Dislocations in Solids Volume 10
This Page Intentionally Left Blank
Dislocations in Solids Volume 10 L12 Ordered Alloys Edited by
E R. N. N ABARRO Department of Physics University of the Witwatersrand Johannesburg, South Africa and
M . S. D U E S B E R Y Fairfax Materials Research Inc. Springfield, VA, USA
1996 ELSEVIER Amsterdam • Lausanne • New York • Oxford • Shannon • Tokyo
ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands
ISBN: 0-444-82370-0 (volume) ISBN: 0-444-85269-7 (set) (~) 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 or 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, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A.- This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands
Preface Volume 10 of this series is the first to appear under the joint editorship of E R. N. Nabarro and M. S. Duesbery. It contains a symposium on the behaviour of dislocations in alloys with the L 12 structure typified by Ni3A1. These structures are of engineering importance because they form an essential constituent, and, in modem alloys, the major constituent, of superalloys. They also present scientific problems in the behaviour of dislocations which are still not fully resolved. Unlike most metals and alloys, they have a flow stress which increases with increasing temperature over a wide range. Unlike most metals and alloys, they behave very differently in tensile tests and in creep tests. The first paper in the L12 symposium is a historically-oriented article by Westbrook, who observed the anomalous temperature dependence in 1967, and remarked that "Although several possible sources of this anomaly were investigated no unequivocal explanation could be established. Further study is indicated". Then Sun and Hazzledine report studies by transmission electron microscopy of the behaviour of dislocations in these alloys, while Caillard goes further in relating the structure of dislocation cores in these and other alloys to the anomalous temperature dependence of the flow stress. Vitek, Pope and Bassani, from the group which first expanded the ideas of Takeuchi and Kuramato on the anomalous temperature dependence into a detailed theory, show how their theory is modified in the light of recent experimental evidence. This theory considers the way in which dislocations in L12 become pinned at certain points, and the way in which these pinning obstacles can be overcome. Chrzan and Mills carry the analysis further by considering not only the individual pinning and unpinning events, but also the way in which the overcoming of one obstacle can trigger the overcoming of neighbouring obstacles. This leads them to an important link with the modern theory of phase transitions. The symposium ends with a rather controversial article by Veyssi~re and Saada, claiming that the model which Vitek, Pope and their colleagues have developed so successfully puts too much emphasis on the anisotropy of the flow stress and too little on other observations. They concentrate on the importance of the viscous motion of dislocations on the cube planes. The first chapter outside the symposium is by Takeuchi and Maeda on the effects of electronic excitation on the plasticity of semiconducting crystals. The effects are well documented, but their mechanism is not clear. The authors support a "phonon kick" interpretation. Finally, Joos examines the role of dislocations in melting, exploiting the old idea that the presence of small dislocation loops in thermal equilibrium reduces the energy required to form more loops, so leading to a first-order phase transition.
This Page Intentionally Left Blank
Contents Volume 10
Preface v Contents vii List of contents of Volumes 1-9
ix
48. J. H. Westbrook
Superalloys (Ni-base) and dislocations
1
49. Y. Q. Sun and P. M. Hazzledine
Geometry of dislocation glide in L12 7 I-phase
27
50. D. Caillard and A. Couret
Dislocation cores and yield stress anomalies
69
51. V. Vitek, D. P. Pope and J. L. Bassani
Anomalous yield behaviour of compounds with Lie structure
135
52. D . C . Chrzan and M. J. Mills
Dynamics of dislocation motion in L12 compounds
187
53. P. Veyssi~re and G. Saada
Microscopy and plasticity of the LI2 "~ phase
253
54. K. Maeda and S. Takeuchi
Enhancement of dislocation mobility in semiconducting crystals 55. B. Jo6s
The role of dislocations in melting
505
Author index 595 Subject index 613
vii
443
This Page Intentionally Left Blank
CONTENTS OF VOLUMES 1-9 VOLUME 1. The Elastic Theory 1979, 1st repr. 1980; ISBN 0-7204-0756-7
1. 2. 3. 4.
J. Friedel, Dislocations - a n introduction 1 A.M. Kosevich, Crystal dislocations and the theory of elasticity 33 J.W. Steeds and J.R. Willis, Dislocations in anisotropic media 143 J.D. Eshelby, Boundary problems 167 B.K.D. Gairola, Nonlinear elastic problems 223
VOLUME 2. Dislocations in Crystals 1979, 1st repr. 1982; ISBN 0-444-85004-x 5. 6. 7.
R. Bullough and V.K. Tewary, Lattice theories of dislocations S. Amelinckx, Dislocations in particular structures 67 J.W. Matthews, Misfit dislocations 461
1
VOLUME 3. Moving Dislocations 1980; 2nd printing 1983; ISBN 0-444-85015-5 8. 9. 10. 11. 12.
J. Weertman and J.R. Weertman, Moving dislocations 1 Resistance to the motion of dislocations (to be included in a supplementary volume) G. SchSck, Thermodynamics and thermal activation of dislocations 63 J.W. Christian and A.G. Crocker, Dislocations and lattice transformations 165 J.C. Savage, Dislocations in seismology 251
VOLUME 4. Dislocations in Metallurgy 1979; 2nd printing 1983; ISBN 0-444-85025-2 13. 14. 15.
16. 17.
R.W. Balluffi and A.V. Granato, Dislocations, vacancies and interstitials F.C. Larch6, Nucleation and precipitation on dislocations 135 P. Haasen, Solution hardening in.fc.c, metals 155 H. Suzuki, Solid solution hardening in body-centred cubic alloys 191 V. Gerold, Precipitation hardening 219 S.J. Basinski and Z.S. Basinski, Plastic deformation and work hardening E. Smith, Dislocations and cracks 363
1
261
VOLUME 5. Other Effects of Dislocations: Disclinations 1980; 2nd printing 1983; ISBN 0-444-85050-3 18. 19. 20. 21. 22.
C.J. Humphreys, Imaging of dislocations 1 B. Mutaftschiev, Crystal growth and dislocations 57 R. Labusch and W. SchriSter, Electrical properties of dislocations in semiconductors F.R.N. Nabarro and A.T. Quintanilha, Dislocations in superconductors 193 M. K16man, The general theory of disclinations 243
127
x 23. 24.
CONTENTS OF VOLUMES 1-9 Y. Bouligand, Defects and textures in liquid crystals 299 M. Kl6man, Dislocations, disclinations and magnetism 349
VOLUME 6. Applications and Recent Advances 1983; ISBN 0-444-86490-3
25. 26. 27. 28. 29. 30. 31. 32.
J.E Hirth and D.A. Rigney, The application of dislocation concepts in friction and wear 1 C. Laird, The application of dislocation concepts in fatigue 55 C.A.B. Ball and J.H. van der Merwe, The growth of dislocation-free layers 121 V.I. Startsev, Dislocations and strength of metals at very low temperatures 143 A.C. Anderson, The scattering of phonons by dislocations 235 J.G. Byrne, Dislocation studies with positrons 263 H. Neuhfiuser, Slip-line formation and collective dislocation motion 319 J.Th.M. De Hosson, O. Kanert and A.W. Sleeswyk, Dislocations in solids investigated by means of nuclear magnetic resonance 441
VOLUME 7 1986; ISBN 0-444-87011-3
33. 34. 35. 36. 37.
G. Bertotti, A. Ferro, F. Fiorillo and P. Mazzetti, Electrical noise associated with dislocations and plastic flow in metals 1 V.I. Alshits and V.L. lndenbom, Mechanisms of dislocation drag 43 H. Alexander, Dislocations in covalent crystals 113 B.O. Hall, Formation and evolution of dislocation structures during irradiation 235 G.B. Olson and M. Cohen, Dislocation theory of martensitic transformations 295
VOLUME 8. Basic Problems and Applications 1989; ISBN 0-444-70515-5
38. 39. 40. 41. 42. 43.
R.C. Pond, Line defects in inte~. aces 1 M.S. Duesbery, The dislocation core and plasticity 67 B.R. Watts, Conduction electron scattering in dislocated metals 175 W.A. Jesser and J.H. van der Merwe, The prediction of critical misfit and thickness in epitaxy P.J. Jackson, Microstresses and the mechanical properties of crystals 461 H. Conrad and A.F. Sprecher, The electroplastic effect in metals 497
421
VOLUME 9. Dislocations and Disclinations 1992; ISBN 0-444-89560-4 44. 45.
46. 47.
G.R. Anstis and J.L. Hutchison, High-resolution imaging of dislocations 1 I.G. Ritchie and G. Fantozzi, Internal .friction due to the intrinsic properties of dislocations in metals: Kink relaxations 57 N. Narita and J.-I. Takamura, Defi)rmation twinning in.fc.c, and b.c.c, metals 135 A.E. Romanov and V.I. Vladimirov, Disclinations in crystalline solids 191
CHAPTER 48
Superalloys (Ni-base) and DislocationsAn Introduction J. H. W E S T B R O O K Brookline Technologies 5 Brookline Road Ballston Spa, N Y 12020 USA
Dislocations in Solids 9 1996 Elsevier Science B.V. All rights reserved
Edited by E R. N. Nabarro and M. S. Duesbeo,
Contents 3 Background Superalloys, their history and nature 3 Dislocations in intermetallics and superalloys 9 Some significant problems and observations 15 4.1. On dislocations in intermetallics 15 4.2. On dislocations in superalloys 21 Future developments and challenges 23 6. Concluding remarks 24 References 24
1. 2. 3. 4.
~
1. Background A superalloy has been defined as "an alloy developed for elevated temperature service, usually based on Group VIIIA elements, where relatively severe mechanical stressing is encountered and where high surface stability is frequently required" (Sims and Hagel [1 ], p. ix). Compositionally superalloys are perhaps the most complex alloys ever developed in that they typically contain eight to twenty controlled alloying elements, the proportions of which must be exquisitely balanced in order to optimize their properties. Structurally, they are equally complex in that, unlike such common alloys as brass or Monel which are single-phase in character, superalloys have multi-phase microstructures in which the identity, size, shape, and disposition of each of the secondary phases must also be carefully managed for the best performance. Still further complications arise from the fact that superalloys typically are required to operate at temperatures of the order of 0.8Tmp or higher in aggressive atmospheres. Under such conditions most microstructures are unstable; new modes of deformation appear; and constituent elements can react with the environment. Small wonder, then, that understanding of how these alloys respond to mechanical stressing at high temperatures has evolved slowly. Most deformation of crystalline solids is now known to involve dislocations; thus, their structure, number, disposition, mutual interaction, and interaction with solute atoms and microstructural features all become critical. A large proportion of the content of the volumes in this series over the past 16 years has been devoted to such topics but generally with reference to other classes of materials. In order to understand how we arrived at the present state of the science of dislocations in nickel-base superalloys, we need to look at the separate historical development of superalloys and of dislocations, and then turn to the interaction of these two themes. Finally, we will attempt to project likely directions of future development, both in understanding and in application.
2. Superalloys, their history and nature Superalloys were not invented de novo but came as the culmination of a series of developments over hundreds of years aimed at improved structural materials for use at high temperatures. Although the fact that metals soften with increasing temperature has been known for thousands of years, for most of this period there was no interest in materials with useful high temperature strength. Only two instances are known of very early attempts to actually use metals at high temperatures. The Chinese (475-221 BC) used cast-iron molds for casting tool shapes from non-ferrous metals (Lu Da [2]). Here it was the relatively high melting point of the mold material (and hence minimal distortion and interaction) that was exploited, rather than high temperature strength per se.
4
J.H. Westbrook
Ch. 48
The next relevant development was the introduction of novel tool steels. With the advent of powered machine tools in the 19th c. and the increased use of iron and steel in machine construction, cutting materials superior to high carbon steel were badly needed. Perhaps the first of these was Robert Mushet's Special, introduced in 1868 and containing 1 wt.% Mn and 8-10 wt.% W. Although, as described by Keown [3], this was not a true high-speed steel, it did permit a 50% increase in cutting speed. The first real break-through in this type of material came in 1898 when Taylor and White [4] developed a tungsten steel with a 3.8% Cr content and a special heat treatment. At its first public demonstration at the Paris World Exposition in 1900, it was observed that both the tool tip and the chips coming from the workpiece were "red hot" at a cutting speed of 150 ft/min. (about 10x that possible with Mushet's Special). It was thus clear that the high temperatures generated in high-speed cutting required a tool material which softened substantially less than did the workpiece. Taylor and White's alloy was the forerunner of the 18W-4Cr-IV tool steels we know today. While casting molds and cutting tools are early instances of the need for high temperature materials, a major demand arose only with the progressive development of heat engines (Burstall [5], Ch. VI and VII, pp. 201-365). Following the invention of the first practical steam engines by Savery and Newcomen in 1700, the further improvements by Watt and Trevithick in the late 18th and early 19th c. ushered in the industrial revolution and the "steam power age". It was appreciated that both the boiler and the engine itself required structural materials which would operate at elevated temperatures - the higher the temperature and pressure, the greater the efficiency. However, the early steam engines were materials-limited, not so much by inadequate high temperature strength as by poor materials quality control, poor construction techniques, and lack of pressure measurement and control. At a later point, Charles A. Parsons in 1884 patented his design for a steam turbine that was an immediate commercial success and a further driver for improved high temperature materials. Design improvements over the next 40 years resulted in steady increases in size and efficiency. Over the same period, internal combustion engines, both gasoline and Diesel, were developed. For both the steam turbine and the internal combustion engine, the increases in power and efficiency that were achieved were aided in no small way by concurrent, evolutionary improvements in alloy steels (Dulieu [6]) 1. A greater incentive to the development of high temperature alloys was provided by superchargers intended for both piston engines and gas turbines (Moss [7]). However, only with the advent of the turbojet aircraft engine in the late 1930s, as a result of independent work by Whittle in England and von Ohain in Germany, did the higher temperatures and stresses encountered demand a truly discontinuous change in materials (Somerscales [8], Sims [9]). In answer to this challenge, metallurgists turned to some specialty alloys, originally developed for quite different purposes as summarized in table 1 (Westbrook [10]). Fifty 1For example, Ni-Cr-Mo steels were introduced in 1924 for steam turbine rotors; about the same time certain stainless steels for blades, and low C, 0.5 wt.% Mo steel for steam tubing with capability to 500~ and after 1930, Cr-Mo-V steel for a variety of applications. It is nonetheless remarkable that neither the ASM Handbook of 1930 nor the SAE Handbook of 1933 contain any data on the temperature dependence of mechanical properties!
w
Superalloys (Ni-base) and dislocations Table 1 Origins of the superalloy families (after Westbrook [10]). Class
Progenitor system
Fe-base
Fe-Cr stainless steel with/without Ni* (1914)
Co-base
Ni-base
Original application cutlery steel
Early superalloy descendant 16-25-6 (1940s)
Co-Cr-W-C**
cutting tool and hard facing
Haynes 23 (1946)
Co-Cr-Mo** (1929)
surgical and dental alloy
Vitallium (1950's)
Ni-Cr*** (1906)
resistance heating element
Nimonic 80 (1940's)
* Truman [ 11] ** Fritzlen [12] *** Marsh [13] 1600 Modem ? 1200
~2
800 ....,
g~ 400 _ Steam engine / ~/ r 0 1900
/
~"
~
~
1930s Air-cooled aircraft engine
~
1
I
I
I
1920
1940 Year
1960
1980
Fig. 1.20th c. increases in engine operating temperature made possible by new materials [14].
years of largely empirical work resulted in the families of superalloys we know today. This has been a truly remarkable achievement! The Ni-base alloys in particular are able to sustain stresses of 170 MPa for thousands of hours, running at temperatures of the order of 0.8Tmp, sometimes even higher. Another measure of the advance in high temperature structural materials since the days of the steam engine is shown in fig. 1. In parallel with materials development, processing of these materials became extremely sophisticated. Scientific understanding of the superalloy family also improved markedly, not just with respect to their phase constituency and the roles of individual alloying
6
J.H. Westbrook
Ch. 48
elements, but also with respect to strengthening mechanisms, deformation mechanisms, and oxidation behavior. In this paper we will restrict our attention to the history of a single family of superalloys, the nickel bases hardened principally by Ni3A1 (3"), not only for reasons of space but also because that family is the focus of most of the other papers in this volume. Although many of these alloys contain very small volume fractions of carbides and borides that play a significant role with respect to mechanical properties, we will concentrate on the characteristics of the Ni3A1 phase and its relation to the solid solution matrix. Similarly, we will not treat oxide dispersion strengthened superalloys, a powder metallurgy product. The fact that aluminum conferred age hardening behavior to a Fe-Ni or Fe-Ni-Cr base composition was deduced very early by Chevenard [15] from dilatometric and hardness tests. These effects and similar ones due to titanium and silicon additions were confirmed by a number of different workers during the 1930s, but results were inconsistent and confusing. In the late 1930s, at the beginning of World War II, when the aircraft gas turbine established an unprecedented requirement for strong high-temperature alloys, a broad systematic attack to generate precipitation-hardenable alloys from a NiCr base was launched at the Mond Nickel laboratories in England (Pfeil et al. [16] and Betteridge and Bishop [17]). This research produced the famous Nimonic series of alloys of which Nimonic 80, introduced in 1941, and later Nimonic 80A, were the first high temperature alloys 2 to be intentionally hardened by precipitation of a Ni3(A1, Ti) compound. Nimonic 80 was thus the forefather of all the "Inconel", "Rend", "Udimet", and similar alloys of today. Yet, while precipitation hardening was undoubtedly occurring, was responsible for the outstanding high temperature properties, and was associated with the Ti and A1 additions, no precipitates could be seen. Ni3A1 or 3" was not identified as the precipitate until about 1950 (Hignett [21]) with the aid of careful X-ray diffraction and electron microscopy, and the phase diagrams of the pertinent regions of the Ni-CrA1-Ti system were not published until 1952 (Taylor and Floyd [22, 23]). The further development of this class of alloys, via variations in composition and heat treatment, has again been largely empirical. Improved processing techniques, most importantly vacuum melting (permitting higher Ti and A1 contents and Nb and Hf additions without formation of deleterious oxides and nitrides) and directional solidification, introduced by Ver Snyder [24] (eliminating grain boundaries transverse to the stress axis or altogether, and allowing higher contents of refractory metal solutes) have brought about a 200~ increase in the permissible operating temperature and greatly improved reliability (10,000 hrs time between overhauls of jet engines). However, the reasons for all of these improvements were often in doubt despite the opportunities offered for "cleaner" experiments by the availability of superalloy single crystals. It has been observed (usually after the development of superior alloys) that: the solution hardening of the matrix was increased, the volume fraction of 3" increased, and the presence of deleterious phases avoided; but the more subtle and perhaps more important a s p e c t - the development of an optimal compositional relationship between matrix and 3" remained poorly understood and hence not realized. 2ICI Metals Division in England, now IMI, developed age-hardenable copper-base alloys, named Kunial, in the early 1930s. The hardening compound was later shown to be Ni3AI [18-20].
w
Superalloys (Ni-base) and dislocations
7
A1
1 Fig. 2. The L12, cP4 structure of Ni3AI. Note that the 5(110) vector, which is the perfect dislocation in the disordered f.c.c. "7 phase, becomes a superpartial dislocation in the ordered 7' phase. Thus movement of a pair of such partials is required to restore order (after Sun [25]).
We must next digress briefly to consider what "),' is and why it is important to Ni-base superalloys. Ni3A1 ('7') possesses a f.c.c, structure (hence the 3' in reference to the usual nomenclature in Fe-based systems) and is ordered (indicated by the prime symbol) as shown in fig. 2. Intermetallic compounds have been recognized as such only since the work of Karsten [26]. The idea of atomic ordering was introduced by Tammann [27] and confirmed by the X-ray diffraction studies of Bain [28]. The first determination of the ordered f.c.c, arrangement of Ni3A1 (L12, cP4) was by Westgren and Almin [29]. In contrast to many other cases (e.g., Cu-Au) where this structure forms on cooling from a disordered solid solution at high temperatures, Ni3A1 remains ordered to its solution temperature or melting point, forming directly by peritectic reaction between the disordered Ni solid solution and the liquid, as shown in fig. 3. 3 This circumstance is itself indicative of the strong bonding between nickel and aluminum atoms at this composition. The surprisingly high hardness characteristic of compounds formed, even between two relatively soft metals, was recognized early on, e.g., Martens [31], but studies of the temperature dependence of mechanical properties of intermetallics were not undertaken until much later. Tammann and Dahl [32] studied over 30 intermetallics and found that while all were brittle in the macroscopic sense at room temperature (albeit, some showed slip lines in individual crystals) macroscale plasticity could always be observed at sufficiently elevated temperatures. In the 1930s Shishokin and co-workers [33-36] studied the temperature and compositional dependence of hardness and flow stress (extrusion pressure) of a number of intermetallics and found a minimum in the temperature dependence of these properties at a composition corresponding to the ideal stoichiometry of any intermetallic formed, thus foreshadowing their potential as high temperature 3The peritectic and eutectic reaction temperatures for the binary are very close as seen in fig. 3 as also are the compositions of 3/ and eutectic liquid. These relationships may well be altered with ternary and higher alloys. There is reason to believe that in practical multicomponent superalloys the eutectic composition lies between 3, and 7', rather than between NiA1 and "3,' as seen in fig. 3.
8
Ch. 48
J.H. Westbrook
Weight Percent Nickel
1450
r,.)
\
1400
v 87Hil A 88Bre 9 90Jia - 91Udol o 91Udo3
L
o
\
~
v
/
v
cD
A1Ni ~ , /
1350
^ • 1360~
'~,.---*'~ o /~
^ ~ 4^ 1362~
\~ ~
76
78
(Ni)
1300 68
70
72
74
80
Atomic percent nickel Fig. 3. Partial A1-Ni phase diagram for the 68-80 at.% Ni region between 1300 and 1450~ (after Okamoto [30]). structural materials. Other aspects of the history of intermetallic compounds have been previously reviewed by the author [10, 37-40]. In the early post-World-War-II period, research in this area took two general paths. On the one hand there was continued empirical development in alloying of superalloys and improvements in their processing, accompanied by increased effort to understand their behavior; and, on the other, some pioneering work was begun to see if intermetallics themselves might be exploited as a new class of high-temperature materials, rather than simply as constituents of more conventional alloys. In connection with the first theme, an appreciation of the uniqueness of Ni3A1 began to be realized. No aluminide with comparable structure occurs in Fe- or Co-based systems. While in the Ni-A1 binary the compositional range of Ni3A1 is quite restricted, it can accommodate considerable solid solution alloying by ternary additions on both the Ni and A1 sites, leading to both significant hardening and the ability to adjust the lattice mismatch between Ni3A1 and the matrix. Thus, the improved scientific understanding led to concentration on Ni-base superalloy development rather than on Fe- or Co-based alloys. With respect to the second theme, single phase intermetallics or alloys with very high volume fractions of intermetallic were increasingly seen as having attractive potential for high temperature applications from a fundamental point of view because of several factors:
w
Superalloys (Ni-base) and dislocations
9
Mechanical: 9 high intrinsic yield strength at high fractions of the melting point; 9 deformability at low homologous temperatures (at least for some compounds, especially as single crystals); 9 high strain hardening rate. Kinetic: 9 low diffusion rates (relative to metals of comparable melting point) contributing to microstructural stability; 9 low diffusion rates that slow degradation by oxidation or corrosion or by diffusive deterioration of coatings; 9 low diffusion rates retarding atomic deformation processes. Matrix compatibility: 9 equilibrium or quasi-equilibrium structures; 9 compositional similarity between compound and matrix; 9 adjustable coherency. In an early instance of this type of observation, Westbrook [41] found that two-phase Ni-Cr-A1 alloys, consisting of a high volume fraction of 3'1 in a 3' matrix, exhibited an unusual combination of low temperature toughness and high temperature strength. We conclude this summary of the history of superalloys by referring the reader to two sets of conference proceedings: the Seven Springs Superalloy Conferences (1968, 1972, 1976, 1980, 1984, 1988 and 1992)sponsored by TMS/ASM cover developments in superalloys, and the Materials Research Society Symposia (1984, 1986, 1988, 1990, 1992 and 1994) treat high temperature ordered intermetallic alloys. More focussed reviews of the mechanical properties of 3'~ are given by Pope and Ezz [42], Stoloff [43], Suzuki et al. [44], and Liu and Pope [45]. Nembach and Neite [46] and Anton [47] have supplied reviews of the role of 3'~ in superalloys. Later we will review some particular observations relevant to the interaction between superalloy structure and dislocations. Now we trace the history of our understanding of dislocations.
3. D i s l o c a t i o n s in intermetallics a n d superalloys There have been several recent histories of the development of dislocation concepts, most notably those of Hirth [48] and Schulze [49], and it would be pointless to recapitulate them here in any detail. A convenient summary is given by tables 2a, b and c, taken from Schulze. As is well known, the modern conception of the dislocation theory dates from the independent publications of Orowan, Polanyi and Taylor in 1934. Their model successfully accounted for the low shear strength of real crystals and for the strain hardening of crystalline materials. A flood of papers appeared over the next 20 years by authors from all over the world elaborating the basic concepts and interpreting a wide variety of experimental results in terms of the generation, motion, and interaction of dislocations, both with themselves and with other microstructural features and atomistic defects.
10
Ch. 48
J.H. Westbrook
Table 2 Key points in the evolution of dislocation theory: a) first period, 1892-1920; b) second period, 1921-1934; c) third period, 1934-1958 (after Schulze [49]). a) First period Event "strain-figure" in a granular medium as a modification in the structure of mechanical aether
Year 1892
Authors C.V. Burton
"intrinsic strain-form" in a medium showing strictly linear elasticity constituting an electron - moving in the stagnant aether
1897
J. Larmor
Observation of "slip steps" on Pb; PRIORITY?
1900
J.A. Ewing, W. Rosenhain
First observation of slip lines (NaC1; CaCO3); origin of terms "slip plane", "direction of easiest translation" and "weak positions" in crystals
1867
EE. Reusch (Poggendorfs An.) (once again brought to view by G.E.R. Schulze)
9 Really first observation of slip lines in metals (Au, Cu .... ); 9 Determination of slip systems; 9 Originator of term "translation"
1899
O. Miigge (once again brought to view by G.E.R. Schulze)
Theory of self-strain states in an elastic continuum ("distorsioni")
1901 1907 1912/15
G. Weingarten, V. Volterra, C. Somigliana
Theory of hysteresis in a lattice of rotating dipoles with slipable regions of abnormal alignment
1907
J.A. Ewing
"I have ventured to call them Dislocations"
1920
A.E.H. Love
b) Second period Event Proposed notch effect by internal cracks in order to overcome the contradiction between theoretical and measured critical shear stress
Year 1921
Authors A.A. Griffith
Systematic measurements of stress-strain curves with Polanyiapparatus using single and polycrystalline metals
twenties
M. Polanyi, E. Schmid, W. Boas, G. Sachs et al.
Shear stress law
1926?
E. Schmid
Model of flexural slip ("Biegegleitung")
1923
G. Masing, M. Polanyi
Interpretation of mechanical hysteresis on account of translation periodicity of crystal lattice
1928
L. Prandtl
Model of "Verhakung"
1929
U. Dehlinger
Observation of decorated dislocations; not interpreted as such
1905 1932 1932
H. Siedentopf, E. Rexer, A. Edner
"Lockerstellen"
about 1930
A. Smekal
Model of local slip
1934
E. Orowan
Superalloys (Ni-base) and dislocations
w
Table 2 (Continued) c) Third period Event Models of edge dislocation
Year 1934
Authors E. Orowan, M. Polanyi, G.I. Taylor
Model of screw dislocation
1939
J.M. Burgers
Idea of partial dislocations
1948
W. Shockley
Influence of screw dislocations on crystal growth
1949
W.K. Burton, N. Cabrera, F.C. Frank
Pictures and energy estimations of small angle boundaries
1949 1953
W. Shockley, W.T. Read
Model of dislocation multiplication (Frank-Read-source)
1950
EC. Frank, W.T. Read
Imaging of an edge dislocation in lattice fringes of platinum phthalocyanine
1956
J.W. Menter
Evidence of real existence of dislocations in metals by electron diffraction contrast
1956
P.B. Hirsch, R.W. Home, M.J. Whelan
1957/58
A. Seeger
First estimation of critical stress
Work hardening theories (I) Glide-zone model (II) Forest-theories
1958/60 [ P.B. Hirsch l Z.S. Basinski
(III) Jog-dragging theory
1960
(IV) Mushroom theory "The dislocation is the elemental residual-stress source"
P.B. Hirsch
1962
D. Kuhlmann-Wilsdorf
1958
E. KrOner
Embarrassingly, as Read [50] noted, not only could any particular result be given a reasonable dislocation interpretation, but often several radically different models appeared to serve equally well. Worse still was the fact that until the end of this period no one had observed a dislocation "in the flesh", although slip lines had been seen as early as 1867 by Reusch [51]. An enormous boost was given the field in the 1950's with the introduction of techniques which permitted the observation of dislocations: etch pitting (Vogel et al. [52]), decoration (Hedges and Mitchell [53]), and especially electron microscopy (Heidenreich [54], Hirsch et al. [55] and Bollman [56]) and field emission microscopy (Muller [57]). Not only were some dislocation configurations now seen to be exactly what had been imagined, but it was finally possible to perform critical experiments, obtain quantitative data such as dislocation mobilities (e.g., Johnston and Gilman [58]), make detailed observations of dislocation core structures (down to resolutions of 1-2 nm), and even, in the case of transmission electron microscopy, observe other dynamic effects by stressing samples in situ in the microscope (Suzuki et al. [59, 60]).
12
Ch. 48
J.H. Westbrook
APB
b
O@O@Q@O @0@0@0 @Q@O@O@ O@O@O@ O@O@Q@Q @O@Q@Q | @ 0 @0@ Q @ Q@ Q@ 0 @ O@O @ O @Q @0 @ 0 @O@ 0 @ O@ m
Superpartial Dislocations Fig. 4. Two superpartial dislocations coupled with a piece of antiphase boundary (after Sun [25]).
The improved atomistic level of understanding of plastic deformation in solids had some particular impacts on interpreting the behavior of intermetallic compounds and superalloys, and conversely experimental findings on the latter contributed to the former. An early problem was that what would have been a unit dislocation in the disordered lattice, in attempting to glide through an ordered lattice, should experience a large resistance due to the disordering effect brought about by its motion; yet some fully ordered alloys, e.g., Cu3Au, deformed quite readily. Koehler and Seitz [61] predicted that the ordered structure of most intermetallics would require a superdislocation, i.e., a pair of partial dislocations separated by a strip of material with the ordering of the atoms exactly out-of-phase with the normal structure, a so-called antiphase boundary (APB) as shown in fig. 4. They reasoned that, in motion of this complex, the order that is destroyed by the leading member of a pair would be immediately restored by the trailing dislocation and hence would encounter much lowered resistance. At that time, however, no one had knowingly seen a dislocation, let alone a closely spaced pair of dislocations. Only a few years later, however, such superdislocations were first revealed by Marcinkowski et al. [62] by electron microscopy. More recently weak beam electron microscopy was employed by Crawford and Ray [63] to demonstrate the expected four-fold dissociation of dislocations in Fe3A1 (D03 structure). Elastic anisotropies, energetic anisotropies of internal surfaces such as APBs and stacking faults, and the often complex core structures of the dislocations themselves lead to still further complications in the geometry and dynamics of dislocations in ordered alloys. One of the most famous of these is the so-called Kear-Wilsdorf lock (Kear and Wilsdorf [64]) shown in fig. 5, caused by a screw superdislocation dissociated into (111) APB and cross-slipped segments on (100) planes. Consideration of the behavior of dislocations in two-phase alloys, e.g., Ni3A1 particles in a Ni solid solution matrix, brought about more complications. For example, compar-
w
Superalloys (Ni-base) and dislocations
Stacking ~, fault ,,
Dissociated screw dislocation
Trace of[lO0]
S
Antiphase boundary
",,r
Trace of[lll]
Cross slip
"
", ",
(a)
13
',, (b)
,, (c)
Fig. 5. The Kear-Wilsdorf lock formed by cross-slip pinning (Kear and Wilsdorf [64]).
Fig. 6. Dislocation structure developed during primary creep at 850~ of a Ni-base single crystal superalloy, CMSX-3. Dislocations appear in the matrix channels between ~,~ particles (horizontal channels, upper part and vertical channels, lower part of the figure) (courtesy of T.M. Pollock). ing the creep behavior at 1000~ of "7, "7~, and a mixture of the two phases, Nathal et al. [65] found for the two-phase alloy a decrease in creep rate of about a factor of 1000 relative to "7' alone. Creep tests of Ni-base superalloy single crystals were coupled with stereo transmission electron microscopy by Pollock and Argon [66]. For their alloy
J.H. Westbrook
14
Ch. 48
10 000
1000
10o Ni-Cr-AI alloys 700~ - 21,200 psi
0.01
0.006
0.002 00.002
0.006
0.01
zXa, (a~,- a~), kX
Fig. 7. Effect of lattice mismatch
(Aa) on creep-rupture life of Ni-Cr-AI alloys (after Mirkin and Kancheev [67]).
with a 3" volume fraction ~ 67% and particle size ~ 0.5 gm (typical of present day superalloy microstructures), they found that for creep at 850~ or less the "T' was essentially dislocation-flee and undeformable. As a result, dislocations are forced to move through the relatively narrow channels between q" particles, ultimately forming complex 3D networks and thereby constituting the principal cause for the high creep resistance. The dislocation structure developed during early stages of primary creep at 850~ and 552 MPa is shown in fig. 6. No dislocations appear within the cuboidal '7~ particles that extend through the foil thickness. The dislocations visible occur in matrix channels between '7' precipitate particles: vertical channels (lower part of the figure) and horizontal channels (upper part of the figure) that are contained within the thickness of the foil. To be effective in practice, where superalloys are stressed for long times at high temperatures, the microstructure must be stable. The coherent interface between "7 and "7' contributes basically to a low interfacial energy, which is further minimized by appropriate alloying. The practical result of this is seen in the work of Mirkin and Kancheev [67] illustrated in fig. 7. Still further contribution to microstructural stability can be obtained by addition of elements such as Re, which dissolves almost exclusively in "7 rather than in "7' (Giamei and Anton [68]). One interpretation of this benefit is that any growing 3,~ particles must drive Re away from the "7/"7' interface, and thus Re diffusion becomes the limiting step. However, solutes not only partition between "7 and "7' but may also segregate to grain boundaries, APBs, or interfaces between phases and contribute to strengthening and microstructural stability through various mechanisms. Furthermore the relative degree of partitioning and segregation may vary with solute concentration and temperature, so a general picture of the contribution of alloying to microstructural stability and strengthening does not readily emerge.
w
Superalloys (Ni-base)and dislocations
15
Modern views of the dislocation behavior of superalloys may be found in Sims et al. [9], Anton [47], and Pollock and Argon [66]; Ardell [69] reviews dislocation/particle interactions in precipitation and dispersion strengthened systems at room temperature and low volume fraction of the dispersed phase. Another historical development that must be remarked on is the use of the computer to perform various simulation "experiments". These may be used either to generate possible structures or to make quantitative or semi-quantitative estimates of dislocation behavior given known or assumed values for various key parameters. Such studies began as early as that of Foreman and Makin [63] and continue to this day. A summary of early work with this approach is given by Puls [71]. The method may be expected to be even more fruitful in the future with better interatomic potential models and as better estimates or experimental measures of key modelling parameters come to hand, e.g., elastic constants, vacancy formation energy, stacking fault and antiphase boundary energies, etc. We conclude this section with a brief citation of some other key reviews. Duesbery and Richardson [72] discuss dislocation core structures in all crystalline materials. Veyssi~re and Douin [73] review dislocations in intermetallic compounds with particular emphasis on core structures and core-related mechanical properties. Sun [25] discusses how defects in intermetallics (including dislocations) differ from those in ordinary metals and disordered solid solutions and the implications these differences have for mechanical behavior at both room and elevated temperatures. Yoo et al. [74] provide a convenient summary of currently unsolved problems. It must finally be noted that there are several aspects of the relevance of dislocations to the formation and behavior of superalloys that are not considered by the other papers in this volume; among these are: low temperature toughness and fracture, solid solution hardening, dislocation interaction with solute atoms and point defects, microstructural stability, grain boundaries and crystal growth. These topics, which are summarized in detail in two reference works (Sims and Hagel [1] and Sims et al. [75]), therefore, receive only incidental mention in this Introduction.
4. Some significant problems and observations 4.1. On dislocations in intermetallics
Anomalous temperature dependence of flow stress. This phenomenon was first reported by Westbrook [76] from hot hardness measurements on Ni3A1 where it was found, as seen in fig. 8, that the hardness increased with increasing temperature in contrast to the behavior of all other intermetallic compounds he studied. 4 He also found that the effect was both intrinsic to the material and temperature-path history independent. This surprising result was soon confirmed (Flinn [79], Davies and Stoloff [80]), and 4It is recognized that more modest increases in strength with increasing temperature for intermetallics were previously reported for CuZn (Barrett [77]) and for Cu3Au (Ardley [78]). These results, however, are readily interpreted as due to change in domain size and/or degree of long range order. Ni3A1does not have a stable domain structure nor does its ordering degree decrease significantly with increasing temperature; hence its strength increase is truly anomalous.
16
Ch. 48
J.H. Westbrook
2000 oO:>.
1000
|
9o,o~ 1 4 9 ooO_ _o
,_
w
9
~
ooo _~
OLeO OOoO0 o u
~ A..~#~....z~%~Y~z~ Z~.~NX
~
"
~
~ , ~ ~ ~
r
~o 500 ~g
o EttS~ oo
[]
mm -mm"ram"
0
9
999
~-~
Cr7C3
,m
9 Cr23C6
,
I
200
,
I
9
Ni3Ti 9 Ni3AI
[]
A Co2Ti(MgCu2 type) 9 Fe2Ti(MgZn2 type)
0
o
[]
~
m',,'m. "
100
9
A~j,A
o
200
~~
0
,'~
,
I
400 600 Temperature, ~
,
I
800
,
1000
Fig. 8. Hardness as a function of temperature for several phases common in superaUoys. The vertical arrows indicate a Th --0.5 (after Westbrook [76]).
many other aspects of the phenomenon documented by Thornton et al. [81]. These are summarized by Vitek et al. in this volume to include: 1) 2) 3) 4) 5) 6)
the yield and/or flow stress increases with increasing temperature; flow stress is temperature-path history independent; failure to obey Schmid's law; different strengths obtain in tension and compression; flow stress is virtually strain rate independent in the anomalous region; slip occurs predominantly on { 111 }(101) system below peak temperature and on {001}(101) above it; 7) predominantly long (101) screw dislocations are seen in the anomalous region; 8) no anomaly occurs in the microstrain range, 10-6-10-5; 9) the temperature dependence of the (low) work hardening rate is unusual in the anomalous region.
Scores of papers over the nearly forty years past have shown that the phenomenon occurs in many other intermetallic structures, yet not all L12 structures exhibit it. Much effort has been expended over this period to develop a model that would satisfactorily include all of the above listed features. The five papers that comprise the bulk of this
w
Superalloys (Ni-base) and dislocations
17
volume present our current level of understanding, which in many respects appears to be very good. In essence, all the proposed models envision the thermally activated locking of superdislocations by cross-slip from the primary plane onto another plane, leading to an APB partly or completely out of the primary plane. They differ in the way they account for secondary characteristics of the phenomenon. Before the problem can be regarded as solved, however, we must have a generic mechanism which can both apply to structures other than L12 and account for the absence of anomalous flow stress behavior in certain L12 compounds. (See especially the chapter by Caillard and Couret.) Effects of stoichiometric deviation on flow stress. It was observed very early (Kurnakov and Zhemchuzhnii [82]) that deviations from ideal stoichiometry resulted in an increase in the strength properties of intermetallics. Later, when measurements could be extended to high homologous temperatures (Th) at least for model compounds, as seen in fig. 9 1000
I
I
I
I
I
I
Homologous temperature T[Tmp
100
/ O
O
off
r~
c~
0.7
-----.a.
0.8
35
I
I
I
40
45
50
I
55 A/o Mg
I
I
60
65
Fig. 9. Influence of defect concentration (deviation from stoichiometry) and homologous temperature on the hot hardness of single phase AgMg alloys (after Westbrook [83]).
18
Ch. 48
J. H. Westbrook
700 LX 23.5 at % A1 600
-
[]
24.1
o
9 25.3
/
,9
500
/
9
/ ./
f.-..
Z
N-
\
\
400
300
200
100
I
I
I
0
200
400
I
I
600 800 Temperature, K
I
I000
1200
Fig. 10. Effect of A1 concentration on yield strength of Ni3AI as a function of temperature (after Noguchi et al. [84]).
from hot hardness studies on B2 AgMg by Westbrook [83], it was found that the reverse situation obtained. Thus, the same defects that impeded slip at low Th enhanced diffusion and lowered diffusion-controlled deformation resistance at high Th. 5 The defect structure of off-stoichiometric AgMg is very simple: substitution of the excess species on atom sites of the other component for both sides of stoichiometry. Marked effects of stoichiometric deviation are also to be found in Ni3A1 as seen from the results of Noguchi et al. [84] in fig. 10. In this case the strength d e c r e a s e s rather than increases with departure from stoichiometry on the Ni-rich side. Whether this result occurs because of interaction of the anomalous yield phenomenon with the point defects or because of a more complex defect structure, the possibilities for which have been studied by Fleischer [85], is not yet clear. In any case, in superalloys the composition of the Ni3A1 particles in equilibrium with the nickel solid solution matrix is always Ni-rich, although its exact composition may vary with the bulk composition of the alloy as well as with temperature. 5Note that both species need to diffuse for dislocation climb to occur. A given type of defect arising from stoichiometric deviation will not necessarily affect the two diffusion rates equally, although both should increase with increasing defect concentration.
w
Superalloys (Ni-base) and dislocations
1200
~A
/ 1000
/
/
~g r~
~: 600 o
O -/ 9
2/Hf O
400 -
-
i/o
0
200
_ -O1
/
,
O
O
/
~o
c5
/o,~ /
Hf
\
d
/"
800
19
o/
/Q /. .,
/ Ni3A,
/
9
200
I
I
I
I
400 600 800 Temperature, K
I
1000 1200
Fig. 11. Effect of a large misfit solute (Hf) on the yield strength-temperature characteristics of Ni3A1 (after Mishima et al. [86, 87]).
Anomalous solid solution hardening at high temperature. Mishima et al. [86, 87] have shown that at room temperature the solution hardening that is observed in Ni3A1 is consistent with the model that this is caused by atom size and modulus interactions with moving dislocations, just as in Ni or other metallic matrices. However, this model would predict a decrease in solid solution hardening at elevated temperatures in contrast to a persistance or even an increase that have been observed in intermetallic compounds. This anomaly obtains for both large misfit solutes such as Hf in Ni3A1 (see fig. 11) 6 or small misfit solutes such as Fe in CoAl (see fig. 12). Whether complications of the nature of defects introduced by alloying or changes in the character of dislocations 6Note that while the magnitude of the solid solution hardening increases at high temperatures, the relative increase is lessened.
20
Ch. 48
J.H. Westbrook
1000
~
)Al
i
E d 100 -
\
(Co,Fe)AI
10
I
I
-200
0
25 A/o Co 25 AJo Fe
i
i
\
i
200 400 600 Temperature, ~
\
1
800
1000
Fig. 12. Effect of a small misfit solute (Fe) on the hot hardness of CoAl (Westbrook, unpublished).
at high temperatures is responsible cannot presently be determined, as discussed by Fleischer [85]. Analogous effects are also seen in the creep resistance of ternary vs. binary stoichiometric B2 aluminides, as shown by Sauthoff [88] in fig. 13. The correlation with the effective diffusion coefficient suggests that this factor must also be taken into account, at least for long-time tests at elevated temperatures. This high temperature solid solution hardening anomaly is both something that must be understood and could well be exploited in practical materials.
Strength of ternary compounds. Many binary compounds have ternary counterparts, not simply solid solutions extending from the binary but true ternary compounds, usually possessing further degrees of order based on the same crystal structure motif as the binary. The few cases that have been examined- ternary sigma phase, Fe36CrlzMo10 vs. FeCr and CoCr (Westbrook [76, 83]), (Co, Ni)3V vs. Ni3V or Co3V (K6ster and Sperner [89]), Ni2TiA1 vs. Ni3A1 (Strutt and Polvani [90]), and Nb(Ni, A1)2 vs. NiA1 (Sauthoff [91-93]) - all show strength increases relative to the binary of as much as a factor of 3 or 4, especially at high temperatures. In the example shown in fig. 14, the ternary compound is a C14 Laves phase (hence better represented as Nb(Ni, A1)2) and is contrasted with binary B2 NiA1 and D022 A13Nb. Not only is this strength benefit desirable in and of itself, but diffusion rates in ternary compounds may be expected to be even lower than in
Superalloys (Ni-base) and dislocations
w
21
300 T = ~--
900~ 10-7
S-I
100 cq
2;
CoAl
~
50-
[] (Nio.sFeo.2)A1 ~n
20-
NiAI
o,,.~ tD
~.8)A1
10L) _
~ F e A 1
o\
10-19
I
10-17
I
10-15
Diffusion coefficient, m2/s Fig. 13. Effect of solid solution hardening of Fe in NiAI and Ni in FeAI on compressive creep resistance at 900~ and correlation with the diffusion coefficient (after Sauthoff [88]). their binary counterparts, thus improving strength in temperature regimes where diffusion control exists as well as contributing to microstructural stability.
Ductilization or embrittlement phenomena in intermetallics. Several different classes of these effects have been reviewed by Stoloff [43, 94] and Westbrook [ 10]: ductilization by macroalloying, ductilization by microalloying, ductilization by grain refinement, ductilization by increase in mobile dislocation density, and embrittlement by stoichiometric excess of the active metal component. Not every effect appears in every system, and dislocation-based models in this area are too poorly developed to permit useful exploitation of those effects beneficial in intermetallics or superalloys. They are not discussed in the other chapters in this volume.
4 . 2 . O n d i s l o c a t i o n s in s u p e r a l l o y s
Peak strengthening in two-phase alloys. While good success has been obtained in modelling the strength of under-aged and over-aged alloys at low temperatures and with low to moderate volume fractions of the ordered phase (Ardell [69]), the same model does not account well for the behavior of alloys aged to peak strength, especially where the intermetallic phase is very strong relative to the matrix. Strengthening by multi-modal particle size distributions. Empirical optimization of strength of practical superalloys by multi-stage heat treatments frequently results in bi- or
22
Ch. 48
J.H. Westbrook
2000 ~= 10-4S-1
A1-Nb-Ni-3 [] 1500 -
A1-Nb-Ni-~ []
eq
AI-N
1000
nl A13Nb AINbNi
r
500 NiA1
\
'1~\~~~0
X.\o 0
!
0
I
,
200 400 600 800 1000 1200 1400 Temperature, ~
Fig. 14. 0.2% compressive proof stress for single phase binary compounds (NiAI and AI3Nb), a ternary compound (Nb(AI, Ni)2, shown as A1NbNi) and several three phase alloys (AI-Nb-Ni 1, 2 and 3) (after Sauthoff [88]).
even tri-modal distributions of particle sizes. In very crude terms this is understandable inasmuch as it is desired that the alloy be strong over an extended range of temperatures, and for short times as well as long times. Strength is favored by small particles at low temperatures and short times and by large particles at high temperatures and long times. More quantitative modelling of the situation has not been generally successful in two respects: different additivity rules must be invoked in different systems and the parameters introduced to weight the individual contributions of each size mode lack physical significance [69].
Directional coarsening.
Under creep conditions, the originally cuboidal Ni3A1 particles in nickel-base superalloys, change their morphology by coarsening into preferentially oriented plates or "rafts". This phenomenon was first studied by Tien and Copley [95] and more recently by Pollock and Argon [96]. The structural change is related to the misfit (both sign and degree) between the intermetallic phase and matrix and by the imposed stress. This rafting morphology is of practical significance because dislocations have a more difficult time passing the laterally elongated particles and hence the creep strength is improved. Quantitative understanding of the process and its contribution to high temperature strength is impeded by the complicated roles of stress, misfit dislocations, temperature and crystallographic orientation. Vall6s and Arrell [97] have had some recent success in computational modelling of rafting behavior.
w
SuperaUoys (Ni-base) and dislocations
23
5. Future d e v e l o p m e n t s a n d c h a l l e n g e s While we may be justifiably proud of the vastly improved understanding we have retrospectively of the high temperature behavior of the present generation of superalloys, our predictive capabilities for inventing new high temperature structural materials or improving existing ones are still inadequate. It is chagrining to note that "enlightened empiricists" (to use Bruce Chalmer's phrase) still continue to produce beneficial practical results which can only be understood after the fact. A few examples may illustrate the capabilities we wish we had.
A rationale for the selection of new compound bases.
As Fleischer [98] has reviewed, the primary selection criteria for an initial screening are now melting point, density, and elastic modulus, supplemented when possible by knowledge of crystal structures, phase equilibria, and environmental resistance. How helpful it would be, either for intermetallics in an appropriate matrix or for single phase intermetallics, if we could predict a priori from fundamental parameters, slip systems, dislocation structures and mobilities, and likely interactions with solutes and lattice defects.
A rationale for alloying to balance compound and matrix properties and stabilize microstructures. We are aware that for optimum behavior of two-phase alloys we need to balance not only their individual mechanical properties (how each responds to the generation, motion, and interaction of dislocations) but we must also maximize the microstructural stability of the system. Addition of certain alloying elements is effective in both respects, but the processes are not parallel and the results often not predictable.
A basis for specifying and achieving the optimum microstructure and dislocation structure of the alloy. Even when composition is fixed, the properties of a superalloy can still be varied substantially. Complex thermomechanical treatments are known to be capable of altering the size and disposition of the second phase particles as well as the dislocation structures, but we neither know what would be optimum nor specifically how to achieve it.
Basis and means for atomistic alloying.
Most alloying performed today is at the macro level of several atom percent and is understood largely in terms of how the bulk properties of matrix and dispersed phase are affected. 7 If we knew how to select solutes, perhaps even at the ppm level, that would preferentially segregate to APBs, stacking faults, particle interfaces, and grain boundaries, and if we further knew what the effects would be of such segregation on dislocation energy, dislocation mobility, stacking fault energy, diffusion suppression etc., we would be in a fair way to advance to a new level of sophistication in alloy design. Alas, this is not yet possible. 7It is recognized that compositional control in the p.p.m, range has been exercised in the past, both for deleterious elements (Se, Bi, Pb) and for beneficial elements (Zr, B, C). These well-known instances are of less consequence now in the era of single crystals, vacuum melting, and virgin melt stock, although B and C segregation to low angle grain boundaries in single crystals may still contribute significant strengthening.
24
J.H. Westbrook
Ch. 48
6. Concluding remarks This article has attempted to sketch briefly how far we have come and how far we have yet to go in understanding and exploiting the behavior of dislocations in Ni-base superalloys and related materials. The five other chapters in this volume concentrate on but a single aspect of that subject, the flow stress anomaly. In reflecting on where we are and what has been accomplished, the author is reminded of the remark by a famous scientist, "I have never encountered a problem, however complex, that after careful study did not turn out to be even more complex". In this sense metallurgists and physicists will find interesting employment with dislocations in superalloys for some time to come.
Acknowledgment Appreciation is gratefully extended to R.L. Fleischer, J.J. Gilman, T.M. Pollock, C.T. Sims and N.S. Stoloff for insightful reading of and critical comments on an early draft of this paper.
References [1] C.T. Sims and W.C. Hagel (eds), The Superalloys (Wiley, New York, 1970). [2] Lu Da, Acta Met. Siniatica (Peking) 9 (1966) 1; translation in: Durrer Festrschrift, Vita pro Ferro, ed. W.M. Guyan (Schaffhausen, 1965) p. 65. [3] S. Keown, Historical Met. 19(1) (1985) 97. [4] F.W. Taylor and M. White, Metal Cutting Tool and Method of Making Same, US Pat. 668,269 (19 Feb. 1901). [5] H.E Burstall, A History of Mechanical Engineering (MIT Press, Cambridge, MA, 1965). [6] D. Dulieu, Historical Met. 19(1) (I 985) 104. [7] S.A. Moss, Trans. ASME 66 (1944) 351. [8] E.EC. Somerscales, in: An Encyclopedia of the History of Technology, ed. I. McNeil (Rutledge, London, 1990) p. 272. [9] C.T. Sims, in: Superalloys II, eds C.T. Sims, N.S. Stoloff and W.C. Hagel (Wiley, New York, 1987) p. 3. [10] J.H. Westbrook, in: Structural Intermetallics, eds R. Darolia, J.J. Lewandowski, C.T. Liu, P.L. Martin, D.B. Miracle and M.V. Nathal (TMS, Warrendale, PA, 1993) p. 1. [ 11] J. Truman, Historical Met. 9(1) (1985) 116. [12] G.A. Fritzlen, in: High Temperature Materials, eds R.F. Hehemann and G.A. Ault (Wiley, 1959) p. 56. [13] J.H. Marsh, UK Patent 2129, US Patent 811,859 (1906). [14] National Research Council, Materials Science and Engineering for the 1990s (Nat. Acad. Press, Washington, DC, 1989) p. 21. [15] P. Chevenard, Compt. Rend. 189 (1929) 846. [16] L.B. Pfeil, N.P. Allen and C.G. Conway, Iron and Steel Inst. Special Report, No. 43 (1952) p. 37. [17] W. Betteridge and J. Bishop, The Nimonic Alloys (Arnold, London, 1974). [18] W.O. Alexander and D. Hanson, J. Inst. Met. 61 (1937) 83. [19] W.O. Alexander, J. Inst. Met. 63 (1938) 163. [20] W.O. Alexander, J. Inst. Met. 64 (1939) 499. [21] H.W.G. Hignett, High Temperature Alloys in British Jet Engines (Int. Nickel Co., New York, 1951). [22] A. Taylor and R.W. Floyd, J. Inst. Met. 81 (1952) 25. [23] A. Taylor and R.W. Floyd, J. Inst. Met. 81 (1952) 643. [24] EL. Ver Snyder, US Patent 3,260,505 (1966).
Superalloys (Ni-base) and dislocations [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
25
Y.Q. Sun, MRS Bull. 20(7) (1995) 29. K. Karsten, Pogg. Ann. Ser. 2 46 (1839) 160. G. Tammann, Z. Anorg. Chemie 107 (1919) 1. E.C. Bain, Chem. Metall. Eng. 28 (1923) 21. A. Westgren and A. Almin, Z. Phys. Chem. B5 (1929) 14. H. Okamoto, J. Phase Equil. 14 (1993) 25. A. Martens, Mitt. K. Tech. Versuchs-Anst. 8 (1890) 216. G. Tammann and K. Dahl, Z. Anorg. Chemie 126 (1923) 104. V.P. Shishokin, Tsvetnye Metalli (November 1930). V.P. Shishokin and V.A. Ageeva, Tsvetnye Metalli 2 (1932) 119. V.P. Shishokin, V.A. Ageeva and V.T. Mikhieva, Metallurg 10 (1935) 81. V.P. Shishokin, Bull. Acad. Sci. URSS, CI. Sci. Math. Natur. (1937) p. 341. J.H. Westbrook, in: Mechanical Properties of Intermetallic Compounds, ed. J.H. Westbrook (Wiley, New York, 1960)p. 1. [38] J.H. Westbrook, in: Ordered Alloys: Structural Applications and Physical Metallurgy, eds B.H. Kear, C.T. Sims, N.S. Stoloff and J.H. Westbrook (Claitor's, Baton Rouge, LA, 1970) p. 1. [39] J.H. Westbrook, Metall. Trans. 8A (1977) 1327. [40] J.H. Westbrook, in: Intermetallic Compounds: Principles and Practice, eds J.H. Westbrook and R.L. Fleischer (Wiley, Chichester, UK, 1994) p. 3. [41] J.H. Westbrook, General Electric Report 55RL1281 (1955). [42] D.P. Pope and S.S. Ezz, Int. Metall. Rev. 29 (1984) 136. [43] N.S. Stoloff, Int. Metall. Rev. 29 (1984) 123. [44] T. Suzuki, M. Ichihara and S. Miura, ISIJ Int. 29 (1989) 1. [45] C.T. Liu and D.P. Pope, in: Intermetallic Compounds: Principles and Practice, Vol. 2, eds J.H. Westbrook and R.L. Fleischer (Wiley, Chichester, UK, 1994) p. 17. [46] E. Nembach and G. Neite, Progr. Mater. Sci. 29 (1985) 177. [47] D.L. Anton, in: Intermetallic Compounds: Principles and Practice, Vol. 2, eds J.H. Westbrook and R.L. Fleischer (Wiley, Chichester, UK, 1994), p. 3. [48] J.P. Hirth, Metall. Trans. 16A (1985) 2085. [49] D. Schulze, Prakt. Metallogr. 26 (1989) 559, 604. [50] W.T. Read, Dislocations in Crystals (McGraw-Hill, New York, 1953). [51] E. Reusch, Ann. Phys. Chem. 132 (1867) 441. [52] EL. Vogel Jr., W.G. Pfann, C.L. Corey and G. Thomas, Phys. Rev. 90 (1953) 489. [53] J.N. Hedges and J.W. Mitchell, Philos. Mag. 44 (1953) 223. [54] R.D. Heidenreich, J. Appl. Phys. 20 (1949) 993. [55] P.B. Hirsch, R.W. Home and M.J. Whelan, Philos. Mag. 1 (1956) 677. [56] W. Bollman, Phys. Rev. 103 (1956) 1588. [57] E.W. Muller, Acta Metall. 6 (1958) 620. [58] W.G. Johnston and J.J. Gilman, J. Appl. Phys. 30 (1959) 129. [59] K. Suzuki, M. Ichihara and S. Takeuchi, in: Proc. 5th Int. Conf. HREM, eds T. Imura and H. Hashimoto (Jpn. Soc. of Electron Microscopy, 1977) p. 463. [60] K. Suzuki, E. Kuramoto, S. Takeuchi and M. Ichihara, Jpn. J. Appl. Phys. 16 (1977) 919. [61 ] J.S. Koehler and F. Seitz, J. Appl. Mech. 14 (1947) A217. [62] M.J. Marcinkowski, R.M. Fisher and N. Brown, J. Appl. Phys. 31 (1960) 1303. [63] R.C. Crawford and I.L.F. Ray, in: Proc. Int. Symp. on Electron Microscopy, Grenoble, 1970, p. 277. [64] B.H. Kear and H.G.F. Wilsdorf, Trans. Metall. Soc. AIME 224 (1962) 382. [65] M.V. Nathal, J.O. Diaz and R.V. Miner, in: High Temperature Ordered Intermetallic Alloys III, eds C.T. Liu, A.I. Taub, N.S. Stoloff and C.C. Koch (MRS, Pittsburgh, PA, 1989) p. 269. [66] T.M. Pollock and A.S. Argon, Acta Metall. Mater. 40 (1992) 1. [67] I.L. Mirkin and O.D. Kancheev, Met. Sci. Heat Treat. No. 1-2, (1967) 10. [68] A.F. Giamei and D.L. Anton, Metall. Trans. 16A (1985) 1997. [69] A.J. Ardell, in: Intermetallic Compounds: Principles and Practice, Vol. 2, eds J.H. Westbrook and R.L. Fleischer (Wiley, Chichester, UK, 1994), p. 257. [70] A.J.E. Foreman and M.J. Makin, Philos. Mag. 14 (1966) 131.
26
J. H. Westbrook
[71] M.E Puls, in: Dislocation Modelling of Physical Systems, eds M.E Ashby, R. Bullough, C.S. Hartley and J.P. Hirth (Pergamon, Oxford, 1981) p. 249. [72] M.S. Duesbery and G.Y. Richardson, CRC Critical Rev. in Solid State and Mater. Sci. 17 (1991) 1. [73] P. Veyssi~re and J. Douin, in: Intermetallic Compounds: Principles and Practice, Vol. 1, eds J.H. Westbrook and R.L. Fleischer (Wiley, Chichester, UK, 1994) p. 519. [74] M.H. Yoo, S.L. Sass, C.L. Fu, M.J. Mills, D.M. Dimiduk and E.P. George, Acta Metall. Mater. 41 (1993) 987. [75] C.T. Sims, N.S. Stoloff and W.C. Hagel (eds), Superalloys II (Wiley, New York, 1987). [76] J.H. Westbrook, Trans. Metall. Soc. AIME 209 (1975) 898. [77] C.S. Barrett, Trans. Metall. Soc. AIME 200 (1954) 1003. [78] G.W. Ardley, Acta Metall. 3 (1955) 525. [79] P.A. Flinn, Trans. Metall. Soc. AIME 218 (1960) 145. [80] R.G. Davies and N.S. Stoloff, Trans. Metall. Soc. AIME 233 (1965) 714. [81] P.H. Thornton, P.G. Davies and T.L. Johnston, Metall. Trans. 1 (1970) 207. [82] N.S. Kurnakov and S.F. Zhemchuzhnii, Z. Anorg. Chemie 60 (1908) 1; translated from J. Russ. Phys. Chem. Soc. 32 (1907) 448. [83] J.H. Westbrook, J. Electrochem. Soc. 104 (1957) 369. [84] O. Noguchi, Y. Oya and T. Suzuki, Metall. Trans. 12A (1981) 1647. [85] R.L. Fleischer, in: Structural Intermetallic Compounds, eds R. Darolia, J.J. Lewandowski, C.T. Liu, P.L. Martin, D.B. Miracle and M.V. Nathal (TMS, Warrendale, PA, 1993) p. 691. [86] Y. Mishima, S. Ochiai and Y.M. Yodagawa, Trans. Jpn. Inst. Met. 27 (1986) 648. [87] Y. Mishima, S. Ochiai and Y.M. Yodagawa, Trans. Jpn. Inst. Met. 27 (1986) 656. [88] G. Sauthoff, Intermetallics (VCH, Weinheim, Germany, 1995). [89] W. Kt~ster and E Sperner, Z. Metallkd. 48 (1959) 540. [90] P.R. Strutt and R.S. Polvani, Scr. Metall. 7 (1973) 1221. [91] G. Sauthoff, Z. Metallkd. 80 (1989) 337. [92] G. Sauthoff, Z. Metallkd. 81 (1990) 855. [93] G. Sauthoff, in: Proc. Int. Symp. on Intermetallic Compounds - Structure and Mechanical Properties, JIMIS-6, ed. O. Izumi (Jpn. Inst. of Metals, 1991) p. 371. [94] N.S. Stoloff, Metall. Trans. 24A (1993) 561. [95] J.K. Tien and S.M. Copley, Metall. Trans. 2 (1971) 219. [96] T.M. Pollock and A.S. Argon, Acta Metall. Mater. 42 (1994) 1859. [97] J.L. Vall6s and D.J. Arrell, Paper M10C6, in: Proc. 14th Int. CODATA Conf., Chamb6ry, France, 1994 (to be published). [98] R.L. Fleischer, in: Intermetallic Compounds: Principles and Practice, Vol. 2, eds J.H. Westbrook and R.L. Fleischer (Wiley, Chichester, UK, 1994) p. 237.
CHAPTER 49
Geometry of Dislocation Glide in L12 7~-phase: TEM Observations Y. Q. SUN Department of Materials University of O~.ord Parks Road, O~.ord, OX1 3PH UK and
R M. HAZZLEDINE UES Inc. 4401 Dayton-Xenia Road Dayton, OH 45432-1894 USA
Dislocations in Solids 9 1996 Elsevier Science B.V. All rights reserved
Edited by F. R. N. Nabarro and M. S. Duesberv
Contents 1. 2. 3. 4.
Introduction 29 Selection of foil orientations 32 Normal slip mode (100){001} 33 Normal slip mode (110){001} 37 4.1. The low mobility of edge and screw dislocations in (110){001} 38 4.2. Observation of dissociations 39 4.3. Slip mechanism 45 5. Dislocations in the anomalous (101) { 111 } slip 45 5.1. Distribution of dislocations 45 5.2. Observation of dissociations 47 5.3. Superkinks and switched superpartials 48 5.4. Increasing activity on the cube cross slip plane with temperature 50 5.5. APB tubes and edge dipoles 51 5.6. Dislocation propagation in (101){111} slip as suggested by TEM observations 6. (101){111} slip" SISF dissociation 55 7. A summary of observations relevant to yielding 58 8. Geometrical aspects of work hardening 65 References 66
53
1. Introduction The only active slip direction in the L12 ordered '7~-phase is the close-packed (110), except at very high temperatures (above ~ 800~ Yet depending on whether that slip occurs on {111} or {001}, the material's response to an external load and to changes in temperature is very different. When the slip plane is { 111 } the yield stress increases with temperature (fig. 1). Such an abnormal variation in the yield stress with temperature is also accompanied by a number of other distinguishing properties- violation of the Schmid law and asymmetry of yield stress in tension and compression, near-zero sensitivity of the flow stress to strain-rate changes, and a yield stress that is reversible with respect to temperature changes, etc. - properties which make the ( 101 ) { 111 } slip in "7' outstanding and anomalous. However, when slip in (110) changes to {001}, occurring when the temperature is above a certain value Tp (fig. 1), the "7~ phase is normal: its yield stress decreases with temperature and increases substantially with strain-rate, the Schmid law is obeyed, and the yield stress is the same in tension and compression. (For a full review of the mechanical properties of "7~ the reader is referred to several review articles, e.g., [1, 2].) The aim of this chapter is to present transmission electron microscopy (TEM) observations of dislocations and other related defects that characterise these two contrasting slip modes in "7~ and, on this basis, to explore the controlling dislocation mechanisms. When the temperature exceeds about 800~ the slip direction changes to (100), the shortest lattice translation vector in L12; the origin for this transition will also be investigated. It was using a transmission electron microscope that Marcinkowski, Brown and Fisher [3] obtained the first direct experimental evidence for an important hypothesis, proposed almost half a century ago by Koehler and Seitz [4], about dislocations in superlattice structures: the dislocations would propagate more readily if they existed in groups coupled by antiphase boundaries (APB). Marcinkowski et al observed in the TEM that dislocations in L12-ordered Cu3Au indeed contained two components or superpartials which were held together by a strip of APB. Today, as exemplified in this volume, understanding of the varied and sometimes intriguing plastic properties of intermetallic compounds, notably the L12 ordered "7~-phase, is based on dislocation structures with complexities that have gone far beyond that of the simple splitting into two superpartials. The further splitting of the superpartials (e.g., each 89 (110} dislocation splitting into two 1 (112) Shockley partials) is known to play an important role in influencing the motion of superdislocations and the properties of plastic flow. In some cases the APB does not represent the major mode of dissociation and faults of other types (e.g., superlattice intrinsic stacking fault SISF) dominate the coupling of the partials. Owing, to a large extent, to advances in TEM techniques, experimental examination of dislocations with complicated internal structures can be performed, often to reveal details approaching atomic resolution; this may be made in parallel with observations of dislocation substructure, from which evidence for the relative mobility of dislocations can be obtained.
30
Ch. 49
Y. Q. Sun and P. M. Haz~ztedine
(/) or) CIJ
t.-
.4--
. _..,,,
TEMPERATURE
[10]'1(111)
--
-------
i []'10](001) ~ [010](001)... [001](010)
[
Fig. 1. A schematic illustration of the yield stress (YS) and work hardening rate (WHR) of 71 and their variations with temperature. For most crystal orientations the active slip system is (103){ 111 } below the peak temperature T o and { 111 } octahedral slip in L12 has been studied most extensively. Experimentally it has been established that, when the yield stress shows an anomalous increase with temperature, the (101) superdislocations are dissociated into two 89 superpartials bordering the APB on { 111 }, i.e., 1
1
~(10T> + APB{lll} + ~(101).
(1)
The 89 (101) superpartials correspond to the unit dislocations in the face-centred cubic 1 (f.c.c.) lattice and were postulated to split further into g(l12) Shockley partials on
{111} [3], 1
-
1
~(101> -~ g1(1- 1 2 ) + C S F + g (2ii),
(2)
where the CSF connecting the Shockleys stands for a complex stacking fault which involves not just a disruption in the stacking sequence but also changes in the bonding between the atoms across the fault. Although the above dissociation mode was initially proposed on the basis of a hard-sphere model, it has been substantiated by atomistic simulations (e.g., [5]) and has received direct experimental confirmation from lattice imaging TEM [6-8]. The above dissociation modes have formed the basis of all the current models for the yield stress anomaly in 7' [9-11 ].
w1
Geometry of dislocation glide in L12 "Y~-phase
31
Another possible mode of dissociation for (110){111} dislocations involves splitting 1 (112), i.e into two super-lattice Shockley partials .~ 1
(3)
(101)_+~-1 (211) + SISF + ~(112),
where the superlattice intrinsic stacking fault SISF contains a fault in the stacking sequence but maintains the bonding condition among the nearest neighbouring atoms. Although analysed theoretically in detail [12-14], this dissociation mode is not observed frequently in "7' and there is a lack of experimental characterisation of the associated deformation properties. This dissociation mode has been suggested to lead to normal temperature dependence of the yield stress, but under the condition of the { 111 } APB _ being fully unstable [13, 14]. In Pt3A1 for which the slip system is (101){111} with a normal temperature dependence of the yield stress, it has been suggested that the dissociation involves SISF [13, 14], but so far there has been no direct experimental evidence, e.g., with TEM, to support this suggestion. When the temperature exceeds Tp (fig. 1), slip is still in a (110) direction but the slip plane changes to {001 }. Experimental observations have shown that when the superdislocation dissociates into two superpartials, the APB is on the glide plane {001 }, i.e.,
1
1
(4)
(110) -+ ~(110) -+- APB{001} + ~(110).
Investigations of the further splitting of the 1(110){001} superpartials have centred on the screw orientation, but edge dislocations have also received some attention. On {001}, screw and edge dislocations are special because their lines are parallel to the intersection of {001} with two {111} planes on which further glide splitting can take place. For screws the further splitting into Shockleys is the same as in eq. (2) but on planes alternating between the two co-zonal {111} planes at distances of b/2, b being the magnitude of ~1 (110), the Burgers vector of the superpartial. This further dissociation renders the dislocation core non-planar. Along the edge orientation a ~1 (110) superpartial on {001} is the equivalent of a Lomer dislocation in f.c.c. [15] and is thus expected to dissociate into a Lomer-Cottrell lock [16, 17] involving two Shockleys on the two { 111 } planes and a stair-rod dislocation, i.e., -
~[1T0]I -+ 61[11~'] + C S F ( l l- l ) + ~I [ l l 0 ] + C S F ( l l l ) +
1
-
g [112].
(5)
A schematic illustration of this structure and its experimental observation will be shown later in fig. 6. The above are the basic dislocation dissociation modes in L12 '7~ which have been studied and observed most extensively; they have formed the basis for the investigation of dislocation behaviour and mechanical properties. This paper demonstrates the experimental verification of the above dissociation schemes and also several of the variations derived thereon. Some of the dissociation schemes, e.g., (1, 2), have been studied
32
Y. Q. Sun and P. M. Ha~ledine
Ch. 49
extensively with atomistic simulations, while others, e.g., (5), proposed on the basis of crystallography and also directly observed, have received little atomistic study. In general TEM observations in recent years have put more dislocation types under scrutiny than atomistic studies (e.g., Lomer-Cottrell locks and B5 locks on which there have been several experimental observations but little atomistic simulation so far). This paper also aims to relate the observed dislocation substructure and, to certain extent, mechanical properties, to the dislocation dissociations.
2. S e l e c t i o n o f f o i l o r i e n t a t i o n s A full account of the experimental details, related to TEM, in single crystal preparation, sample testing and foil thinning is beyond the scope of this chapter, but the selection of foil orientation from deformed single crystal samples for TEM observations deserves an appropriate exposition. TEM micrographs recorded from foil specimens, usually 1000 or less in thickness as needed for sufficient electron wave transmission, inevitably offer a very limited view of the overall dislocation distribution in the bulk, and it is thus imperative that the thin foils be prepared parallel to selected crystallographic planes, including the macroscopic slip plane. In single crystals deformed by a uniaxial load, very often only one slip system, i.e., the primary slip system, is activated and, to observe the arrangement of dislocations in that slip system, the foil plane is often selected to be parallel to the macroscopic slip plane which is identified by optical microscope observations of surface slip markings. Hence many of the foils selected for the study of {101){111} and (110){001} slips are parallel to the primary slip planes, i.e., (111)and (001), with the loading axis in the 001-011-111 unit triangle. Such samples enable the distribution of dislocations to be viewed over a field tens of microns in width. In the ~"-phase the extent of cross slip of screw dislocations is an important experimental feature to observe and this can be revealed either by tilting the sample around the screw dislocation line direction to detect any line curvature off the slip plane, or by imaging foils which are parallel to selected crystallographic planes co-zonal with the slip direction or Burgers vector, or a combination of both. In these samples, the relative mobilities of dislocations are deduced from the arrangement of dislocations and those which dominate the structure and for which the dominance is not caused by self energies are thought to be the least mobile. Samples for the above observations are in general deformed by a small amount, typically 1-3 %. The observation of the dissociation structures of dislocations can be made on similarly oriented samples using the weak-beam method which is capable of resolving partials that are 15 A apart. When the partial separation is close to the resolution limit, the experimental error makes it unreliable to determine the dissociation plane, which in many cases is not the same as the glide plane, by trace analysis. Performing lattice resolution HREM observations on the basis of the weak-beam result has been found to be useful in providing further details of small-scale dissociations. For HREM observations, the foil orientation is selected such that the dislocations to be observed are normal to the foil plane so that they can be examined end-on. Also, owing to the required large magnification (~ 106 times), a high dislocation density is needed, thus necessitating a o
w
Geometry of dislocation glide in L12 ")/'-phase
33
relatively large amount of plastic strain (up to 10%). In the "/phase, as established by weak-beam observations, the substructure of deformed samples is often dominated by one type of dislocation (screws) in the case of (101){ 111 } slip, or two types (screw and edge dislocations) in (110){001} slip; dislocations with mixed characters have a very low density. This property ensures that foils containing the majority dislocations end-on are not influenced much by other dislocations. Image simulations will not be addressed here and references will be given when they are of concern. It is generally found that the identification of the Burgers vector of dislocations in 3,~ can be performed quite straightforwardly from their visibility at different 9" b or 9" b x u values, and the effects of elastic anisotropy do not appear to introduce significant ambiguity in the determination of Burgers vector b (9 represents the diffraction vector used to form the image and u is the line direction). Image calculation has been found necessary when the exact partial separation is required for the accurate measurement of fault energies (e.g., [18-20]) or when the relative image intensity becomes a factor for differentiating two dissociation schemes [21]. For the lattice imaging work, comprehensive image matching has not been thought to be feasible because the experimental imaging condition apparently change significantly within short intervals; this was reflected by the change of image contrast owing perhaps to the buckling of foils or surface contamination when the sample is subjected to a highly condensed electron beam. The combination of two tools, i.e., of weak-beam and lattice imaging, is the authors' approach for an improved identification of the dissociation mode of dislocations.
3. N o r m a l slip m o d e ( 1 0 0 ) { 0 0 1 } (100} is the shortest lattice translation vector in the L12 structure. On the basis of the isotropic elasticity theory of dislocations, (100) dislocations have smaller self energies than (110) dislocations. Using anisotropic elasticity and taking the dissociation of (110) 's into account, Hazzledine et al. [22] made a comparison of the line energies of (100) and (110) dislocations for Ni3A1. They found that except for a small angular range in which (100) dislocations are elastically unstable, (100) dislocations have lower line energies than (110) dislocations. Although the self energy is the lowest among the known dislocation systems in 3/, (100){001 } slip does not operate until the temperature is very high, above about 800~ higher than the temperature range of (110){001 } slip [23, 24]. This suggests that (100){001 } dislocations probably have the least mobility and require the highest thermal activation to operate. TEM observations have demonstrated that the low mobility of (100){001} dislocations is associated with the non-planar dissociation structures on 45 ~ dislocations, or B5 locks. (100) dislocations were first noted in a polycrystalline Ni3A1 deformed at 600~ [25], and they were in the form of short segments linked to (110) dislocations; they were thus thought to be the result of dislocation reactions between simultaneously active (110){001 } dislocations, which are likely in a polycrystalline sample. Later it was found in both Ni3Ga and Ni3A1 deformed at around 850~ that substantial numbers of (100) dislocations existed in the deformed samples [2, 24]. At such temperatures, (100) dislocations are in the form of long segments or loops and thus clearly contributed to the
34
Y. Q. Sun and P. M. Hazzledine
Ch. 49
plastic strain. (110) dislocations that dominate the plastic strain at low temperatures are still present at the high temperatures but with a much lower density. Figure 2 shows a TEM micrograph of Ni3A1 deformed at 900~ showing dislocations with (100) Burgers vectors. The occurrence of (100) slip is accompanied by a rise in the work hardening rate (WHR, fig. 1). The high work hardening rate is consistent with multiple slip. There are always at least two (010){001 } slip systems that possess the same highest Schmid factor. This makes the work hardening associated with (100){001 } slip essentially the same as in stage-II deformation of f.c.c, metals where the high work hardening rate is caused by the activation of a secondary slip system. There is as yet no quantitative modelling for the work hardening rate of multiple (100){001 } slip. Further details will be given in section 8. In the L12 structure a {001} plane has a smaller interplanar spacing than {111} and thus on {001} a lower dislocation mobility is generally expected. The fact that (100){001) requires a higher temperature to operate than (110){001} suggests that (100) dislocations are less mobile than (110)'s, considering that the former are energetically more favourable. In experiment, (100) dislocations on {001) were found to have particularly low mobilities along (110) directions along which they are 45 ~ in character. The low mobility of (100){001} dislocations was found to lie in the non-planar core structures formed on dislocations along these line orientations. From the curvature of the dislocations in fig. 2, the active glide plane has been identified as {001 }. The dislocation labelled D in fig. 2 has the Burgers vector (111) and lies exactly along the cube edge (100). It is thought to be formed by the reaction (110){001} + (001){010) -+ (111), with line direction parallel to (100). This is a locked dislocation because its allowed glide plane is (111) x (100) - {011} which is a high friction plane. The most distinguishing feature, however, is that most of the (100) dislocations are preferentially aligned along (110) 45 ~ directions. These observations led to the suggestion [23] that, since (110) is the intersection line of two { 111) planes with the {001 ) glide plane, a 45 ~ dislocation could split on the close-packed planes to lower its self energy, rendering the core nonplanar. The resultant structure is in fact the equivalent of B5 locks in f.c.c. [27]; the difference is that in f.c.c. B5 locks, like Lomer-Cottrell locks, are formed as a result of a special reaction between (110){111} dislocations, whereas in 1,/ they arise from the operation of a single (100){001} slip system. As in f.c.c., 45~ } dislocations are expected to dissociate to form non-planar structures subtending either an acute or an obtuse angle, fig. 3(a). The lattice imaging observations have shown the existence of non-planar cores on 45 ~ (100){001 } dislocations and an example is given in fig. 3(b). This image was recorded in the lattice imaging mode from a Ni3A1 sample deformed at 850~ The foil is parallel to (110) in which 45 ~ dislocations in the [010](001) primary system lie end-on. From the closure failure of a Burgers circuit built around the entire dislocation, the component of the Burgers vector orthogonal to the electron beam is found to be 1 [~ 10], consistent with the observed dislocation being a 45 ~ [010]. The three-fold non-planar structure observed in fig. 3(b) is thought to represent the following dissociation scheme [010] ~ ~1 [112.] + C S F ( l l l ) +
1
-
1
[031](stair-rod) + CSF(111) + ~ [i21] .
M 'w
Fig. 2. A weak-beam image to show 45°(010){001} dislocations in a Nij(A1, Ti) sample deformed at 900OC. Dislocations marked A have [loo] Burgers vector and the dislocation marked B has [OIO]. A and B dislocations are preferentially oriented along 45' (1 10) directions. Dislocations marked C are [I101 edges. Two (010) dislocations with orthogonal Burgers vectors have reacted to form a (110) dislocation lying in the edge orientation. Dislocation labelled D lies along cube edge and has Burgers vector (1 1 I ) . (From P.M. Hazzledine, M.H. Yo0 and Y.Q. Sun [22], by courtesy of Pergamon Press PLC).
VI w
36
Ch. 49
Y. Q. Sun and P. M. Hazzledine
(a)
x"
(OOl)
b,
Fig. 3. (a) B5 locks on 45 ~ (010){001 } dislocations. (b) A lattice imaging TEM picture showing the internal structure of a 45 ~ (010){001} dislocation lying end-on in the thin foil. The net Burgers vector normal to the electron beam is given by the closure failure of the Burgers circuit. The non-planar core structure is responsible for the low mobility of 45 ~ (010){001 } dislocations.
w
Geometry of dislocation glide in L12 "y~-phase
37
The locking of 45~ dislocations is similar to that of (111) screw dislocations in some b.c.c, metals because, in both cases, the extended non-planar structure must constrict to make the dislocation movable, which requires the assistance of thermal fluctuations, and the mobile state is unstable to the locking transformation. However, unlike (111) slip in b.c.c., in a given {001} plane a (010) dislocation may be locked along two (orthogonal) (110) directions, causing the entire dislocation loop to be trapped along these two directions. In b.c.c, only the screw dislocations are locked and the plastic flow can still be carried by the non-screw dislocations which remain mobile (at least during the early stage of plastic deformation). We shall see in the next section that the same property (i.e., loop trapping)is also possessed by (110){001} slip.
4. Normal slip mode (110){001 } Except in samples oriented very close to [001] (within approximately 5 ~ around [001]), the active slip system above the peak temperature Tp is the primary (110){001} slip; below Tp the slip direction is also (110) but on a { 111 } plane. Tp has been found to vary substantially with sample orientation and strain rate [2]. Since the onset of (110){001} slip with rising temperature marks the termination of the preferred (110){111} slip mode (preferred technologically because it raises the strength with increasing temperature), pushing the peak strength of Ni3A1 to still higher temperatures would require the suppression of the (110){001} glide, dislocation mechanisms for which need to be identified. The behaviour of (110){001} dislocations is important also because it is a necessary element in the current models for the yield stress anomaly of (110){ 111 } slip [9-11 ]. A feature common in these models is that the pinning of the dislocations, initially gliding on {111}, starts with some screw segments cross slipping onto {010}. But the models differ in the extent to which such cross slip events need to occur in order to become effective barriers (see, e.g., Vitek, Pope and Bassani chapter in this volume for a full description of these models). When the APB is completely on the cube cross slip plane, the screw dislocation is known as a Kear-Wilsdorf lock [28]. An understanding of the dislocation processes controlling pure (110){001 } slip is thus also relevant to the interpretations of the anomalous regime associated with (101){111} slip. (For a single crystal sample the cube cross slip [10i](010) (associated with the primary octahedral system [101](111))is not the most highly stressed {110){001} system, thus pure cube slip is possible only with the primary system [110](001).) A pair of 89 screw superpartials coupled by the APB on {010} have a lower energy than if the APB is on { 111}, and this is the fundamental driving force for { 111 } -+ {010} cross slip, another key element common to the models for the yield stress anomaly. This energy anisotropy for the screws is caused by a combination of two effects: a lower APB energy on {010} than on {111} [29] and a torque force, arising from elastic anisotropy, which acts in the direction of forcing the dislocation pair onto {010} [30]. As a result, an APB-dissociated (110) screw is bi-stable with dissociations on {111} and {010} as the two stable states [30, 31] although {010} is usually the most stable dissociation plane (i.e., dissociation on { 111 ) is metastable) even if "~010/> 3'111 (3'010 and 3'111 are APB energies). Despite being favoured by lower
38
Y Q. Sun and P M. Hazzledine
Ch. 49
self energies, (110){001 } slip requires substantially higher temperatures to operate than (110){111}; this is usually attributed to the low mobility of these dislocations. This section is aimed at identifying the origin of the low mobility of (110){001 } dislocations.
4.1. The low mobility of edge and screw dislocations in (110){001} Slip-line observations [32] have shown that the primary (110){001} cube slip is planar with no indication of extensive cross slip. In keeping with this conclusion, TEM observations (fig. 4) also show that in (110){001 } slip dislocation propagation is confined to the {001} glide plane, which is in contrast to the anomalous (101){111} slip in which screw dislocations are seen to cross slip frequently off the macroscopic slip plane {111} onto {010} (see section 5). Glide on {001 } is, in general, expected to experience stronger frictional resistance than on { 111 } because of the smaller interplanar spacing (d001 < dill). Experiments have shown that the mobility is particularly low for screw and edge dislocations. As shown in fig. 4, a weak-beam image of dislocations in a Ni3Ga sample deformed at 700~ ( ~ 100 ~ above the yield stress peak Tp), the majority of the dislocations lie along screw and edge orientations with very short lengths of dislocations having mixed characters. In fig. 4 the specimen foil is parallel to the macroscopic slip
Fig. 4. A weak-beam image to show the distribution of dislocations in the primary (110){001 } slip in a sample deformed just above the yield stress peak. The plane of the foil sample is (001). The substructure is dominated by screw (S) and edge (E) dislocations, with very few dislocations having mixed characters. Such observations indicate that both screw and edge (110) dislocations are relatively immobile on {001}. (From Y.Q. Sun, EM. Hazzledine, M.A. Crimp and A. Couret [17], by courtesy of Taylor Francis Ltd.)
w
Geometry of dislocation glide in L12 .yt_phase
39
plane {001} and the dislocations can be traced over long distances, indicating that they are lying within the foil, and thus their propagation is largely planar. The edge dislocations exhibit a pronounced rectilinear morphology while the screw dislocations, which jointly dominate the substructure with the edges, are slightly curved. Using anisotropic elasticity Douin et al. [33] analyzed the effect of line energy on dislocation loop shape using the method of the inverse Wulff plot. The study showed that, without the energy reduction due to Lomer-Cottrell dissociation, on {001} near-screw (110) dislocations are favoured, but the pure edge orientation should be (elastically) unstable. The presence of rectilinear edge dislocations therefore suggests that they are less mobile than other dislocations, consistent with the locked structures observed. The fact that the screw dislocations are able to retain their curvature without an external load suggests that they also experience a high frictional force. The scarcity of mixed dislocations shows that dislocations with intermediate characters are mobile.
4.2. Observation of dissociations
The low mobilities of screw and edge (110){001} dislocations have been found to be caused by their non-planar dissociation structures and this is demonstrated by the TEM observations with a combination of weak-beam and lattice imaging methods. There is a general agreement on the screw dislocations being immobile as a result of the superpartials' further splitting into Shockleys on the intersecting { 111 } planes. This can be envisaged on the basis of a hard sphere model and has been substantiated by the atomistic simulations of Yamaguchi et al. [12] which, although concerned with {111} -+ {010} cross slip, can be applied to pure (110){001} cube slip. The dissociation of the (110) superdislocation on {001} is described by eq. (4)in which the (110) superdislocation 1 first splits into two ~(110) superpartials with the APB on the glide plane {001}. The further splitting of the screw superpartials into Shockleys, according to eq. (2), may occur on two {111 } planes which are co-zonal with {001 } and alternate between the two at lattice positions separated by b/2, as shown in fig. 5(a). Superpartial configurations at positions in between are not known and can be assumed to be fully constricted [10]. 1 However, C16ment et al. [34] hypothesized a planar dissociation of the ~(110) screw superpartials on {001} with a view to accounting for the in-situ observations and an anomaly in the flow stress for (110){001} slip in a Ni-based 3'/7' two-phase superalloy. Such a core has not been substantiated by atomistic simulations, nor was there any direct experimental verification. Direct experimental evidence for the dissociated structure of screw dislocations in (110){001} slip has been provided by lattice resolution TEM, shown in fig. 5(b)[6, 7]. In this picture a (110) screw dislocation lies normal to the thin foil and parallel to the electron beam, i.e., it is viewed end-on. The image is from a Ni3A1 sample deformed at 400~ at which, although the primary (110){001} slip has not yet fully started, the large extent of cross slip onto {010} permits the imaging of screw (110) dislocations with the APB on {010}. The image shows that the extended cores of the !(110) screw superpartials are aligned along {010} ' showing that the APB is on {010} " 2 The displacement of the APB is not visible since it is parallel to the direction of the
40
Ch. 49
Y. Q. Sun and P. M. Hazzledine
(a) ,,,, ~
APB
.
b/.,/ .
.
.
Fig. 5. (a) End-on schematic of the Kear-Wilsdorf lock on the screw (110){001} dislocation. For glide on {001}, the locking effect derives from the out-of-the-plane splitting into CSF-coupled Shockleys. (b) An HREM image showing a screw (110){001 } dislocation imaged end-on in a thin foil, showing the predicted Kear-Wilsdorf structure. Positions of the superpartials are indicated by the arrows.
Geometry of dislocation glide in L12 "y~-phase
w
41
electron beam. The possible supplementary displacement normal to the fault plane is not revealed in the HREM pictures. The two superpartials are spread on two parallel { 111 } planes. Splittings on different { 111 } planes have also been observed and they directly confirm the dissociation structure illustrated in fig. 5(a). Since the dissociation of the screw superpartials is off the glide plane, the screw dislocation is locked against motion on {001 }. The dislocation becomes mobile if the non-planar core is constricted and this requires thermal activation; this is the origin for the increased activity of (110) screws on {001} with increasing temperature. The screw dislocations in (110){001} therefore experience essentially the same type of lattice friction as 45~ } dislocations or as screw (111) dislocations in some b.c.c, alloys. Experimental observations have shown that the edge (110){001} dislocations should feel the same type of lattice friction since their core structures, though different from those of screws, are of the same nature, i.e., their mobile, constricted core structure is unstable and is driven to transform into a non-planar configuration. For the edge or near edge dislocations, several mechanisms have been proposed to explain their low mobility. Veyssi~re [35] noted that dislocation loops on {001} were segmented with screw dislocations showing smooth curvature and edge and near-edge dislocations having a rectilinear image. For the edge orientation Veyssi~re suggested a climb dissociation 1 mode in which the (110) superdislocation first splits into two 7(110) partials which then separate by a conservative climb dissociation on { 110} perpendicular to {001 }, i.e., [110]-+ ~1[110] + APB(il0) + 1 [~ 10] This is followed by a further decomposition of the superpartials by glide 1 ~[110]-+ ~1[111]+
1
[112]
Here between the Frank ( 89(111)) and Shockley partials is a strip of stacking fault which may be intrinsic or extrinsic depending on the direction of motion of the Shockleys. However, owing to the limitations of the weak-beam method, it was not possible to identify experimentally the dissociation mode in further detail. It was subsequently found that climb dissociation could also occur on {310} planes [36]. The pronounced lirrearity of edge dislocations was later confirmed by Sun and Hazzledine [23] in both Ni3A1 and Ni3Ga and by Korner [37] in Ni3(A1, Ti) and was also revealed later by in-situ experiments [38]. Different immobilization mechanisms were 1 proposed, however [22, 17, 24]. It was noted [23] that a 7(110) edge superpartial on {001 } was the equivalent of the Lomer dislocation in f.c.c., although in 7' such dislocations were a consequence of cube slip and not formed as a result of dislocation reaction as they are in the original models of Lomer [15] and Cottrell [16]. An edge 1 (110)(001} superdislocation dissociating into two 7(110) partials was called a double Lomer [17, 23], to which the dissociation mode proposed by Cottrell for f.c.c, can be extended to form a double Lomer-Cottrell lock, described by eq. (5) and illustrated in fig. 6(a).
42
Ch. 49
Y. Q. Sun and P. M. Hazzledine
(a)
B~
~ J.
ar
s
APB
~A
Bo~
~A
Fig. 6. (a). The double Lomer-Cottrell lock on the edge (110){001} dislocation. The dislocation is split into two edge superpartials each of which is the equivalent of the Lomer dislocation in f.c.c.. The entire lock consists of two ordinary Lomer-Cottrell barriers. (b) An HREM picture of a double Lomer-Cottrell lock in Ni3Ga deformed at 600~ (From Y.Q. Sun, P.M. Hazzledine, M.A. Crimp and A. Couret [17], by courtesy of Taylor and Francis Ltd). In the experiments on Ni3Ga, edge (110){001} dislocations in the double-Lomer configuration were observed by the weak-beam method in samples deformed at around 600~ Weak-beam observations showed that edge (110){001 } dislocations in the double Lomer configuration were less mobile than those of other angular characters. Limited by the resolution power of the weak-beam method, no further splitting can be resolved on the individual Lomer superpartials, although the Lomer-Cottrell dissociation provided the most straight-forward explanation for their low mobility [23]. Lattice resolution microscopy was used to reveal the further splittings by viewing the edge dislocations end-on in specially oriented foils. Figure 6(b) shows an example of a Ni3Ga sample
w
Geometry of dislocation glide in LI 2
,3[ ! -phase
43
parallel to (110) with the edge dislocations in the primary cube system [110](001) lying end-on. The dislocation is seen to consist of two A-shaped components lying on the trace of the (001) plane, consistent with their being two superpartials each of which has dissociated according to scheme (5). The dissociation into Lomer-Cottrell locks is allowed only at lattice positions separated by b. At intermediate positions there is no apparent crystallographic reason for the edge superpartials to split on {001} and it is most likely that they are effectively constricted, as a result of which there is no resistance to the locking dissociation. At higher temperatures, dissociation by climb was observed on edge (110){001} dislocations, but the mode of dissociation, as revealed by lattice imaging TEM, is different from that initially proposed by Veyssi~re et al. [35, 36]. With the weak-beam method, as shown in fig. 7(a) for a Ni3Ga sample deformed at 750~ the pure edge dislocations are seen to have split into three partials while the non-edge dislocations are split into two. The transition from pure edge to non-edge is sharp, as pointed to by arrows in fig. 7(a), and all the partials recombine at the junctions. An undissociated (110){001} edge dislocation is called a super-Lomer since its Burgers vector is twice as long as that of a Lomer in f.c.c. [ 17]. At the temperature where super-Lomers are observed, the substructure is still dominated by screw and edge dislocations, but with the edges having visibly larger proportions than the screws, as noted in a number of studies [24, 35-38]. The screw dislocations are curved smoothly and thought to be the major contributor to the plastic strain. Observations such as fig. 7(a) suggest that at this temperature the super-Lomers are probably effectively immobilised by their non-planar core structures. It proved to be impossible to index the Burgers vectors of the three partials involved with the weak-beam method, and lattice resolution TEM was thus used which showed that the dissociation has taken place by climb, as shown in fig. 7(b). Here the sample was deformed under conditions similar to those for fig. 7(a) except that a much higher dislocation density was introduced by straining to about 10%. The foil is parallel to (110) so that the edge dislocations in the primary [110](001) slip are lying normal to the foil plane. The dislocation in fig. 7(b) has been identified to be a (110){001 } edge dislocation by constructing a Burgers circuit around the entire dislocation and indexing its closure failure, giving b - [110] [17]. The dislocation is seen to split into three partials, two on the intersecting { 111 } planes ((1 i 1) and (111)) and the third, the stair-rod dislocation, at their junction. The two partial dislocations at the extremities of the V-shaped structure are connected to the stair-rod dislocation by stacking faults on { 111 }. By viewing along the traces of the two { 111 } planes, it is seen that the stacking faults on both { 111 } planes are characterised by the removal of a { 111 } layer of atoms and thus the stacking faults 1 are intrinsic and the two extreme partials are therefore Frank partials with b = .~ (111). This observation is explained by the following dissociation scheme, -
1
1
1
[110] -+ ~ [ l i l ] + S I ( E ) S F ( l i l ) + 5[li0] + SI(E)SF + g[liT], where the choice of SISF or SESF depends on the climb direction of the Frank partials. In fig. 7(b) SISFs are involved, but many of the dislocations observed contain an SISF on one arm and an SESF on the other [17]. In fig. 7(b) further dissociation on the intersecting (111 } planes can be seen on the two Frank partials and such dissociation is
Fig. 7. (a) A weak-beam image showing the three-fold splitting on edge (110){001} dislocations. The image plane is (001). The three images constrict at the sharp junctions with the screw orientation at which the ordinary two-fold splitting is observed. (b). An HREM picture showing a super-Lamer-Cottrell lock on an edge (1 10){001} dislocation in Ni3Ga. The dissociation involves climb and the structure is illustrated in (c). Here ‘in.’ stands for superlattice intrinsic stacking fault (SISF). (From Y.Q. Sun, P.M. Hazzledine, M.A. Crimp and A. Couret [17], by courtesy of Taylor and Francis Ltd.)
9 %
Geometry of dislocation glide in L12 "y~-phase
w
45
thought to be of the same type as the formation of a stacking fault tetrahedron from a prismatic Frank dislocation loop [39], i.e., -
1
-
1
-
1[111]--9 ~ [ l l 2 ] + C S F ( 1 T i ) + ~[110] A schematic illustration of the climb dissociation of the super-Lomer dislocation is given in fig. 7(c). 4.3.
Slip mechanism
(110){001} slip has the property that dislocations are locked along two directions, the Kear-Wilsdorf lock on screws and Lomer-Cottrell locks on edges. For a given slip system, if the dislocations are locked along only one orientation, slip can still proceed through the propagation of the mobile dislocations and the deformed structure consists of a high density of dislocations locked in the specific orientation. This is the case in some b.c.c, metals in which (111) screw dislocations are locked in the three-fold non-planar core structures, and it is also true of the anomalous (110){111) slip in L12 (see later) for which locking also occurs along the screw orientation. If locking occurs on two or more orientations, the entire dislocation loop may be blocked along these orientations, forming rhomboidal or polygonal loops. The locking of both screw and edge dislocations in (110){001} is essentially the same as that of the screws in b.c.c, in that the mobile state is unstable to the locking dissociation. Continuation of loop expansion requires its unlocking and thus at a given stress level (110){001} slip starts when the temperature is sufficiently high to assist the recombination of the partials involved in the non-planar dissociations.
5. Dislocations in the anomalous ( 1 0 1 ) { 111 } slip 5.1.
Distribution
of dislocations
A well established fact concerning dislocations in samples that have undergone (101){111} slip with the yield stress anomaly is that the substructure is dominated almost entirely by screw dislocations with very few edge segments present. This was noted in the early TEM observations of deformed Cu3Au and led to a mechanism, proposed by Kear and Wilsdorf [28], in which pinning was caused by the screw dislocation cross slipping onto a cube plane. When the screw dislocation is fully dissociated on the cube plane, the structure is known as a Kear-Wilsdorf lock. This microstructural feature was subsequently observed in Ni3A1 and Ni3Ga containing various alloying additions where the dominance of the screw dislocations was more pronounced and occurred in the entire temperature range of the yield stress anomaly [9, 40-45]. Figure 8 shows a typical example of dislocations in a Ni3Ga single crystal sample deformed at 200~ which is within the temperature region of the yield stress anomaly and at which the active slip system is (101){111}. The foil plane in fig. 8 is parallel to the macroscopic { 111 } slip plane and the 202. diffraction vector used to form the weak-beam images is m
46
Y. Q. Sun and P. M. Hazzledine
Ch. 49
Fig. 8. A dislocation loop in a Ni3Ga sample deformed at 200~ The loop on the whole is elongated along the screw orientation. A superkink is marked SK. The four pictures were taken while the specimen was tilted around the screw dislocation line. In (a) the electron beam direction (viewing direction) is close to [111] and in (d) the beam direction is very close to [010]; for (b) and (c) the beam is at intermediate orientations. The pictures show that the edge parts of the loop, marked E in (a), are on the (111) plane, but some of the screw parts, marked C, are curved on (010).
w
Geometry of dislocation glide in LI 2 "7~-phase
47
parallel to the Burgers vector [101]. This microstructural feature, showing dominating screw dislocations in the substructure, characterises the entire regime of the yield stress anomaly in many L 12 compounds. Although the macroscopic slip plane is { 111 }, as has been confirmed frequently by slip-line observations (e.g., [32]), the screw dislocations are dissociated with the APB on the cube cross slip plane {010}, i.e., they are in the Kear-Wilsdorf configuration. In this experiment the dissociation plane of the screws is determined by tilting the sample around the screw dislocation line direction and measuring the separation of the superpartials when they are imaged along different directions. In fig. 8 the four TEM images were taken while the sample was tilted around the screw dislocation line direction to four different orientations relative to the incident electron beam. The top picture (fig. 8(a)) views the sample near the [111] direction and the bottom picture (fig. 8(d)) views the sample along [010]; figs 8 (b) and (c) are in between. The predominance of screws in the deformed microstructure is usually taken to indicate that the edge dislocations are relatively more mobile than the screws. The slip system (101){111} is known to possess a yield stress which does not obey the Schmid law and differs in tension and compression [32, 46, 47]. These macroscopic properties have been interpreted on the basis of the effects of the uniaxial load on the further dissociation of the screw superpartials [10, 32]. 5.2. Observation of dissociations
Dissociation of (101) superdislocations into two APB-coupled superpartials on {111} was first observed in L12 ordered Cu3Au by Marcinkowski et al. [3] using the TEM bright-field imaging method. Development of the weak-beam method [48, 49] has greatly enhanced the resolution capability of the TEM for dislocations and partials separated by around 15 ,~ can be resolved. The method has been used extensively to identify dislocation dissociations in ordered alloys, including the -y'-phase. Edge dislocations on { 111 } are expected to exhibit wider dissociations owing to their stronger elastic interactions and resolving the dissociation for the (101){l 1 l) edge dislocations using the weak-beam method has been found to be relatively straight forward; although in practice their rarity in some alloys is more of a problem than resolving the partials. The full dissociation mode of edge (101){111} dislocations, including the dissociation into Shockleys, has been observed together with a systematic measurement of dissociation widths and fault energies [ 19, 20, 50, 51 ]. A certain degree of splitting of the superpartials has been reported frequently on edge dislocations with the weak-beam method, but this requires a very large deviation from the Bragg diffraction condition and as a result the exposure time can be very long. The most systematic study using the weak-beam method is perhaps that of Hemker and Mills [20] who observed, on near-edge dislocations, the further splitting of the superpartials and also performed corrections for diffraction effects to obtain the true dissociation widths. The width of dissociation into Shockley partials was found to be around 2 nm on the near-edge dislocations in a binary Ni3A1 compound, corresponding to a CSF energy of approximately 200 mJ/m 2. Of particular relevance to the understanding of this slip system and other microstructural features, to be outlined in the following sections, is the elucidation of the dissociation mode of the screw dislocations and the extent to which the superpartials are able to
Y Q. Sun and P M. Hazzledine
48
Ch. 49
split further into Shockley partials. Weak-beam experiments conducted on a number of alloys have confirmed that along the screw orientation the superpartials are on the cube cross slip plane; the dissociation of the cross slipped screws is therefore the same as that of the screws in the pure (110){001} slip. With the weak-beam method, resolving the further splitting into Shockley partials on screws has proved to be difficult. Along the screw orientation the expected separation between the Shockleys is around 0.7 nm based on the CSF energies measured on the edge dislocations, and this is beyond the resolution limit of the weak-beam technique. Using the lattice imaging method [6-8] the CSF-coupled splitting was resolved by imaging the screw dislocations end-on in thin foils, as shown in fig. 5(b). Owing to the surface effects, the observed core extension representing the dissociation into Shockley partials may not represent the true dissociation width in the bulk, but the observed small core extension is in keeping with the weak-beam experiments. The fact that the CSFs on the two superpartials of the same superdislocation are sometimes on different { 111 } planes suggests that the CSF dissociation plane is able to change under thermal activation, perhaps through the well-developed kink models [52, 53].
5.3. Superkinks and switched superpartials Whereas it was established that the screw dislocations dominated the substructure in samples deformed in the anomalous regime, TEM observations in recent years have shown a number of other important features that also characterise the anomalous (101){111} slip. Of particular importance are the superkinks, screws bowing-out on the cube plane, and APB tubes and short edge dipoles; these features are dealt with in the next three sections and their interpretations will be given in section 5.6. Weak-beam TEM observations have shown in a number of anomalous L12 alloys that although the screw dislocations dominating the substructure are dissociated on {010}, they are not perfectly straight but contain many steps, known as superkinks, which are lying on the {111} slip plane [43, 45, 54-56]. One of such superkinks can be seen in fig. 8 marked by SK. That the superkinks lie on, and are therefore dissociated in, the { 111 } slip plane, is determined by noting the change in the projected kink height while the sample is viewed along different crystallographic orientations. Most of the superkinks have heights many times the APB width on screw dislocations, but smaller kinks were also noticed in early studies [54, 56]. Couret et al. [57, 58] carried out a statistical measurement of kink distribution (i.e., kink population vs. kink height) in Ni3Ga samples deformed at various temperatures. They found that, except for the kinks with height corresponding approximately to the APB width in samples deformed at low temperatures, the kink number N(h) follows an exponential variation with kink height h. The results for 20, 200 and 400~ are shown in fig. 9. This shows that the superkink distribution can be written in the following form,
N(h)-
h N'exp ( - ~),
where the parameter N ' is related to the total number of kinks and 1/h is the slope in the In N - h plot, h being the statistical average of kink height. The mean kink height
Geometry of dislocation glide in L12 ,yt_phase
w
(a)
20 ~
(b_)
~
In
49
200=(3
114
In80
4
4
"i _=
=,.
,
I
100
a
200
I
300 h(.~)
I
400
I
500
100
(c)
200
300 h(A)
400
500
400"(3 41
i.-=1
i.-,.=1
I
100
I
200
I
300 h(A)
I
400
_
,
500
Fig. 9. The logarithm of kink numbers (N) plotted against kink height (h) to show experimental measurements of the distribution of superkinks in Ni3Ga samples deformed at three different temperatures. The distribution follows an exponential relation except for the kinks with height scaling approximately with the APB width. The average kink height decreases with increasing temperature. (From A. Couret, Y.Q. Sun and EB. Hirsch [58], by courtesy of Taylor and Francis Ltd.)
50
Y Q. Sun and P. M. Hazzledine
Ch. 49
Fig. 10. A TEM weak-beam micrograph to show switched superpartials (indicated by an arrow). At some places, e.g., those marked C, the two partials are combined and full switching can not be identified. is found to decrease with temperature, with h - 170 A, h - 120 ,~, and h - 80 ,h,, at the above three temperatures. This study also shows that in samples deformed at 20 and 200~ there is an exceptionally large number of kinks which have heights approximately equal to the width of the APB (50-100 ,~); in fig. 9 for 20~ and 200~ such kinks are represented by data points significantly off the exponential distribution. Bontemps and Veyssi~re [56] have shown that there are also many kinks on which only one of the partials is stepped and the other remains straight. Another important feature characterizing the superkinks is that the two APB-coupled superpartials are frequently seen to have changed partners across the superkink. Figure 10 shows an example where the switched partials are indicated by an arrow. By tilting the sample inside the TEM over a wide angular range, it has been confirmed that the switching of the partials is real and is not a deception due to projection [24, 58]. There are also numerous cases where the full switching is not shown in the images, but instead the images of the superpartials are seen to have constricted; examples of such constrictions are marked C in fig. 10. It is possible that these images of constrictions also represent switching of the partials (fig. 10).
5.4. Increasing activity on the cube cross slip plane with temperature TEM evidence of significant activity on the cross slip cube plane below Tp was noted in early studies by Staton-Bevan and Rawlings [42] who found many curved dislocations on the cube cross slip plane in samples deformed at temperatures near but still below the yield stress peak. Later investigations showed that in some alloys dislocation activity on the cross slip cube plane existed at much lower temperatures, e.g., room temperature [43, 59]. While most of the screw dislocations locked in the Kear-Wilsdorf configuration are straight (apart from the superkinks shown above), some of the screw dislocations have been found to bow out on their dissociation plane, i.e., the cube cross slip plane which is not the dominant macroscopic slip plane. Such examples can be
w
Geometry of dislocation glide in L12 "g'-phase
51
seen in fig. 8 where the screw segments marked C are nearly straight in fig. 8(a) when the sample is viewed along [111], but are curved when viewed along [010], fig. 8(d). Such observations show that whereas the edge dislocations have propagated on { 111 }, numerous screw segments have propagated on {010} as well as being dissociated on that plane. The fact that the screw dislocations are able to maintain their curvature on {010} after the removal of the load indicates that the lattice frictional force on the screw dislocations is high. Recently, Karnthaler et al. [60] compared the degree of activity on the cube cross slip plane for two samples deformed along the > G1, in such a way that the movement of screw dislocations is mainly controlled by the unlocking process, i.e., by cross slip from the basal plane onto the prismatic plane. The same result is obtained if the dislocation velocity is set as v = APul, considering that the mean jump length A varies only slowly with temperature. It has been shown recently [16, 17] that this locking-unlocking mechanism is not fundamentally different from the Peierls mechanism, since the Peierls mechanism is a limiting case of the locking-unlocking mechanism, when the jump distance between locked positions decreases and scales with interatomic distances. The locking-unlocking mechanism has been evidenced in several h.c.p, metals, with either normal or anomalous stress-temperature relationships [17, 19]. The transition between locking-unlocking and Peierls mechanisms has been studied in titanium [20]. Since the velocity of screw dislocations is controlled by a positive activation energy, Gul - G1, the anomalous behaviour of beryllium cannot result solely from the lockingunlocking mechanism. A model for the yield stress anomaly has thus been proposed by the present authors [15], on the basis of their in-situ measurements of screw dislocation velocities. The main assumption is that, since the deformation is controlled by the cross slip of screw dislocations from the basal plane onto the prismatic plane in the whole temperature range, and since the probability of this cross-slip process is shown to be constant at constant strain rate in spite of a simultaneous increase of stress and temperature in the
80
D. Caillardand A. Couret
Ch. 50
domain of yield stress anomaly (table 1), there is necessarily an increase in the difficulty of cross slip at a constant stress, as the temperature is increased. Such a behaviour corresponds to an evolution of the dislocation core structure with temperature in basal and/or prismatic planes. Assuming dislocations in basal planes are split into two Shockley partials separated by a stacking fault (in spite of their very narrow spreading), this evolution can be described by a decrease in the stacking fault energy with increasing temperature, which may be accounted for by the change in the relative stabilities of h.c.p, and f.c.c. phases with temperature [ 15, 21 ]. On the basis of their own in-situ observations, Beuers et al. [14] proposed another model where the CRSS is controlled by the movement of salient points on edge dislocations, resulting from double cross-slip processes between basal and prismatic planes. Under such a condition, an increasing density of salient points (which are in fact macrokinks) with increasing temperature may hinder dislocation movements and give rise to a flow stress anomaly. This model is incomplete due to lack of details, and the role of screw dislocations has probably been underestimated on account of the special orientation of the microsamples.
5. Cube glide in L12 intermetallic alloys 5.1. Experimental results
Alloys with the L12 structure are deformed by the glide of pairs of dislocations with 7(110) Burgers vectors (similar to those in f.c.c, metals) separated by an antiphase boundary (APB) ribbon which is most often on octahedral ({111}) and cube ({100}) 1 planes. Each 7(110) dislocation is called a superpartial dislocation, and one pair with a (110) Burgers vector is a superdislocation. Superdislocations can glide on octahedral and cube planes, however with very different characteristic features. The existence of an anomalous temperature dependence of the CRSS of cube glide was pointed out after an in-situ study of the glide mechanism in the "7' phase of a nickelbased superalloy (C16ment, Couret, Caillard [22] and C16ment, Mol6nat, Caillard [23]). Experiments with a (110) tensile axis for which octahedral and cube glides have the same Schmid factor (0.35-0.4) reveal that both cube and octahedral glides are activated extensively in a wide temperature range down to 140 K. Since the two glide systems have different Burgers vectors, it is clear that dislocations are able to multiply and glide over large distances in both slip systems. This behaviour is rather surprising when the flow stress anomaly in octahedral planes is compared with the normal stress increase which is expected to occur in cube planes with decreasing temperatures (fig. 7). On the basis of this comparison, cube glide should be inactive below about 500 K, because its CRSS should become much higher than the CRSS of octahedral glide. The CRSS of cube glide seems thus to be closer than expected to the CRSS of octahedral glide between 140 K and 500 K in this material. This hypothesis has been directly verified by measuring the local stress necessary to move screw dislocations in the cube plane of "7' single crystals, as a function of temperature (fig. 8). In spite of rather large uncertainties, the curve clearly exhibits a
w
81
Dislocation cores and yield stress anomalies
"%
O-
r-' K Fig. 7. Schematic description of the CRSS versus temperature curves of cube and octahedral glide in Ni3A1. Dotted lines correspond to the behaviour of cube glide expected at low temperature.
600
f 400
in
\
situ(t')
Latt
oL.INi3( AL,Nb))
-. ~ ~ "'~ - + ~i.._'~" ' ~ Umokoshiet al.. ,''""~ ]---~',"d~" (Ni3(At,Ta))
1
~
"~'."%
~q,,
q"q',~
%
to o~ L~ 200 !
anomalous behaviou r 9
200
9
9
400
Saburi et aL. ( Ni3 (At,W)) ,
9
9
600
,
800
~
9
r
T(K)
Fig. 8. Temperature variation of the flow stress of cube glide in several nickel-based intermetallic alloys (LI2 structure). Dotted lines refer to macroscopic deformation experiments, and the full line refers to in-situ measurements in the 3,t phase. From [23-26]. plateau or a stress anomaly between 140 K and 600 K, and the CRSS of cube glide remains close to the CRSS of octahedral glide at temperatures as low as 140 K. This unexpected result is confirmed by the close inspection of the macroscopic results of Lall, Chin and Pope [24] in Ni3(A1, Nb), Umakoshi, Pope and Vitek [25] in Ni3(A1, Ta), Saburi, Hamana, Nenno and Pak [26] in Ni3(A1, W). These authors reported that, with tensile axes close to a (111) direction, extensive slip occurs on cube planes above room temperature. The corresponding CRSS values are also plotted in fig. 8. They exhibit the same temperature variation as those measured in situ, with the same small stress anomaly. A similar positive temperature dependence of the CRSS of cube glide has been measured in Pt3A1, with the same L 12 structure, by Wee, Pope and Vitek [27]. Unlike Ni3A1,
82
Ch. 50
D. Caillard and A. Couret
80 0
0
70
x
60
o
50
.IC
1D >
0 v)
40
t...
-0 U
30
4.-
20
"C] ~C].~.
(001)[~10]
( 010 ) [ ~011 10
l
I
200
400
1
.
600
Temperoture
I
800
1 1000
( K )
Fig. 9. Temperature dependence of the CRSS of octahedral and cube glide in Pt3A1 (filled symbols correspond to measurements on the same specimen). From Wee et al. [27].
the CRSS of octahedral glide is not anomalous and exhibits a rapid increase with decreasing temperature. Here, cube glide is the primary glide system at all temperatures for tensile axes far away from (100). Its CRSS decreases rapidly from 20 to 500 K, and then increases from 500 to 1100 K (fig. 9). Conversely, the temperature dependence of the flow stress is always negative in Fe3Ge which has been shown recently to deform by cube glide [28]. Several post-mortem observations in Ni3A1, Ni3(A1, Ti), and Ni3Ga [29-31] above 673 K (in the domain of negative stress-temperature dependence) indicate that dislocation loops in cube planes have straight portions parallel to edge and screw orientations. Screw dislocations are dissociated in two superpartials separated by an APB in the cube plane. Each superpartial is dissociated in one of the two intersecting octahedral planes, which leads to a sessile configuration. Edge segments are dissociated in "super Lomer-Cottrell" or "double Lomer-Cottrell" type non-planar configurations. In-situ observations in Ni3A1 and 7' show that edge and screw segments move steadily or jerkily, which indicates that the glide is controlled either by a Peierls mechanism or by a locking-unlocking mechanism [22, 32]. Most post-mortem observations have been made on alloys which do not exhibit slip traces on cube planes at low temperatures. Only Korner [33] mentioned that many dislocations have glided in the primary cube plane after deformation of a Ni3(A1, Ti) alloy at room temperature.
w
Dislocation cores and yield stress anomalies
83
Fig. 10. Glide of rectilinear screw dislocations in a cube plane of a 7 t nickel based alloy (L12 structure). In-situ experiment at 140 K. From Cl6ment et al. [22]. In-situ observations in 3,t reveal that in the cube plane rectilinear screw dislocations move jerkily in the whole temperature range 120 K - 1000 K, as in beryllium, and edge dislocations are sometimes sessile (fig. 10) [23]. Dislocation movements are proceeded by avalanches in the temperature range 300-673 K (unpublished result). No microstructural observation is available on Pt3A1. Superdislocations gliding in cube planes are however likely to be dissociated into two superpartials separated by an APB, since SISF are restricted to octahedral planes.
5.2. Interpretations Considering that the glide of screw dislocations is controlled by a Peierls-type mechanism in the whole temperature range, the stress versus temperature curve associated with cube glide in -),t (which has been obtained at constant velocity of screw dislocations) and Ni3A1 is truly anomalous for the following reason: For all thermally activated controlling mechanisms, the gradient of the stress versus temperature curve is: 0~*] OT
-H v = bAT'
(1)
where or* = o" - o- i (o- i is the athermal internal stress), H is the activation enthalpy, A is the activation area, v is the velocity of superdislocations (screws in the present case), and b is their Burgers vector.
84
D. Caillard and A. Couret
Ch. 50
When only orders of magnitude are concerned, the following approximation can usually be made: H ~ G ~ c k T , where G is the activation energy, and
c =
In go ln~
(2)
is effectively constant (c ~ 25 usually). This leads to: ~)cr*], ~
ck
OT ,~
bA"
(3)
In the case of a Peierls type frictional force, A is low and decreases as cr increases. This indicates that cr should increase rapidly at low temperatures. The experimental curve is thus unambiguously anomalous. The same conclusion would be derived even in the case of a zero or small negative temperature variation of the flow stress below 600 K. Under such conditions, and since the core structures of superpartials are similar to those of dislocations in beryllium ({ 111 } planes correspond to the basal plane, and { 100} planes to prismatic planes), the stress anomaly has been interpreted in the same way, i.e., it may result from an increase in the energy necessary to recombine superpartials dissociated in octahedral planes, as the temperature is increased. Such an increase may be explained by a short-range diffusional lowering of the stacking fault energy, as suggested by Ahlers [34]. Short-range diffusional processes indeed take place in disordered alloys at similar homologous temperatures, leading to a Portevin-Le Chatelier effect (section 3). Regarding Pt3A1, it should be noted that the same explanation was proposed earlier, in 1984, by Wee, Pope and Vitek [27]. It was not however mentioned again in subsequent publications on the same material [35].
6. Octahedral glide in L12 alloys Glide in octahedral planes in alloys with the L12 structure exhibiting an anomalous yield stress-temperature dependence is clearly one of the best documented situations in the study of plastic properties of materials. Many intermetallic compounds have been found to possess such anomalous behaviour [36]. The most extensive studies, however, have been conducted on nickel-based alloys, especially Ni3A1 and Ni3(A1, X) alloys. Important work has also been done on Ni3Ga and Ni3Si and will be mentioned in what follows. Nickel-based alloys have very similar properties, which are also apparently representative of the properties of L12 alloys with anomalous behaviour. We shall thus focus in the following on the properties of nickel-based alloys. Other L12 alloys are treated in section 6.5.
w
85
Dislocation cores and yield stress anomalies
6.1. Macroscopic results in n i c k e l - b a s e d alloys The main macroscopic properties of Ni3(A1, X) alloys in the domain of stress anomaly where deformation is achieved by octahedral glide are summarized in several review articles (e.g., [37-41 ]). We only mention below the main characteristics which are thought to be of importance to the understanding of the microscopic mechanisms in the domain of yield stress anomaly. (i) There is a strong effect of orientation and sense of the applied stress on the CRSS of octahedral glide. These effects, known as the violation of the Schmid law (fig. 11) and the tension-compression asymmetry, are described in many articles [25, 42-45]. They indicate that the CRSS is related to cross-slip processes, in which the forces acting on the Shockley partials have to be taken into account, according to Escaig [46]. We can only remark that this effect is difficult to explain quantitatively, since its amplitude as a function of the orientation of the applied stress varies significantly from one material to another. (ii) Below the peak temperature the strain rate sensitivity 1 0
S-- TSln~
I
T
has a rather low value in Ni3A1, decreasing slowly as the temperature is increased (fig. 12, from Thornton, Davies and Johnston [48]). The corresponding activation area has been measured from relaxation experiments in Ni3(A1, 1 at.% Ta) and has a high but finite value (Bonneville, Baluc and Martin [49]; Baluc et al. [50]). It is however not yet clear 4o0
~.,
t Orientation10~ I"1 Orientation[ i23] A Orientation[ i 11]
300
A V
200
lOO
0
200
400
600
800
I000
1200
1400
T e m p e r a t u r e (K)
Fig. 11. CRSS of octahedral glide in Ni3(AI, 0.25at.% Hf)as a function of temperature, for different orientations of the compression axis. From Bontemps [47].
D. Caillard and A. Couret
86
25
-
Ch. 50
t
STRAIN RATE SENSITIVITY S I I j Iog~ ~
A /92.5
Ni3AI 20
/ /
INCREASE --~--- DECREASE
/
/
%
v
I0
0
-
0
" ,,, -..
,,s
1. . . . .
200
1
..
.1
400 600 TEMPERATURE, "C
..
I
800
IOOO
Fig. 12. Strain-rate sensitivity of Ni3A1as a function of temperature. From Thornton et al. [48]. whether the activation areas vary monotonically or not with temperature, after being corrected for the effect of strain hardening [50]. Ni3(Si, 11 at.% Ti) is exceptional because S reaches values about 10 times larger than in Ni3A1 (Takasugi et al. [44]). All these results clearly indicate that thermal activation plays an important role in the mechanisms which control deformation in the domain of the stress anomaly. (iii) The variation of the CRSS is largely reversible upon temperature changes. This property was found for the first time by Davies and Stoloff [51] who showed that a sample prestrained at a high temperature (high CRSS value) and subsequently strained at a low temperature had the same yield stress as a virgin sample deformed only at low temperature. In fact, more recent experiments of Yoo and Liu [52], Dimiduk [39], Hemker [53], Couret, Sun and Hirsch [54] show that the process is a little bit more complex: the second flow stress is identified with the CRSS of a virgin sample at low temperature, plus the strain hardening of the high temperature predeformation. This indicates that if the motion of dislocations in the perfect crystal is fully reversible upon temperature changes, the strain hardening due, in particular, to the accumulation of sessile dislocations and debris is not eliminated by cooling the samples. (iv) The CRSS is rather ill defined. It has already been shown by Thornton, Davies and Johnston [48] that the yield stress anomaly appears for very small amounts of plastic deformation, as small as e - 10 -5. The deformation stress increases with increasing strain, in such a way that the amplitude of the stress anomaly also increases in the same proportion. Experiments of Baluc [55] show that this homothetic increase goes on up to a 25% shear strain. Creep experiments of Hemker et al. [56] also show that about 0.5% deformation can be achieved by glide in { 111 } planes at a stress much lower than that needed for the 0.2% strain which is usually taken as the CRSS. This stress has however the same
w
87
Dislocation cores and yield stress anomalies
10000
0
(MPa)
P~
8000
,,,
6000
----
1' = 7xlO-Ss "~
-
1' = 7 x l O ' ~ s "1
4000
//
2000
'~
\o
\~
!
I
-2000 -4000
i
200
,
400
,
I
;
~"
/
o
/
~'~'~'~~
i
600
9
1
800
-
i
i
1000
1200
(K) -
i
1400
Fig. 13. Strain-hardening coefficients at 0.2% deformation in Ni3(A1, 1 at.% Ta) as a function of temperature. From Baluc [55].
anomalous temperature dependence as the CRSS. As pointed out by Hemker et al., all these observations (including those of Thornton et al.) show that { 111 } glide is not controlled by the movement of the edge portions of dislocation loops when it is activated at stresses lower than the 0.2% proof stress. Hemker concludes that there is no critical stress for the motion of screw dislocations which controls plastic deformation. This conclusion may however be modified by recent results of Sp~itig, Bonneville and Martin [57] who show that activation areas versus strain curves exhibit an inflexion point at e ~ 0.2%, typical of a macroscopic elastic limit. (v) The strain hardening coefficient is high, even in single slip, especially for load axes close to (111 ). It also exhibits an anomalous temperature dependence, which indicates that strain hardening is associated with the same cross-slip process that controls the movement of dislocations (fig. 13) [47, 55]. Sp~itig et al. [57] have shown that strain hardening during relaxation tests is even higher than strain hardening during constant strain rate deformation.
6.2. Microscopic observations in nickel-based alloys Extensive microscopic observations have been made, giving indications of many dislocation mechanisms. It is thus especially difficult to determine which are the most important in explaining the origin of the yield stress anomaly. A review of microscopic observations has been made by Veyssi~re [58]. Only those which are considered to be of some importance by most authors are briefly described below. (i) The most striking characteristics, reported after the first TEM observations of Kear and Hornbecker [59] and Thornton, Davies and Johnston [48] is the presence of long and straight screw superdislocations in deformed samples. This indicates that screws are much less mobile than non-screws, as confirmed later by several in-situ deformation experiments [60-62]. Evidence of locking along 60 ~ orientations at low temperatures (below the domain of strong positive temperature dependence of the flow stress) has also been given in [63, 64].
88
D. Caillard and A. Couret
(Ill)
1/2[i01] ~
~
Ch. 50
~
APB
(121) Fig. 14. Scheme of an incomplete Kear-Wilsdorf lock. The apparent APB plane is shown by the dotted line.
(ii) Superdislocations are mostly dissociated into two superpartial dislocations with the same 71 (110) Burgers vector, separated by an APB ribbon. According to several authors (see a discussion in [39]), dissociations into super Shockley dislocations separated by superlattice intrinsic stacking fault (SISF) are not frequently observed in the domain of strong positive temperature dependence of the yield stress. The first weak-beam observations showed that screw superdislocations were mostly in the Kear-Wilsdorf configuration, with the dissociation into the cube cross-slip plane. More recent in situ and post-mortem observations however reveal that straight screw superdislocations are sometimes dissociated in the octahedral glide plane at low temperatures and room temperature [47, 65]. Intermediate situations have also been found, the APB ribbon being partly in the octahedral and partly in the cube cross-slip plane [66-68] (fig. 14). Lastly, further in-situ investigations in the octahedral plane have shown that when rectilinear screws are seen to be dissociated in the octahedral plane the APB is also probably slightly extended in the cube plane [62]. (iii) Many macrokinks along straight screw dislocations can be observed post mortem. They are lying in the primary octahedral plane, and detailed analyses allow them to be classified into three categories [41]: regular kinks on superdislocations, switched over kinks reversing the order of superpartials, and simple kinks affecting one superpartial only. Statistical measurements of macrokink heights indicate a small decrease with increasing temperature and increasing stress. The mean height of macrokinks has been measured by Couret, Sun and Hirsch in Ni3Ga [69], and Bontemps in Ni3(A1, Hf) [47]. The results obtained vary from 18-20 nm at 300 K to 10-12 nm at 673 K. Similar measurements performed by Dimiduk [64] indicate that the height of the macrokinks varies as the inverse of the strength of the alloys (the corresponding data are however not given). The distribution of macrokink heights, measured at different temperatures by Couret et al., exhibits an exponential decrease (fig. 15) similar to that observed in beryllium (section 4, table 1). This has been interpreted in the same way, i.e., in terms of a thermally activated locking process. An abnormally large number of macrokinks with 5-10 nm height is however observed after deformation at room temperature and 200~ It is discussed in point (v) below. (iv) In the higher temperature range of the yield stress anomaly (and in some cases above room temperature [33]), dislocations cross slip and glide in the cube plane as
w
Dislocation cores and yield stress anomalies
80
4
70
log 9 N
89
400"C
60 5O 4O 30 2O
I
lo.
~
0
(a)
5
100 2OO
66o
Z.00
h(~)
h(~)
~oo 2oo
4oo
, ~
log N ,.._,i"log 11/.
,--,
200"C
4 2 0
(b)
\ lOO 26o
"
~o
9
6~
h(A}
Fig. 15. Statistical measurements of macrokink heights in Ni3Ga deformed at 200~ and 400~ Note the abnormally large number of macrokinks of 5-10 nm height after deformation at 200~ From Couret et al. [69, 166]. shown in fig. 16. This result is deduced from both post-mortem [33, 58, 63, 70] and in-situ [71, 72] observations. Macrokinks in octahedral planes act as anchoring points, in such a way that dislocations appear as a series of adjacent segments bowing out in the cube plane. The amount of deviation in the cube plane is higher when the tensile axis is far from (100), and at high temperatures. Post-mortem observations in Ni3Ga [70] and in-situ observations in Ni3A1 [72] however indicate that the reverse crossslip process from cube onto octahedral planes is also operating at high temperatures (fig. 17). (v) In-situ observations in Ni3(A1, 0.25 at.% Hf) show that straight screw dislocations move jerkily. When they are dissociated in the octahedral plane in the lower temperature range of the stress anomaly (cf. point (ii)), they jump over distances often scaling with their dissociation width [62] (fig. 18), in accordance with the observation of an abnormally large number of kinks with height 5-10 nm in Ni3Ga (fig. 15) (cf. point (iii)). In the higher temperature range of the yield stress anomaly, octahedral glide proceeds by bursts [60, 61, 71-73] with indications of double cross-slip between octahedral and cube planes [72], in accordance with the post-mortem observations described above (point (iv)). All dislocations are dissociated in cube planes just after the bursts.
90
D. Caillard and A. Couret
Ch. 50
Fig. 17. Dislocation D curved in a cube plane and cross-slipping (segment S) onto an octahedral plane in Ni3Ga deformed at 400~ (a), (b) and (c) - Observations under different tilting conditions; (d) - Scheme of the dislocation structure. From Mol6nat et al. [70].
w
Dislocation cores and yield stress anomalies
91
Fig. 18. Screw dislocation dissociated in a (111) plane and gliding by jumps over a distance equal to its dissociation width, in Ni3(A1, 0.25at.%Hf). In-situ experiment at room temperature. From Mol6nat and Caillard [62].
6.3. A discussion of microscopic dislocation processes Although microscopic observations reveal rather complex dislocation behaviour in the domain of the yield stress anomaly, post-mortem and in-situ observations provide several consistent results which form a suitable basis for discussing the different models of anomalous mechanical properties.
6.3.1. Sessile configurations The deformation is controlled by the movement of screw superdislocations, which tend to cross slip into the cube plane. The driving force for cross slip is a combination of Yoo's force couplet [75] and a lower APB energy in cube planes. Kear-Wilsdorf locks are formed at high temperatures. They tend to glide in the cube plane especially for loading
92
Ch. 50
D. Caillard and A. Couret
t CSF9 AP,
"~---
>.C~ (111),
_ ..~
_-__
9
P
Ca)
:
(b)
-
unlocking -- - " ~ ~ --
-',K P
(c)
~]
II
I
(~
i
-
-
(d)
",!
II II
,~
(e)
Q
Q
R
R
Q
(0
R
't ,
(g)
(h)
I
Fig. 19. Schematic description of the glide mechanism deduced from in situ observations in Ni3A1 at 300 K. w is the width of the APB ribbon lying in the cube plane. Cycle B corresponds to jumps over a length scaling with the dissociation width. From [62]. axes far away from (100) directions. Cross slip however appears to be less pronounced than expected at lower temperatures, as indicated by the observation of incomplete KearWilsdorf locks, and screw dislocations dissociated in octahedral planes. A model has been set up with the aim of explaining jumps of rectilinear screw dislocations dissociated in octahedral planes over a length equal to their dissociation width, and the formation of macrokinks of height 5-10 nm [62]. It is based on the short range cross slip of the leading dislocation into the cube plane over its whole length (fig. 19). This process, which is different from the local pinning process described by Paidar, Pope and Vitek [76], locks the superdislocation until the trailing superpartial cross slips in turn and follows the same path as the leading one. Two situations may arise according to whether or not the leading superpartial cross slips again when the superdislocation is locked. Jumps over the width of the APB are obtained in the second case, in agreement with the observations. This observation provides evidence of unlocking by cross slip. These results show that screw dislocations can take many non-planar configurations of different energies, especially at low temperatures. Calculations of Saada and Veyssi~re [77] show that incomplete Kear-Wilsdorf locks cannot exist without a strong frictional force in the cube plane. Such a frictional force is obviously present as discussed in section 5. However, in order to explain observations of incomplete Kear-Wilsdorf locks, it is also necessary to assume that the ratio of APB energies in octahedral and cube planes, ")'lll/T10o, is closer to 1 than expected from weak beam measurements of dissociation widths. According to Saada and Veyssi~re, stable configurations with the APB lying either in the octahedral plane or partly in octahedral and cube plane have very high energies for 7111/3'100 much larger than 0.9 in Ni3A1. The stability of incomplete Kear-Wilsdorf locks has been calculated under dynamical conditions as a function of the applied stress and as a function of the frictional force in the cube plane by Paidar, Mol6nat and Caillard [78] and Chou and Hirsch [79]. The results show that incomplete Kear-Wilsdorf locks can be stable under stress. They may however be transformed into complete locks upon unloading.
w
Dislocation cores and yield stress anomalies
93
6.3.2. Locking process and kink motion There is a general agreement on the origin of macrokinks, which result from thermally activated glissile-sessile transitions. This can be deduced from the exponential distribution of macrokink heights, as discussed in [69]. This is also in a good agreement with the observed decrease of macrokink heights with increasing temperature. However, it should be noticed that the height of macrokinks can be written as: h = vFt, where t is the life time in the glissile configuration before locking and is temperature dependent, and VF is the stress-dependent free dislocation velocity in the glissile configuration. Life time t is expected to decrease with increasing temperature, thus decreasing h. However, the CRSS increases at the same time, thus increasing VF. This means that the macrokink height results from an equilibrium between these two opposite contributions. There is little data concerning the mobility of kinks along screw dislocations. Insitu experiments indicate that this velocity is high. Most jumps indeed correspond to nucleations and propagations of macrokinks which are too fast to be recorded. Theoretical estimates of the kink velocity also yield high values [80]. The average kink velocity may however be controlled by interactions with obstacles which are not seen in thin foils. 6.3.3. Unlocking processes No definitive conclusions about the existence and the nature of unlocking processes can be deduced from the experimental observations. Unlocking may be impossible under normal deforming conditions. It could also be initiated either at the non-screw glissile portions of dislocation loops like macrokinks, or at screw sessile portions (a general discussion about these two mechanisms is given in section 9). Experimental results however show that initiation of unlocking by cross slip at sessile screw portions does exist at least in two situations: when dislocations dissociated in cube planes are seen to cross-slip back into the octahedral plane (section 6.2 point (iv)), and when short-range double cross slip is shown to occur, in the case of jumps over the dissociation width (section 6.3 point i).
6.4. Discussion of models of yield stress anomaly in nickel-based alloys The different models have been simplified and classified according to their main hypotheses. Some of them are compared on table 2.
6.4.1. Accumulation of locked dislocations This model has been developed mainly by Greenberg and co-workers [81]. It is very similar to the model developed earlier by R6gnier and Dupouy [11] for beryllium. A general expression for the flow stress is: O" -- k v / - N -[- o-f,
(4)
where N is the density of dislocations, and O'f is a frictional force acting on mobile dislocations.
Table 2 Principles of several models of the stress anomaly in L I Z alloys. Thick lines correspond to locked dislocations. Note that I, h, and G,I may have different meanings (see text).
I
TakcuchiKurmo~oI821
I
G
2 ,
kT
"..--wcp-'
I
I
SOdaniVitek [851
I
dc Bussac et al. [89]
P
v=hlP,l v = vocxp
-5 kT
3
v = voexp -
GUI-TGI kT
? 0-
VP
o-cxp-
a=@
G
,14 G xcxp-' kT
,
I.
0
P VI
0
w
Dislocation cores and yield stress anomalies
95
Three cases have been defined for the behaviour of screw dislocations: - When the glissile-sessile transition by the formation of Kear-Wilsdorf locks is thermally activated, but the temperature is too low to activate the reverse sessile-glissile transition, N increases rapidly with increasing plastic strain. The storage rate of sessile dislocations also increases with increasing temperature, leading to an increase of cr at constant strain, ~cr/~T > O. At higher temperatures, the glissile-sessile transition becomes athermal and the reverse sessile-glissile transitions become thermally activated. Then, O~r/OT < 0. As shown in [16, 17, 20], this situation corresponds to a Peierls mechanism. Intermediate situations may be found when both transitions are thermally activated. This situation which has been developed in section 4 corresponds to the lockingunlocking mechanism for which Ocr/OT < 0 (higher temperature range in the model of R6gnier). However, according to Greenberg, O~r/OT > 0 may also result from a very low unlocking rate leading to the first situation. -
-
This model agrees with microscopic observations only qualitatively. Variations in the dislocation density N as a function of deformation and temperature are indeed too low to explain both high values of strain-hardening and the positive temperature dependence of the yield stress. This model may explain the high strain-hardening rate and the tension-compression asymmetry. Thermal activation may result from the term O'f although there is no experimental evidence of such a frictional force. The model however is inconsistent with the reversibility of the flow stress after a temperature change.
6.4.2. Local pinning models A local pinning model was first proposed by Takeuchi and Kuramoto (TK) [82]. The movement of dislocations results from a balance between two processes acting in opposite directions. The first is the thermally activated locking by cross slip, which starts at some points, and tends to extend along dislocations. The second is the break-away from this local cross slip, through the rapid glide of dislocation segments which are not yet locked (table 2). In other words, it is a competition between the velocity of the extension of the locked configuration along the dislocation line, and the velocity of parts of the dislocations which are not yet locked. In this model, dislocations glide at the minimum velocity which prevents the extension of locking over their whole length. This velocity is proportional to the probability of locking for a length l, P~ o( l exp[-Gl/(kT)], where GI denotes the enthalpy for locking. Namely, v o( hlexp[-G1/(kT)]. Since l and h are shown to be inversely proportional to stress, and since Gl depends on the shear stress in the cube plane, the dislocation velocity is v o( (1/crZ)exp[-Gl(cr)/(kT)]. On the other hand, since v is a free glide dislocation velocity (v = VF), it determines the stress ~r through a relation of the type VF = K o typical of viscous glide. This leads to cr3 c~ exp[-Gl(O)/(kT)] and ~r o( exp
3kT
"
D. Caillard and A. Couret
96
Ch. 50
When there is no applied stress in the cube plane (for (100) tensile axes), Gl does not depend on tr, and v oc (1/cr2)exp[-Gl/(kT)], whence cr oc exp[-G/(3kT)]. This model has been improved by Paidar, Pope and Vitek (PPV) [76] in order to account for the orientation effects and the tension-compression asymmetry. These two models explain successfully the reversibility of the flow stress upon temperature changes. Moreover, since mobile dislocations slowing down just below their minimum stable velocity are immediately trapped into Peierls valleys, a high strain-hardening rate is expected. These models are rather difficult to prove by microscopic observations because the density of mobile dislocations would be very low, of the order of 0.04 cm -2 according to Hirsch [83, 84]. Post-mortem and in-situ observations may thus concern only immobile dislocations and dislocations with unimportant movements. The main objection against these models is however their stress-strain dependence. If the dislocation velocity is considered to be controlled by free flight glide, i.e., v = vF cx Kcr, it will lead to ~ = pmbKcr. Unless Pm varies inversely proportional to the stress, low values of the strain rate dependence of the flow stress (S') cannot be explained according to Hirsch [84]. It can however be considered that (at least when there is no stress in the cube plane) cr is determined only by the temperature in such a way that the dislocation velocity is a constant at a given temperature. Strain-rate changes are thus necessarily accommodated by changes in the mobile dislocation density Pm, and the strain-rate sensitivity must be zero. A new model has been formulated by Sodani and Vitek [85] to account for the low strain-rate sensitivity. According to this new model, the probability of locking no longer determines the velocity of dislocations, but the distance between pinning points: 1 cx exp[G1/(kT)]. This can be compared with the TK model, which yields: 1 ~x l/or cx exp[Gl/(3kT)]. The two models are thus different. Another illustration of this difference is that v o ( 1 / o "2exp
( c,) -~
,
within the TK model, and
v = vo exp
_ Gui
where v0 is a constant and Gul denotes the activation energy of unpinning, according to Sodani and Vitek. v is lower than in the TK and PPV models since it includes waiting times to recombine pinning points. Since the latter relation implies: Gul = kT In vo v
w
Dislocation cores and yield stress anomalies
97
and Gul - G u l o - crbAul, where Aul denotes the activation area for unlocking, Aul = lb cx exp[Gl/(kT)], the stress is"
cr oc co(Gulo - kTln V~~) eXp ( - -~T ).
(6)
The increase in stress is due to a decrease in the activation area of the unlocking process with increasing temperature. This model gives higher values of the density of mobile dislocations (Pm - 1.6 • 106 cm-2). However, the distance between obstacles at 1000 K seems too large to be realistic [86]. It should also be noted that according to Hirsch [84] there is no physical basis for assuming l oc exp[Gl/(kT)], since Gl controls the rate at which cross-slip occurs, not the steady state concentration. Moreover, Gul is taken as 3 eV which is quite large. Lastly, the model assumes that Gul < Gl, which is inconsistent with in-situ observations indicating that locking is easier than unlocking, and with estimates based on a lower energy of locked dislocation segments. The model has been then further modified by Khantha, Cserti and Vitek [87, 88] in order to explain the observed discontinuities in activation area versus temperature curves [49, 50]. The modification relies on the assumption that dislocations move after release of single pinning points at low temperatures, while double or multiple release is necessary at higher temperatures. It is however not certain that corrected values of activation areas are really discontinuous (cf. section 6.1 point (ii)). Another model has been proposed by de Bussac, Webb and Antolovich [89]. In contrast with the above models, the density of pinning points in this model is assumed to depend only on an activation energy of formation G1 and an activation energy of vanishing Gul. With this assumption, it is possible to justify that an equilibrium density can be reached. Its value 1 is [1 + e x p [ ( G l - Gul)/(kT)]] -1, which is to be compared with 1-1 - exp[-G1/(kT)] in the model of Sodani and Vitek. The flow stress is then" ere( co[1 + e x p Gl - Gul] -1 k~ '
(7)
where co is a drag force per cross-slipped segment. It is concluded that cr increases with increasing temperature, provided Gl > Gul. The term co is however not discussed. It is clear that it should be related to Gul, and thus should decrease as the temperature is increased. In fact, the origin of the stress anomaly is the same as in the model of Sodani and Vitek, since it results from a decrease in the driving force per pinning point, i.e., a decrease in the activation area, as the temperature is increased. The main objections are the inequality Gul < GI (as in the previous models) and the stability of pinning points which must tend to expand or to shrink easily. l A modified expression is proposed in [90]" [1 + (1/n) exp(AHn/ (kT))]--1, where AHn is the "enthalpy change associated with the formation of a cross-slip segment of length nb". The main conclusions are the same.
98
D. Caillard and A. Couret
Ch. 50
Fig. 20. Simulation of the expansion of a dislocation loop in Ni3AI, for a temperature of 300~ and a stress of 240 MPa. The pinning points are represented by blackened points. From Mills and Chrzan [91]. Computer simulations of Mills and Chrzan [91, 92] allow the local pinning model to be tested under more realistic hypotheses, namely using a statistical distribution of pinning points and a free flight velocity of dislocations including line tension effects. Since unpinning is athermal, these simulations reflect dislocation behaviour in TK and PPV models (dynamic break away). In particular, the same very low density of mobile dislocations is obtained. It is interesting to note that owing to the line tension effects, more or less rectilinear screw dislocations connected by macrokinks are obtained, as observed in TEM (fig. 20). Such computations should be improved by taking into account the lateral extension of cross slipped segments (Hirsch [79]). The same stress-strain rate dependence as found in the TK and PPV models is obtained a priori. The authors propose that strain-rate changes are almost completely accommodated by changes of mobile dislocation density, whence a low stress-strain rate dependence.
6.4.3. Sources at macrokinks The basic idea of this type of model was proposed first by Mills, Baluc and Karnthaler [93]. Since the critical stress for multiplying dislocations from Orowan-type sources is inversely proportional to the separation between anchoring points, a simple explanation for the yield stress anomaly is obtained if two hypotheses are verified: - The junction between macrokinks and screw segments (points A and B in Table 2) is either pinned or difficult to move. - The height of macrokinks decreases as the temperature is increased. The model proposed by Hirsch [83] can be summarized as follows, with some simplifications. The height of macrokinks, h, is related to the probability of locking per unit time, Pl, by the relation: h - vF/PI, where VF is the free flight velocity of glissile dislocations, proportional to the stress or. Since the probability of locking is P~ cx l exp[-G~/(kT)], where 1 is the distance between macrokinks, and is assumed to vary as the reverse of the stress or, the kink height is: h ~ cr2 exp[-Gl/(kT)]. Since macrokinks are very mobile along screw dislocations [79], strong pinning points must be formed by complex cross-slip processes, the details of which are not reproduced here. In this model, the stress necessary to emit dislocation loops at macrokinks is cr -O'i -~-O'*, where O"i is close to the Orowan stress # b / h (internal stress) and or* is the effective stress necessary to cut pinning points with the help of thermal activation.
w
Dislocation cores and yield stress anomalies
99
The average dislocation velocity is thus: v o~ hPul. It can be written: v - vo exp
-
Gul - G l ) kT
'
where vo is a constant, Gul = Gulo -o*bAul, and Gul > Gl. This yields: O"-
O"i
nt- O'*
#b
- -h-- +
Gu2o - k T In ~V - G! haul
Since Aul o( hb and h o( a 2 exp[Gl/(kT)], the stress is:
1[
GI ---Gl]exp(-~--~),
cr o( ~-~ #b 3 + Gulo - k T In VOv whence o" o(
~ b 3 -+- Gulo - k T In --v~ _ G1 v
]1/3
exp ( -
Gi
3---~)
(8)
This model has many common properties with the model of Sodani et al. The temperature variation of cr is however weaker, owing to the factor 3 in the exponential term (as in the TK model). In addition, the main contribution to stress is the Orowan stress (term #b 3) which is not included in the model of Sodani et al. The activation area decreases with increasing temperature. Assuming that two anchoring mechanisms operate at low and high temperatures, namely a short range cross-slip at low temperatures, and the formation of Kear-Wilsdorf locks at high temperatures, Hirsch obtains two domains in the activation area versus temperature curve, in agreement with the measurements of Bonneville [49, 50]. This model is in good agreement with experimental observations of macrokinks, incomplete Kear-Wilsdorf locks, rapid glissile-sessile transitions. The mechanism leading to the formation of pinning points has however to be confirmed experimentally. In addition, the average kink height is far from being inversely proportional to the stress: it is found to decrease by only 40% when the yield stress increases by a factor of 3 between 300 K and 673 K in Ni3A1 and Ni3Ga (section 6.2 point iii). 6.4.4. Models based on the activation o f cube glide
The early model of Thornton, Davies and Johnston [48] is based on the strong hardening due to the interaction between dislocations gliding in the octahedral plane and another dislocation family gliding in the cube plane, which is more and more easily activated as the temperature is increased. Microstructural observations do not fit with this hypothesis, since the density of dislocations in cube planes is far from being high enough to explain the stress increase, at least with a (100) deformation axis. Moreover, the reversibility of the CRSS upon temperature changes is at variance with this model, since a high density of dislocations in the cube plane cannot be eliminated by cooling the sample.
100
D. Caillard and A. Couret
Ch. 50
Saada and Veyssi~re [94] have proposed a model of yield stress anomaly based on the more and more pronounced curvature of screw dislocations in cube planes, which is observed as the temperature is increased. The movement of screws is assumed to be controlled by the lateral motion of macrokinks which are always produced at a sufficient rate. This process is quite easy at low temperatures, since screw superdislocations are straight. It may however be more difficult at higher temperatures, since the bowing of screw superdislocations in cube planes should inhibit the lateral movement of superkinks which allow screw superdislocations to glide in octahedral planes. The total frictional force on kinks has been computed by Saada and Veyssi~re and is given by: O'f -- O'fo -]
O'fc
(q0c O" -- O'fc)
o*[ 1- (~,~a--O'f~)~],/2,
(9)
where O'fo and O'fc are the frictional forces in the octahedral and cube planes, respectively, cr is the applied stress, CPc is the Schmid factor on the cube plane, O'c and o-* are constants depending on the height and the separation of kinks. Provided macrokinks are mobile, it is, however, likely that the segments with larger bowing will grow at the expense of the shorter ones, leading to the coalescence of macrokinks, a favoured process since it decreases the total energy by increasing the total work done by the resolved shear stress in the cube plane. Dipole formation may also inhibit this process [70, 71]. The violation of the Schmid law may be explained by this model. However, according to Saada and Veyssi~re, the stress anomaly does not result straightforwardly from this mechanism, unless the height of kinks decreases, and their distance increases, as the temperature is increased. A small decrease of the kink height has been measured. However, no data is available on their distance.
6.4.5. Double cross-slip mechanisms Here, dislocation movements are assumed to be controlled by a cross-slip process leading to the nucleation and the rapid extension of a glissile loop on a rectilinear screw dislocation. This dislocation loop is rapidly locked by another cross-slip event, which leads to the formation of a macrokink pair of height h. Macrokinks are assumed to glide further rapidly along the screw dislocation and move it over the distance h. In this model, dislocations move by series of thermally activated locking and unlocking processes by double cross slip, similar to those described in the case of prismatic glide in beryllium, with Gul > Gl (section 4). Such a suggestion was also made by Sun and Hazzledine [95] for the 3" phase of superalloys. As in the case of beryllium, the dislocation velocity can be set equal to: v = hPul, where Pul = Pulo exp[-Gul/(kT)] is the probability of unlocking per unit time and unit length for each screw dislocation, and h -1 c~ ~ c~ exp[-Gl/(kT)], which yields v -- v0 e x p [ - ( G u l - Gl)/(kT)], where v0 is a constant. Since h does not vary very much with stress and temperature, the velocity of dislocations is controlled by the unlocking process. Nabarro has proposed a model of stress anomaly based on this double cross-slip process, however with some modifications [96]. The main assumption is that after the
w
101
Dislocation cores and yield stress anomalies
8.
0.6 l o~
l
i
"t"p
7a
04
90.2
0 2
/ MPo
lO0
i/lo
0 0" 9u
~
! 10
b.
! 20
wtb
Fig. 21. Stress (or) necessary to unlock incomplete Kear-Wilsdorf locks, for various values of the extent of the APB in the cube plane, w, and frictional stress in the cube plane, trfc. The dotted line refers to the situations where the driving stress trfc on the trailing superpartial is equal to the driving stress crfc t on the leading superpartial. From Mol6nat et al. [97]. jump the dislocation reaches an equilibrium position illustrated in table 2, which means that the macrokinks thus formed are not mobile 9 Accordingly, the area swept by each unlocking process is A oc (1/F!3/2), instead of A ~ h oc (1/Pl) in the above model. This yields a dislocation velocity 9 v = v 0 e x p [ - ( G u l - 3 G l ) / ( k T ) ] (instead of v v0 e x p [ - ( G u l - G l ) / ( k T ) ] in the above model). A yield stress anomaly is thus obtained if 3Gl > 2Gul. Since the kinetics of dislocation movements indicate that G1 < Gul, both conditions are satisfied if ~Gl > Gul > Gl. This model is however incompatible with observations of highly mobile macrokinks. Since the model developed for beryllium explains the experimental observations most satisfactorily, the yield stress anomaly can only arise from an increased difficulty of crossslipping from sessile to glissile configurations (as already concluded in sections 4 and 5). Sun and Hazzledine proposed that the formation of a pair of macrokinks is hindered by an increased tendency of superpartials to dissociate onto the cross-slip octahedral plane, as the temperature is increased [95]. Another possibility has been suggested by Mol6nat et al. [97]. Following the calculations of Paidar et al. [78], the stress cr necessary to unlock incomplete Kear-Wilsdorf locks (fig. 21 (b)) has been calculated as a function of the extension w of the APB ribbon in the cube plane, and as a function of the stress Crfc necessary to move the trailing
D. Caillard and A. Couret
102
Ch. 50
dislocation in the cube plane (the frictional stress in the cube plane) (fig. 21(a)). ~r is shown to increase with increasing w and Crfc. The dotted line is the critical stress crc for which the total driving force for glide in the cube cross-slip plane is the same for leading and trailing superpartials (tree - Cr~c). According to [78, 79, 97], this situation results in jumps over a distance equal to the dissociation width in the octahedral plane (see fig. 19). For low values of w, the critical stress crc has been calculated by Paidar et al. [78] and Hirsch [79, 83], and is given by Crc- o71( -7-- 1
70 1 +(2/A))31/2 "71
)
where A is the elastic anisotropy ratio, ~'0 is the APB energy in the cube plane and 71 is the APB energy in the octahedral plane. For the maximum value of w, the critical stress is
,,( 1 ,o3'1 31/2,)
Oc---b-
"
It corresponds to the stress level necessary to destroy a Kear-Wilsdorf lock under slightly different hypotheses, according to Saada and Veyssi~re [77]. The dynamic behaviour of incomplete Kear-Wilsdorf locks is considered next. Chou and Hirsch [79] have shown that incomplete Kear-Wilsdorf locks evolve rapidly towards complete Kear-Wilsdorf locks at stresses lower than crc. Stable movements of incomplete Kear-Wilsdorf locks by locking and unlocking processes are thus possible only for O ' ~ O " c.
Recently, Mol6nat et al. [97] have postulated that octahedral glide may be controlled by the repeated locking and unlocking of incomplete Kear-Wilsdorf locks, with values of w increasing with temperature. This model is supported by several microscopic observations (sections 6.2 and 6.3). At low temperatures, the yield stress should be of the order of or slightly larger than Crc(w-b)-b-(1
701+(2/A)) '71
31/2
"
At high temperatures, the yield stress should be of the order of: Crc(W = d0) = bl- (1
"Yl 3 1 / 2
'
where do is the dissociation width in the cube plane. A general expression is thus:
[
3'1 cr ~ -b- 1
( 2 w,T> 0)1 .
1 3'0 1 + - 31/2 "71 A
do
(10)
w
Dislocation cores and yield stress anomalies
103
Three parameters are important: "71 (or "70), "70/"71, and the CSF energy which determines the temperature variations of w (larger values of the CSF energy makes cross slip from {111} onto {100} easier, whence higher values of w at a given temperature). Orientation effects and tension-compression asymmetries could also be explained by variations of w. The stress can be shown to be very sensitive to the ratio "70/"71, with "70/"71 ~> 1, as already pointed out by Saada and Veyssi~re [76]. The yield stress can also be expressed as cr = o i a t- o * , where O'i is an internal stress, and or* is an effective stress associated with the frictional stress acting on superpartials in the cube plane (fig. 21b). For "70 = 150 mJ/m 2 and "70/"71 -- 0.9, estimations give a ~ 100 MPa for w = b, and cr ,-~ 250 MPa for w = do. These values fit with yield stresses at 300 K and 673 K in Ni3Ga, where clear indications of double cross slip with corresponding low and maximum values of w have been obtained. Further developments are however needed in order to test other properties, e.g., the stress-strain rate dependence and the reversibility of the flow stress upon temperature changes. Three-dimensional configurations including the formation of pairs of macrokinks also need to be considered.
6.5. Other L12 alloys Many other intermetallic alloys with the L 12 structure exhibit the yield stress anomaly to different degrees (Wee et al. [36]). Nevertheless only those on which substantial information is available on mechanical properties, glide systems and/or dislocation structures will be discussed here. 6.5.1. Co3Ti
Mechanical properties of Co3Ti have several common properties with those of nickelbased alloys (Takasugi et al. [98]). Deformation is produced by octahedral slip, strong orientation effects can be observed in the domain of positive temperature dependence of the yield stress (fig. 22), and the stress-strain rate sensitivity is low. Cube slip is however more difficult to activate than in nickel-based alloys, since it is observed only for a (111 ) tensile axis, above the peak temperature. TEM observations reveal rectilinear screw superdislocations dissociated into two superpartials separated by an APB ribbon probably in the cube plane, after deformation in the anomalous domain (873 K) (Liu et al. [99]). The origin of the stress anomaly is thus likely to be the same as in nickel-based alloys. In the low-temperature regime, the normal flow stress-temperature dependence has been attributed to a dissociation of superdislocations into two super Shockley dislocations, separated by a ribbon of SISF [99]. 6.5.2. Cu3Au
The mechanical properties of Cu3Au exhibit important differences with respect to those of nickel-based alloys (Kuramoto and Pope [100]). The yield stress indeed increases steeply up to the order-disorder transition temperature and then sharply decreases (fig. 23). In addition, orientation effects are much smaller than in nickel-based alloys in the whole temperature range investigated.
104
Ch. 50
D. Caillard and A. Couret I
250 -
w
I
r-"
22 at'/', Ti 3.Sx10-4s-1
1
,
', 9
d'~,a__ 1s 200 o A v--
150 "
t"
o
o5 v5 to
100 -
50-
ff"E closed
0
0
2; 0
' 400
6;o
' BOO
Temperature I K
marks 10011[il O] ' 1000
' 1200
Fig. 22. Temperature and orientation dependence of the CRSS for octahedral slip of Co3Ti. From Takasugi et al. [98].
TEM observations indicate that a high density of rectilinear screw dislocations are present in deformed samples (Kear and Wilsdorf [101], Sastry and Raswamany [102]). The origin of the stress anomaly seems to be strongly correlated with the orderdisorder transition, and it may thus be different from that in nickel-based alloys and Co3Ti. According to Pope [103], it could result from an interaction between dislocations and local regions of disorder. This hypothesis does not however fit with microscopic observations, since wavy dislocations with no specific orientation would be observed in this case. Yamaguchi and Umakoshi [40] suggested that Flinn's model may also apply. This model is based on the change of the APB plane from {111} onto {100} by diffusion (or climb dissociation). This hypothesis is also in conflict with microscopic observations, since frictional forces would act on all dislocation characters, not just screw. The same remark also holds true for Brown's model [104]. The principle of Brown's mechanism is schematized on fig. 24. The APB is assumed to lie in the dislocation glide plane. The leading dislocation creates a fresh APB of energy ")/1 which relaxes rapidly by diffusion and lowers its energy (~,[). Then, the trailing dislocation does not restores the perfect crystal, but creates a fault of energy ~,{', which subsequently vanishes also by a diffusional process. The stress required for moving the dislocation is then: (7" -- ("Yl -t- "Yl' -- 7 [ ) / ( 2 b ) .
w
105
Dislocation cores and yield stress anomalies
Cu3Au o
/:
v
u') bJ
5
l-V'}
rr L.
f
w
o "'
I
,
I
IAA
3
0 w nr _j
9 e,/
2.
O---A e--- B A---C X--- D
u
l
.t
o0
.
100
t
I
200
300
.I
400
. z
-, L J. ,
500
600
'700
800
TEMPERATURE (K) Fig. 23. Temperature dependence of the CRSS for octahedral slip of Cu3Au. From Kuramoto and Pope [100].
I
1. w z
7 "_I 7? t~
I
. . . . . . . . . .
%1 ,
I
I
,
==>
Fig. 24. Schematic description of the Brown mechanism.
This model may explain the yield stress anomaly close to the order-disorder transition temperature Tc. Diffusional effects could indeed strongly modify the structure and the energy of APB's above 0.75Tc in L12 alloys, according to Sanchez, Eng, Wu, and Tien [ 105]. Models based on the formation of Kear-Wilsdorf locks would explain microscopic observations better. It is not clear, however, why orientation effects should be so small if cross slip is controlling the strain-rate. Combining cross-slip and Brown's mechanisms may explain both the observation of rectilinear screw dislocations and the relation between the yield stress anomaly and the order-disorder transition. The APB of immobile Kear-Wilsdorf locks may indeed relax substantially even at intermediate temperatures. As pointed out by Morris [106], during the cross-slip process of Kear-Wilsdorf dislocations onto octahedral planes, a fresh APB is thus created in the octahedral plane whereas a relaxed APB is erased in the cube plane,
106
Ch. 50
D. Caillard and A. Couret
300 R ~260 ,~ CH
c5
-2 k
S
(? I
220
t
.O
/
,o
t:)
100
1 -0.5 -0
180 140
1.5
I
i / d
A
"13 05 m
--1
t~ (0.2%)
--1.5
. . 2 6 d ' :46d' i3bd'i36d "l'0bb 2
Temperature (~
Fig. 25. Temperature dependence of the yield stress tr and the strain-rate sensitivity S in a AI62,5Ti24,5Cu13 alloy. From Potez et al. [108].
resulting in a more difficult unlocking process of screw dislocations as the temperature is increased. The yield stress anomaly cannot however be discussed in more detail without further microscopic observations.
6.5.3. Al3Ti Titanium trialuminides may be changed into the Lie structure by adding some ternary elements. The corresponding yield stress versus temperature curves exhibit in several cases a weak anomaly between 300~ and 500-700~ (Morris, Gunther and Lerf [ 107], Potez, Lapasset and Kubin [108]) (fig. 25). In both studies, strain instabilities such as the Portevin-Le Chatelier effect have been clearly identified in the domain of the flow stress anomaly (fig. 26). In one study, these instabilities are shown to be the cause for a dip in the strain rate sensitivity versus temperature curve, which becomes negative (fig. 25). Activation areas are small at room temperature (,-~ 25b a) and large at 600~ (~ 300b 2) [1091. Microscopic observations show that deformation is due to (110) superdislocations gliding in octahedral planes. These dislocations are seen to be dissociated (within the resolution of weak beam) only above 500~ and they do not exhibit any strong directionality. Intensive cross-slip from octahedral onto cube planes is observed at 700~ A Peierls mechanism may explain the small values of activation areas at 20~ although no strong directionality is seen at this temperature [ 109]. The yield stress anomaly was first explained on the basis of the formation of KearWilsdorf locks [ 109]. The more recent studies however indicate that the stress anomaly may be coupled to the Portevin-Le Chatelier effect, owing to dynamic strain ageing processes as in dilute alloys (section 3) [107, 108]. Static age-hardening due to the precipitation of the A12Ti phase may also contribute to the increase of yield stress with increasing temperature [107]. Precipitates of A12Ti phase have been observed in dislocation cores in [ 110].
w
Dislocation cores and yield stress anomalies
107
I ioo M~d 300 oC
jr
40ON::
Strain Fig. 26. Serrations on stress-strain curves of a AI-Ti-Fe alloy. From Morris et al. [ 107].
6.6. Discussion on cross slip in L12 alloys
Except for A13Ti, and to a less extent Cu3Au, most experimental results show that the yield stress anomalies are related to an increased propensity of superdislocations to cross slip from octahedral onto cube planes, as the temperature is increased. As already discussed in many review articles, cross slip can be primarily induced by a lower APB energy on cube planes. Lower APB energies in cube planes are related to a lower stability of the L12 phase with respect to D022 or D019 phases (Wee et al. [36]). The lower the stability of the L12 phase, the larger is the tendency for a yield stress anomaly (fig. 27). A good example of this correlation is the progressive vanishing of the stress anomaly of (Ni, Fe)3Ge alloys, as the amount of nickel decreases, and as the phase stability of the L12 structure with respect to the D022 structure simultaneously increases (Suzuki, Oya, Wee [ 111 ]) (fig. 28). The driving force due to the difference in APB energies is however (at least in most alloys) not high enough to promote cross slip, whence the important role of the torque introduced by Yoo [75]. This torque depends on the elastic anisotropy factor A, which varies from 3.33 in Ni3A1, to 1.30 in Pt3A1 and 0.90 in A13Sc which do not exhibit stress anomalies (Yoo [112]). Saada and Veyssi~re [41 ] pointed out that the very small splitting width of superpartials is inconsistent with cross-slip mechanisms which remain active over several hundreds of degrees. As mentioned in section 4, cube glide, which is also controlled by the splitting of superpartials, is also thermally activated over a large temperature range (900~ in -y', 700~ in Ni3(A1, Ta) and Ni3 (A1, Nb) - cf. fig. 8). An explanation for this inconsistency may lie in the temperature dependence of the core of superpartials, as discussed in section 4. However, all the models presented in this section either suffer from serious discrepancies from experimental results, or are not developed systematically. Further developments are thus needed, especially on the kinetics of macrokinks along screw superdislocations.
108
Ch. 50
D. CaiUard and A. Couret
A1 S1 Gc] fie In Sn Sb Pb
Nls Pt,
Fig. 27. Part of the periodic table, showing the increasing trend of the magnitude of the anomalous strength behaviour when changing the combination of elements in A3B alloys. From Wee et al. [36].
(a)
I
1200
(b)
I
--"-v ---~ . . . . . . .~.
,
v
I
1000
lOOO
"-,~
-9
Q.
~
""
,'~
Fo3o
I
\
uo
I
o
~I i
200
400
;! \
"
I / / / / o ' , Fe25,'~ /v /
Ni3Ge _ ~ /' ] /FelOat'l,
\ --
600
Temperature [ K )
800
600
".
I
A
o-
400
"
300 0
/~.~
E
d
, "~
i
,. 800
..o~1_:...--i--o-~,.~_
9
.
1000
200 0
200
400
600
800
1000
Temperature (K)
Fig. 28. Yield stress versus temperature curves for various (Ni, Fe)3Ge L12 alloys. From Suzuki et al. [111].
7. TiAl (Llo structure) Crystals with the L10 structure can be regarded as f.c.c, crystals with alternating layers of each element parallel to one face of the cube. In this crystal structure there are two types of perfect dislocations, with Burgers vectors 89 (110] and (011] respectively (following 1 the notation introduced by Hug et al. [18]). g( 110] ordinary dislocations are comparable to g1 (110) dislocations in the f.c.c, structure, while (011] superdislocations are similar to (011 ) superdislocations in the L 12 structure, although the detailed core structures are different.
w
109
Dislocation cores and yield stress anomalies
7.1.
Mechanical
properties
Many deformation experiments have been made on single phase 3' (L10) polycrystals and two-phase "7 + c~2 polycrystals and single crystals. The most significant results on deformation mechanisms involving the yield stress anomaly have however been obtained on single crystals of several A1 rich single-phase TiA1 alloys by Kawabata, Kanai and Izumi [113] and more recently by Stuke, Dimiduk and Hazzledine [114]. The following conclusions have been given by Kawabata et al.: - Yield stress versus temperature curves exhibit pronounced anomalies and similar stress levels when either ordinary 89 (110] or super (011] dislocations have the highest Schmid factor (fig. 29(a)). - In both cases the flow stress-strain rate sensitivity is low in the anomalous domain and goes through a minimum close to zero in the middle of this domain (fig. 29(b)). - The strain hardening coefficient increases with increasing temperature when ordinary dislocations are thought to be activated. It decreases normally when superdislocations are thought to be activated. - Twinning is observed in a few orientations in connection with either ordinary or superdislocations. In addition, experiments of Stuke et al. revealed that when ordinary dislocations are likely to be activated, the flow stress is not reversible when samples are predeformed at a high temperature and restrained at room temperature (fig. 30). This behaviour is (a)
(b) 400 %
o
300
-
...
m m
~D
200
~zll
'
~
"
~~o
x
--
)' "~
"~
-
.~:o
100
o
Temperoture
1000 (K)
1
1500
/
o~o
50
E "6
500
'd
150
~9
IO0
|
i
1 / e l n o" \
~I
0 o
9
500 Temperoture
1000
1500
(K)
Fig. 29. Yield stress (a) and strain-rate sensitivity (b) versus temperature curves in single phase TiA1. Solid and open marks refer to orientations with high Schmid factors on superdislocations and on ordinary dislocations, respectively. From Kawabata et al. [113].
110
Ch. 50
D. Caillard and A. Couret
500
400 296K ~"
=_..
300 20o
lOO
0.5
1.0
Total
1.5
Strain
2.0
2.5
(%)
Fig. 30. Schematic description of the stress-strain curve for a single room-temperature test and a twotemperature compression test in TiA1. From Stucke et al. [114]. at variance from that observed in Ni3A1 (section 6.1 point (iii)). The strain hardening coefficient (which is found to increase with increasing temperature, in agreement with Kawabata's results) does not depend on the prestraining conditions, contrary to Ni3A1.
7.2. Microscopic observations Microscopic observations reveal substantial differences in the dislocation microstructure of different deformed single-phase and two-phase alloys [115-124]. Ordinary dislocations, super-dislocations and twinning are however always observed with the following common properties: (i) Ordinary dislocations are observed most frequently after deformation at all temperatures. They are rectilinear along the screw orientation in the anomalous domain, and contain some jogs which act as anchoring points (fig. 31). In-situ straining experiments [125, 126] indicate that rectilinear screws move more slowly than edges, by a series of jumps over short distances often clearly related to nucleations and propagations of macrokink pairs (fig. 32). Jogs are seen to originate from double cross slip. They significantly slow down the screw dislocations. Individual dislocation movements are observed at room temperature, whereas deformation is produced by bursts of dislocation movements at 200~ and 400~ In the latter case, dislocations are immobilized in the screw orientation and cease to move upon subsequent deformation, in which new dislocations bursts are activated. Intensive double cross slip has been observed between octahedral and cube planes at all temperatures [125, 126]. (ii) Superdislocations are observed less frequently than ordinary dislocations. They can be absent either at low or high temperatures, depending on the alloys investigated.
w
Dislocation cores and yield stress anomalies
111
Fig. 32. Glide of ordinary dislocations in TiAI. Note the jerky movement of rectilinear screw segments. In situ experiment at 20~ From Farenc and Couret [125].
112
Ch. 50
D. Caillard and A. Couret
Direction of projection:
[100l [100]
~10] ~ 0 ]
[010] ~[010]
f Fig. 33. Scheme of screw superdislocations viewed along several directions in TiAI deformed at high temperature. From Hug [18]. They are very rectilinear along the screw orientation. Rectilinear screws have complex dissociations, which are shown to lie in the octahedral glide plane at low temperatures and in the cube cross-slip plane at high temperatures [18, 121, 122] (fig. 33). No cube glide seems to be possible for superdislocations. In situ observations have shown that rectilinear screw superdislocations move jerkily with an average velocity much smaller than that of edge segments. As in the case of macroscopic deformation tests, these dislocations are dissociated in the { 111 } glide plane at room temperature, whereas the APB straddles {111} and {100} planes at higher temperatures [125, 126]. (iii) Twinning has been observed extensively both post mortem and in situ (for a review, see for instance [127]). No significant change in the twinning processes has been detected between low and high temperature deformation experiments. Twinning proceeds by steady movements of Shockley partial dislocations on adjacent { 1 ! 1} planes, which leads to formation of almost perfect twins. Several twin-twin and twin-dislocation interactions have been reported.
7.3. Discussion of microscopic dislocation processes
The experimental results on TiA1 are more fragmentary than those on Ni3A1 alloys. Dislocation mechanisms can however be proposed on the basis of consistent results obtained post mortem and in situ by different authors. Glide of ordinary dislocations is possible at low and high temperatures. It is probably controlled by a Peierls-type frictional force on screw segments. Jerky movements however indicate that a double cross-slip (locking-unlocking) mechanism may operate instead of a pure Peierls mechanism, as in beryllium (section 4). In such cases, pinning points may not control directly the velocity of dislocations. The origin and the exact role of pinning points has however to be investigated more carefully.
w
Dislocation cores and yieM stress anomalies
113
Two distinct origins for the Peierls-type frictional forces have been proposed: a nonplanar extended core as in prismatic glide of beryllium (section 4), cube glide of Ni3A1 (section 5), and b.c.c, metals, or covalent bonding as in semiconducting materials [ 116, 128]. The second hypothesis has been formulated after calculations of charge densities around Ti atoms, showing that directional bonding develops between Ti atoms. However a detailed analysis of the kinetics of screw dislocations indicates that the Peierls forces are better explained by an extended core [129], although an extended core has not been predicted by atomistic calculations [130] and edge dislocations observed in high resolution TEM have a compact core [ 131 ]. - Glide of superdislocations is also possible, although not observed in all cases. Several observations report planar dissociations at room temperature and non-planar dissociations (more or less amount of cross slip into the cube plane) at higher temperatures. This behaviour is similar to that observed in Ni3A1, although with more complex dissociations, and no cube glide has been observed in TiAI. - Twinning is frequently observed at all temperatures. The exact role of twinning in the anomalous variation of yield stress with temperature is however unclear.
7.4. Models of the yield stress anomaly When superdislocations are activated, it is generally assumed that the yield stress anomalies in TiA1 and L12 alloys arise from a common origin. This hypothesis is based on measurements of similar orientation effects and values of stress-strain rate sensitivity and observations of cross slip into the cube plane. Greenberg et al. [132] considered that many locking processes can occur on superdislocations, e.g., local pinning, formations of roof type or Kear-Wilsdorf type barriers (see also [133]) which are all thermally activated with higher frequencies as the temperature is increased. The stress anomaly is thus interpreted within the frame of the models detailed section 6.4. Hug came to the same conclusions on the basis of his own observations [18]. It is much more difficult to account for the yield stress anomaly when ordinary dislocations are activated. Greenberg has proposed a model based on the existence of strong covalent frictional forces (deep Peierls valleys) along Ti rows of { 111 } planes [ 128]. For ordinary dislocations, this corresponds to the screw orientation. Ordinary dislocations are thus assumed to be more likely to be locked in the screw orientation as the temperature is increased. The thermally activated process leading to dislocation locking is however not described in detail. In addition, unlocking must be impossible, or at least very difficult in the domain of the anomaly. This model agrees with post-mortem observations. However, it conflicts with results of in-situ experiments which show that rectilinear screw dislocations have a jerky movement, for which unlocking is always possible and controls deformation. In addition, the kinetics of glide of ordinary dislocations does not correspond to the occurrence of covalent friction forces, as mentioned in section 7.3. Since the movement of dislocations is similar to those observed on prismatic planes of beryllium and cube planes of the -,/' phase (L12 structure), the origin of the yield stress anomaly could be the same, namely an increase in the width of non-planar screw dislocation cores as the temperature is increased. It would however be surprising if the yield stress
114
Ch. 50
D. Caillard and A. Couret
anomalies associated with ordinary and superdislocations should have different origins, as they have so similar properties (the same stress level, the same values of strain-strain rate sensitivity).
8. /3-CuZn and other B2 alloys The B2 structure is based on the body-centred cubic (b.c.c.) lattice. Accordingly, superdislocations with Burgers vectors (111) made of two superpartials with Burgers vec1 tors 7(111) (as in b.c.c, metals) separated by an APB ribbon have been found in some materials. However, ordinary dislocations with (100) Burgers vectors have also been found in some B2 alloys. Among b.c.c, type intermetallic alloys,/3 brass with the B2 structure has been studied most extensively. Materials with other superlattice structures based on the b.c.c, lattice are also treated in section 8 (subsections 8.4 and 8.5).
8.1. M a c r o s c o p i c
results on fl-CuZn
The flow stress in fl-CuZn increases with increasing temperature up to a peak temperature of about 200~ which is far below the order-disorder transition temperature (Tc = 460~ (fig. 34). 20--
1
I
1
1
I
I -
_
o A X=28 *
_
9
_
r D X
\
,,-17
.//"
_
I0
_
It,~
_
y/
"10
T5
-
001
1
50
I
I00
1
i50
Tempero-t'ure,
011
I
200
I
250
1
300
*C
Fig. 34. Yield stress of fl-CuZn as a function of temperature. From Umakoshi et al. [134].
w
115
Dislocation cores and yield stress anomalies
A c c o r d i n g to U m a k o s h i et al. [134] and N o h a r a et al. [135], slip lines are parallel to
{110} planes in the domain of the yield stress anomaly. {112} slip is also observed at low temperature ( - 196~ and above the peak [ 135]. In addition to the yield stress anomaly, the yield stress-temperature curves exhibit strong dependence on orientation, as first shown by Umakoshi et al. using compression tests (fig. 34). Later, Nohara et al. showed that there is also a tension-compression asymmetry. When samples are prestrained at room temperature in tension and further deformed at different temperatures in compression, the peak temperature is the same as if the crystal was only deformed in tension, and vice-versa [136]. These orientation effects cannot be explained as in L12 crystals, since //
o
.~.,__/o.-.-----"-0
r
0
-
1
~
1 200
1
o.,
_1 4O0
1
,
o----o
1 6OO
_
,o-s
'-'' 1
I BOO
-~ I-o
1
I000
TEMPERATURE. "C
Fig. 5. Dependence of the flow stress of Ni3A1 on temperature and plastic deformation offset (Thornton et al. [33]).
dislocations and other crystal defects (Vitek [40, 67], Veyssiere [68], Duesbery [41], Duesbery and Richardson [69]). Hence, in the following section we summarize our present understanding of the dislocation cores in L12 alloys. This includes the usual dislocation splitting into partial dislocations which we can regard as a special form of the dislocation core.
3. Dislocation dissociation and dislocation core structure 3.1. Anti-phase domain boundaries and stacking faults The distribution of atoms in the (111) planes is shown schematically in fig. 6. Three layers, marked successively A, B and C as in the f.c.c, case, are shown here. Small circles represent the majority atoms (e.g. Ni) and large circles the minority atoms (e.g. A1). Generally, it is assumed that three distinct metastable planar faults, whose displacement vectors are shown in fig. 6, can be formed on (111) planes. They are the APB with the displacement 71[]-01] , the complex stacking fault (CSF) with the displacement ~ []-]-2] and 1 [~11]. While the the superlattice intrinsic stacking fault (SISF) with the displacement _~ metastability of such faults has often been assumed a priori, it does not follow from the symmetry of the L12 lattice and needs to be assessed carefully. Stacking fault-type defects can, in general, be studied using the concept of the 7surface (Vitek [70]). We cut the crystal along the given crystal plane and rigidly displace
w
145
Anomalous yield behaviour of compounds with L12 structure
|
|
@ 9
(?) @
G |
9 ~ ~ ~ ~ @--~ | @-~ | ]L12
@
@ | 9 @ | 9 | @ 9 | | G| | 9 @ |
Fig. 6. The distribution of atoms in the three adjacent (111) planes. the upper part with respect to the lower part by a vector t, parallel to the plane of the cut. The energy of such a fault, "/(t), can be evaluated, at least in principle, using the chosen description of atomic interactions. Repeating this calculation for various vectors t within the repeat cell of the crystal plane, we obtain an energy-displacement surface which is called the -,/-surface. The local minima on this surface determine possible metastable stacking faults on the crystal plane considered and certain faults can be anticipated using the crystal symmetry. In particular, the -)'-surface possesses extrema (minima, maxima or saddle points) for such displacements for which there are at least two non-parallel mirror planes in the ideal lattice; the first derivative of the "/-surface is then zero in two different directions (Yamaguchi et al. [71]). Whether any of these extrema correspond to minima, and thus metastable faults, can often be decided by considering the first nearest neighbour interactions, in particular possible overlaps of the neighbours. Hence, by analysing the symmetry properties of a ")'-surface one can assess, in advance, the possible existence of metastable stacking faults. Such stacking faults are then common to all materials crystallizing in a given structure. However, for a particular material other minima than those associated with symmetry-dictated extrema may exist, but these cannot be anticipated on crystallographic grounds. The existence of these extrema depends on the details of the atomic interactions. As seen in fig. 6, there is only one mirror plane for the displacement corresponding to the APB. It is parallel to the [121] direction and its trace is marked m l . Hence, the 89
146
Ch. 51
V. Vitek et al.
APB on the (111) plane may be, but need not be, a metastable fault depending on the details of atomic bonding. Furthermore, even when it is metastable, the corresponding displacement vector may not be exactly 89 []-01] but may possess a component parallel to the [121] direction. The same applies in the case of the 1 []--f2] CSF as there is also only one mirror plane for this displacement. It is parallel to the [112] direction and marked m3 in fig. 6. Thus, in addition to APB, the CSF may be, but need not be, metastable on (111) planes, depending on the details of atomic bonding. Indeed, both pair potentials and many body potentials were constructed for which the CSF and APB are unstable (Tichy et al. [72], Vitek et al. [44]). On the other hand for the displacement !3 [211] corresponding to the SISF, there are three mirror planes perpendicular to the (111) plane. Their traces, parallel to the directions [121], [211] and [112], respectively, are marked ml, m2 and m3 in fig. 6. The ")'-surface, therefore, must have an extremum for this displacement and since the separations and stoichiometry of the first and second nearest neighbours remain the same as in the perfect lattice, it is likely to be a minimum. Hence, the SISF is a metastable fault which may exist in any material crystallizing in the L 12 structure. For the {010} planes in the L12 structure the situation is very simple. There are two mirror planes perpendicular to the (010) plane and parallel to the [101] and [101] directions, respectively, for the displacement 89 [TO1]. This implies that the 7-surface must have an extremum for this displacement, and since the separations and stoichiometry of the first nearest neighbours remain the same as in the perfect lattice (Flinn [3]), it is likely to be a minimum. Hence, the l I-f01] APB on (010) planes is a metastable fault in any L 12 alloy.
3.2. D i s l o c a t i o n
dissociation !
Generally, it has been assumed that in L 12 alloys the [101] superdislocations may dissociate on both (111) and (010) planes according to the reaction [-1 0 1 ] - ~1[101] +
1
(3.1)
[101]
with an APB between the l [TO1] superpartials. However, as explained above, a metastable !2 []-01] APB can always be formed on {010} planes but not necessarily on {111} planes Hence, the above dissociation (3.1) may, but need not, be possible on (111) planes. Another common assumption is that 89 superpartials may further dissociate into []-]-2] and g1 [~11] Shockley partials separated by CSF on (111) planes. Again, the CSF may be, but need not be, metastable and, therefore, the dissociation into Shockley partials may, but need not, occur, depending on the material. On the other hand the SISF is always a possible metastable fault on (111) planes, and thus [101] superdislocation may always dissociate on these planes according to the reaction m
1
[-1 0 1 ] - ~1 [211] + ~ IT1-2].
(3.2)
w
147
Anomalous yield behaviour of compounds with L12 structure
This splitting will occur either when the APB on (111) planes is unstable or its energy is very high (Yamaguchi et al. [73]).
3.3. Core structure of (101) screw dislocations
Screw dislocations are frequently the most important dislocations when considering core effects. The reason is that they usually lie parallel to low index directions in which several low index crystallographic planes intersect and thus cross slip type spreading of the core into these planes can occur (Vitek [40]). Such cores are sessile and hinder the movement of screw dislocations which then control the deformation process owing to their low mobility. The most familiar examples are 89 (111) screw dislocations in b.c.c. materials, the core of which spreads into several {110} and {112} planes (Vitek [39], Duesbery [41]). In the L12 structure two {111} planes and one {010} plane intersect along any {110) direction, providing an opportunity for the screw dislocations to spread into several non-parallel crystallographic planes. Atomistic studies of the screw dislocation cores in L12 alloys were originally made using pair-potentials (Paidar et al. [42], Yamaguchi et al. [43]) and more recently employing central force many body potentials of the Finnis-Sinclair type (Finnis and Sinclair [74], Vitek et al. [28, 44], Vitek [40]) as well as the embedded atom method (Farkas and Savino [75], Yoo et al. [76], Pasianot et al. [77, 78]). The results of these calculations are very similar and here we summarize the findings of these studies by presenting the core structures calculated using the many-body potentials. Details of these potentials and corresponding atomistic studies can be found in [28, 40, 44]. The first set of many-body potentials, constructed to reproduce a number of equilibrium properties of Ni3A1, leads to metastable APB and CSF on the { 111 } plane with respective energy 189 and 226 m J m -2 and to metastable APB on {010} planes and SISF on {111} planes with energy 53 and 11.4 mJ m -2, respectively. The atomistic studies were carried out for [101] superdislocations dissociated according to (3.1), either on the (111) or on the (010) plane. Two alternate core configurations of the 1[]-01] superpartial, found when the APB is on the (111) plane, are shown in figs 7 (a) and (b). In this and the following e. - o. -. e. - .o .- e. -.o.- e. -. o. - e. .. o. - .e -. o. -.e .. o. -.o. o. . e. . .o .- O. -.o .- e. - .o -. O. . .o .. e.
.
.
.
.
.
.
.
.
.
.
o. -.O. - o.
.
. .
. .
.
9o
.
.
o . O . o - e . o . O . o - e - o . O . o . e - o . O . o e
= oeo
o
= oi
,
. o . o .
o.
.
O
.
.
.
.
o
-
O
.
.
.
.
.
.
.
.
.
e
-
o
.
.
.
.
.
.
.
o
-
e
.
.
.
.
.
e
-
o
-
-
O
-
-
o
.
O
-
o
.
.
.
.
o e.o
.
.
.
.
.
.
.
.
.
.
.
.
.
.
l.o-e.o.t.o.e.o-t.o.e-o
(a)
.
.
.
.
.
.
.
=.o.e.0...o.e.o.,.o.e.o.,.0.e
9O e o
.
.
.
.
.
.
.
.
.
.
.
.
9
.
.
.
.
.
.
.
.
.
.
.
.
.
.
o o o o o . % ~ ~ ~ , ~ ~
9
9. o. . O. . o. - e. -.o -. O. - .o -. e.- o. - .O .. o. . e. - .o .
. o
.
o'O'.b'o'.o'e'-o'-o.oQo .
+'o'i'-o'.i'.o'.e'-O'.='-o'.i'.o'..'.o'-i' .
9 o.@
b'-o'.b'-o'-b'-o'-b'e-0 + ' b i . o ' =
. '
.
b
.'o'.0"o'.o'-o'-0"-o'.o'oO0= 0 e
o . o . o . i . b '~./ ~+1 ~' . o ' . 2 " . . " . . . . ( ~ )
. . . . . . .
.
o
= o'e'-o'-o.ooo
[email protected]@.o.e.o.@-o
e
o'. O'- o'. o'. o'- O'- o" " ~ . . ~ - . e . . t . . e - O
9
.
..e .- (111)
9o o o o" o ' ~ ' o r ~ ' o ~ 0 6 . - o e o , o e ~ . ~ ~ ot .o. .e. . o. . . . . . . 9. 0 . . . . . . . . . . . . . . . .
o.O.o.e.o.O.o.e.o-o.o.e.o.O.o.e
o'.o'.O'.o'.*'.o'-O'-~'-o~o'.e'.o'-o'.o'-i
o ooOooo.Oo'.='b.i'oo
(b) ~TI)
0
Fig. 7. Two alternate core configurations of the ] [T01] superpartial found using the potentials for Ni3A1when the APB is on the (111) plane: (a) glissile core, (b) sessile core.
148
Ch. 51
V. Vitek et al.
,/~0~
0)
O'o.o.o.o.o.o.o...l~-~..
o.o.o.o.o
o . o.0.o.,.o.~, , o o.o.,.0.~,
o, o, o o, o, o,
.. o. o. o-.., o. e. o...t~. -..o. o-..-, o-. i . o 9" o. 0. o'. o'. o ' - ~ - ~ - o - o . 0 - o . o. o. o. o ;" ." o" e" b'. ~,'-.~'.k b" ,'. o" 0" o" ,," o" .
.
oo-o.o-o-o--o.o.0".b'.k,'.b'-0.b.o" .
.
.
.
.
9 o O .
o .
.
.
.
.
.
.
.
.
9 o .
9 o .
.
.
.
.
.
9 o O
.
.
.
.
o .
.
.
O .
.
.
.
.
.
.
p
,
,
.
.
.
.
.
9 o-O-o-o-o-O-o
.
o
.
.
.
9
.
.
.
.
.
.
.
.
,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
o.o-O-o.o-o.O.o-ap.o.O.o .
.
.
e-o-o.o.o.o.O.o
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
9 o .
.
.
.
.
.
O
o
o.o .
o
o.o.o-O-o.o.o.O.o.o.o.O.o
111) Fig. 8. The core configurations of the 1[]-01] superpartial found using the potentials for Ni3A1when the APB is on the (010) plane; the core is spread onto (111) plane. figures depicting the dislocation cores the atomic arrangement is shown in the projection onto the (101) plane. Small circles represent the Ni atoms, and large circles the A1 atoms. Two consecutive (101) planes are always shown and distinguished as darkly and lightly shaded circles. The [101] (screw) component of the relative displacement of the neighbouring atoms are represented by arrows, the length of which is proportional to the 1 []-01]1. The rows of arrows of magnitude of the displacement and normalized modulo 1~ constant length represent APBs and/or SISFs. The core shown in fig. 7(a) is planar, spread in the plane of the APB, and in this form the dislocation is glissile, while the core shown in fig. 7(b), is non-planar and extends not only in the (111) but also onto the (111) plane. The dislocation possessing the latter core configuration is sessile. Two symmetry related configurations were found when the APB is on the (010) plane with the core spread onto (111) and (111) plane, respectively. The latter configuration is shown in fig. 8. The core spreads onto the (111) plane and it is thus sessile. An equivalent core spread onto the (111) plane also exists and is sessile but no glissile configuration with the core spread onto the (010) plane was found. The common feature of all the four configurations is the spreading of the dislocation core into { 111 } planes. This core spreading can be depicted as dissociation into ] (112) type Shockley partials, but it must not be considered as the usual splitting, since the width of the core is of the order of three lattice spacings, too small to be regarded as well defined separate partial dislocations. When the sessile core structure shown in fig. 7(b) is compared with that displayed in fig. 8, a close resemblance of the core displacements is apparent. Therefore, this structure can be interpreted as consisting of a narrow strip of (010) APB, about 1~[101]1 wide (a is the lattice parameter), which connects the APB on the (111) plane with the dislocation spread onto the (111) plane. The assumption that such a sessile core configuration exists and is energetically favoured over the glissile configuration, is the basic assumption of m
Anomalous yieM behaviour of compounds with L12 structure
w
149
(010) o.e.o.o.o.e.o.o.O
et-~' 9 o
o ~..o.o o ,.o o ~ ~ , o , o. o.,.o, o ~, o.0 o. , . o. ,,. o. 0. o..,~..o..o , . o o . . , o. , . o..', o - , ~ . . . . , ..o .0. o
(111)
0. b. i . b. 0'. b'. k--~. 0. o. o. o 0 .
.
.
.
.
.
o.O.o,
.
.
.
,
~
,r
~,,~
~
9
.
.
.
.
.
e-o-O-o--o-o-O-o-
.
.
.
9
.
o o b. e . b . o , b -e" b'.o, b. 0 o .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
O. o- o- o . 0 - o - o - o - O - o - o - o - O .
.
.
.
.
.
.
o.O-o, .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
o.o-O.o-o.o-O-o.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
o.
[email protected] .
.
.
o-o
.
Fig. 9. The core structure of the 1 [i01] superpartial terminating the APB on the (010) plane found using the model potentials for which the CSF on { 111 } planes is unstable.
o.o.O.o-o.o.Q.o.o-o-O-o-o.o-Q.o O.,.d.,.o:o-d.o-O-,.O.o-o.o.O e.o.e.O-e.o-o.O-o-o-e-O-o-o.o- 9 o.o.O.o.o.o-Q-o-o-o-Q-o-o.o.@-o
o, o,
o.
, .o .,-o
o
. * . o . . . co ~, ,,,_~- ~ - *,.~ - ~ . . *- . . *- . . *-. . 0,0,d..'d.o-o-o.6.'..d.d. .
.
.
.
.
.
.
.
.
.
.
.
9
,
l
,
~
.
~
*
,
9
.
,
.
.
.
.
.
.
.
.
.
.
.
0 9o - o . O - o - o - e - Q - o - o - o - O . e . o e.o.e.O.e.o-o-O-o-o-e-| .
.
.
.
.
.
.
9
9
9
.
~
9
.
.
.
.
.
.
.
.
.
.
.
.
d...o.,.d.o.6.'o.o.o.|
"0 .'. .'o .'. .'0 .'. .'o .'. .'| .'. .'o .'. .6 .", ."d
Fig. 10. The core structure of the 89 superpartial bounding the SISF on the (111) plane found using the model potentials for which the CSF on { 111 } planes is unstable.
the model for the anomalous temperature dependence of the yield stress, in particular the PPV model, described below. The second set of many-body potentials was not constructed to describe any one specific material but rather a class of L 12 alloys with an unstable CSF. The APB on { 111 } planes is still metastable but its energy is so high that on these planes the super-dislocation 1 dissociates into .~(112)superpartials (3.2)rather than into 89 (3 1). On the (010) plane only dissociation (3.1) exists, as before, and the core structure of the ![T01] superpartial terminating an APB on this plane is shown in fig. 9. Its core is not 2 confined to the (010) plane but spreads simultaneously into both the (111) and (111) 1[~11] superpartial bounding planes and thus its core is sessile. The core structure of the .~ the SISF on the (111) plane is shown in fig. 10. Since this superpartial possesses an edge m
150
v. Viteket al.
Ch. 51 u
component, significant displacements perpendicular to the [101] direction exist in its core but only the screw component of the displacements is shown here. These displacements are spread simultaneously into the (111) plane, one layer above the plane of the SISE and to the (111) plane. The edge component of the displacements remains in the plane of the SISE Thus the core of this superpartial is again sessile and its screw component has practically the same form as the core of the 89 superpartial shown in fig. 9. These results suggest that there are L12 compounds in which no mobile screw dislocations exist. In these compounds APB on { 111 } plane is either unstable or possesses such a high energy that dissociation (3.2) is favoured over dissociation (3.1). The screw 1 superdislocations dissociated into g(l12) superpartials and gliding in such compounds on { 111 } planes will be equally difficult to move as the superdislocations dissociated into 89 superpartials on {001 } planes. The Peierls stress of such dislocations is very high but at non-zero temperatures the motion of sessile screw dislocations can be aided by thermal activation. This results in a strong temperature dependence of the yield stress at low temperatures. Such a temperature dependence of the yield stress has, indeed, been observed in Pt3A1 (Wee et al. [20, 21], Heredia et al. [22]), and recently in the L12modified A13Ti (Kumar and Pickens [79], Wu et al. [24-26], Kumar et al. [80], Kumar and Brown [81 ]). The plastic behaviour of L12 alloys of this type is not described in this chapter since we concentrate here only on those L 12 compounds which display the yield anomaly and, as explained below, this cannot occur in the alloys in which no glissile form of screw dislocations exists.
4. Theory o f the yield anomaly In the L 12 alloys displaying the yield anomaly the yield (flow stress) at low temperatures is much lower than at high temperatures and is practically temperature independent below room temperature (see figs 1-3). This implies that there must be fully mobile dislocations available in these materials at low temperatures. Since the slip system operating at these temperatures is { 111 }(101) the dislocations carrying the slip on the (111) plane are the [-101 ] superdislocations which dissociate according to (3.1) into two 1 [T01] superpartials separated by the APB as first proposed by Koehler and Seitz [82] and observed in the field ion microscope by Taunt and Ralph [83]. Each of the 1 IT01] superpartials either splits into two Shockley partials or, at least, their cores are planar, i.e. of the type shown in fig. 9(a). Hence, in these materials both the APB and CSF on { 111 } planes must be metastable. Otherwise no mobile screw dislocations would exist and the yield stress would be rapidly increasing with decreasing temperature.
4.1. The T a k e u c h i - K u r a m o t o m o d e l m
The first model employing a mechanism of immobilization of [101] screw dislocations at high temperatures was proposed by Takeuchi and Kuramoto [11, 12]. It is based on the original suggestion of Thornton et al. [33] that immobilization involves cross slip of screw dislocations from (111) to (010) planes, which is a variant of the work-hardening
w
Anomalous yield behaviour of compounds with L12 structure
151
Fig. 11. Schematic pictures showing the formation of the pinning points on the [101] dislocation moving on the (111) plane and its bowing-out around them.
theory of Kear and Wilsdorf [37]. The screw dislocations on (010) planes are, indeed, sessile owing to the spreading of their cores into (111) or (111) plane as demonstrated in the atomistic studies (see fig. 8). This cross slip process can be thermally activated and is therefore more frequent at higher temperatures. The driving force is provided by the anisotropy of the APB energy (Flinn [3]) and by the resolved shear stress (RSS) on (010)[ 101 ] system; using linear elasticity the splitting (3.1) is favoured on (010) planes, APB , where 7 APB and _ APB are the energies of APBs on {111} and provided 7~1]B > v Jfbl0 "~'010 {010 } planes, respectively. However, a permanent immobilization of dislocations would lead to a work-hardening type increase of the yield stress which would not be reversible upon decrease of temperature (see section 2, characteristic (2)). For this reason Takeuchi and Kuramoto [12] assumed that the cross slip occurs only locally along short segments of the dislocations, leading to the formation of pinning points. This is a reasonable assumption since these cross slips (in fact core transformations) occur on dislocations gliding on { 111 } planes and, as the transformed immobile part of the dislocation is being formed, the dislocation in the glissile form bows out around it away from the screw character. This prevents the transformation into the sessile form from spreading along the entire length of the dislocation and, therefore, only a small segment of the sessile dislocation, i.e. a pinning point, is formed. These pinning points then act as obstacles to the dislocation motion. This is shown schematically in fig. 11. An athermal release of the dislocations from the pinning points is then assumed to take place so that a steady state, corresponding to the creation of a constant average density of pinning points, is attained at a given temperature. Since the density of the pinning points increases with increasing temperature, a positive temperature dependence of the yield stress ensues. At the same time when the temperature is decreased a new steady state develops and the yield stress decreases accordingly. If the activation enthalpy for the formation of the pinning points is Hp, and their extent along the screw dislocation is go, the frequency of formation of the pinning points per unit length of the dislocation is (Friedel [84]) n
fp--~exp
-~
,
(4.1)
where Vd is the Debye frequency, b the Burgers vector of the dislocation, k the Boltzmann constant and T the temperature. At steady state one new pinning point is created for
152
V. Vitek et al.
Ch. 51
every pinning point annihilated by the release of the dislocation. Hence, if tu is the time the dislocation is, on average, trapped at a pinning point and L is the average separation between the pinning points, then
tuLfp = 1.
(4.2)
This is then the equation determining L. Similarly as in the case of solid solution hardening (see, e.g., Friedel [84]), we assume that the dislocation is bowing out between the pinning points, and unpinning occurs when the force acting on the pinning point reaches a critical value, which can be written as 2T sin 0c, where r is the line tension and 0c is the critical angle at which the unpinning occurs. The RSS in the (111) glide plane, apb (p implies primary slip plane, b implies in the direction of the Burgers vector), can be written as --
(4.3)
O'pb,
where the superscripts 0 and T distinguish the athermal and thermal components of the stress, respectively. O'pb 0 corresponds to the RSS at low temperatures and no motion of the dislocation occurs for smaller stresses. Hence, the bowing out of a dislocation is determined by O'pb x and, therefore, the radius of curvature of the bowed out dislocation R - 7/(aTpbb). It follows from obvious geometrical considerations that sin 0c - L / ( 2 R ) , and the critical stress in the primary plane needed to tear the dislocation away from the pinning point is then apTb = 2T sin 0c bL "
(4.4)
At this point the dislocation has bowed out to the distance d
-
-
Oc)
-
-
cos
Oc)
apTb
9
(4.5)
Since unpinning is athermal, the time tu is the time needed to move the dislocation to this distance by gliding in the (111) plane, i.e. tu = d/v, where v is the dislocation velocity. If a power law dependence of the velocity on the RSS is assumed, v - v0(apb) rn, where m is a positive exponent, eqs (4.1), (4.2) and (4.5) yield L-
(g0)2V0kvpb] exp "r(1 - cOS0c)Ud
( ) Hp
~
9
(4.6)
Inserting this value of L into (4.4) we obtain
O'pb = O"0 exp
- (m + 2 ) k T
,
(4.7)
w
Anomalous yield behaviour of compounds with L12 structure
153
where ] 1/(m+2) 2r 2 sin 0c ( 1 - cos 0c) Ud "0
be2vo
--
Takeuchi and Kuramoto [12] assumed m - 1, i.e. a Newtonian viscous motion of glissile dislocations in the (111) plane. Furthermore, the only stress component aiding the formation of pinning points was assumed to be the RSS on the (010) plane in the direction of the Burgers vector, Crcb, since glissile-sessile transformation is considered as a cross slip into this plane (c implies cross slip plane, b implies in the direction of the Burgers vector). Hence, they wrote the activation enthalpy for the formation of the pinning points as np
-
n ~ -
(4.8)
Vpl~cbl
where H ~ is a constant and Vo is the stress independent activation volume. Note that Hp does not depend only on the thermal components of the stress but on the total applied stress since no dislocation motion takes place during the glissile-sessile transformation. Furthermore, Hp does not depend on the sign of Crcb since both positive and negative shear stress in the (010) plane aids the cross slip into this plane. For a given applied uniaxial stress (tension or compression) Crcb and O'pb are related geometrically and a geometrical parameter N, depending on the orientation of the sample with respect to the tensile/compressive axis, can be introduced (Lall et al. [13]) such that
N-
O'pb
I
The implicit relation
Crpb -- o'0 exp
[ --
3kT
,410
then determines both the temperature and orientation dependence of the yield stress in the anomalous regime for temperatures lower than the peak temperature. Owing to the dependence on the orientation factor N eq. (4.10) predicts a breakdown in the Schmid law since O'pb, the CRSS for the (111)[101] slip system, will be different for different orientations of the tensile/compressive axis. The results of the early experimental studies of the orientation dependence of the CRSS, measured in compression for Ni3Ga (Takeuchi and Kuramoto [12]), Ni3Ge (Pak et al. [85]) and Ni3A1 (Saburi et al. [86], Aoki and Izumi [87], Kuramoto and Pope [88]) were in agreement with this model. However, these tests were flawed by the fact that only very limited number of sample orientations were tested. When tests were carried out for more orientations (Lall et al. [13], Ezz et al. [14], Umakoshi et al. [15]) significant discrepancies with the model were found. For example, the relative strength of samples having nearby orientations is, in some cases, opposite to the prediction of the model. However, most
154
V Vitek et al.
Ch. 51
significant is the observed tension/compression asymmetry (see section 2) which cannot be described in the framework of the Takeuchi-Kuramoto model because the activation enthalpy, Hp, does not depend on the sign of the applied stress. A comprehensive description of the yield anomaly requires, therefore, a detailed understanding of the transformation from the glissile to sessile form which takes fully into account the core structures of the corresponding dislocations. Such an analysis was carried out by Paidar et al. [38].
4.2. The PPV model
The starting points of this model are the same as in the case of Takeuchi-Kuramoto model. First it is assumed that there are two possible configurations of the 1 []-01] screw superpartials bounding an APB on the (111) plane" a glissile configuration with its core spread in the (111) plane and a sessile configuration with a non-planar core spread on the (111) plane. These two types of the cores are shown in figs 7(a) and (b), respectively. The second assumption is that the glissile screw dislocations transform with the aid of thermal activations into sessile ones since the sessile core is energetically favoured over the glissile core. Furthermore, as in the Takeuchi-Kuramoto model, it is assumed that these transformations occur locally on dislocations gliding on { 111} planes and this leads to the formation of the pinning points which hinder the dislocation motion (see fig. 11). As discussed in section 3.3, in the glissile core (fig. 7(a)) the Burgers vector is distributed in the similar way as if the dislocation were dissociated into two 1 (112) type Shockley partials on the (111) plane. The sessile core (fig. 7(b)) can be regarded as a narrow strip of the (010) APB, I~[101]1 wide, terminated by the dislocation spread onto the (111) plane. The Burgers vector is distributed in this plane again as if the dislocation dissociated into two ~(112 / type Shockley partials. Hence, in both cases the core is spread in the planes of the { 111 } type and only a narrow ribbon of the APB on the (010) plane occurs in the sessile form. For this reason, the core transformation is best described in three steps" (i) Constriction of the glissile core on the (111) plane, (ii) movement of the constricted dislocation along the (010) plane by the distance I~[101]1, and (iii) spreading of the core onto the (111) plane. A schematic picture showing the core of the 1 [101] screw superpartial before and after the glissile-sessile transformation is shown in fig. 12. This process is significantly different when compared with the model of Takeuchi and Kuramoto because this model only includes step (ii) since it is assumed that the transformation corresponds to the cross slip onto the (010) plane. The activation enthalpy, Hp, for the transformation process defined above is then composed of two parts (Paidar et al. [38]). The part associated with the steps (i) and (iii) consists of the energy of constrictions, Wc, formed on the dislocation cores spread into the (111) and (111) planes, respectively. The part associated with the step (ii) has been evaluated by regarding this step as the formation of a pair of kinks of the height I~[101]1 on the (010) plane. After reaching a critical separation,/c, corresponding to the maximum of the enthalpy when considered as a function of kink separation, the kinks m
w
Anomalous yield behaviour of compounds with L12 structure
w
155
)- . . . . . . . . . . . . . . . . . . . . r
1 [101"] SUPERPARTIALS 2
(olo)
"~
Fig. 12. Schematic picture showing the core of the l [T01] screw superpartial before and after the glissile-sessile transformation.
move apart and the dislocation has been displaced by the distance corresponding to the kink height. This part of the enthalpy is (Paidar et al. [38])
.o,0
1/2
27r
2 -
G
/
'
(4.11)
where c is the normalized self-energy of the kink, approximately equal to 0.5, and AE is the energy difference per unit length of the dislocation, between the sessile and glissile core configuration; this energy gain is the driving force of the transformation. The critical length of the transformed segment is ] ~/2 gc = b
G 167r - ~ +
(4.12)
We assume here that as the sessile segment of the screw dislocation is being formed the dislocation bows out around it away from the screw character which prevents the transformation into the sessile form from spreading along the entire length of the dislocation and only a pinning point is formed. The extent of this sessile segment will then be close to gc so that in (4.6), go can be identified with gc. In the original PPV model step (ii) was treated analogously to cross slip onto the (010) plane, as originally proposed by Kear and Wilsdorf [37]. This process is driven (a) by lowering of the APB energy owing to the decrease of the APB on the (111) plane and formation of a new segment of the APB on the (010) plane and (b) by the torque arising due to the elastic anisotropy, as proposed by Yoo [89-91]. The saddle point corresponds
156
Ch. 51
V. Vitek et al.
then to the formation of the pair of kinks of height b/2 having a critical width gc. AE was then evaluated using the standard elastic theory of dislocations. In this approximation
m E --
b
(1 + f X/2)
where f = v ~ ( A -
_ APB] - "/010 J
(4.13)
1)/(A + 2) arises from the anisotropic torque term (Yoo [89, 901);
A = 2 C 4 4 / ( C l l - C12) is the Zener anisotropy ratio for cubic structures.
However, it should be emphasized that while this analysis relates AE to the energies of APBs and the elastic anisotropy of the material, it must still be regarded as a model calculation of the core transformation. Indeed, the width of the ribbon of the APB on the (010) plane is only b/2 and the decrease of the extent of the APB on the (111) plane is of the same order, i.e. both are of atomic dimensions. These transformations are, therefore, not equivalent to the formation of Kear-Wilsdorf locks. Hence, either AE or "/1A1PBand "Y010 - APB have to be regarded as disposable parameters and the measured values of the energy of APBs (see e.g. Douin et al. [92], Baluc et al. [93], Dimiduk et al. [94], Hemker and Mills [95]) can only serve as guidelines for a judicious choice of these parameters. Assuming that b2lCrcbl 0 the CRSS in tension (apb > 0) is smaller than in compression since Hp(tension) > Hp(compression) and thus fewer pinning points are formed in tension. On the other hand the CRSS in tension is larger than in compression when K < 0 since Hp(tension) < Hp(compression). When K = 0 the term responsible for the tension/compression asymmetry vanishes and thus there is always a value of N, dependent on the value of ~;, for which there is no tension/compression asymmetry. It should be noted that Q -4 0 does not necessarily imply that K (and also ase) converge to zero. The reason is that Q and N are related: for example, on the [001]-[011] boundary of the unit triangle Q
__
3N-
x/~
x/'3(x/~- N) and on the [011 ]-[111] boundary 3NQ=
x/~
x/~(v/-~ + N)
Thus when Q --4 0, N ~ 1/x/~, and appropriate limits have to be taken in (4.18) and (4.19). The predictions of the tension/compression asymmetry for different orientations of the tensile/compressive axis, based on the calculated dependence of the parameter K on the orientation factor N, are shown in fig. 15. Independent of the value of ~;, K is negative near the [001] corner and it reaches a maximum near the [111] corner. The orientations for which there is no asymmetry, i.e. K - 0, depend on the value of ~; and, in general, they lie on the line which deviates significantly towards the [001 ] corner from the [012]-[113] line for which Q - 0. In this calculation n was set equal to 0.3 [38]. These predictions are in an excellent agreement with observations of the orientation
160
Ch. 51
V. Vitek et al.
111
TENSION To the movement of the sessile screw dislocations can occur via a thermally activated process of formation of pairs of kinks as, for example, in the case of screw dislocations in b.c.c, metals (Duesbery and Foxall [102], Duesbery [41 ]). The yield stress then decreases in the usual way with increasing temperature. Since in this regime the screw dislocations are sessile on both {111} and {010} planes, the slip on the latter planes will be preferred if possible, because the dislocations gliding on {010} planes possess a lower energy owing to the lower energy of the APB on these planes.
4.3. Strain-rate effects
As mentioned in section 2, one of the remarkable characteristics of the anomalous regime is a very small strain-rate sensitivity,
below the peak temperature (Thornton et al. [33], Takeuchi and Kuramoto [12], Miura et al. [34], Bonneville and Martin [35], Sp~itig et al. [36]). Nevertheless, the strain-rate sensitivity and/or the activation volume V* = kT/3 have recently been measured by several authors using stress relaxation techniques (Baluc et al. [57], Bonneville et al. [58, 62], Bonneville and Martin [35], Spatig et al. [36, 59, 60]) and in strain-rate jump tests (Ezz and Hirsch [61], Ezz et al. [63]). While the strain-rate sensitivity can always be measured, the activation volume has a good physical meaning only if the motion of dislocations is associated with a thermally activated process aiding the dislocation glide. In this case V* = -OHg/OCrapp,where Hg is the activation enthalpy of this process and Crapp is the applied stress. The origin of such a thermally activated process in the anomalous regime has not yet been established unequivocally, but two alternatives have been proposed: the first
164
V. Vitek et al.
Ch. 51
possibility is that the thermally activated process is not directly related to the mechanism of the anomalous yielding. This has been recently proposed by Hirsch and co-workers [61, 63] who concluded that, at least in Ni3(A1, Hf), the strain rate sensitivity of the flow is associated with work-hardening and converges to zero as the strain, and thus the work-hardening related part of the yield stress, approaches zero. The second possibility is that a thermally activated process is aiding the release of the screw dislocations which have been pinned by the transformation into the sessile form. Several such mechanisms have been proposed recently. In the model of Hirsch [103-107], it is reasoned that the jogs created by the PPV cross-slip mechanism are actually highly glissile and can therefore lead to the formation of long cross slipped segments rather than to the formation of pinning points. These long segments are then stabilized either by a second cross slip step of the leading superpartials resulting in the formation of strong edge dipole barriers or the cross slip continues until Kear-Wilsdorf locks are formed. In the former case, the barriers are unlocked by the movement of the superkinks of edge character which link the adjacent cross slipped segments. This motion is a thermally activated process with a large athermal component. The small thermally activated contribution is introduced in this model to give rise to a small strain-rate sensitivity. A different thermally activated process of release is proposed for the Kear-Wilsdorf locks that also comprises a significant athermal contribution but gives rise to a larger activation volume than the previous process. This mechanism is suggested to explain the observed jump in the activation volume vs. temperature dependence (Bonneville and Martin [35], Sp~itig et al. [36]). A dynamical simulation of the motion of [101] superdislocations was carried out by Mills and Chrzan [108, 109] and is described by these authors in detail in this Volume. In this study the evolution of a dislocation discretized into segments was studied by imposing a certain probability of the formation of PPV-type pinning points at each discrete unit, depending on its orientation. The localized barriers can dissolve athermally when the adjacent segment is bowed-out to a critical angle. A large propensity for the formation of long cross slipped segments terminated by superkinks was found. The motion of the sessile dislocations then occurs by the glide of unpinned superkinks of edge character. Fluctuations in the population of superkinks along a given superdislocation can lead to their exhaustion and thus immobilization of the dislocation. It is then suggested that the small strain-rate sensitivity may be due to the strong dependence of the density of mobile superkinks on the applied stress. However, the unpinning process may be aided by thermal agitations directly, as is the pinning process. A model which considers thermally assisted unpinning from individual pinning points has been suggested in [29, 110, 111]. The unpinning is assumed to trigger a major breakaway, i.e. release from many pinning points at the same time, and the unpinned dislocation then glides freely before being pinned again. Yet, it was recognized by Khantha et al. [45, 46] that release from a single pin is rate controlling only at low temperatures, when the mean separation between the pinning points is large. However, as the temperature and the stress increase and the mean separation between the pinning points (also formed with the aid of thermal activations) decreases, major breakaway becomes unlikely after the release from a single pin since re-pinning becomes more probable than further unpinning. The release is then controlled by more difficult, m
w
165
Anomalous yield behaviour of compounds with L12 structure
(11) .
.
.
.
.
.
liB.
.
.
.
.
.
(32)
(21)
(31)
(33)
(22)
Fig. 17. Equilibrium configurations of a dislocation with two pinning points. but alternate, reaction paths that involve multi-pin activations. A similar situation arises when considering release of dislocations pinned at impurities, and it was investigated in detail in conjunction with studies of internal friction (Teutonico et al. [112, 113], Blair et al. [ 114], Granato and Lticke [ 115]). Combining these results with the analysis of the pinning process via the PPV mechanism, Khantha et al. [45-48] developed a strain-rate dependent model of the anomalous yielding which is summarised below. As the first step in the development of this model we consider the thermally assisted release of a dislocation from two pinning points separated by a distance L. There are six non-equivalent equilibrium configurations of such a dislocation which are shown in fig. 17. The notation (ij) ( i , j - 1,2,3) refers to the displacement states at the two pins with zero displacement at the ends: the pinned, saddle-point and unpinned states are denoted as 1, 2 and 3, respectively. The configurations (11), (31) and (33) correspond to energy minima, (21) and (32) to saddle-points and (22) to an energy maximum. The saddle-point (21) involves activation at one pin while the saddle-point (32) involves simultaneous activation at both pins. Major breakaway can result from unpinnings which occur either via the path (11)-(21)-(31)-(32)-(33) or the path (11)(32)-(33) (Blair et al. [114]). The (21) saddle-point is rate-controlling for transitions via the first path (since the subsequent passage over the state (32) is easier) while the (32) saddle-point controls the activation energy for the second path. For large values of L, the release via the second path is unlikely since simultaneous activation over two pins is difficult to achieve. The transition via the second path becomes likely only when L is small and it will be favoured if the transition via the (easier) first path is inhibited owing to the backward jumps. However, when the backward jumps become easier than the forward jumps, the controlling path becomes the simultaneous unpinning from three or more pins. The path (21) is, therefore, rate controlling provided the activation enthalpy for unpinning, Hu, is smaller than the activation enthalpy, Hb, for the backward movement which leads to re-pinning. However, as the value of L decreases, a situation when Hu ~> Hb will ensue at a certain stress. When this happens, the release of the dislocation via the (21) path is unlikely if the attempt frequencies for unpinning and re-pinning are equal. Major breakaway now becomes possible only via the second path that takes the dislocation directly from (11) to (32), bypassing (31) [114]. Thus a transition from one reaction path to another occurs at those values of L, for which Hu ~ Hb. General expressions for the activation enthalpies Hu and Hb as functions of the applied stress, L, and displacements at the pins in the saddle point and unpinned states, respectively, have been derived in [46, 114] assuming a reasonable dependence of the dislocation-pin interaction energy on the distance from the pin.
v. Viteket al.
166
Ch. 51
After the major breakaway from the pinning points, the dislocation is assumed to glide freely on the (111) plane with a velocity v, traveling a distance d until enough pinning points have nucleated on the unpinned segment that it again becomes sessile. To determine a steady state we consider a random distribution of pinning points separated by a distance, L, on average, along a segment of length Ls of a screw dislocation. Let Np(t) be the number of pinned segments at time t. Np(t) changes as follows: (i) it decreases when the dislocation is released from a certain number of pinning points resulting in the major breakaway and (ii) it increases when new pinning points are formed on the unpinned gliding segments. Clearly, the pinning process must dominate over the unpinning process to provoke the anomalous temperature dependence of the yield stress. In order to attain a steady state in which both pinned and unpinned segments are present, a gain-loss rate equation must couple the dominant (pinning) and weak (unpinning) processes. Otherwise, the dominant process wins which in the present case would imply a complete transformation of screw dislocations into the sessile form. The rate equations of predator-prey models, logistic equation and autocatalytic reactions (Haberman [116], Reichl [117]) are examples in which strong and weak processes are coupled in a non-linear manner. A similar but simpler approach was adopted by Khantha et al. [46]. The gain term is considered to be proportional to the product of the number of unpinned segments that can glide freely at time t, Np(t) e x p ( - H u / ( k T ) ) , and the number of pinning points formed during the free flight, equal to v gc exp
( Ho
Here udb/g c is the attempt frequency for pinning, as in (4.1) and it can be approximated as Ud since gc ~ b. The rate equation for Np(t) can then be written as
_
where Ls/s c is the number of nucleation sites for pinning and Ls/L the number of nucleation sites for unpinning on the dislocation length Ls and Hp, the activation enthalpy for the nucleation of pinning points, is again given by (4.16). In the steady state when dNp/dt = 0, it is reasonable to take d ~ L for a random distribution of pinning points [117]. It follows then from (4.21) that in the steady state
/ gCbv L-
Hp
3V---~d exp 3kT
which is analogous to (4.20).
(4.22)
w
Anomalous yield behaviour of compounds with L12
structure
167
The rate of thermally assisted unpinning of screw dislocations moving on the (111) plane determines the strain rate which can be written as = e0exp
- ~
(4.23)
with e0 = p b A N ( v d b / L ) , where p is the density of mobile dislocations, A the area swept by the unpinned dislocation, and N the number of nucleation sites for unpinning per unit length of the dislocation. Since A is proportional to L 2 and N ~ 1/L, ~o can be treated as a constant. The RSS on the (111) plane is given by (4.3) and the activation enthalpy Hu is a complicated function of (rpTb and L which is given by (4.22) (see [46]). The activation volume related to the strain-rate sensitivity is in this case V* - --0Hu/0CrpTb. A discontinuity in V* will occur whenever there is a change in the dominant unpinning path; V* is, in general bigger for simultaneous release from (m + 1) pins than from m pins. T can be obtained by solving For a given strain rate the temperature dependence of (rpb eq. (4.23) for Crpb. T This can only be done numerically. Such calculations have been made by Khantha et al. [46] for binary Ni3A1 and ternary Ni3(A1, Ta) assuming that the principal difference between these two alloys is a different core width of the screw superpartials, i.e. in terms of dissociations, different energies of APBs and CSF. Furthermore, the effect of the ternary addition (Ta) was also taken into account via the parameter/3c, as described in section 4.2.1. In these calculations Hp and t?c were calculated using (4.16) and (4.12), respectively. In these equations the constriction energy, We, has been determined employing the complete formulas derived by Escaig [96, 97] and using the anisotropic elasticity to estimate the widths of the cores [45]. Following the recent measurements (Baluc et al. [93, 118]), the values of the APB energy on the (010) and (111) planes were taken as 100 and 120 m J m -2 in binary Ni3A1 and 200 and 237 mJm -2 in Ni3(A1, Ta), respectively. The CSF energy on the (111) planes was taken in these two materials as 200 and 290 mJ m -2, respectively; this choice leads to reasonable values of We for which the observed temperature dependence of the CRSS is well reproduced. The energy gain, AE, has not been evaluated explicitly but an order of magnitude estimate, 0.01 J m -1, was used in both Ni3A1 and Ni3(A1, Ta). The factor, 13c, is zero in the binary Ni3AI and in Ni3(A1, Ta) it was estimated to be -0.09 on the basis of the orientation for which no tension/compression asymmetry was observed (Heredia and Pope [56]). For the free flight velocity, v, the Leibfried's expression, v - lOcr~bb4Vd/(3kT), was used (Leibfried [119]). Finally, the maximum value of the dislocation-pin interaction energy, U0, was taken as 3.2 eV and 2.8 eV in Ni3A1 and Ni3(A1, Ta), respectively and eo - 1012 s -1. The sensitivity of the results to changes in U0, e0, AE and/3c have been investigated. Even a 50% change in the values of U0, AE and/3c and three to four orders of magnitude change in eo does not produce any significant difference in crpTb. The results are most sensitive to the value of the CSF energy which is to be expected since small changes in the constriction energy and hence, Hp, produce large changes in L. The calculations have always been made for unpinning from one, two or three pinning points concomitantly (m - 1,2, 3), and in each case the activation enthalpy for the backward movement,
168
Ch. 51
V. Vitek et al.
400
9
I
0
9
9
I
9
I
Compression(Exp.)
D
3OO
I
Tension (Exp.)
] ] ~, 9
Compression
I
400
9
E?
I
9
I
9
(a)
....,
200
I
600 800 1000 Temperature T (K)
D /
"
9
/.~fO
100
,,
200
Compression(Exp.)
300
100
0
9
0
.. ;O
TensionS:" / ~'200
400
9
9
1200
0
200
~-./ ~
~ m=l 9
!
9
400
m=~ '~> !
600
9
1
800
j
,
|
.
1000
1200
Temperature T (K)
(b)
Fig. 18. The calculated and measured (Heredia and Pope [55]) CRSS vs. T dependence in the binary Ni3A1 at the strain rate of g = 1.3 • 10 -3 s -l" (a) [3220] orientation, (b) [1922] orientations. In the case (b) the squares correspond to values calculated for the strain rate g = 1.3 • 10 -2 s -1
Hb, has also been calculated. The transition from one reaction path to another was then determined according to the criterion Hu ~ Hb. Figure 18(a) shows the calculated temperature dependence of the RSS in binary Ni3A1 for the [12 20] orientation of the tensile/compressive axis at g - 1.3 x 10 -3 s -1 along with the experimental values (Heredia and Pope [55]). Following the experimental data 0 was chosen as 59 MPa. For this orientation the RSS in tension is higher than in Crpb compression, which is borne out by the calculations. There are three distinct temperature regimes with different rate controlling paths. The path corresponding to the release from one pinning point (m - 1) is controlling from 300 K to ~., 600 K. Between 600 K and 650 K, when L ..~ 120b, Hb becomes smaller than Hu signaling a change in the reaction path to simultaneous activation over two pinning points (m = 2). When this path starts to dominate, the activation volume exhibits a jump from ~ 400b 3 to 2000b 3. The transition temperature is very close to 670 K, the temperature at which an anomaly was observed in strain-rate jump experiments performed on polycrystalline binary Ni3A1 (Thornton et al. [33], Bonneville et al. [58]). From 650-850 K, the m - 2 path is controlling. Between 850 K and 900 K, when L ~ 50b, there is another transition to the path corresponding to the concomitant release from three pinning points (m = 3). However, the jump in the activation volume is now much smaller. The peak temperature, Tp, determined from the condition that L ~ gc, formulated in section (4.2.2), is approximately 1150 K which agrees very well with observations [55]. The calculated temperature dependence of the RSS in binary Ni3A1 for the [19 22] orientation of the tensile/compressive axis at g - 1.3 x 10 -3 s -1 (Crp ~ - 69 MPa) along with the measured values [55] are presented in fig. 18(b). For this orientation the tension/compression asymmetry is almost absent and thus only the compression results are shown. The RSS calculated for the strain rate 1.3 x 10 -2 s -1 is also presented and this demonstrates the very low strain-rate sensitivity below Tp. Again different paths D
w
Anomalous yield behaviour of compounds with L12 structure 300
9
I
"
0
I
"
I
"
I
"
I
"
/I
/O
169
"
#
Compression (Exp.).. l
200 /
;b
/o
/ ~
(~"
100
0
9
I
,
I
,
I
.
m=3
I
i
I
,
I
.
200 300 400 500 600 700 800 900
Temperature T (K) Fig. 19. The calculated and measured (Heredia and Pope [56]) CRSS in compression vs. T dependence in Ni3(A1, Ta) at the strain rate of g = 1.3 x 10 -3 s -1 for the orientation [123].
3000
9
2500
I
O
"
I
E
1000
I
Volume(Exp.)
v 2000
~9
"
m=2\O
m=I"~k.~~,,, ~~.......,..(~Om, :~ 9
I
~
I
200 300 400
9
I
9
I
.
I
i
I
,
500 600 700 800 900
Temperature T (K) Fig. 20. The calculated and measured (Bonneville et al. [58]) activation volume vs. T dependence in Ni3(A1, Ta).
control the unpinning process in different ranges of the temperature, leading to jumps in the activation volume. Figure 19 shows the calculated temperature dependence of the RSS (in compression) in the ternary Ni3(A1, Ta) for the [123] orientation of the tensile/compressive axis at - 1.3 x 10 -5 s -1 (o-p ~ - 40 MPa) along with the experimental values (Bonneville et al. [58]). There are again three distinct rate-controlling paths and discontinuities in V* occur between 450-500 K and at 650 K. The calculated peak temperature Tp ~ 800 K. The variation of the activation volume with temperature is shown for this case in fig. 20 along with the values of the apparent activation volume measured in stress relaxation
170
V. Viteket al.
Ch. 51
experiments [58] 2. The jump in the V* observed at 470 K agrees very well with the present calculation, but the second jump, around 600 K, suggested by the calculations, is too small to be observable.
o
Continuum theory of crystal plasticity of L12 compounds in the anomalous regime
From the point of view of the continuum theory of crystal plasticity the most interesting deformation properties, observed in the L12 compounds with anomalous temperature dependence of the yield stress, are the orientation dependence of the CRSS and the tension/compression asymmetry. These represent so-called non-Schmid behaviour, i.e. the Schmid law (Schmid [16], Schmid and Boas [17]) which states that only the RSS on the slip plane in the direction of the Burgers vector governs slip on that system, does not apply. The continuum theory of plastically deforming crystals, with its origins dating back to the work of G.I. Taylor [120, 121], was originally based on the Schmid law and the recent monograph by Havner [122] provides a historical perspective and develops the mathematical theory of finitely-strained crystals that display Schmid-type behaviour. Recently this theory has been extended to include non-Schmid effects (Qin and Bassani [18, 49], Bassani [50]), and certain aspects of the theory are reviewed in this section, particularly those that relate to the anomalous behaviour of L 1z alloys. Relatively simple multi-slip yield criteria are formulated on the basis of the microscopic analysis of the anomalous yielding (PPV model), and it is shown that the observed tension/compression asymmetry at initial yield in single crystals and its orientation dependence are well characterized using these criteria. Furthermore, strain localization in single crystals and overall hardening of polycrystals are predicted to be significantly influenced by non-Schmid effects of the type observed in L12 alloys in the anomalous regime. Since non-Schmid stresses are introduced into the yield criteria, this theory is of a non-normality or non-associated-flow type. In contrast, the constitutive model for L12 crystals recently proposed by Cuitifio and Ortiz [123] takes the Schmid stress to be the driving force for slip (a normality-type relation) but accounts for non-Schmid effects through hardening alone. The latter is not in accord, for example, with characteristic (2) of section 2.
5.1. Flow behaviour and the Schmid stress
In the framework of the continuum crystal plasticity the plastic flow in the single crystal is assumed to result from continuous shearing or slip on well defined lattice planes in well defined directions, and the underlying lattice is assumed to be unaffected by these plastic slips. With superscript c~ denoting a particular slip system, where cx ranges from 2Recently Spiitig et al. [36, 59, 60] introduced the effective activation volume which excludes the effect of changes in the mobile dislocation density and in the internal stresses. The values of the effective activation volume are lower than those of the apparent activation volume but the functional dependence on the applied stress and/or temperature remains the same.
w
Anomalous yield behaviour of compounds with L12 structure
171
1 to 12 for {111}(110) slip systems in f.c.c.-based lattices, the rates of slip .~c, on well defined crystallographic planes with normals n a in directions m '~ in the deformed configuration ( m '~ is in the direction of the Burgers vector and m c' 9n '~ = 0) give rise to the plastic part of the rate of stretching (see Rice [124], Hill and Rice [125], Hill and Havner [ 126])
Dip = y ~ ~;y~(m~n; + m;n~).
(5.1)
OL
In the small-strain version, Dij = Di~ + Dip corresponds to a strain rate (see, e.g. Hill [127]); at finite strains there is a subtle difference between the rate of stretching, Dij, and strain rate, but for the present treatment this difference is not important. Since the lattice is assumed to be unaffected by slip, the slip vectors are taken to convect with the elastic (lattice) deformation which also determines the stress via an elastic strain-energy function. In order to determine the slip rates, "~'~, a slip-system flow rule is required that, in general, depends on the stress in the crystal, the hardness of each system, and possibly the stress rate. In a time-independent theory this involves a yield criterion and hardening rule for each slip system. For a slip system to be potentially active the stress must be at yield. For a potentially-active system to be active, the rate of stressing must keep up with the rate of hardening. In a time-dependent theory typically the same dependence on stress appearing in the yield criterion is assumed to determine the slip rate, and in this case there is no stress-rate dependence (Asaro [128]). In conventional crystal plasticity the Schmid stress on a particular system determines the corresponding slip-rate. The Schmid stress apb '~ is defined such that ap'~-~'~ is the rate of plastic working (per unit reference volume) due to slip on system a; under multiple slip, the rate of plastic working, with (5.1), is
SijDip = ~
l ~ s i j (m~'n~ + mj'~n'~), -~
(5.2)
ot
where Sq is the (Kirchhoff) stress acting on the crystal. Consequently, the Schmid stress for slip system a is defined as 1
(5.3a)
or since Sij = Sji
apb~ -- Sijm ianj.a
(5.3b)
In the setting of a time-independent flow theory of plasticity, Schmid law states that a slip system is potentially active when the RSS on the slip plane attains the value of the critical resolved shear stress (CRSS). When the rate of shear stressing equals the rate of hardening of the CRSS then the slip system is said to be active. Conversely, if ~c~ # 0,
172
Ch. 51
V. Vitek et al.
then the Schmid stress equals the CRSS. Two important consequences of Schmid law for well-annealed cubic crystals are that the CRSS is independent of orientation of the uniaxial loading axis with respect to the lattice and there is no tension/compression asymmetry.
5.2. Yield criterion with non-Schmid stresses and hardening To include non-Schmid behaviours, Qin and Bassani [18] proposed that slip system c~ is potentially active when a yield function that equals the Schmid stress plus a linear combination of other components of stress attains a critical value (which in general is different from the CRSS). For cross slip phenomena in f.c.c, materials (Escaig [96, 97], Bonneville and Escaig [99]) and the anomalous yielding of L 12 intermetallic compounds (see section 4.2) these additional non-Schmid stresses are also shear stresses. Therefore, the yield function for a given slip system depends on the stress in the crystal and a set of pairs of lattice vectors denoting plane normals and directions for each shear stress just as r~~ and rn ~ determine the Schmid (shear) stress. The additional shear stresses entering the PPV theory of the anomalous yielding in L12 compounds (see (4.15)) include the shear stress on the primary slip plane, (111), in the direction perpendicular to the Burgers vector of the superpartial (ape), the shear stress on the secondary slip plane, (111), in the direction perpendicular to the Burgers vector of the corresponding superpartial (Crse), and the shear stress on the (010) plane in the direction of Burgers vector (Crcb). Because of the core configuration of the superpartials, the effects of O'pe and Crse on dislocation motion depend on their sense as described in detail in section 4.2. These stresses are also shear stresses and, therefore, are given in terms of the stress Sij acting on the crystal by equations analogous to (5.3). A table of the planes and directions for each slip system that define these shear stresses which enter the yield criterion given below for L12 crystals is found in [18] and [50]. Slip system yield criteria for L12 intermetallic compounds that involve a linear combination of ffpb, O'pe, O'se and Crcb can be deduced from the knowledge of the activation enthalpy, Hp, for the glissile-sessile transformation given by eq. (4.15). As explained in section 4.2, assuming that b2[Crcb[ ~ 1. It is shown in section 3.2.3 that this scaling relation holds for the cellular automata model described below. Using eqs (10), (12) and (14), one can show that r / - c~/~;- 1. The physics succinctly stated in eq. (17) is remarkable. Consider two dislocations mobilized at the beginning of the creep test. Suppose that one of these dislocations is destined to exhaust at late times, and one is destined to exhaust almost immediately.
w
Dynamics of dislocation motion in LI 2 compounds
219
Equation (17) states that, on average, the areal-velocity profiles of the two dislocations are related by a trivial scaling transformation. It is this fact that allows the spanning of the time-scales to produce a quantitative theory of the mechanical properties of these compounds. One calculates the velocity scaling function for the dislocation events that exhaust rapidly, and hence are amenable to computer modeling. The scaling transformation is then used to produce v(t, t ~) for the much larger values of t ~ relevant to macroscopic strain measurements. Given this understanding of the scaling properties of ~(t) and v(t, t'), and eq. (13), one can show that the creep-rate at the critical point becomes d3' = b~,(1)tl+n_ ( dt V
s
9(x) dz. z2+'7-~
(18)
Thus the creep-rate time dependence reduces to a simple power-law, provided the integral on the right-hand side of eq. (18) is finite. The assumption ofscale invariance at a critical stress leads directly to the conclusion that the creep-rate shows a simple power-law dependence on the time. It is important to note that the above analysis holds strictly only at the critical stress, Tc. However, as reasoned above in section 3.2.1, mechanical tests are not always performed at the critical stress. It is, therefore, important to analyze the behavior of the system for stresses 7- -r 7-c. For 7- < 7-c, n(a, t) is no longer scale invariant. Instead, there is a characteristic time beyond which the number of mobile dislocations decays exponentially. It is reasonable to assume that most mechanical tests are performed at stresses near the critical stress. For stresses much less than "re, the characteristic time is too short to allow for measurable plastic strain. In many systems displaying scale invariance at a critical point, the function which becomes scale-invariant at "re assumes the Ornstein-Zernike form [39] near the critical stress. The following, therefore, is conjectured to hold near to and below the critical stress:
~(t) = ~tt-~e -t/t~
(19)
with ~t a normalization constant and to, the characteristic exhaustion time, diverging at Tc according to
to - Alrc - 7.1- ~ ,
(20)
with g' a critical exponent of the transition and A a constant. The value of ~t is determined by requiring that the integral from some initial time (nonzero, to avoid the singularity at t = 0) to infinity of ~(t)dt be the total number of dislocations mobile at the initial time. Note that no analytical theory has been developed to support the conjecture in eq. (19). However, it has been demonstrated to hold analytically in related models [43]. Furthermore, the data generated from the cellular automata studied below is consistent with this expression, as shown in the next section.
D. C Chrzan and M. J. Mills
220
Ch. 52
Assuming that the scaling form for v(t, t ~) holds near the critical stress, and also assuming ~(t) is of the form of eq. (19) the creep-rate becomes
d__~_~= bfi,t tl+n_ ~ dt V
L1
9(x)e
-t/(xto)
X 2+rl-~
dx
(21) "
Note that the integral in this equation is finite as along as 9(0) is not infinite. The finiteness of 9(0) is insured because the physics of motion of a finite length dislocation does not allow an infinite areal velocity. It should also be noted that the above expression does not reduce to a simple power-law time dependence. Finally, it is noted that eq. (21) is based on the assumption that v(t, t ~) is described well by the scaling form of eq. (17) even for stresses less than "rc. Equation (17) is likely to be accurate for t ~ < to, but certainly cannot be applied to times t > to. Thus application of eq. (21) to experimental data must be restricted to times less than to. Equation (21) is of central importance to the theory. First, it is the predicted constitutive law for the creep transient based on the postulated pinning/depinning transition. In addition, it embodies the direct connection between the areal-velocity scaling function and the creep-rate during the primary creep transient. The interpretation of an experimental creep curve using eq. (21) allows one to extract directly information about the dynamics of dislocation motion.
3.2.3. Discrete model results and test of the scaling hypothesis In this section the scaling properties of the proposed transition are explored fully. The continuum-based simulations presented above are too numerically intensive to allow for detailed numerical study. Therefore, a discrete version of the simulations has been developed. A detailed description of the discrete model is presented in Appendix B. The discrete model is designed to mimic as closely as possible, the physical processes thought to be important for a proper description of dislocation dynamics. These processes are concerned mostly with the superkink spawning, annihilation and scattering processes described in section 2 of this chapter. The correlations between pinning points discussed in connection with fig. 5 are determined by both the pinning frequency and the dislocation velocity. It is reasonable to expect that the probability for creating an additional immobile dislocation segment increases with the temperature and decreases with the net stress (including the bowing stress) on the free dislocation segment. Therefore, the probability of creating an immobile dislocation segment in the discrete model depends on the following ratio @ Tn~
(22)
where O is a parameter which increases with increasing in temperature to reflect a higher probability of pinning, -r is a parameter which represents the applied stress (including the internal stress) and t~ is a parameter reflecting the bowing stress and related to the local configuration of the dislocation. See Appendix B for more details. There are two goals for the following calculations. Most importantly, because the possibility that the dislocation pinning/depinning transition may display critical behavior
w
Dynamics of dislocation motion in L12 compounds
221
has not been appreciated until recently [37], the physical phenomena associated with the transition is unknown. It is hoped that study of a simplified version of dislocation dynamics will allow a detailed understanding of the general physics expected from such a transition. The second goal of these calculations is to make quantitative predictions (if possible) for the critical exponents of the transition, enabling calculation of the form of the velocity scaling function and the creep transient through eq. (21). In the next subsection, the best estimates for the critical exponents are given, and a brief discussion of finite-size effects is provided. For the interested reader, a detailed test of the proposed scaling hypothesis for the pinning/depinning transition is discussed in subsection 3.3.3.2. Others may wish to skip subsection 3.3.3.2. 3.2.3.1. Summary of the results from the discrete model. Figure 15 displays the dislocation configurations resulting from the discrete model described in Appendix B. The black lines represent the position of the dislocation upon exhaustion of its motion. The white patches represent the area swept out by the dislocation before exhaustion. Panel (a) contains the successive configurations generated at a stress parameter "r well below the critical stress parameter rc. Note that the white areas are all relatively small, suggesting that there is some upper limit to the area swept out by a single dislocation. Panel (b) contains the successive configurations of a dislocation moving under an applied stress near T = Tc. Note that the white areas are distributed over a broad range of sizes, and one cannot identify easily a "typical" area. This panel displays directly the scale invariance of the exhaustion event distribution. Panel (c) is the result for a net applied stress above the transition stress. An interesting feature of this panel is that dislocations do exhaust, but only after sweeping out very large areas. The exhaustion of dislocations at an applied stress above the critical stress is a finite size effect, which is expected in the experiments as well. The discrete model is used to determine the scaling exponents appearing in eqs (14) through (21). The results are best described using the exponents c~ = - 1 / 2 and 5 - 1 / 3 . This choice of c~ and 5 imply that ~ = 3/2 and r / = 1,/2. These represent a selfconsistent set of values for the exponents that are deduced from analyzing the results of the discrete model in several different ways. It should be noted that in contrast to thermodynamic phase transitions, there is no theory for dynamic systems stating that the critical exponents are "universal" [38]. In this context, "universal" means that the exponents are determined by overall symmetries of the system rather than the detailed physical mechanisms. Thus there is no guarantee that the simple model described in Appendix B is governed by the same set of scaling exponents as the actual dislocations in the solid. Nevertheless, it is hoped that the simple model studied can provide accurate estimates of the exponents. Most importantly, however, the discrete model establishes that the pinning/depinning transition does occur for this model system, that the transition displays critical behavior, and that much can be learned about the transition through the study of this example. One important result from the study the model is the observation of the effect of the finite size of the simulated system. These finite-size effects manifest themselves, for example, in the deviation of the simulated results for ~(t) from the Ornstein-Zernike form of eq. (19). The deviation occurs for times of the order of those required for a
D. C. Chrzan and M. J. Mills
222
Ch. 52
(a)
(o)
(c)
Fig. 15. The exhausted configurations of a single dislocation as calculated from the discrete model described in section 6.1 (Appendix B). The dislocations are composed of 100 segments. The temperature parameter is fixed to be 69 = 6.5. (a) The stress, 7" = 10, is well below the critical stress. (b) The stress, 7" -- 12, is near the critical stress. (c) The stress, 7" -- 15, is well above the critical stress. At low stresses, the events are all small in duration and area (the white patches correspond to the areas of the events). At high stresses, all events are large. At stresses near the critical stress, it is difficult to assign a characteristic size to the events. This is a manifestation of the scale invariance associated with the critical point.
Dynamics of dislocation motion in L12 compounds
w
223
superkink to pass over the entire length of the dislocation, and are thought to arise from the interaction with and loss of mobile superkinks to the ends of the dislocation. Evidence is also presented in the next section suggesting that the collective dynamics of these "large events" are different from the smaller events. These large events are expected in the real world, dictated by the size of the crystal or grain being deformed. Therefore, the presence of the large events may be important to consider when interpreting the experimental creep curves. Unfortunately, as of this writing, characterization of the collective dynamics of these large events is incomplete. This means that the comparison of the experimental creep transient with that predicted using eq. (21) should strictly be valid only for stresses near to the critical stress, but not so near as to include many events which interact directly with the edge of the crystal. As it will be shown in section 3.2.4, using eq. (19) and the values of the critical exponents listed above, good agreement is obtained for these stresses. It is hoped that accurate prediction of the entire creep transient will be possible when the dynamics of the large events are fully understood.
3.2.3.2. Test of the scaling hypothesis.
The starting point for the analysis is evidence that the pinning/depinning transition exists in the discrete model. Equation (20) is one of the fundamental assumptions of the theories treating such transitions. The creep curves calculated from the automata are fit to eq. (21) [using a = - 1 / 2 and ~ = - 1 / 3 ] and a value of to is determined. (The quality of those fits is addressed below.) This value of to is plotted as a function of the applied stress (at fixed temperature) in fig. 16. The solid line is a fit to the form of eq. (20) with ~b = 2.14 and "rc = 97.3 This result is consistent with the divergence of the characteristic dislocation exhaustion time at a critical stress. 6000 5000 4000
t.__.a
3000 2000 I000
80
85
90
I
95
!
100
Fig. 16. The characteristic time, to, as a function of the applied stress for a dislocation containing 10000 segments and a temperature parameter 6) -- 50. The circles are the values of to measured directly from the simulations (as described in the text) and the solid line is a fitted curve of the form of eq. (20). The critical stress is ~-c -- 97.3 and the exponent characterizing the divergence is ~ --- 2.14. These values must be viewed only as approximate as placing error bars on the measurementis difficult.
D. C. Chrzan and M. J. Mills
224 1.0
Ch. 52
-
0.8
,~ ~l/,~" ~ ~[~
~
0.6
~ ~ %
e
0.4
0.3. For the sake of comparison, Fe0.275and Feo4 are common to (a) and (b). Co3Ti and L12 trialuminides are among the few alloys in which the strain-rate sensitivity of the flow stress has been addressed in domain A. In Co3Ti, it has been determined that the flow stress is strain-rate independent from 77 K up to the peak (Liu et al. [25]). By contrast, in Mn-doped L12 A13Ti, activation volumes measured at 210 K are rather small (Morris et al. [13, 14]), indicative of significant strain-rate sensitivity. By analogy with similar behaviour encountered in a number of materials, such as b.c.c, crystals, some semiconductors and ceramics, the negative TDFS at low temperature of L12 systems should originate from lattice friction (section 3.3), which is thermally activated. However, there should be several microscopic origins for this behaviour since the operating slip system is not unique in domain A. It may differ from one L 12 crystal to another and one given crystal may exhibit two nonequivalent slip systems, both showing a negative TDFS: (i) Pt3A1 single crystals loaded near [001] deform by octahedral slip, whereas those oriented in the near-J111] and in the near-[ 123] orientations deform on the primary cube plane. The cube slip system exhibits the lowest CRSS (fig. 3) which, in addition, is independent of the orientation of the external stress (Wee et al. [27]). (ii) It seems that Fe3Ge deforms principally by cube slip (Ngan et al. [31, 32]). In Co3Ti on the other hand, slip occurs on octahedral planes - in both domains A and B - regardless of the orientation tested (Liu et al. [25]). In this latter alloy, the CRSS on (110){ 111 } is claimed to be orientation-dependent, but the reported variations are so modest that one may wonder whether they are actually meaningful (section 2.2.2). (iii) In L12 Fe- and Cr-doped A13Ti single crystals, the CRSS on (110){111} is independent of load orientation (Wu and Pope [15]).
Microscopy and plasticity of the LI 2 "7' phase
w tl:l
600
r-I Pt3A I (octa.)
s
o~
O~ O~
,-r it) -o >
[] Ni3(AI,Nb )
9Pt3AI (cube)
500
259
O Ni3(AI,Ti )
400 300
o
n-"
100 0
I 0
200
I 400
I
I
I
I
600
800
1000
1200
1400
Temperature (K) Fig. 3. Selected positive temperature dependencies of the shear stress resolved on (I01){111} in several L12 alloys deformed in compression in the neighbourhood of the [123] direction, except for Ni3(Si, Ti) and for Pt3AI in octahedral slip which were deformed near [111] and [001], respectively. Data are taken from the following studies: Ni3(AI, Nb): Lall et al. [28]; Ni3(Si, Ti): Takasugi and Yoshida [29]; Pt3AI: Wee et al. [27]" Ni3(AI, Hf)" Bontemps-Neveu [23]; Ni3(A1, Ti): Staton-Bevan and Rawlings [30]; Co3Ti (actual composition Co74Ni3Ti23): Liu et al. [25]. For the sake of clarity, the points plotted in this graph do not correspond to the temperatures at which the original measurements were conducted.
2.2. Domain B
Fu et al. [33] have recently pointed out that the first indication of a flow stress anomaly in L12 alloys was reported in Ni3Si by Lowrie [34]. It is nevertheless a study of the hardness dependence of Ni3A1 upon temperature (Westbrook [35]) that drew the attention of materials scientists to the anomalous plastic behaviour of L 12 alloys. However, because of the nature of a hardness test, it was not then possible to discriminate between the variations of the flow stress and those of work hardening. After 35 years of investigation, the ambiguity is still not fully elucidated in this category of alloys (see however [69, 71, 93, 280]). The domain of temperatures over which domain B takes place varies from one alloy to the other (fig. 3). This will be largely illustrated throughout this section and information on this subject can be found more consistently in the excellent review of Suzuki et al. [16], in the original works of Curwick [36], of Heredia [37] and in the paper of Heredia and Pope [38]. One considers in general the TDFS to be only slightly positive at 77 K. The case of Ni3(Si, Ti) should be mentioned since the flow stress for these materials increases continuously from 4.2 K to a temperature located between 600 and 800 K, depending upon sample orientation (Takasugi and Yoshida [39]). The mechanical behaviour is already anomalous at 4.2 K, as indicated by the fact that the flow stress is already orientation-dependent (see section 2.2.6). In this respect, the situation is in fact quite confusing in the low-temperature part of domain B, since all sorts of mechanical
260
P. VeyssiOreand G. Saada
Ch. 53
behavior can be encountered. Some alloys, such as Ni3(A1, Nb), that show quite a steep positive TDFS, have little orientation dependence (Ezz et al. [40]). On the other hand, amongst alloys that exhibit a small positive TDFS at 77 K, some, such as Ni3(A1, Ta), show pronounced orientation dependence all over domain B from 77 K to the peak, while others, such as Ni3(A1, Zr) and Ni3(A1, B), are orientation-insensitive at 77 K [37]. The most significant macroscopic features that characterize the positive TDFS (domain B) are analyzed in this section, but before addressing the possible origins of the positive TDFS, we find it important to shed some light on compositional effects. 2.2.1. Generalities on compositional effects
Our knowledge of the positive TDFS of L12 alloys relies essentially on measurements carried out on Ni3Al-based alloys. However, since binary Ni3A1 single crystals are difficult to grow near the stoichiometric composition, it should be kept in mind that most of the properties discussed below refer to off-stoichiometric and ternary compounds. One of the earliest indications of the role of composition seems to be due to Thornton et al. [41] 1 who pointed out niobium as a potent strengthener. The unpublished work of Curwick [36] is the first extensive study of compositional effects ever carried out on single crystals. In practice, composition adds to the overall complexity of core-controlled situations since modest changes, even at the level of one percent or less, may effect mechanical properties of Ni3Al-based alloys quite dramatically (Mishima et al. [42], Suzuki et al. [16], Khan et al. [43], Heredia and Pope [38], Dimiduk and Rao [44]). In binary alloys, it is known that above room temperature, the larger the atomic fraction of aluminium, the steeper the positive TDFS (Noguchi et al. [45]). According to Dimiduk [46], the composition dependence of the flow stress is even more pronounced in single crystals. On the other hand, Mishima et al. [47] have shown that for solute additions which substitute for Al-sites, transition metal elements induce extra hardening as compared to solute hardening obtained by additions of B-subgroup elements. Heredia [37] has in addition determined that composition strengthening differs according to whether ternary atoms are in substitutional or interstitial positions. The sensitivity of the flow stress to a deviation from stoichiometry, further complicated by the fact that site occupancy is not always identified in these alloys, is probably one of the reasons which make rationalizing compositional effects in ternary alloys so uncertain (Dimiduk et al. [48]). An illustration of these difficulties may be found, for instance, in the work of Heredia [37] in the case of Ta-doped Ni3A1 where the strengthening does not depend monotonically on the Ta content (70.2(2 at.% Ta) > 70.2(1 at.% Ta) at low temperatures, whereas above 800 K, T0.2(2 at.% Ta) > 7"0.2(5 at.% Ta), Tp = 1000 K). At the present time, it is not yet clearly elucidated whether chemistry acts intrinsically on core structure through modifications of the energy of planar defects, or else extrinsically, by the usual solution hardening process. Heredia [37] has concluded to the predominance of a classical solution-hardening effect at low temperatures, though he 1We would like to draw the reader's attention to the experimental work of Thornton, Davies and Johnston [41], in which almost every basic property was addressed very early and sometimes in great detail. Since then, this pioneering work has appeared instrumental to the understanding of the flow stress anomaly of L12 alloys.
w
Microscopy and plasticity of the L12 .yt phase
261
relates the mechanical behaviour at higher temperatures to dislocation core transformations. In view of several major contributions (Mishima et al. [42, 47], Dimiduk [46, 49], Heredia [37], Hemker and Mills [50]), it seems in fact reasonable to consider that the composition dependence of the flow stress stems from a combination of both solution hardening and core transformation. A discussion of these effects, about which still very little is known theoretically, is beyond the scope of the present contribution. Compositional effects will nevertheless be addressed occasionally in the following when they may explain apparently inconsistent findings. In conclusion for this question, it appears that detailed comparisons between the mechanical properties of alloys with different solute contents should be considered with some care since no systematic trend has been established. For further and extensive details on solute effects, the reader should consult the studies of Curwick [36], Mishima et al. [42, 47], Miura et al. [51-53], Suzuki et al. [16], Dimiduk et al. [46, 48], Heredia [37] and Heredia and Pope [38].
2.2.2. Experimental uncertainties Besides difficulties which arise from compositional changes, mechanical tests are not always as reliable as suggested in the literature. Experimental uncertainties are not limited to the accuracy in measuring the conventional flow stress at 0.2% from the engineering stress-strain curves, which one may estimate to about 1% to 2%. For instance, in view of the large work-hardening rate at 0.2% of permanent strain (section 2.2.3), the experimental determination of the flow stress is inherently inaccurate (fig. 4) and one may wonder whether a difference of less than say 5% between two flowstress measurements allows one to claim significant differences in mechanical behaviour. This is particularly important in the discussion of the tension-compression asymmetry of the flow stress (section 2.2.6). The adverse effects of large work-hardening rates are also encountered in stress-relaxation experiments where resulting corrections to derive true activation volumes may represent a considerable fraction of the uncorrected signal (section 2.2.8.2). Although the appropriate precautions are usually taken, a further source of uncertainty may arise from sample preparation, since surface damage introduced during sample machining can increase the flow stress by a factor as large as two (Bontemps et al. [55]). The tension-compression asymmetry (section 2.2.6) is however unaffected. Still regarding boundary conditions, Mulford and Pope [22] have pointed out that prestraining at 77 K does minimize end effects such as plastic flow due to stress concentrations at the loading faces. Recent work has shown the importance of sample end effect under compression [286]. It should be kept in mind that since most straining devices exhibit significant thermal inertia, information derived from temperature transients should be regarded with care. Thermal stability is of critical importance in stress relaxation tests. Measurements depend strongly upon single crystal perfection, upon the accuracy in orientating single crystals and upon maintaining axiality during deformation. For instance, in order to explain a difference of approximately 20 MPa in the level of the CRSS at 77 K of a sample under one particular orientation (sample #5 in Ezz et al. [56]), with respect to the value expected from the value determined in every other orientations tested (~ 75 MPa), Ezz et al. [56] were prompted to invoke some experimental
262
Ch. 53
P. Veyssidre and G. Saada
(a)
"a" n
600
o~ 500 ,e (/}
4--" O3
400
300 200
100
0
0
1
2
Strain (%)
3
600
(b) (II 13_
O3 (1)
9231
[] V a
13
2500
9Vh 9 Veff
2000 -
c
1500
l c (or/o < /,~), a twofold configuration transforms into a KW (outer ellipse) whereas when lc < l~' (or l,, > l,]), the eventual configuration is planar in the octahedral plane. It can be remarked that for/Jcump > Ao/x/r3, a screw superdislocation should remain split in the cube plane./lcUmp is defined in fig. 32(d).
Microscopy and plasticity of the L12 ~ff phase
w
371
4.2.2.2. The unstressed case (Saada and VeyssiOre [257]). Consider the twofold configuration of fig. 52(a) in which L and T slip on the octahedral and on the cube plane, respectively. Of course, L and T play entirely symmetrical roles in the unstressed case, but this notation is nevertheless kept for consistency with the stressed situation. Equilibrium occurs when the relations
r
(alo +/c)Ao = 0,
(16)
r
(lo + lcV )r
(17)
= o
are simultaneously satisfied, that is, when the two corresponding ellipses intersect one another within the first quadrant of the {lo, lc} representation (fig. 53). Three distinct situations may occur according to the value of ff relative to c~. (i) ff < c~: there is no ellipse intersection (fig. 53(a)) and the only stable situation is the complete KW. A superdislocation initially extended in the octahedral plane, crossslips spontaneously in the cube plane under the action of the force FcL which is written as
F~ = 7o(O~ - ~).
(18)
(ii) ff > x/~: again the ellipses do not intersect (fig. 53(b)). Hence, the only stable situation is the planar octahedral configuration. A superdislocation extended in the cube plane cross-slips in the octahedral plane under the action of the force FoL given by Fo
-
"yo(r -
.
(19)
(iii) c~ < ff < x/~: the two ellipses intersect (fig. 53(c)) at the point S with coordinates {lo, 1c }, where
lo -
Ao ~-2 _ 2 o ~ + c~x/3'
~--O~
Ic - Ao ff2 _ 2o~ + c~v~"
(20a)
(20b)
The incomplete KW is an equilibrium solution but, as a saddle point of the energy, its equilibrium is unstable. A planar superdislocation extended in either plane is metastable. However, the activation energy to transform one planar configuration into the other is in general prohibitive, so that the planar configuration cannot cross-slip without the help of an applied stress. The dissociated screw superdislocation can thus be regarded as a bistable system. So far every L12 alloy that has been found to correspond to case (i) does exhibit a positive TDFS. Unfortunately, the APB energies have not been measured for L12 alloys which do not exhibit a positive TDFS. The transformation from a planar configuration located in the octahedral plane into a Kear-Wilsdorf lock is represented graphically in fig. 54.
372
Ch. 53
P. VeyssiOre and G. Saada
o Ic A ; L e n A "
TO___ )~o L ................................. "......... 0 Xo ::.::::::::::::::::::::::::::::::::::::::::::::::::::::::
Fc = 0
(b)
O
.:i: .i5:
"r
Io
.i!'
A '0
...............! ! ;..... ! ! : " ' :..... "?
3.0 o ~ c (a)
O .:~: #? 3-' .:v
B , O .....................:.~:
? ;57
:;i: .5?;"
C
:?"0 .:iF
i 61o
,q: :?:"
d
/!;i: ~"C .iif
Xo o~ ~c
.ii:
C O
(c) Fig. 54. Graphical analysis of a possible evolution of a dissociated screw superdislocation, the intermediate configurations represented end-on in (b). Each of the schemes (O, A, B, A t, B t, A ' , C t and C) relates to a representative point in the chart of (a). In (b) heavy lines represent the difference between the situation under consideration and the preceding one located immediately on top of it in figure (b). We consider a "usual" L12 crystal, i.e., such that ~ < c~, in which a screw configuration is initially dissociated in the O configuration represented by the point O{Ao, 0} on L'o. In this case Fc~ = 0 and FcL > 0, which promotes the cross slip of L in the cube plane. Suppose the position of T is fixed momentarily, then L cross-slips onto the cube plane over some distance until the representative point of the twofold configuration hits the outer ellipse, Zc, and this defines a point A{Ao, lc } which is located outside Z,,. At this stage, F~ = Fcx = 0 and FoT > 0. If, for the sake of simplicity, it is now assumed that L can be fixed in turn and T is freed, then T moves in the positive direction and the representative point shifts horizontally until it hits the inner ellipse, S,,, at B{lo, ld}. The screw segment has now adopted a twofold shape and the signs of the forces are the same as for the initial configuration O. Configuration C{0, Ac }, which is the only stable one in the crystal, can be reached after several A -+ B --~ A t -+ B t steps, whose number depends on the relative positions of the two ellipses. (c) A more realistic, yet oversimplified, path for the transformation of (a): when it is considered that lattice friction is large in the cube plane and negligible in the octahedral plane, the excursion 61c, of the representative point off the inner ellipse Zo is modest.
w
Microscopy and plasticity ~( the LI 2 ,.yt phase
373
4.2.2.3. The effect of an applied stress (Saada and Veyssikre [243]). We focus on alloys where ~ < a, that is, such that the KW configuration is the most stable in the absence of an external stress. The effect of an applied stress can be treated along the same lines as in the above development. It should first be noted that when both superpartials are allowed to glide freely in the same plane, the equilibrium width /~i of the planar superdislocation is maintained during its motion. The fact that friction is by far the larger in the cube plane provides a simple priority rule for slip in which octahedral slip can be regarded as the fastest, while cube slip is delayed. Nevertheless, since friction is thermally activated, there is a finite probability for cross slip on the cube plane to occur at any stage, which we shall however consider only when the appropriate force F J is positive. In their simulations Chou and Hirsch [199] have chosen to adopt strict geometrical (crystallographic) constraints whereby the spreading symmetry of superpartial core may hinder slip in one or other of the octahedral planes; the choice of the forbidden plane is determined by the atomic environment of the (110) atomic row where the superpartial is sitting (fig. 51). We have preferred to relax these constraints, conforming to the above assumptions on superpartial cores (section 4.2.1.1) and on lattice friction (section 4.2.1.2). Furthermore, ignoring core symmetries agrees better with the idea of a dynamical transformation wherein mobile cores are not necessarily localized with precision in planes. We now consider the transformation schemes of fig. 55. In the course of the motion of a superdislocation in the primary octahedral plane, one of its superpartials, say L, cross slips over the distance/jump in the cube plane before it may be regarded as immobilized by viscosity (step 1). lJcump can be of varied magnitude, consistent with the analysis of lattice friction made in section 4.2.1.2. When T has reached its equilibrium position in the octahedral plane, the force on L becomes FoL - 2rob, which implies that the configuration in step 1 is in principle always unstable. In view of the above priority rule between slip planes, and although the formation of a complete KW by motion of L all the way through in the cube plane (step KW) cannot be totally excluded, L would most likely cross-slip onto the octahedral plane and then glide over some distance; T is then provisionally freed in the octahedral plane and both superpartials are now simultaneously mobile on two parallel octahedral planes (step 2). As soon as it has erased all the APB in its own octahedral slip plane, T becomes locked and is subjected directly to the driving force from the APB in the cube plane; meanwhile, L proceeds until FoL vanishes (step 3), the APB width on the octahedral plane, lot, is given by solving FoL{/JcUmp}-- 0 (expression (1 l a)). It should be noted that the distance L has slipped in the octahedral plane may then be larger than Ao (i.e., for small/jump, the slip distance is approximately [Ao/(1 - s)]). The screw segment has thence achieved a so-called APB jump by a double cross slip of L. In three dimensions, this process is accompanied by the nucleation of a pair of elementary kinks (EKs, section 3.4.3.4) at both ends of the cross-slipped segment (fig. 32). At step 3, what matters is the force F L that pushes L onto the cube plane:
- If F L > 0, L may cross-slip before T has erased the APB on the cube plane. The smaller s (i.e., the distance between T and L) and the larger N (i.e., the external shear
374
P. Veyssidre and G. Saada
Ch. 53
stress resolved in the cube plane), the larger FeE. As a result both superpartials glide in the cube plane (fig. 55, step 4) until either L is stopped by friction, and/or T has completely erased the APB on the cube plane, at which stage the system transforms into a configuration similar to that of fig. 55, step 1. Provided the motions of L and T are adequately synchronized, this sequence (fig. 55, step 1 to step 4) may then repeat itself several times giving rise to the process of repeated APB jumps and, in three dimensions, to multiple pairs of EKs with opposite signs (fig. 32). If FcL < 0, L is ascribed to the octahedral plane until the configuration becomes planar by motion of T in the cube plane. Then the superdislocation is freed in the octahedral plane where it automatically assumes its equilibrium width under no stress, )~o (step 3 to step 0). It is then liable to undergo a jump of unpredictable length. In three dimensions, this motion results in the formation of a simple macrokink (fig. 56, step (a), then step (e) and section 4.2.3.2). Note that, just like an EK, a simple MK requires that the leading superpartial has undergone a double cross slip. The sign of FcL is not only dictated by the magnitude of the stress in the cube plane, but also by that in the octahedral plane. This is because, given/jump, the interaction force between superpartial is a function of lo, which is itself affected by the resolved shear stress %. FcL is always positive, including small stresses, when N > (, which, in the unit triangle, corresponds to the [111] side of the (111) great circle. When, on the other hand, N < (, for Fct" to be positive, s has to be smaller than a critical stress s' given by n
s'= c~-(
oe-N"
(21)
This implies that, in this particular range of orientations, repeated APB jump, will occur for small stresses (see section 4.2.3.2.2). When now N < ( still and s' < s < s", where
x/3-r , v~-N
s" = ~
(22)
the sign of F~ depends on the extension of the incomplete KW in the cube plane,/jump, relative to a stress-dependent critical length, 1c {s, N}, (fig. 53(c)). FeE is negative when lieump < lc{s , N } and otherwise positive. The critical length lc{s , N } is obtained from eq. (20b) by substituting ff for the adequate effective APB energy ratio, which in this particular case, expresses as
~=
~-Ns 1-s
(~ - (, for s = 0). Finally, when s > s", F~ is always positive.
(23)
Microscopy and plasticity of the L12 ~/~ phase
w
(~
\
X
"
i ..................... o ......................
i
.~0
Icjump i ~"
~
1st APB jump
,j'"
375
(IKW)
.:i!ii:
(0)
2nd APB jump
Fig. 55. Transformations of a superdislocation initially glissile and mobile in the octahedral plane (step 0). At step (1), the leading dislocation is being immobilized by friction after it has cross-slipped in the cube plane over a distance, vc/Jump, which is controlled by thermally-activated lattice friction. The trailing dislocation proceeds to a distance Lom which is smaller than the equilibrium dissociation distance in the octahedral plane under no stress, )~o. Then the configuration may evolve towards an incomplete or a complete KW configuration (IKW and KW, respectively) or else L can be freed in the octahedral plane (step 2). Then both partials glide simultaneously on parallel though distinct octahedral planes until T meets the cube cross-slip plane where an APB had been created by the leading partial (step 3). At this stage the position of the leading superpartial is fixed at LoM > /~o, which is dictated by the applied stress. Again, there is a choice in the further evolution of the configuration. If the trailing superpartial erases the strip of APB on the cube plane before the leading itself cross-slips on a second cube plane, the superdislocation can become fully glissile in the primary octahedral plane (step 0). This part of the superdislocation will then slip over a distance dictated by the probability of cross slip of L on the cube plane, resulting in a MK. If now the leading superpartial has jumped in the cube /.jump plane over vc , which is not necessarily the same as in step (1), then the process of APB jump can repeat itself. At step (3), the choice between step (0) and step (4) is biased by the applied stress. This can be roughly understood as follows. If the stress is small, LoM is small, thus the force F L on the leading dislocation in the cube plane is positive. There is a driving force for the leading dislocation to pursue its motion in the cube plane. If, on the other hand, the stress is large, LoM becomes too large for the interaction between superpartial to help the leading dislocation to cross-slip in the cube plane.
376
Ch. 53
P. Veyssidre and G. Saada
(111
(olo) ! (c)
(olo)
~~,~
i
i
(o~o)
zU
(e)
~
j
,j~~
(z zz
{010) Fig. 56. Comparison between the process of a repeated APB jump (a) to (d) and that of the formation of a simple MK of height hMK (a) and (e). In fact, fig. 56(a) is the next step of the last configuration of fig. 32, represented in fig. 32(f) after the short central KW segment is unzipped. Both the repeated APB jump and the simple macrokink are formed by double cross slip of L and T. The end-product depends on the velocity of T relative to L.
Microscopy and plasticity of the L12 "7~ phase
w
377
S t
S!
ffl t__
"0 N
~>~/3
I
"
-
E 0
z
S III
I ,
o
i
N*
i ,
I
r~
~
~3
Orientation factor N Fig. 57. Orientation dependence of the critical stresses s ~, s" and s H~. Repeated APB jumps may occur for stresses below s/, while they cannot occur above s H. When s ~ < s < s', the occurrence of APB jumps will depend on the value of lc. For N larger than N*, the microscopic saturation stress, s H~, is always smaller than s ~/, so that KW destruction occurs first. When N < N*, s ~ is smaller than s~; in this domain of stress and orientation, the process of APB jumps is the most favourable. Obviously, the same formalism can be used in order to calculate the stress, s Ht, required to destroy a complete KW
s'" =
v/3-~ v~+N
(24)
which may be considered as a microscopic saturation stress (sections 4.2.2.3 and 4.2.3.5) above which KW locks all become unstable, that is, the strongest obstacles against slip are all destroyed. The stress s' is larger than s'" when the load orientation is such that N passes a critical value N* of the order of 0.4. The orientation dependencies of the stresses s', stt and s H~ are represented in fig. 57, which maps the domains of operation of the different mechanisms of transformation of a screw segment. Finally, let us consider L le alloys where screw superdislocations are bistable, that is, alloys such that c~ < ~ < v ~ , if any. The above formalism predicts that a KW lock may be stabilized as well; however, the transformation is no longer spontaneous (Saada and Veyssi~re [243, 257]). In this case, metastable KWs can be nevertheless stabilized by adjusting the adequate effective APB energy ratio ( to a value exceeding v/3. This can be actually achieved by means of an appropriately oriented external load. Considering a superdislocation gliding in the octahedral plane in the bistable situation, the smaller the shear stress in the cube plane, the more difficult the creation of a KW lock. In these conditions, L12 alloys such that a c~ < ~ < v/3, if they exist, should exhibit almost no flow stress anomaly in a region of the unit triangle located in the neighbourhood of the [001] direction. On the other hand, it is even possible to adjust the applied load so as to obtain ( > v ~ , in which case an incomplete KW should always tend towards the octahedral configuration. Since superdislocations are then no longer liable to form locks, such alloys, if any, should not exhibit a flow stress anomaly.
378
P Veyssidreand G. Saada
Ch. 53
This simple reasoning thus predicts a category of L12 alloys (~ < ~ < v/3) whose TDFS might appear and disappear according to the orientation of the load axis. 4.2.3. The motion of an individual superdislocation in an octahedral plane This section is restricted to analysing the elementary mechanisms that give rise to a single kinked superdislocation (at this stage, no hypothesis is made on the nature of superdislocation sources). For this aim, we first consider the formation of KW segments (section 4.2.3.1). We then detail different mechanisms by which this screw superdislocation eventually becomes kinked (section 4.2.3.2), and we explore the implications of the hypothesis that macrokinks may be moved in the octahedral plane by application of an external load (section 4.2.3.3). The relationship between the nature of macrokinks and planarity of slip is discussed in section 4.2.3.4. Finally, the strength of KW locks as well as its influence on mechanical properties is analysed in section 4.2.3.5. The expansion of a superdislocation within a pre-existing microstructure, including source operation, is analysed in section 4.3. 4.2.3.1. The locking in the screw orientation and the formation of KW locks. As mentioned earlier, TEM observations consistently show that locked segments are by far l o n g e r - at least by one order of magnitude - than implied in the pinning point models. Pope [156] has offered a tentative rationale of this contradiction. His explanation is based on the transformation by relaxation of the microstructure into a succession of KWs and MKs (fig. 50). It relies essentially on a scheme that assumes that the pinning points are rigorously aligned along the screw orientation over the extent of the future relaxed KW segment 17. Although unlocking controls the motion of an isolated superdislocation, we have found it instrumental to understand the conditions of formation of incomplete KW segments, that is, the conditions of locking, since they dictate the further evolution of the superdislocation in its near-screw orientation. The following shows that the locking is not a completely trivial process. Let us consider first a circular dislocation loop of radius R and a chord AB of length X, parallel to its Burgers vector (fig. 58). The chord is located at a distance ( R - h) from the centre of the loop. Provided h is small enough, say of the order of the Burgers vector of a complete superdislocation (2b ~ 0.5 nm), it can be assimilated to a screw segment. It can be considered as undertaking a favourable situation for cross slipping in the cube plane when its length X is long enough to hinder the backward effects of line tension that are exerted by the lateral jogs (CJ). When h lc{S , N}, the system has a finite probability to undergo the transition from the incomplete to the complete
w
Microscopy and plasticity of the L12 "y~phase
383
KW (step (3) to step (4)) in fig. 55). On the other hand, when/jump < l c { s , N } , the incomplete KW cannot evolve towards a complete KW, which implies that it will undergo an APB jump sooner or later. We expect the cross-slipped distance/jump to increase as lattice friction on the cube plane weakens (sections 4.2.1.2 and 4.2.2), i.e., as temperature is raised. At low temperatures, /jump is a small number of interatomic distances, which favours the production of APB jumps. As the temperature is increased, /jump becomes eventually larger than 1c {s, N}, at which stage the twofold configuration should evolve towards the KW lock. Since cross slip is a thermally activated process, there is a finite probability for L to cross-slip before T has erased the APB on the cube plane, and there is still a finite probability, however smaller than for a single APB jump, for the sequence (1) to (4) in fig. 55 to occur several consecutive times. In other words, APB jumps together with repeated APB jumps should be regarded statistically rather than deterministically, as has been done elsewhere. The origin of the discrepancy between the present results and others stems from the fact that the synchronization of companion superpartials is not reduced to the equality between the force applied onto the leading and the trailing superpartials. In favour of a statistical point of view is the observation that the number of sequences of repeated APBs which involve a given number of EKs, decreases with this number (Couret et al. [72]). It may be remarked in addition, that though sequences of EKs are observed in the largest quantities after deformation at modest temperatures, yet a few examples of successive EKs in Ni3(A1, Hf) have been found after deformation at 650~ (Bontemps-Neveu [23]).
4.2.3.2.3. The polarization of elementary kinks. The EK distribution is almost systematically polarized (figs 36 and 37). By kink polarization, we mean a situation where kinks of both signs are not equiprobable on the same dislocation line. A microstructure where EKs of both signs are interspersed along the same screw superdislocation would attest to low EK mobility. What occurs in L12 alloys is that in the course of superdislocation expansion in the primary octahedral plane and because of the favourable elongation along the screw direction, the distribution of mixed parts - in particular of E K s - self-organizes so as to accommodate the mean local curvature imposed by local stresses (fig. 38). Hence, the polarization of the EK distribution is a local property of line tension dictated to a large extent by the applied stress; it originates from the ease of EK propagation along the screw superdislocation. Experimentally, EK mobility is further supported by room temperature in-situ experiments wherein very long portions of screw superdislocations are seen to undergo the APB jump process "as a whole", that is, within the time period of a video frame 18 (Caillard et al. [190]). 4.2.3.2.4. Remarks on kink distribution. As a screw superpartial slips in the closepacked octahedral plane, it has a finite probability to deviate (over llcump) into the cube plane. Whatever its formation mechanism, i.e., by double cross slip or bypassing, the 18EKs are in general not seen in situ since standard recording does not allow to resolve features moving faster than approximately 250 ~m s-l under the magnifications that are usually employed under weak-beam conditions.
384
P. Veyssidre and G. Saada
Ch. 53
height of a macrokink, hMK, depends directly on the free flight distance of a screw segment in the octahedral plane (fig. 56; steps (a) then (e)). Let us consider a straight superpartial segment of length L, let fcs be the cross-slip probability per unit time and per unit length (eq. (9)). The probability per unit time of cross slip of the dislocation segment is given by p = fcsL. Therefore, provided the time required to nucleate cross slip is much larger than the flight time, the number nMK of segments whose free flight distance is hMK can be written as an exponential decay nMK{T , N, s} = no exp {
p
-- ~oohMK
}
(26)
where Vo is the dislocation velocity in the octahedral plane. This form of the kink distribution is exemplified in fig. 35. As was first verified experimentally in Ni3Ga deformed at 20~ 200~ and 400~ by Couret et al. [72, 202], p increases with increasing temperature. However, the analysis of MK distribution may remain quite uncertain when it is affected by subsequent interactions. (i) As for EK polarization (section 4.2.3.2.3), there is a possibility that two neighbouring MKs, heights h~K and h~K, will coalesce and leave an MK whose resulting height, hMK -- h~K + h~K, is not related to the thermally-activated process of locking. If kink dragging takes place to some significant extent (section 4.2.3.3), this process should result in noticeable flattening of the MK distribution and the question as to why histograms fit an exponential decay so well remains open. We show in the following that the coalescence rate of kinks is probably quite significant, at variance from what the agreement between the experimental distribution and expression (26) indicate. (ii) Suppose that screw superdislocations undergo repeated APB jumps only. As the superdislocation continues to expand in the octahedral plane, more EKs are pushed to both extremities of the screw segment, where the deviation from screw orientation increases as the average distance between EKs gets gradually smaller. At some stage, consecutive EKs should eventually coalesce and yield macrokinks with a height equal to an integer number of elementary heights hEK. The macrokink of height mhEK that is formed after the coalescence of m EKs differs in structure from a simple MK of the same height by the fact that it contains ( m - 1) jogs in the cube plane (CJ), regularly distributed along its line. Such a macrokink (e.g., a multiple EK), is exemplified in figs 36 and 37 and schematized in fig. 61. In the micrographs, a succession of cusps whose length scales with that of the APB jumps can be distinguished along the mixed segment. The process of kink coalescence offers a general explanation of why some mixed segments often show considerable cusping on a fine scale. Cusped MKs must indeed be significantly less mobile than the coplanar simple MK because of the distribution of the jogs that would slip exclusively in the cube plane. The mobility of such MKs is in addition lessened by the process of trailing of APB tubes [285] (fig. 40). Hence, the coalescence of EKs and simple MKs may explain the presence in thin foils of a number of mixed segments whose length is much larger than the critical length lc -- c~#b/rc and also explain why the mixed segments may remain heavily bowed out after the applied load is removed. It should be noted that because coalesced switch-over MKs are planar while coalesced simple MKs or EKs are jogged, the latter should be much less mobile than the former (sections 4.3.1 and 4.5).
Microscopy and plasticity of the LI 2 ~! phase
w
385
(a)
9IcJump on (01 O)
(b)
Fig. 61. Formation of a jogged MK upon coalescence of EKs. (a) The individual EKs are pushed by the applied stress along the screw direction. (b) The leading EKs have coalesced to form a jogged MK, five EKs long. Since the mobility of jogs in the cube plane is less than that of dislocation kinks in the octahedral plane, the MK adopts a cusped aspect with a periodicity that scales with the amplitude of an APB jump. (iii) By contrast with the distribution of EKs, that of MKs does not seem to be clearly polarized (figs 26 and 27), which may imply that MKs are less mobile than EKs. This could also originate from the preceding argument on the jogged (sessile) fine structure of some MKs. It can therefore be concluded that kinks are not straightforward features. Their properties are far from being understood satisfactorily. Some theoretical work remains to be done in order to understand the relationship between the distances between kinks and their height. Many more determinations of MK (which would discriminate between jogged and unjogged MKs) and EK distributions- heights and d i s t a n c e s - are needed before we have a clear understanding of the experimental situation. So far, the dynamic simulations of Mills and Chrzan [241 ], constitute the only attempt at predicting kink distribution, based on three-dimensional properties of superdislocations in the L 12 structure. This work will be analysed in more detail in section 5.7.1.
4.2.3.3. The viscous sliding of macrokinks (Saada and Veyssik.re [206]).
The applied force on a given MK is proportional to its length hMK. On the other hand, it is subjected to a drag force, itself being the resultant of two contributions:
386
Ch. 53
P. Veyssidre and G. Saada
~
~
o
Fo Fig. 62. The viscous sliding of MKs upon bending of KWs. At low temperatures, superdislocations are not significantly bent in the cube plane (a) so that MKs (examplified here as a simple MK) slide freely on the octahedral plane (b). As a result KWs are transported conservatively by a zipping/unzipping mechanism. At higher temperatures, thermal activation becomes enough to allow for some bending of the KWs in their dissociation plane (c). Then for MKs (examplified here as a switch-over MK) to undergo the same zipping/unzipping mechanism, the bent superdislocation has to be brought back at the intersection between the cube and the octahedral planes ((d) dashed line). Work has to be produced against the force Fc which tends to how out the KW segment in the cube plane. - the friction force in the octahedral plane which is proportional to hMK and which is small, - the friction force due to the bending in the cube plane of the KW segment (fig. 62), as explained below. This latter force is independent of hMK and is large. Since the longer a MK, the larger the force that is applied onto it, MKs of varied heigths should slip at different velocities under a given applied stress (the longer a MK, the faster). This should result in a finite coalescence rate especially in regions of the dislocation line where the curvature has to increase. When two MKs with the same sign coalesce, a longer kink is formed that may operate as a source when its length is larger than a critical height, provided the resulting kink is coplanar (section 4.3.1). It is thus important to discuss the conditions under which MKs may migrate along KW segments. We have seen in section 3.4.3.5 that as the test temperature is increased, sessile KWs tend to deviate from the screw orientation by bending themselves in the cube plane. Accordingly, Mills et al. [200] postulated that the bending prevents MKs from pursuing their displacement in the octahedral plane. We have nevertheless explored the possibility that MKs can still move by pulling the curved KW segment back to the screw orientation (fig. 62); this takes place against the Peach-Koehler force that is responsible for their curvature (Saada and Veyssi~re [206]). Hence the applied stress, which forces MKs to move in the primary slip plane, is also responsible for an additional resistance against M K motion by promoting cube bending. This is formally equivalent to some extrinsic friction exerted onto mixed dislocations during octahedral slip. Calculations indicate that (fig. 63): - MKs may slide and KW segments can thus be transported as long as the applied axial stress is less than a critical stress cra,
w
387
Microscopy and plasticity of the L12 "7t phase
GF II
IV
|
"Cfc
r
IJ a
O'a*
~fc + ~c
%
Fig. 63. Dependence of the effective friction stress O'F that is experienced by a mobile kink upon the applied axial stress cra, under the assumption that bent KW segments oppose kink slip (Saada and Veyssi~re [206]). qo,, and 99c are the Schmid factors on the octahedral and on the cube plane, respectively. When O'a is larger than a critical stress cra, the stress resolved in the octahedral plane, qo,,cra, is lower than crF. The diagram consists of four domains. In region I, KW configurations cannot bend in the cube plane. In region II, the applied stress is responsible for the viscous glide of kinks. In region III, which starts at a stress given by eq. (27), kinks are locked and octahedral slip is exhausted. In region IV, deformation proceeds by multiplication of Frank-Read sources in the cube plane. - above that stress and below another critical stress given by 2#b qOcaa = "rfc + ~ . LKW
(27)
KWs are sufficiently bent in order to exert a drag stress larger than the external resolved shear stress here Cpc is the Schmid factor in the cube cross-slip plane. MKs are then immobilized (LKw is the length of a K W segment between two consecutive M K s and "rfc represents the friction in the cube plane); -above the stress given by expression (27), bent KWs act as dislocation sources in the cube cross-slip plane. At this stage cube slip may compete with octahedral slip: when the test temperature is increased, the average M K height decreases whereas the bending is increased. As a result, M K motion is unfavoured so that new sources are more difficult to form by M K coalescence, whereas cube slip is facilitated as friction is reduced. The drag stress increases with decreasing kink height and increasing K W length. It is in addition partially reversible with temperature and it exhibits a non-Schmid dependence on the orientation of the external stress, which roughly conforms to experimental observations [206]. The viscous motion of macrokinks constitutes one of the numerous effects that may contribute to work harden L12 alloys, but it alone cannot explain the positive T D E It is
388
Ch. 53
P. Veyssidre and G. Saada
S,,,
/Sfc(T )
(a)
S
(b)
Fig. 64. Analysis of the peak stress. (a) The critical resolved shear stress on the octahedral plane becomes larger than the stress required to activate slip on the cube plane sfc(T) (considered as roughly controlled by lattice friction), before the microscopic saturation stress (s'" in eq. (28)) is reached. (b) Lattice friction on the cube plane is still larger than the microscopic saturation stress at the temperature it is intersected by the ascending CRSS on the octahedral plane.
not a model in itself. It is in particular unable to account for the low strain-rate sensitivity of L12 alloys, whose interpretation might be more appropriately related to the operation of MKs as sources (section 4.3.1). 4.2.3.4. The microscopic saturation stress. Other than being transported by moving MKs, KWs can be eliminated mechanically upon application of a sufficiently high stress (section 4.2.3.4, expression (24)). In this case, KWs are eliminated as soon as they are created which, in the absence of further strengthening mechanism, should correspond to the ultimate mechanical resistance of a given L12 alloy deformed in domain B. In fact, the stress at the peak "rp could originate from two different mechanisms (fig. 64). On the one hand, the applied shear stress is large enough to overcome lattice friction on the cube plane (fig. 64(a)) and the KW segments proceed preferentially on these planes. This agrees with properties derived from slip line analysis (section 3.4.1), except in the [001] orientation. In addition, it seems quite evident that this scenario applies to creep tests carried out below the peak temperature (section 2.2.9). As to the operation of cube slip, note however the significant difference between creep and constant strain-rate tests. When cube slip dominates in a test of the latter type, sources operate in the primary cube plane, whereas under creep deformation proceeds in the cube cross-slip plane, once primary octahedral slip is exhausted (sections 2.2.9 and 2.2.10). On the other hand, if friction on the cube plane cannot be overcome below the critical stress, s'", to destroy a KW (eq. (24)), the so-called microscopic saturation stress (section 4.2.2.3), the theoretical upper strength of these obstacles is reached. Instead of a peak (fig. 64(a)), this process should in principle give rise to a plateau (fig. 64(b)). Based on the analysis outlined in section 4.2.2.3, we have shown that both the peak flow stress 7-p and the peak temperature Tp should decrease with increasing N (Saada
Microscopy and plasticity of the L12 "y~phase
w
389
and Veyssi~re [261]). The predicted orientation dependence of the peak stress is given by the following expression
s'"{~,N}-s'"{~,O}~
,/3 N+V~'
(28)
which agrees surprisingly well with some experimental data taken from a few binary or weakly alloyed L12 compounds. However, it disagrees with many more results obtained not only in alloys containing significant amounts of ternary additions but also in a binary Ni3A1 compound (see for example figs 8, 9 and 11). Possible causes of these discrepancies have been discussed by Saada and Veyssibre [70].
4.3. The collective behaviour of dislocations in domain B
We now aim to show that the organization of superdislocations and their interactions in the slip plane might well be as important as the behaviour of individual dislocations. Neither the strain rate that can be expected to result from APB jumps nor the strain resulting from the motion of screw superdislocations assumed to slip over some reasonable mean free path appears sufficient to achieve significant compressive ductility under usual deformation rates (i.e., larger than 10 -5 s -1) as is the case in L12 alloys. As expected, some superdislocation multiplication must occur. On the other hand, the tremendously high values of the WHR, in particular below 2% of permanent strain, together with the wide extension of the microplastic stage [286] further suggested by the strain dependencies of the WHR and of the activation volume (sections 2.2.3 and 2.2.8.2)- imply that multiplication of dislocations is difficult and/or that the annihilation rate is high. This is confirmed by the exhaustion of octahedral slip during primary creep (Hemker et al. [79], section 3.4.2), which indicates that as time proceeds the rate of superdislocation multiplication in octahedral slip is much smaller than the overall locking rate. From this point of view, the situation in L12 alloys closely resembles the stage of microplasticity in a regular alloy where strain stems from the exhaustion of mobile dislocations before multiplication has yet taken place.
4.3.1. Dislocation multiplication From the observed microstructure, it is reasonable to follow Mills et al. [200] and to consider that every MK is a potential source of new dislocations which contain in turn a fresh distribution of kinks, that is, of potentially operating new sources (fig. 65). A MK will operate as a source when its length becomes larger than a critical length given by
#b
h~K ~ - - . To
(29)
Following the results of Sun et al. [124] (section 3.1.1.2), Hirsch [99] postulated that the closing jogs in a KW segment (CJ in fig. 30) transform into a sessile Lomer-Cottrell
P. Veyssi~re and G. Saada
390 i
Ch. 53
j
h g
e
f
c
d
a
i -k.
J
h
k~l
e
a
g.,
h
f e
-,qyf
c
6
a
e,
f
b
c
d
a
5 c a
4 c.~-d
a
3 KW
a
b
b
2
b Fig. 65. Development - the figure reads from bottom to top - of a deformation microstructure from a kink M K (1) whose height has become large enough to operate as a source (2-3) is sitting at every junction between a kink and a K W portion such as at a and b. At some stage, which is governed by the probability of cross slip in the cube plane, the screw part starts to transform into a KW lock, which is marked with a star between c and d in (3). Then the kinks ac and db slip apart. The former is too small to operate as a source, whereas bd is large enough. The same process as in (2-3) repeats in different places, giving rise to kinked KWs.
w
Microscopy and plasticity of the L12 "y~ phase
391
configuration (see however section 5.6.2.2.2 and fig. 74), thus anchoring the MK at both extremities (assuming that KWs are not bowed too much, Hirsch also proposed that the critical stress above which a MK is unstable should be smaller, by a factor of 6, than that for the operation of a Frank-Read source). Note that a MK could also operate as a source when the closing jogs (located at a, b, c, d in fig. 65) are significantly less mobile than the MK segment itself. Worthly of consideration is that when a kink operates as a source, the lateral part of the expanding dislocation is very often placed in dipolar situation with the adjacent outer immobile segments. It interacts, in particular, with its nearest own pre-existing branches that are located within or in the close vicinity of its slip plane. We shall see in the following section that this results in significant annihilation and/or immobilization of mobile superdislocation with immobile segments (section 4.3.2, see figs 36(b) and 66). As a consequence, a given kink cannot operate as a source more than once, by contrast with a regular source such as a Frank-Read or a pole source. It is remarkable that, in the microstrain stage (~ ..~ 10 -6 to 10-5), Ni3A1 deforms plastically at stresses which are one or two orders of magnitude less than the conventional flow stress "r0.2, while the WHR is dramatically high. This suggests that at a stress of say 10 -2 7-0.2, the amount of mobile dislocations which can be produced from the available one-cycle sources is not sufficient and that the microstructure exhausts its reservoir of mobile segments more rapidly than it generates kinks with appropriate lengths; still a large fraction of the strain is of elastic origin. In Ni3Al-based alloys 7-0.2 ranges between 10 -3 # and 10 -2 # in domain B. Thus multiplication should be restricted to those mobile segments whose length hMK is above 300 nm at room temperature and 30 nm at the peak. Within a freshly formed kinked screw line there must be some provision of adequate kinks for new sources, but it is not ensured that this provision is enough (that would explain the strong tendency for a primary octahedral slip to exhaust during primary creep). In order for the sample to deform plastically at a given strain rate, the stress must be raised to such a level that the rate of production of mobile segments is positive. We now suppose that MKs have some mobility in the primary octahedral plane as discussed in section 4.2.3.3 thus conferring the microstructure an additional degree of freedom. Then MKs larger than the critical length h~K are formed permanently by coalescence of pre-existing kinks of sub-critical size and multiplication is enhanced (we consider the coalescence of switch-over MKs which is the only one to yield fully glissile kinks, section 4.2.3.2.1). Therefore, increasing the external load at a given temperature enhances the mobility of kinks and favours the creation of new sources. As the temperature is increased, the bending introduces a larger effective friction against MK motion in the octahedral plane (section 4.2.3.3) and multiplication by kink coalescence is decreased. We conclude for this section that, even though the number of potential sources is finite, deformation is difficult because dislocation multiplication is largely inhibited in L12 alloys. Multiplication does not occur by bursts of superdislocations originating from the same source which itself operates repeatedly. Instead the operation of a source is restricted to the formation and expansion of one arc of superdislocation whose expansion in the slip plane is in addition hindered by strong interactions with the neighbouring superdislocations that belong to the same slip system (this is actually at variance from in-situ experiments where the avalanche production of dislocations has been reported
Ch. 53
P. V e y s s i k r e a n d G. S a a d a
392
(a)
simple MKs
switch-over MKs ii
........2- ' - t
IJ"
(b)
~_~sped
MK )
source 4~ mobile MK
(c)
.......................................................................................................................................................................................
i
i
i
source
!
_.r_.F a
(d)
-
................................................................ i
\
!
source
~i
dipoles --e
I
(e)
iii~.~
APB tubes & mixed dipoles I'--'~ ~-'l l ....
.!}
~ .... r-
r . .~ ypa
....
.l->-
~ i ! i
w
Microscopy and plasticity of the L12 ,yt phase
393
to occur at high temperatures (section 3.5). This discrepancy is another indication that in-situ experiments do not always reproduce the two-dimensional expansion of a source in its slip plane to a sufficiently long free-flight distance). 4.3.2. Dislocation annihilation That the density of KW locks per unit volume does not appear to increase dramatically with strain attests to some significant annihilation rate during deformation. There are at least three potential mechanisms that are liable to confer some mobility to a KW segment and/or to help its annihilation: (i) By mechanical transformation into a glissile superdislocation, itself entirely contained in the octahedral plane. This requires a level of stress (the microscopic saturation stress, s m, eq. (24)) which is comparable to the flow stress peak; (ii) By means of the lateral slip of MKs (Veyssi~re [3], Saada and Veyssi~re [206]). KW zipping, which is schematized in fig. 62, offers the advantage to avoid the prerequisite destruction of the KW lock into a form that would be glissile in the octahedral plane. In this mechanism of MK sliding, a length of KW equal to the distance over which the kink has slipped is simply transported over a distance equal to the MK height; (iii) By interaction with a mobile superdislocation with opposite Burgers vector. These two superdislocations need not belong to the same octahedral plane and the interaction is governed by their approach distance, that is, the distance between the two parallel octahedral planes. Mechanisms (i) and (ii) have already been addressed in section 4.2.3.3 and section 4.2.3.4, respectively. We now concentrate on mechanism (iii). Consider a mobile superdislocation that originates from a MK, operating itself as a source. As schematized in fig. 66, the expanding loop is subjected to a number of attractive interactions with neighbouring segments, including the adjacent kinked immobile parts of the parent superdislocation. The subsequent transformations are governed by varied approach distances (for one double cross-slip event involving an excursion over /jump in the cube plane, the approach distance for self-annihilation is ljumP/V/-3). In fact, depending upon the approach distance and on the applied stress, an immobile kinked KW line may be
- bypassed by the expanding superdislocation, totally annihilated, -
+___ Fig. 66. Multiplication and annihilation. (a) For the sake of clarity, it is assumed that switch-over and simple MKs are not formed on the same side of the expanding loop. They all move from left to right under the effect at the applied stress. (b) Cusped and coplanar MKs are formed upon coalescence of simple and switchover MKs, respectively. Only the latter can expand as a source, provided of course it is longer than the critical height h~K. (c) As the first source expands, it produces new kinked screw segments and sources. (d) and (e): Depending upon the shear stress and upon the cross-section of interaction (determined by/jump), dislocations may just proceed or annihilate totally or partially with the surrounding microstructure, and give rise to APB tubes as proposed by Chou et al. [208], and to mixed dipoles. The geometry of sessile segments implies that consecutive mixed dipoles exhibit aligned extremities (represented by dashed lines in (e)). In the bottom part of figure (e), the mobile segment, which is too far from the sessile KW, is by-passed. Some similarities between this figure and fig. 9 by Jumojni et al. [73] should be noted.
394
P. Veyssidre and G. Saada
Ch. 53
bound to the expanding superdislocation in the form of a stepped dipole, when the approach distance differs from 0. Screw parts are still liable to annihilate by cross slip. This annihilation may occur either completely or incompletely; in the latter case, the product takes the form of APB tubes according to a mechanism similar to that proposed by Chou et al. [208, 211] for B2 alloys, but which remains to be clarified for L lz alloys (see section 4.3.3.2). Hence, as the mobile segment expands, the prior distribution of obstacles that surrounded the kink source is replaced in part by a morphologically identical succession of screw and mixed dipolar branches (figs 65 and 66). Annihilation debris are shown in figs 36(b), 41 and 46. They have been recently studied at length by Shi et al. [284, 287]. Since a screw segment has a finite probability per unit length to undergo a double cross slip, both the mobile segment and the obstacles should be stepped over several parallel octahedral planes (section 4.2.3.2). Hence, the approach distance varies from place to place. Along the line of a given outer immobile superdislocation, mixed dipoles together with APB tubes and screw superdislocation dipoles - now of varied widths should again be formed in places. Elsewhere, the approach distance is too large for the mobile dislocation to be arrested. This is represented in fig. 66. As to the characteristic dimensions of mixed dipolar loops, it can be roughly inferred that the larger the applied stress, the narrower the dipoles. Only very narrow mixed superdislocation dipoles are stable against the levels of stress applied in domain B. In NiaAl-based alloys, the maximum distance between parallel octahedral slip planes, below which edge superdislocations dipoles are stable against the applied stress, is about 30 nm and 10 nm at 300 K and Tp, respectively. At room temperature, for instance, this implies that it is only when a superdislocation has achieved a total path in the cube plane of more than about 100b (as a result of several double cross-slip events, each involving a varied length/Jcurnp in the cube plane) that this part of the dislocation can proceed without being arrested by the segments of the same dislocation that belong to the octahedral plane where the motion of this superdislocation was initiated. The probability of immobilization by attractive interaction with pre-existing kinked KWs is increased as bending in the cube plane starts to proceed, that is, as the test temperature is increased in domain B 1. KWs that would be ineffective for the locking if they were rectilinear become part of the active obstacles as the approach distance is slightly varied by the effect of bending in the cube plane. However, when the bending gets pronounced, the obstacles are no longer nearly parallel to the octahedral slip plane but steeply inclined to this plane, so that interactions between the mobile dislocation and the surrounding obstacles become much weaker and gradually resemble those of a classical forest mechanism. 4.3.3. Further remarks on deformation debris
The nature of debris, especially their Burgers vectors, is representative of the interactions that are taking place during deformation. We shall restrict our discussion to debris resulting from the annihilation process within the primary slip system described in the previous section, that is, to mixed superdislocation dipoles and APB tubes. It is first worth noting that the presence of mixed dipoles suggests that cross slip has operated but deformation microstructures containing dipoles are not specific to L12
w
Microscopy and plasticity of the L12 .[t phase
395
alloys. In fact, mixed dipoles should be present in any material where cross slip and in particular double cross-slip may operate, since then a given dislocation is liable to share segments over several not too distant parallel slip planes for these segments to interact mutually. Regarding L12 alloys, every model of the flow stress anomaly involving a double cross slip - i.e., except point pinning models where dislocations are not transferred to a parallel neighbouring slip plane - predicts implicitly a distribution of dipolar loops as well as of APB tubes. Hence, the presence of either of these cannot be invoked in order to validate the details of a particular model. 4.3.3.1. Mixed superdislocation dipoles. Because of the particular microstructure of L12 alloys deformed in domain B, the distribution of superdislocation dipoles scales with that of pre-existing kinks (section 3.4.3.7.1). However, neither the structure of these dipoles which should be that of a prismatic superdislocation loop, nor their distribution, has ever been checked systematically by TEM (see however Shi et al. [287]). The mechanism of dipole pinching-off into loop rows may occur at relatively low temperatures since it requires pipe diffusion only. Point defects injected into the crystal as a result of dipole annihilation may help promote non-conservative processes among surrounding superdislocations, such as climb dissociation and APB dragging. As the test temperature is raised in domain B 1, mixed dipoles may still be formed, but they should disappear more readily from the microstructure, simply because diffusion becomes faster. Note that the extensive formation and annihilation of dipoles and the resulting production of point defects in large quantities explain the tendency of Ni3AI alloys towards disordering in localized shear bands [239]. 4.3.3.2. APB tubes. L12 alloys contain a distribution of narrow defects elongated along the screw orientation of the primary slip system, that show a typical faint contrast under specific diffraction vectors (section 3.4.3.6). So far, it is envisaged that these defects are result from the fact that the trailing superpartial does not follow the same path as the leading during the annihilation process of a mobile superdislocation with a locked screw segment (Chou et al. [208, 211], Sun [189], Hirsch [100]). In fact, the situation in L12 alloys is not as simple as that schematized for B2 alloys, since in order for the Chou et al.'s annihilation process to take place the two superdislocations should be lying on parallel habit planes according to the scheme of fig. 67. In the case of the annihilation of an incomplete or of a complete KW segment by interaction with a superdislocation gliding in an octahedral plane, one expects that in most cases annihilation occurs favourably by direct annihilation of the leading superpartials (fig. 67(c)), rather than through the double leading/trailing annihilation that is required to form an APB tube (fig. 67(d)). There is however a possible contribution of torque forces arising from anisotropic elasticity. The fact that APB tubes have also been observed in Ni3Fe (Ngan et al. [188]) raises additional questions as to the formation process of these defects. This particular alloy deforms essentially under cube slip with little evidence, if any, of cross slip and of the resulting locking of screw segments (Ngan et al. [31, 32]). According to Ngan et al. [188], the presence of APB tubes in Ni3Fe indicates that screw superdislocations may nevertheless annihilate by a local and limited cross slip (which, in this case, must
P. VeyssiOre and G. Saada
396 APB
(a)
Ch. 53 APB
X
\ APB
APB
w,,.
(b) ,A~
APB
APB
W
APB
A
K
i
APB
IP'
(c)
lw'
% % I % % % % - - D " - % %
~,L IP'
APB
IP'
% % % ,t
APB #
Fig. 67. The annihilation of screw superdislocation dipoles and the formation of APB tubes (after a mechanism designed for B2 alloys by Chou et al. [208]). (a) Two dissociated screw superdislocations are in position to annihilate mutually such that the leading superpartial of one meets with the trailing of the other, by cross slip directly on the appropriate plane. (b) In a more realistic mechanism, annihilation is allowed to occur through multiple cross slip; the path is determined by computer simulations according to priority rules for slip in the B2 lattice (for details, see [208]). In the L12 lattice, the annihilation proceeds onto two octahedral planes and one cube plane, with markedly different dislocation mobilities. When annihilation occurs between a mobile superdislocation and a KW lock, APB tubes may not be formed systematically. (c) In this configuration of the slip plane of the incident dislocation, the trailing superpartial should follow the same path as its companion and annihilation should be total. (d) In this configuration, the attractive force between the trailing superpartials may be larger than the surface tension that tends to drive them in the wake of the leading superpartials. Recent work by Hirsch [279] and by Paidar and Veyssi~re (unpublished) point to the role of elastic anisotropy in determining the trajectory of superpartials during annihilation, showing that APB tubes tend to form more systematically than expected from the above schemes.
w
Microscopy and plasticity of the L12 "y' phase
397
occur on octahedral planes). However, in the absence of long preexisting locked screw segments, the origin of rectilinear APB tubes, a few ~tm long, is difficult to explain based on the above cross-slip annihilation process. Concerning APB tubes, it is worth pointing out that according to the mechanism of formation proposed by Chou et al. [208], the mean APB tube and KW lengths should scale and they should be interrupted or terminated by mixed dipolar configurations, themselves scaling with the MKs in size and distribution. According to our experience, this is rarely the case: it is very rare to spot a mixed dipole that would unambiguously connect two finite APB tubes shifted with respect to one another. The features named APB tubes usually appear as extremely rectilinear. They are at least one order of magnitude longer than the average length of KW segments located in their immediate neighbourhood (section 3.4.3.6). This suggests that APB tubes might not formed as a result of the annihilation of a mobile superdislocation onto a sessile kinked KW lock but as a particular dragging process (see fig. 36 and the recent work of Shi et al. [286]). Finally, the APB tube width has been observed to coincide, at least within the resolution of the weak-beam method, with that of SISF dipoles to which they seem to be parallel (fig. 44). This suggests that those elongated SISF dipoles that lie in the primary slip plane, which are seen in large quantities in the deformation microstructure at moderate temperatures (section 3.4.3.8), may be more important than initially anticipated in order to understand the organization of the deformation microstructure. Unfortunately, SISF dipoles, which are intrinsically unstable, are still poorly documented. To date, APB tubes have only been analysed in terms of their properties of contrast. Their actual shape is not known and their distribution is not documented. Their mechanism of formation is not elucidated (see however [287, 288]).
4.4. Work hardening We may now address in a preliminary form the question of the unusually large workhardening rates that are measured in domain B (section 2.2.3). In f.c.c, metals, for instance, dislocation interactions result in WHRs of at most 5 x 1 0 - 3 # under a wellorganized multiple slip, that is, the most stringent mechanical conditions for these materials. This is to be compared to WHRs of the order of # / 1 0 at the peak of WHR in Ni3Al-based alloys (section 2.2.3). In the latter alloys, deformation is ensured by the operation of one-step sources at kinks, a process by which a kinked KW line destroys itself permanently after moving some finite distance. As deformation proceeds and because of the pronounced planarity of slip, such an expanding superdislocation will eventually encounter an obstacle with which it interacts in a manner that is specific to L12 alloys. This obstacle consists in a distribution of provisionally immobile kinked superdislocations that are distributed in the slip plane or in a plane parallel to the slip plane and located in its immediate neighbourhood. Since the obstacle consists in a continuous superdislocation line, the mobile superdislocation has a large probability to be blocked, and even to be annihilated over large fractions of its length by interactions with segments with opposite Burgers vector (fig. 66).
398
P. Veyssidreand G. Saada
Ch. 53
This dynamical behaviour can be contrasted to the situation met in "usual" alloys, where a collection of dislocations originating from the same source interact with trees interspersed some distance apart in the crystal; the mean resistance that arises from the microstructure is then inversely proportional to the inter-tree distance, which lies typically between 0.2 and 1 ~tm. Increasing deformation increases the forest density, that is, the stress that opposes dislocation motion. When the applied stress is maintained at a constant level, the density of mobile dislocations decreases. One may thus consider the work hardening rate as resulting from the increase in the internal stress due to the dislocation forest, and it is well known that the estimated work-hardening rate is in this particular case in good agreement with experiments. The situation in L12 alloys is quite different from the preceding: the mean free path of the superdislocation is rather small. In fact, it can be inferred from TEM observations that the blocking interaction should occur after a free flight distance of a few microns. This may be due either to the transformation of screw dislocations into KW, or, as has been pointed out above, by local interactions with neighbouring superdislocations. Then the exhaustion rate, therefore the work-hardening rate, is not the result of the increase of the long-range stress field of the deformation substructure. Note that, because of the occurrence of double cross-slip processes, a kinked screw superdislocation is in fact stepped over parallel octahedral slip planes with two consecutive KW segments being connected by a pair of jogs CJ whose length is of the order of /Jcump. When a mobile superdislocation interacts with a locked superdislocation consisting in a succession of KW segments, its expansion should be in principle hindered by two categories of interactions: (i) Dipolar interactions, with a strength of about #b/(27rl jump) per unit length of dislocation; (ii) Tree interactions with the jog pairs that are separated from the next by a distance equal to the KW length. Since both /jump and the density of jog pairs are small, it can be reasonably anticipated that the tree interaction is much less potent that the former. As the test temperature is increased, however, KWs tend to bend gradually in the cube plane (section 3.4.3.5). In a first stage, the bending is slight so that the tree interactions are not significantly modified but the cross section of blocking by dipolar interactions should increase. This would result in an increased WHR, conforming to the experimental observations (section 2.2.3). However, as the test temperature is increased further, that is, beyond a given mean radius of curvature, KW segments tend, on average, to escape in the cube plane, which results in a microstructure that resembles the usual discrete forest (sections 3.4.3.5 and 4.2.3.3), conforming to the "exhaustion" hardening model of Thornton et al. [41] (see section 5.3). Accordingly, the strength of interactions in octahedral slip should decrease, gradually adopting a magnitude corresponding to that of a forest mechanism. It is recalled that the peak of WHR does not coincide with the flow stress peak but is shifted towards lower temperatures, approximately at the level of the inflection point of r(T). The present qualitative description of dislocation interactions in domain B relies on two microstructurally-based properties: - the development of very potent dipolar obstacles in the immediate vicinity of each source;
w
Microscopy and plasticity of the L12 "[~ phase
399
the rapid decay of the density of mobile dislocations, itself due to the fact that sources can operate only once. In the absence of a quantitative analysis, it is however too early to conclude on the validity of the present tentative explanation of the origins of the work hardening in domain B. It has been also suggested [285] that the dragging of APB tubes by jogged mixed segments (fig. 40) may contribute quite significantly to the work-hardening rate. The drop of WHR at high temperature in domain B would coincide with the disappearance of APB tubes, known to occur at these temperatures (section 3.3.3.6). This description is in addition compatible with the property of flow stress reversibility (section 2.2.5), although this may not be obvious at first inspection since TEM observations indicate that the bent dislocations that have been formed during the prestraining stage do not disappear during the cooling stage (section 3.4.3.5). That, upon restraining, superdislocations do not experience the strengthening interactions with the bent superdislocations may in fact originate from the property that after modest permanent strain deformation has not proceeded evenly, so that superdislocations are not homogeneously distributed in the sample, even to the point that one encounters significant fractions of crystal containing no dislocations. Upon restraining, deformation would start at the edges of sheared zones and may expand more or less freely with respect to the high-temperature strengthening obstacles allowing only a limited effect of these, except maybe during some transient when the self-strengthening microstructure is not yet established in the free zones. When then prestrain amounts to a very large magnitude, samples are now filled with dislocations and the flow stress is no longer reversible.
4.5. Summary
and concluding remarks
As long as the radius of curvature of the nearly screw parts of an expanding superdislocation is less than a critical value of about 80 nm, these segments have a finite probability to cross slip onto the cube plane and to nucleate KW locks. Depending essentially on local stress conditions, the cross-slipped segment may be (i) Locked in an incomplete KW configuration that is given enough time to evolve gradually towards a complete one. The transition and the evolution of screw superdislocations towards the KW configuration depend critically on several factors, such as the relative value of the crystal parameters c~ and ff (section 4.2.2), a large lattice friction in the cube plane, - the ability to cross-slip in the cube plane, which is a consequence of the finite core size of superpartials.
-
Since the stress required to destroy a complete KW lock is of the order of the peak stress (section 4.2.3.4), the cross-slipped segment is permanently immobilized at any temperature within domain B. As the connecting mixed part slips in the octahedral plane, the locked screw segment increases its length by a zipping process, but it can be bypassed by this mixed part (fig. 59). This process of forming locked screw segments repeats itself as superdislocations expand, to give rise to a kinked KW configuration with varied kink heights. The macrokink height decreases with the probability of cross-slip
400
P. Veyssi~re and G. Saada
Ch. 53
into the cube plane at a given temperature. In this process where a double cross slip of screw segments is impeded, the macrokinks, named switch-over MKs, are coplanar, that is, the entire superdislocation expands in one single atomic octahedral plane (fig. 60(b)). Under the applied stress, these MKs may slide and coalesce to form larger MKs, which are themselves fully planar. The new MKs should move somewhat faster since the force applied to them is increased. (ii) Locked momentarily, that is, liable to undergo a double cross-slip process corresponding to an APB jump. This mechanism produces a pair of elementary kinks (EKs) with opposite signs bordering an incomplete KW. If the external stress is small enough, there is a finite probability for an APB jump to occur repeatedly. Otherwise, the superdislocation is freed in the octahedral plane. It then slips over some distance in this plane until it resumes cross slip. This produces simple MKs whose heights depends on the probability of cross slip in the cube plane at a given temperature. The higher the probability, the smaller the kink height. The cross-slipped distance is itself controlled by lattice friction: the larger the temperature, the longer/jump. The further evolution of KW segments is determined by the distance the leading superpartial has jumped in the cube plane, /Jcump, or equivalently by the distance between the two octahedral planes. Multiplication occurs at MKs longer than a critical length which is inversely proportional to the applied stress. Although each site of multiplication operates only once, superdislocation expansion may give rise to a number of kinks with a sufficient length (hMK > h~K) each of which can be activated as a source, which can operate only once, but multiplication still may not exceed exhaustion. This behaviour is similar to a microplastic deformation, where irreversible motion of dislocations occurs but where exhaustion is efficient enough in order to maintain the density of mobile dislocations at a low level. Another means for the microstructure to form new sources is by the coalescence of several mobile switch-over MKs with sub-critical heights. MK mobility is enhanced by the applied stress, the larger To, the larger the coalescence rate. This mechanism may help multiplication to overcome exhaustion under constant strain rate, but not under creep conditions. By contrast, the coalescence of two simple MKs does not contribute to producing new sources since the resulting MK is non-planar, exhibiting a pair of jogs at the points of impingement of the parent MKs. Note also that long mixed segments should be formed in order to provide the strain. It is likely that such segments exhaust rapidly. They are by nature included in kink-based models. As the temperature is raised, the bending of KW segments in the cube plane hinders MK mobility, which implies that the stress at which multiplication takes over exhaustion is increased. Beyond a certain temperature and/or a certain stress KWs are bent so far into the cube plane that kink coalescence does not operate any more. Under creep, this tends to favour slip on the cube cross-slip plane, while under constant strain-rate, this results in an increase of the applied stress so that kinks of smaller dimensions can be activated as sources. Meanwhile, the KW locks become less stable under stress, and multiplication may further occur upon destruction of incomplete KWs unless they have bent too far in the cube cross-slip plane, at which point octahedral slip is exhausted. As it stands, the present description of the possible microstructural evolution is not complete. It nevertheless helps understand the following puzzling experimental facts: (i) Since the multiplication rate is small and annihilation efficient, the microplastic stage is unusually extended [286].
w
Microscopy and plasticity of the L12 "7~ phase
401
(ii) The small strain-rate sensitivity in domain B. In fact, the dislocations that carry most of the strain are neither the screw parts, which can be considered as locked up to the peak, nor the MKs, whose motion can be significantly viscous (section 4.2.3.3), but the longest mobile mixed segments formed in the course of source activation at MKs. These mixed segments are intrinsically mobile (fig. 26) and can respond to a strain-rate jumps in the same way as dislocations do in usual f.c.c, metals. (iii) The fact that slip is localized implies that the rate of locking/annihilation with the surrounding microstructure is large it can provide some hint for the large work-hardening rate in domain B, that could nevertheless decrease as a forest tends to take over in the crystal, by gradual exhaustion of the KWs in the cube plane. The possible contribution of APB tube dragging to the WHR should nevertheless not be underestimated [285].
5. Analysis of theoretical contributions on the positive temperature dependence of the flow stress This section is aimed at discussing several models of the flow stress anomaly in L12 alloys which have been selected for their representativeness of the variety of theoretical approaches in this domain. In doing this we have deliberately chosen to account for the arguments of the various authors. As a consequence, this section may seem redundant, contradictory, or even inconsistent, since it reflects the outcome of about 30 years of imaginative work by many scientists on various hypotheses. A patient reader will find however an expression of our views at the end of each section.
5.1. Introduction
Despite the wealth of theories that have been designed in the recent years, the flow stress anomaly and related properties are far from being fully explained from a theoretical standpoint. This frustrating though challenging situation results from the repeated attempt to adapt to situations encountered in L12 alloys methods and reasonings that had been successful so far in the case of more usual crystals. Reference to the much better known case of f.c.c, crystals - which in addition we shall use to discuss the mechanism of cross slip of a superpartial- may help clarify this remark. In f.c.c, crystals, the fundamental microscopic mechanisms are well identified and the relevant physical parameters satisfactorily documented - except for the actual core structure of dislocations in high stacking-fault energy systems. The microscopic properties of f.c.c, crystals can be thus regarded as conceptually understood. On the other hand, thermally activated glide is described at a macroscopic level by means of a simple equation that, under some sound physical assumptions, represents the process and captures qualitatively its phenomenological behaviour. However, due to our insufficient knowledge of the core structure of dislocations and of its behaviour under stress, the precise calculation of the activation parameters is still not achieved. Nevertheless, its is quite generally acknowledged that the macroscopic plastic behaviour of f.c.c, crystals
402
P VeyssiOre and G. Saada
Ch. 53
is consistent with microscopic observations (dislocations interaction and distribution, insitu experiments, and slip-line distribution). In order to go one step further, that is, to model quantitatively the plastic flow of f.c.c, crystals, a complex dynamical problem has to be solved. With regards to the number of parameters, i.e., to the different types of interactions that are involved, a rigorous mathematical analysis of the phenomena seems beyond reach. Nevertheless, one may try to model the phenomenon with the help of numerical simulation. Despite some promising achievements (Devincre and Kubin [262]), a comprehensive modelling of the behaviour of f.c.c, crystals at a mesoscopic scale is still not available. For the following discussion on L12 alloys, it is worth noting in addition that in attempting to understand the mechanisms of plastic deformation of f.c.c, metals, the field has been obscured by a few inadequate methods such that using concepts beyond their limit of validity. This point is well illustrated by the crossslip models initially designed within the frame of linear elasticity and subsequently applied as a quantitative theory to dislocations exhibiting a splitting width of less than say 1 nm (Saada and Douin [137]; see also section 5.2.1); - fitting a single macroscopic observation to a microscopic analysis in view of establishing a correlation between theory and experiment. This argument can be exemplified by the several theoretical attempts at correlating the stress for the onset of stage III to the occurrence of cross-slip e v e n t - the so-called 7"ii I - - that have been made but only with limited success (Saada [263]); - failing to recognize that the proper function of experiments is to test theory, not to verify it (Popper [264]). -
Finally, it happens in many fields of research that uncertain analyses become legitimated by being inadequately quoted in subsequent papers. With regard to L 12 alloys, most of the debate on the causes of the flow stress anomaly has focused on properties of the KW configuration, that is, on the analysis of cross slip in the L12 structure. In a number of cases, the screw segments retain some mobility and the flow stress anomaly is assumed to be controlled by a pinning/unpinning process, itself governed by the local cross-slip locking of one of its superpartials. The presence of elongated though kinked KW segments and the organization of these, which characterize the deformation microstructure of L12 alloys in domain B so well, may be largely ignored or just regarded as dead debris (section 4.1, for a review see Pope [156]). In other analyses, some priority is given to accounting more closely to the deformation microstructure; the difficulty then consists in discriminating between the features that are relevant to the rate-controlling mechanism and the random debris. However, prior to getting to these analyses, we need to outline the way the orientation dependence of the flow stress has been modelled.
5.2. O n t h e e l e m e n t a r y
cross-slip process
There is ample experimental evidence that, at temperature below Tp, the flow stress in tension differs from that in compression. However, contrary to early beliefs, the
Microscopy and plasticity of the L12 7 ~ phase
w
403
orientation dependence of the flow stress is so much affected by composition that the properties that should be actually modelled are hard to define (section 2.2.6). To date, there has been a general consent that the TC asymmetry originates from the subsplitting of superpartials into two Shockley partials separated by a complex stacking fault. This subsection is aimed at questioning the theoretical foundations of this assumption that we have already criticised on experimental grounds in section 4.2.1.1.
5.2.1. The case of fc.c. crystals In f.c.c.-related structures, the elementary process of cross-slip between two octahedral planes is generally regarded as resulting from the creation of a constriction locally along a screw dislocation, then allowed to expand in the cross-slip plane (fig. 68(a)). Microscopic cross-slip mechanisms have been proposed by Schoeck and Seeger [265] and by Friedel [266]. The latter process, which appears to be the more realistic one, has been further developed in an analytical form by Escaig [267]. It is orientation-dependent, with its activation enthalpy differing according to whether the sample is deformed in tension or in compression. Under its Escaig formulation, the Friedel cross-slip model is a central ingredient that has been incorporated by Paidar, Pope and Vitek [86] to their modification of the original model of Takeuchi and Kuramoto [81], in order to predict the TC asymmetry of L12 alloys deformed in domain B. Obviously, deriving a correct estimate of the activation enthalpy is the critical step. As we shall see, this goal has not been achieved. In the Escaig formulation, notwithstanding the detailed process under study, the subtle and critical part of the calculation of the activation enthalpy requires a determination of the constriction energy Wc. This is achieved based on a general method outlined by (a)
Fig. 68. Cross-slip processes of dislocations with 51 (110) Burgers vector. (a) The Friedel mechanism in f.c.c. crystals. (b) In L12 alloys, the superpartial has to slip over b/2 in the cube plane before it reaches an atomic row where it is liable to spread in the cross-slip octahedral plane. The configuration is then highly asymmetric and this is even reinforced by the application of a stress.
404
P. VeyssiOreand G. Saada
Ch. 53
Stroh [268]. The analysis of the cross-slip process relies on a few approximations that are worth recalling (Saada [263], Duesbery et al. [269]): (i) In the calculation of Wc, the configuration under study does not have to be symmetric as assumed in Escaig's calculations. The corresponding uncertainty on the activation enthalpy associated with the asymmetry has been evaluated to about 40%. (ii) Wc cannot be calculated precisely for small separations, since it is critically affected by the choice of the cut-off radius. The uncertainty in this case may be as large as 100%. The smaller the dissociation width, the larger the error. (iii) The double constriction requires an energy Wd ~ Wc(1 -or/Ors), where Os has to be evaluated numerically [263]. (iv) How the theoretical cross-slip rate compares with the experimental plastic behaviour of crystals is not clearly established. Since constrictions may pre-exist on dislocations, the activation enthalpy may vary between Wd and Wd + Wc, that is, by a factor of about 2. Since the activation enthalpy enters in the calculations as the argument of an exponent, the theory can be regarded as quite flexible, to say the least. 5.2.2. L12 alloys Considering L12 alloys, it should first be noted that superpartials exhibit a core width of at most 2 to 3b, which implies a very small activation enthalpy even at low temperatures. In addition, predicting activation parameters for cross slip from linear elasticity, appears in this case very uncertain (Saada [263], Saada and Douin [137]). The elementary crossslip configuration envisaged in the PPV model differs indeed quite significantly from Friedel's mechanism in that the cross-slipped segment does not actually lie along the same atomic row as the rest of the screw segment but one atomic row farther, b/2, in the cube plane (fig. 68). In view of the above remark on the role of shape effects, one may anticipate that the intrinsic asymmetry of cross slip in L12 alloys should result in significant though unpredictable corrections to the activation enthalpy. Hence, utilizing Escaig's or an equivalent formalism in order to study the cross-slip transformation of a dissociated screw superdislocation in L 12 alloys that exhibit a positive TDFS, may at most provide a hint of what the actual cross-slip process could consist of. Such calculations are intrinsically quite flexible. We recall that the successive attempts at predicting the experimental orientation dependence of the flow stress through appropriate orientation factors, i.e., N, Q a n d / ( (eqs (2), (3) and (4), respectively) reveal themselves more or less inadequate to account for the wide variety of experimental responses (section 2.2.6). The fact that some correlation could be found between experiments and predictions based for instance on Q or K suggests that the hypothesis of stress-induced core deformations may be partly relevant in order to explain the orientation dependence of the flow stress but, in the present state, it appears impossible to tell which part of this hypothesis might be correct. The poor quality of the fits led Khantha et al. [101 ] to give up the idea of introducing a refinement such as the Friedel process in the description of cross slip, and to return to the treatment of Lall et al. [28]. Hence, designing a detailed theoretical model of this property turns out to be more complex than initially anticipated; it might well be pointless. Further theoretical effort is needed for this question and it might be worth exploring alternative mechanisms, not necessarily based on core effects. In this light, anomaly-related
w
Microscopy and plasticity of the L12 7 ! phase
405
macroscopic mechanical properties other than the violation of the Schmid law are worth considering in priority; these properties include the dramatic magnitude of the workhardening rate (section 2.2.3), the absence of strain-rate sensitivity (section 2.2.8), the partial reversibility of the flow stress with temperature (section 2.2.5) .... Finally, further calculations of core-controlled properties in L 12 alloys that have been conducted under linear elasticity should be mentioned. These have been achieved by Hirsch in order to compare the double jog and the critical bow-out mechanisms for cross slip and for unpinning (appendices 1, 2 and 3 in Hirsch [99]), by Schoeck [247] on a set of infinitely long dislocations; in this case the theoretical analysis was conducted within the frame of anisotropic elasticity in order to account for torque effects. In view of the above objections, Hirsch's results are evidently uncertain. On the other hand, Schoeck's work provides some unexpected information on properties of pinned configurations. According to his analysis, it is only when the separation between Shockley partials is less than about 1.3b that the critical part of the transformation of a glissile configuration into a PPV lock (e.g.,/Jcump - b/2, with the subsplitting occurring in the cross-slip octahedral plane, that is, from site "7 to site /3 in fig. 51) would be energetically favoured. Schoeck acknowledges, however, that in the particular case of a modest dissociation width the concept of linear elasticity whereon his calculation is founded clearly breaks down. Interestingly, when the subsplitting exceeds atomic dimensions, that is, when elasticity calculations become reliable, the PPV lock is always energetically, unfavourable. Hence, cross slip on the cube plane is still possible but by a step height/lcump = 2b/2, so that the superpartial subsplitting would not spread into the cross-slip octahedral plane (direct jump from 7 to 3', in fig. 51).
5.3. Miscellaneous models
Flinn [172] was the first author to propose a microscopic explanation of the increasing strength of L12 alloys. Though the premise of Flinn's model was r i g h t - e.g., a favourable APB energy on some atomic plane such as the cube plane favours the subsequent locking of superdislocations out of the primary octahedral p l a n e - the development of the model was incorrect. This is because the explanation was applied very generally to superdislocation lines with any character, so that climb had to be implied in their dissociation process. For dislocation motion to proceed, Flinn proposed a diffusion-controlled mechanism involving APB dragging. This mechanism is inappropriate because of its pronounced strain-rate sensitivity, its independence on thermal history, and of the fact that the positive TDFS is already significant at temperatures too low for diffusion processes to be important (Davies and Stoloff [74]). It is nevertheless interesting to note that (i) Flinn's analysis inspired the concept of the Kear-Wilsdorf lock, which is actually contained in Flinn's paper as a particular case of the dissociation out of the octahedral slip plane (in fact, the lowest APB energy was determined by Flinn to be a conservative APB in the cube plane, that is, that of a KW configuration),
406
P. VeyssiOre and G. Saada
Ch. 53
(ii) climb dissociation and APB dragging have been identified in L 12 alloys deformed in domain B2 (section 3.1.2.6 and 3.4.3.6). Davies and Stoloff [74] and Johnston et al. [75] suggested that the lattice opposes an increasing resistance to dislocation motion as temperature is raised, but they did not address the actual origin of lattice/dislocation interaction. Subsequently, Copley and Kear [67] proposed that this latter effect could originate from the subsplitting of superpartials in the primary slip plane which, as a result of the temperature dependence of lattice constants, would decrease in magnitude as temperature is increased. In the frame of a Peierls-Nabarro model of lattice friction, the narrowing of dislocation cores implies an increasing slip difficulty. Copley and Kear's hypothesis has been ruled out quite elegantly by the experiment on the strain dependence of the flow stress anomaly performed by Thornton et al. [41] (section 2.2.4, fig. 6). At variance from the ideas and concepts that were prevailing at that time and in order to account for differences observed especially in strain-rate jump transients (section 2.2.8.1), the latter authors proposed the first two-stage mechanism for the positive TDFS. Below about 400~ that is, approximately the temperature of the inflection point of r ( T ) , the density of mobile dislocations decreases as a result of the increasing production of KW locks. Above 400~ where cube-slip activity becomes significant, dislocations of the primary slip plane exhibit an increasing frequency of intersections with trees that belong to the cube cross-slip plane. These low and high temperature mechanisms are known under the names of "exhaustion" and "debris" hardening mechanisms, respectively [41 ]. The "debris" hardening mechanism was later criticized by Staton-Bevan and Rawlings [30, 89], on the grounds that cross slip on the cube plane is not expected in samples deformed along [001] 19 and along []-23]. The "debris" hardening mechanism has nevertheless been supported since by Korner [204] as a result of careful weak-beam TEM observations. In fact, in order for the hardening intersection mechanism to take place, cube slip need not operate extensively, and it seems difficult to rule out the idea of a "debris" hardening based on the argument that cube slip is suppressed on a macroscopic scale. Rather, significant KW bending appears to be sufficient to provide a large density of trees, as observed in L 12 alloys deformed in domain B2. However, there remains the problem of accounting for the anomalous work-hardening rate in domain B 1 and of its peak at the temperature where the flow stress undergoes an inflection. Almost every subsequent approach to the flow stress anomaly is based on some a priori point of view on the motion of a dissociated screw superdislocation in view of its eventual elimination in the crystal either at a free surface or else by annihilation with another superdislocation. This is what we shall examine in the following sections. 19Some controversy will probably remain as to how accurately a given orientation such as [001] can be controlled in a real deformation experiment. Furthermore, it cannot be ensured that the stress tensor does remain the same throughout the sample. In fact, whatever the precision of sample orientation, local rotations due to crystal imperfections or after some permanent strain, due to end-effects (section 2.2.2) or else to local stress concentrations (section 3.4.2) should be enough in order to promote some bending in the cube plane. It is almost hopeless to suppress local cube slip by accurately orientating samples along [001], but it can be significantly inhibited.
w
Microscopy and plasticity of the L12 7' phase
407
S.4. Local pinning models Early theories of the positive TDFS (Takeuchi and Kuramoto [81], Lall et al. [28], Paidar et al. [86]) rely on the hypothesis that plastic deformation is controlled by the locking rate of a screw dislocation, whose motion is arbitrarily forced to be steady (section 5.4.1). Based on otherwise reasonable assumptions, these pinning models were aimed at reproducing the variation of the flow stress with temperature, the violation of the Schmid law and facets of the tension/compression asymmetry. However, as we shall see in the following (sections 5.4.1 and 5.4.2), none of these early theories accounts for the actual strain-rate sensitivity of the flow stress (Hirsch [98, 99], Nabarro [250]) nor do they explain transient tests (Hemker et al. [54]). More recently, Vitek and Sodani [246] and Khantha et al. [88, 101,260] have re-visited the TK and PPV theories in an attempt to circumvent their most serious conflicts with macroscopic properties (sections 5.4.3 and 5.4.4). We now outline and discuss this family of models based on the point-pinning mechanism; several of the following arguments are inspired by the papers of Dimiduk [49], of Hirsch [98-100] and of Nabarro [250]. 5.4.1. Takeuchi and Kuramoto (1973) The Takeuchi and Kuramoto (TK) model [81 ], which is the first comprehensive statement of the local pinning models, can be summarized as follows. Consider a screw superdislocation containing a distribution of pinning points located a distance g apart according to fig. 69. Under the effect of the applied stress To, the dislocation bows out at a velocity (30)
V = "rob/B
until its centre is displaced by a distance d, at which point breakaway occurs. Since d scales with g, which itself scales with the breakaway stress To, the time d / V required to reach the breakaway position scales with (To)2. Assuming that the pinning process is steady allows one to write s ~ d/V = 1
(31)
where u is the frequency per unit length of dislocation of pinning point formation by cross slip. The TK model is founded on the assumption that breakaway is steady. In its original form, it does not address the question of the strain-rate sensitivity. Equation (31) simply states that the yield stress is the stress required to create a steady density of mobile dislocations, e.g., one pinning point is formed over the length t during the time required to travel over d. It implies in turn that the frequency u scales with ('to) 3. As a result, when it is assumed that the point pinning is thermally-activated, then
% -- A exp
HI } 3kT
(32)
408
Ch. 53
P. Veyssidre and G. Saada
I iiii,,'"'"'"'"'"'"'""'"'"'"'"'"'"'"'""'"'"'"'"'"'"'"", iiiIII,'"'"'"'"'"'"""""'"'"'"'""'"'"'"'"'"' '/ '"'""'""'"'"'tlll till',,' ill!o,'""'"'"'"'"'"'"""'"'"'"" r v Y
-'V"-
,,
'
|~
g
A
-X
;
i1,,,'-|
Fig. 69. The dynamical breakaway mechanism proposed by Takeuchi and Kuramoto [81]. The black and dotted grey lines represent the dislocation positions at two consecutive steps of the breakaway process.
where A is a constant that has been estimated by Takeuchi and Kuramoto (TK), and H1 is the activation energy for pinning 2~ which, in general is expressed as H1 -- H ~ - - T c V . From their experimental data, Takeuchi and Kuramoto obtained sound values for the enthalpy (H1~ - 0.3 eV) and for the activation volume (v = 6b3). Given the temperature and the shear stress in the cube cross-slip plane, HI determines a distribution of equidistant pinning points. It is the fact that the frequency of cross-slip events, that is, the density of pinning points per unit length of screw dislocation, increases with temperature that causes the the flow stress increase. Hemker et al. [54] argued that expression (32) implies the existence of a critical stress for octahedral slip, To. Below To dislocation motion is automatically impeded by KW formation while, above that stress, dislocations break away from the pinning points with no difficulty and they move subsequently at the free flight velocity. Hemker et al. [54] pointed out that the TK model is inherently inconsistent with the absence of a critical stress for octahedral slip in domain B as can be deduced from the shape of stress/strain curves and from the fact that primary creep on the primary octahedral plane can be activated at stresses below 7o.2. In view of the apparent activation energies listed by Liang and Pope [270], Nabarro [250] argued that, e v e n at r o o m t e m p e r a t u r e , all segments will readily jump between sessile and glissile configurations, with the majority in the low-energy sessile state. This prediction, deduced from the TK model, is clearly at odds with the premise of the model. Even more serious is Hirsch's remark [98] that taking B - 5 x 10 -5 Pas, To - 200 MPa and a strain-rate of 10 -4 s -1 yields a density of mobile dislocations of approximately Pm ~ 2 x 10 .2 cm -2. As pointed out by Hirsch [98], the stress dependence of the dislocation velocity postulated by Takeuchi and Kuramoto (eq. 30) implies that the strain rate depends linearly on stress, which conflicts severely with the fact that the flow stress is actually insensitive to strain-rate changes (section 2.2.8). Note that the failure of Flinn's explanation to account for the absence of strain-rate sensitivity of the flow stress has been regarded as sufficient to rule out this explanation (Pope and Ezz [271 ]) but, for some reason, this argument has not been used against the Takeuchi and Kuramoto theory, nor against the refinements of this theory achieved by Lall et al. [28] and by Paidar et al. [86]. 2~ should be aware that in most subsequent experimental studies of the mechanical behaviour of L12 alloys deformed in domain B, the activation energy for pinning has been estimated through eq. (32).
Microscopy and plasticity of the L12 "y~phase
w
409
Finally, the pinning point model requires implicitly that the cross-slipped segments exhibit some stability against stress; the stability in question is measured by the critical angle q~ beyond which breakaway occurs (fig. 69). This question has been discussed quantitatively by Hirsch [100]. In view of the predicted distances between pinning points and of the levels of stress that are applied in practice over domain B, the idea that the local pinning event could be strong enough to provide the desired strengthening appears to be unplausible.
5.4.2. Paidar, Pope and Vitek (1984) Paidar et al. [86] have made a gallant attempt at using what was available on crossslip calculations (section 5.2.1) in order to re-evaluate the activation enthalpy of the TK model. By introducing the asymmetrical cross-slip behaviour of a screw dislocation dissociated into Shockley partials, these authors were able to account for the TC asymmetry that was documented at that time. Paidar et al.'s method follows that introduced by Lall et al. [28] wherein the details of the pinning process are modified. As discussed at length earlier, the fact that examples of a significant disagreement between the PPV predictions and TC asymmetry experiments are at least as numerous as those of agreement (section 2.2.6) should be taken into consideration. The PPV model suffers from the same drawbacks as the original TK model in that it cannot account satisfactorily for the kinetic aspects (section 5.4.1). The PPV expression of the activation enthalpy has been subsequently corrected for anisotropic elasticity by Yoo [255]. In view of the arguments developed in section 5.2 on the validity of elastic calculations of the cross-slip process in the limit of small splittings, the claimed agreement between the orientation dependence of the flow stress predicted by the PPV model and experimental observations is certainly not as convincing as one would expect considering the widespread acceptance of the model. 5.4.3. Vitek and Sodani (1991) Several new theories including some of the most recent approaches the flow stress anomaly have followed the suggestion that the pinning process should be very fast compared to the unpinning process (de Bussac et al. [245]). The model proposed by Vitek and Sodani [246] is one of the earliest of this series. These authors have designed an analytical formulation, again relying on the pinning point principle, that is aimed at accounting for the absence of strain-rate sensitivity. In doing so they ignored, without justification, the in-situ observations of Caillard et al. [190] that indicate quite clearly that the motion of a screw superdislocation does not proceed through a steady mechanism of local pinning/unpinning. The Vitek and Sodani form of the TK-PPV model relies on the following assumptions (i) as postulated in TK-PPV, the distance g between two consecutive pinning points is a decreasing function of temperature of the form -goexp
~
,
(33)
where go is of the order of 2 to 3b. In fact, this expression is equivalent to assuming that the density of pinning points is under thermal equilibrium, which offers the advantage
410
P. VeyssiOreand G. Saada
Ch. 53
of implicitly accounting for the thermal reversibility of the flow stress (section 2.2.5). As emphasized by Hirsch [98], this assumption cannot however be justified by any thermodynamic consideration. (ii) the unpinning process is thermally activated, with an activation enthalpy Hu Hu
-
H ~ -
~-[b2e,
(34)
where ToT is the temperature-dependent part of the stress. The strain rate is given by the usual expression g = io exp
-
gu }
(35)
and "r[ = k T l n { ~ / ~ o } + H ~
bZgoexp { ~ }
(36)
The model reproduces the positive TDFS in compression (the orientation dependence of the flow stress is by nature similar to that of the PPV model) and the small strain-rate sensitivity, up to 800 K. As for the TK model, expression (36) implies the existence of a critical stress for octahedral slip. Using the expression of the activation enthalpy for pinning derived by Paidar et al. [86], but corrected for the torque effect, Vitek and Sodani [246] published a value of Hi~ = 4.9 x 10 -19 J, at zero stress. This yields a distance between consecutive obstacles that is far too large. This was a misprint which was corrected in a subsequent paper (Sodani and Vitek [272]) where, with an activation enthalpy for pinning ten times smaller than in the previous work, the distance between pinning points below 500-600 K is still too large to make sense (> 10 gm). 5.4.4. Khantha, Cserti and Vitek (1992, 1993)
Khantha et al. [88, 101, 260] have produced a model in which "the deformation is mainly controlled by the mobile screw dislocations moving on the (111) planes" and which relies on the existence of localized pinning points. This model differs from those of Paidar et al. [86] and of Sodani and Vitek [272] in many respects. In order to correct for discrepancies between the predictions of the PPV model and experimental measurements of the orientation dependence of the flow stress (section 2.2.6), Khantha et al. [88, 101] postulated that both the pinning and the unpinning mechanisms are thermally activated. Regarding the pinning process, the general trend is about the same as in the PPV model with the following modifications: (i) The derivation of the constriction energy and in particular, that of the saddle-point configuration, involves only the constriction in the primary slip plane. In other words, the role of the subsplitting in the octahedral cross-slip plane, which constitutes one of the most remarkable differences between the model of Paidar et al. [86] and that of Lall et al. [28], is no longer included.
w
Microscopy and plasticity of the L12 "y' phase
411
(ii) The effect of ternary additions is accounted for through an ad-hoc adjustable parameter/3c, whose value is zero for binary alloys. As to the unpinning mechanism, Khantha et al. [88, 101] consider that the breakaway process may take place simultaneously over more than one pinning point (fig. 70(a)). The technique of calculation is inspired from methods developed in the treatment of internal friction. The method is based on the idea that the release through a different path is necessitated whenever, along a given path, repinning by backward motion from the saddle point configuration becomes more likely than the forward transition. Since the breakaway path depends on the applied stress, different processes are liable to occur according to the applied stress, or, equivalently, according to the test temperature. As a result, the activation volume exhibits discontinuities at each transition between one breakaway path and the next (fig. 70(b)). The breakaway process is still postulated to occur steadily, which introduces a correlation between the processes of pinning and unpinning. In practice, this is inserted in the equations through the condition d = g, which, at high temperatures, is quite arbitrarily replaced by d = g/3 for a better fit. Not only did Khantha et al. obtain a good fit with the data on activation volumes but, by inserting these into the rate eq. (35), they were able to derive a variation of the flow stress with temperature in excellent accord with experimental data in Ni3A1, Ni3(A1, Ta) and Ni3Ga (although in the latter, the chosen value of the APB energy on the cube plane of 20 mJ m -2 is certainly much too small). So far, Khantha et al.'s model represents the most elaborate version within the pointpinning theories. It relies on the assumption of a steady-state breakaway process as the result of the "balance between pinning and unpinning rates" although it is "governed by two thermally-activated processes with different waiting times". It is stated that "the pinning process ... is dominant over the unpinning process," but then these authors force the rate of pinning to be equal to that of unpinning in order to avoid exhaustion. It is finally considered that the flight distance scales with the distance between pinning points, which seems rather arbitrary. Regarding comparison with experiments, it must be noted that the in-situ observations of Caillard et al. [190] together with those of Mol6nat and Caillard [196] are ignored. In Khantha et al.'s work, the comparison with experiments focuses on the activation volume (Khantha and Vitek [260]) which, in Ni3(A1, Ta) does show some anomalous behaviour between domains B1 and B2 (section 2.2.8.3). At variance from the PPV and the SV point-pinning models, the necessity of accounting for the orientation dependence of the flow stress is no longer given top priority in Khantha et al.'s thesis. As a matter of fact, the ultimate expression of the activation enthalpy is now much more similar to the Lall et al. form [28]. This enthalpy is flexible in order to account for the wide diversity of mechanical behaviour as alloy composition is varied (section 2.2.6). It should be noted that, as in the previous point pinning models, that of Khantha et al. implies the existence of a critical stress for an octahedral slip within a given sequence of major breakaway which, as emphasized by Hemker et al. [54], is in marked contradiction with experiments (especially with the activation of an octahedral slip in primary creep below to.2 within he range of temperature of domain B, section 2.2.9). Finally, the very recent work of Sp~itig [282] suggests that there is no discontinuity of the activation volume in Ni-rich Ni3A1 and Ni3(A1, Hf), so that introducing the multiple breakaway concept loses partly its relevance.
412
Ch. 53
P. Veyssidre and G. Saada
(a) !
S
......
-~- ...............
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(13)
(b)
30OO
E 0
>
2000
-
O
m=l
> O
tr, Consid~re criterion) and this represents an intrinsic limitation to ductility. - We have no systematic knowledge of the effects of alloying and of the deviation from the stoichiometric Ni/A1 ratio. These effects have been shown to be indeed significant and difficult to correlate quantitatively with macroscopic quantities relevant to the flow stress anomaly such as the flow stress, the peak stress and with the orientation dependence of these. It could be that the motion of EKs or CJs would be affected by their interaction with point-like obstacles (point defects or impurity atoms), but the extent of this effect is unpredictable. - A s it stands, the kink-based interpretation presently proposed for L12 alloys might be generalized to situations of a positive TDFS in other ordered structures, when the locking of screw segments is similarly quite irreversible. It appears quite clear that, contrary to a belief that dominated research on L12 alloys over the eighties, the questions of the orientation dependence of the flow stress and of the tension-compression asymmetry are not of primary importance in the understanding of the positive TDFS. - It is probably time that several hypotheses on which reliance has been placed, such as a steady dislocation behaviour, a mechanism based on point-pinning, a quasi reversible locking-unlocking process, a critical stress for octahedral glide, be carefully reconsidered. -
-
Acknowledgements
We are glad to acknowledge the contributions of Drs J. Oliver, A. Korner and M. Mills for having provided us with illustrations, Dr X. Shi for having allowed us to use part of her PhD results, K. Hemker, D. Dimiduk, J. Douin and F.R.N. Nabarro for careful and critical review of the manuscript and B. Willot for invaluable technical help.
Microscopy and plasticity of the L12 7 ~ phase
435
References [1] V. Vitek, in: Dislocations in Real Materials, ed. M.H. Loretto (The Inst. of Metals, London, 1985) p. 30. [2] M.S. Duesbery, Dislocations in Solids, Vol. 8, ed. F.R.N. Nabarro (North-Holland, Amsterdam, 1989) p. 67. [3] P. Veyssi6re, MRS Symp. Proc. 213 (1989) 175. [4] P. Veyssi6re, ISIJ Int. 31 (1991) 1028. [5] P. Veyssi6re, in: Intermetallic Compounds- Structure and Mechanical Properties, ed. O. Izumi (Japan Inst. of Metals, Sendai, 1991) p. 745. [6] H. Saka and Y.M. Zhu, Philos. Mag. A 51 (1985) 629. [7] Y.M. Zhu and H. Saka, Philos. Mag. A 59 (1989) 661. [8] A. Nohara, in: Intermetallic Compounds- Structure and Mechanical Properties, ed. O. Izumi (Japan Inst. of Metals, Sendai, 1991) p. 561. [9] G. Dirras, P. Beauchamp and P. Veyssi6re, Philos. Mag. A 65 (1992) 815. [10] M. Yoshida and T. Takasugi, Philos. Mag. A 68 (1993) 401. [11] A. Franqois and P. Veyssi6re, Intermetallics 2 (1994) 9. [12] K.S. Kumar, in: Int. Symp. on Structural Intermetallics, Champion, USA, eds R. Darolia, J.J. Lewandowski, C.T. Liu, P.L. Martin, D.B. Miracle and M.V. Nathal (The Minerals, Metals and Materials Soc., Warrendale, 1993) p. 87. [13] D.G. Morris and S. GUnther, Philos. Mag. A 68 (1993) 81. [14] D.G. Morris, S. GUnther, R. Lerf and J.C. Joyce, in: Int. Symp. on Structural Intermetallics, Champion, USA, eds R. Darolia, J.J. Lewandowski, C.T. Liu, P.L. Martin, D.B. Miracle and M.V. Nathal (The Minerals, Metals and Materials Soc., Warrendale, 1993) p. 97. [15] Z.I. Wu and D.P. Pope, in: Int. Symp. on Structural Intermetallics, Champion, USA, eds R. Darolia, J.J. Lewandowski, C.T. Liu, P.L. Martin, D.B. Miracle and M.V. Nathal (The Minerals, Metals and Materials Soc., Warrendale, 1993) p. 107. [16] T. Suzuki, Y. Mishima and S. Miura, ISIJ Int. 29 (1989) 1. [17] D.-M. Wee, O. Noguchi, Y. Oya and T. Suzuki, Trans. JIM 21 (1980) 237. [18] T. Suzuki and Y. Oya, J. Mater. Sci. 16 (1981) 2737. [19] E.M. Schulson and J.A. Roy, Acta Metall. 26 (1978) 29. [20] T. Suzuki, Y. Oya and S. Ochiai, Metall. Trans. A 15 (1984) 173. [21] D.-M. Wee and T. Suzuki, Trans. JIM 22 (1981) 163. [22] R.A. Mulford and D.P. Pope, Acta Metall. 21 (1973) 1375. [23] C. Bontemps-Neveu, Th6se de Doctorat (Univ. de Paris-Sud, Orsay, France, 1991). [24] T. Suzuki, Y. Oya and D.-M. Wee, Acta Metall. 28 (1980) 301. [25] Y. Liu, T. Takasugi, O. Izumi and S. Ono, Philos. Mag. A 59 (1989) 401. [26] T. Suzuki, Y. Oya and S. Ochiai, Metall. Trans. A 15A (1980) 173. [27] D.-M. Wee, D.P. Pope and V. Vitek, Acta Metall. 32 (1984) 829. [28] C. Lall, S. Chin and D.P. Pope, Metall. Trans. A 10 (1979) 921. [29] T. Takasugi and M. Yoshida, Philos. Mag. A 65 (1992) 613. [30] A.E. Staton-Bevan and R.D. Rawlings, Phys. Status Solidi A: 29 (1975) 613. [31] A.H.W. Ngan, I.P. Jones and R.E. Smallman, Philos. Mag. A: 65 (1992) 1003. [32] A.H.W. Ngan, I.P. Jones and R.E. Smallman, Mater. Sci. Eng. A 153 (1992) 387. [33] C.L. Fu, Y.-Y. Ye and M.H. Yoo, MRS Symp. Proc. 288 (1993) 21. [34] R. Lowrie, Trans. MetaU. Soc. AIME 194 (1952) 1093. [35] J.H. Westbrook, Trans. Metall. Soc. AIME 209 (1957) 898. [36] Curwick, PhD Thesis (Univ. of Minnesota, USA, 1972). [37] H.F. Heredia, PhD Thesis (Univ. of Pennsylvania, USA, 1990). [38] H.F. Heredia and D.P. Pope, Acta MetaU. Mater. 39 (1991) 2027. [39] T. Takasugi and M. Yoshida, Philos. Mag. A 67 (1993) 447. [40] S.S. Ezz, D.P. Pope and V. Paidar, Acta Metall. 30 (1982) 921. [41] P.H. Thornton, R.G. Davies and T.L. Johnston, Metall. Trans. 1 (1970) 207. [42] Y. Mishima, S. Ochiai, M. Yodogawa and T. Suzuki, Trans. JIM 27 (1986) 41.
436
P. VeyssiOre and G. Saada
[43] T. Khan, E Caron and S. Naka, in: High Temperature Aluminides and Intermetallics, Vols 35-36, eds S.H. Whang, C.T. Liu, D.P. Pope, J.O. Stiegler (The Minerals, Metals and Materials Soc., Warrendale, 1990) p. 219. [44] D.M. Dimiduk, S. Rao, MRS Symp. Proc. 213 (1991) 499. [45] O. Noguchi, N. Oya and T. Suzuki, Metall. Trans. A 12 (1981) 1647. [46] D.M. Dimiduk, PhD Thesis (Carnegie-Mellon University, Pittsburg, USA, 1989). [47] Y. Mishima, S. Ochiai, N. Hamao, M. Yodogawa and T. Suzuki, Trans. JIM 27 (1986) 648. [48] D.M. Dimiduk, A.W. Thompson and J.C. Williams, Philos. Mag. A 67 (1993) 675. [49] D.M. Dimiduk, J. Phys. (Paris) III 1 (1991) 1025. [50] K.J. Hemker and M.J. Mills, Philos. Mag. A 68 (1993) 305. [51] S. Miura, Y. Mishima and T. Suzuki, Z. Metall. 80 (1989) 48. [52] S. Miura, Y. Mishima and T. Suzuki, Z. Metall. 80 (1989) 157. [53] S. Miura, Y. Mishima and T. Suzuki, Z. Metall. 80 (1989) 164. [54] K.J. Hemker, M.J. Mills and W.D. Nix, J. Mater. Res. 7 (1992) 2059. [55] C. Bontemps, P. Veyssi6re and G. Saada, Scr. Metall. Mater. 24 (1990) 2385. [56] S.S. Ezz, D.P. Pope and V. Vitek, Acta Metall. 35 (1987) 1879. [57] Y. Umakoshi, D.P. Pope and V. Vitek, Acta Metall. 32 (1984) 449. [58] A.E. Staton-Bevan, Philos. Mag. A 47 (1983) 939. [59] A.C. Arko and Y.H. Liu, Metall. Trans. 2 (1971) 1975. [60] E. Kuramoto and D.P. Pope, Philos. Mag. 34 (1976) 593. [61] P. Sp~itig, J. Bonneville and J.-L. Martin, J. Phys. (Paris) III 4 (1994) 1017. [62] Y.Q. Sun, PhD Thesis (Univ. of Oxford, 1990). [63] B.H. Kear and H.G.F. Wilsdorf, Trans. Metall. Soc. AIME 224 (1962) 382. [64] B.H. Kear, Acta Metall. 12 (1964) 555. [65] B.H. Kear, Acta Metall. 14 (1966) 659. [66] M.J. Marcinkowski, Phys. Status Solidi A: 90 (1985) 621. [67] S.M. Copley and B.H. Kear, Trans. Metall. Soc. AIME 239 (1967) 977. [68] N. Baluc, J. Bonneville, K.J. Hemker and J.L. Martin, R. Schatiblin and P. Sptitig, Mater. Res. Eng. A 164 (1993) 379. [69] S.S. Ezz, Y.Q. Sun and P.B. Hirsch, in: High Temperature Ordered Intermetallics VI, MRS Symp. Proc. 364 (1995) 695. [70] G. Saada and P. Veyssi6re, in: Int. Symp. on Structural Intermetallics, Champion, USA, eds R. Darolia, J.J. Lewandowski, C.T. Liu, P.L. Martin, D.B. Miracle and M.V. Nathal (The Minerals, Metals and Materials Soc., Warrendale, 1993) p. 379. [71] S.S. Ezz and P.B. Hirsch, Philos. Mag. A 69 (1994) 105. [72] A. Couret, Y. Sun and P.B. Hirsch, Philos. Mag. A 67 (1993) 29. [73] K. Jumojni, S. Ueta, M. Kato and A. Sato, Philos. Mag. A 69 (1994) 1111. [74] R.G. Davies and N.S. Stoloff, Trans. Metall. Soc. AIME 233 (1965) 714. [75] T.L. Johnston, A.J. McEvily and A.S. Tetelman, in: High Strength Materials, ed. V.E Zackay (Wiley, New York, 1965) p. 363. [76] P. Spfitig, J. Bonneville and J.-L. Martin, MRS Symp. Proc. 288 (1993) 429. [77] M.H. Yoo and C.T. Liu, J. Mater. Res. 3 (1988) 845. [78] W.E. Dowling and R.G. Gibala, MRS Symp. Proc. 133 (1989) 209. [79] K.J. Hemker, W.D. Nix and M.J. Mills, Acta Metall. Mater. 39 (1991) 1901. [80] D.M. Dimiduk and T.A. Parthasarathy, Philos. Mag. Lett. 71 (1995) 21. [81] S. Takeuchi and E. Kuramoto, Acta Metall. 21 (1973) 415. [82] K. Aoki and O. Izumi, Acta Metall. 26 (1978) 1257. [83] T. Saburi, T. Hamana, S. Nenno and H.-R. Pak, J. Appl. Phys. 16 (1977) 267. [84] E. Kuramoto and D.P. Pope, Acta Metall. 26 (1978) 207. [85] T. Takasugi, S. Watanabe, O. Izumi and N.K. Fat-Halla, Acta MetaU. 37 (1989) 3425. [86] V. Paidar, D.P. Pope and V. Vitek, Acta Metall. 32 (1984) 435. [87] H.F. Heredia and D.P. Pope, J. Phys. (Paris) III 1 (1991) 1055. [88] M. Khantha, J. Cserti and V. Vitek, Scr. Metall. Mater. 27 (1992) 481. [89] A.E. Staton-Bevan and R.D. Rawlings, Philos. Mag. 32 (1975) 787.
Microscopy and plasticity of the L12 ~/i phase
437
[90] J. Bonneville and J.-L. Martin, MRS Symp. Proc. 213 (1991) 629. [91] S.J. Basinski and Z.S. Basinski, in: Dislocations in Solids, Vol. 4, ed. F.R.N. Nabarro (North-Holland, Amsterdam, 1979) p. 261. [92] S.S. Ezz and P.B. Hirsch, in: High Temperature Ordered Intermetallics VI, MRS Symp. Proc. 364 (1995) 35. [93] J.D. Meakin, Can. J. Phys. 45 (1967) 1121. [94] B. Escaig, J. Phys. (Paris) 35 (1974) 151. [95] P. Groh and R. Conte, Acta Metall. 19 (1971) 895. [96] P. Sp~itig, J. Bonneville and J.-L. Martin, in: Strength of Metals, ICSMA 10, eds H. Oikawa, K. Maruyama, S. Takeuchi and M. Yamaguchi (Japan Inst. of Metals, Sendai, 1994) p. 353. [97] N. Baluc, H.P. Karnthaler and M.J. Mills, in: Institute of Physics Conf. Series, Vol. 2, No. 93, (Institute of Physics, Bristol, 1988) p. 463. [98] P.B. Hirsch, J. Phys. (Paris) III 1 (1991) 989. [99] P.B. Hirsch, Philos. Mag. A 65 (1992) 569. [100] P.B. Hirsch, MRS Symp. Proc. 288 (1993) 33. [101] M. Khantha, J. Cserti and V. Vitek, Scr. Metall. Mater. 27 (1992) 487. [102] P. Sp~itig, J. Bonneville and J.-L. Martin, MRS Symp. Proc. 364 (1995) p. 713. [103] G. Saada, in: Plasticity of Metals and Alloys, The Engineering Materials, Vols 97-98 (Trans. Tech. Publications, Aedermansdorf, 1995) p. 275. [104] J. Bonneville, P. Sp~itig and J.-L. Martin, MRS Symp. Proc. 364 (1995) p. 369. [105] D.L. Anton, D.D. Pearson and D.B. Snow, MRS Symp. Proc. 81 (1987) 287. [ 106] P.M. Hazzledine and J.H. Schneibel, Scr. Metall. 23 (1989) 1887. [107] J.H. Schneibel and P.M. Hazzledine, in: Ordered Intermetallics - Physical Metallurgy and Mechanical Behaviour, NATO ASI Series, eds C.T. Liu, R.W. Chan, G. Sauthoff (Kluwer, Dordrecht, 1992) p. 565. [108] J.R. Nicholls and R.D. Rawlings, J. Mater. Sci. 12 (1977) 2456. [109] J.H. Schneibel and J.A. Horton, J. Mater. Res. 3 (1988) 651. [110] K.J. Hernker, PhD Thesis (Standford Univ., USA, 1990). [111] K.J. Hemker and W.D. Nix, MRS Symp. Proc. 133 (1989) 481. [112] N. Cl6ment, A. Couret and D. Caillard, Philos. Mag. A 64 (1991) 669. [113] N. Cl6ment, G. Mol6nat and D. Caillard, Philos. Mag. A 64 (1991) 697. [114] G. Mol6nat and D. Caillard, Philos. Mag. A 65 (1992) 1327. [115] T. Takasugi, S. Hirakawa, O. Izumi, S. Ono and S. Watanabe, Acta Metall. 35 (1987) 2015. [116] A. Komer, Philos. Mag. Lett. 65 (1992) 195. [117] H.-R. Pak, T. Saburi and S. Nenno, Trans. Jpn Inst. Met. 18 (1977) 617. [118] L. Potez, G. Lapasset and L.P. Kubin, Scr. Metall. Mater. 26 (1992) 841. [119] A.K. Korner, D.J.H. Cockayne and Y.Q. Sun, Philos. Mag. A 68 (1993) 993. [120] P. Veyssi~re and J. Douin, in: Intermetallic Compounds: Principles, Vol. 1, eds J. Westbrook and R.L. Fleischer (Wiley, New York, 1994) Ch. 22, p. 519. [121] B.H. Kear and J.M. Oblak, J. Phys. (Paris) Suppl. 35 (1974), C7-35. [122] M.A. Crimp, Philos. Mag. Lett. 60 (1989) 45. [123] Y.Q. Sun, M.A. Crimp and P.M. Hazzledine, Philos. Mag. A 64 (1991) 223. [124] Y.Q. Sun, M.A. Crimp, P.M. Hazzledine and A. Couret, Philos. Mag. A 64 (1991) 311. [125] F. Ducastelle, C. Ricolleau and A. Loiseau, in: Reactive formation of phases at interfaces and diffusion processes, Mat. Sci. Forum, Vols 155-156, eds J.L. Bocquet and Y. Limoge (Trans. Tech. Publ. Lausanne, 1994) p. 367. [126] M.A. Crimp and P.M. Hazzledine, MRS Symp. Proc. 133 (1989) 131. [127] P.M. Hazzledine, H.P. Karnthaler and E. Winter, Philos. Mag. 32 (1975) 81. [128] P. Veyssi~re, Philos. Mag. A 50 (1984) 189. [129] J. Douin, P. Beauchamp and P. Veyssi~re, Philos. Mag. A 58 (1988) 923. [130] J. Bonneville, N. Baluc and J.-L. Martin, in: Intermetallic Compounds, ed. O. Izumi (Japan Inst. of Metals, Sendai, 1991) p. 323. [131] D. Caillard, N. C16ment and A. Couret, in: Int. Symp. on Structural lntermetallics, Champion, USA, eds R. Darolia, J.J. Lewandowski, C.T. Liu, P.L. Martin, D.B. Miracle and M.V. Nathal (The Minerals, Metals and Materials Soc., Warrendale, 1993) p. 409.
438
P. VeyssiO.re and G. Saada
[ 132] J. Douin and E Veyssi~re, Philos. Mag. A 64 (1991) 807. [133] A. Komer, Philos. Mag. Lett. 60 (1989) I03. [134] A. Korner, in: Int. Symp. on Structural Intermetallics, Champion, USA, eds R. Darolia, J.J. Lewandowski, C.T. Liu, EL. Martin, D.B. Miracle and M.V. Nathal (The Minerals, Metals and Materials Soc., Warrendale, 1993) p. 419. [135] N. Baluc, Th~se de Doctorat (Univ. of Lausanne, Switzerland, 1990). [136] A.E Giamei, J.M. Oblak, B.H. Kear and W.H. Rand, in: Proc. 29th Annual Meeting of EMSA, ed. J. Arceneaux (Claitor's Publishing Division, Baton Rouge, 197 l) p. 112. [ 137] G. Saada and J. Douin, Philos. Mag. Lett. 64 (1991) 67. [138] E Veyssi~re, in: Ordered Intermetallics - Physical Metallurgy and Mechanical Behaviour, NATO ASI Series, eds C.T. Liu, R.W. Chan and G. Sauthoff (Kluwer, Dordrecht, 1992) p. 165. [139] J. Douin, E Veyssi~re and E Beauchamp, Philos. Mag. A 54 (1986) 375. [140] E Veyssi~re, J. Douin and E Beauchamp, Philos. Mag. A 51 (1985) 469. [ 141 ] E Veyssi~re and D.G. Morns, Philos. Mag. Lett. 67 (1993) 491. [142] N. Baluc, R. Schatiblin and K.J. Hemker, Philos. Mag. A 64 (1991) 327. [143] N. Baluc, H.E Kamthaler and M.J. Mills, Philos. Mag. A 64 (1991) 137. [144] S.-W. Chiang, C.B. Carter and D.L. Kohlstedt, Philos. Mag. A 42 (1980) 103. [145] A. Korner and H.E Karnthaler, Philos. Mag. A 52 (1985) 29. [146] EB. Hirsch, A. Howie, R.B. Nicholson, D.W. Pashley and M.J. Whelan, in: Electron Microscopy of Thin Crystals (Butterworths, London, 1965) p. 166. [147] J. Oliver, Th~se de Doctorat (Univ. de Paris-Sud, Orsay, France, 1992). [148] Y. Liu, T. Takasugi, O. Izumi and H. Ohta, Philos. Mag. Lett. 58 (1988) 81. [149] Y.Q. Sun and EM. Hazzledine, Philos. Mag. A 58 (1988) 603. [150] M. Yoshida and T. Takasugi, Philos. Mag. A 65 (1992) 41. [151] S.M.L. Sastry and B. Ramaswami, Philos. Mag. 33 (1976) 375. [152] A. Komer, H.E Kamthaler and C. Hitzenberger, Philos. Mag. A 56 (1987) 73. [153] A. Korner, Philos. Mag. A 63 (1991) 117. [154] Y.Q. Sun and EM. Hazzledine, in: Ordered Intermetallics- Physical Metallurgy and Mechanical Behaviour, NATO ASI Series, eds C.T. Liu, R.W. Chan and G. Sauthoff (Kluwer, Dordrecht, 1992) p. 177. [155] A. Korner and E Veyssi~re, in: Electron Microscopy, Vol. 2A, eds B. Jouffrey and C. Colliex, (Les Edditions de Physique, Les Ulis, France, 1994) p. 77. [156] D.E Pope, in: High Temperature Intermetallics, eds S.H. Whang, C.T. Liu, D.E Pope and J.O. Stiegler (TMS, New York, 1990) p. 51. [157] M. Yamaguchi, V. Paidar, D.E Pope and V. Vitek, Philos. Mag. A 45 (1982) 867. [158] C. Tichy, V. Vitek and D.E Pope, Philos. Mag. A 53 (1986) 467. [159] C. Tichy, V. Vitek and D.E Pope, Philos. Mag. A 53 (1986) 485. [ 160] R. Lerf and D.G. Morris, Acta Metall. Mater. 39 (1991) 2419. [161] V.K. Vasudevan, R. Wheeler and H.L. Fraser, MRS Symp. Proc. 133 (1989) 705. [162] E.E George, J.A. Horton, W.D. Porter, D.E Pope and J.H. Schneibel, J. Mater. Res. 5 (1990) 1639. [163] D.G. Morris, J. Mater. Res. 7 (1992) 303. [164] G. Hu, S. Chen, X. Wu and X. Chen, J. Mater. Res. 6 (1991) 957. [165] H. Inui, D.E. Luzzi, W.E Porter, D.E Pope, V. Vitek and M. Yamaguchi, Philos. Mag. A 65 (1992) 245. [166] Y. Gao, J. Zhu and S.E. Mikolla, Philos. Mag. A 68 (1993) 517. [167] Y. Liu, T. Takasugi, O. Izumi and T. Takahashi, Philos. Mag. A 59 (1989) 423. [168] Y. Liu, T. Takasugi, O. Izumi and S. Ono, Philos. Mag. A 59 (1989) 437. [169] Y. Liu, T. Takasugi, O. Izumi and T. Takahashi, Acta Metall. 36 (1988) 2959. [170] E Veyssi~re, D.L. Guan and J. Rabier, Philos. Mag. A 49 (1984) 45. [171] E Veyssi~re, J.A. Horton, M.H. Yoo and C.Y. Liu, Philos. Mag. Lett. 57 (1988) 17. [172] EA. Flinn, Trans. Metall. Soc. AIME 218 (1960) 145. [173] E Beauchamp, J. Douin and E Veyssi~re, Philos. Mag. A 55 (1987) 565. [174] J.M. Sanchez, S. Eng, Y.E Wu and J.K. Tien, MRS Symp. Proc. 81 (1987) 57. [175] R. Kikuchi and J.W. Cahn, Acta Metall. 27 (1979) 1337. [176] V. Paidar, Acta Metall. 33 (1985) 1803. [177] E Veyssi~re, M.H. Yoo, J.A. Horton and C.Y. Liu, Philos. Mag. Lett. 59 (1989) 61.
Microscopy and plasticity of the L12 7 ~ phase [178] [179] [180] [181] [182] [183]
439
H.E Karnthaler, EM. Hazzledine and M.S. Spring, Acta Metall. 20 (1972) 459. N. Baluc, J. Stoiber, J. Bonneville and J.-L. Martin, Israel J. Technol. 24 (1988) 269. A. Komer, Philos. Mag. A 58 (1988) 507. K. Suzuki, M. Ichihara and S. Takeuchi, Acta Metall. 27 (1979) 193. B. Tounsi, Th~se de Doctorat (Univ. de Poitiers, France, 1988). M. Yoshida and T. Takasugi, in: Intermetallic Compounds - Structure and Mechanical Properties, ed. O. Izumi (Japan Inst. of Metals, Sendai, 1991) p. 403. [184] A. Korner, Philos. Mag. Lett. 59 (1989) 1. [ 185] A. Korner and G. Schoeck, Philos. Mag. A 61 (1990) 909. [ 186] G. Schoeck and A. Korner, Philos. Mag. A 61 (1990) 917. [187] A. Lasalmonie, B. Chenal, G. Hug and E Beauchamp, Philos. Mag. A 58 (1988) 543. [188] A.H.W. Ngan, I.E Jones and R.E. Smallman, Philos. Mag. A 66 (1992) 55. [189] Y.Q. Sun, Philos. Mag. A 65 (1992) 287. [190] D. Caillard, N. C16ment, A. Couret, E Lours and A. Coujou, Philos. Mag. Lett. 58 (1988) 263. [ 191 ] E Veyssi~re and J. Douin, Philos. Mag. Lett. 51 (1985) 1. [192] A. Korner, in: Intermetallic Compounds, ed. O. Izumi (Japan Inst. of Metals, Sendai, 1991) p. 319. [193] H.E Karnthaler, C. Rentenberger and E.Th. Mtilbacher, MRS Symp. Proc. 288 (1993) 293. [194] M.V. Nathal, J.O. Diaz and R.V. Miner, MRS Symp. Proc. 133 (1989) 269. [195] C. Bontemps and E Veyssi~re, Philos. Mag. Lett. 61 (1990) 259. [196] G. Mol6nat and D. Caillard, Philos. Mag. A 64 (1991) 1291. [197] A. Korner, Philos. Mag. Lett. 66 (1992) 141. [ 198] A. Couret, J. Crestou, S. Farenc, G. Mol6nat, N. Cl6ment, A. Coujou and D. Caillard, Microsc. Microanal. Microstruct. 4 (1993) 153. [199] C.T. Chou and EB. Hirsch, Philos. Mag. A 68 (1993) 1097. [200] M.J. Mills, N. Baluc and E Karnthaler, MRS Symp. Proc. 133 (1989) 203. [201] B. Tounsi, P. Beauchamp, Y. Mishima, T. Suzuki and P. Veyssi~re, MRS Symp. Proc. 133 (1989) 731. [202] A. Couret and D. Caillard, J. Phys. (Paris) III 1 (1991) 885. [203] A. Couret, Y. Sun and EM. Hazzledine, MRS Symp. Proc. 213 (1991) 317. [204] A. Komer, Philos. Mag. A 63 (1991) 407. [205] A. Korner, in: Proc. 10th European Congress on Electron Microscopy, Vol. 2, eds A. Rios, L. MegiasMegias and A. Lopez-Galindo (1992) p. 279. [206] G. Saada and E Veyssi~re, Philos. Mag. Lett. 64 (1991) 365. [207] A.E. Vidoz and L.M. Brown, Philos. Mag. 7 (1962) 1167. [208] C.T. Chou, EM. Hazzledine, P.B. Hirsch and G.R. Anstis, Philos. Mag. A 56 (1987) 799. [209] C.T. Chou and P.B. Hirsch, Philos. Mag. A 56 (1981) 799. [210] P.M. Hazzledine and P.B. Hirsch, MRS Symp. Proc. 81 (1987) 75. [211] C.T. Chou, P.B. Hirsch, M. McLean and E. Hondros, Nature 300 (1982) 621. [212] A.H.W. Ngan, I.P. Jones and R.E. Smallman, MRS Syrup. Proc. 288 (1993) 269. [213] A.H.W. Ngan, I.P. Jones and R.E. Smallman, Philos. Mag. B 67 (1993) 417. [214] N. Jian and Y.Q. Sun, Philos. Mag. Lett. 68 (1993) 107. [215] C.T. Chou and P.B. Hirsch, Proc. R. Soc. London Ser. A: 387 (1983) 91. [216] A. Korner, Philos. Mag. A 66 (1992) 103. [217] G. Saada, Th~se (Paris, 1960). [218] G. Saada, Physica 27 (1961) 657. [219] S.A. Court, J.P.A. LtJfvander, M.H. Loretto and H.L. Fraser, Philos. Mag. A 61 (1990) 109. [220] J. Friedel, Dislocations (Pergamon Press, Oxford, 1964) p. 104. [221] K. Jumojni, T. Shinohara, M. Kato and A. Sato, Philos. Mag. A 67 (1993) 627. [222] W. Yan, I.P. Jones and R.E. Smallman, Scr. Metall. 21 (1987) 1511. [223] B.H. Kear, J.M. Oblak and A.E Giamei, Metall. Trans. 1 (1970) 2477. [224] A.E Oblak and B.H. Kear, in: Electron Microscopy and the Structure of Materials, eds G. Thomas and J. Washburn (Univ. of California Press, Berkeley, 1971) p. 566. [225] I. Baker and E.M. Schulson, Phys. Status Solidi A: 89 (1985) 1963. [226] L.M. Howe, M.H. Rainville and E.M. Schulson, J. Nucl. Mater. 50 (1974) 139. [227] J. Douin, Philos. Mag. Lett. 63 (1990) 109.
440
[228] [229] [230] [231 ]
P. Veyssidre and G. Saada
E Caron, T. Khan and E Veyssii~re, Philos. Mag. A 60 (1989) 267. G. Molrnat and D. Caillard, Philos. Mag. A 69 (1994) 939. R. Pasianot, D. Farkas and E.J. Savino, J. Phys. (Paris) III 1 (1991) 997. K. Suzuki, M. Ichihara and S. Takeuchi, in: Proc. 5th Int. Conf. HVEM, eds T. Imura and H. Hashimoto (Japan Soc. Electron Microscopy, Kyoto, 1977) p. 463. [232] K. Suzuki, M.E. Kuramoto, S. Takeuchi and M. Ichihara, Jpn. J. Appl. Phys. 16 (1977) 919. [233] M. Nemoto, J. Echigoya and H. Suto, in: Proc. 5th Int. Conf. HVEM, eds T. Imura and H. Hashimoto (Japan Soc. Electron Microscopy, Kyoto, 1977) p. 467. [234] A. Couret and D. Caillard, Philos. Mag. A 59 (1989) 7831. [235] A. Couret and D. Caillard, Philos. Mag. A 59 (1989) 801. [236] V. Paidar, G. Molrnat and D. Caillard, Philos. Mag. A 66 (1992) 1183. [237] G. Molrnat, D. Caillard, A. Couret and V. Paidar, MRS Symp. Proc. 288 (1993) 287. [238] G. Molrnat, Thrse de l'Universit6 Paul Sabatier (Toulouse, France, 1991). [239] I. Baker, J.A. Horton and M.H. Yoo, Philos. Mag. Lett. 55 (1987) 3. [240] J.A. Horton, I. Baker and M.H. Yoo, Philos. Mag. A, 63 (1991) 319. [241] M.J. Mills and D.C. Chzran, Acta Metall. Mater. 40 (1992) 3051. [242] P.B. Hirsch, Prog. Mater. Sci. 36 (1992) 63. [243] G. Saada and P. Veyssirre, Philos. Mag. A 70 (1994) 925. [244] J. Oliver and E Veyssirre, Philos. Mag. Lett. 63 (1991) 141. [245] A. de Bussac, G. Webb and S.D. Antolovitch, MRS Syrup. Proc. 213 (1991) 235. [246] V. Vitek and Y. Sodani, Scr. Metall. Mater. 25 (1991) 939. [247] G. Schoeck, Philos. Mag. Lett. 67 (1993) 193. [248] M.H. Yoo, M.S. Daw and M.I. Baskes, in: Atomistic Simulations of Materials, eds V. Vitek and D.J. Srolovitz (Plenum, New York, 1989) p. 401. [249] R. Pasianot, D. Farkas and E.J. Savino, Scr. Metall. Mater. 24 (1990) 1669. [250] F.R.N. Nabarro, in: Solid State Phenomena, Vol. 35-36, eds J. Rabier, A. George, Y. Brrchet and L. Kubin (Scitec Publications, Zug, Switzerland, 1994) p. 19. [251] M. Carrard and J.L. Martin, Philos. Mag. A 5 (1987) 391. [252] M. Carrard and J.L. Martin, Philos. Mag. A 58 (1988) 491. [253] J. Bonneville, D. Caillard, M. Carrard and J.-L. Martin, Rev. Phys. Appl. 23 (1988) 461. [254] J.J. Gilman, J. Appl. Phys. 36 (1965) 3195. [255] M.H. Yoo, Scr. Metall. 20 (1986) 915. [256] G. Schoeck, Philos. Mag. Lett. 59 (1989) 251. [257] G. Saada and E Veyssirre, Philos. Mag. A 66 (1992) 1081. [258] C.L. Fu and M.H. Yoo, Philos. Mag. Lett. 58 (1988) 199. [259] C.L. Fu and M.H. Yoo, Philos. Mag. Lett. 67 (1993) 179. [260] M. Khantha and V. Vitek, in: Int. Symp. on Structural Intermetallics, Champion, USA, eds R. Darolia, J.J. Lewandowski, C.T. Liu, EL. Martin, D.B. Miracle and M.V. Nathal (The Minerals, Metals and Materials Soc., Warrendale, 1993) p. 399. [261] G. Saada and E Veyssi~re, in: Fundamental Aspects of Dislocation Interactions: Low-Energy Dislocation Structures, eds G. Kostorz, H.A. Calderon and J.L. Martin (Elsevier, Lausanne, 1993) p. 281. [262] B. Devincre and L.P. Kubin, Modell. Simul. Mater. Sci. Eng. 2 (1994) 559. [263] G. Saada, Mater. Sci. Eng. A 137 (1991) 177. [264] K.R. Popper, Logik des Forschung (Springer, 1934). [265] G. Schoeck and A. Seeger, Defects in Crystalline Solids (Physical Soc., London, 1955) p. 340. [266] J. Friedel, Les Dislocations (Gauthier Villars, Paris, 1956); see also: Dislocations and Mechanical Properties of Crystals (Wiley, New York, 1957) p. 330. [267] B. Escaig, J. Phys. 155 (1969) 274. [268] A.N. Stroh, Proc. Phys. Soc. London Sect. B 67 (1954) 427. [269] M.S. Duesbery, N.E Louat and K. Sadananda, Acta Metall. Mater. 40 (1992) 149. [270] S.-J. Liang and D.E Pope, Acta Metall. 25 (1977) 845. [271] D.P. Pope and S.S. Ezz, Int. Metall. Rev. 29 (1984) 136. [272] Y. Sodani and V. Vitek, in: Intermetallic Compounds- Structure and Mechanical Properties, ed. O. Izumi (Japan Inst. of Metals, Sendai, 1991) p. 27.
Microscopy and plasticity of the LI 2 ,),t phase [273] [274] [275] [276] [277]
[278] [279] [280] [281 ] [282] [283] [284] [285] [286] [287] [288] [289]
441
H.J. Frost and M.E Ashby, J. Appl. Phys. 42 (1971) 5273. B.A. Greenberg and M.A. Ivanov, Mater. Sci. Eng. A 153 (1992) 356. A.M. Cuitino and M. Ortiz, Mater. Sci. Eng. A 170 (1993) 111. N. Baluc, M.J. Mills, J. Bonneville and J. Martin, in: 9th Int. Conf. on Strength of Metals and Alloys, eds D.Y. Brandon, R. Chaim and A. Rosen (Freund Publishing House, London, 1991) p. 593. P.B. Hirsch and G. Taylor, in: Int. Symp. on Structural lntermetallics (poster session), eds R. Darolia, J.J. Lewandowski, C.T. Liu, P.L. Martin, D.B. Miracle and M.V. Nathal (The Minerals, Metals and Materials Soc., Warrendale, 1993). H.P. Karnthaler, E. Mtilbacher and C. Rentenberger, Acta Metall. Mater. 44 (1995) 547. P.B. Hirsch, Philos. Mag. A 74 (1996), in press. S.S. Ezz and P.B. Hirsch, Philos. Mag. A 72 (1995) 383. S.S. Ezz and P.B. Hirsch, MRS Symp. Proc. 364 (1995) 719. P. Sp~itig, These de Doctorat (Univ. de Lausanne, Suisse, 1995). X. Shi, Th~se de Doctorat (Univ. de Paris-Nord, France, 1995). X. Shi, G. Saada and P. Veyssii~re, MRS Symp. Proc. 364 (1995) p. 701. X. Shi, G. Saada and P. Veyssi~re, Philos. Mag. A 73 (1995) 1159. X. Shi, G. Saada and P. Veyssi~re, Philos. Mag. A 73 (1995) 1419. X. Shi, G. Saada and P. Veyssi~re, Philos. Mag. Lett. 71 (1995) 1. J.P. Hirth and J. Lothe, in: Theory of Dislocations (Krieger, Malabar, USA, 1982) p. 265. H. Yasuda and M. Koiwa, J. Phys. Chem. Solids 52 (1991) 723.
This Page Intentionally Left Blank
CHAPTER 54
Enhancement of Dislocation Mobility in Semiconducting Crystals by Electronic Excitation K. MAEDA Department of Applied Physics Faculty of Engineering The University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113 Japan and
S. TAKEUCHI Institute for Solid State Physics The University of Tokyo Roppongi, Minato-ku, Tokyo 106 Japan
Dislocations in Solids 9 1996 Elsevier Science B.V. All rights reserved
Edited by F. R. N. Nabarro and M. S. Duesbery
Contents 1. 2.
Introduction 445 Dislocation motion in semiconducting crystals 447 2.1. Dislocation types in semiconducting crystals 447 2.2. Characteristics of dislocation motion in semiconducting crystals 449 2.3. Two schemes of the Peierls mechanism 451 2.4. Discrimination of the two schemes 456 3. Effects of electronic excitation on dislocation glide 461 3.1. Velocity measurements 462 3.2. Macroscopic plasticity 473 3.3. Miscellaneous facts 477 4. Mechanism of the REDG effect 481 4.1. Phonon-kick (recombination enhanced) mechanism 482 4.2. Interpretation of experimental results by the phonon-kick mechanism 487 4.3. Other mechanisms 492 5. Summary 500 References 500
1. Introduction Ordinary factors that govern the plasticity of crystalline matter are temperature, the content of foreign atoms and the density of defects. The roles of these metallurgical factors in dislocation glide are well understood not only in metals but also in non-metallic crystals. However, it is commonly known that in non-metals an additional factor, the electronic state of the crystals, significantly affects the plasticity of the crystal: doping of electronically active impurities alters the mobility of dislocations in semiconductors (doping effect [1]), application of an electrical field induces change in the crystal strength (electromechanical effect [2, 3]), and electronic excitation such as that caused, for instance, by light illumination brings about remarkable effects on the dislocation motion in the crystals (photomechanical effect [4-15]). These non-metallurgical effects are considered to be related to the fact that dislocations in non-metals are usually electrically active and their electronic states are quite sensitive to external electrical disturbances. However, the microscopic mechanisms of these effects have rarely been elucidated. Soon after Kuczinski [4] reported as early as 1957 the photomechanical effect (softening under illumination) on Ge for the first time, strong objections were raised to the presence of the effect [16-18]. After many careful tests were repeated, the effect was denounced to be erroneous [17] or due to nothing other than heating of the crystals by strong illumination [18]. Thus the photomechanical effect appeared to have been overlooked for many years, although some experiments [13] were left unexplained. Meantime, similar optoplastic effects caused by light illumination were discovered also in alkali halide crystals [ 19], in various IIb-VIb compound semiconductors [20], in I-VII crystals of CuC1 [21] and CuBr [22], and even in organic crystals as anthracene [23]. The effects, called the photoplastic effect (PPE), were, however, qualitatively different from the photomechanical effect in elemental semiconductors in the point that in most cases illumination induces hardening of the crystal rather than softening. The PPE is characterized by the reversible hardening on illumination, the specific wavelengths inducing the effect, the significant hardening by rather weak illumination and the saturation of the effect with increasing illumination intensity. Since dislocation motion in these crystals of relatively high ionicity is dominated by point obstacles, the entity that is primarily affected by electronic excitation due to illumination is considered to be the point defects (or impurities). Hence, the central problem is thought to be in the state of the point defects rather than dislocations themselves. We will not devote much space to the PPE in this article but will only briefly touch on the effect in relation to the main topic of this article, the radiation enhanced dislocation glide (REDG). The readers who are interesting in the PPE are referred to review articles [24-26]. The REDG effect treated in this article is an enhancement of the dislocation velocity caused by an electronic excitation of the crystal with electron beam irradiation or with light illumination. The phenomenon is considered to be due essentially to a modification
446
K. Maeda
and S. Takeuchi
Ch. 54
of the Peierls mechanism and hence intrinsic in nature, in contrast to the photoplastic effect in alkali-halide crystals which is undoubtedly the modification of defect hardening and therefore extrinsic in nature. The REDG effect is a softening effect and is at variance with the generally observed hardening effects or the positive PPE in II-VI crystals. Independently from the academic activities mentioned above, some researchers in industries who were developing optoelectronic devices such as light-emitting diodes and semiconductor lasers had noticed the presence of a phenomenon similar to the photomechanical effect [27-33]. One of the technological difficulties that was most serious in the early stage of the development of these devices was rapid degradation of the devices that occurred immediately after they began to operate. It was soon clarified that the degradation is caused by anomalously rapid multiplication of dislocation networks that is induced concurrently with the forward biasing which is necessary for device operation [34]. In most cases the anomalous dislocation multiplication was achieved by dislocation climb that was enhanced by forward biasing; however, in some cases evidence was found for dislocation multiplication in the glide mode [30, 35, 36]. Later, when the entire features were revealed by systematic experiments, it turned out that some of the photomechanical effects are the same phenomenon as the REDG effect. To elucidate the mechanism of the REDG effect, we need full understanding of the fundamental mechanism of dislocation glide. Although it is beyond doubt that the Peierls potential is the main obstacle to dislocation motion in highly covalent crystals, there is controversy on the elementary processes: which process, kink nucleation or kink migration, controls the dislocation mobility? What are the magnitudes of the formation energy of a kink and of the activation energy for kink migration? To these basic questions no unanimous answer has been obtained up to date. In this article, however, a discussion is given about the mechanism of the REDG effect based on our own position on the above problems. The present article is organized in the following way. General features of the dislocation motion in semiconducting crystals are reviewed in section 2, where empirical dislocation mobility equations, doping effects and polarity effects are described. Then, the theory of the Peierls mechanism for a dislocation with high Peierls potentials both of the first and the second kinds is reviewed. According to the theory, there exist two schemes of motion of the dislocation depending on the magnitude of the migration energy of the kinks [37]: one scheme is that the kink velocity is so high that the dislocation migrates without kink collision, and the other scheme is that the dislocation migrates with the generated kinks colliding to annihilate with those of the opposite sign moving in the opposite direction. It is impossible to discriminate the two cases only by the mobility measurements, but the discrimination is essential in understanding the mechanism of REDG effects. Hence, in the final part of section 2, the present authors' view is presented on the elementary process of dislocation motion in the semiconducting crystals under consideration, based mainly on the results of the recent novel experiments for dislocation motion in strained thin films conducted by the group of one of the present authors (Maeda) [38]. Section 3 is the main part of the present article, in which experimental results are presented on REDG effects in III-V compounds of GaAs [39-42], InP [43] and GaP [44], those in elemental semiconductors of Si [45, 46] and Ge [46], those in the II-VI compound ZnS [47] and those in the IV-IV compound SiC [48, 49]. It is shown
w
Enhancement of dislocation mobility in semiconducting crystals
447
that the effect not only depends on the compound species but varies greatly with the type of the crystal, i.e. n-type or p-type, and also with the type of the dislocation. However, it has been clarified that the temperature and excitation-intensity dependencies of the effect can be described by a common formula for every case, strongly indicating that the mechanism of the effects is in common for all the crystals. The negative PPE is also described in this section, and is interpreted in relation to the REDG effect. Section 4 is devoted to the discussion of the mechanisms of the excitation enhancement of the dislocation mobility. At first, the so-called phonon-kick mechanism, the most plausible mechanism for the REDG effect, is briefly reviewed and then it is shown how the experimental results are interpreted by the model. Subsequently, other possible mechanisms for the REDG effect are also presented and critically discussed. Section 5 summarizes the previous sections.
2. Dislocation motion in semiconducting crystals 2.1. Dislocation types in semiconducting crystals Dislocations in covalent crystals differ from those in metallic crystals in various respects [50, 51 ]: (1) Along the core of the dislocation having an edge component, dangling bonds appear at every atomic distance; they may be reconstructed to form pairs of bonds. (2) Localized electronic states may accompany the atoms at the core of the dislocations; they can produce deep energy levels in the bandgap. (3) The Peierls potential of the first kind and also that of the second kind are high due to the covalency in the interatomic bonding. (4) In compound crystals, a variety of dislocation types exists depending on the atomic structure of the core. These peculiarities in covalent crystals yield characteristic dislocation behavior which is substantially different from that in metallic crystals. Some examples are doping and crystal polarity effects on the dislocation mobility, as described later. The materials treated in this article are crystals tetrahedrally coordinated due to sp 3 hybridization, i.e. elemental semiconductors of Si and Ge with the diamond structure, III-V and II-VI compounds with the zincblende or the wurtzite structure and SiC with the 6H hexagonal structure. Although the degree of the covalent bonding, or the ionicity, varies from crystal to crystal, dislocation behavior in these crystals has many common features: (1) The mobility of the dislocation, except at high temperature, is controlled by the Peierls mechanism. (2) Dislocations are generally dissociated into Shockley partial dislocations [52, 53]. (3) Two types of dislocations (c~- and t-dislocations) in compound crystals have different mobilities [54, 55]. These results suggest that the mechanism of the dislocation glide in these crystals is largely the same. Before going into detail, we describe here the nomenclature concerned with the dislocations in tetrahedrally coordinated crystals. The glide plane of dislocations is { 111 } in cubic crystals with the diamond or the zincblende structure and (0001) in hexagonal crystals with the wurtzite or 6H structure. The Burgers vector of the dislocations 1 1 corresponds to the shortest lattice vector; i.e. ~(110) in cubic crystals and g(1210) in hexagonal crystals. The close-packed glide planes do not stack equally-spaced but consist
448
K. Maeda and S. Takeuchi
Ch. 54
of alternate narrowly-spaced and widely-spaced planes; the ratio of these spaces is 1:3. It is established [56-58] that dislocations in any tetrahedrally coordinated crystals are dissociated into partial dislocations on the glide plane in their stable state, in the same manner as the dislocations in face-centered-cubic metals and in hexagonal-close-packed metals. The dissociation reactions are
1
~ ( 1 1 0 ) - 4 g1(121)-- -4- (211)
(2.1)
and
1 ~{1210)--+ ~1(1]00) +
1 (0]-10},
(2.2)
creating an intrinsic stacking fault between the partials. The plane of the stacking fault is in the narrowly spaced close-packed planes, and hence the glide plane of the dissociated dislocation must be in the narrowly-spaced (glide) planes, not in the widely-spaced (shuffle) planes [59]. It is also known that at low temperatures the glide dislocations tend to lie along close-packed directions due to deep troughs of Peierls potentials along these directions, i.e. in cubic crystals along (110) directions and in hexagonal crystals along (1210) directions. Thus, the total Burgers vector of a glide dislocation is either parallel or at 60 ~ to the dislocation line. The former dislocation is a screw dislocation and the latter dislocation is called a 60~ The Burgers vectors of both partial components of the screw dislocation make 30 ~ to the dislocation line and hence they are called 30~ dislocations or simply 30 ~ The Burgers vector of one partial dislocation of the 60~ makes 30 ~ and that of the other 90 ~ to the dislocation line, and hence they are called the 30 ~ and 90 ~-partials, respectively. Since the zincblende and wurtzite structures do not have a centre of symmetry, there exists a polarity for {111} and {0001} surfaces. As a result, two edge dislocations having the same Burgers vector but with dissimilar signs are not equivalent. In III-V and II-VI compounds, in the direction perpendicular to { 111 } and {0001 } planes are oriented pairs of unlike atoms, group III and V atoms or group II and VI atoms. The {111} or {0001} surface in the direction of the group V or VI atom is defined as the plus surface and that in the direction of the group III or II atoms as the minus surface. An edge dislocation whose extra-half plane resides on the side of the minus surface is conventionally called an a-type dislocation and that having the extra-half plane on the side of the plus surface is a /3-type dislocation. For any non-screw dislocations having an edge component, there exist a-types and/3-types in zincblende and wurtzite crystals. Thus, we have 60~ and 60~ 30~ and 30 ~ -partial, 90~ -partial and 90~ -partial. For dissociated dislocations, the extra half-plane ends at the narrowly-spaced plane; therefore, an c~-type dislocation ends at group V or group VI atoms at the dislocation core, and that of/3-type at group III or group II atoms. Hence, dangling bonds associated with a dislocation having an edge component are on the metalloid element in a-type dislocations and on the metallic element in/3-type dislocations. In fig. 1 we show an expanding hexagonal dislocation loop in the diamond structure (a) and that in the zincblende or the wurtzite structure
w
Enhancement of dislocation mobility in semiconducting crystals
,,
90" 30"
30"
30" 90"
30"
i
30"
30~
90"a"
30"13
449
''-~\\~ 30"a
30"
30"
30"~ ' - ' % \ \ \ ~ (a)
\ 90"1~
(b)
Fig. 1. Expanding hexagonal loops of dissociated dislocations in diamond structures (a) and in zincblende and wurtzite structures (b). (b), consisting of screw segments and 60~ segments each dissociated into partial dislocations. In every dissociated dislocation segment in (b), the combination of the leading partial and the trailing partial differ from that in any others, and hence we have six different types of gliding dislocations in compound crystals. In most of the experimental situations, screw dislocations, c~-dislocations and /3-dislocations are distinguished but no distinction is made for the sequence of the partials, e.g. between the dislocation with 30~ leading partial and 90~ trailing partial and vice versa. When we specify the sequence, we use the convention, e.g., 30c~/90c~ for the above case. It should be noted that in real crystals the dislocation mobilities are different for different dislocation types, and hence the shape of the dislocation loops is generally elongated in one direction.
2.2. Characteristics of dislocation motion in semiconducting crystals The dislocation motion in Si, Ge and III-V compound crystals is smooth and continuous, indicating a large lattice friction. The velocity has been measured as functions of stress and temperature by the double etching technique, or the X-ray Lang method [60]. The velocity is represented experimentally, in most cases, by the following equation which is the product of a stress term and a temperature term [50]. V-
Vo
7-
exp
( ) - ~Q
9
(2.3)
Here, 7- is the applied shear stress, Q is the activation energy, V0 and 7-o are constants having dimensions of the velocity and the stress, respectively. The stress exponent rn ranges between one and two. The activation energy varies from crystal to crystal and
K. Maeda and S. Takeuchi
450
>
2.5
.
E
o
1.5
._>
(9
..'"
InP I Ga Ge .
"'"
o-'"0I~
InAs (~ " " ~ ' 0 -
inSb
_
O
0
r
o 13_
Xc
"~
Kink-PairSeparation
Fig. 4. The potential energy of a kink-pair as a function of the pair separation in the point obstacle
model [72, 73].
with Ed, the activation energy required for a kink to surmount the point obstacle, and denoting the average separation of the point obstacles on the dislocation. The kink migration velocity in this model is given by Vk--Uk~exp
--~
,
(2.11)
where Uk is the vibrational frequency of kink. Hence, one obtains from eq. (2.6) the dislocation velocity for the kink-collision case as
V "~ 2 h i ~ U~ukf (T) e x p ( - F k p ( r ) / 2 + E d ) k T
for
L >> ~,
(2.12)
and for the kink-collisionless case as V ~ L haus f (~) exp (
Fkp(r)]~ kT /
for
L Lb.
(2.19)
For the kink-collisionless case, from eqs (2.7) and (2.16), one obtains
kT
uoexp
-
kT
for
L > Lb.
(2.20)
The critical dislocation length Lb that bounds the two regimes is given by Lb--2aexp
~F k ) ,
(2.21)
which is equivalent to the average separation of thermal kinks. Thus also in the kink diffusion model, we have two schemes for the dependence of dislocation velocity on the length of dislocations as illustrated in fig. 6: in the dislocations shorter than Lb, the velocity increases linearly with the segment length L, and in the dislocations longer than Lb, the velocity is saturated to a value independent of L. A substantial difference from the point-obstacle model is that the dislocation velocity depends linearly on stress in both regimes.
456
K. Maeda and S. Takeuchi
Ch. 54
2.4. Discrimination of the two schemes
The elementary process of dislocation glide in semiconductors is a subject of a longlasting controversy which has not been settled satisfactorily as yet. The first experimental fact that should be compared with theories is the dependence of dislocation velocity on the dislocation segment length. The velocity of dislocations with macroscopic lengths is apparently independent of the dislocation length [63], which appears to rule out the kinkcollisionless case in ordinary experimental conditions. The length-independent velocity, however, does not necessarily mean that dislocations proceed with kink collision as discussed above. We shall return to this problem shortly. The second fact of substantial importance is the stress dependence of dislocation velocity, which was most intensively studied in Si [63, 67, 75, 76]. In situ X-ray topographic measurements by Imai and Sumino [75] showed clearly that the dislocation velocity in bulk Si single crystals decreases with decreasing stress quite non-linearly at low stresses and below a critical stress around 10 MPa dislocation motion is completely suppressed. At such low stresses the activation energy also exhibits an anomalous increase with decreasing stress [50, 63]. Since such non-linear stress dependence is often observed in presumably impure samples, the most probable cause of this starting stress is thought to be the presence of impurity atoms that block dislocation motion [64]. In more ionic crystals such as II-VI compounds for which the stress levels used in experiments are relatively low, the effect of point obstacles should be more pronounced. Thus as a general statement, dislocation mobility at low stresses is quite likely to be affected by point defects and hence the point-obstacle model described above may not be too far from reality. However, at high stress levels, this non-linearity disappears and instead the dislocation velocity exhibits a fairly linear dependence on the stress [75] 1. As pointed out above, the kink diffusion model and the point-obstacle model differ in the stress dependence of dislocation velocity, linear in the former and non-linear in the latter. Therefore at moderate stresses as used in ordinary experiments for covalent crystals, the kink diffusion model seems more relevant than the point-obstacle model. If we assume that the microscopic process of dislocation glide in semiconductors is described by the kink diffusion model and the dislocations proceed with kink collision and annihilation, the glide velocity should be described by eq. (2.19). This is the most conventional picture that has been accepted by majority of researchers in this field [77-81]. Nevertheless, as remarked by Louchet and George [82] and Heggie and Jones [83], there is a serious discrepancy in the magnitude of the pre-exponential factor between the theory and experiments: the theoretical value calculated from eq. (2.19) is always several orders of magnitude smaller than the experimental values as listed in [50]. An attempt was made to explain this discrepancy by considering an entropy factor [84]: precisely, the activation energy Fk + E m must be replaced by the activation free energy 1Velocity measurements in a wider stress range using the double etching method [76] showed that the stress exponent ranges from 1 to 2 depending on the dislocation type and the temperature. The disagreement with the results of X-ray measurements by Imai and Sumino [75] probably arose from the use of high stresses by Alexander et al. that must reduce the kink-pair formation energy and hence enhance the apparent stress exponent.
Enhancement of dislocation mobility in semiconducting crystals
w
457
Fk + Em - (Sk + Sm)T, where Sk and Sm denote the entropies of kink formation and kink migration, respectively; therefore, the pre-exponential factor in eq. (2.19) must be multiplied by a factor of exp{(S'k + Sm)/k}. Using a Keating-type potential, Marklund obtained S'k ~ 0.5k and Sm as large as 5k for 90 ~ partials. No calculation was made, however, for 30 ~ partials which are considered to control the overall motion of perfect dislocations. The magnitude of Sk + Sm that is required to account for the experimental pre-factors is as large as ~ 8k for the velocity data of Si obtained by Imai and Sumino [75] that reproduced the linear stress dependence compatible with eq. (2.19). Furthermore, another fact that is incompatible with the kink-collision picture was revealed by recent measurements, the experimental principle of which was first devised by Nikitenko et al. [85], of dislocation velocity in Si [86, 87] and GaAs [88, 89]. A remarkable fact was that dislocations would not move when the crystal was subjected to intermittent loading with pulse durations shorter than time required for a dislocation to advance by one period of the Peierls relief. This intermittent loading effect is difficult to explain by the kink collision picture. In the kink collision regime, the mean density of kinks Ck = 1/(2A) -- 1/Lb under a stationary load. This may be combined with eqs (2.18), (2.19) and (2.21) to give the expression for the dislocation velocity in terms of Ck
V = 2Ckhvk.
(2.22)
As long as the kink density is determined by kink collisions, this expression should still hold even under a time-varying load as well. Since the drift motion of kinks is caused by applied stress, the kink velocity Vk will quickly attain a steady value on loading and will diminish on unloading. In contrast, the kink density Ck will not respond to loading conditions immediately but takes some time to relax to the steady levels. The intermittent loading effect arises from the fact that the relaxation time for kink generation differs from that for kink annihilation. Since the kink migration energy controls the kink generation and annihilation, the magnitude of Em can be deduced from the time scale in which the intermittent loading effect is observed. A quantitative analysis [89], however, shows that a large value of Ern ~ 1.9 eV or a small value of Fk ~ 0.1 eV is required for Si to interpret the actual effect experimentally observed [85]. The value of Fk ~ 0.1 eV is unphysically too small. An atomistic calculation using the Keating potential for Si [90] showed that the formation energy of a double kink is at least 1 eV on the 30~ For b.c.c, metals and ionic crystals, the smooth kink model is considered to be a good approximation. This is supported by the fact that the glide activation energy evaluated by macroplastic deformation tests is in good agreement with the kink-pair formation energy calculated based on the observed Peierls-Nabarro stress [61]; that is, in these crystals, Q ,~ Fkp = 2Fk and Em ~ 0. In the smooth kink model with the assumption of a constant line tension, the energy of an isolated kink is given by [74]
[;'k '~ - - ( 2 Wp Eo ) 1/2 "~ - 7r 7r
2
Tp EO
(2.23)
458
K. Maeda and S. Takeuchi
Ch. 54
In semiconductors, however, this relation would not hold because the kinks are abrupt. Nevertheless, the kink formation energy should not be smaller than the value simply calculated from eq. (2.23) with appropriate parameters. Substituting E0 ~ 0.5Gb 2 and -rp ~ 0.1G for semiconductors, one obtains Fk ,~ 0.2Gb 3. If one considers that dislocation motion is controlled by the least mobile partial, the Burgers vector b in the above discussion should be replaced by that of the partial dislocation bp. Hence for Si (G 68 GPa, bp ~ 0.22 nm), we have Fk ,~ 0.2Gb 3 ~ 1.0 eV. An atomistic calculation of Fk by Marklund [189] for a kink on 30 ~ partial dislocations in Si showed that Fk ~ 0.5 eV O. 1Gb~. Therefore, the kink formation energy in Si should be reasonably around Fk ~ (0.1 ~ 0.2)Gb~.
(2.24)
Incidentally, it may be worth noting that the glide activation energy Q exhibits an experimental correlation with a material parameter Kb2h (~ 1.2Gb3). The overall fitting is achieved by Q - 0.25Kb2h (~ 0.3Gb~) as shown in fig. 2, which together with eq. (2.24) suggests that also in semiconductors Q ,~ 2Fk and Em O .m
tl:l O NO
,,r-
10
~
s 6HSiC
0
0
I
I
I
I
I
20
40
60
80
100
120
Dislocation Length (10-9 m)
Fig. 22. The velocity of the 90~
marked Dl in fig. 21 measured as a function of the segment length [49].
The activation energy of enhanced glide in ZnS was assessed from the temperature dependence with appropriate corrections taking into account the intensity dependence of the velocity and the effect of stress deduced from the curvature of the dislocation line [99]. The experimental values are listed also in table 1 together with the value of activation energy in darkness evaluated by extrapolating the data to zero intensity.
3.2. Macroscopic plasticity In comparison with the large number of materials exhibiting the positive PPE in II-VI and ionic crystals, observations of the negative PPE at the level of plastic behavior are limited to a few materials. Apart from the old data mainly obtained by indentation tests [4, 13, 15], the negative PPE has been observed in the form of the temporary reduction in the flow stress during light illumination in CdS [108, 109], CdTe [110, 111], CdSe [112] and GaAs [95]. Figure 23 shows the results for GaAs [95]. An analysis of the stressstrain curve in the negative PPE in CdTe [ 111] showed that the increase of dislocation velocity under illumination was several hundreds percent. The excitation spectra of the negative PPE have been obtained for CdS [109], CdTe [111] and GaAs [95] (fig. 24). The peak in the excitation spectra corresponds to the intrinsic absorption in the respective crystal, which indicates that the negative PPE is brought about by the electronic excitation that generates electron-hole pairs, as
474
Ch. 54
K. Maeda and S. Takeuchi
175 150
125 I ~" i o o ~ 5o
.
.
.
.
K 25
0 e (%)
--
Fig. 23. The negative photoplastic effect observed in GaAs excited by interband light illumination (Mdivanyan and Shikhsaidov [95]). 1.5 CdS
CdTe
GaAs
0.5 t~
7
0
-0.5 400
600
800
1000
1200
/~ ( n m )
Fig. 24. The excitation spectra of the negative photoplastic effect in CdS [109], CdTe [111] and GaAs [95]. The arrows indicate the band edge positions of the respective crystals. in the positive PPE [25, 26]. The decrease of the effect at wavelengths shorter than the band edge is due to the extremely strong absorption that blocks bulk excitation necessary for the effect. The spectra in CdS and CdTe have long tails extending to low energies with some structure, which resembles that obtained in the positive PPE. It is known that the positive PPE can be reduced by simultaneous illumination of infrared (IR) light in addition to the fundamental excitation [108, 109, 113, 114]. An interesting fact is
Enhancement of dislocation mobility in semiconducting crystals
w
I~Ron
I•off
1 13.. v
b
475
o
Interband Illumination on -2
Interband Illumination off Time
Fig. 25. The effect of additional infrared (IR) illumination on the positive photoplastic effect caused by interband illumination in CdS in the prismatic slip orientation (Shikhsaidov and Osip'yan [109]).
that, in some cases, the IR illumination even reverses the sign of the PPE as shown in fig. 25. Disappearance of the negative component on cessation of gap illumination indicates that the interband excitation is necessary for the negative PPE to occur under the IR illumination. The IR light alone induces only a minor PPE [109]. The negative PPE is induced by more intense light than that inducing the positive PPE. The light intensity inducing the negative PPE in CdTe (typically 0.1 W/cm 2 [111 ]) is not very different from that inducing the REDG effect in Si (,-., 1 W/cm 2 [45]), but if one takes into account the difference in the excitation depth (3 ~ 4 mm ~ the crystal size in CdTe [111] and 50 ~tm in Si [45]), the excitation densities per unit volume are much different. Although it is not easy to compare directly the electronic excitation level due to light illumination with that due to electron irradiation, a rough estimate of electron-hole generation rate density based on appropriate parameters involved in the excitation process [40] indicates that the light intensity used in the study of the REDG effect in Si [45] is about one order of magnitude weaker than the electron beam intensity used in studies of the REDG effects in GaAs [40]. Nevertheless, in the study of GaAs by Mdivanyan and Shikhsaidov [95], they presumably used the same light source (a monochromatic focused light from a 200 W xenon lamp with unknown intensity) to successfully observe the negative PPE and the REDG effect of almost the same magnitude as that induced by electron irradiation [40]. Generally, however, the excitation intensity in the bulk samples in the studies of PPE is much smaller, probably by a factor of 103~5, than in those inducing the REDG effect studied for velocity measurements. One may wonder whether the negative PPE due to the REDG effect is observable at such low excitation levels. Generally the macroscopic deformation is controlled by the least mobile dislocation c~mponent. For fl-dislocations, least mobile in n-GaAs, the critical temperature Tc at the light intensity used by Mdivanyan and Shikhsaidov [95] can be estimated from the velocity data available [40]: knowing the experimental results that the REDG effects for c~-dislocations are almost the same in the two experiments
476
K. Maeda and S. Takeuchi
Ch. 54
using electron beam [40] and light [95], one can estimate Tc for/3-dislocations at the same light intensity to be around 600 K, which is well above 423 K at which the negative PPE was observed. At a temperature T __ o
0.58) is a common feature. The origin of this anomalously large exponent is not clear. Is an additional ordering such as herringbone responsible as has been suggested [203]? Other compounds, such as PHOAB [202], have the first order SmA-HexB transition. Mixtures of PHOAB with 3(10)OBC, a compound exhibiting a continuous transition, show the same anomalies in c~ for concentrations of PHOAB (< 20% wt). For higher concentrations (> 40% wt) the hexatic phase is stable over a large temperature range [204] and the transition is first order. This argues against the presence of herringbone fluctuations near the transition which should be continuous. In bulk tilted smectics asymmetric heat capacity peaks have been found [205] in the SmC-SmI transition and the situation is, not surprisingly, even more complicated. In a beautiful set of experiments Geer et al. (1992) [206] and Stoebe et al. (1992) [207] made detailed heat capacity measurements on a series of free standing films ranging from thicknesses of 300 layers down to 2 layers to see the transition to 2D behaviour. The compound used 75OBC exhibits the following sequence of transitions: I (81~ SmA ( 6 5 ~ HexB (59~ - C r y E . The HexB-CryE transition involves a change in symmetry and is first order (see fig. 20). The SmA-HexB transitions appear continuous, but the heat capacity profiles as one goes to the very thin films does converge to the expected form for a KT transition. It has a sharp peak with c~ = 0.28, as for a three state Potts model [324], the symmetry of the 2D herringbone order, the model for spins on a lattice with three possible orientations, and also, as we shall see, for the most widely studied phase of adsorbed rare gases on graphite (see section 3.3.4) when one adsorption site out of three is occupied. The above exponent 0.28 was observed for He 4 on graphite. Equally intriguing is that the heat capacity profile is identical to the variation of dp/dT, where p is the in-plane molecular density (see fig. 21). The two observations suggest some additional ordering at the transition than the type involved in a KT transition.
546
Ch. 55
B. Jo6s
Interior HexB Transition
4.5
750 BC
4.0 o
~
i
3.5
i~ Surface CryE t ~ 3.0
Transition
.
.
.
AdjacentHexB Transition
]
64 .
.
~
.
6'6
Surface HexB Transition ~
. . . . .68 . . Temperature ('C)
7 ' 0 '
'
72 '
"
Fig. 20. Heat capacity Cp/A per unit area against temperature of a ten-layer free-standing 750BC film. The relevant phase transitions are indicated in this figure [205, 193]. This is only a very small sample of the work which has been done on phase transitions in smectics (for a recent review and details we refer to Huang and Stoebe [ 195] or Huang [196]). The focus was on transitions from the SmA or SmC phases to their corresponding hexatic phases. In summary, the ideas in the KTHNY theory and their generalizations to anisotropic molecules has greatly helped unravel the basic classifications of smectic phases and hint at the role of dislocations and disclinations in the ordering process but a lot more is going on here than those theories can explain. Heat capacity measurements reveal that the liquid to hexatic transitions in the smectics are not exactly XY. These differences must be caused by additional degrees of freedom, namely interactions between the layers and coupling between BOO and tilt, aspects which have not been explored theoretically sufficiently with respect to their consequences on the heat capacity. The richness in the microscopic structure of the defects (a reference to it has already been made in the previous section) is also expected to play a role. The study of these defects is a fascinating subject in itself (see for instance [190-192]). Hints at the complex role played by defects can be found in such works as by Benattar et al. (1979) [209] and Sirotta et al. [210] in which it is shown that the diffusive peak in the SmF phase and the Bragg peaks in the CryG phase have been observed to be coincident in high resolution X-ray diffraction studies. This indicates that the CryGSmF transition does not change the short range structure and that the SmF phase is a faulted version of the CryG phase. Furthermore, chemical changes can also play a role in such transitions [211].
The role of dislocations in melting
w
~" 2.05eq
3(10)OBC
l
2.00-
547
- 5.06
-5.04
r..)
-5.02 >~
~Z 1.95
. o~q' , o o~ o~176176176e~ ~ i
!
73
,
I
,
I
% ~ o % 0 o' ,
,
,
I
i
,
74 75 TEMPERATURE (~
,
I
5.00
76
Fig. 21. Heat capacity (circles) and in-plane molecular density (+) near the SmA-HexB transition of a twolayer film of 3(10)OBC. The data are obtained from the simultaneous measurements of both heat capacity and reflectivity [206, 193].
More theoretical understanding is required. What complicates the picture is the extra degrees of freedom act differently in each class of system enriching the observed behaviour. But overall the transition from SmA to HexB exhibits the expected continuous behaviour of a liquid to hexatic phase transition in the limit of thin films.
3.3.2. Electrons on the surface of liquid helium This 2D system is realized by setting up an electric field perpendicularly to the surface of liquid helium with a capacitor immersed in the liquid. The perpendicular motion of the electrons is frozen into an isolated quantum level while the parallel motion is classical in the sense that T >> Tfermi because of the very low density of electrons. The interaction is pure Coulomb e2/r up to several hundred interparticle spacings. A liquid-solid transition is induced as the temperature is lowered. This was first observed by Grimes and Adams (1979) [212]. Because the electron Boltzmann factor is given by e x p ( - V / ( k B T ) ) , where V is a sum of pair potentials, the phase diagram can be expressed in terms of one combined surface density Ns and temperature T parameter s - ~x/-~--~se2/(kBT). For/-' < 1 the kinetic energy dominates and the system behaves like an electron gas. At intermediate densities 1 < F < 100 the electron motions become highly correlated or liquid-like. For large values of /-' a transition to a solid phase is expected as the Coulomb interactions dominate. No direct structural information is available showing a dislocation unbinding mechanism. The predictions on BOO have not been verified because they are not attainable. The location of the melting point has been used as the key test of the KT theory. Driving the electron lattice with an a.c. voltage, mobility [213] and power absorption [214] measurements were used to identify a melting point in accord with the KT stability criterion (eq. 23), and estimates from
B. Jo6s
548
D-'~oo
9
o o o o o
Ch. 55
9 ...... 9
9
9
Fig. 22. Principle of confinement of colloidal crystals between two plane glass boundaries (the real wedge angle is ~ 5 x 10-4) [228]. defect studies [167] and from Monte Carlo simulations [215]. A value of s of about 124 i 4 is found. Shear moduli obtained from coupled helium-electron modes show also a variation in accord with the KT theory [216]. So do thermodynamic measurements for the heat capacity although Glattli et al. [217] did not reach Tin. Their maximum reported value is for T/Tm < 0.85. There is additional information on this system from computer simulations which is discussed in section 3.4.4.
3.3.3. Monolayers of micron and sub-micron size objects 3.3.3.1. The systems. In contrast to the previously discussed system, the ones in this section can be viewed by video imaging with visible light so that one can obtain direct visual evidence of the lattice defects and their dynamics, and monitor BOO in real space with no limits on time scales. The main systems in question are suspensions of monodisperse (all of the same size) colloidal spherical particles of submicron size immersed in a solvent and usually confined between glass plates (see fig. 22). Such particles have been used for a while as model systems in condensed matter physics [218]. The ones most used in 2D experiments are polystyrene latex spheres which range from ,-~ 0.1-10 ~tm in diameter (for a review of the experiments see Murray [219]). The spheres are formed by emulsion polymerization techniques [220] in which polar species, most often carboxylate or sulphate groups, are appended to the polymer ends on the surface of each sphere. When immersed into a polar solvent such as water, these polar groups dissociate, leaving one electron surface charge and one oppositely charged screening ion in solution. The surface charge density can be very h i g h - as large as one electronic charge per 100 /~2 on the sphere surface. For high surface charge density spheres the total surface charge for a 0.1 #m diameter sphere can be typically 103 to 10a electronic charges. For a diameter range between 0.1 and 1 gm, the colloidal spheres in a solvent suspension exhibit Brownian motion due to collisions with the suspension medium. The collision time between particles is ~ 30 ms for a 0.3 ~tm diameter sphere [221]. The space between spheres is roughly 1 lxm. The Brownian motion permits thermodynamic equilibrium and the definition of a temperature. Both space and times are very amenable to observation. These are not the only advantages of these systems. It is possible to
w
The role of dislocations in melting
549
create a perfectly smooth substrate and to confine the spheres to a plane, all this using electrostatic forces as we shall discuss below. The interaction between spheres is basically a screened Coulomb potential due to the excess charge on each sphere and the opposite charges in the solvent but there are other contributions, van der Waals, hard sphere and many body hydrodynamic [222]. In spite of the expected complexity of the interactions, molecular dynamics simulations using Yukawa (or Debye-Htickel) potentials with an effective charge much smaller than the actual titratable charge [223] reproduce the essential features of the 3D phase diagram [224]. In addition, a different kind of experiment can be made [225] when the solvent is replaced by a a ferrofluid made of much smaller particles (100-1000 ~,). The latex spheres then create a lattice of magnetic dipole "holes" [226]. Some other systems are discussed in section 3.3.3.4 which have similar advantages as the colloidal particles. 3.3.3.2. Experiments on colloidal spheres between flat plates in wedge geometry. Two groups have mapped out the complex phase diagram of these colloidal spheres between two flat smooth and repulsive glass plates [227-230]. The trick used is to make a small angle wedge between the glass plates in order to produce a gradual controlled density change along the wedge and to achieve simultaneous equilibration of the two-dimensional fluid, two-dimensional crystal, and a three dimensional crystalline reservoir directly in contact with each other (see fig. 22). One is then able to sample equilibrated phases of colloid at various average densities. As the system is in direct contact with both thermal and particle reservoir, constant chemical potential and temperature are maintained along the wedge in equilibrium. In 1987, Murray and Van Winkle performed a series of experiments on wedges with angles more than an order of magnitude smaller than the ones previously used and which allowed them a detailed study of the 2D melting transition [231-233,219]. Murray and Van Winkle used polystyrene sulphate spheres of d = 0.305 ~tm diameter with a a titratable surface charge of ~ 2 x 104 electrons. The average separation between spheres near the 2D melting transition was 2.5d. The wedge angle producing the density gradient was ~ 5 x 10 -4 rad. The gradient in density was 1% per image (of size ~ 30 sphere separations). In the other direction the wedge angle was 5 times smaller. The major disadvantage of a wedge geometry is the imposed density gradient in the direction of the wedge, which could smear out a density jump associated with the first order transition. The density gradient also favours the creation of dislocations. A 1% gradient produces on average a density of about 3 x 10 -4 dislocations per image, with a Burgers vector perpendicular to the gradient. In addition the two dimensional crystal nearby serves as an orientation boundary condition for a possible hexatic phase in direct contact. The main advantage is equilibration in a statistical sense of all phases simultaneously. In the experiments a charge-coupled device camera was used to image a region of size 25 x 38 ~tm2 with between 1000-2000 particles. Digital imaging was used to locate the particles. From the accumulated data, accurate determinations of 2D sphere density, instantaneous and time-resolved pair correlation functions, structure factors and orientational correlation functions were done. Figure 23 (this and following figures taken from [231]) gives the density variation vs. position along the wedge identifying the locations of the change in PO and BOO. Figure 24 (p. 158 of [219]) shows the change
B. Jo6s
550
Ch. 55
Yl !
0.2
i
i
t
t
t
i
i
I
n2--nl N I
(/1 tY i,i
I
i
i
i
Y2
I 1 i
i
i
i
I
b\ ~ , . ~ ~ ~ ' ~
o\
0.15
i
\
C
----
I
i
~
i
~
~
I
g
f
bJ =E pO3 Z IJJ
0.1
t (y) 9
J~ J
J
9
9
9
9
9 9 9 9 00000
I I
i
I
I
i
Y-" i
i
I
i
I
I
I
I
I
2O0 y (MICRONS)
=
=
I
I
~
40O
t
i
Fig. 23. Two-dimensional density of colloidal spheres as a function of translation along the wedge in the Murray and Van Winkle experiment [231]. Insets schematic of the wedge (not to scale).
with density in the structure factor S(k), the pair distribution function g(r) and the BO correlation function g6(r). With increasing density n one sees first, for n > nl a large change in BOO with a slight change in PO, then, as n > n2, a large increase in PO while g6(r) becomes fiat. At that point the first peak in S(k) reaches 5, the limit at 2D freezing predicted by Ramakrishnan [171], giving a clear indication of the freezing transition. The variation in the correlation lengths is given in fig. 25. The observed behaviour of the translational and orientational correlation functions is remarkably similar to the predictions of the KTHNY theory with a definite two-stage melting or freezing scenario. The limiting ~76 obtained from power law fits to g6(r) was consistent with the predicted 1/4 value. But the imposed density gradient and accompanying lack of density resolution made the determination of the exponents imprecise. It should be noted that the intermediate hexatic phase occurs along the wedge over a distance ~ 4~, the correlation length making it hard to confidently characterize this phase. Later, time resolved correlation functions were calculated [232, 219]. They do show quite nicely how BOO decays much more slowly than PO in the intermediate hexatic phase. To complete the picture a Voronoi polyhedron analysis and the dual representation of Delaunay triangulations (see section 3.2.5.3) were used to study defect statistics and configurations in the three phases. Figures 26 show the instantaneous positions of particle centers for various average densities very close to melting.
The role of dislocations in melting
w
s(~')
4 5 ! ~ , - r ~ T - - " - - - -
~
g(r)
551
'
,o_
0 = 0
,
gs(r) r
.
10 20 K (mic ro n s - t )
0
~
t0 r (m~crons)
20
0
.
.
'
.
.
.
10 r (microns)
20
Fig. 24. Static correlation functions for various densities of monolayers of monodisperse colloidal spheres, in the Murray and Van Winkle experiment for densities shown in fig. 23 [231]. S(k) is the structure factor, g(r), the pair distribution function, and g6(T) the bound-orientational correlation function. There is a large change in orientational order between curves c and d, with a slight increase of translational order there. Between curves e and f, the translational order exhibits a dramatic increase, while g6(r) becomes flat.
In the Delaunay triangulation of bonds shaded areas mark regions with coordination numbers different from six. These are regions of disclinations. A dislocation is a nearest neighbour pair of 5 and 7 coordinated disclinations. These can be seen in fig. 26(a), the crystal phase close to melting. At n2, the melting point, the concentration of five- and seven-fold disclinations (an upper bound for dislocations) is ~ 5% and the concentration of vacancies is ~ 1%. Dislocation motion has been observed at that point. In the intermediate hexatic phase the defect structure and motion is very similar to but more complex and clustered than in the crystal (figs 26 (b) and (c)). As the density is decreased into the intermediate region below n2 there is a rapid increase in fiveand seven-fold disclinations followed by a smooth rise to a concentration of ~ 8% at nl. The defect clusters have become more numerous and have a wide distribution of separation. They tend to form elongated strings which however do not close up into grain boundary loops. Most dislocations appear free. In the high density fluid defect strings completely encircle the ordered regions. When this happens BOO is completely lost. Very close to freezing the 2D fluid is over 80% six-fold coordinated. This has also been observed in simulations (see section 3.4 and [234, 235]). This is basically a
552
Ch. 55
B. Jo6s
n1
n2
100 o uJ .J
-
-
z 0
J
w
5O I
!
0 0
/
0.16 0.18 DENSITY (DIAMETERS-2) Fig. 25. Translational and orientational correlation lengths corresponding to the correlation functions in fig. 24 [219].
consequence of dimensionality. As mentioned earlier, there is a prevalence of BOO in 2D systems where the average number of neighbours in the liquid and the solid are the same, contrary to 3D systems [114]. There is no evidence of a melting process driven by grain boundary melting. There are loosely bound disclination pairs and unbound disclinations which are not included in Saito's model [174]. It is hard to see a two-phase coexistence either. In this Coulombic repulsive system, at the melting point the defect clusters are about l a - 2 a in size, they then become branched and by the density at which both BOO and PO is lost the clusters have percolated across the images. More recently Murray, Sprenger and Wenk [221 ] performed another set of experiments with spheres of the same size but five times the amount of charge. With stronger repulsive forces melting occurred at a third of the earlier densities. The purpose was to compare 3D and 2D melting behaviours, in particular to show the absence of two-phase separation or of grain-boundary like behaviour in the 2D system. Phase separation on the contrary is quite clearly seen in the 3D systems (see fig 27, also figs in [221]). The greater extent of BOO in 2D is also quite evident in this study (see fig. 28).
w
The role of dislocations in melting
553
Fig. 26. Delaunay triangulations (with the nearest neighborhoods of disclinations in the lattice darkened) for four densities ranging from crystalline to liquid phases: (a) crystal at density n -- 1.03nl, (b) hexatic at density n = nl + 0.6(n2 - nj).
554
B. Jo6s
Ch. 55
Fig. 26 (continued). Delaunay triangulations: (c) hexatic at density n = nl + 0.33(n2 - h i ) , and (d) liquid at density n = 0.98nl. Dislocations are tight 5-7 disclination pairs. Dislocation pairs are quadruplets of 5-7.7-5 disclinations and vacancies are usually eightfold disclinations with two fivefold neighbours [219].
w
The role of dislocations in melting
555
Fig. 27. Defect maps comparing 3D and 2D melting behaviour in experiment of Murray, Sprenger and Wenk [221]. Sphere centers are represented as vertices. Defects or non-sixfold coordinated neighborhoods are shaded. Left: three dimensional run. Density from top to bottom: crystal n -- 0.0599 -- nc; and intermediate region, n = 0.98nc, 0.965ne and 0.953nc. Right: two dimensional run. Density from top to bottom: crystal, n = 0.0606 - nc, and intermediate region, n = 0.974nc and 0.93nc.
B. Jo6s
556
Ch. 55
e
eg(r)
10 b-2
0
100 200 r ( pixels )
0
300
r ( pixels ) ~lW11111111 llil~1~l Wll Wllill
1.0
1.0
g6(r) 0.1
g(r)l
g6(r) .1
I
:
0.01
0
100 200 r ( pixels )
300
.01
0
b
50
100 150 200 r ( pixels )
-
250
3( 300
Fig. 28. Correlation functions for the Murray, Sprenger and Wenk experiment [221] shown in fig. 27. Top panels: g(r) and bottom panels: 9n (r). The arrows on each curve mark the translational correlation length for that density. Left: three dimensional run. The densities for the curves are as follows: (a) fluid at n -- 0.0560, (b) fluid at n - 0.0561, (c) intermediate region at n - 0.0571, (d) intermediate region at n -- 0.0578, (e) intermediate region at n = 0.0587, and (f) crystal at n - 0.0599. Right: two-dimensional run. The densities for the curves are as follows: (a) fluid at n = 0.519, (b) fluid at n -- 0.535, (c) intermediate region at n - 0.0563, (d) intermediate region at n = 0.0569, (e) intermediate region at n = 0.0590, and (f) crystal at n = 0.0606.
3.3.3.3. Expansions with uniform density.
Two other p r o m i s i n g g e o m e t r i e s h a v e b e e n used in studies with colloidal m o n o d i s p e r s e spheres: m o n o l a y e r s on the surface of water by A r m s t r o n g , M o k l e r and O ' S u l l i v a n [236] and freely e x p a n d i n g m o n o l a y e r s b e t w e e n flat parallel plates by Tang and the above authors [237]. In the first of these studies [236] m o n o l a y e r s of p o l y s t y r e n e m i c r o s p h e r e s are trapped by surface tension on the surface of water in a L a n g m u i r trough. T h e t r o u g h was m o u n t e d on an isolation box and had barriers used to sweep the surface clean and c o m p r e s s the m o n o l a y e r . T h e entire apparatus is installed on a m i c r o s c o p e . T h e spheres at the surface exhibit strong optical contrast so they could easily be i m a g e d due to the very different ratios of the index of refraction, 1.6 b e t w e e n those of p o l y s t y r e n e and air, c o m p a r e d to 1.2 b e t w e e n those of p o l y s t y r e n e and water. Being able to c o m p r e s s the m o n o l a y e r gives
w
The role of dislocations in melting
557
it an advantage over the wedge geometry, but this is offset by the difficulty of keeping impurities out of the trough. Contamination of the surface produces aggregate particles and impurity induced free dislocations. Two microsphere sizes were considered, 1.01 ~m and 2.88 l.tm. The smaller sized spheres showed a proliferation of defects even in the solid phase and have a first order melting transition in which the impurity induced defects coalesce into a close network of grain boundaries completely disordering the system. The larger sphere system had a more typical two-stage melting sequence with the application of the pressure. The authors also comment on the tendency of defects to cluster and of multi-dislocation interactions and pairing. The correlation lengths ~ 10a-20a and the multiparticle defects raise here also the role of contaminants. The second experiment [237] with expansion of the monolayer of 1.01 ~tm particles between flat parallel plates circumvents the problem of contaminants, which was rather difficult to avoid in an exposed surface. By varying the separation of the plates a density driven melting could be produced in a system which still had uniform density. A twostage melting consistent with a KTHNY melting is observed. But the defect kinetics was nothing like a dislocation unbinding. In the intermediate phase there was spontaneous creation of clusters consisting of dislocations and dislocation pairs, or small dislocation loops that are characteristic of grain boundaries. Vacancies were also generated by the annihilation of two off-one-row dislocations with opposite Burgers vector. The relationship between dislocations and vacancies is discussed in [165]. These local vacancies tend to act as nuclei for the melt. 3.3.3.4. Magnetic or electric holes, bubbles, and flux lines. Helgesen and Skjeltorp [225] have made experiments on the lattice of magnetic dipole "holes" [226] obtained by suspending the spheres in a ferrofluid made of much smaller particles (1001000 A). In their experiment an in-plane rotating magnetic field HII binds the spheres in rotating pairs whereas a perpendicular field Ha_ produces a repulsive force between the pairs which form a triangular lattice. With only the rotating HII the pair-pair interaction averaged over time will give a net attractive potential between the pairs. On the other hand with only Ha_ the force between the pairs will be repulsive. The competition between the two fields creates a minimum in the pair-pair interaction. The equilibrium lattice parameter is determined by the ratio Ha_/HII; a ratio of ~ 0.3 would correspond to a soft core system and a ratio of ~ 1.0 to a hard core system. The solid-liquid transition is driven by the frequency of the rotating field. Increasing that frequency is equivalent to heating the pairs. A microscope connected to a video camera is used to observe the spheres. For a given value of the frequency the position of each sphere can be recorded as a function of time and analysis of positional and orientational correlations performed and diffraction patterns constructed as for the suspensions in polar solutions. Two-stage melting is observed but grain boundary melting is proned. There is also an interesting run showing melting nucleated at a grain boundary (see fig. 29). In a similar experiment by Kusner et al. [238] a 3.75 M.Hz oscillating electric field is applied to a monolayer of 1.6 ~m spheres confined between glass plates. The electric field polarizes the water but not the polystyrene spheres which act as "electric holes". Two-stage melting is proned on the basis of the variation of the correlation functions. No evidence of grain boundaries is found. The dislocations in the hexatic phase close to
B. Jo6s
558
Ch. 55
(a)
~:o -o b~ ~,o_o~o~~o~=o_-o-Jo~OO o ~ ~ ~ oO,O~o~OOo ~ ~ ' a : o ~ o ~ ~o%O2o2~%%oo2o'~,.p~o_'-6~,.To'-8~,o. ooVUUOO o o V U O O O ~ r Oo 0 0 ~ 0 0 0 ( 30
~ ~ ~ o
oob'~ u o~':'o%O0~ oooqo(o oooO~o ooo ffdoo
~oo~ oo
:OY~oOXO oby~g,q T, ~,-,_o6_~o~o&"~9 ~o
~ % o ~ _ ~ o o ~ ~ ~o~'1 ~000 "~uO0 "uo "uo ~vO0 Ooo o Ooo ~ ~ o O o o o ,~oo,,]
,_~o~o~'oo~o~o~o~'~aoo~o~'~
L~..q~'q~uao~qoB'cF,goowoo~,~-u(
v u o oo ~ u o ?Oo v u O r ~'oOoI oo u~ uOb'?~O )~".I o6O~oAq~o~q~o~qd~,q6c t'~ob_%%oxqbyo%%qpi'o%9,o..~_T::,%~c o o bY~u PEo'~176176 ~~ Pj~ ~O,2o~%O-d"da~g ai%~ o" uuo uoO0 "000 u .O~-O~VO~ ooo - " ~'-' ( p_o-J I._~ o ,..o;o_~ o o ,..o~o_~o ~o?~ o Oo.o
I~o
UO0 uoO0 v vOO'?,.-,uO 0
~ - 0 0 0 ~ u o ~ u o ) b_~u O O h~V u o ~vu 0 v o 0 ,"-'-uI ' ~176 o o ~o~ o o Ao~o~ o Ao~T~ oAo~
OO'P'~ ".~-~o u OOoo u Ooo0 0 o c D~'I O 0Oo O,QT~o o 00.-,-'-~ .Ooo O0 ( Oo
~_~~o~ 6 o ~
O ;~'UOo~VUOObVUoO ,-,uO, ,o oo o~ r~2b%@#o~o~@~p~%o2o ~ ~o2o~ ~ o ~oo
t::4
p ~ ,~%_~ ~ ~ o ~ ,o
oo ,~a: o 8o~o.','o0"~ o ~':~_~o,,,o~c~ c oo-JOb"
l~{ivOOo ~ ' u u o v u o O v u O u U ( 9.7.o~%g.b~~ AOO O.O~Oo~O2doh%9~O~~ O "~'~ u O 0 g O v o O ~ u O o
U~
~01
O O o o O O O o u O'p" v o uO uo u v O O o ,uO O o q oo oo oo o b~ 0 b_'~OuOo0 * O O 0 O O 0 0 0Oo ~ oO 0 00 )O0
Loob%~ob~oo..ob~qo.~o
_o ~oo_~8o( ~;-J
PLo 8~176
(b) ~oO~~ ~@g-o6~oOMoOO o76,_; .&~,~-~c~o:o-.-o'm:.,_oa'..oo~r -," 9
,' 9 o r .
oo 9
*
ur
o,,,
6~9.
roq, ~
~:To~U * u O o ~ U O o
o-,vo2
~176176 o* m ~
1
, *o
-,.,0
*~176176176176oo
oo"o-'9~.~162 .,-,o~'~,.,oo '-'oo ,.,o
t-~..'~,~~176176176176
C;
o oooo~Ooo~0o~ F; oO o oo o ~ oOooooo l
9, ~,~ ,-*=bob ~" _ g ooo Ooo uo~%CZo~'-'oobvoo6VOo uoo~-~_OoyUO ooo oo Oo* *o Ooo Ooo oo~,9o%0oo Oo Ooo Ooo oo oo. oo *
[ o 6%qb a ~ ~ ~ .*au~~ U_Oo-%uo_8%qA~_"" ~ O d O vuo
b'{ T M
u
uO~O v u O
v
~?..ob'~Od~u~oo ;..p..o o ~u -II
~_-:~JOo~vO v u ~ u O ~_uO0~U
o ~_~o~ "~o9ooao~o~, O o o O ~ ~~ooZW~oo o :J t~.~oo o~o~' ~?. o ~ oO~oOO~oo. ~ ~,o~o~O~o ilU 0 0 g O O 0 ~ , ~ u O ' - ~ u O ~ V O ~ u O ~ v O ~ O 0~)"~"0 0 ~ o 0"~ u O ; , " u O
.....
.....
_
,,..V,,,..v......e
,. ..=v..
V O o " . u 41
...
"*"
(c) Q "*@
~'0
*t1[*
o
-uu-~llQ
xg"
~,~.
u.~v_o0~ ~
Woo
I'-
-
* *
o
,
@011~ Q * Q W U -
*
A,
*-88,
o
0--1 i-
I
0
~I-ou
OOQ
oo o * , , - o ~ . d d , ~ - ~
o
;~,
o
O~)~, O 0
o
:,'~.~
[:~ o ~ O ~ O o O ~ i r_ o~,..
,.,6,9.,Lo(3~_d'o'8~ ,.,,
~g,d,g,,~o~_~O5oOo r_Oo~ %"_,,1
rx ~ a ~ ~o~H5.*~v~o_,4T~~ ~ o,~,
.u
e". Xa..,*.:, 9 ~...(:r,"o~ *_o o89Xoa2~__o"d'~,O~l
,, l~dn'~::~,ooV~VoO-~oiSJ ~r
oO'o*a
Oao
oo 9 " oo,.,O~od,dg~L~O-o--~ , u , u F. ~o Xo 0~o~X~r1~3:~,o~q-~ g Oo~q~o%~o o*~_e_;_'av, o:~ , , ^U~,O , o O , o u~c~o_~ o,, 2 " ~. ' A ,~oooz~Yt~ l v *Y ~Uuu O O~= ,
1.
-Oo . o _ . o _ . C>c~_ .......
~o'-'. . . .
......o.
:,,o. * * , , _ . . ~ 4
5OO ;um Fig. 29. Defect plots in the Hegelsen and Skjeltrop experiment on magnetic holes [225]. First set (a) to (c) started with a perfect lattice. Second set (d) to (e) with a grain boundary: (a) or (d) fH -- 0.43 Hz, (b) or (e) fH = 0.47 Hz, and (c) and (f) fH -- 0.60 Hz. Sixfold coordinated molecules are depicted as circles, sevenfold as asterisks, and fivefold as triangles.
The role of dislocations in melting
w
559
(d) "6"oo ~6" ~ .fi"~ "o ' ~ . ' 6 "~o~o -o . .~ ..o . . .-.o. .~ o _ o-_". ~, o~ "~' ~~6 -o~ a~_o%~ o
~O o OXOo ~~~a~go~o~@oOo~; ~,~,O
I~ ~~
;
o ~
xo~O~O'-'.
I
:
~
~ o
'~ o
;
g ~
~ , o ~u6o~'--'o~
o
~
o
0
0000~0
o~ o oo o o o ~ o g ~ o o
g o
9. - oo~ . . . . . -ooo-oo. . 9t
uuOOOOOOOO
|
o
uuu,.,uu_u
o
oo
:~
%
I o o o o o o o o _oo_o~_o-oR
o
~
69oo ~, o: ~ ; ~ g ~ n ~c~! !~uSo"'~Uo~ ~ O o ~ ~6~o* , ,O o O o O O o O,cyco O~c~d,o,:,oV0Vv, o . 1 ,.,,.,oooo N5~2_~_~.~-'~ ~" 6,.,o~U ooo oo
1:~8~o~_~~~~.m.1 I:o~ o oO~o ~j o _o o~-.'~ o "
0 u
--- u
0
0u
~O-"~Q
0 u~-'~
u
0u
o'='_ ,,,
0u
0
~ ' C ) P ~ c ~ Q O-u~'~k
....
u
-
uooo0000
~~176 1 rR~,Oooo -v"r'o~o~.
o u;
00]
0
uuuuooou
oo o ooO o o ~~ oooooooooo o o o o o, o ooo oR~o ooo ~ ~ _ o ~o oO. ooooo~do oO . o o ~ ~ ~~ ~~ a oo ooO ooO oo oob%oooo oO , ' ~ o0~O~0 O O o Oo , o O O o O O o oo~O~_,oooo OO O O 0 oCo~~O,O, ,e oO_~o~_oOO~ ~ ~ o oO~o o ~o~O~;~o~O~O;Oo o ~d o o o o
, ~ o. 0 ~oo o . ~. % ~. ~. O ~oo ~ o ~oi~Oao ~.. . .o.o.o.m . O;.Oo-O~O .o.;O .po. . , (e)
,~.~..~o~~;~O~o~ooo oooo1
NN
N
N!gN .,,
~
c?oo
)
o8_~
8~ ~
Fo.~.~.T~.: ~.~..-~.
o
,'r
~0,(0
N ~
:
0000000
o
e
o.
O000e
b"o_"o~db~d~oo o o ocro'-',-;~1
o ~ ~ o ~ O o O o @ooo ; ooo oo oo o o o. 9 9 o 0 1 b "oooo ;....,, ~oo ~-
(f)
9
~)00_(~000000
~ '
!
500 Fig. 29 (continued).
560
B. Jo6s
Ch. 55
the solid phase show a number of isolated dislocations. Even well into the intermediate phase there are no elongated strings as have been observed by Murray et al. In an earlier experiment by the same group [239], a small density (1%) of larger spheres was added. The larger spheres act as pinning sites for the dislocations. The last experiment described in this section on macroscopic objects is about magnetic bubbles in a garnet, which can also be viewed by optical microscopy. In this experiment Seshadri and Westervelt [240] used a bismuth substituted iron garnet film with material composition: Fe3.91Y1.20Bil.09Gdo.95Gao.76Tmo.09012. The film had a strong growth induced magnetic anisotropy with easy axis perpendicular to the film plane. The anisotropy supports domains with magnetization parallel or antiparallel to the applied field HB. The domain walls are narrow (N 0.1 ~tm) compared to domain sizes (~ 10 ~tm). There is an interval of HB for which small domains called bubbles are energetically favorable. The bubbles move without substantial deformation in the lattice. Bubble size and density is controlled by varying the field. The techniques have been extensively studied since these bubbles are of high technological importance [241,242]. Arrays of ,.., 12000 bubbles with densities ~ 4000 mm -2 were produced. The bubbles form a lattice with a dipolar repulsive force 1/r 3. They can be viewed by optical microscopy using the Faraday rotation of transmitted polarized light. Questions about equilibration time make it hard to argue about defect configuration. A Lindemann criterion has been found to describe well the melting point: melting occurs when the root mean square of the difference between the displacements of adjacent bubbles equals ,.., 10% of the average spacing between bubbles. A related system which will only be mentioned here is the superconducting flux lattice. This is a lattice of tiny current vortices, induced in the mixed state of a sample in the presence of an applied magnetic field. They are sometimes called Abrikosov vortex lattices after the name of their inventor. There have been early suggestions that this lattice can undergo a solid-to-liquid transition, i.e., melt. A continuous transition has been been observed in the melting of the vortex glass in crystal samples with many defects [243]. It has been explained as the transition between a vortex liquid and a vortex glass [244, 245]. The transition has been shown both theoretically [245] and experimentally [243, 246, 247] to become sharper as the amount of disorder decreases. In clean samples the transition is first order [248, 249] according to theoretical suggestions [250]. In such samples the vortices can form a 2D vortex glass with hexatic order [251,252], and its melting is thought to involve dislocations [253]. Overall these experiments give similar results showing a two-stage melting process. The beauty of these experiments is that the kinetics of the melting can be followed in real time. Disagreements lie in the details of the kinetics of melting or the arrangements of the defects in the hexatic like phase near melting. In the latest measurements by Kusner et al. [238] defect configurations and BOO correlations follow KTHNY behaviour. Murray and co-workers see patterns not in disagreement with the KTHNY theory albeit noticeable clustering of the defects which is not a feature of the 3.3.3.5. Discussion.
w
The role of dislocations in melting
561
theory [219]. The other groups [236, 237, 225] argue that the kinetics of the defects do not fit the dislocation unbinding mechanism and lean towards a grain boundary mechanism. The difference in perspective is mostly a matter of the degree of clustering. But from a thermodynamic point of view, as seen from Chui's theory [168], this can make the difference between a first and a second order transition. Grain boundary melting is always first order although weakly so for large core energies. In the Murray and van Winkle [231] experiment the core energy lies between values for which the strong first order and weakly first order transition is predicted. The core energy argument is however not a strong one. Other properties of dislocations such as mobility are equally important [165]. It has been argued [219] that when the systems are not well equilibrated grain boundary melting is favored. This may explain the different observed behaviours. It should be noted that these experiments do not provide the possibility of thermodynamic measurements. Interestingly whatever arguments are put forward to explain the different observations the question remains that all these systems, except for the small spheres on the contaminated surface, exhibit from a structural standpoint two-stage melting, with persistence of BOO in the intermediate hexatic phase, whether well equilibrated or not. 3.3.4. Physisorbed and intercalated monolayers This class of systems has been the most studied in relation to two-dimensional melting. They have also been the most controversial. The KTHNY theory was applied to it very soon after the Halperin and Nelson papers came out in 1978-1979 [2]. By then experimental techniques to study surface structures had made considerable progress and the groundwork was laid for tackling the 2D melting problem. In May 1980, a major conference on "Ordering in Two Dimensions"[115] reported early results. For over a decade following these promising beginnings, the subject was intensely studied by a variety of techniques; synchrotron X-ray diffraction, neutron scattering, heat capacity measurements, to mention the most important. The systems of choice were the rare gas monolayers adsorbed on graphite (Gr). But other systems were also considered, in particular some simple molecules on the same substrate, some intercalates also in graphite, and rare gases on metallic substrates such as Ag. We will focus on monolayers adsorbed on graphite, rare gases and some molecular systems. Additional references to other systems will be made at the end of the section. 3.3.4.1. Rare gases and graphite. The rare gas monolayers on graphite were attractive because they formed nearly ideal model systems (for a recent review see [254]). Rare gases are chemically inert and interact weakly through a van der Waals type of interaction which is accurately known [255-257]. The popularity of graphite is due to the marked mechanical stability of its basal plane and its chemical inertness. Little surface preparation is required compared to metallic surfaces. This is important in view of the large areas required for thermodynamic and scattering measurements. A common form of graphite which has been used in experiments is the exfoliated crystal. The brutal process of exfoliation leaves large basal plane areas intact. The estimated coherence length ranges from ~ 200-400 A for the Papyex and Grafoil to several thousand for the single crystal exfoliated UCAR ZYX graphite [258] (see fig. 30). More recently experiments on single crystal surfaces have been made with coherence lengths of at least
562
Ch. 55
B. Jo6s
(a)
(b)
SHEET OF GRAFOIL
L
P(O)
/ Fig. 30. Schematic diagram of a Grafoil sheet. L indicates the typical coherence length of the adsorbed layers while P(O) describes the distribution arising from the non-parallelism of the substrate surface [117]. 10000 ,~ [259]. This increase in coherence length comes at the expense of the specific area of adsorption. It is as high as 30 m2/g for Grafoil down to 2.8 m2/g for the UCAR ZYX samples, and still smaller in the single crystal. The area of the single crystal in the Specht et al. experiments [259] was about 2 • 3 mm 2 orders of magnitude smaller than a UCAR ZYX sample. The exfoliated graphites greatly stimulated the study of physisorbed systems because their high specific area provided a large number of surface atoms ( ~ 1021 g - l , for the case of Grafoil). An added advantage for thermodynamic studies is that overlayer excitations are generally much lower in frequency than those of the substrate. Also, graphite is nearly transparent to X-rays and neutrons. On the other hand the adsorption potential for rare gases is sufficiently strong to produce a well defined monolayer, of the order of 100 meV for Ar, Kr and Xe (to convert to Kelvins multiply by 11.6). An extensive bibliography of physical adsorption potentials can be found in [257]. And the "corrugation", the amplitude of the periodic part of the potential, is smooth enough in some cases to produce a legitimate 2D solid but strong enough in others to have dramatic consequences - commensurate (C) phases and highly incommensurate (IC) domain wall lattices (DWL). This corrugation creates a wealth of different behaviour which enriches the physics but can complicate comparisons with the KTHNY theory of melting. In discussing melting the effects of the motion of the adsorbed atoms perpendicular to the substrate and the exchange of these atoms with the gas phase have also to be considered. 3.3.4.2. Commensurate and incommensurate phases. The easiest way to understand the types of phases produced by the modulation in the substrate potential is to consider the simple one dimensional model known as the Frenkel-Kontorowa [260] or Frank and van der Merwe [261] model, in which a chain of atoms interacting with springs, and
w
The role of dislocations in melting
563
Fig. 31. A chain of particles in a periodic potential.
having equilibrium separation a, is subjected to a sinusoidal substrate potential of period b (see fig. 31). Its classical Hamiltonian is: /z
a) 2 i
W(
1-cos
27rxi) b
(35)
i
where xi is the position of the ith atom. This model is known in the field of dislocation theory as the simplest model for dislocation motion. If a r b, there is a competition between the order preferred by the 1D lattice and that preferred by the substrate. The equilibrium configurations are given by the set of difference equations: r
- 2r + r
7rW = ~ sin 2"ar tto"
(36)
where r is the displacements of atom from some equilibrium periodic configuration in registry with the substrate (a commensurate state). The location by xi = b ( i n + r If the change in separation from atom to atom is small, this set of equations can be approximated by a differential equation which is none other than the familiar sineGordon equation with soliton-like solutions: d2r di 2
71-W
= ~
/Lb2
sin 27rr
(37)
The solutions to the difference equations or the sine-Gordon equation yield two types of equilibrium configurations for the chain of atoms, the already mentioned commensurate (C) and the incommensurate (IC). In a C phase the atoms lie at minima in the substrate field, the substrate wins out. In an IC phase, soliton-like domain walls (DW) separate C regions (see fig. 32). If the density of DW is low, each wall and each C region is well resolved, we have what is known as a domain wall lattice (DWL). If the domain walls overlap, a weakly modulated "smooth" structure is obtained which structurally retains the essential features of the original lattice. This is usually what is obtained when the corrugation is very weak relative to the strength of interparticle interaction, or the two lattice constants are very different. This simple 1D model predicts an exponentially decaying repulsive interaction between walls [263, 262]. The approximation leading to the sine-Gordon equation is equivalent to assuming a continuous mass density in the
564
B. Jo6s
Ch. 55
(a)
no
(b)
q,
/-- / J
_./
Fig. 32. The displacement ~b of an adsorbate atom from a commensurate site as a function of atom number n in the 1D Frenkel-Kontorowa model; (a) an isolated soliton-like domain-wall, (b) a periodic domain-wall structure.
chain; the discreteness is lost. In the continuum model the DW are free to slide in the lattice. If the lattice discreteness is restored, a pinning potential exists, the Peierls potential, which decays exponentially with the width of the walls [262]. Basically if the walls are a few lattice constant wide, the substrate pinning force is negligible. In generalizing to two dimensions, the essential physics remains the same. The interplay between the preferred substrate spacing and that preferred by the monolayer can lead to soliton like DW and the existence of C and IC phases. The 2D model was developed by Shiba [264] for the graphite substrate which has hexagonal symmetry. In the IC phase two types of wall ordering can be observed on a graphite surface, a striped phase, with walls parallel to each other, and a honeycomb phase with the walls forming hexagons intersecting at three-pronged vertices (see fig. 33). DWL form a fascinating subject. Extensive reviews of their properties exist, in particular for striped phases (see reviews [263,268-270, 121]). The honeycomb phase is now also fairly well understood, studied in some detail by Shrimpton et al. [271-274]. From energetic considerations, striped phases are expected to be favored at low DW densities over honeycomb phases because of the compression or expansion at the vertices which usually makes wall crossing energies positive [275, 276]. At higher DW densities, the larger average distance between DW reduces the repulsion energy between the walls in a honeycomb array making that phase more favorable. In reality, however, as Villain [263] has shown, entropy makes the honeycomb phase favored in isotropic systems even at low DW densities. Consider the diagram of such a phase in fig. 34(a). The total DW length and the total number of intersections does not change if we shrink one of the honeycombs in the network. This configuration can fluctuate into the one shown in fig. 34(b) by repeated breathings of the different honeycombs in the network.
w
565
The role of dislocations in melting
(a)
(b)
(c)
(d)
Fig. 33. Schematic pictures showing honeycomb array of domain walls (a) (expanded) and (c) (compressed), and striped arrays (b) (expanded) and (d) (compressed) on a graphite substrate when the walls are unrelaxed or very narrow. Expansions and compressions lead naturally to the walls shown, which have been given the name super-light and super-heavy respectively. The presence of DW means an excess or a shortage of atoms in the monolayer with respect to the relevant C phase. In the systems that we will be mostly discussing, the C phase of interest is the phase known as V~ x v~. This means that the nearest distance between atoms is v/3 times the distance between nearest adsorption sites and it corresponds to a filling factor of one third (see fig. 35). Figure 35 also shows the different possible wall geometries in IC phases close to a v/3 x v/3 C phase. Light and super light DW correspond to decreased densities, and heavy and superheavy DW to increased densities. Light and heavy DW are actually stacking faults in the triangular lattice and for this reason are not favored for spherical adsorbates. This has been demonstrated by computer simulation [275, 277] and experiment [278] for Kr/Gr. In this system DW
566
Ch. 55
B. Jo6s
(o1
(b)
Fig. 34. Honeycomb domain wall networks: (a) before breathing; (b) after breathing.
heovy
light
super heavy
super light
Fig. 35. Possible types of domain walls (DW) for an adsorbed monolayer on a graphite substrate. This a schematic representation based on infinitely narrow DW. The thick solid line corresponds to a super heavy domain DW, the double thin lines to a heavy DW. The thick dashed line is a superlight DW, while the double thin dashed lines a light wall [254]. have been shown by the same authors to be fairly wide, 5 to 10 Kr atomic rows, making them very mobile. Heavy DW are only observed in a hexagonal IC phase of the highly quantum D2 [321]. A great deal of attention has been given to the C-IC transition. Theories based on Fermi statistics have been particularly useful in understanding this transition in striped phases [270]. The configurational meanderings of the striped DW are analogous to the motions of a 1D lattice of fermions. DW energies replace fermion mass; line tensions, fermion hopping probabilities. Pauli exclusion principle represent the hard core repulsion of the wall. Within this framework Pokrovsky and Talapov showed that the C to striped IC transition should be continuous [280]. Arguments based on the entropy associated with the configurational meanderings of the DW reproduced their results [281-283]. The transition is continuous because collisions between the DW reduce the total entropy of the system.
The role of dislocations in melting
w
567
In contrast the C-IC transition is predicted to be first-order in the honeycomb DWL. The entropy associated with the breathing of the DW network is sufficient to reduce the total free energy at the onset of the transition [263, 282]. A heat capacity study of CO on graphite seems to confirm this prediction [285]. The DW networks in both the striped and honeycomb phases can be treated as renormalizable objects with elastic properties [281-283,273,274]. A C phase is not an elastic medium because of the exponential decay of any particle displacement. The DW in an IC phase restore its elasticity to the lattice. The low energy excitations of the lattice are now those of the DW network. In the striped phase it is dominated by the meandering and the repulsion of the DW. In the honeycomb phase, a more complex set of DW excitations can be observed, in addition to the lowest breathing mode. These are similar to the vibrational modes of a network of strings tied at three pronged vertices as has been shown by Shrimpton and co-workers [271-273]. The elastic properties are mediated by the repulsive interactions between the vertices carried by the connecting DW. This vertex repulsion produces in Kr on graphite a variation of the misfit e with chemical potential p, = A l l z - pc(T)l b,
(38)
where b ,-., 0.33 and A = 8 • 10-4K -b [272]. The misfit e is defined as a ~ ac]
= ~ ,
ac
(39)
where a is the average lattice constant of the monolayer and ac the lattice constant of the closest C phase. Equation (38) is in agreement with the experimentally observed relationship found at much higher temperatures [284]. In a striped phase a 1/2 power law has been predicted by Pokrovsky and Talapov [280] and observed in Br intercalated in graphite [286, 287]. As any lattice, DWL are prone to defects and they can melt. The elementary topological defects are the dislocations, diagrams of which are shown in fig. 36 for striped and honeycomb DWL. A prime example of DWL melting is the intermediate DW fluid phase (DWF) between a C and an hexagonal IC phase (HIC) which has been predicted by Coppersmith et al. [282]. As the average DW separation increases the DWL weakens, reflected by an increased meanderings of the walls until a mechanical instability occurs which can be represented by the KTHNY theory. This instability is not temperature driven, so the re-entrant liquid phase produced could extend all the way to zero temperature (fig. 37). The DWF phase has been observed in several systems, including Kr on graphite, which are discussed in the next section. In the limit of strongly overlapping DW, the smooth IC monolayer is found. It is in this limit that another substrate effect exists which has been predicted by Novaco and McTague [145], known as "orientational epitaxy". Novaco and McTague showed that the orientation of the adsorbed monolayer need not match that of the substrate, i.e., that of the C phase. Since shears are usually energetically more favorable than compressions, more adsorbed atoms can be placed close to adsorption sites if the monolayer rotates with
568
Ch. 55
B. Jo6s
b
13
(Q)
(b)
Fig. 36. Domain wall dislocation dipoles; (a) in a striped phase, and (b) in a honeycomb domain wall lattice. When the dislocations appear unpaired, a domain wall fluid is obtained. Compare figure (b) with fig. 12.
%
LIQ UID
,,,
COMM~I~NCOMM SOLID V
SOLID
| Fig. 37. Possible phase diagram for an adsorbate with C and IC phases, as a function of temperature T and chemical potential ~. respect to the substrate. The degree of rotation depends on the misfit of the monolayer e. The onset of the rotation is a function of the strength of the corrugation [264], the density modulations within the monolayer [272] and temperature since the energy variation with respect to rotation is small. In the opposite limit lies the C phase. It is not an elastic medium and its melting is more accurately viewed as a disordering process in a lattice gas (see for instance [288, 289]). When considering the role played by dislocations in the melting of the adsorbate lattice, the range of lattices going from smoothly modulated to DW-like will be considered.
w
The role of dislocations in melting
569
3.3.4.3. Melting of monolayers adsorbed on graphite.
In most experiments the graphite sample is placed into a cell in which gas is pumped. The pressure in the cell controls the chemical potential of the gas, and hence the adsorption on the surface and the lateral pressure on the 2D solid. Together with temperature it is the key thermodynamic variable. Using the equation of state for the gas and the known number of adsorption sites on the graphite sample (obtained using the C phase of krypton), a coverage can be calculated and used instead of the pressure as the other thermodynamic variable. In terms of coverage, C and IC phases are easier to characterize. One way to classify the many adsorbed monolayer films on graphite is to group them according to the general features of their coverage vs temperature phase diagrams, because these features are a very good measure of the effect of substrate corrugation. If we follow this principle, developed in detail by Moses Chan [129] for the adsorbed monolayers on graphite, four categories are found: (i) Monolayers of Ne, Xe, CD4 and CH4 which are incommensurate at melting and have phase diagrams resembling those of their bulk counterparts (see fig. 38), (ii) monolayers of Ar and C2D4 similar to (i) but with a very narrow or non-existent solid-liquid coexistence region, (iii) the monolayers Kr, N2 and CO which have a very stable x/~ x x/~ C phase persisting up to temperatures higher than the "expected" liquid-vapour critical temperatures, and a C-IC phase proceeding via intermediate DW fluid regions (see fig. 39), and (iv) light molecules such as 3He, 4He, H2 and D2 similar to (iii) but with a less stable C phase and a C-IC transition which can involve a striped DW phase (see fig. 43). We will not attempt an exhaustive review of each of these categories but focus on some well characterized examples. Our interest is in the melting kinetics of the elastic lattices, i.e., of the IC phases. /5, it
;I
,?
sltl I S
~I i s s l sS /
YF+ S. . . . i
I
I
LI +, V,
S+V
""
i
i
F
I
] L.. S i
i
T Fig. 38. Schematic coverage n versus temperature T phase diagram of a physisorbed monolayer exhibiting first order melting such as Xe on graphite. S, L, V and F stand for solid, liquid, vapour and fluid phases respectively. Dashed lines stand for first order phase boundaries.
570
B. Jo6s
Ch. 55
IC
f / !
C+F
,
! !
I
] /
/
I
T Fig. 39. Phase diagram of Kr, N2 and CO on graphite. The tricritical and critical melting points are shown as solid triangle and solid circle respectively. Solid and dashed lines stand for continuous and first order phase boundaries. The nature of the transition along the dotted line is less certain [129].
(i) The most studied example in this category is Xe. Xe favors a lattice spacing of 4.40 A, noticeably larger than 4.26 A, the lattice spacing of the ~ x v/3 C phase on graphite. Upon adsorption, the Xe monolayer has a tendency to compress. This compression preserves essentially its harmonicity [290] and the properties of its topological defects [163]. Its phase diagram (see fig. 38)closely resembles that of a bulk solid [129]. Its superlight DW are wide (~ 100 A) and consequently atomic separations on the walls are close to those preferred in the ideal monolayer [291]. At submonolayer coverages Xe patches exhibit typical first order melting behaviour [292, 293]. In the compressed monolayer region, the melting temperature increases rapidly from the triple point at 100 K. High resolution X-ray scattering experiments claim to find a crossover from the first order to continuous melting behaviour with increasing pressure near 125 K [294]. The continuous melting regime has been touted as an example of KTHNY melting and has been the subject of a number of careful X-ray scattering experiments [ 144, 293-297], some on single crystal substrates [144, 294]. These experiments find correlation lengths o up to 2000 A in the solid phase. The variation of the BOO and PO has been argued to be as predicted by the KTHNY theory and not to be due to the substrate field [296], in particular since similar results are observed on a much smoother substrate Ag(111) [298]. The transition is however fairly sharp [293]. A similar crossover from first order to continuous has been observed in vapour pressure isotherm measurements, albeit at a higher temperature, 155 K in one experiment [299, 300] and 147 K in another [301]. A combined heat capacity and vapour pressure isotherm experiment on much larger graphite samples finds that the melting transition is always first order [302]. This conclusion is based on the persistent sharp heat capacity peak that appears at melting. So thermodynamically the melting is first order but there is persistence of BOO. Abraham has argued
w
The role of dislocations in melting
571
on the basis of molecular dynamics (MD) studies that the third degree of freedom gives the appearance of a continuous transition; there is a continuous exchange of atoms between the first and second layer producing near melting a small temperature interval of solid-liquid coexistence [303, 304]. In computer simulations (see section 3.4.2) it is now believed that the ideal Lennard-Jones (LJ) monolayer has a first order transition, although also some form of intermediate phase may exist (a Xe monolayer is for all practical purposes a LJ monolayer) [305]. (ii) Ar is a much smaller atom than Xe. Its preferred interatomic spacing on graphite is 3.8 ,~, significantly smaller than the 4.26 A of the nearest C phase. Ar is always IC and its phase diagram also resembles that of an ideal monolayer. The difference with Xe is that it does not seem to have a solid-liquid coexistence region. A series of intriguing results on this system show a unique melting behaviour. A heat capacity study finds a small sharp peak beside a broad anomaly, at the melting temperature predicted by X-ray scattering studies [307]. The small peak is consistent with an "abrupt" density change of about 0.2% over a temperature range of 0.3 K. But scattering studies of melting reveal no evidence of density discontinuity at melting, just a continuous change in correlation length from 900 A to 200 A [308] in one study and 1500 ,~ to 400 ,~ in the other [309]. In addition, although the Novaco-McTague orientational epitaxy present in the solid phase decreases as the solid melts, some of it persists into the liquid phase [309]. This behaviour is consistent with the KTHNY theory where the creation of dislocation dipoles destroys the PO but maintains the BOO of the solid. Zhang and Larese [310] have recently managed to conciliate to a degree the two types of studies. From vapour pressure isotherm studies they found evidence of two-stage melting. They observe a narrow peak in the temperature derivative of the lattice constant. It is not clear whether this peak makes the transition weakly first order or can be explained within the context of a KTHNY scenario. The intermediate phase between the solid and isotropic liquid exhibits short range PO and solid-like properties such as the persistence of the orientational epitaxy, phonons and a compressibility anomaly. What seems quite clear is that this unique behaviour is the result of the substrate corrugation, and may reflect the existence of two competing types of ordering, that of an atomic lattice and of the DWL. The adsorbed Ar monolayer is, after all, an IC lattice with DW, even if they are fairly dense. In monolayers expanding upon adsorption the DW maintain separations close to those favored by the adsorbate while within domains the lattice is expanded. DW tend for this reason to be more important than in monolayers compressing upon adsorption. The intermediate phase could be a defected DWL. Shrimpton et al. [311] illustrated this point using MD simulations. Argon's melting behaviour bridges the floating monolayers with the well resolved DWL such as Kr which is discussed in the following section. The melting of C2D4 or C2H4 shows all evidence of being continuous [312, 313]. The computer evidence, however, shows that it is not of the KTHNY type but is associated with the gradual tilting of the long molecular axis towards a direction perpendicular to the surface [314]. The reduced local density produces a gradual disordering of the monolayer. Similar explanations have been given for the continuous melting of hexane as opposed to butane, a similar molecule, which has a first order melting transition (chemical formula CH3(CH2)n_2CH3, n = 4 for butane and n = 6 for hexane). Hexane, the longer molecule, goes from a trans- to a cis- or gauche-configuration [315, 316].
572
B. Jo6s
Ch. 55
(iii) The Kr monolayer is probably the most interesting of the adsorbed rare gases because it illustrates a lot of the physics discussed in section 3.3.4.2 in particular in relation to the C-IC transition and DWL. It has a very stable C phase, so stable actually that its melting temperature is comparable to the critical temperature Tc of the liquid gas coexistence region. Its phase diagram is described as having an incipient triple point [317, 259] (see fig. 39). This C phase has been used to determine the number of adsorption sites on graphite samples, using the fact that the amount of adsorbed gas is easily determined from the gas pressure in the cell and the number of moles introduced into the cell [258]. In the C phase the Kr atoms are separated well beyond their equilibrium lattice parameter of 4.02 ,~, to 4.26 ,~ as mentioned earlier. So, although this phase is very stable, by increasing the gas pressure more atoms can be easily adsorbed. These additional atoms form mobile DW. Close to the C phase their interactions are too weak to stabilize them into a regular lattice and they form a DWF as explained above. As the density of DW increases they solidify into a honeycomb array of walls. Kr shows clearly the predicted fluid phase between the C and IC phases [259]. Upon further compression the DW merge and the monolayer rotates with respect to the substrate [ 147] exhibiting the orientational epitaxy discussed in section 3.2.2.5. X-ray scattering finds the melting of the DWL to be continuous. The DWL seems a prime candidate for a dislocation unbinding melting mechanism [162, 163,274]. The IC solid-DWF boundary was mapped out by Shrimpton and Jo6s applying the KTHNY stability criterion of eq. 32 to the string model discussed in the previous section. At lower coverages or lower misfits, the IC solid melts into the DWF which, with increased temperature, evolves into a conventional fluid. At higher coverages the IC to fluid transition is still continuous. These X-ray scattering results are confirmed by heat capacity measurements on a very similar system, CO on graphite [285]. The entire submonolayer and monolayer phase diagram of CO on graphite is surprisingly similar to that of Kr on graphite. A statistical model called the "Potts model", a generalization of the Ising model (for a review on Potts models see [318]), has been successfully used by Caflish, Berker and Kardar [320] to map out the overall phase diagram of the Kr on graphite system. A given C phase can be in any one of three sublattices, a, b or c (see fig. 40). If the adsorption sites are split into triplets of sites, an order parameter can be defined which specifies which type of site is occupied by an adsorbed atom. The relevant Potts model is therefore the three-state Potts model. In the submonolayer regime, a "dilute Potts model" is used with an order parameter determining whether a triplet is occupied [319]. In the IC phase, hexagonal domains are used as units, and they can be in any of the three sublattices, a, b or c. The interaction energy between two domains will depend upon the type of wall separating them (see fig. 40). It is actually most effective to consider triplets of domains because the vertex configurations determine the energy of the lattice and they depend on the three adjacent domains. To correctly account for this topology "chirality" has to be introduced in the Potts model, hence the name "helical Potts model". "Chirality" is the dependence of the interaction energy between the three nearest neighbour domains on the order of appearance of the sublattices: U(a, b, c) = U (b, c, a) ~: U (a, c, b). Renormalization group techniques are used to solve for the phase diagram. This last model is very interesting from the perspective of this review because
w
573
The role of dislocations in melting
Dislocation
(
~, subl.c occupied
0
Super2ovy
Heavy__..l crossing
crosstng
subl.a
t
Heavy wall
subl. b
t
subl.a occupied
Super-heavywall
Fig. 40. Structures in a compressed adsorbed layer on graphite in the IC phase near the x/~ x x/~ C phase. It shows the possible wall crossings in the honeycomb phase and the neighboring occupied sublattices (subl). [320]. it takes into account dislocations in a natural way (see fig. 40). The full phase diagram of Caflish, Berker and Kardar [320] for Kr on graphite is shown in fig. 41. Figure 42 shows one of their fully dislocated configurations which occurs in the DWF phase. The alternance of superheavy and heavy wall segments reflects the dissociation of the dislocations in the DWL into two partials separated by a stacking fault, the heavy wall, as discussed in section 3.2.3 and shown in fig. 10. This picture complements the "string model" of Shrimpton et al. discussed in the previous section. (iv) The light molecules 3He, 4He, H2 and D2 adsorbed on graphite have been the subject of a great deal of interest partly because of their highly quantum character. Their phase diagrams (see fig. 40) are exceedingly similar to each other (for the most recent studies and references see [321,322], and for a review [129]). They are also dominated by the x/~ x x/~ C phase (see fig. 40). The melting of the C phase into the fluid (F) phase is a vacancy driven continuous transition of the 3-state Potts model type with its characteristic 1/3 specific heat exponent [323, 325-327]. With increasing coverage, the C monolayer becomes incommensurate (IC); at low temperature, first forming a striped phase evolving into a hexagonal IC (HIC) phase [321, 322]. At higher temperature, the C-IC transition proceeds with a hexagonal DWF which solidifies with increasing density into a HIC phase. The transitions from the DWF to the HIC and striped phases are continuous [129] This is very similar to Kr on graphite, with the difference that Kr does not show a striped IC phase as we have just seen earlier. For Kr high entropic effects produced by wide mobile DW favoured the HIC phase. Theoretical studies support the observation of the striped phase in the monolayers of light molecules [328-330]. The domain walls are fairly narrow although the pinning stress appears to be still fairly small [330].
574
Ch. 55
B. Jo6s
60 / '
Temperature(K) I00 ' ' ~~
b 0]- Commensurate
'
'
..aox~ d
%
-
o-5~- , ~
I
-
Fluid - ~
I/,
,
,,
, ,........
1
-
I~1
,,
I
50 Temperature (K)
"
i
I00
0
"", "
Incommensurate / / Commensurate 9J solid// solid 2 /
-
0
l/
400
v
(1)
i
200
if) o3
~' (3..
" 0
Fig. 41. Reentrant phase diagram (curves) in the pressure and temperature variables calculated by Caflish, Berker and Kardar for Kr on graphite within the helical Potts-lattice gas model. Also are shown experimentally determined phase transition points (for details see [320]).
The above overview is far from being exhaustive. A number of systems have not been mentioned such as CF4 on graphite [331 ] which exhibit interesting phase diagrams with similar physics. 3.3.4.4. 2D melting on other substrates. Graphite has not been the only substrate used to study 2D melting. We noted already earlier that Xe has been adsorbed on Ag(111) to check the effect of the substrate corrugation on the melting transition. Other substrates involve a range of transition metals with rare gas adsorbates, such as Kr [335] and Xe [336] on Pt(111), and alkali metal adsorbates, in particular K and Cs on a wide range of substrates, Ru(001), Pt(111), Ni(111), Rh(100), etc. Adsorption in these systems is induced by the high degree of electroposivity of the alkali-metal atoms (for reviews, see [332-334]). The alkali atom has an induced dipole as it draws close to the surface which is of opposite direction to the surface dipole which produces the work function of the surface. A variety of phase diagrams has been observed. The ones of interest to us are those exhibiting significant incommensurate phases such as K [337] and Cs/Cu(111) [338], K on Ni(111) [339] and on Ni(100) [340] which are all on substrates of hexagonal symmetry. The melting of the incommensurate phases of these systems has been studied in the references quoted. The literature has been reviewed by R. Diehl, the leader of one of two groups reporting work on this subject. LEED measurements show evidence of a gradual loss in ordering evidenced by the increased widths of the diffraction peaks in the first three of these systems. So far it appears to be the only evidence available for
w
The role of dislocations in melting
575
Fig. 42. One of the many fully-dislocated configurations which can occur in the zero-temperature dense fluid. Heavy and superheavy-wall segments are drawn with thin and thick lines respectively. Walls meander with such segments alternating between dislocations. They are maximally packed but never cross [320].
a continuous melting transition. There is not enough detail to argue whether these are more like DWL or floating monolayers. In the last system, desorption occurred before disordering could be observed. 3.3.4.5. Intercalates. We mentioned earlier Br intercalated into Gr as an example of a striped DWL [286, 287] and that in situ high resolution X-ray scattering experiments have found the 1/2 power law for the variation of the misfit with temperature. The melting transition from the X-ray scattering perspective is found to be continuous, consistent with the picture of a striped DWL disordering in a KTHNY way. But Mochrie et al. [341 ] find that the decay of positional correlations as measured by a sine wave order parameter has a power law exponent evolving continuously from 0.04 to ~ 2.0 contrary to KTHNY theory. For the hexagonal DWL of K and Rb layers intercalated in graphite, the melting transition is found to be smooth with a power law decay with temperature of the elastic constant C44 [342] as observed by neutron scattering. With nitric acid-intercalated graphite the dislocation density was followed by relating it to the variation of the correlation length and the diffusion constant [343]. 3.3.5. Discussion In summary, the best candidates for dislocation mediated melting are the IC systems, where a DWL melts continuously, in a manner consistent with a scenario involving a mechanical instability, similar to the one forming the basis of the KTHNY theory. This should not come as too much of a surprise. DWL, in contrast to the actual adsorbed monolayers, form an ideal 2D elastic medium. The adsorbed monolayers have to contend with the effects of the third degree of freedom (effectively turning the whole system into a surface), and substrate effects (modulation in the adsorption potential, most notably).
576
B. Jo6s
Ch. 55
3.4. Computer simulations on systems of particles 2D melting offers a major challenge to computer simulation studies. At first sight it seems like a simple enough problem, to set up a lattice, heat it up and follow either particle trajectories in molecular dynamics (MD) or thermodynamic quantities in Monte Carlo (MC) simulations, but quickly the limitations of the size of the system, the boundary conditions and the length of time of the runs raised questions about the significance of the results. There have been nearly as many simulations done on 2D melting as experiments. A number of extensive reviews have been published on different aspects of these calculations (see [118-120, 173]) and we will refer to those for more details, in particular for those not directly relevant to the focus of this review. 3.4.1. The methods
Two methods of computer simulation are used: the Monte Carlo (MC) and the molecular dynamics (MD) methods. In both methods results for quantities of interest are obtained as averages over configurations generated by the repeated application, of basically simple algorithms, which govern the motion of the particles. The physics is classical. Quantum effects, except for those incorporated into the interparticle interactions or the interactions with a substrate are ignored. In the MD method a classical hamiltonian is set up and the particle motions are determined as a discretized integration of the equations of motion. The time increments are called time steps and they are a key parameter in ensuring a reliable solution. Various algorithms have been set up for different statistical ensembles with the conserved quantities incorporated into the Hamiltonian or added as additional constraints. Considerable developments in the field have occurred over the last two decades. The melting studies have used mainly two types of algorithm: (i) a canonical ensemble with the system coupled to a heat bath (some care is required in the choice of constraint to ensure that a canonical distribution is maintained [344, 345, 118]), and (ii) a constant pressure ensemble usually implemented by coupling the system to a "piston" [346]. The mass of the piston is arbitrary but the algorithm works well if it is chosen so as to have area fluctuations on the same time scale as the time it takes for sound to propagate through the sample [346]. In the MC method [347], as its name implies, particle motion is random. It is, however, governed by a probability distribution which in the simplest case of the canonical ensemble is the Boltzmann distribution. In this case particle moves are determined by a random number between 0 and 1. If this number is lower than exp(-/3AE) the move is rejected. In principle the MC method offers considerably more flexibility in implementation than the MD method. It is not restricted, as the MD method is, to obeying the time evolution imposed by the equations of motions. Phase space can be sampled much faster by a clever choice of moves. For example a constant pressure ensemble can be implemented simply by using volume changes, as attempted moves with the appropriate statistical weight. Thermodynamic quantities are therefore more easily calculated. The loss of dynamical information is however not trivial.
w
The role of dislocations in melting
577
The two methods are complementary. It is hard to discriminate against one or the other. When discussing the effectiveness of each method in the study of melting, there are some crucial considerations discussed below. 1. Choice of the ensemble. A constant density ensemble gives the opportunity to investigate whether or not coexistence of the liquid and solid phases can be achieved, a signature of the first order transition, or whether a hexatic phase with long range BOO is possible. Particle trajectories have been invoked to demonstrate coexistence [348-350]. True separation is however not observed but an alternance of solidlike and liquidlike patches. In the 1/r 12 system at the apparent coexistence temperature the solid part was found by Broughton et al. [368] to be stable but to melt and reform during the simulation indicating a low interracial energy between the solid and liquid phases. In very long runs on the same system Naidoo, Schnitker and Weeks [353] find similar results but they temper their conclusions with arguments of limitations of time, sensitivity to boundary conditions, and finite size effects. In both calculations the energy versus density diagrams show a "van der Waals" type loop characteristic of a first order transition (see fig. 44). In principle the long distance behaviour of the BOO should be able to distinguish hexatic and two-phase coexistence. However in a finite system both may exhibit persistence of BOO, and long range behaviour due to the periodic boundary conditions may be elusive. In a constant pressure ensemble there is no coexistence region. If no hexatic phase is found, then none exists. But the width of the transition could be quite narrow in the path chosen to cross the transition. The order of the transition is not easily determined either if the transition region is narrow. Implementing the constant pressure may induce an artificial effect from pressure waves as argued by Toxvaerd [351 ]. But Toxvaerd and Abraham and Koch [352] disagree on this issue. 2. Choice of boundary conditions. Periodic boundary conditions are usually chosen. They eliminate edge effects but they favour superheating by stabilizing the solid and preventing the nucleation of defects such as vacancies [349, 118]. In a recent study on the 1/r 12 system, Naidoo et al. [353] have used different boundary conditions and found significant differences in the variation of the energy with density when changes in boundary conditions were made (see fig. 44). 3. Finite-size effects. It is hard to believe that these would not be important especially in systems where the dislocation cores are large such as the Lennard-Jones system [162, 163]. In spite of this the subject has been explored in only a few studies. Toxvaerd had observed significant size dependence on defect density and elastic constants [354, 355]. Udink and van der Elsken have also come to a similar conclusion when looking at PO and BOO in the Lennard-Jones system [356]. And so have Zollweg, Chester and Leung [235], and Zollweg and Chester [360] on the hard-disk and the 1/r 12 systems. All these studies indicate that the first order behaviour is weakened as the system size is increased. But, is that any proof of a continuous transition? Larger systems will have larger kinetic barriers to transform into the liquid phase, leading to the next question, the effects of finite time. Ultimately what is required is finite size scaling, such as was done by Lee and Strandburg [357] on the hard disk system. 4. Effects offinite time. The greatest frustration to any one involved in computer simulations is the constant reminder that MD runs correspond typically to picoseconds and at
578
Ch. 55
B. Jo6s
HIC
i~lO-.---~,
9 o 9 o,,,,,
.....................
,..,
C+F / t
0
[="
t
i s
qFig. 43. Typical phase diagram for a light molecule adsorbed on graphite, namely 4He, 3He, H2 and D2, SIC and HIC stand for unaxial incommensurate with stripes of domain walls and a hexagonally symmetric incomensurate phase respectively. C, F and DWF stand for the ~ • v ~ commensurate (C) phase, fluid and domain wall fluid phases. Solid lines stand for continuous transition, dashed lines for first order, and dotted line, uncertain. Solid triangle and circle locate positions of 3-state-Potts critical and 3-state-Potts tricritical points [129].
best nanoseconds, rather short times compared to what is available to experimentalists. Novaco and Shea [370] observed in the 1/r 5 system relaxation times longer than their longest runs (100000 time steps). Naidoo et al. [353] found evidence of kinetic bottlenecks which prevented the break-up of the 1/r 12 system set up in an unstable hexatic phase, even after five million time steps. This suggests that long times on a small system may be better than a large size without good enough statistics, which takes us back to the need for finite size scaling. Having made all these precautionary statements let us now look at what has been found for some key systems. 3.4.2. H a r d core p o t e n t i a l s
In hard core potential systems a repulsive core keeps the particles apart. We will present results on the hard disk system, the Lennard-Jones monolayer, the 1/r 12 potential and the Weeks-Chandler-Andersen potential (WCA) [ 184] systems (the last potential is just the repulsive part of the LJ potential). Weeks [358] has shown that for central potentials the steeper the repulsive part the larger the density discontinuity in a first order transition. For this reason we will discuss the systems in increasing order of the smoothness of the interparticle interactions. (i) The melting of the hard disk system has been the first studied, even before the predictions of the KTHNY theory were published. In 1962 MD simulations were performed by Alder and Wainwright [359]. From the variation of the density versus pressure
The role of dislocations in melting
w
579
2.9
2.7
E!
F4
2.5
,:,
U
0
0
2.3
2.1
1.9
9
I
0
oO
9 9
13
m
9 t
0.950
t
t I I 0.970
t
, 1 t 0.990
,
• I , 1.010
,
,
I , 1.0"50
t
, 1.050
Fig. 44. E n e r g y - d e n s i t y d i a g r a m , for l i q u i d - s o l i d traverse of periodically replicated s y s t e m P (filled circle), of s y s t e m F1 (open circle) w i t h s u r r o u n d i n g solid-like walls, and s y s t e m F 4 (squares) w i t h s u r r o u n d i n g l i q u i d - l i k e walls [353].
curve they deduced a first order transition. There have been a few other studies since, although the attention moved more towards the other hard core systems. Recently it has been given renewed attention to elucidate the controversy concerning the order of the transition in 2D systems [235, 360, 357]. Being the system with the steepest potential interaction, evidence of continuous melting in this system would be a strong indication of similar behaviour in the other hard core systems. The simplicity of the potential also allows for the study of large systems or long times, or both. Zollweg and Chester [360] showed a significant decrease in the first order character when going from system size 1024 to 16 384 and even up to 65 536 in their MC simulations. They observe in the pressure density curve a shorter tie-line than had been previously observed by Alder and Wainwright in their 870 disk system [359]. The supposition is that for even larger systems it will vanish. The intermediate phase has not been identified. They contest on the grounds of finite size effects the contention by Lee and Strandburg [357] that the transition is first order. However, the jump in density between the two phases may decrease with system size but that does not mean it will go to zero in the thermodynamic limit. Lee and Strandburg use an elegant approach developed by Lee and Kosterlitz [361]. It involves a finite size scaling argument based on a MC calculation of the bulk free energy barrier between the liquid and solid phases. Lee and Strandburg arrive at the opposite conclusion to the previous authors. The energy barrier increases with increasing size of the sample. The method only requires small systems, much smaller than the correlation length at the transition and therefore long runs with good statistics can be achieved. The method has been tested with success on the 3-state Potts model [362]. It is however not clear that, even the largest system studied, is sufficiently large for the melting kinetics. (ii) The Lennard-Jones monolayer received a great deal of attention not only because it is the most physical of the simple potentials but because it is readily identifiable with the rare gas monolayer. And as we know, this class of monolayers has been the subject of
580
Ch. 55
B. Jo6s
extensive experimental investigations. A study by Frenkel and Mctague [363] following soon after the KTHNY theory, reported observation of the hexatic phase. But quickly the evidence in favour of a first order transition began accumulating [364, 365, 349, 118]. At this stage the situation is less clear but quite interesting and revealing about two dimensional melting. The first evidence of unusual behaviour came with the observation of a low interface free energy in the coexistence region [364]. This was followed by the observation of noticeable size dependence which indicated weakening of the first order transition with increasing sample size [354, 355]. The coexistence region proved the most revealing. It does seem to exist. Particle trajectory pictures of Abraham [348-350] show patches of solidlike and fluidlike regions. His constant pressure simulations show hysteresis [349, 118]. Loops were observed in the variation of the pressure with density [366, 364, 354] (see fig. (45)). Such loops are characteristic of a first order transition. In spite of this Udink and Frenkel find the solid-liquid coexistence region to be a well defined homogeneous state with a BOO transition [305]. The hexatic-like phase has the predicted properties of a KTHNY state but it seems rich in defects. The KTHNY theory has definite predictions about the elastic constants specifically at melting. K defined in eq. 23 should be equal to 167r. Using K, Abraham [349] had early proposed that the KTHNY melting point was a point of mechanical instability and hence the melting point of a superheated solid. He based his conclusion on free-energy calculations that placed the transition at a lower temperature than the point where K 167r. The more recent calculations of Udink and Frenkel [305] find melting temperatures for which K ~ 167r. Matters are complicated by evidence that the shear modulus depends sensitively on the size of the system, and hence so does K [355]. The LJ system seems to be in an intermediate situation with some possibility of hexatic like character but thermodynamically with the first order melting transition. A van der Waals equation of state argument when applied to this system would still prefer the first order transition [367]. (iii) The work of Broughton et al. [368] on the 1/r 12 system shows coexistence but also finds a small interface energy between the solid and liquid phases. Recent studies further confirm the very weak nature of the first order phase transition and the existence of large correlations in the coexistence region" Zollweg et al. [235, 360] with their study of size dependence and Naidoo et al. [353] with their extensive studies on the effects of boundary and simulation time. Naidoo et al. [353] find in the intermediate region very similar defect patterns as have been observed in the photographs of the colloidal spheres discussed previously (see section 3.3.3.2). There is a greater clustering of the defects than expected in an hexatic phase, a point to which we will return later. (iv) The WCA potential is a repulsive potential obtained by truncating the LJ potential at its minimum and shifting it upwards by an amount c equal to the depth of the LJ potential. As a result, both the potential and its derivative are continuous at the cut-off. The WCA potential is given by VWCA(r) =
{ 4e[(~r/r) 12 - ( o r / r ) 6] + e 0
for r < rc, for r ~> rc.
(40)
Glaser and Clark have carried out extensive calculations on the WCA system. [140, 172, 173]. Their thermodynamic studies on 896-, 3584-, and 8064-particle systems show the typical first order melting transition. The most interesting aspect of their
w
581
The role of dislocations in melting ,
i
I
I
i
lad
i
i
i
'
i
i
I
'
I
i
1
'
/
// /
I
I
/
tr"
/ /"
IM IZ.
n
/'/'/
t~ la r =}
I
./1
I,LI
I
0.80
!
,
I
l
I
0.8,5
REDUCED
I
i
I
0.90
DENSITY
Fig. 45. Pressure (unit e/tr 2) as a function of density (unit cr-2 for the Lennard-Jones potential system at a temperature T* : 1.0 [354]. The full line and full circles for a 3600 particle system, and the dotted line and empty circles for the smaller 256 particle system. Also shown is the extension of the fluid pressure and the line in the small system (dotted line) together with the virial expansion pressure and its tie line (dotted and dashed line). work is their study of the defect structures in the liquid. It starts from the Voronoi polygon construction which Glaser and Clark used to investigate in detail the defect structure through the melting transition. They also used the dual construction, the so-called Delaunay triangulation where all nearest neighbour pairs are connected by "bonds", which therefore focuses more on the bond lengths and the bond angles. They observe a condensation of defects at the transition. The same is observed in the dual construction. The defect structures show a great deal of variation at the transition. The general impression of polycrystalline disorder is in particular apparent in the Delaunay triangulation. This observation has lead Glaser and Clark to set up the defect mediated melting theory, where the focus is on geometrical rather than topological defects, which was discussed earlier in section 3.2.5.
582
B. Jo6s
Ch. 55
3.4.3. Intermediate strength interactions There have been a few studies on intermediate strength interactions such as the potentials 1/r 6 [369], 1/r 5 [370, 371] and 1/r 3 [372-374]. These studies also point to a weak first order transition. But concerns about long time fluctuations and pre-transitional effects left Allen et al. unwilling to draw a conclusion on their study of the 1/r 6 system [369]. Novaco and Shea [370] arrived at similar conclusions. 3.4.4. Soft core interactions Using a scaling property inherent to 1/r n potentials, which he applied to the ClausiusClapeyron equation, Weeks [358] showed that when n < d (where d is the dimensionality of the system) the system upon melting cannot have a density discontinuity. It is expected that this would apply to any similar long range potential with a soft core, such as the Coulomb interaction. It, however, does not exclude the possibility of an entropy driven first order transition but it would soften its character. We have already considered the experimental realization of such a system in the 2D electron lattice on the surface of liquid helium (see section 3.3.2) and reported good evidence of a continuous transition of the KTHNY type. There is however no agreement among the various computer simulations. First order transitions have been reported which agree in the location of the transition with experiment [375] or in the entropy jump [376, 373] with the density wave theory of Ramakrishnan [171]. Locating the transition is not itself proof of predicting the correct nature of the transition especially in view of the small difference in free energies between the liquid and solid phases. The theory of Ramakrishnan is a mean field treatment so its reliability as to the order of the transition is not clear. Cheng et al. [377] calculated by MD simulations pressure-area isotherms and observed hysteresis leading them to conclude also to the first order transition. The discontinuities in the isotherms are almost imperceptible in accord with Weeks' predictions. These first order predictions remain challenged by the studies of Morf [ 167,378,379] and more recently Naidoo and Schnitker [380]. In the Fisher et al. paper [167], the dislocations were found to obey elastic behaviour down to very close separations. On the other hand, because of the long range of the potential interaction, the dislocations are expected to be mobile. These two properties favour dislocation mediated melting. The most revealing results are however those of Naidoo and Schnitker [380] who made extensive simulations on the Yukawa system in an attempt to explain the experiments on the colloidal suspensions. Their simulations similar to the ones mentioned above on the 1/r 12 system [353] were run for very long times, often millions of time steps. Their study, however, relies on the absence of a volume change on melting, a conclusion based on the similarity of the Yukawa potential with the 1/r potential, and on a calculated equation of state. The absence of a volume change permits a search for the hexatic phase by varying the density. By identifying the onset of BOO and PO with the densities at which the orientational correlation length exceeds 30a0 (where a0 is the lattice parameter) and the positional correlation length 15a0, in a similar way as Murray and Van Winkle [231] did for the colloidal suspensions, they find an intermediate phase ~ 1% wide in density which may be a hexatic phase. The decay in the correlation functions Cd(~' ) and C6(~) is algebraic as predicted by the KTHNY theory although not with the same exponents. The discrepancy is in particular large for ~Td, more than a factor of 2 (0.7 instead of
w
The role of dislocations in melting
583
1 / 4 - 1/3). The analysis of defect configurations shows in general a very complex structure usually with clustering of dislocations as has been observed in some other simulations [353, 381] and experiments on the colloidal suspensions (see section 3.3.3.2). Interestingly the "clumping" of defects is more similar to that of Tang et al. [237] than that seen by Murray et al. [219]. Murray [219] argued that the clumping was the result of a lack of equilibration and that it takes hours for dislocation climb to be completed. This time is not reasonably possible in computer simulations. It should also be mentioned that the sphere sizes are not the same in the Murray et al. and Tang et al. experiments, 0.305 ~tm in the former and 1.01 ~tm in the latter. 3.4.5. The nearest neighbour piecewise linear force interaction This is also a soft core potential. But in addition it is only nearest neighbour. It is defined in eq. 33. Combs [382] considered the melting of the monolayer with this interaction. He finds in his MD studies first order grain boundary type of melting similar to what had been proposed in the theory of Chui [168]. Runs were fairly short, ~ 4007- after equilibration. 3.4.6. Synthesis Computer simulations, although having some limitations in particular of time, boundary conditions, and system size, point at a 2D melting behaviour which is different from bulk melting, and does not appear universal. The nature of the interparticle interactions is important. Weeks [358] seems to have been the first to draw attention to this fact, when he argued that the density discontinuities decreased with decreasing n for 1/r n interactions. The sensitivity of the dislocation properties to the range of the interactions [165] also bears evidence to this fact. Computer simulations show a gradual evolution of behaviour as one goes from the hard disk system to the soft core potentials. As the core softens recent evidence points more and more towards the existence of an intermediate phase. It is not seen in the hard disk system, but BOO is found in the solid liquid coexistence region of the Lennard-Jones monolayer [305]. In the 1/r 12 system, and in the screened Coulomb system [380], defect patterns similar to those seen in the colloidal suspensions are observed. The calculated patterns show some evidence of not being totally thermalized. Kinetic bottlenecks require simulation times still beyond present capabilities. The observed intermediate phase is not as expected from the hexatic phase of the KTHNY theory. There is more defect clumping and an overall higher density of them. That the intermediate phase is not identical to the hexatic phase proposed by the KTHNY theory should come as no surprise since this theory assumes elastic dislocation dipoles with point-like cores. Clumping of the defects may at least partially be due to extended cores, which have been demonstrated to exist in the Lennard-Jones monolayer [ 162, 165]. The theory also excludes the possibility of aggregation of dislocations as it is derived within the dilute limit. This aggregation could lead in some cases to the first order grain boundary transition predicted by Chui [ 168]. The evidence for such melting behaviour is the strongest for the nearest neighbour piecewise linear force interaction system. But some key features of the hexatic phase predicted by the KTHNY theory are seen in the smoother potential systems: the persistence of BOO and algebraic decay of correlation functions.
584
B. Jo6s
Ch. 55
Computer simulations overall still tend to favour a first order melting with loop like energy-density and pressure density variations, but the homogeneous hexatic-like intermediate phases found in the smoother systems including the LJ system indicates behaviour quite different from that observed in bulk systems. This intermediate phase is only seen in constant density simulations. The clue in the melting dynamics lies in this coexistence region which needs further probing with attention to the boundary conditions.
4. Conclusion It seems that the melting of bulk systems is dominated by the surface, and defect cores. In the latter category belong dislocations, but especially grain boundaries. There is no strong evidence of dislocations being involved in the kinetics of melting. Imaging techniques are progressing rapidly and new insight may be available in the near future if a renewed interest in this subject occurs. The DTM appears useful mainly as an approximate theory of the liquid state near melting. In 2D systems, the situation is quite different. Increased thermal fluctuations will tend to wash away sharp phase boundaries. And topological constraints brought about by dimensionality favour the persistence of BOO into the liquid phase. As discussed by Collins [114] the average number of neighbours in both the liquid and solid phases is six in a triangular lattice [114]. These two facts favour a continuous phase transition. The persistence of BOO may however in itself not be a sufficient proof of a dislocation unbinding process. It is also compatible with a grain boundary process [168]. There is sufficient evidence to suggest that the loss of translational order can be in many cases attributed to the appearance of topological defects. Whether they form clusters, grain boundaries or remain dilute as assumed by the KTHNY theory, will affect the width of the intermediate phase, and will have a great influence on the thermodynamic signature of the transition. There is experimental evidence of continuous melting, in micron and submicron size colloidal suspensions, with screened Coulombic interactions, and similar systems (magnetic or electric holes, magnetic bubbles ...), electrons confined on the surface of liquid helium, incommensurate adsorbed monolayers with DWL character. Liquid crystals may be added to this list. Computer simulations, on the other hand, do not yet offer a clear picture. Do they reveal the present limitations of computer simulations? The melting behaviour seems dependent on the potential interaction. The nature of the melting transition in 2D is still not very well understood. Profound differences from bulk behaviour are found. In addition to dependence on the interaction potential, the melting kinetics may also be more sensitive than bulk systems to boundary conditions, imperfections, and overall the way the 2D system is created.
Note a d d e d in p r o o f There are two recent computer simulation studies further confirming the special character of 2D melting: ref. [383] on the 1/r 12 system (increased evidence of a hexatic phase),
w
The role of dislocations in melting
585
and ref. [384] on the hard disk system (one stage continuous transition). The occurrence of a hexatic phase is predicted in refs [385] and [386] for systems with solid-solid transitions. On the experimental front, ref. [387] reasserts the persistence of BOO through the melting transition of Xe on graphite, while refs [388] and [389] present two new systems with hexatic phases, some Langmuir-Blodgett films, and "plasma crystals" respectively. ("Plasma crystals" are layered materials made of colloidal particles introduced into a charge neutral plasma.)
Acknowledgments This work has been supported by the Natural Sciences and Engineering Research Council of Canada. Helpful discussions are acknowledged with M.S. Duesbery and B. Grossmann.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
N.F. Mott and R.W. Gurney, Trans. Faraday Soc. 35 (1939) 364. B.I. Halperin and D.R. Nelson, Phys. Rev. Lett. 41 (1978) 121. D.R. Nelson and B.I. Halperin, Phys. Rev. B: 19 (1979) 2456. A.P. Young, Phys. Rev. B: 19 (1979) 1855. J.M. Kosterlitz and D.J. Thouless, J. Phys. C: 5 (1972) L124. J.M. Kosterlitz and D.J. Thouless, J. Phys. C: 6 (1973) 1181. V.L. Berezinskii, Sov. Phys. JETP 34 (1972) 610. V.L. Berezinskii, Sov. Phys. JETP 32 (1971) 493. L. Landau, Phys. Z. Sovjet. 11 (1937) 26, 545. L.D. Landau and E.M. Lifshitz, Statistical Physics, 3rd edn (Pergamon, New York, 1980). For a review of the theoretical situation, see: D.R. Nelson, in: Phase Transitions, Vol. 7 (Academic Press, London, 1983)p. 1. A.R. Ubbelohde, The Molten State of Matter: Melting and Crystal Structure (Wiley, Chichester, 1978). L.L. Boyer, Phase transitions 5 (1985) 1. J.H. Bilgram, Phys. Rep. 153 (1987) 1. R.W. Cahn and W.L. Johnson, J. Mater. Res. 1 (1986) 724. S.R. Phillpot, S. Yip and D. Wolf, Computers in Physics (Nov/Dec 1989) 20. M. Born, J. Chem. Phys. 7 (1939) 591. M.A. Durand, Phys. Rev. 50 (1936) 449. J.L. Tallon, Philos. Mag. A 39 (1979) 151. For a compilation of elastic constants, see: G. Simmons and H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties (MIT Press, Cambridge, MA, 1971). K.F. Herzfeld and M. Goeppert-Mayer, Phys. Rev. 46 (1934) 995. F.A. Lindemann, Phys. Z. 11 (1910) 609. J.N. Shapiro, Phys. Rev. B: 1 (1970) 3982. C.J. Martin and D.A. O'Connor, J. Phys. C: 10 (1977) 3521. N.P. Gupta, Solid State Commun. 13 (1973) 69. W.L. Bragg, in: Symposium on Internal Stresses (Inst. of Metals, London, 1947) p. 221. R.M.J. Cotterill and M. Doyoma, Phys. Rev. 145 (1966) 465. W. Shockley, L'Etat Solide (Inst. International de Phys. Solvay, Brussels, 1952) p. 431. S. Mizushima, J. Phys. Soc. Jpn 15 (1960) 70. A. Ookawa, J. Phys. Soc. Jpn 15 (1960) 2191.
586 [31] [32] [33] [34] [35] [36]
B. Jo6s
D. Kuhlmann-Wilsdorf, Phys. Rev. 140 (1965) A1599. F.R.N. Nabarro, Theory of Crystal Dislocations (Oxford Univ. Press, Oxford, 1967). T. Ninomiya, J. Phys. Soc. Jpn 44 (1978) 263. T. Ninomiya, J. Phys. Soc. Jpn 44 (1978) 269. S.E Edwards and M. Warner, Philos. Mag. A40 (1979) 257. A.A. Maradudin, in: Dynamical Properties of Solids, Vol. 1, eds G.K. Horton and A.A. Maradudin (North-Holland, Amsterdam, 1974) p. 1. [37] M. Siol, Z. Physik 164 (1961) 93. [38] J. Holder and A.V. Granato, Phys. Rev. 182 (1969) 729. [39] H. Kleinert, Physica B 127 (1984) 332. [40] H. Kleinert, Gauge Fields in Condensed Matter (World Scientific, Singapore, 1989). [41 ] A. Ferraz, Solid State Commun. 54 (1985) 413. [42] I.A. Kotz6 and D. Kuhlmann-Wilsdorf, Appl. Phys. Lett. 9 (1966) 96. [43] D. Turnbull, J. Appl. Phys. 21 (1950) 1022. [44] W.T. Read, Dislocations in Solids (McGraw-Hill, New York, 1953). [45] R.M.J. Cotterill, W. Damgaard Kristensen and E.J. Jensen, Philos. Mag. 30 (1974) 245. [46] R.M.J. Cotterill and J.U. Madsen, Nature 288 (1980) 467. [47] P.R. Couchman and W.A. Jesser, Philos. Mag. 35 (1977) 787. [48] H. Suzuki, in: Topological Disorder in Condensed Matter, eds F. Yonegawa and T. Ninomiya (SpringerVerlag, Berlin, 1983). [49] J.P. Poirier, Geophys. J. R. Astron. Soc. 85 (1986) 315. [50] J.P. Poirier and G.D. Price, Phys. Earth Planetary Interiors 69 (1992) 153. [51] W.D. Kristensen, J.K. Kristensen and E. Pr~estgaard, Phys. Scr. 30 (1984) 421. [52] E Lund, J. Non-Cryst. Solids 156-158 (1993) 536. [53] J. Broughton, Phys. Rev. B: 46 (1992) 2523. [54] D.P. Woodruff, The Solid-Liquid Interface (Cambridge Univ. Press, 1973). [55] J.G. Dash, Contemporary Phys. 30 (1989) 89. [56] J.G. Dash, in: Solvay Conference on Surface Science, Springer Series in Surface Sciences, Vol. 14, ed. EW. de Wette (Springer, Berlin, 1988) p. 142. [57] J.F. van der Veen, B. Pluis and A.W. Denier van der Gon, in: Chemistry and Physics of Solid Surfaces VII, eds R. Vanselow and R.E Howe (Springer-Verlag, Berlin, 1989) p. 455. [58] E. Tosatti, in: Structure of Surfaces II, eds J.E van der Veen and M.A. van Hove (Springer-Vedag, Berlin, 1988) p. 535. [59] J.W.M. Frenken, J.P. Toennies, C. W611, B. Pluis, A.W. Denier van der Gon and J.E van der Veen, in: Structure of Surfaces II, eds J.E van der Veen and M.A. van Hove (Springer-Verlag, Berlin, 1988). [60] H. L6wen, Phys. Rep. 237 (1994) 249. [61] J.W.M. Frenken and J.E van der Veen, Phys. Rev. Lett. 54 (1985) 134; J.W.M. Frenken, P.M. Maree and J.E van der Veen, Phys. Rev. B: 34 (1986) 7506. [62] B. Pluis, J.M. Gay, J.W.M. Frenken, S. Gierlotka, J.E van der Veen, J.E. MacDonald, A.A. Williams, N. Piggins and J. Als-Nielsen, Surf. Sci. 222 (1989) L845. [63] H.M. Pinxteren and J.W.M. Frenken, Surf. Sci. 275 (1992) 383. [64] R.L. Cormia, J.D. Mackenzie and D. Turnbull, J. Appl. Phys. 34 (1963) 2239. [65] D.R. Uhlmann, J. Non-Cryst. Solids 41 (1980) 347. [66] G. Ainslie, J.D. Mackenzie and D. Turnbull, J. Phys. Chem. 65 (1961) 1718. [67] M. Volmer and O. Schmidt, Z. Phys. Chem. Abt. B: 35 (1937) 467. [68] J.D. Mackenzie and R.L. Cormia, J. Chem. Phys. 39 (1963) 250. [69] G.J. Abbaschian and S.F. Ravitz, J. Cryst. Growth 28 (1975) 16. [70] I. Gutzow and E. Pacheva, Krist. Tech. 11 (1976) 793. [71] M.M. Martynyuk and V.D. Lyakhovets, Sov. Phys. Solid State 16 (1974) 1180. [72] N. Maeno, Can. J. Phys. 46 (1968) 313. [73] Ph. Buffat and J.-P. Borel, Phys. Rev. A: 13 (1976) 2287. [74] J.-P. Borel and A. Chatelain, Helv. Phys. Acta 58 (1985) 443. [75] M. Eldrup and J.H. Evans, J. Phys. F: 12 (1982) 1265. [76] C.J. Rossouw and S.E. Donnelly, Phys. Rev. Lett. 55 (1985) 2960.
The role of dislocations in melting
587
[77] S.E. Donnelly and C.J. Rossouw, Nucl. Instrum. & Methods B 13 (1986) 485. [78] A. vom Felds, J. Fink, Th. Mtiller-Heinzerling, J. Pfliiger, B. Scheerer and G. Linker, Phys. Rev. Lett. 53 (1984) 922. [79] J. Daeges, H. Gleiter and J.H. Perepezko, Phys. Lett. A 119 (1986) 79. [80] G.K. Celler, K.A. Jackson, L.E. Trimble, McD. Robinson and D.J. Lischner, in: Energy Beam-Solid Interactions and Transient Thermal Processing, MRS Symp. Proc. 23 (1984) 409. [81] K.T. Aust, in: Progress in Materials Science, Chalmers Anniversary Volume, eds J.W. Christian, P. Haasen, and T.B. Massalski (Pergamon, Oxford, 1981), p. 27. [82] E.W. Hart, in: The Nature and Behaviour of Grain Boundaries, ed. P. Hu (Plenum, New York, 1972) p. 155. [83] J.E. Hilliard and J.W. Cahn, Acta Metall. 6 (1958) 772. [84] J.W. Cahn and J.E. Hilliard, J. Chem. Phys. 28 (1958) 258. [85] U. Erb and H. Gleiter, Scr. Metall. 13 (1979) 61. [86] S.W. Chan, J.L. Liu and R.W. BaUuffi, Scr. Metall. 19 (1985) 1251. [87] R.W. Balluffi and R. Maurer, Scr. Metall. 19 (1985) 1251. [88] R.W. Baluffi and T.E. Hsieh, J. Phys. (Pads) Colloq. 49 (1988) C5-337. [89] J.E Lutsko, D. Wolf, S.R. Phillpot and S. Yip, Phys. Rev. B: 40 (1989) 2841. [90] S.R. Phillpot, J.E Lutsko, D. and S. Yip, Phys. Rev. B: 40 (1989) 2831. [91] G. Ciccotti, M. Guillop6 and V. Pontikis, Phys. Rev. B: 27 (1983) 5576. [92] G. Besold and O.G. Mouritsen, Phys. Rev. B: 50 (1994) 6573. [93] E Sato, M. Matsui and J. Chikawa, in: Inst. Phys. Conf. Ser. No. 79 (Institute of Physics, Bristol, 1986) p. 193. [94] E Sato, M. Matsui and J. Chikawa, Bull. Acad. Sci. USSR Phys. Ser. 51 (1987) 99. [95] J. Chikawa and S Shirai, J. Cryst. Growth 39 (1977) 328. [96] J. Chikawa and E Sato, in: Inst. Phys. Conf. Ser. No. 59 (Institute of Physics, Bristol, 1980) p. 95. [97] G. Grange, C. Jourdan, A.L. Coulet and J. Gastaldi, J. Cryst. Growth 72 (1985) 748. [98] Y. Hiki, in: Dislocations in Solids: Proc. IX Yamada Conf., ed. H. Suzuki (VNU Science, Utrecht, 1985) p. 607. [99] G.A. Saunders, N.G. Pace and T. Alper, Nature 216 (1967) 1298. [100] H. Kamioka, J. Phys. Soc. Jpn 52 (1983) 2433; 53 (1984) 1349. [101] M.A. Awal and H.Z. Cummins, J. Phys. Chem. Solids 46 (1985) 1037. [102] P.R. Okamoto, L.E. Rehn, J. Pearson, R. Bhadra and M. Grimditch, in: Proc. Conf. on Solid State Amorphization Transformations, Los Alamos, August 10-13, 1987, J. Less-Common Met. 140 (1988) p. 231. [103] A. Voronel, S. Rabinovich, A. Kisliuk, V. Steinberg and T. Sverbilova, Phys. Rev. Lett. 60 (1988) 2402. [104] N.Q. Lam and P.R. Okamoto, MRS Bulletin, XIX(7) (July 1994) 41. [105] D. Wolf, P.R. Okamoto, S. Yip, J.F. Lutsko and M. Kluge, J. Mater. Res. 5 (1990) 286. [106] H. Moil, H. Fujita and M. Fujita, Jpn. J. Appl. Phys. 22 (1983) L94. [107] H. Mori and H. Fujita, in: Dislocations in Solids: Proc. IX Yamada Conf., ed. H. Suzuki (VNU Science, Utrecht, 1985) p. 563. [108] W.J. Meng, P.R. Okamoto, L.J. Thompson, B.J. Kestel and L.E. Rehn, Appl. Phys. Lett. 53 (1988) 1820. [109] Proc. of the Conf. on Solid State Amorphization Transformations, Los Alamos, August 10-13, 1987, J. Less-Common Met. 140 (1988). [110] R. Peierls, Ann. Inst. Henri Poincar6 5 (1935) 177. [111] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133. [112] N.D. Mermin, Phys. Rev. 176 (1968) 250. [113] J.D. Brock, in: Bond Orientational Order in Condensed Matter Systems, ed. K.J. Strandburg (SpringerVerlag, New York, 1992) p. 1. [114] R. Collins, in: Phase Transitions and Critical Phenomena, Vol. 2, eds C. Domb and M.S. Green (Academic Press, London, 1972) p. 271. [115] S.K. Sinha (ed.), Ordering in Two Dimensions (North-Holland, New-York, 1980). [116] T. Riste (ed.), Ordering in Strongly Fluctuating Condensed Matter Systems (Plenum, New York, 1980). [ 117] G. Venkataraman and D. Sahoo, Pramana 24 (1985) 317. [118] EE Abraham, Adv. Phys. 35 (1986) 1.
588
B. Jo6s
[119] K.J. Strandburg, Rev. Mod. Phys. 60 (1988) 161; 61 (1989) 747(E). [120] K.J. Strandburg (ed.), Bond Orientational Order in Condensed Matter Systems (Springer-Vedag, New York, 1992). [121] I. Lyuksyutov, A.G. Naumovets and V. Pokrovsky, Two Dimensional Crystals (Academic Press, San Diego, 1992). [122] H. Taub, G. Torzo, H.J. Lauter and S.C. Fain Jr. (eds), Phase Transitions in Surface Films 2, NATO ASI Ser., Vol. 267 (Plenum, New York, 1991). [123] See for instance, H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford Univ. Press, New York, 1971). [124] J.G. Dash, Phys. Today 38(12) (1985) 26. [125] D.M. Butler, J.A. Litzinger and G.A. Stewart, Phys. Rev. Lett. 44 (1980) 466. [126] R.J. Birgeneau, E.M. Hammons, P. Heiney and P.W. Stephens, in: Ordering in Two Dimensions, ed. S.K. Sinha (North-Holland, New-York, 1980). [127] R.H. Willens, A. Kornblit, L.R. Testardi and S. Nakahara, Phys. Rev. B: 25 (1982) 290. [128] H.K. Kim and M.H.W. Chan, Phys. Rev. Lett. 56 (1984) 170; H.K. Kim, Q.M. Zhang and M.H.W. Chan, Phys. Rev. B: 34 (1986) 4699. [129] M.H.W. Chan, in: Phase Transitions in Surface Films 2, NATO ASI Ser., Vol. 267 (Plenum Press, New York, 1991)p. 1. [130] J.M. Kosterlitz, J. Phys. C: 7 (1974) 1046. [131] A.N. Berker and D.R. Nelson, Phys. Rev. B: 19 (1979) 2488. [132] S.A. Solla and E.K. Riedel, Phys. Rev. B: 23 (1981) 6008. [133] J.E Fem(mdez, M.E Ferreira and J. Stankiewicz, Phys. Rev. B: 34 (1986) 292. [134] D.J. Bishop and J.D. Reppy, Phys. Rev. Lett. 40 (1978) 1727; G. Agnolet, D.C. McQueency and J.D. Reppy, Phys. Rev. Lett. 52 (1984) 1325; D. Finotello and EM. Gasparini, Phys. Rev. Lett. 55 (1985) 2156. [135] A.E Hebard and A.T. Fiory, Phys. Rev. Lett. 50 (1983) 1603; A.T. Fiory, A.E Hebard and W.I. Glaberson, Phys. Rev. B: 28 (1983) 5075. [136] K.J. Strandburg, Phys. Rev. B: 35 (1987) 7161. [137] B. Jancovici, Phys. Rev. Lett. 19 (1967) 20. [138] R.L. Elgin and D.L. Goodstein, in: Monolayer and Submonolayer Helium Films, eds J.G. Gaunt and E. Lerner (Plenum, New York, 1973). [139] K.J. Strandburg, J.A. Zollweg and G.V. Chester, Phys. Rev. B: 30 (1984) 2755. [140] M.A. Glaser and N.A. Clark, Phys. Rev. A: 41 (1990) 4585. [141] G.E Voronoi, J. Reine. Angew. Math. 134 (1908) 198. [142] F.P. Preparata and M.I. Shamos, Computational Geometry, an Introduction (Springer-Verlag, New York, 1985). [143] G. Aeppli and R. Bruinsma, Phys. Rev. Lett. 22 (1984) 2133. [144] S.E. Nagler, P.M. Horn, T.E Rosenbaum, R.J. Birgeneau, M. Sutton, S.G.J. Mochrie, D.E. Moncton and R. Clarke, Phys. Rev. B: 32 (1985) 7373. [145] A.D. Novaco and J.P. McTague, Phys. Rev. Lett. 38 (1977) 1286. [146] J.P. McTague and A.D. Novaco, Phys. Rev. B: 19 (1979) 5299. [147] K.L. D'Amico, D.E. Moncton, E.D. Specht, R.J. Birgeneau, S.E. Nagler and P.M. Horn, Phys. Rev. Lett. 53 (1984) 2250. [148] J.D. Brock, R.J. Birgeneau, J.D. Litster and A. Aharony, Cont. Phys. 30 (1989) 321. [149] A. Aharony, R.J. Birgeneau, J.D. Brock and J.D. Litster, Phys. Rev. Lett. 57 (1986) 1012. [150] M. Paczuski and M. Kardar, Phys. Rev. Lett. 60 (1988) 861. [151] A. Aharony and M. Kardar, Phys. Rev. Lett. 61 (1988) 2855. [152] J.D. Brock, A. Aharony, R.J. Birgeneau, K.W. Evans-Lutterodt, J.D. Litster, P.M. Horn, G.B. Stephenson and A.R. Tajbakhsh, Phys. Rev. Lett. 57 (1986) 98. [153] J.D. Brock, D.Y. Noh, B.R. McClain, J.D. Litster, R.J. Birgeneau, A. Aharony, P.M. Horn and J.C. Liang, Z. Phys. Abt. B: 74 (1989) 197. [154] F. Lund, Phys. Rev. Lett. 69 (1992) 3084. [155] F. Lund, J. Non-Cryst. Solids 156-158 (1993) 536. [156] M. Cheng, J.T. Ho, S.W. Hui and R. Pindak, Phys. Rev. Lett. 59 (1987) 1112.
The role of dislocations in melting [157] [158] [159] [160] [ 161 ] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172]
589
M. Cheng, J.T. Ho, S.W. Hui and R. Pindak, Phys. Rev. Lett. 61 (1988) 550. I.R. Peterson and V.M. Kaganer, Phys. Rev. Lett. 73 (1994) 102. W.G. Hoover, N.E. Hoover and W.C. Moss, J. Appl. Phys. 60 (1979) 829. M.S. Duesbery and B. Jo6s, Philos. Mag. 54 (1986) 145. M.S. Duesbery and B. Jo6s, Philos. Mag. 54 (1986) 165. B. Jo6s and M.S. Duesbery, Phys. Rev. Lett. 55 (1985) 1997. B. Jo6s and M.S. Duesbery, Phys. Rev. B: 33 (1986) 8632. B. Grossmann, B. Jo6s and M.S. Duesbery, Phys. Rev. B: 39 (1989) 7917. B. Jo6s, Q. Ren and M.S. Duesbery, Surf. Sci. 302 (1994) 385. A.J.C. Ladd and W.G. Hoover, Phys. Rev. B: 26 (1982) 5469. D.S. Fisher, B.I. Halperin and R. Morf, Phys. Rev. B: 20 (1979) 4692. S.T. Chui, Phys. Rev. Lett. 48 (1982) 933; Phys. Rev. B: 28 (1983) 178. H. Kleinert, Phys. Lett. A 95 (1983) 493. T.V. Ramakrishnan and M. Yussouff, Phys. Rev. B: 19 (1979) 2775. T.V. Ramakrishnan, Phys. Rev. Lett. 48 (1982) 541. M.A. Glaser and N.A. Clark, in: Geometry and Thermodynamics, ed. J.C. Toledano (Plenum, New York, 1990) p. 193. [173] M.A. Glaser and N.A. Clark, Adv. Chem. Phys. 83 (1993) 543. [174] Y. Saito, Phys. Rev. Lett. 48 (1982) 1114; Phys. Rev. B: 26 (1982) 6239. [175] Y. Saito and H. Muller-Krumbhaar, in: Applications of the Monte Carlo Method in Statistical Physics, ed. K. Binder, Topics in Current Physics, Vol. 36 (Springer-Verlag, New York, 1984) p. 223. [176] D.R. Nelson, Phys. Rev. B: 26 (1982) 269. [177] K.J. Strandburg, Phys. Rev. B: 34 (1986) 3536. [178] K.J. Strandburg, S.A. Solla and G.V. Chester, Phys. Rev. B: 28 (1983) 2717 [179] H. Kleinert, Phys. Lett. A 89 (1982) 294. [180] W. Janke and H. Kleinert, Phys. Lett. A 105 (1984) 134. [181] W. Janke and H. Kleinert, Phys. Lett. A 114 (1986) 255. [182] W. Janke and D. Toussaint, Phys. Lett. A 116 (1986) 387. [183] W. Janke and H. Kleinert, Phys. Rev. B: 41 (1990) 6848. [184] J.D. Weeks, D. Chandler and H.C. Andersen, J. Chem. Phys. 54 (1971) 5237. [ 185] G.W. Gray and J.W. Goodby, Smectic Liquid Crystals: Textures and Structures (Leonard Hill, Glasgow, 1984). [186] R.J. Birgeneau and J.D. Litster, J. Phys. (Paris) Lett. 39 (1978) L-399. [187] D.E. Moncton and R. Pindak, Phys. Rev. Lett. 43 (1979) 701. [188] R. Pindak, D.E. Moncton, S.D. Davey and J.W. Goodby, Phys. Rev. Lett. 46 (1981) 1135. [189] P.S. Pershan, Structure of Liquid Crystal Phases (World Scientific, Singapore, 1988). [190] W.E Brinkman and P.E. Cladis, Phys. Today 35(5) (May 1982) p. 48. [ 191 ] H. Pleiner, Philos. Mag. A 54 (1986) 421. [192] P.S. Pershan, J. Appl. Phys. 45 (1974) 1590 [193] D.R. Nelson and B.I. Halperin, Phys. Rev. B: 21 (1980) 5312. [194] R. Bruinsma and D.R. Nelson, Phys. Rev. B: 23 (1981) 402. [195] C.C. Huang and T. Stoebe, Adv. Phys. 42 (1993) 343. [196] C.C. Huang in: Bond Orientational Order in Condensed Matter Systems, ed. K.J. Strandburg (SpringerVerlag, New York, 1992) p. 78. [197] S.B. Dierker, R. Pindak and R.B. Meyer, Phys. Rev. Lett. 56 (1986) 1819. [198] C.C. Huang, J.M. Viner, R. Pindak and J.W. Goodby, Phys. Rev. Lett. 46 (1981) 1289. [199] T. Pitchford, G. Nounesis, S. Dumrongrattana, J.M. Viner, C.C. Huang and J.W. Goodby, Phys. Rev. A: 32 (1985) 1938. [200] J.C. LeGuillon and J. Zinn-Justin, J. Phys. (Paris) Lett. 46 (1985) L-137. [201] R. Mahmood, M. Lewis, R. Biggers, V. Surrendranath, D. Johnson and M.E. Neubert, Phys. Rev. A: 33 (1986) 519. [202] C.C. Huang, G. Nounesis and D. Guillon, Phys. Rev. A: 33 (1986) 2602. [203] R. Bruinsma and G. Aeppli, Phys. Rev. Lett. 48 (1982) 1625. [204] C.C. Huang, G. Nounesis, R. Geer, J.W. Goodby and D. Guillon, Phys. Rev. A: 39 (1989) 3741.
590 [205] [206] [207] [208] [209] [210] [211] [212] [213] [214] [215] [216] [217] [218] [219] [220] [221] [222] [223] [224] [225] [226] [227] [228] [229] [230] [231] [232] [233] [234] [235] [236] [237] [238] [239] [240] [241] [242] [243] [244] [245] [246]
B. Jo6s
J.M. Viner and C.C. Huang, Phys. Rev. A: 27 (1983) 2763. R. Geer, T. Stoebe, C.C. Huang and J.W. Goodby, Phys. Rev. A: 46 (1992) R6162. T. Stoebe, C.C. Huang and J.W. Goodby, Phys. Rev. Lett. 68 (1992) 2944. M.J. Tejwani, O. Ferreira and O.E. Vilches, Phys. Rev. Lett. 44 (1980) 152. J.J. Benattar, J. Doucet, M. Lambert and A.M. Levelut, Phys. Rev. A: 20 (1979) 2505. E.B. Sirota, P.S. Pershan, L.B. Sorensen and J. Collett, Phys. Rev. Lett. 55 (1985) 2039; Phys. Rev. A: 36 (1987) 2890. F. Moussa, J.J. Benattar and C. Williams, Mol. Cryst. Liq. Cryst. 99 (1983) 145. C.C. Grimes and G. Adams, Phys. Rev. Lett. 42 (1979) 795. R. Mehrotra, B.M. Guenin and A.J. Dahm, Phys. Rev. Lett. 48 (1982) 641. C.J. Guo, D.B. Mast, R. Mehrotra, Y.-Z. Ruan, M.A. Stan and A.J. Dahm, Phys. Rev. Lett. 51 (1983) 1461. R.H. Morf, Helv. Phys. Acta 56 (1983) 743. F. Gallet, G. Deville, A. Valdi~s and F.I.B. Williams, Phys. Rev. Lett. 49 (1982) 212. D.C. Glattli, E.Y. Andrei and F.I.B. Williams, Phys. Rev. Lett. 60 (1988) 420; D.C. Glattli, G. Deville, V. Duburcq, F.1. Williams, E. Paris, B. Etienne and E.Y. Andrei, Surf. Sci. 229 (1990) 344. Pa. Pieranski, Contemp. Phys. 24 (1983) 25. C.A. Murray, in: Bond Orientational Order in Condensed Matter Systems, ed. K.J. Strandburg (SpringerVerlag, New York, 1992) p. 137. W.B. Russell, D.A. Saville and W.R. Schowalter, Colloidal Dispersions (Cambridge Univ. Press, Cambridge, 1989). C.A. Murray, W.O. Sprenger and R.A. Wenk, Phys. Rev. B: 42 (1990) 688. W. Hess and R. Klein, Adv. Phys. 32 (1983) 173. M.O. Robbins, K. Kremer and G. Grest, J. Chem. Phys. 88 (1988) 3286. E.B. Sirota, H.D. Ou-Yang, S.K. Sinha, P.M. Chaikin, J.D. Axe and Y. Fujii, Phys. Rev. Lett. 62 (1989) 1524. G. Helgesen and A.T. Skjeltorp, Physica A 170 (1991) 488. A.T. Skjeltorp, Phys. Rev. Lett. 51 (1983) 2306; J. Magn. Magn. Mat. 65 (1987) 195. Pa. Pieranski, L. Strzlecki and B. Pansu, Phys. Rev. Lett. 50 (1983) 900. B. Pansu, P. Pieranski and L. Strzelecki, J. Phys. (Paris) 44 (1983) 531; B. Pansu, Pi. Pieranski and Pa. Pieranski, J. Phys. (Paris) 45 (1984) 331. D.H. Van Winkle and C.A. Murray, Phys. Rev. A: 34 (1986) 562. D.H. Van Winkle and C.A. Murray, J. Chem. Phys. 89 (1988) 3885. C.A. Murray and D.H. Van Winkle, Phys. Rev. Lett. 58 (1987) 1200. C.A. Murray and R.A. Wenk, Phys. Rev. Lett. 62 (1989) 1643. C.A. Murray, D.H. Van Winkle and R.A. Wenk, Phase Trans. 21 (1990) 93. A.F. Bakker, C. Bruin and H.J. Hilhorst, Phys. Rev. Lett. 52 (1984) 449. J.A. Zollweg, G.V. Chester and P.W. Leung, Phys. Rev. B: 39 (1989) 9518. A.J. Armstrong, R.C. Mockler and W.J. O'Sullivan, J. Phys. Cond. Matter. 1 (1989) 1707. Y. Tang, A.J. Armstrong, R.C. Mockler and W.J. O'Sullivan, Phys. Rev. Lett. 62 (1989) 2401; Phase Trans. 21 (1990) 75. R.E. Kusner, J.A. Mann, J. Kerins and A.J. Dahm, Phys. Rev. Lett. 73 (1994) 3113. R.E. Kusner, J.A. Mann, J. Kerins, W.J. Benton and A.J. Dahm, J. Phys. (Paris) Colloq. 3 (1993) C2-159. R. Seshadri and R.M. Westervelt, Phys. Rev. B: 46 (1992) 5142; R. Seshadri and R.M. Westervelt, Phys. Rev. B: 46 (1992) 5150. A.H. Eschenfelder, Magnetic Bubble Technology (Springer-Verlag, New York, 1981). A.P. Malozemoff and J.C. Slonczweski, Magnetic Domain Walls in Bubble Materials (Academic Press, New York, 1979). R.H. Koch, V. Foglietti, W.J. Gallagher, G. Koren, A. Gupta and M.P.A. Fisher, Phys. Rev. Lett. 63 (1989) 1511; P.L. Gammel, L.F. Schneemeyer and D.J. Bishop, Phys. Rev. Lett. 66 (1991) 953. M.P.A. Fisher, Phys. Rev. Lett. 62 (1989) 1415. D.S. Fisher, M.P.A. Fisher and D.A. Huse, Phys. Rev. B: 43 (1991) 130. R.H. Koch, Phys. Rev. Lett. 63 (1989) 1511; R.H. Koch, V. Foglietti and M.P.A. Fisher, Phys. Rev. Lett. 64 (1990) 2586.
The role of dislocations in melting
591
[247] H. Safar, Phys. Rev. Lett. 68 (1992) 2672. [248] H. Safar, EL. Gammel, D.A. Huse, D.J. Bishop, J.E Rice and D.M. Ginsberg, Phys. Rev. Lett. 69 (1992) 824. [249] W.K. Kwok, S. Fleshier, U. Welp, V.M. Vinokur, J. Downey, G.W. Crabtree and M.M. Miller, Phys. Rev. Lett. 69 (1992) 3370. [250] E. Brezin, D.R. Nelson and A. Thiaville, Phys. Rev. B: 31 (1985) 7124. [251] C.A. Murray, EL. Gammel, D.J. Bishop, D.B. Mitzi and A. Kapitulnik, Phys. Rev. Lett. 64 (1990) 2312. [252] D.G. Grier, C.A. Murray, C.A. Bolle, EL. Gammel, D.J. Bishop, D.B. Mitzi and A. Kapitulnik, Phys. Rev. Lett. 66 (1991) 2270; [253] H. Ma and S.T. Chui, Phys. Rev. Lett. 67 (1991) 505; S.T. Chui, Europhys. Lett. 20 (1992) 535. [254] N.D. Shrimpton, M.W. Cole, W.A. Steele and M.H.W. Chan, in: Surface Properties of Layered Structures, ed. G. Benedek (Kluwer, The Netherlands, 1992) p. 219. [255] G.C. Maitland, M. Rigby, E.B. Smith and W.A. Wakeham, Intermolecular Forces (Clarendon Press, Oxford, 1981). [256] R.A. Aziz, in: Inert Gases, ed. M.L. Klein (Springer-Verlag, Berlin, 1984). [257] G. Vidali, G. Ihm, H.-Y. Kim and M.W. Cole, Surf. Sci. Rep. 12 (1991) 134; review: L.W. Bruch, Surf. Sci. 125 (1983) 194; Rev. Mod. Phys. 60 (1988) 67. [258] R.J. Birgeneau, P.A. Heiney and J.P. Pelz, Physica B 109-110 (1982) 1785. [259] E.D. Specht, A. Mak, C. Peters, M. Sutton, R.J. Birgeneau, K.L. D'Amico, D.E. Moncton, S.E. Nagler and EM. Horn, Z. Phys. Abt. B: 69 (1987) 347. [260] J. Frenkel and T. Kontorowa, Phys. Z. Sowj. 13 (1938) 1. [261] EC. Frank and J.H. van tier Merwe, Proc. R. Soc. London Ser. A: 198 (1949) 205. [262] R. Hobart, J. Appl. Phys. 36, 1944 (1965); B. Jo6s, Solid State Commun. 42 (1982) 709. [263] J. Villain, Surf. Sci. 97 (1980) 219. [264] H. Shiba, J. Phys. Soc. Jpn 46 (1979) 1852; 48 (1980) 211. [265] P. Bak, Rep. Prog. Phys. 45 (1982) 587. [266] V.L. Pokrovsky and A.L. Talapov, Theory of Incommensurate Crystals (Harwood Academic Publishers, New York, 1984). [267] M. den Nijs, in: Phase Transitions and Critical Phenomena, Vol. 12, eds C. Domb and J.L. Lebowitz (Academic Press, London, 1988) p. 219. [268] P. Bak, Rep. Prog. Phys. 45 (1982) 587. [269] V.L. Pokrovsky and A.L. Talapov, Theory of Incommensurate Crystals (Harwood Academic Publishers, New York, 1984). [270] M. den Nijs, in: Phase Transitions and Critical Phenomena, eds C. Domb and J.L. Lebowitz (Academic Press, London, 1988), p. 219. [271] N.D. Shrimpton, B. Bergersen and B. Jo6s, Phys. Rev. B: 34 (1986) 7334. [272] N.D. Shrimpton, B. Jo6s and B. Bergersen, Phys. Rev. B: 38 (1988) 2124. [273] N.D. Shrimpton and B. Jo6s, Phys. Rev. B: 40 (1989) 10564; Can. J. Phys. 68 (1990) 587. [274] N.D. Shrimpton and B. Jo6s, Phys. Rev. B: 41 (1990) 4764. [275] R.J. Gooding, B. Jo6s and B. Bergersen, Phys. Rev. B: 27 (1983) 7669. [276] M.B. Gordon and E Lanccon, J. Phys. C: 18 (1985) 3929. [277] EE Abraham, W.E. Rudge, D.J. Auerbach and S.W. Koch, Phys. Rev. Lett. 52 (1984) 445. [278] P.W. Stephens, E Heiney, R.J. Birgeneau, P.M. Horn, D.E. Moncton and G.S. Brown, Phys. Rev. B: 29 (1984) 3512. [279] J. Cui and S.C. Fain Jr., Phys. Rev. B: 39 (1989) 8628. [280] V.L. Pokrovsky and A.L. Talapov, Phys. Rev. Lett. 42 (1979) 65. [281] M.E. Fisher and D.S. Fisher, Phys. Rev. B: 25 (1982) 3192. [282] S.N. Coppersmith, D.S. Fisher, B.I. Halperin, EA. Lee and W.E Brinkman, Phys. Rev. Lett. 46 (1981) 549. [283] D.A. Huse and M.E. Fisher, Phys. Rev. B: 29 (1984) 239. [284] P.W. Stephens, E Heiney, R.J. Birgeneau and P.M. Horn, Phys. Rev. Lett. 43 (1979) 47; R.J. Birgeneau, E.M. Hammonds, E Heiney, EW. Stephens and EM. Horn, in: Ordering in Two Dimensions, ed. S.K. Sinha (North-Holland, New York, 1980) p. 29. [285] Y.E Feng and M.H.W. Chan, Phys. Rev. Lett. 64 (1990) 2148.
592
B. Jo6s
[286] A. Erbil, A.R. Kortan, R.J. Birgeneau and M.S. Dresselhaus, Phys. Rev. B: 28 (1983) 6329. [287] S.G.J. Mochrie, A.R. Kortan, R.J. Birgeneau and P.M. Horn, Z. Phys. Abt. B: 62 (1985) 79. [288] T.U Einstein, in: Chemistry and Physics of Solid Surfaces VII, eds R. Vanselow and R.F. Howe (Springer, Berlin, 1988) p. 307. [289] P. Piercy, K. De'Bell and H. Pfntir, Phys. Rev. B: 45 (1992) 1869. [290] N.D. Shrimpton, B. Bergersen and B. Jo6s, Phys. Rev. B: 29 (1984) 6999. [291] B. Jo6s, B. Bergersen and M.L. Klein, Phys. Rev. B: 28 (1983) 7219. [292] J.A. Litzinger and G.A. Stewart, in: Ordering in Two Dimensions, ed. S.K. Sinha (North-Holland, New York, 1980). [293] P.A. Heiney, P.W. Stephens, R.J. Birgeneau, P.M. Horn and D.E. Moncton, Phys. Rev. B: 28 (1983) 6416. [294] P. Dimon, P.M. Horn, M. Sutton, R.J. Birgeneau and D.E. Moncton, Phys. Rev. B: 31 (1985) 437. [295] P.A. Heiney, R.J. Birgeneau, G.S. Brown, P.M. Horn, D.E. Moncton and P.W. Stephens, Phys. Rev. Lett. 48 (1982) 104. [296] T.E Rosenbaum, S.E. Nagler, P.M. Horn and R. Clarke, Phys. Rev. Lett. 50 (1983) 1791. [297] E.D. Specht, R.J. Birgeneau, K.L. D'Amico, D.E. Moncton, S.E. Nagler and P.M. Horn, J. Phys. Lett. 46 (1985) L561. [298] N.Greiser, G.A. Held, R. Frahm, R.L. Greene, P.M. Horn and R.M. Suter, Phys. Rev. Lett. 59 (1987) 1706. [299] N.J. ColeUa and R.M. Suter, Phys. Rev. B: 34 (1986) 2052. [300] R. Gangwar, N.J. Colella and R.M. Suter, Phys. Rev. B: 39 (1989) 2459. [301] C. Tessier, DSc thesis (L'Universit6 de Nancy I, 1983). [302] A.J. Jin, M.R. Bjurstrom and M.H.W. Chan, Phys. Rev. Lett. 62 (1989) 1372. [303] EE Abraham, Phys. Rev. Lett. 50 (1983) 978. [304] F.E Abraham, Phys. Rev. B: 29 (1984) 2606. [305] C. Udink and D. Frenkel, Phys. Rev. B: 35 (1987) 6933. [306] A.D. Migone, Z.R. Li and M.H.W. Chan, Phys. Rev. Lett. 53 (1984) 810. [307] J.P. McTague, J.Als-Nielsen, J. Bohr and M. Nielsen, Phys. Rev. B: 25 (1982) 7765. [308] M. Nielsen, J. Als-Nielsen, J. Bohr, J.P. McTague, D.E. Moncton and P.W. Stephens, Phys. Rev. B: 35 (1987) 1419. [309] K.L. D'Amico, J. Bohr, D.E. Moncton and G. Gibbs, Phys. Rev. B: 41 (1990) 4368. [310] Q.M. Zhang and J.Z. Larese, Phys. Rev. B: 43 (1991) 938. [311] N.D. Shrimpton, H.-Y. Kim, W.A. Steele and A. Cheng, unpublished. [312] H.K. Kim, Q.M. Zhang and M.H.W. Chan, Phys. Rev. Lett. 56 (1986) 1579. [313] J.Z. Larese, L. Passell, A. Heideman, D. Richter and J.P. Widested, Phys. Rev. Lett. 61 (1988) 432. [314] M.A. Moiler and M.L. Klein, J. Chem. Phys. 90 (1989) 1960. [315] EY. Hansen and H. Taub, Phys. Rev. Lett. 69 (1992) 652. [316] EY. Hansen, J.C. Newton and H. Taub, J. Chem. Phys. 98 (1993) 4128. [317] D.M. Butler, J.A. Litzinger, G.A. Stewart and R.B. Griffiths, Phys. Rev. Lett. 42 (1979) 1289. [318] F.Y. Wu, Rev. Mod. Phys. 54 (1982) 235. [319] A.N. Berker, S. Oslund and EA. Putnam, Phys. Rev. B: 17 (1978) 3650. [320] R.G. Caflisch, A.N. Berker and M. Kardar, Phys. Rev. B: 31 (1985) 4527. [321] J. Cui and S.C. Fain Jr., Phys. Rev. B: 39 (1989) 8628, and references therein. [322] H.J. Lauter, H. Godfrin, V.L.P. Frank and H.P. Schildberg, Physica B 165-166 (1990) 597. [323] M. Bretz, Phys. Rev. Lett. 38 (1977) 501. [324] M.J. Tejwani, O. Ferreira and O.E. Vilches, Phys. Rev. Lett. 44 (1980) 152. [325] R.E. Ecke and J.G. Dash, Phys. Rev. B: 28 (1983) 3738. [326] S.B. Crary and D.A. Fahey, Phys. Rev. B: 35 (1987) 2101. [327] R.E. Ecke, Q.S. Shu, T.S. Sullivan and O.E. Vilches, Phys. Rev. B: 31 (1985) 448. [328] T. Halpin-Healy and M. Kardar, Phys. Rev. B: 34 (1986) 318. [329] J.Q. Broughton and F.E Abraham, J. Phys. Chem. 92 (1988) 3274. [330] J.M. Gottlieb and L.W. Bruch, Phys. Rev. B: 40 (1989) 140. [331] Q.M. Zhang, H.K. Kim and M.H.W. Chan, Phys. Rev. B: 33 (1986) 5149.
The role of dislocations in melting
593
[332] R.D. Diehl, in: Phase Transitions in Surface Films 2, NATO ASI Ser., Vol. 267, eds H. Taub, G. Torzo, H.J. Lauter and S.C. Fain (Plenum, New York, 1991) p. 97. [333] T. Aruga and Y. Murata, Prog. Surf. Sci. 31 (1989) 61. [334] H.P. Bonzel, A.M. Bradshaw and G. Ertl (eds), Alkali Adsorption on Metals and Semiconductors (Elsevier, Amsterdam, 1989). [335] K. Kern, P. Zeppenfeld, R. David and G. Comsa, Phys. Rev. Lett. 59 (1987) 79. [336] K. Kern, P. Zeppenfeld, R. David and G. Comsa, Phys. Rev. B: 35 (1987) 8265. [337] W.C. Fan and A. Ignatiev, Phys. Rev. B: 37 (1988) 5274. [338] W.C. Fan and A. Ignatiev, J. Vac. Sci. Technol. A 6 (1988) 735. [339] S. Chandavarkar and R.D. Diehl, Phys. Rev. B: 40 (1989) 4651. [340] D. Fisher and R.D. Diehl, Phys. Rev. B: 46 (1992) 2512. [341] S.G.J. Mochrie, A.R. Kortan, P.M. Horn and R.J. Birgeneau, Phys. Rev. Lett. 58 (1987) 690. [342] H. Zabel, S.E. Hardcastle, D.A. Neumann, M. Suzuki and A. Magerl, Phys. Rev. Lett. 57 (1986) 2041. [343] Ch. Simon, I. Rosenman, F. Batallan, C. Lartigue and J.F. Legrand, Phys. Rev. B: 45 (1992) 2694. [344] S. Nos6, Mol. Phys. 52 (1984) 255. [345] S. Nos6, J. Chem. Phys. 81 (1984) 511. [346] H.C. Andersen, J. Chem. Phys. 72 (1980) 2384. [347] K. Binder (ed.), Monte Carlo Methods in Statistical Mechanics, Topics in Current Physics, Vol. 36 (Springer-Verlag, New York, 1979); Monte Carlo Simulation in Statistical Physics, An Introduction, Springer Series in Solid State Science, Vol. 80 (Springer-Verlag, Berlin, 1988). [348] EE Abraham, Phys. Rev. Lett. 44 (1980) 463. [349] EE Abraham, Phys. Rev. B: 23 (1981) 6145. [350] S.W. Koch and EF. Abraham, Phys. Rev. B: 27 (1983) 2964. [351] S. Toxvaerd, Phys. Rev. B: 29 (1984) 2821. [352] EF. Abraham and S.W. Koch, Phys. Rev. B: 29 (1984) 2824. [353] K.J. Naidoo, J. Schnitker and J.D. Weeks, Mol. Phys. 80 (1993) 1. [354] S. Toxvaerd, Phys. Rev. A: 24 (1981) 2735. [355] S. Toxvaerd, Phys. Rev. Lett. 51 (1983) 1971; Phys. Rev. Lett. 53 (1984) 2352. [356] C. Udink and J. van der Elsken, Phys. Rev. B: 35 (1987) 279. [357] J. Lee and K.J. Strandburg, Phys. Rev. B: 46 (1992) 11 190. [358] J.D. Weeks, Phys. Rev. B: 24 (1981) 1530. [359] B.J. Alder and T.E. Wainwright, Phys. Rev. 127 (1962) 359. [360] J.A. Zollweg and G.V. Chester, Phys. Rev. B: 46 (1992) 11, 186. [361] J. Lee and J.M. Kosterlitz, Phys. Rev. Lett. 65 (1990) 137. [362] J. Lee and J.M. Kosterlitz, Phys. Rev. B: 43 (1991) 1268. [363] D. Frenkel and J.P. Mctague, Phys. Rev. Lett. 42 (1979) 1632. [364] S. Toxvaerd, Phys. Rev. Lett. 44 (1980) 1002. [365] J.A. Barker, D. Henderson and EF. Abraham, Physica A 106 (1981) 226. [366] E Tsien and J.P. Valleau, Mol. Phys. 27 (1974) 177. [367] J.D. Weeks and J.Q. Broughton, J. Chem. Phys. 78 (1983) 4197. [368] J.Q. Broughton, G.H. Gilmer and J.D. Weeks, Phys. Rev. B: 25 (1982) 4651. [369] M.P. Allen, D. Frenkel, W. Gignac and J.P. McTague, J. Chem. Phys. 78 (1983) 4206. [370] A.D. Novaco and P.A. Shea, Phys. Rev. B: 26 (1982) 284. [371] A.D. Novaco, Phys. Rev. B: 35 (1987) 8621. [372] R.K. Kalia and P. Vashishta, J. Phys. C: 14 (1981) L643. [373] P. Vashishta and R.K. Kalia, in: Melting, Localization and Chaos, eds R.K. Kalia and P. Vashishta (North-Holland, New York, 1982) p. 43. [374] V.M. Bedanov and G.V. Gadijak, Phys. Lett. A 92 (1982) 400. [375] R.C. Gann, S. Chakravarty and G.V. Chester, Phys. Rev. B: 20 (1979) 326. [376] R.K. Kalia, P. Vashishta and S.W. de Leeuw, Phys. Rev. B: 23 (1981) 4794. [377] H. Cheng, P. Dutta, D.E. Ellis and R. Kalia, J. Chem. Phys. 85 (1986) 2232. [378] R.H. Morf, Phys. Rev. Lett. 43 (1979) 931. [379] R.H. Morf, in: Physics of Intercalation Compounds, eds L. Pietronero and E. Tosatti, Springer Series in Solid State Sciences, Vol. 38 (Springer-Verlag, New York, 1981) p. 252.
594 [380] [381 ] [382] [383] [384] [385] [386] [387] [388] [389]
B. Jo6s
K.J. Naidoo and J. Schnitker, J. Chem. Phys. 100 (1994) 3114. V.M. Bedanov, G.V. Gadiyak and Y.E. Lozovik, Sov. Phys. JETP 61 (1985) 967. J.A. Combs, Phys. Rev. Lett. 61 (1988) 744; Phys. Rev. B: 38 (1988) 6751. K. Bagchi, H.C. Andersen and W. Swope, Phys. Rev. Lett. 76 (1996) 255. J.F. Fernandez, J.J. Alonso and J. Stankiewicz, Phys. Rev. Lett. 75 (1995) 3477. P. Bladon and D. Frenkel, Phys. Rev. Lett. 74 (1995). T. Chou and D.R. Nelson, Phys. Rev. E: 53 (1996) 2560. W.J. NuttaU, D.Y. Noh, B.O. Wells and R.J. Birgeneau, J. Phys. Cond. Matter 7 (1995) 4337. R. Viswanathan, L.L. Madsen, J.A. Zasadzinski and D.K. Schwartz, Science 269 (1995) 51. H.M. Thomas and G.E. Morrill, Nature 379 (1996) 806 (commented by D.G. Grier, ibid. p. 773); A. Melzer, A. Homann and A. Piel, Phys. Rev. E: 53 (1996) 2757.
Author Index
Abbaschian, G.J., 586 Abraham, EE, 587, 591-593 Abraham, EE, s e e Barker, J.A., 593 Abraham, F.E, s e e Broughton, J.Q., 592 Abraham, F.E, s e e Koch, S.W., 593 Ackland, G.J., s e e Vitek, V., 183 Adams, G., s e e Grimes, C.C., 590 Aeppli, G., 588 Aeppli, G., s e e Bruinsma, R., 589 Ageeva, V.A., s e e Shishokin, V.P., 25 Agnolet, G., 588 Agullo-Lopez, F., 503 Aharony, A., 588 Aharony, A., s e e Brock, J.D., 588 Ahlers, M., 131 Ahlquist, C.N., s e e Bush, F., 503 Ahlquist, C.N., s e e Carlsson, L., 503 Ainslie, G., 586 Akita, K., s e e Ueda, O., 503 Alder, B.J., 593 Aleinikova, I.N., s e e Deryagin, B.V., 501 Alexander, H., 502, 503 Alexander, H., s e e Gottschalk, H., 502 Alexander, H., s e e Kuesters, K., 501 Alexander, H., s e e Wessel, K., 502 Alexander, W.O., 24 Allen, C.W., s e e Kuczinski, G.C., 501 Allen, M.P., 593 Allen, N.P., s e e Pfeil, L.B., 24 Almin, A., s e e Westgren, A., 25 Alonso, J.J., s e e Fernandez, J.F., 594 Alper, T., s e e Saunders, G.A., 587 Als-Nielsen, J., s e e McTague, J.P., 592 Als-Nielsen, J., s e e Nielsen, M., 592 Als-Nielsen, J., s e e Pluis, B., 586 Andersen, H.C., 593 Andersen, H.C., s e e Bagchi, K., 594 Andersen, H.C., s e e Weeks, J.D., 589 Andrei, E.Y., s e e Glattli, D.C., 590 Androussi, Y., s e e Caillard, D., 504 Anisimov, V.I., s e e Greenberg, B.A., 133 Anstis, G.R., s e e Chou, C.T., 68, 439 Antolovich, S., s e e Webb, G., 132 Antolovich, S., s e e de Bussac, A., 132, 251,440 Anton, D.L., 25, 133, 437 Anton, D.L., s e e Giamei, A.F., 25 Antonova, O.V., s e e Greenberg, B.A., 133 Aoki, K., 184, 436 Ardell, A.J., 25 Ardley, G.W., 26
Argon, A.S., s e e Pollock, T.M., 25, 26 Arko, A.C., 436 Armstrong, A.J., 590 Armstrong, A.J., s e e Tang, Y., 590 Arrell, D.J., s e e Vall6s, J.L., 26 Aruga, T., 593 Asaro, R.J., 185 Ashby, M.F., s e e Frost, H.J., 252, 441 Auerbach, D.J., s e e Abraham, EF., 591 Aust, K.T., 587 Awal, M.A., 587 Axe, J.D., s e e Sirota, E.B., 590 Aziz, R.A., 591 Bacon, D.J., 131 Bagchi, K., 594 Bain, E.C., 25 Bak, E, 591 Baker, I., 132, 439, 440 Baker, I., s e e Horton, J.A., 440 Baker, I., s e e Munroe, P.R., 134 Bakker, A.E, 590 Ballufii, R.W., 587 Balluffi, R.W., s e e Chart, S.W., 587 Baluc, N., 67, 132, 183, 184, 436--439, 441 Baluc, N., s e e Bonneville, J., 132, 183, 437 Baluc, N., s e e Mills, M.J., 67, 132, 251,439 Baluc, N.L., 251 Barker, J.A., 593 Barnby, J.T., 131 Barrett, C.S., 26 Basinski, S.J., 437 Basinski, Z.S., s e e Basinski, S.J., 437 Baskes, M.I., s e e Yoo, M.H., 184, 440 Bassani, J.L., 183 Bassani, J.L., s e e Qin, Q., 182, 183 Bassani, J.L., s e e Wu, T.-Y., 185 Batallan, E, s e e Simon, Ch., 593 Beardmore, P., 131 Beauchamp, E, 134, 438 Beauchamp, P., s e e Dirras, G., 134, 435 Beauchamp, E, s e e Douin, J., 67, 184, 437, 438 Beauchamp, E, s e e Lasalmonie, A., 439 Beauchamp, E, s e e Tounsi, B., 132, 439 Beauchamp, E, s e e Veyssi~re, E, 68, 438 Bedanov, V.M., 593 Behrensmeier, R., s e e Kung, J., 503 Belyavskii, V.I., 504 Benattar, J.J., 590 Benattar, J.J., s e e Moussa, F., 590 595
596
Author
index
Borel, J.-E, 586 Benton, W.J., s e e Kusner, R.E., 590 Borel, J.-P., s e e Buffat, Ph., 586 Berezinskii, V.L., 585 Born, M., 585 Bergersen, B., s e e Gooding, R.J., 591 Bouard, A.D., s e e Fortini, A., 501 Bergersen, B., s e e Jo6s, B., 592 Bourgoin, J.C., 504 Bergersen, B., s e e Shrimpton, N.D., 591,592 Boyer, L.L., 585 Berker, A.N., 588, 592 Bradshaw, A.M., s e e Bonzel, H.E, 593 Berker, A.N., s e e Caflisch, R.G., 592 Bragg, W.L., 585 Besold, G., 587 Brechet, Y., s e e Louchet, E, 502 Betteridge, W., 24 Bretz, M., 592 Beuers, J., 131 Brezin, E., 591 Beuers, J., s e e Jonsson, S., 131 Brinkman, W.E, 589 Bhadra, R., s e e Okamoto, P.R., 587 Brinkman, W.E, s e e Coppersmith, S.N., 591 Biggers, R., s e e Mahmood, R., 589 Britun, V.E, s e e Pilyankevich, A.N., 501 Bilgram, J.H., 585 Brock, J.D., 587, 588 Binder, K., 593 Brock, J.D., s e e Aharony, A., 588 Birgeneau, R.J., 588, 589, 591 Broughton, J., 586 Birgeneau, R.J., s e e Aharony, A., 588 Broughton, J.Q., 592 Birgeneau, R.J., s e e Brock, J.D., 588 Broughton, J.Q., s e e Weeks, J.D., 593 Birgeneau, R.J., s e e D'Amico, K.L., 588 Brown, G.S., s e e Heiney, P.A., 592 Birgeneau, R.J., s e e Dimon, P., 592 Brown, G.S., s e e Stephens, EW., 591 Birgeneau, R.J., s e e Erbil, A., 592 Brown, L.M., s e e Vidoz, A.E., 68, 439 Birgeneau, R.J., s e e Heiney, P.A., 592 Brown, N., 133, 182, 185 Birgeneau, R.J., s e e Mochrie, S.G.J., 592, 593 Brown, N., s e e Marcinkowski, M.J., 25, 66 Birgeneau, R.J., s e e Nagler, S.E., 588 Brown, ED., s e e Loginov, Y.Y., 502 Birgeneau, R.J., s e e Nuttall, W.J., 594 Brown, S.A., s e e Kumar, K.S., 184 Birgeneau, R.J., s e e Specht, E.D., 591,592 Bruch, L.W., s e e Gottlieb, J.M., 592 Birgeneau, R.J., s e e Stephens, P.W., 591 Bruin, C., s e e Bakker, A.E, 590 Bishop, D.J., 588 Bruinsma, R., 589 Bishop, D.J., s e e Gammel, P.L., 590 Bruinsma, R., s e e Aeppli, G., 588 Bishop, D.J., s e e Grier, D.G., 591 Brtimmer, O., 504 Bishop, D.J., s e e Murray, C.A., 591 Buffat, Ph., 586 Bishop, D.J., s e e Safar, H., 591 Burstall, H.E, 24 Bishop, J., s e e Betteridge, W., 24 Burta-Gapanovich, L.N., s e e Deryagin, B.V., 501 Bishop, J.F.W., 185 Bush, E, 503 Bjurstrom, M.R., s e e Jin, A.J., 592 Butler, D.M., 588, 592 Blackburn, M.J., s e e Shechtman, D., 133 Bladon, P., 594 Blair, D.G., 184 Caflisch, R.G., 592 Blanchard, C., s e e Levade, C., 504 Cahn, J.W., 587 Boas, W., s e e Schmid, E., 182 Cahn, J.W., s e e Hilliard, J.E., 587 Bohr, J., s e e D'Amico, K.L., 592 Cahn, J.W., s e e Kikuchi, R., 438 Bohr, J., s e e McTague, J.P., 592 Cahn, R.W., 585 Bohr, J., s e e Nielsen, M., 592 Caillard, D., 131-134, 437, 439, 504 Bolle, C.A., s e e Grier, D.G., 591 Caillard, D., s e e Bonneville, J., 440 Bollman, W., 25 Caillard, D., s e e Cl6ment, N., 67, 131, 132, 437 Bondarenko, I.E., 502 Caillard, D., s e e Couret, A., 131, 133, 439, 440 Bonneville, J., 67, 132, 182-184, 252, 437, 440 Caillard, D., s e e Farenc, S., 131, 134 Bonneville, J., s e e Baluc, N., 132, 183, 436, 439, 441 Caillard, D., s e e Fnaiech, M., 502 Bonneville, J., s e e Spfitig, P., 132, 183, 436, 437 Caillard, D., s e e Levade, C., 503 Caillard, D., s e e Louchet, F., 504 Bonneville, J., s e e Stoiber, J., 252 Caillard, D., s e e Mol6nat, G., 67, 131-133, 251,252, Bontemps, C., 67, 131, 132, 251,436, 439 437, 439, 440 Bontemps-Neveu, C., 435 Caillard, D., s e e Paidar, V., 132, 440 Bonzel, H.P., 593 Caillard, D., s e e Vanderschaeve, G., 501 Booker, G.R., s e e Titchmarsh, J.M., 504
Author
Car, P., 504 Carlsson, L., 503 Cames, C.P., s e e Kimerling, L.C., 504 Caron, P., 440 Caron, P., s e e Khan, T., 436 Carrard, M., 440 Carrard, M., s e e Bonneville, J., 440 Carter, C.B., s e e Chiang, S.-W., 438 Carter, C.B., s e e Kuesters, K., 503 Castaldini, A., s e e Cavallini, A., 501 Catlow, C.R.A., s e e Agullo-Lopez, F., 503 Cavallini, A., 501 Celler, G.K., 587 Celli, V., 502 Chaikin, P.M., s e e Sirota, E.B., 590 Chakravarty, S., s e e Gann, R.C., 593 Champier, G., s e e George, A., 502 Chan, M.H.W., 588 Chan, M.H.W., s e e Feng, Y.P., 591 Chan, M.H.W., s e e Jin, A.J., 592 Chan, M.H.W., s e e Kim, H.K., 588, 592 Chan, M.H.W., s e e Migone, A.D., 592 Chan, M.H.W., s e e Shrimpton, N.D., 591 Chan, M.H.W., s e e Zhang, Q.M., 592 Chan, S.W., 587 Chandavarkar, S., 593 Chandler, D., s e e Weeks, J.D., 589 Chatelain, A., s e e Borel, J.-P., 586 Chaudhuri, A.R., s e e Patel, J.R., 500 Chen, C.Q., s e e Zhang, Y.G., 133 Chen, S., s e e Hu, G., 438 Chen, X., s e e Hu, G., 438 Chenal, B., s e e Lasalmonie, A., 439 Cheng, A., s e e Shrimpton, N.D., 592 Cheng, H., 593 Cheng, M., 588, 589 Chester, G.V., s e e Gann, R.C., 593 Chester, G.V., s e e Strandburg, K.J., 588, 589 Chester, G.V., s e e Zollweg, J.A., 590, 593 Chevenard, P., 24 Chiang, S.-W., 438 Chikawa, J., 587 Chikawa, J., s e e Sato, F., 504, 587 Chin, A.K., 501 Chin, S., s e e Lall, C., 67, 131, 182, 435 Chinone, N., s e e Kishino, S., 501 Chinone, N., s e e Nakashima, H., 501 Choi, S.K., 502, 503 Chou, C.T., 68, 132, 439 Chou, T., 594 Choyke, W.J., s e e Dean, P.J., 503 Christian, J.W., 185 Chrzan, D.C., 132, 184, 251 Chrzan, D.C., s e e Mills, M.J., 132, 184, 251,440
index
597
Chui, S.T., 589 Chui, S.T., s e e Ma, H., 591 Ciccotti, G., 587 Cladis, P.E., s e e Brinkman, W.F., 589 Clark, N.A., s e e Glaser, M.A., 589 Clarke, R., s e e Nagler, S.E., 588 Clarke, R., s e e Rosenbaum, T.F., 592 Cl6ment, N., 67, 131, 132, 437 Cl6ment, N., s e e Caillard, D., 131-133, 437, 439, 504 Cl6ment, N., s e e Couret, A., 131,439 Cockayne, D.J.H., 67 Cockayne, D.J.H., s e e Gom6z, A., 502 Cockayne, D.J.H., s e e Korner, A., 67, 437 Cockayne, D.J.H., s e e Ray, I.L.F., 502 Cole, M.W., s e e Shrimpton, N.D., 591 Cole, M.W., s e e Vidali, G., 591 Colella, N.J., 592 Colella, N.J., s e e Gangwar, R., 592 Collett, J., s e e Sirota, E.B., 590 Collins, R., 587 Combs, J.A., 594 Comsa, G., s e e Kern, K., 593 Conte, R., s e e Groh, P., 437 Conway, C.G., s e e Pfeil, L.B., 24 Cooman, B.C.D., s e e Kuesters, K., 503 Copley, S.M., 182, 251,436 Copley, S.M., s e e Tien, J.K., 26 Coppersmith, S.N., 591 Corbett, J.W., s e e Bourgoin, J.C., 504 Corey, C.L., s e e Vogel Jr., EL., 25 Cormia, R.L., 586 Cormia, R.L., s e e Mackenzie, J.D., 586 Costa, P., s e e Naka, S., 134 Cotterill, R.M.J., 585 Cottrell, A.H., 67, 251 Couchman, P.R., 586 Couderc, J.J., s e e Levade, C., 503 Couderc, J.J., s e e Vanderschaeve, G., 501 Coujou, A., s e e Caillard, D., 132, 439 Coujou, A., s e e Cl6ment, N., 132 Coujou, A., s e e Couret, A., 131, 133, 439 Coujou, A., s e e Lours, P., 132 Coulet, A.L., s e e Grange, G., 587 Coulomb, P., s e e Lours, P., 132 Couret, A., 67, 131-133, 251,436, 439, 440 Couret, A., s e e Caillard, D., 131-134, 437, 439, 504 Couret, A., s e e Cl6ment, N., 67, 131,437 Couret, A., s e e Farenc, S., 131, 133, 134 Couret, A., s e e Fnaiech, M., 502 Couret, A., s e e Levade, C., 503 Couret, A., s e e Mol6nat, G., 132, 133, 252, 440 Couret, A., s e e Sun, Y.Q., 67, 131,437 Court, S.A., 133, 439 Crabtree, G.W., s e e Kwok, W.K., 591
598
Author
Crary, S.B., 592 Crawford, R.C., 25, 134 Crestou, J., s e e Couret, A., 131,439 Crimp, M.A., 66, 67, 437 Crimp, M.A., s e e Sun, Y.Q., 67, 68, 131,437 Cserti, J., s e e Khantha, M., 132, 183, 251,436, 437 Cserti, J., s e e Vitek, V., 182, 183 Cui, J., 591,592 Cuitifio, A.M., 185, 441 Cummins, H.Z., s e e Awal, M.A., 587 Curwick, H., 435 Czernuszka, J.T., 504 D'Amico, K.L., 588, 592 D'Amico, K.L., s e e Specht, E.D., 591,592 Dadras, M.M., s e e Morris, D.G., 134 Daeges, J., 587 Dahl, K., s e e Tammann, G., 25 Dahm, A.J., s e e Guo, C.J., 590 Dahm, A.J., s e e Kusner, R.E., 590 Dahm, A.J., s e e Mehrotra, R., 590 Damgaard Kristensen, W., s e e Cotterill, R.M.J., 586 Darinskii, B.M., s e e Belyavskii, V.I., 504 Dash, J.G., 586, 588 Dash, J.G., s e e Ecke, R.E., 592 Davey, S.D., s e e Pindak, R., 589 David, R., s e e Kern, K., 593 Davidson, S.M., 504 Davies, R.G., 26, 132, 182, 251,436 Davies, R.G., s e e Beardmore, P., 131 Davies, R.G., s e e Stoloff, N.S., 134 Davies, R.G., s e e Thornton, P.H., 26, 67, 132, 182, 251,435 Daw, M.S., s e e Yoo, M.H., 184, 440 de Bussac, A., 132, 251,440 de Bussac, A., s e e Webb, G., 132 de Leeuw, S.W., s e e Kalia, R.K., 593 De'Bell, K., s e e Piercy, P., 592 Dean, P.J., 503 DeAngelis, H.M., s e e Kimerling, L.C., 504 den Nijs, M., 591 Denier van der Gon, A.W., s e e Frenken, J.W.M., 586 Denier van der Gon, A.W., s e e van der Veen, J.F., 586 Deryagin, B.V., 501 Deville, G., s e e Gallet, E, 590 Deville, G., s e e Glattli, D.C., 590 Devincre, B., 440 Diaz, J.O., s e e Nathal, M.V., 25, 439 Diehl, R.D., 593 Diehl, R.D., s e e Chandavarkar, S., 593 Diehl, R.D., s e e Fisher, D., 593 Dierker, S.B., 589 Dimiduk, D.M., 67, 131, 132, 182, 184, 251,436 Dimiduk, D.M., s e e Simmons, J.P., 133
index
Dimiduk, D.M., s e e Sriram, S., 133 Dimiduk, D.M., s e e Stucke, M.A., 133 Dimiduk, D.M., s e e Yoo, M.H., 26 Dimon, P., 592 Ding, E.J., 251 DiPietro, M.S., s e e Kumar, K.S., 184 Dirras, G., 134, 435 Dirras, G., s e e Beauchamp, P., 134 Donnelly, S.E., 587 Donnelly, S.E., s e e Rossouw, C.J., 586 Doucet, J., s e e Benattar, J.J., 590 Douin, J., 67, 184, 437, 438, 439 Douin, J., s e e Beauchamp, P., 438 Douin, J., s e e Saada, G., 438 Douin, J., s e e Veyssi~re, P., 26, 67, 68, 437-439 Dowling, W.E., 436 Downey, J., s e e Kwok, W.K., 591 Doyoma, M., s e e Cotterill, R.M.J., 585 Dresselhaus, M.S., s e e Erbil, A., 592 Drose, K., s e e Loginov, Y.Y., 502 Duburcq, V., s e e Glattli, D.C., 590 Ducastelle, F., 437 Duesbery, M.S., 26, 133, 134, 183, 184, 435, 440, 589 Duesbery, M.S., s e e Grossmann, B., 589 Duesbery, M.S., s e e Jo6s, B., 589 Dulieu, D., 24 Dumrongrattana, S., s e e Pitchford, T., 589 Dupouy, J.M., s e e R6gnier, P., 131 Durand, M.A., 585 Dutta, P., s e e Cheng, H., 593 Dzhafarov, T.D., 503 Echigoya, J., s e e Nemoto, M., 132, 440 Ecke, R.E., 592 Edwards, S.F., 586 Einstein, T.L., 592 Ekwall, R.A., s e e Brown, N., 185 Eldrup, M., 586 Elgin, R.L., 588 Ellis, D.E., s e e Cheng, H., 593 Eng, S., s e e Sanchez, J.M., 133, 438 Erb, U., 587 Erbil, A., 592 Erofeev, V.N., 502 Erofeev, V.N., s e e Bondarenko, I.E., 502 Erofeeva, S.A., 502 Ertl, G., s e e Bonzel, H.P., 593 Escaig, B., 67, 131, 184, 437, 440 Escaig, B., s e e Bonneville, J., 67, 184 Escaravage, C., s e e George, A., 502 Eschenfelder, A.H., 590 Esquivel, A.L., 504 Estrin, Y., s e e Kubin, L.P., 131 Etienne, B., s e e Glattli, D.C., 590
Author
Evans, J.H., s e e Eldrup, M., 586 Evans-Lutterodt, K.W., s e e Brock, J.D., 588 Ezz, S.S., 67, 68, 131, 182-184, 251,435-437, 441 Ezz, S.S., s e e Pope, D.P., 25, 66, 131, 182, 440 Fahey, D.A., s e e Crary, S.B., 592 Fain Jr., S.C., s e e Cui, J., 591,592 Fain Jr., S.C., s e e Taub, H., 588 Fan, W.C., 593 Farber, B.Y., 502 Farber, B.Y., s e e Nikitenko, V.I., 502 Farenc, S., 131, 133, 134 Farenc, S., s e e Caillard, D., 131 Farenc, S., s e e Couret, A., 131, 133, 439 Farenc, S., s e e Faress, A., 503 Faress, A., 503 Faress, A., s e e Levade, C., 503 Faress, A., s e e Vanderschaeve, G., 501 Farkas, D., 134, 183 Farkas, D., s e e Pasianot, R., 184, 440 Fat-Halla, N.K., s e e Takasugi, T., 131,436 Faust, W.L., s e e Williams, R.T., 504 Feibelman, J., s e e Knotek, M.L., 504 Feng, Y.P., 591 Fem,Sndez, J.E, 588, 594 Ferraz, A., 586 Ferreira, M.E, s e e Femfmdez, J.F., 588 Ferreira, O., s e e Tejwani, M.J., 590, 592 Fink, J., s e e vom Felds, A., 587 Finnis, M.W., 183 Finotello, D., 588 Fiory, A.T., 588 Fiory, A.T., s e e Hebard, A.F., 588 Fisher, D.S., 251,589, 590, 593 Fisher, D.S., s e e Coppersmith, S.N., 591 Fisher, D.S., s e e Fisher, M.E., 591 Fisher, D.S., s e e Narayan, O., 251 Fisher, M.E., 591 Fisher, M.E., s e e Huse, D.A., 591 Fisher, M.P.A., 590 Fisher, M.P.A., s e e Fisher, D.S., 590 Fisher, M.P.A., s e e Koch, R.H., 590 Fisher, R.M., s e e Marcinkowski, M.J., 25, 66 Flannery, B.P., s e e Press, W.H., 252 Fleischer, R.L., 26 Fleshier, S., s e e Kwok, W.K., 591 Flinn, P.A., 26, 67, 182, 251,438 Floyd, R.W., s e e Taylor, A., 24 Fnaiech, M., 502 Foglietti, V., s e e Koch, R.H., 590 Forbes, K.R., s e e Hernker, K., 132, 251 Foreman, A.J.E., 25 Fortini, A., 501 Fourdeux, A., s e e Kubin, L.P., 134
index
Foxall, R.A., s e e Duesbery, M.S., 184 Frahm, R., s e e Greiser, N., 592 Frangois, A., 435 Frank, F.C., 591 Frank, V.L.P., s e e Lauter, H.J., 592 Fraser, H.L., s e e Court, S.A., 133, 439 Fraser, H.L., s e e Vasudevan, V.K., 438 Freeland, P.E., s e e Patel, J.R., 502 Frenkel, D., 593 Frenkel, D., s e e Allen, M.P., 593 Frenkel, D., s e e Bladon, P., 594 Frenkel, D., s e e Udink, C., 592 Frenkel, J., 591 Frenken, J.W.M., 586 Frenken, J.W.M., s e e Pinxteren, H.M., 586 Frenken, J.W.M., s e e Pluis, B., 586 Friedel, J., 184, 439, 440 Filsch, H.L., 502 Filtzlen, G.A., 24 Frost, H.J., 252, 441 Fu, C.L., 435, 440 Fu, C.L., s e e Yoo, M.H., 26 Fujii, Y., s e e Sirota, E.B., 590 Fujita, H., s e e Moil, H., 587 Fujita, K., s e e Yamashita, Y., 501 Fujita, M., s e e Moil, H., 587 Fujiwara, T., 501,503 Fukatsu, S., s e e Yamashita, Y., 501 Gadijak, G.V., s e e Bedanov, V.M., 593, 594 Gal'vides, N.M., s e e Deryagin, B.V., 501 Gallagher, P.C.J., 131 Gallagher, W.J., s e e Koch, R.H., 590 Gallet, E, 590 Galligan, J.M., s e e Kung, J., 503 Gammel, P.L., 590 Gammel, P.L., s e e Grier, D.G., 591 Gammel, P.L., s e e Murray, C.A., 591 Gammel, P.L., s e e Safar, H., 591 Gangwar, R., 592 Gann, R.C., 593 Gao, Y., 438 Gaspailni, EM., s e e Finotello, D., 588 Gastaldi, J., s e e Grange, G., 587 Gay, J.M., s e e Pluis, B., 586 Geer, R., 590 Geer, R., s e e Huang, C.C., 589 George, A., 502 George, A., s e e Louchet, F., 502 George, E.P., 438 George, E.P., s e e Yoo, M.H., 26 Giamei, A.E, 25, 68, 438 Giamei, A.E, s e e Kear, B.H., 439 Gibala, R.G., s e e Dowling, W.E., 436
599
600
Author
Gibbs, G., s e e D'Amico, K.L., 592 Gierlotka, S., s e e Pluis, B., 586 Gignac, W., s e e Allen, M.P., 593 Gilman, J.J., 440 Gilman, J.J., s e e Johnston, W.G., 25 Gilman, J.J., s e e Westbrook, J.H., 500 Gilmer, G.H., s e e Broughton, J.Q., 593 Ginsberg, D.M., s e e Safar, H., 591 Glaberson, W.I., s e e Fiory, A.T., 588 Glaser, M.A., 589 Glattli, D.C., 590 Gleiter, H., s e e Daeges, J., 587 Gleiter, H., s e e Erb, U., 587 Godfrin, H., s e e Lauter, H.J., 592 Goeppert-Mayer, M., s e e Herzfeld, K.E, 585 Goldheim, D.L., s e e Westwood, A.R.C., 501 Gomer, R., s e e Menzel, D., 503 Gom6z, A., 502 Gondi, P., s e e Cavallini, A., 501 Goodby, J.W., s e e Geer, R., 590 Goodby, J.W., s e e Gray, G.W., 589 Goodby, J.W., s e e Huang, C.C., 589 Goodby, J.W., s e e Pindak, R., 589 Goodby, J.W., s e e Pitchford, T., 589 Goodby, J.W., s e e Stoebe, T., 590 Gooding, R.J., 591 Goods, S.H., 251 Goods, S.H., s e e Mills, M.J., 251 Goodstein, D.L., s e e Elgin, R.L., 588 Gordon, M.B., 591 Gorid'ko, N.Y., 501 Gornostyrev, Yu.N., s e e Greenberg, B.A., 132, 133 Goto, K., s e e Sato, E, 504 Gottlieb, J.M., 592 Gottschalk, H., 502, 503 Gottschalk, H., s e e Alexander, H., 503 Granato, A.V., 184 Granato, A.V., s e e Holder, J., 586 Granato, A.V., s e e Teutonico, L.J., 184 Grange, G., 587 Gray, G.W., 589 Greenberg, B.A., 132, 133, 441 Greene, R.L., s e e Greiser, N., 592 Greiser, N., 592 Grest, G., s e e Robbins, M.O., 590 Grier, D.G., 591 Griffiths, R.B., s e e Butler, D.M., 592 Grimditch, M., s e e Okamoto, P.R., 587 Grimes, C.C., 590 Griscom, D.L., 504 Griscom, D.L., s e e Tsai, T.E., 504 Groh, P., 437 Grossmann, B., 589 Guan, D.L., s e e Veyssi6re, P., 438
index
Guedon, J.Y., s e e Kubin, L.P., 134 Guenin, B.M., s e e Mehrotra, R., 590 Guillon, D., s e e Huang, C.C., 589 Guillop6, M., s e e Ciccotti, G., 587 Guiu, E, 252 G0nther, S., s e e Morris, D.G., 133, 435 Guo, C.J., 590 Gupta, A., s e e Koch, R.H., 590 Gupta, N.P., 585 Gurney, R.W., s e e Mott, N.F., 585 Gutmanas, E.Y., 503 Gutzow, I., 586 Haasen, P., 131,502, 504 Haasen, P., s e e Gutmanas, E.Y., 503 Haasen, P., s e e Jendrich, U., 502 Haberman, R., 184 Hagel, W.C., s e e Sims, C.T., 24, 26 Hahn, Y.D., s e e Whang, S.H., 133 Haken, H., 251 Hall, R.N., 501 Halperin, B.I., 585 Halperin, B.I., s e e Coppersmith, S.N., 591 Halperin, B.I., s e e Fisher, D.S., 589 Halperin, B.I., s e e Nelson, D.R., 585, 589 Halpin-Healy, T., 592 Hamana, T., s e e Saburi, T., 131,436 Hamao, N., s e e Mishima, Y., 436 Hammonds, E.M., s e e Birgeneau, R.J., 588, 591 Hamona, T., s e e Saburi, T., 184 Hanada, S., 134 Hanada, S., s e e Takasugi, T., 134 Hanneman, R.E., 501 Hansen, F.Y., 592 Hanson, D., s e e Alexander, W.O., 24 Hardcastle, S.E., s e e Zabel, H., 593 Harding, W., s e e Titchmarsh, J.M., 504 Harker, A.H., s e e Itoh, N., 504 Hart, E.W., 587 Hartig, C., s e e Schr6er, W., 134 Hartman, R.L., s e e Petroff, P., 501 Haussermann, E, 502 Havner, K.S., 184 Havner, K.S., s e e Hill, R., 185 Hayes, W., 503 Hazzledine, P.M., 67, 68, 437, 439 Hazzledine, P.M., s e e Chou, C.T., 68, 439 Hazzledine, P.M., s e e Couret, A., 67, 251,439 Hazzledine, P.M., s e e Crimp, M.A., 66, 437 Hazzledine, P.M., s e e Karnthaler, H.P., 439 Hazzledine, P.M., s e e Schneibel, J.H., 182, 437 Hazzledine, P.M., s e e Stucke, M.A., 133 Hazzledine, P.M., s e e Sun, Y.Q., 67, 68, 131, 133, 251,437, 438
Author
Hebard, A.E, 588 Hebard, A.E, s e e Fiory, A.T., 588 Hedges, J.N., 25 Heggie, M.I., 502, 504 Heggie, M.I., s e e Jones, R., 504 Heideman, A., s e e Larese, J.Z., 592 Heidenreich, R.D., 25 Heiney, P.A., 592 Heiney, P.A., s e e Birgeneau, R.J., 588, 591 Heiney, P.A., s e e Stephens, P.W., 591 Held, G.A., s e e Greiser, N., 592 Helgesen, G., 590 Hemker, K.J., 67, 132, 133, 184, 251,436, 437 Hemker, K.J., s e e Baluc, N., 67, 132, 184, 436, 438 Henderson, D., s e e Barker, J.A., 593 Henry, C.H., 503, 504 Heredia, EE., 131, 182, 183 Heredia, H.E, 435, 436 Hersh, H., 504 Herzfeld, K.F., 585 Hess, W., 590 Heuser, F.W., s e e Teutonico, L.J., 184 Hignett, H.W.G., 24 Hiki, Y., 587 Hilhorst, H.J., s e e Bakker, A.F., 590 Hill, R., 185 Hill, R., s e e Bishop, J.EW., 185 Hilliard, J.E., 587 Hilliard, J.E., s e e Cahn, J.W., 587 Hirakawa, S., s e e Takasugi, T., 133, 437 Hirsch, P.B., 25, 67, 68, 132, 184, 251,437, 438, 440, 441,502 Hirsch, P.B., s e e Chou, C.T., 68, 132, 439 Hirsch, P.B., s e e Couret, A., 67, 132, 251,436 Hirsch, P.B., s e e Ezz, S.S., 68, 183, 436, 437, 441 Hirsch, P.B., s e e Gom6z, A., 502 Hirsch, P.B., s e e Hazzledine, P.M., 439 Hirsch, P.B., s e e Silcox, J., 67 Hirth, J.P., 25, 67, 441,501,502 Hitzenberger, C., s e e Korner, A., 438 Ho, J.T., s e e Cheng, M., 588, 589 Hobart, R., 591 Hochman, R.E, s e e Kuczinski, G.C., 501 Holder, J., 586 Holt, D.B., s e e Yacobi, B.G., 502 Homann, A., s e e Melzer, A., 594 Hondros, E., s e e Chou, C.T., 68, 439 Hoover, N.E., s e e Hoover, W.G., 589 Hoover, W.G., 589 Hoover, W.G., s e e Ladd, A.J.C., 589 Horn, P.M., s e e Birgeneau, R.J., 591 Horn, P.M., s e e Brock, J.D., 588 Horn, P.M., s e e D'Amico, K.L., 588 Horn, P.M., s e e Dimon, P., 592
index
601
Horn, P.M., s e e Greiser, N., 592 Horn, P.M., s e e Heiney, P.A., 592 Horn, P.M., s e e Mochrie, S.G.J., 592, 593 Horn, P.M., s e e Nagler, S.E., 588 Horn, P.M., s e e Rosenbaum, T.E, 592 Horn, P.M., s e e Specht, E.D., 591,592 Horn, P.M., s e e Stephens, P.W., 591 Hornbecker, M.E, s e e Kear, B.H., 67, 132, 183 Home, R.W., s e e Hirsch, P.B., 25 Horton, J.A., 440 Horton, J.A., s e e Baker, I., 132, 440 Horton, J.A., s e e George, E.P., 438 Horton, J.A., s e e Schneibel, J.H., 437 Horton, J.A., s e e Veyssi~re, P., 438 Horton, J.A., s e e Yoo, M.H., 132, 184 Howe, L.M., 439 Howie, A., s e e Hirsch, P.B., 438 Hsieh, T.E., s e e Baluffi, R.W., 587 Hsiung, L.M., 68 Hu, G., 438 Huang, C.C., 589 Huang, C.C., s e e Geer, R., 590 Huang, C.C., s e e Pitchford, T., 589 Huang, C.C., s e e Stoebe, T., 590 Huang, C.C., s e e Viner, J.M., 590 Hug, G., 131, 133 Hug, G., s e e Lasalmonie, A., 439 Hui, S.W., s e e Cheng, M., 588, 589 Huse, D.A., 591 Huse, D.A., s e e Fisher, D.S., 590 Huse, D.A., s e e Safar, H., 591 Hutchinson, J.W., 185 Hutchinson, T.S., s e e Blair, D.G., 184 Ichihara, M., s e e Maeda, K., 502, 503 Ichihara, M., s e e Suzuki, K., 25, 68, 132, 439, 440 Ichihara, M., s e e Suzuki, T., 25, 66 Ichihara, M., s e e Takeuchi, S., 251 Ignatiev, A., s e e Fan, W.C., 593 Ihm, G., s e e Vidali, G., 591 Imai, H., s e e Fujiwara, T., 501,503 Imai, M., 502 Imura, T., s e e Nohara, A., 133 Indenbaum, V.N., s e e Greenberg, B.A., 133 Inui, H., 438 Iqbal, M.Z., s e e Davidson, S.M., 504 Ishida, K., s e e Kamejima, T., 501 Isozumi, S., s e e Ueda, O., 503 Ito, R., s e e Kishino, S., 501 Ito, R., s e e Nakashima, H., 501 Itoh, H., s e e Murakami, K., 503 Itoh, N., 503, 504 Iunin, Y.L., s e e Farber, B.Y., 502 Iunin, Y.L., s e e Nikitenko, V.I., 502
602
Author
Ivanov, M.A., s e e Greenberg, B.A., 132, 441 lwamoto, M., 501 Iwanaga, H., s e e Takeuchi, S., 501,502 Iyer, K.R., s e e Kuczinski, G.C., 501 Izumi, M., s e e Nohara, A., 133 Izumi, O., s e e Aoki, K., 184, 436 Izumi, O., s e e Hanada, S., 134 Izumi, O., s e e Kawabata, T., 133 Izumi, O., s e e Liu, Y., 133, 435, 438 Izumi, O., s e e Takasugi, T., 131, 133, 134, 436, 437 Jackson, K.A., s e e Celler, G.K., 587 Jancovici, B., 588 Janke, W., 589 Jendrich, U., 502 Jenkins, M.L., s e e Cockayne, D.J.H., 67 Jensen, E.J., s e e Cotterill, R.M.J., 586 Jesser, W.A., s e e Couchman, P.R., 586 Jian, N., 439 Jiang, N., 68 Jin, A.J., 592 Johnson, D., s e e Mahmood, R., 589 Johnson, W.D., 503 Johnson, W.L., s e e Cahn, R.W., 585 Johnston, T.L., 436 Johnston, T.L., s e e Beardmore, P., 131 Johnston, T.L., s e e Thornton, P.H., 26, 67, 132, 182, 251,435 Johnston, W.D., 501 Johnston, W.D., s e e Chin, A.K., 501 Johnston, W.G., 25 Jones, I.P., s e e Ngan, A.H.W., 68, 131,435, 439 Jones, I.P., s e e Yan, W., 439 Jones, R., 502, 504 Jones, R., s e e Heggie, M., 502, 504 Jonsson, S., 131 Jonsson, S., s e e Beuers, J., 131 Jo6s, B., 589, 592 Jo6s, B., s e e Duesbery, M.S., 589 Jo6s, B., s e e Gooding, R.J., 591 Jo6s, B., s e e Grossmann, B., 589 Jo6s, B., s e e Shrimpton, N.D., 591,592 Jorgensen, P.J., s e e Hanneman, R.E., 501 Jourdan, C., s e e Grange, G., 587 Joyce, J.C., s e e Morris, D.G., 133, 435 Jumojni, K., 436, 439 Kabler, M., s e e Celli, V., 502 Kaganer, V.M., s e e Peterson, I.R., 589 Kalia, R.K., 593 Kalia, R.K., s e e Cheng, H., 593 Kalia, R.K., s e e Vashishta, P., 593 Kamejima, T., 501
index
Kamioka, H., 587 K a n ~ f i , T . , s e e Kawabata, T., 133 Kancheev, O.D., s e e Mirkin, I.L., 25 Kapitulnik, A., s e e Grier, D.G., 591 Kapitulnik, A., s e e Murray, C.A., 591 Kardar, M., s e e Aharony, A., 588 Kardar, M., s e e Caflisch, R.G., 592 Kardar, M., s e e Halpin-Healy, T., 592 Kardar, M., s e e Paczuski, M., 588 Karkina, L.E., s e e Greenberg, B.A., 133 Karnthaler, H.P., 67, 439, 441 Karnthaler, H.P., s e e Baluc, N., 67, 184, 437, 438 Kamthaler, H.P., s e e Hazzledine, P.M., 437 Karnthaler, H.P., s e e Korner, A., 438 Karnthaler, H.P., s e e Mills, M.J., 67, 132, 251 Karnthaler, P., s e e Mills, M.J., 439 Karsten, K., 25 Kasami, A., s e e Iwamoto, M., 501 Kashyap, B.P., 131 Kato, M., s e e Jumojni, K., 436, 439 Kawabata, T., 133 Kawabata, T., s e e Takasugi, T., 134 Kear, B.H., 25, 67, 132, 133, 183, 436, 437, 439 Kear, B.H., s e e Copley, S.M., 182, 251,436 Kear, B.H., s e e Giamei, A.E, 68, 438 Kear, B.H., s e e Oblak, A.E, 439 Kelly, P.J., s e e Car, P., 504 Keown, S., 24 Keramidas, V.G., s e e Chin, A.K., 501 Kerins, J., s e e Kusner, R.E., 590 Kern, K., 593 Kestel, B.J., s e e Meng, W.J., 587 Khan, T., 436 Khan, T., s e e Caron, P., 440 Khantha, M., 132, 183, 251,436, 437, 440 Khantha, M., s e e Vitek, V., 182 Kikuchi, M., 501 Kikuchi, R., 438 Kim, H.-Y., s e e Shrimpton, N.D., 592 Kim, H.-Y., s e e Vidali, G., 591 Kim, H.K., 588, 592 Kim, H.K., s e e Zhang, Q.M., 592 Kimerling, L.C., 504 Kimerling, L.C., s e e Weeks, J.D., 503 Kimura, K., s e e Maeda, N., 501 Kishino, S., 501 Kishino, S., s e e Nakashima, H., 501 Kisielowski-Kemmerich, C., s e e Alexander, H., 502 Kisliuk, A., s e e Voronel, A., 587 Klassen, N.V., 503 Klein, M.L., s e e Jo6s, B., 592 Klein, M.L., s e e Moiler, M.A., 592 Klein, R., s e e Hess, W., 590 Kleinert, H., 586, 589
Author
Kleinert, H., s e e Janke, W., 589 Kluge, M., s e e Wolf, D., 587 Knotek, M.L., 504 Koch, R.H., 590 Koch, S.W., 593 Koch, S.W., s e e Abraham, EE, 591,593 Koehler, J.S., 25, 66, 184 Kohlstedt, D.L., s e e Chiang, S.-W., 438 Koiwa, M., s e e Yasuda, H., 441 Kojima, K., 501 Komiya, S., s e e Ueda, O., 503 Kontorowa, T., s e e Frenkel, J., 591 Koren, G., s e e Koch, R.H., 590 Kornblit, A., s e e Willens, R.H., 588 Korner, A., 67, 68, 131, 132, 251,437-439 Korner, A., s e e Schoeck, G., 439 Kortan, A.R., s e e Erbil, A., 592 Kortan, A.R., s e e Mochrie, S.G.J., 592, 593 K6ster, W., 26 Kosterlitz, J.M., 585, 588 Kosterlitz, J.M., s e e Lee, J., 593 Kotz6, I.A., 586 Kravchenko, V.Y., s e e Vardanyan, R.A., 504 Kremer, K., s e e Robbins, M.O., 590 Kristensen, J.K., s e e Kristensen, W.D., 586 Kristensen, W.D., 586 Krug, J., 252 Kubin, L.P., 131, 134 Kubin, L.P., s e e Devincre, B., 440 Kubin, L.P., s e e Naka, S., 134 Kubin, L.P., s e e Potez, L., 133, 437 Kubo, A., s e e Maeda, K., 501 Kuczinski, G.C., 501 Kuesters, K., 501,503 Kuhlmann-Wilsdorf, D., 586 Kuhlmann-Wilsdorf, D., s e e Kotz6, I.A., 586 Kulkarni, S.B., 502 Kumar, K.S., 184, 435 Kung, J., 503 Kuramoto, E., 133, 182, 184, 436 Kuramoto, E., s e e Suzuki, K., 25 Kuramoto, E., s e e Takeuchi, S., 67, 132, 182, 251, 436 Kuramoto, M.E., s e e Suzuki, K., 440 Kurnakov, N.S., 26 Kusner, R.E., 590 Kuz'menko, P.P., s e e Gorid'ko, N.Y., 501 Kwok, W.K., 591 Kyser, D.E, s e e Wittry, D.B., 504 Ladd, A.J.C., 589 Laird, C., s e e Wu, T.-Y., 185 Lall, C., 67, 131, 182, 435 Lam, N.Q., 587
index
Lambert, M., s e e Benattar, J.J., 590 Lan~on, E, s e e Gordon, M.B., 591 Landau, L.D., 585 Lang, D.V., s e e Henry, C.H., 504 Lang, D.V., s e e Kimerling, L.C., 504 Lapasset, G., s e e Potez, L., 133, 437 Larese, J.Z., 592 Larese, J.Z., s e e Zhang, Q.M., 592 Lartigue, C., s e e Simon, Ch., 593 Lasalmonie, A., 439 Lasalmonie, A., s e e Naka, S., 134 Lauter, H.J., 592 Lauter, H.J., s e e Taub, H., 588 Lee, J., 593 Lee, P.A., s e e Coppersmith, S.N., 591 Lefebvre, A., s e e Caillard, D., 504 Legrand, B., 131 Legrand, J.E, s e e Simon, Ch., 593 LeGuillon, J.C., 589 Leibfried, G., 184, 251 Left, R., 133, 438 Left, R., s e e Morris, D.G., 133, 435 Leung, C.H., s e e Williams, R.T., 504 Leung, P.W., s e e Zollweg, J.A., 590 Levade, C., 503, 504 Levade, C., s e e Faress, A., 503 Levade, C., s e e Louchet, F., 504 Levade, C., s e e Vanderschaeve, G., 501 Levelut, A.M., s e e Benattar, J.J., 590 Lewis, M., s e e Mahmood, R., 589 Li, J.C.M., 251 Li, Z.C., 133 Li, Z.C., s e e Whang, S.H., 133 Li, Z.R., s e e Migone, A.D., 592 Li, Z.X., 133 Li, Z.X., s e e Whang, S.H., 133 Liang, J.C., s e e Brock, J.D., 588 Liang, S.-J., 440 Lifshitz, E.M., s e e Landau, L.D., 585 Lin, D., s e e Wen, M., 132 Lin, W.N., s e e Esquivel, A.L., 504 Lindemann, EA., 585 Linker, G., s e e vom Felds, A., 587 Lipsitt, H.A., s e e Shechtman, D., 133 Lischner, D.J., s e e Celler, G.K., 587 Litster, J.D., s e e Aharony, A., 588 Litster, J.D., s e e Birgeneau, R.J., 589 Litster, J.D., s e e Brock, J.D., 588 Litzinger, J.A., 592 Litzinger, J.A., s e e Butler, D.M., 588, 592 Liu, C.T., 25 Liu, C.T., s e e Yoo, M.H., 132, 184, 436 Liu, C.Y., s e e Veyssi6re, P., 438 Liu, J.L., s e e Chan, S.W., 587
603
604 Liu, Y., 133, 435, 438 Liu, Y.H., s e e Arko, A.C., 436 L6fvander, J.P.A., s e e Court, S.A., 439 Loginov, Y.Y., 502 Loiseau, A., s e e Ducastelle, F., 437 Loiseau, A., s e e Hug, G., 133 Lomak, N.V., s e e Negrii, V.D., 503 Lomer, W.M., 67 Loretto, M.H., s e e Court, S.A., 439 Lothe, J., s e e Hirth, J.P., 67, 441,501,502 Louat, N.P., s e e Duesbery, M.S., 440 Louchet, F., 502, 504 Lours, P., 132 Lours, P., s e e Caillard, D., 132, 439 Lours, P., s e e Cl6ment, N., 132 L6wen, H., 586 Lowrie, R., 435 Lozovik, Y.E., s e e Bedanov, V.M., 594 Lu Da 24 Lu, Y.N., s e e Ding, E.J., 251 Lticke, K., s e e Granato, A.V., 184 Lticke, K., s e e Teutonico, L.J., 184 Lund, F., 586, 588 Lutsko, J.F., 587 Lutsko, J.F., s e e Phillpot, S.R., 587 Lutsko, J.F., s e e Wolf, D., 587 LUty, F., 504 Luzzi, D.E., s e e Inui, H., 438 Lyakhovets, V.D., s e e Martynyuk, M.M., 586 Lye, R.G., s e e Westwood, A.R.C., 501 Lyuksyutov, I., 588 Ma, H., 591 MacDonald, J.E., s e e Pluis, B., 586 Mackenzie, J.D., 586 Mackenzie, J.D., s e e Ainslie, G., 586 Mackenzie, J.D., s e e Cormia, R.L., 586 Madsen, J.U., s e e Cotterill, R.M.J., 586 Madsen, L.L., s e e Viswanathan, R., 594 Maeda, K., 501,502, 503 Maeda, K., s e e Nakagawa, K., 504 Maeda, K., s e e Nakano, K., 501 Maeda, K., s e e Takeuchi, S., 501,502 Maeda, K., s e e Yamashita, Y., 501 Maeda, N., 501,503, 504 Maeda, N., s e e Maeda, K., 502 Maeno, N., 586 Magerl, A., s e e Zabel, H., 593 Mahajan, S., s e e K. Chin~ A., 501 Mahmood, R., 589 Maitland, G.C., 591 M a k , A . , s e e Specht, E.D., 591 Makin, M.J., s e e Foreman, A.J.E., 25 Malozemoff, A.P., 590
Author
index
Mann, J.A., s e e Kusner, R.E., 590 Maradudin, A.A., 586 Marcinkowski, M.J., 25, 66, 436 Maree, P.M., s e e Frenken, J.W.M., 586 Marklund, S., 502, 504 Marklund, S., s e e Jones, R., 504 Marsh, J.H., 24 Martens, A., 25 Martin, C.J., 585 Martin, J.L., s e e Baluc, N., 132, 183, 436, 439, 441 Martin, J.L., s e e Bonneville, J., 132, 182, 183, 252, 437, 440 Martin, J.L., s e e Carrard, M., 440 Martin, J.L., s e e Sp~itig, P., 132, 183, 436, 437 Martin, J.L., s e e Stoiber, J., 252 Martynyuk, M.M., 586 Mast, D.B., s e e Guo, C.J., 590 Masuda, K., s e e Murakami, K., 503 Matsui, J., s e e Kamejima, T., 501 Matsui, M., 503 Matsui, M., s e e Sato, F., 587 Matsumoto, A., 133 Maurer, R., s e e Balluffi, R.W., 587 McClain, B.R., s e e Brock, J.D., 588 McEvily, A.J., s e e Johnston, T.L., 436 McLean, M., s e e Chou, C.T., 68, 439 McQueency, D.C., s e e Agnolet, G., 588 McTague, J.P., 588, 592 McTague, J.P., s e e Allen, M.P., 593 McTague, J.P., s e e Frenkel, D., 593 McTague, J.P., s e e Nielsen, M., 592 McTague, J.P., s e e Novaco, A.D., 588 Mdivanyan, B.E., 502 Meakin, J.D., 437 Mecking, H., s e e Schr6er, W., 134 Mehrotra, R., 590 Mehrotra, R., s e e Guo, C.J., 590 Melzer, A., 594 Meng, W.J., 587 Menzel, D., 503 Mera, Y., s e e Yamashita, Y., 501 Mermin, N.D., 587 Merzhanov, K.I., s e e Deryagin, B.V., 501 Meyer, R.B., s e e Dierker, S.B., 589 Migone, A.D., 592 Mihara, M., s e e Choi, S.K., 502, 503 Mikhieva, V.T., s e e Shishokin, V.P., 25 Mikolla, S.E., s e e Gao, Y., 438 Miller, B.I., s e e Johnston, W.D., 501,503 Miller, M.M., s e e Kwok, W.K., 591 Mills, M.J., 67, 132, 184, 251,439, 440 Mills, M.J., s e e Baluc, N., 67, 184, 437, 438, 441 Mills, M.J., s e e Chrzan, D.C., 132, 184, 251
Author
Mills, M.J., s e e Hemker, K.J., 67, 132, 133, 184, 251, 436 Mills, M.J., s e e Yoo, M.H., 26 Miner, R.V., s e e Nathal, M.V., 25, 439 Miracle, D.B., s e e Farkas, D., 134 Mirkin, I.L., 25 Mishima, Y., 26, 435, 436 Mishima, Y., s e e Miura, S., 182, 436 Mishima, Y., s e e Suzuki, T., 131, 182, 435 Mishima, Y., s e e Tounsi, B., 132, 439 Mitchell, J.W., s e e Hedges, J.N., 25 Mitzi, D.B., s e e Grier, D.G., 591 Mitzi, D.B., s e e Murray, C.A., 591 Miura, S., 182, 436 Miura, S., s e e Suzuki, T., 25, 66, 131, 182, 435 Mizushima, S., 585 Mochrie, S.G.J., 592, 593 Mochrie, S.G.J., s e e Nagler, S.E., 588 Mockler, R.C., s e e Armstrong, A.J., 590 Mockler, R.C., s e e Tang, Y., 590 Mol6nat, G., 67, 131-134, 251,252, 437, 439, 440 Mol6nat, G., s e e Caillard, D., 131, 134 Mol6nat, G., s e e C16ment, N., 131,437 Mol6nat, G., s e e Couret, A., 131,439 Mol6nat, G., s e e Paidar, V., 132, 440 Moiler, M.A., 592 Moncton, D.E., 589 Moncton, D.E., s e e D'Amico, K.L., 588, 592 Moncton, D.E., s e e Dimon, P., 592 Moncton, D.E., s e e Heiney, P.A., 592 Moncton, D.E., s e e Nagler, S.E., 588 Moncton, D.E., s e e Nielsen, M., 592 Moncton, D.E., s e e Pindak, R., 589 Moncton, D.E., s e e Specht, E.D., 591,592 Moncton, D.E., s e e Stephens, P.W., 591 Monemar, B., 501,504 Montoya, K., s e e Kumar, K.S., 184 Morf, R.H., 590, 593 Morf, R.H., s e e Fisher, D.S., 589 Morrill, G.E., s e e Thomas, H.M., 594 Moil, H., 587 Morris, D.G., 133, 134, 435, 438 Morris, D.G., s e e Left, R., 133, 438 Morris, D.G., s e e Veyssi6re, P., 67, 438 Morris, M., 133 Morris, M.A., s e e Morris, D.G., 134 Moss, S.A., 24 Moss, W.C., s e e Hoover, W.G., 589 Mott, N.F., 585 Mouritsen, O.G., s e e Besold, G., 587 Moussa, F., 590 Muchy, D.C., s e e Louchet, F., 502 Mtilbacher, E.Th., s e e Kamthaler, H.P., 67, 439, 441 Mulford, R.A., 68, 183, 435
index
Muller, E.W., 25 Mtiller-Heinzerling, Th., s e e vom Felds, A., 587 Muller-Krumbhaar, H., s e e Saito, Y., 589 Munroe, P.R., 134 Murakami, K., 503 Murakami, K., s e e Umakoshi, Y., 133 Murata, Y., s e e Aruga, T., 593 Murayama, Y., s e e Maeda, K., 501 Murray, C.A., 590, 591 Murray, C.A., s e e Grier, D.G., 591 Murray, C.A., s e e Van Winkle, D.H., 590 Nabarro, F.R.N., 133, 440, 586 Nadeau, J.S., 501 Nadgoryni, E., 502 Nagler, S.E., 588 Nagler, S.E., s e e D'Amico, K.L., 588 Nagler, S.E., s e e Rosenbaum, T.F., 592 Nagler, S.E., s e e Specht, E.D., 591,592 Naidoo, K.J., 593, 594 Naka, S., 134 Naka, S., s e e Khan, T., 436 Nakagawa, K., 504 Nakagawa, K., s e e Takeuchi, S., 501 Nakahara, S., s e e Willens, R.H., 588 Nakamura, N., 134 Nakano, K., 501 Nakashima, H., 501 Nakashima, H., s e e Kishino, S., 501 Namba, Y., s e e Umakoshi, Y., 133 Narayan, O., 251 Nathal, M.V., 25, 439 Naumovets, A.G., s e e Lyuksyutov, I., 588 Nazmy, M., s e e Anton, D.L., 133 Negrii, V.D., 503 Neite, G., s e e Nembach, E., 25 Nelson, D.R., 585, 589 Nelson, D.R., s e e Berker, A.N., 588 Nelson, D.R., s e e Brezin, E., 591 Nelson, D.R., s e e Bruinsma, R., 589 Nelson, D.R., s e e Chou, T., 594 Nelson, D.R., s e e Halperin, B.I., 585 Nembach, E., 25 Nembach, E., s e e Mol6nat, G., 134 Nemoto, M., 132, 440 Nenno, S., s e e Pak, H.R., 68, 184, 437 Nenno, S., s e e Saburi, T., 131, 184, 436 Neubert, M.E., s e e Mahmood, R., 589 Neumann, D.A., s e e Zabel, H., 593 Newton, J.C., s e e Hansen, F.Y., 592 Ngan, A.H.W., 68, 131,435, 439 Nicholls, J.R., 437 Nicholson, R.B., s e e Hirsch, P.B., 438 Nielsen, M., 592
605
606
Author
Nielsen, M., s e e McTague, J.P., 592 Nikitenko, V.I., 502 Nikitenko, V.I., s e e Bondarenko, I.E., 502 Nikitenko, V.I., s e e Erofeev, V.N., 502 Nikitenko, V.I., s e e Farber, B.Y., 502 Ninomiya, T., 586 Ninomiya, T., s e e Celli, V., 502 Ninomiya, T., s e e Choi, S.K., 502, 503 Nishitani, S.R., s e e Yamaguchi, M., 182 Nix, W.D., s e e Hemker, K.J., 132, 251,436, 437 Noguchi, O., 26, 436 Noguchi, O., s e e Wee, D.M., 131, 182, 435 Noh, D.Y., s e e Brock, J.D., 588 Noh, D.Y., s e e Nuttall, W.J., 594 Nohara, A., 133, 134, 435 Northrop, D.C., s e e Davidson, S.M., 504 Nos6, S., 593 Notkin, A.B., s e e Greenberg, B.A., 133 Nounesis, G., s e e Huang, C.C., 589 Nounesis, G., s e e Pitchford, T., 589 Novaco, A.D., 588, 593 Novaco, A.D., s e e McTague, J.P., 588 Novikov, N.N., s e e Gorid'ko, N.Y., 501 Nuttall, W.J., 594 Oberg, S., s e e Jones, R., 504 Oblak, A.E, 439 Oblak, J.M., s e e Giamei, A.F., 68, 438 Oblak, J.M., s e e Kear, B.H., 437, 439 Ochiai, S., s e e Mishima, Y., 26, 435, 436 Ochiai, S., s e e Miura, S., 182 Ochiai, S., s e e Suzuki, T., 435 O'Connor, D.A., s e e Martin, C.J., 585 Ohta, H., s e e Liu, Y., 438 Okamoto, H., 25 Okamoto, P.R., 587 Okamoto, P.R., s e e Lam, N.Q., 587 Okamoto, P.R., s e e Meng, W.J., 587 Okamoto, P.R., s e e Wolf, D., 587 Oliver, J., 438, 440 Ono, S., s e e Liu, Y., 435, 438 Ono, S., s e e Takasugi, T., 133, 134, 437 Ookawa, A., 585 Orlov, A.N., s e e Rybin, V.V., 502 Ortiz, M., s e e Cuitino, A.M., 185, 441 Oshiyama, A., s e e C a r , P., 5 0 4 Osip'yan, Y.A., 501,503 Osip'yan, Y.A., s e e Erofeeva, S.A., 502 Osip'yan, Y.A., s e e Klassen, N.V., 503 Osip'yan, Y.A., s e e Negrii, V.D., 503 Osip'yan, Y.A., s e e Vardanyan, R.A., 504 Oslund, S., s e e Berker, A.N., 592 O'Sullivan, W.J., s e e Armstrong, A.J., 590 O'Sullivan, W.J., s e e Tang, Y., 590
index
Osvenskii, V.B., s e e Erofeev, V.N., 502 Ou-Yang, H.D., s e e Sirota, E.B., 590 Ourmazd, A., s e e Hirsch, P.B., 502 Oya, N., s e e Noguchi, O., 436 Oya, Y., s e e Miura, S., 182 Oya, Y., s e e Noguchi, O., 26 Oya, Y., s e e Suzuki, T., 133, 435 Oya, Y., s e e Wee, D.M., 131, 182, 435 Pace, N.G., s e e Saunders, G.A., 587 Pacheva, E., s e e Gutzow, I., 586 Paczuski, M., 588 Paidar, V., 67, 132, 183, 251,436, 438, 440 Paidar, V., s e e Ezz, S.S., 67, 131, 182, 251,435 Paidar, V., s e e Mol6nat, G., 133, 440 Paidar, V., s e e Yamaguchi, M., 66, 183, 438 Pak, H.R., 68, 184, 437 Pak, H.R., s e e Saburi, T., 131, 184, 436 Pansu, B., 590 Pansu, B., s e e Pieranski, Pa., 590 Pantelides, S.T., s e e C a r , P., 5 0 4 Paris, E., s e e Glattli, D.C., 590 Parthasarathy, T.A., s e e Dimiduk, D.M., 436 Pashley, D.W., s e e Hirsch, P.B., 438 Pasianot, R., 184, 440 Pasianot, R., s e e Farkas, D., 134 Passell, L., s e e Larese, J.Z., 592 Patel, J.R., 500, 502 Patel, J.R., s e e Frisch, H.L., 502 Pearson, D.D., s e e Anton, D.L., 437 Pearson, J., s e e Okamoto, P.R., 587 Peierls, R., 587 P61issier, J., s e e Louchet, E, 502, 504 Pelz, J.P., s e e Birgeneau, R.J., 591 Perepezko, J.H., s e e Daeges, J., 587 Pershan, P.S., 589 Pershan, P.S., s e e Sirota, E.B., 590 Peters, C., s e e Specht, E.D., 591 Peterson, I.R., 589 Petrenko, V.E, s e e Osip'yan, Y.A., 501,503 Petroff, P., 501 Petzer, G., s e e Gottschalk, H., 502 Petzow, G., s e e Beuers, J., 131 Peyrade, J.P., s e e Louchet, E, 504 Pfann, W.G., s e e Vogel Jr., EL., 25 Pfeil, L.B., 24 Pfliiger, J., s e e vom Felds, A., 587 Pfn•r, H., s e e Piercy, P., 592 Phillpot, S.R., 585, 587 Phillpot, S.R., s e e Lutsko, J.F., 587 Pickens, J.R., s e e Kumar, K.S., 184 Piel, A., s e e Melzer, A., 594 Pieranski, P., s e e Pansu, B., 590 Pieranski, Pa., 590
Author
Pieranski, Pa., s e e Pansu, B., 590 Pieranski, Pi., s e e Pansu, B., 590 Piercy, P., 592 Piggins, N., s e e Pluis, B., 586 Pilyankevich, A.N., 501 Pindak, R., 589 Pindak, R., s e e Cheng, M., 588, 589 Pindak, R., s e e Dierker, S.B., 589 Pindak, R., s e e Huang, C.C., 589 Pindak, R., s e e Moncton, D.E., 589 Pinxteren, H.M., 586 Pirouz, P., s e e Hirsch, P.B., 502 Pitchford, T., 589 Pleiner, H., 589 Pluis, B., 586 Pluis, B., s e e Frenken, J.W.M., 586 Pluis, B., s e e van der Veen, J.F., 586 Poirier, J.P., 586 Pokrovsky, V., s e e Lyuksyutov, I., 588 Pokrovsky, V.L., 591 Pollock, T.C., 131 Pollock, T.M., 25, 26 Polvani, R.S., s e e Strutt, P.R., 26 Ponomarev, M.V., s e e Greenberg, B.A., 133 Pontikis, V., s e e Ciccotti, G., 587 Pooley, D., 504 Pope, D.P., 25, 66, 131, 133, 182, 438, 440 Pope, D.P., s e e Ezz, S.S., 67, 131,182, 184, 251,435, 436 Pope, D.P., s e e George, E.P., 438 Pope, D.P., s e e Heredia, EE., 131, 182, 183 Pope, D.P., s e e Heredia, H.E, 435, 436 Pope, D.P., s e e lnui, H., 438 Pope, D.P., s e e Kuramoto, E., 133, 182, 184, 436 Pope, D.P., s e e LaU, C., 67, 131, 182, 435 Pope, D.P., s e e Liang, S.-J., 440 Pope, D.P., s e e Liu, C.T., 25 Pope, D.P., s e e Mulford, R.A., 68, 183, 435 Pope, D.P., s e e Paidar, V., 67, 132, 183, 251,436 Pope, D.P., s e e Tichy, C., 438 Pope, D.P., s e e Tichy, G., 67, 183, 185 Pope, D.P., s e e Umakoshi, Y., 67, 131, 182, 436 Pope, D.P., s e e Wee, D.M., 131, 182, 435 Pope, D.P., s e e Wu, Z.I., 435 Pope, D.P., s e e Wu, Z.L., 182 Pope, D.P., s e e Yamaguchi, M., 66, 67, 183, 438 Popper, K.R., 440 Porter, W.D., s e e George, E.P., 438 Porter, W.P., s e e Inui, H., 438 Potemski, R.M., s e e Monemar, B., 504 Potez, L., 133, 437 Pr~estgaard, E., s e e Kristensen, W.D., 586 Pratt, N., s e e Czernuszka, J.T., 504 Pratt, P.L., s e e Guiu, E, 252
index
607
Preparata, EP., 588 Press, W.H., 252 Price, G.D., s e e Poirier, J.E, 586 Puls, M.P., 26 Puschl, W., s e e Couret, A., 131 Putnam, EA., s e e Berker, A.N., 592 Qin, Q., 182, 183 Rabier, J., s e e George, A., 502 Rabier, J., s e e Veyssi~re, P., 438 Rabinovich, S., s e e Voronel, A., 587 Rainville, M.H., s e e Howe, L.M., 439 Ralph, B., s e e Taunt, R.J., 184 Ramakrishnan, T.V., 589 Ramaswami, B., s e e Sastry, S.M.L., 133, 438 Rand, W.H., s e e Giamei, A.F., 68, 438 Rao, S., s e e Dimiduk, D.M., 436 Rao, S.I., s e e Simmons, J.P., 133 Ravitz, S.F., s e e Abbaschian, G.J., 586 Rawlings, R.D., s e e Nicholls, J.R., 437 Rawlings, R.D., s e e Staton-Bevan, A.E., 67, 183, 251, 435, 436 Ray, I.L.F., 502 Ray, I.L.F., s e e Crawford, R.C., 25, 134 Ray, I.LF., s e e Cockayne, D.J.H., 67 Read, W.T., 25, 586 Redhead, P.A., 503 R6gnier, P., 131 Rehn, L.E., s e e Meng, W.J., 587 Rehn, L.E., s e e Okamoto, P.R., 587 Reichl, L.E., 184 Ren, Q., s e e Jo6s, B., 589 Rentenberger, C., s e e Karnthaler, H.P., 67, 439, 441 Reppy, J.D., s e e Agnolet, G., 588 Reppy, J.D., s e e Bishop, D.J., 588 Reusch, E., 25 Reynaud, F., s e e Fnaiech, M., 502 Rice, J.P., s e e Safar, H., 591 Rice, J.R., 185 Rice, J.R., s e e Asaro, R.J., 185 Rice, J.R., s e e Hill, R., 185 Richardson, G.Y., s e e Duesbery, M.S., 26, 183 Richter, D., s e e Larese, J.Z., 592 Ricolleau, C., s e e Ducastelle, F., 437 Riedel, E.K., s e e Solla, S.A., 588 Rieu, J., s e e Kubin, L.P., 134 Rigby, M., s e e Maitland, G.C., 591 Riste, T., 587 Rivaud, G., s e e Levade, C., 504 Robbins, M.O., 590 Robinson, McD., s e e Celler, G.K., 587 Roccasecca, D.D., s e e Chin, A.K., 501
608
Author
Rogers, D.H., s e e Blair, D.G., 184 Rosenbaum, T.E, 592 Rosenbaum, T.F., s e e Nagler, S.E., 588 Rosenman, I., s e e Simon, Ch., 593 Rtisner, H., s e e Mol6nat, G., 134 Rossouw, C.J., 586 Rossouw, C.J., s e e Donnelly, S.E., 587 Roy, J.A., s e e Schulson, E.M., 435 Ruan, Y.-Z., s e e Guo, C.J., 590 Rudge, W.E., s e e Abraham, EE, 591 Russell, W.B., 590 Rybin, V.V., 502 Saada, G., 67, 131, 132, 134, 251,436--440 Saada, G., s e e Bontemps, C., 131,436 Saada, G., s e e Shi, X., 441 Saburi, T., 131, 184, 436 Saburi, T., s e e Pak, H., 68 Saburi, T., s e e Pak, H.-R., 437 Saburi, T., s e e Pak, H.R., 184 Sadananda, K., s e e Duesbery, M.S., 440 Safar, H., 591 Sahoo, D., s e e Venkataraman, G., 587 Saito, M., s e e Kikuchi, M., 501 Saito, Y., 589 Saka, H., 435 Saka, H., s e e Matsumoto, A., 133 Saka, H., s e e Nohara, A., 133 Saka, H., s e e Zhu, Y.M., 133, 134, 435 Sakamoto, K., s e e Maeda, K., 501 Sakka, Y., s e e Nakamura, N., 134 Sanchez, J.M., 133, 438 Sass, S.L., s e e Yoo, M.H., 26 Sastry, S.M.L., 133, 438 Sato, A., s e e Jumojni, K., 436, 439 Sato, E, 504, 587 Sato, E, s e e Chikawa, J., 587 Sato, M., s e e Maeda, K., 501 Sato, T., s e e Hanada, S., 134 Saunders, G.A., 587 Sauthoff, G., 26 Savchenko, I.B., s e e Osip'yan, Y.A., 501,503 Saville, D.A., s e e Russell, W.B., 590 Savino, E.J., s e e Farkas, D., 134, 183 Savino, E.J., s e e Pasianot, R., 184, 440 Sch~iublin, R., s e e Baluc, N., 67, 132, 184, 436, 438 Schaumburg, H., 501 Schaumburg, H., s e e Haussermann, E, 502 Scheerer, B., s e e vom Felds, A., 587 Schildberg, H.P., s e e Lauter, H.J., 592 Schmid, E., 182 Schmidt, O., s e e Volmer, M., 586 Schneemeyer, L.E, s e e Gammel, P.L., 590 Schneibel, J.H., 182, 437
index
Schneibel, J.H., s e e George, E.E, 438 Schneibel, J.H., s e e Hazzledine, P.M., 437 Schnitker, J., s e e Naidoo, K.J., 593, 594 Schoeck, G., 439, 440 Schoeck, G., s e e Couret, A., 131 Schoeck, G., s e e Korner, A., 439 Schowalter, W.R., s e e Russell, W.B., 590 Schreiber, J., s e e BrOmmer, O., 504 Schr0er, W., 134 Schrtiter, W., s e e George, A., 502 Schr/Ster, W., s e e Schaumburg, H., 501 Schulson, E.M., 435 Schulson, E.M., s e e Baker, I., 132, 439 Schulson, E.M., s e e Howe, L.M., 439 Schulze, D., 25 Schwartz, D.K., s e e Viswanathan, R., 594 Seeger, A., s e e Schoeck, G., 440 Seitz, E, s e e Koehler, J.S., 25, 66, 184 Sen, S., s e e Esquivel, A.L., 504 Seshadri, R., 590 Shamos, M.I., s e e Preparata, EP., 588 Shang-Keng Ma 251 Shapiro, J.N., 585 Shea, EA., s e e Novaco, A.D., 593 Shechtman, D., 133 Sheinkman, M.K., 504 Shi, X., 441 Shiba, H., 591 Shikhsaidov, M.S., 503 Shikhsaidov, M.S., s e e Klassen, N.V., 503 Shikhsaidov, M.S., s e e Mdivanyan, B.E., 502 Shikhsaidov, M.S., s e e Osip'yan, Y.A., 503 Shinohara, T., s e e Jumojni, K., 439 Shirai, S., s e e Chikawa, J., 587 Shirai, Y., s e e Yamaguchi, M., 182 Shiraki, Y., s e e Yamashita, Y., 501 Shishokin, V.E, 25 Shockley, W., 585 Shrimpton, N.D., 591,592 Shu, Q.S., s e e Ecke, R.E., 592 Silcox, J., 67 Simmons, G., 585 Simmons, J.P., 133 Simon, Ch., 593 Sims, C.T., 24, 26 Sinclair, J.E., s e e Finnis, M.W., 183 Sinha, S.K., 587 Sinha, S.K., s e e Sirota, E.B., 590 Siol, M., 586 Sirota, E.B., 590 Sitch, P., s e e Jones, R., 504 Skjeltorp, A.T., 590 Skjeltorp, A.T., s e e Helgesen, G., 590 Slonczweski, J.C., s e e Malozemoff, A.P., 590
Author
Small, M.B., s e e Monemar, B., 504 Smallman, R.E., s e e Ngan, A.H.W., 68, 131,435, 439 Smallman, R.E., s e e Yan, W., 439 Smimov, L.V., s e e Greenberg, B.A., 133 Smith, E.B., s e e Maitland, G.C., 591 Snow, D.B., s e e Anton, D.L., 133, 437 Sodani, Y., 132, 184, 440 Sodani, Y., s e e Vitek, V., 68, 182, 184, 251,440 Solla, S.A., 588 Solla, S.A., s e e Strandburg, K.J., 589 Somerscales, E.F.C., 24 Song, A.K., 503 Song, K.S., s e e Williams, R.T., 504 Sorensen, L.B., s e e Sirota, E.B., 590 Sp~itig, P., 132, 183, 436, 437, 441 Sp~itig, P., s e e Baluc, N., 132, 436 Sp~itig, P., s e e Bonneville, J., 183, 437 Specht, E.D., 591,592 Specht, E.D., s e e D'Amico, K.L., 588 Sperner, E, s e e K~ster, W., 26 Spohn, H., s e e Krug, J., 252 Sprenger, W.O., s e e Murray, C.A., 590 Spring, M.S., s e e Karnthaler, H.P., 439 Sriram, S., 133 Stan, M.A., s e e Guo, C.J., 590 Stankiewicz, J., s e e Femfmdez, J.E, 588, 594 Stanley, H.E., 251,588 Staton-Bevan, A.E., 67, 68, 183, 251,435, 436 Staubli, M., s e e Anton, D.L., 133 Steele, W.A., s e e Shrimpton, N.D., 591,592 Steinberg, V., s e e Voronel, A., 587 Stephens, P.W., 591 Stephens, EW., s e e Birgeneau, R.J., 588, 591 Stephens, EW., s e e Heiney, P.A., 592 Stephens, P.W., s e e Nielsen, M., 592 Stephenson, G.B., s e e Brock, J.D., 588 Sternbergh, D.D., s e e Hemker, K., 132, 251 Stewart, G.A., s e e Butler, D.M., 588, 592 Stewart, G.A., s e e Litzinger, J.A., 592 Stoebe, T., 590 Stoebe, T., s e e Geer, R., 590 Stoebe, T., s e e Huang, C.C., 589 Stoiber, J., 252 Stoiber, J., s e e Baluc, N., 183, 439 Stokes, R.J., s e e Cottrell, A.H., 251 Stoloff, N.S., 25, 26, 68, 134 Stoloff, N.S., s e e Davies, R.G., 26, 132, 182, 251,436 Stoloff, N.S., s e e Hsiung, L.M., 68 Stoloff, N.S., s e e Sims, C.T., 26 Stoneham, A.M., 503 Stoneham, A.M., s e e Hayes, W., 503 Stoneham, A.M., s e e Itoh, N., 504 Strandburg, K.J., 588, 589 Strandburg, K.J., s e e Lee, J., 593
index
609
Stroh, A.N., 184, 440 Strudel, J.L., 131 Strutt, ER., 26 Strzelecki, L., s e e Pansu, B., 590 Strzlecki, L., s e e Pieranski, Pa., 590 Stucke, M.A., 133 Sullivan, T.S., s e e Ecke, R.E., 592 Sumi, H., 504 Sumino, K., 503 Sumino, K., s e e Imai, M., 502 Sumino, K., s e e Yonenaga, I., 503 Sun, Y., 251 Sun, Y., s e e Couret, A., 251,436, 439 Sun, Y., s e e Mol6nat, G., 132, 252 Sun, Y.Q., 25, 67, 68, 131, 133, 134, 251,436-439 Sun, Y.Q., s e e Couret, A., 67, 132, 251 Sun, Y.Q., s e e Ezz, S.S., 68, 183, 436 Sun, Y.Q., s e e Hazzledine, P.M., 67, 68 Sun, Y.Q., s e e Hirsch, P.B., 68 Sun, Y.Q., s e e Jian, N., 439 Sun, Y.Q., s e e Jiang, N., 68 Sun, Y.Q., s e e Korner, A., 67 Sun, Y.Q., s e e Komer, A.K., 437 Surrendranath, V., s e e Mahmood, R., 589 Suter, R.M., s e e Colella, N.J., 592 Suter, R.M., s e e Gangwar, R., 592 Suter, R.M., s e e Greiser, N., 592 Suto, H., s e e Nemoto, M., 132, 440 Sutton, M., s e e Dimon, P., 592 Sutton, M., s e e Nagler, S.E., 588 Sutton, M., s e e Specht, E.D., 591 Suzuki, H., 586 Suzuki, K., 25, 68, 132, 439, 440 Suzuki, K., s e e Maeda, K., 502, 503 Suzuki, K., s e e Takeuchi, S., 251,502 Suzuki, K., s e e Yamashita, Y., 501 Suzuki, M., s e e Zabel, H., 593 Suzuki, T., 25, 66, 131, 133, 182, 435 Suzuki, T., s e e Mishima, Y., 435, 436 Suzuki, T., s e e Miura, S., 182, 436 Suzuki, T., s e e Noguchi, O., 26, 436 Suzuki, T., s e e Takeuchi, S., 502 Suzuki, T., s e e Tounsi, B., 132, 439 Suzuki, T., s e e Wee, D.M., 131, 182, 435 Sverbilova, T., s e e Voronel, A., 587 Sviridov, V.V., s e e Belyavskii, V.I., 504 Swalski, A.T., s e e Alexander, H., 502 Swope, W., s e e Bagchi, K., 594 Taggart, K., s e e Kashyap, B.P., 131 Tajbakhsh, A.R., s e e Brock, J.D., 588 Takagi, N., s e e Fujiwara, T., 501,503 Takahashi, T., s e e Liu, Y., 133, 438 Takasugi, T., 131, 133, 134, 435-437
610
Author
Takasugi, T., s e e Liu, Y., 133, 435, 438 Takasugi, T., s e e Yoshida, M., 134, 435, 438, 439 Takeuchi, S., 67, 132, 182, 251,436, 501-503 Takeuchi, S., s e e Maeda, K., 501-503 Takeuchi, S., s e e Maeda, N., 501,503, 504 Takeuchi, S., s e e Nakagawa, K., 504 Takeuchi, S., s e e Nakano, K., 501 Takeuchi, S., s e e Suzuki, K., 25, 68, 132, 439, 440 Takeuchi, S., s e e Suzuki, T., 131 Takita, K., s e e Murakami, K., 503 Takusagawa, M., s e e Fujiwara, T., 501 Takusagawara, M., s e e Fujiwara, T., 503 Talapov, A.L., s e e Pokrovsky, V.L., 591 Tallon, J.L., 585 Taluts, G.G., s e e Greenberg, B.A., 133 Tammann, G., 25 Tang, Y., 590 Tangri, K., s e e Kashyap, B.P., 131 Taub, H., 588 Taub, H., s e e Hansen, EY., 592 Taunt, R.J., 184 Taylor, A., 24 Taylor, F.W., 24 Taylor, G., s e e Hirsch, P.B., 441 Taylor, G.I., 184 Tejwani, M.J., 590, 592 Tessier, C., 592 Testardi, L.R., s e e Patel, J.R., 502 Testardi, L.R., s e e Willens, R.H., 588 Tetelman, A.S., s e e Johnston, T.L., 436 Teukolsky, S.A., s e e Press, W.H., 252 Teutonico, L.J., 184 Thiaville, A., s e e Brezin, E., 591 Thibault-Desseaux, J., s e e Louchet, E, 502 Thomas, G., s e e Vogel Jr., EL., 25 Thomas, H.M., 594 Thompson, A.W., s e e Dimiduk, D.M., 67, 184 Thompson, L.J., s e e Meng, W.J., 587 Thomson, N., s e e Loginov, Y.Y., 502 Thomson, R., s e e Celli, V., 502 Thornton, P.H., 26, 67, 132, 182, 251,435 Thouless, D.J., s e e Kosterlitz, J.M., 585 Tichy, G., 67, 183, 185, 438 Tichy, G., s e e Heredia, EE., 131, 182 Tien, J.K., 26 Tien, J.K., s e e Sanchez, J.M., 133, 438 Titchmarsh, J.M., 504 Toennies, J.P., s e e Frenken, J.W.M., 586 Tomizuka, A., s e e Takeuchi, S., 501 Toporov, Y.P., s e e Deryagin, B.V., 501 Torzo, G., s e e Taub, H., 588 Tosatti, E., 586 Tounsi, B., 132, 439 Toussaint, D., s e e Janke, W., 589
index
Townsend, ED., s e e Agullo-Lopez, E, 503 Toxvaerd, S., 593 Toyozawa, Y., 504 Travitzky, N., s e e Gutmanas, E.Y., 503 Trimble, L.E., s e e Celler, G.K., 587 Troxell, J.R., 504 Truman, J., 24 Tsai, T.E., 504 Tsien, F., 593 Tsurisaki, K., s e e Takasugi, T., 134 Tully, J.C., s e e Weeks, J.D., 503 Turnbull, D., 586 Turnbull, D., s e e Ainslie, G., 586 Turnbull, D., s e e Cormia, R.L., 586 Ubbelohde, A.R., 585 Udink, C., 592, 593 Ueda, O., 503 Ueda, O., s e e Maeda, K., 501 Ueta, S., s e e Jumojni, K., 436 Uhlmann, D.R., 586 Umakoshi, Y., 67, 131, 133, 182, 436 Umakoshi, Y., s e e Yamaguchi, M., 131, 134, 183 Umebu, I., s e e Ueda, O., 503 Umerski, A., s e e Jones, R., 504 Usami, N., s e e Yamashita, Y., 501 Vald6s, A., s e e Gallet, E, 590 Valleau, J.E, s e e Tsien, E, 593 VaU6s, J.L., 26 van der Elsken, J., s e e Udink, C., 593 van der Merwe, J.H., s e e Frank, EC., 591 van der Veen, J.E, 586 van der Veen, J.E, s e e Frenken, J.W.M., 586 van der Veen, J.E, s e e Pluis, B., 586 Van Vechten, J.A., 504 Van Winkle, D.H., 590 Van Winkle, D.H., s e e Murray, C.A., 590 Vanderschaeve, G., 501 Vanderschaeve, G., s e e Caillard, D., 504 Vanderschaeve, G., s e e Faress, A., 503 Vanderschaeve, G., s e e Levade, C., 503, 504 Vanderschaeve, G., s e e Louchet, E, 504 Vardanyan, R.A., 504 Vashishta, P., 593 Vashishta, P., s e e Kalia, R.K., 593, 594 Vasudevan, V.K., 438 Vasudevan, V.K., s e e Court, S.A., 133 Vasudevan, V.K., s e e Sriram, S., 133 Vechten, J.A.V., s e e Monemar, B., 504 Veilin, V.M., 501 Vekilov, Y.K., s e e Veilin, V.M., 501 Venkataraman, G., 587
Author
Ver Snyder, EL., 24 Vetterling, W.T., s e e Press, W.H., 252 Veyssi~re, P., 26, 67, 68, 131, 132, 183, 251, 435, 437-439 Veyssi~re, P., s e e Beauchamp, P., 134, 438 Veyssi~re, P., s e e Bontemps, C., 67, 131, 132, 251, 436, 439 Veyssi~re, P., s e e Caron, P., 440 Veyssi/~re, P., s e e Dirras, G., 134, 435 Veyssii~re, P., s e e Douin, J., 67, 184, 437, 438 Veyssi~re, P., s e e Francois, A., 435 Veyssi~re, P., s e e Hug, G., 133 Veyssii~re, P., s e e Korner, A., 438 Veyssii~re, P., s e e Oliver, J., 440 Veyssi~re, P., s e e Saada, G., 67, 131, 132, 134, 251, 436, 439, 440 Veyssi~re, P., s e e Shi, X., 441 Veyssi~re, P., s e e Tounsi, B., 132, 439 Vidali, G., 591 Vidoz, A.E., 68, 439 Vignia, B., 133 Viguier, B., s e e Hemker, K.J., 133 Vilches, O.E., s e e Ecke, R.E., 592 Vilches, O.E., s e e Tejwani, M.J., 590, 592 Villain, J., 591 Viner, J.M., 590 Viner, J.M., s e e Huang, C.C., 589 Viner, J.M., s e e Pitchford, T., 589 Vinokur, V.M., s e e Kwok, W.K., 591 Viswanathan, R., 594 Vitek, V., 68, 182-184, 251,435, 440 Vitek, V., s e e Ezz, S.S., 131, 184, 436 Vitek, V., s e e Gom6z, A., 502 Vitek, V., s e e Heredia, EE., 131, 182 Vitek, V., s e e Inui, H., 438 Vitek, V., s e e Khantha, M., 132, 183, 251,436, 437, 440 Vitek, V., s e e Paidar, V., 67, 132, 183, 251,436 Vitek, V., s e e Pope, D.P., 182 Vitek, V., s e e Sodani, Y., 132, 184, 440 Vitek, V., s e e Tichy, G., 67, 183, 185, 438 Vitek, V., s e e Umakoshi, Y., 67, 131, 182, 436 Vitek, V., s e e Wee, D.M., 131, 182, 435 Vitek, V., s e e Wu, Z.L., 182 Vitek, V., s e e Yamaguchi, M., 66, 67, 183, 438 Vogel Jr., EL., 25 Volmer, M., 586 vom Felds, A., 587 Voronel, A., 587 Voronoi, G.E, 588 Wagner, H., s e e Mermin, N.D., 587 Wainwright, T.E., s e e Alder, B.J., 593 Wakao, K., s e e Ueda, O., 503
index
611
Wakeham, W.A., s e e Maitland, G.C., 591 Wang, H., s e e Simmons, G., 585 Warner, M., s e e Edwards, S.F., 586 Watanabe, S., s e e Takasugi, T., 131, 133, 436, 437 Watanabe, W., s e e Hanada, S., 134 Watkins, G.D., 504 Watkins, G.D., s e e Troxell, J.R., 504 Webb, G., 132 Webb, G., s e e de Bussac, A., 132, 251,440 Wee, D.M., 131, 182, 435 Wee, D.M., s e e Suzuki, T., 133, 435 Weeks, J.D., 503, 589, 593 Weeks, J.D., s e e Broughton, J.Q., 593 Weeks, J.D., s e e Naidoo, K.J., 593 Wells, B.O., s e e Nuttall, W.J., 594 Welp, U., s e e Kwok, W.K., 591 Wen, M., 132 Wenk, R.A., s e e Murray, C.A., 590 Wessel, K., 502 Westbrook, J.H., 24-26, 182, 435, 500 Westervelt, R.M., s e e Seshadri, R., 590 Westgren, A., 25 Westwood, A.R.C., 501 Whang, S.H., 133 Whang, S.H., s e e Li, Z.C., 133 Whang, S.H., s e e Li, Z.X., 133 Wheeler, R., s e e Vasudevan, V.K., 438 Whelan, M.J., s e e Hirsch, P.B., 25, 438 White, M., s e e Taylor, F.W., 24 Whittenberger, J.D., s e e Kumar, K.S., 184 Whitworth, R.W., s e e Osip'yan, Y.A., 501 Widested, J.P., s e e Larese, J.Z., 592 Wight, D.R., s e e Titchmarsh, J.M., 504 Willens, R.H., 588 Williams, A.A., s e e Pluis, B., 586 Williams, C., s e e Moussa, E, 590 Williams, EI.B., s e e Gallet, F., 590 Williams, F.I.B., s e e Glattli, D.C., 590 Williams, J.C., s e e Dimiduk, D.M., 67, 184 Williams, P.M., 504 Williams, R.T., 504 Williams, R.T., s e e Song, A.K., 503 Williams, W.S., s e e Kulkami, S.B., 502 Wilsdorf, H.G., s e e Pollock, T.C., 131 Wilsdorf, H.G.E, s e e Kear, B.H., 25, 67, 133, 183, 436 Winter, E., s e e Hazzledine, P.M., 437 Wittry, D.B., 504 Wittry, D.B., s e e Esquivel, A.L., 504 Wolf, D., 587 Wolf, D., s e e Lutsko, J.F., 587 Wolf, D., s e e Phillpot, S.R., 585 W611, C., s e e Frenken, J.W.M., 586 Woodruff, D.P., 586
612
Author
Woolhouse, G.R., s e e Monemar, B., 501,504 Wu, F.Y., 592 Wu, T.-Y., 185 Wu, X., s e e Hu, G., 438 Wu, Y.P., s e e Sanchez, J.M., 133, 438 Wu, Z.I., 435 Wu, Z.L., 182 Xu, Q.,
see
Zhang, Y.G., 133
Yacobi, B.G., 502 Yakovenko, L.I., s e e Greenberg, B.A., 132 Yamaguchi, A., s e e Ueda, O., 503 Yamaguchi, M., 66, 67, 131, 134, 182, 183, 438 Yamaguchi, M., s e e Inui, H., 438 Yamaguchi, M., s e e Paidar, V., 183 Yamaguchi, M., s e e Umakoshi, Y., 133 Yamakoshi, S., s e e Ueda, O., 503 Yamashita, Y., 501 Yamashita, Y., s e e Maeda, K., 502 Yan, W., 439 Yasuda, H., 441 Ye, Y.-Y., s e e Fu, C.L., 435 Yip, S., s e e Lutsko, J.F., 587 ~ p , S . , s e e Phillpot, S.R., 585, 587 Yip, D., s e e Phillpot, S.R., 587 Yip, S., s e e Wolf, D., 587 Yodagawa, Y.M., s e e Mishima, Y., 26 Yodogawa, M., s e e Mishima, Y., 435, 436
index
Yoffe, A.D., s e e Williams, P.M., 504 Yokoyama, T., s e e Matsui, M., 503 Yonenaga, I., 503 Yoo, M.H., 26, 67, 132, 133, 184, 251,436, 440 Yoo, M.H., s e e Baker, I., 440 Yoo, M.H., s e e Fu, C.L., 435, 440 Yoo, M.H., s e e Hazzledine, P.M., 67 Yoo, M.H., s e e Horton, J.A., 440 Yoo, M.H., s e e Veyssi6re, P., 438 Yoshida, M., 134, 435, 438, 439 Yoshida, M., s e e Takasugi, T., 134, 435 Yoshinaga, H., s e e Suzuki, T., 131 Young, A.P., 585 Yussouff, M., s e e Ramakrishnan, T.V., 589 Zabel, H., 593 Zaretskii, A.V., s e e Osip'yan, Y.A., 501 Zasadzinski, J.A., s e e Viswanathan, R., 594 Zeppenfeld, P., s e e Kern, K., 593 Zhang, Q.M., 592 Zhang, Q.M., s e e Kim, H.K., 588, 592 Zhang, Y.G., 133 Zhemchuzhnii, S.F., s e e Kumakov, N.S., 26 Zhu, J., s e e Gao, Y., 438 Zhu, Y.M., 133, 134, 435 Zhu, Y.M., s e e Saka, H., 435 Zinn-Justin, J., s e e LeGuillon, J.C., 589 Zollweg, J.A., 590, 593 Zollweg, J.A., s e e Strandburg, K.J., 588
Subject Index abrupt kink 454, 458 accepting mode 485, 486 activation energy 449, 461 for kink migration 446 for pinning 364, 408 activation enthalpy 154, 411 activation volume - apparent 277, 282, 424 -discontinuity 283, 411,412, 432 - effective 281-283 - strain dependence 284 408, 412, 424 adiabatic approximation 481 potential 481 Ag2MgZn 119 A13Ti 106, 290, 305, 317 L12-stabilized 258, 306 alignment of dipolar loops 305, 336, 343, 344, 393, 394 alkali halides 445, 495 alloy: - B2 alloy 114 - L12 alloy 80, 84, 103, 107 amorphization 517 amorphous silicon 500 anomalous slip mode 45 anomalous temperature dependence of flow stress 15-17 anthracene 445 anti-bonding (AB) state 481,496 anti-phase defect 491 anti-phase boundary (APB) 12, 29, 30, 59, 127, 138, 144 dissociated 62 -dragging 342, 352, 353, 399, 405 - energy 37, 316 composition dependence 314, 315 -energy ratio 299, 369-371,374 energy temperature dependence 315 - j u m p 323, 329-331,334-336, 373, 385, 389 relaxation 315-317, 340 - t u b e 49, 51, 55, 65, 318, 339, 342, 348, 349, 393-395, 397, 418, 433 - tube contrast 325, 340 wetting 291, 315 s e e anti-phase boundary applied shear stress 449 Ar 571 Arrhenius type of temperature dependence 464
athermal relaxation 484 athermal behavior 493, 494 atomistic alloying 23 studies 147 atoms at the dislocation core 448 Auger process 500
-
-
-
-
-
t
h
e
o
r
e
t
i
c
a
B2 alloy 114, 119, 122, 255 bandgap energy 468, 480 basal slip 476 bending in the cube plane 337 Be 358, 413 beryllium 75 bifurcation 176 bond orientational order (BOO) 508, 521 structure factor 530 -correlation function 527, 529, 582 bond switching 496 bond-weakening mechanism 498 bonding (B) state 496 Br intercalated into graphite 575 breakaway 164, 378, 408, 411,412 Brown mechanism 117 Brown's model 104 bulk crystal 461 Burgers vector 447
l
-
-
-
-
capture of minority carriers 486 carrier capture 484, 486 cartier diffusion length 492 cathodoluminescence (CL) 462 C2D4 on graphite 571 C2H4 on graphite 571 CdS 473, 476--478, 487 CdSe 473 CdTe 473, 475, 477, 480, 487, 450 celular automata 220, 245 centre: - F-centre 496 - FA-centre 494 H-centre 496 Vk-centre 496 centre of symmetry 448 charge state 492 mechanisms 492 CL: s e e cathodoluminescence climb 446 closing jog (CJ) 326, 328, 335, 339, 356, 380, 381, 389, 398, 418, 419
-
-
e
n
e
r
g
y
-
-
-
-
-
A
P
B
-
:
613
614
Subject index
clusters 518 CO on graphite: phase diagram 570 CoHf 122, 255 Co3Pt 291 collision of kinks 451 colloidal suspensions 546, 548, 553-555 between flat plates wedge geometry 549 with uniform density 556 commensurate (C) phase 559, 562, 563 complex stacking fault (CSF) 30, 144 composition effects 260, 266, 364, 433 compound - I-VII 445 - II-VI 446--448, 456, 473 - IIb-VIb 445 - III-V 446--448, 451 - IV-IV 446 compounds 137 computer simulations 15, 421,576 - boundary conditions 577 continuum-based 193, 243 disrete model 220, 245 of finite time 577 finite-size effects 577 Lennard-Jones monolayer 579 configuration coordinate 481 diagram 481,484 constitutive model 170 constrictions 154 continuum theory of plastic deformation 170 core -configuration 260, 273, 274, 291,292, 294, 309, 314, 404, 429 - of configuration 362 structure of dislocation 138, 147 correlation functions 541 suspensions 549, 551 Yukawa system 583 CoTi 122 Co3Ti 103, 256-259, 289, 309, 314, 317, 318, 350, 354, 369 Co3(Ti, Ni) 259, 316 Cottrell-Stokes experiment 207, 268 Coulomb explosion mechanism 499 Coulomb system - colloidal particles 546 -dislocation properties 534 on liquid helium surface 547 Yukawa potential simulation 582 covalency 447 covalent crystals 446 CoZn 122 CoZr 255 -
- e x p a n s i o n s
-
-
- e f f e c t s
-
-
o
n
-
-
- c o l l o i d a l
-
- e l e c t r o n s
-
creep exhaustion 389, 433 - inverted (inverse) 286, 296, 337 - primary 389 rate temperature dependence 287 285, 388, 426, 431 critical dislocation length 455 resolved shear stress 140 stress 214 - temperature of RADG effect 464, 467, 491 cross slip 88, 107, 126, 138, 366, 402 -distance in cube plane 366, 375, 382, 383, 393, 394, 398, 400, 414, 416, 419 -double 78, 89, 93, 100, 128, 331,358, 373, 376, 381,394, 398, 400, 414, 418, 420, 421 - mechanism 189, 314, 361,364, 366, 379, 402 - transformation 327 Cs on alkali metals 572 CSF: see complex stacking fault 30 Cu-A1 alloy 354 Cu3Au 103, 263, 301,314, 316, 339 cube - cross slip 286, 306, 319, 325, 341,358 glide 80, 99 - plane 257 cube slip - primary 258, 276, 286, 289, 297, 319 strain rate dependence 288 CuBr 445, 450 CuCI 445, 450 cumulated strain 263 fl-CuZn 114, 212, 255 -
-
- t e s t
-
-
-
-
-
2D electron lattice on liquid helium surface 582 D2 on graphite 573 2D-XY model 522 dipole dislocation 325 dangling bond (DB) 447, 471,492, 496 -state 491,497 dark contrast 462, 487 debris 394 deep level 447, 492 defect - charge 481 mediated melting theory 536 molecule 482 deformation microstructure 317, 320, 359 degradation of electronic devises 446, 480 Delaunay triangulation 538, 550, 581 - colloidal suspensions 553-555 delay effect in PPE 477 density fluctuations 535 density wave theory of melting (Ramakrishnan) 535, 550, 582 -
-
Subject index
dependence of REDG on bandgap energy 480 length 455, 458, 460, 470, 490 - dislocation type 467 intensity 464, 490, 492, 498 - material 467 stress 465 temperature 464 desorption 499 diamond structure 447 diffusion 255, 286, 325, 352 dilute alloys 73 dipolar loops alignment 302, 335, 343, 347, 393 dipole dislocation 100, 322, 323, 522 directional coarsening 22 disclination 527, 535 dislocation 29 - (100) 320 - 6 0 ~ 302, 310, 323, 325, 352, 355, 448 30 ~ partial 448 90 ~ partial 448 - annihilation 281,339, 389, 393, 416, 428, 435 -bistable 371,377, 382 -bypassing 339, 416, 418-421 charge 493 collective behaviour 193, 212, 391 contrast 296, 299 - core 144 280, 284, 325, 408, 431 - dipole 340, 343 dissociation 146 -dynamical behaviour 354 - exhaustion 202, 203 - forest 66, 281,347, 398, 428 - j e r k y motion 356-358, 360 - kinked 322 - length 452 - locking 296, 297, 312, 357, 378, 416, 417, 423 - loops 449 mobility 446, 451 multiplication 279, 361,389, 393, 400, 426, 428, 433 -organization 320, 322, 361 pile-up 359 sub- 410 - t y p e 447 - a - t y p e 447, 448, 451 - ~ - t y p e 447, 448, 451 -unlocking 357, 416, 423 - vector systems 536 dislocation theory history of 9-15 of melting (DTM) 522 - of melting 2D, see KTHNY theory 520 -
-
d
i
s
l
o
c
a
-
e
x
c
i
t
a
t
i
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
d
e
n
s
i
t
t
y
i
o
o
n
n
615
of melting 3D 511 dislocation velocity 152, 202, 335, 356, 366, 425427 - fluctuations 202 intensity dependence 464 - length dependence 455, 458, 460 linear stress dependence 456 non-linear stress dependence 456 temperature dependence 464 disordering 317, 359, 395 dissociation 447, 491 composition dependence 297 - c l i m b 295, 309, 315, 352, 395, 405 dipole 321 - mode I 291,301,305, 311,312, 317, 320 - mode II 291,301,305, 309 reaction 448 - sub- 271,294, 295, 297, 300, 354, 361 D03 structure 119, 120 dodecahedral slip 352 domain A 257 domain B 257, 259, 285, 315 domain C 257, 264, 286, 325 domain wall (DW) 563-566 domain-wall lattice (DWL) 529, 534, 562 Ar on graphite 571 Br intercalated into graphite 567 dislocations 532, 568 - honeycomb 567 - honeycomb phase 563, 564 - Kr on graphite 572 - striped phase 563, 564, 573 doping effect 445-447, 451,465, 492, 493, 496 double cross slip 78, 89, 93, 100, 128 double etching technique 449 double kink formation energy 457 drag stress 387 ductilization 21 DWL: see domain wall lattice dynamical simulations 192 continuum-based 193, 243 -discrete model 220, 245, 250 -
-
-
-
-
-
-
-
-
-
-
-
edge dipoles 51, 55, 100 efficiency factor 484 Einstein relation 455 elastic - coefficients c~ 368, 369 constants 580 - torque 379, 395 elasticity - anisotropic 299, 367, 380, 409 298, 300, 404, 413 electric holes monolayers 557 -
-
l
i
n
e
a
r
616
Subject index
l~tromechanical effect 445 electron beam REDG excitation 443, 469 electron-hole generation rate density 475 electron-hole pair 473, 499 recombination 486 electron lattice 2D electron lattice on liquid helium surface 582 coupling 485, 491 system 482 electron-stimulated defect motion (ESDM) 481, 492 electron-stimulated desorption 482 electrons on liquid helium surface 547 electrostatic instability 496 elementary process 446 embrittlement 21 energized state 483 energy distribution 483 energy factor 450 entropy - DWL honeycomb 564 - DWL striped 566 of dislocations 512-514 - of kink formation 457 of kink migration 457 - vortices 524 ESDM: see electron-stimulated defect motion Eshelby twist 294 excitation - of REDG 490 - of PPE 473 excited state 481 mechanisms 494 excitonic mechanism 496 exhaustion 266, 283, 286, 337, 398, 400, 431,433 experimental uncertainties 261,267, 270, 271,273, 274, 279, 290, 298, 300, 320, 329, 357, 359, 380, 406 extrinsic photoplastic effect 446 -
-
-
-
-
-
-
face-centered-cubic metal 448 faulted dipole 300 f.c.c, crystal 364, 365, 379, 397, 401 Fe3A1 120 FeCo 119 Fe3Ga 257 Fe3Ge 256--258, 314, 317, 318 Fermi level 451,465, 492 flow stress 139, 473, 476 - anomaly 259, 269, 401 -conventional strain 267, 283 of stoichiometric deviation 17 in compression 269 in tension 269 -
-
-
e
f
f
e
c
t
s
- orientation dependence 258, 260, 270, 271,274, 277, 286, 317, 377, 425 - reversibility 264, 267, 286, 399 strain rate dependence 258, 266, 280, 401,407, 432 - temperature dependence 255, 257, 258, 359 dependence anomalous 15-17 flow stress peak 265, 276 orientation dependence 389 - temperature 276, 325 forest dislocation 66 formation energy of a double kink 457 of a kink 446 of a kink pair 452 of a soliton 491 forward biasing 446, 480 Frank-van der Merwe model 562 Frenkel-Kontorowa model 562 Frenkel pairs 496 fresh dislocations 487 -
-
t
e
m
p
e
r
a
t
u
r
e
-
-
-
-
-
GaAIAs/GaAs 478, 499 GaAIAsP/GaAs 478 GaAs 446, 457, 459, 462-469, 473, 479, 480, 484, 487, 491,494, 499 GaP 446, 450, 462, 465, 466, 469, 478, 487 Ge 300, 445-447, 450, 451, 459, 462, 464, 466, 479 Giamei lock 58, 64 glide 446, 448 activation energy 450 - plane 447 glissile 138 grain boundaries 519 in 3D melting 518, 519 grain boundary melting 552, 557 - theory of 535, 561,583 grain boundary pre-melting 519 -
-
6H hexagonal structure 447 H2 on graphite 573 6H-SiC 470, 480 Halperin, Nelson and Young theory, see KTHNY theory hard core potentials 578 hard disk system 578 hardening by illumination 445 3He on graphite 573 4He on graphite 573 heat capacity 525 Ar on graphite 571 - CO on graphite 572 -
Subject index - 2D-XY model 525 smectic liquid crystals 544-547 Xe on graphite 570 hetero-epitaxial films 458 hexagonal DWL 573 hexagonal-close-packed metal 448 hexatic phase 521,530 colloidal suspensions 551 - electric holes 557 LJ monolayer 580 Yukawa system 583 hexatic transition 526 HgSe 450 Hirth-Lothe theory 453, 489 honeycomb phase 567 HREM 32, 291,296, 302 hybridization: sp 3 hybridization 447 -
o
f
617
ionization 481 IR 474
-
-
-
-
InAs 450 incommensurate (IC) phase 559, 562, 563 indentation test 473 infrared (IR) light 474 InGaAsP/InGaP 478 lnGaAsP/InP 479 inner shell 499 - excitation 481,500 InP 446, 450, 462, 466, 469, 479 InSb 450, 459, 462, 464, 466, 479, 499 in-situ deformation experiments 73, 89, 116, 122, 312, 330, 335, 340, 354, 409, 411, 413, 415, 424, 430 instability - mechanical 509 - mechanical DWL 567 of the electron-lattice system 481 - shear 509, 510 - theory DTM 512, 515 - thermodynamic 508 thermoelastic 510 intensity dependence of REDG 464, 490, 492, 498 mterband excitation of PPE 475 intercalated monolayers 556 intercalates in graphite 575 intermediate region 580 mtermetallic compounds 7 mtermittent loading effect 457, 480 internal stress 277, 281,433 interstitial boron 492 intrinsic REDG 446 - absorption 473 stacking fault 448 ionic crystal 456, 473, 499 ionicity 445, 446 -
-
-
jerky flow 73, 83, 89, 113, 116, 129, 289 jog dragging 345 K on alkali metals 574 Kear-Wilsdorf (KW) configuration 47, 88, 326 Kear-Wilsdorf (KW) lock 12, 37, 40, 64, 91, 95, 105, 138, 190 - as sources for cube slip 339, 387 bending 386 -bending in cube plane 321, 336, 337, 341, 358, 365, 387, 400, 423, 430 -by-passing 380, 381, 393, 413, 417, 422, 429, 430 -complete 309, 311,322, 326, 328, 329, 331,361, 363, 373, 395, 399, 419, 421 - destruction 377, 414 - formation 328, 375 - incomplete 88, 92, 101,306, 309, 311,321,322, 326, 328, 329, 331, 357, 361, 363, 365, 368, 371,395, 399, 415 - kinked 390, 397, 399, 433 - transport 386 unlocking 413 - unzipping 376, 386 zipping 380, 381,386, 399, 415 Keating-type potential 457 kink - as source 389, 390, 397, 433 -coalescence 339, 384-387, 391, 400, 423, 425, 430, 433 collision 446 case 453, 455 regime 452, 461,490 -collisionless case 453, 455 -collisionless regime 452, 461,490 - configuration 330 -cusped/jogged/stepped 329, 339, 342, 345, 385, 393, 421,433 diffusion constant 454 model 453, 454, 487 -distribution 334, 335, 381,383, 385, 426, 431 - double 326, 373 dragging 397 -elementary (EK) 335, 339, 358, 381,383, 418 formation 381 formation energy 446 height 335, 384, 391,451 - macro- 327, 330, 381 - mean free path 451,461,492 - migration 416, 446, 451,487, 489, 490 -
-
-
-
- c o l l i s i o n
- c o l l i s i o n
-
- d i f f u s i o n
-
-
-
-
618
Subject index
migration energy 446 mobility 386 nucleation 446 pair formation 154, 451 - pair formation energy 452, 457 pair nucleation rate 451,452, 488 - polarization 335, 383 - simple 332, 376, 381,382, 384, 386, 393, 425 site 487, 488 -sliding 339, 385-387, 393, 414, 415, 433 - switch-over 332, 381, 382, 384, 387, 393, 400, 420, 425 -velocity 451 width 451 kink-based models 414 Kosterlitz and Thouless (KT) theory 520, 522 correlation function 522, 523 stability criterion: electrons on liquid helium surface 547 stability criterion: ideal 2D system 527, 580 stability criterion with a substrate 529, 572 - vortices 523 Kr 574 Kr on graphite - dislocations 532, 534 - experiment 572 - phase diagram 570, 573 KT: see Kosterlitz and Thouless KTHNY theory 520, 521 - b u l k systems 530 -correlation function, BOO 527 function, PO 526 - for ideal 2D systems 527 hexatic phase 528 of melting 2D 520 with molecular tilt 542 with substrate field 529 -
-
-
-
-
-
-
-
-
-
-
-
c
o
r
r
e
l
a
t
i
o
n
-
-
-
-
Llo structure 108 L12 alloy 80, 84, 103, 107 L 12 ordered 7 t phase 29 L12 structure 7, 71, 108, 137ff., 255ff. Laplacian roughening model 537 laser annealing 481,499 lattice friction 449 - o n cube plane 325, 337, 362, 365, 372, 379, 383, 399 - Peieds-Nabarro model 258, 365, 406 lattice heating 481 lattice matching 480 leading partial 449 length dependence of dislocation motion 455, 458, 460, 470, 490
Lennard-Jones -monolayers: dislocation properties 532 system clusters 518 system with grain boundaries 519 level occupation 497 lifetime of kinks 451 light illumination 445 light intensity for PPE and REDG 475 light-emitting diodes 446 light-emitting double heterostructures 480 Lindemann rule 520, 560 line energies 59 linear stress dependence of dislocation velocity 456 liquid crystal 540 liquid helium: electrons on the surface 547 local pinning 95, 407 local vibration 483 localization of slip 176 localized electronic states 447 locking process 93, 294, 312 locking-unlocking mechanism 79, 82, 413 Lomer 59, 294, 296 Lomer-Cottrell lock 31, 41, 59, 295, 365, 389, 419423 long dislocation 461 long range - order 138 motion 462 low temperature - mechanical properties 257-259, 269, 272, 288, 290 - microstructural properties 260, 302, 305, 306, 310, 319, 321,330, 363 -
-
-
macrokink, see also superkink 75, 80, 88, 98, 100 macroscopic properties 139, 156 magnetic bubbles, monolayers 560 magnetic dipole holes, monolayers 557 magnetic holes 557 many-body potential 147 MC method 576 MD method 576 mean field theories 514, 535 mechanical instability 509 melting rules 509, 560 Born's melting rule 509 Lindemann's melting rule 511 Mg 415 MGR mechanism 482 microdeformation 266, 281,389, 400, 432 microstructural stability 14, 23 minority carrier injection 480, 484, 492, 494, 495 mobility of dissociated dislocation 451 model: 2D-XY model 522
-
-
Subject index molecular dynamics (MD) method 576 Monte Carlo (MC) method 576 Monte Carlo (MC) simulations 576 multi-modal particle size distribution 21 multiphonon emission 484 narrowly-spaced plane 448 nearest neighbour PLFP dislocation properties 533 PLFP computer simulations 583 negative PPE 473, 476, 493 negative TDFS 317, 343 negative-U (Anderson) 491 new compound bases 23 Ni3AI 33, 41, 51, 84, 256, 259, 274, 276-278, 291, 296, 297, 302, 314, 316, 348, 352, 354, 369, 411 Ni3A1 ('7~) 6-8, 137ff. Ni3A1Ni-rich binary 270, 274, 289, 294, 315, 316, 411,432 Ni3(A1, B) 260, 274, 275, 277, 316 Ni3(AI, Cr) 278 Ni3(A1, Hf) 259, 262, 263, 265, 268, 274, 275, 284, 288, 289, 302, 305, 306, 316, 322, 323, 329, 330, 332, 334, 336, 337, 339, 341, 342, 344-349, 351, 352, 355, 359, 383, 411,432 Ni3(AI, Hf, B) 264, 278, 280, 284, 285, 287, 289, 296, 353 Ni3(AI, Mo) 289 Ni3(A1, Nb) 259, 272, 273, 277, 289 Ni3(AI, Sn) 316 Ni3(A1, Ta) 260, 264, 272, 273-277, 281,283, 286, 288, 289, 296, 316, 411,432 Ni3(A1, Ti) 263, 264, 271,276, 278, 289, 296, 300, 316, 320, 325, 330, 352, 354, 369 Ni3(A1, V) 316 Ni3(AI, W) 266, 271,288, 289 Ni3(A1, Zr) 260, 271,274, 275 nickel-based alloys 84, 87 "7t phase 80 Ni3Fe 263, 290, 301, 316, 317, 369, 395 Ni3Ga 41, 84, 88, 264, 269, 273, 275, 277, 290, 291, 296, 301, 302, 305, 316, 334, 348, 354, 369, 411 Ni3Ge 257, 289, 296, 321,369 Ni3Ge-Fe3Ge 256-258 Ni3Mn 369 Ni3Si 256, 312, 316, 337, 354, 369 Ni3(Si, Ti) 259, 272, 290, 301,316, 321,330, 352, 363, 369 non-glide components 140 non-linear stress dependence 456 non-planar 148 -
-
-
619
non-radiative recombination 484, 487, 491,492 non-Schmid effects 139, 170, 269 normal slip mode 33 n-type 447 octahedral - cross slip 272, 306, 319, 403 glide 84 - slip primary 258, 286, 319, 325, 357 order-disorder transition 15, 114, 121 ordered alloys 137 orientation factor - K 272, 404 - N 271,276, 373, 374, 404 - N t 276 - Q 271,275, 404 orientational epitaxy 530, 567 Ar on graphite 571 -
-
partial Shockley 271,291,301 peak strengthening 21 peak temperature 30, 115, 119, 122, 139, 162, 255, 276 Peierls mechanism 76, 79, 82, 446, 487 schemes 451 Peierls potential 446 of the first kind 446, 447 - o f the second kind 446, 447, 454 Peierls stress 122 Peierls-Nabarro stress 454, 457 Peierls-type frictional forces 112, 129 phase: - 3' phase 7 - 3,t phase 80, 288, 330 -),t phase L 12 ordered 29 L12 phase 137 phonon kick energy 485, 487 -mechanism 447, 481,482 phonon softening mechanism 499 photoplastic effect (PPE) 445, 473, 476, 493 photomechanical effect 445 physisorbed monolayers 561 pinning 138 -correlations 195 pinning/depinning transition 192 pinning obstacle 487 planar defects 314 plasma 499 annealing model 499 PLFP (piecewise linear force potential) - dislocation properties 533 computer simulations 583 -
-
-
-
-
-
-
620
Subject index
PO correlation function (Yukawa) 582 point obstacles 444, 452, 456 point-obstacle model 452, 456, 494 point pinning 356, 357, 380, 407, 426, 428 polarity 448 -effect 446, 447, 451 polygon tiling model 538 polysterene latex suspensions 548 Portevin-Le Chatelier (PLC) effect 74, 106, 129 positional order (PO) 508, 521,525 correlation function 526 positive TDFS 473 deformation microstructure 317, 319 inflection point in 274, 278, 281,289, 406 models 401 - strain dependence 266 post-mortem TEM observations 75, 82, 91, 116, 121,340, 430 Potts model 572 - helical 572 PP: see Peierls potential PPE: see photoplastic effect pre-exponential factor 461 precursor effects 510 to melting 519 prestraining 264, 267, 286 primary creep transient 143, 216, 229 principle of corresponding states 509, 511 prismatic slip 476 - in Be 429 Pt(111) 574 Pt3A1 31, 81,256-259, 318, 369 Pt3Cr 256 Pt3Ga 256 Pt3In 256 Pt4Sb 256 Pt3Sn 256 Pt3Ti 256 Pt3X 255 p-type 447 excited state 494 -
-
-
-
-
-
radiation enhanced dislocation glide (REDG) 445 radiative recombination 462 rafting 22 rare gas monolayers - adsorbed on graphite 561,569 - dislocation properties 532 reaction coordinate 482 mode 486 recombination enhanced mechanism 482 reconstructed kink 492 reconstruction of dangling bonds 447, 491 -
-
REDG: see radiation enhanced dislocation glide reduction of activation energy in REDG 484, 486, 487, 490, 491 relaxation process 480 repeated APB jump 335, 336, 340, 357, 358, 374, 376, 377, 382, 400, 414, 423 restricted slip 175 reversibility of PPE and REDG 445, 462, 481 roof type barriers 113 saddle-point mechanism 493, 494 saturation of PPE and REDG 445, 470, 486, 490 regime of dislocation motion 458 - stress in L12 278, 379, 390 scale invariance 214 areal-velocity scaling function 218 scanning electron microscopy (SEM) 462 schemes of Peierls mechanism 451 Schmid factor 269, 358 Schmid law violation 85, 269, 273, 429 screw dislocation 82, 112, 121, 139, 148, 448 see smallest double kink self-interstitial atoms 493 self-trapped exciton (STE) 496 self-trapped hole 496 SEM: see scaning electron microscopy SEM-CL 462, 463 semiconductor lasers 446 serrated flow, see also jerky flow 278, 289 sessile 138 shear band 176 shear instability 509, 510 shear modulus 450 Shockley partial 31, 146 - dislocation 447 shuffle planes 448 Si 446, 447, 450, 451, 456--458, 462, 464, 466, 469, 475, 479, 480, 491--493 SiC 446, 447, 478, 480, 491 - 6H-SiC 470, 480 Sio.9Geo. l 460 Si0.9Geo. 1 alloy 458 simulations - atomistic 305, 317, 354, 363 - deformation 378 sine-Gordon equation 563 SiO2 496 SISF 31, 43, 55, 59, 144 SISF dipole 302, 305, 348, 349, 351 SISF-dissociated 62 slip lines 72, 122, 317, 319 - localization 347, 351 - multiple 263, 288, 313, 347 -
-
S
-
D
K
:
Subject index
-planarity 378, 381,382 single 263 smallest double kink (SDK) 487 formation 489, 491 smectic liquid crystal 540 - BOO structure factor 543 - defects 546 - phases 540, 541,542 studies of BOO 543 - thermodynamic studies of BOO 545 smooth kink 454 - model 457 softening by REDG 446 soliton 491 formation energy 491 site 491 solution hardening 19, 260 sp 3 hydralization 447 specific heat, s e e heat capacity splitting 138 stacking fault 144, 533, 573 - CSF 291,295, 298, 316 - energy 298 - SISF 291,343, 347 stainless steel 73 starting stress 450, 456 STE: s e e self-trapped exciton steady velocity 451 straight dislocation 489 487, 489, 491 strain - hardening 87, 110, 191,207 localization 139 strain rate 139 - constant 215, 432 - j u m p s 278, 279, 280 -sensitivity 86, 109, 121, 140, 163, 206, 258, 269, 277, 282, 408, 432 strength of coupling 486 stress -exponent of REDG effect 449, 450, 456, 465 increment in Cottrell-Stokes tests 268 - increment in strain-rate jump experiments 279 - relaxation 234, 267, 279, 281,434 stress-strain curve 473 string approximation 453 striped phase 564, 567 structure: L 10 structure 108 LI2 structure 7 structure factor 531 - 2D solid 525 - hexatic phase 530 s-type ground state 494 -
-
-
s
t
r
u
-
-
-
-
-
s
i
t
e
c
t
u
r
a
l
621
superalloys 137, 143 - history of 3-9 superconducting flux lattice 560 superdislocation 12, 146, 290 - APB-coupled 313, 318, 326, 352 - dipole 343 - SISF-coupled 352, 359, 317 superheating 515-517 solids 515 superkink, s e e a l s o macrokink 48, 55, 190 distribution of lengths 198 superlattice extrinsic stacking fault (SESF) 45 superlattice intrinsic stacking fault (SISF) 29 superpartial 146 1 - 5(110) 291,300, 354 -
-
1 - 5(110) dipole 343, 345, 346 1 - .~(112) 296, 302, 313, 317, 351 surface: "),-surface 144, 145 surface damage 261 melting 515, 516 switched partials 49, 53 system - l / r 12 577, 580, 582 - l / r 5 578 -
-
T o, peak temperature 30, 139 temperature dependence 15, 93, 113, 123, 124, 138, 168, 237, 250, 255, 258, 464 tension--compression (TC)asymmetry 140, 158, 173 - composition dependence 273 - general 270, 364, 409 - in cube slip 287, 317 - maximum 273, 275 - neutrality or inversion line 273-275, 277 - temperature dependence 275 ternary compounds 20, 162 tetrahedral coordination 447 theory of Celli et al. 452 thermal - activation 237, 245, 250 depth of trap 486 kink 455 - reversibility 139, 207 thermodynamic instability 508 thermoelastic instability 510 thermomechanical treatment 23 threshold stress for octahedral slip 267, 285, 289, 408, 410, 416, 430 TiA1 108 TiAI (Llo) 255 -
-
622
Subject index
Ti3AI (D019) 347 TiO2 500 trailing partial 449 transformation ofthe core 138 transients 277, 284 transmission electron microscopy (TEM) 29 twinning 109 type: - n-type 447 - p-type 447 unimolecular reaction 483 unlocking processes 93 unreconstructed kink 492
work hardening rate 65, 263, 268, 279, 283, 389, 397, 401 - composition dependence 263, 264, 276 - drop 342 orientation dependence 262, 264, 280 - peak 265 reversibility in temperature 264 work softening 264 wurtzite 447 -
-
Xe 574 Xe on Ag(111) 570, 574 Xe on graphite - dislocations 532, 532 - experiment 569, 570 phase diagram 569 X-ray Lang method 449 yield 139 - anomaly 137, 189, 205 - criteria 170 - drop 279 function 172 - strength anomaly 189, 205 - stress anomaly 30, 45, 71, 93 -
vacancies 536 valence excitation 481 velocity - fluctuations 202 - measurement 462 Voronoi construction 538, 550 WCA - liquid 539 potential 536, 580 system 538 weak-beam image shift 298, 299 method 32 multiple images 300 resolution 298 - TEM 48 Weeks-Chandler-Anderson (WCA) potential 538 wide gap semiconductors 480, 491 widely-spaced plane 448 work hardening 143, 280, 397 -
-
-
-
-
-
Zener anisotropy parameter A 369, 379 zincblende 447 ZnO 487 ZnS 446, 469, 470, 478, 480, 490 ZnSe 470, 478, 487 Zr3A1 256, 263, 352
-
/3-CuZn 114, 212, 255 "7 phase 7, 71 "7 surface 144, 145 "7~, see Ni3A1, L 12 6H structure 447, 470, 480
