARO[1975]) will be discussed in section 9.2. When penetrable particles with K / 2 6 < 1 are distributed randomly in a plane, dislocations sample them and penetrate them without the need to bow to semi-circular configurations. In fact, as the particle strength decreases, the mean distance between obstacles being sampled by the dislocation along its length gradually increases steadily above the mean spacing 1. This mean distance, A, known as the Friedel sampling length, is given by (for a=+)
Fig. 13. Orowan loops left around impenetrable particles in Cu. (HIRSCHand HUMPHREYS[1%9], Courtesy of MIT Press.)
Ch. 21, 94
Mechanical pmperties of single phase crystalline media
1901
An example of moving dislocations in MgO sampling iron oxide precipitates is shown in fig. 14. With this the threshold plastic resistance of the alloy with penetrable particles becomes
Comparison of eq. (50) with eq. (48) shows that it is modified primarily by the normalized obstacle strength K / 2 6 raised to the power +. Thus formally, as the particles become impenetrable the precipitate-governed particle resistance goes over smoothly into the expression given by eq. (48). For many mechanisms the particle strength R / 2 6 depends on particle size. This dependence is often linear and introduces a technologically important possibility of maximizing the plastic resistance at constant volume fraction of particles by particle size control. Thus, in the range of precipitation strengthening, where 2?/2&=K'rJ26, eq. (50) transforms into
where K'(= dk/dr,,) is the coefficient of the particle size-strength connection. Equation (5 1) states that in the range 8 / 2 6 < 1 the plastic resistance rises in proportion to r,'" for "L
Fig. 14. Transmission electron micrograph of dislocations in a MgO crystal pressed against unresolvable iron oxide particles of approximatepeak strength 0.5 (APPELet al., [1976], Courtesy of F'hys. Stat. Sol.).
me=
References: p . 1951.
1902
A.S. Argon
a.21.04 1
a constant volume fraction c of particles. The peak strength ( K ' / b ) ( 3 c / 2 ~ ) 5is then achieved at a particle size, r+= 2&K' when the particles become just impenetrable. This rise and fall of the precipitate resistance with increasing particle size, for a variety of alloys, shown in fig. 15, indicates that the form of the interaction is correct, but there is often a significant departure from the expected magnitudes (for a discussion see ARDELL[1985]). The thermally assisted penetration of a field of shearable precipitate obstacles with a threshold plastic shear resistance ?, at a stress a
