Basics of Thermodynamics and Phase Transitions in Complex Intermetallics (Book Series on Complex Metallic Alloys)

Book Series on Complex Metallic Alloys – Vol. 1 BASICS OF THERMODYNAMICS AND PHASE TRANSITIONS IN COMPLEX INTERMETALLI...

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Book Series on Complex Metallic Alloys – Vol. 1

BASICS OF THERMODYNAMICS AND PHASE TRANSITIONS IN COMPLEX INTERMETALLICS

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Book Series on Complex Metallic Alloys – Vol. 1

BASICS OF THERMODYNAMICS AND PHASE TRANSITIONS IN COMPLEX INTERMETALLICS

edited by

Esther Belin-Ferré Laboratoire de Chimie Physique-Matiere et Rayonnement Centre National de la Recherche Scientifique, Université Pierre et Marie Curie, France

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BASICS OF THERMODYNAMICS AND PHASE TRANSITIONS IN COMPLEX INTERMETALLICS Series on Complex Metallic Alloys — Vol. 1 Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-279-058-3 ISBN-10 981-279-058-6

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7/28/2008, 10:36 AM

FOREWORD

This volume assembles the texts of the lectures delivered at the first Euro-School on Materials Science, organised in Ljubljana, Slovenia, from 22 to 27 of May 2006 by the European Network of Excellence Complex Metallic Alloys (CMA) under contract NMP3-CT-2005500145. The central objective of the CMA Euro-School is to provide a lecture-style background to students graduating in the field of materials science, in particular in the physics of metals and metallurgy. Four annual sessions of the CMA-Euro-School are foreseen, from which a series of four books will be issued. It is my great pleasure to introduce here the first volume in the series. During the first session of the CMA Euro-School, emphasis was on the basics of thermodynamics, phase transitions, crystallography and electronic properties of complex metallic alloys. These subjects were accounted for owing to two types of lectures. On the one hand, plenary lectures referred to basic topics and on the other hand, shorter lectures reported on tutorial topics. The book begins with a general introduction to the field of CMAs. Each of the following chapters refers to a distinct lecture: chapters 2 to 6 to basic lectures and chapters 7 to 13 to tutorial lectures. The European Commission is warmly acknowledged for financial support. Special thanks go to all authors. They made editing this volume possible. Paris, April 2007

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CONTENTS

Foreword

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Chapter 1: An introduction to complex metallic alloys and to the CMA network of excellence Jean-Marie Dubois

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Chapter 2: Thermodynamics and phase diagrams Livio Battezzati

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Chapter 3: Permanent magnets and microstructure Paul McGuiness

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Chapter 4: Solidification Peter Gille

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Chapter 5: Diffusive phase transformations Yves Bréchet

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Chapter 6: Diffusionless transformations C. Duhamel, S. Venkataraman, S. Scudino J. Eckert

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Chapter 7: Intermetallics: characteristics, problems and prospects Gerhard Sauthoff

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Chapter 8: An introduction to electronic structure methods D. A. Papaconstantopoulos

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Chapter 9: Crystallography of complex metallic alloys Walter Steurer and Thomas Weber

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Contents

Chapter 10: Electronic properties of alloys Östen Rapp

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Chapter 11: Electron transport properties of complex metallic alloys Uichiro Mizutani

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Chapter 12: Chemical bonding and crystallographic features Yuri Grin

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Chapter 13: Plasticity of complex metallic alloys M. Feuerbacher

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CHAPTER 1

AN INTRODUCTION TO COMPLEX METALLIC ALLOYS AND TO THE CMA NETWORK OF EXCELLENCE Jean-Marie Dubois Institut Jean Lamour (FR 2797 CNRS-INPL-UHP), Nancy Université, Ecole des Mines, Parc de Saurupt, Nancy, F-5404 E-mail: [email protected] Now that new tools are available to solve the crystallographic structure of complex compounds in metallic alloy systems, a vivid interest manifests itself to discover new compounds in multi-constituent alloys. Several are yet known to contain hundreds or more atoms per unit cell. In the meantime, real efforts are made for better understanding of the properties of these compounds and the mechanisms that underpin the progressive loss of metallic character when the size of the unit cell increases. This introductory chapter focuses at a few examples of this atypical behavior of complex metallic alloys, including quasicrystals as the ultimate state of structural complexity in a crystal made of metals. Examples are transport properties, surface electronic structure, surface energy, wetting and friction. All examples show the same trend, namely apparent localization of electronic states, loss of conductivity, opening of gaps, softening with no work hardening, etc. All phenomena are reminiscent of what is observed in nanostructured metals, but together with the increase of the size of the unit cell. This effect is therefore coined “inverse nanostructuration” by the author who argues that complex metallic alloys help us revisit ancient problems in metal physics, while in parallel potential applications may be sorted out.

1. Introduction This introductory chapter aims at a short overview of the crystallographic peculiarities and physical properties of Complex Metallic Alloys (CMAs hereafter). Most of the experimental data presented here was obtained quite some time ago and has provided the basis that was used by K. Urban, L. Schlapbach and the author to file an application within the 6th 1

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Framework Program of the European Communities for granting a socalled Network of Excellence (NoE). This successful application came into force in July 2005 and allowed financing the first session of the European School in Materials Science of the CMA NoE1 to which the present book is associated. The focus of the 2006 school was on general aspects of phase transitions in materials, and more specifically on our current knowledge of the transformations taking place in CMAs, from their synthesis to their possible applications. Other sessions will be organized in the coming years, with emphasis on more specific aspects of CMAs. The chapters that follow in this book will give an impressive account of the great progress that was achieved in recent years on CMAs, and also of the many questions that remain only partially solved, or fully open. CMAs are crystalline compounds of the family of intermetallics that are characterized by a) large unit cells, containing up to thousands atoms, b) the occurrence of well-defined clusters, most often of icosahedral symmetry and c) some disorder, essentially due to the fact that icosahedra do not fill Euclidian 3-dimensional space. Therefore, quasicrystals belong to the family of CMAs, but clathrates also do so. The properties of CMAs are surprising, although they cannot be claimed unique1. In Al-based CMAs, electron transport properties (conductivity, Seebeck coefficient, etc.) are governed by the formation of a pseudo-gap (when not a real gap) in the Al 3p partial density of states at the Fermi energy, which results of a combination of Hume-Rothery scattering of electron waves and sp-d hybridization effects2, 3. It turns out that surface properties like surface energy, solid-solid friction or wetting also reflect the depth of the pseudo-gap balanced by the presence of d states at the Fermi energy. As a result, it turns out that the surface energy of highly complex intermetallics is so much smaller than that of the metallic constituents of the alloy4 that reduced friction or wetting against polar liquids was for long taken as the best examples of potential applications of CMAs1, 5.

a

For simplicity in this chapter, italics represent the CMA network, whereas normal capital letters (CMA) are for the complex metallic alloys themselves.

Introduction to CMAs and to the CMA Network of Excellence

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Quite a few more properties show the same trend, namely apparent localization of electronic states, loss of conductivity, opening of gaps, mechanical softening with no work hardening, etc. All phenomena are reminiscent of what is observed in nanostructured metals, but take place with increasing the size of the unit cell. This effect is therefore coined “inverse nanostructuration” by the author6. In the end, it was admitted by a sufficiently large number of European scientists that the unexpected properties displayed by the few CMAs known so far are fascinating enough, and of strong enough potential interest for technology, to deserve the creation of a new field dedicated to complexity in metallic alloys7. The existence of the field was recognized by the foundation of the NoE labelled according to the same acronym CMA and funded by the Commission of the European Communities as briefly described at the end of this chapter. 2. Complexity in reciprocal and real space 2.1. A definition of CMAs An essential question to address at the beginning of this book is to know what we call a CMA. First, it is a compound, or a phase, or an alloy, essentially made of metals. This does not mean that the alloy is a metal itself, or an alloy characterized by metallic properties, because most often the metallic character of the alloy species has become poor or much weaker than in the pure metal constituents. In scarce cases like Al2Ru, it has turned to semi-conducting. It simply means that the major part of the constituents belongs to sp or d metals (Al, Ga, Sn, Fe, Ni, Pd, W, Rh, Re, etc.), possibly alloyed with semi-conductors (Ge, Si), chalcogenides (Se) and/or rare earths. In few cases, the situation is reversed: the major constituent is the semi-conductor like in clathrates. Oxides, although some may be structurally very complex, are excluded from the CMA family because they present no metallic behaviour whatsoever (except in few cases at very high temperature). The broad variety of chemical combinations that may be synthesized out of about 80 metals in the periodic table participates to the complexity of the compounds

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considered in this book. This also means that the potential for discovering new ternary, quaternary, etc. CMAs is huge. Second, it is often anticipated that complexity is expressed by a difficulty to describe the crystal lattice arrangement due to the large number of independent atomic positions in the unit cell. In many compounds, such a (weak) definition would be acceptable. However, many CMAs do not require lots of independent positions to be accounted for. Very frequently instead, a distribution of occupancy factors must be considered in order to match the chemical disorder inherent to the compound. This is the case for instance of the superstructures of the βCsCl –type cubic phase that forms in Al-Cu alloys8. The basic unit cell is the 2-atom, body-centred cubic unit cell of parameter 0.29 nm. Depending on the Al/Cu composition ratio, substitution vacancies order in the lattice and increase dramatically the size of the unit cell. The largest superstructure known so far forms at composition Al36Cu48V12 (V = vacancy), with a unit cell volume 47.7 larger than that of the conventional β-phase. The example of the superstructures of the β-cubic phase points out the difficulty to describe accurately complexity in real space due to the need to introduce a function that adequately fits the chemical disorder in the lattice, although atomic positions may be easily accounted for by a simple Bravais lattice like body-centred cubic. It is more relevant to call 'complex' an alloy, or a compound, essentially made of metals as presented above, whose reciprocal space exhibits complexity within the Jones zone. The Jones zone is the Brillouin zone constructed with the most important Fourier components in reciprocal space. For a simple crystal, it fits with the first Brillouin zone. When complexity arises, it corresponds to that zone which is built by taking into account the most intense diffraction peaks. A measure of complexity of the crystal is then supplied by the number of peaks that fall inside the Jones zone or equivalently, by the inverse of the reciprocal distance between the first diffraction peak and the origin of reciprocal space: the most complex CMA in a series presents a diffraction peak that falls at the shortest distance to the origin of reciprocal space. In many Al-based compounds, this is a quasicrystal (with in principle 0 distance of the first diffraction peak from the origin of reciprocal space). As an immediate consequence,


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metallic glasses are not considered to belong to the group of CMAs, although it is often observed that metallic glasses of specific composition crystallise in a CMA, because they show no sharp Fourier component is their diffraction pattern. The Jones zone is ill defined by the broad main halo in the diffraction pattern and the pre-peak, when it exists, does not fall that close to the origin of reciprocal space. Their physics however is often very much reminiscent of that of CMAs9. As an example of a typical CMA, Fig. 1 illustrates the case of triclinic Al11Mn4 (unit cell parameters a = 0.5087 nm; b = 0.8848 nm; c = 0.5052 nm; α =89.72°; β =100.54°; χ =105.37°; unit cell volume: 0.215 nm3). The major diffraction peaks fall in the vicinity of the wavevector q=30 nm-1 (q=4π sin θ/λ with θ the Bragg angle and λ the wavelength) whereas many other peaks of variable intensity are observed inside the range [0, q]. The average distance between opposite centres of the facets of the Jones zone is often labelled KP, so that here we have KP ≈ 30 nm-1.

Fig. 1. X-ray diffraction pattern (λ=KαMo) of triclinic Al11Mn4. The x-axis is labelled according to the wave vector q as explained in text. The position of 2kF is shown by a vertical bar, assuming valences of +3 for Al and -3 for Mn. (Courtesy of Dr M. Feuerbacher, CMA).

The number of peaks in the vicinity of q=1/2 KP reflects the degree of symmetry of the Jones zone. The larger the symmetry of the Jones zone,

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the closer the shape will be from a sphere. Such a resemblance to a sphere is actually achieved in many CMAs, for instance, quasicrystals but also γ-brass phases, etc. Furthermore, electronic concentration is very often naturally selected so that a close matching between Jones zone and Fermi surface is observed, which fulfils the Bragg condition KP = 2 kF (where kF is the Fermi vector). Such a selection is responsible for the opening of Hume-Rothery gaps and therefore for the enhanced stability of the compound. In Al11Mn4, the Fermi vector amounts to kF = 14.25 nm-1 if the contribution to the valence band by Al is taken equal to +3 electrons and that of Mn is assumed negative like in many other transition metals such as Fe, Ru, Re (but not Cu or Pd, see ref. 1 for more information on this point) and equal to –3 electrons according to its position in the periodic table along the 3d-metal series. Hence, it is observed that 2kF ≈ KP, a result which is indeed traditionally associated with the formation of a CMA in a given system. 2.2. The example of Al-Cr(-Cu)-Fe alloys After the pioneer contributions of Pauling10 and his successors, the Shoemakers, Samson and others, very little was published on Al-based CMAs until the discovery of quasicrystals re-launched the interest for such crystals. Driven by the need to find a quasicrystalline or approximant material offering high corrosion resistance against acids, we considered addition of Cr to Al-Cu-Fe icosahedral crystals5, 11. Above a small concentration in Cr species, the icosahedral phase is no longer stable and is replaced, depending on the Cu/(Cr+Fe) ratio, by orthorhombic or monoclinic compounds of large to very large unit cell. The same trend is observed in Al-Cr-Fe alloys, in which the icosahedral phase is metastable and can be formed only by rapid cooling of the melt12. Space is too limited here to give a brief description of all compounds discovered in Al-Cu-Fe-Cr alloys (for more details, see ref.1). In the following, I shall concentrate on one single compound, namely the O1-orthorhombic compound of lattice parameters a =3.25 nm, b =1.22 nm, c =2.37 nm, which contains 600 atoms per unit cell (117 independent atomic positions13) and has a hierarchical structural


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relationship with the more simple rhombohedral γ-AlCrFe brass-type phase12. A sketch of the structure is shown in Fig. 2. It consists of a stacking of planar and puckered layers (top left side of the figure). A projection of the planar and puckered layers, respectively, is shown in the bottom part of the figure. Using pentagonal and flattened hexagonal units, a tiling that although periodic is closely related to a Penrose tiling, is superimposed on the drawing. Close examination of the atom positions reveals the constitutive icosahedral units (Fig. 3) and a large amount of close-packed atomic planes that are interspaced by a distance of 0.43 nm. Such planes were shown by Mizutani3 to play the most important role in transport properties of CMAs because they establish a resonance by propagating electron waves with a Fermi wavelength of λF = 0.43 nm whereas the major Fourier lattice components are located at KP ≈ 28 - 30 nm-1 in reciprocal space (i.e. KP/2 ≈ 2π/λF). The very same situation is observed in the previous example of orthorhombic O1-AlCuFeCr (Fig. 3).

Fig. 2. Sketch of the orthorhombic O1-AlCuFeCr atom structure. The stacking structure of flat (bottom left) and puckered (bottom right) atom layers is shown in the top left part of the structure. Flat layers are very close to perfect pentagonal tiling. (Courtesy V. Demange, CMA).

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+

Fig. 3. Formation of icosahedral clusters in O1-AlCuFeCr and their arrangement that shows spacing by 0.43 nm. (Courtesy V. Demange, CMA).

3. The essential property of CMAs 3.1. The pseudo-gap at the Fermi energy This section is for Al-based CMAs and more specifically for the partial density of states (DOS) of aluminum that may be investigated by a large number of techniques, but preferably in the context of the present chapter by emission (XES) and absorption (XAS) X-ray spectroscopies (see 1 and 2 and references therein). As just mentioned above, so-called Hume-Rothery (HR) compounds (i.e. compounds for which it is observed that 2kF ≈ KP) are electronically stabilized. We shall restrict ourselves to such compounds in this section and the followings. The stabilization mechanism induces a depletion at the Fermi energy (EF) in the DOS, the so-called Hume-Rothery pseudogap14. Using the XES and XAS techniques, a series of Al-Cu(-Fe) HR compounds of different atomic structures and accordingly different Jones zones were studied in order to investigate the HR pseudo-gap. The valence band (VB) of all these compounds was analyzed together with the Al p and Cu d (-Fe d) conduction bands. In γ-Al35Cu65 as well as in


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all Al-Cu(-Fe) samples, the Al 3s-d and Al 3p sub-bands split in two distinct parts located on each side of the maximum of the Cu 3d distributions, around 4 eV15. Accordingly, the shapes of the Al 3s-d and Al 3p distribution curves depart dramatically from that in pure Al, which is a typical parabolic curve distorted by experimental artifacts and experimentally induced many-body effects (not shown here, see reference2). This result emphasizes that the free-electron model is no longer valid for HR alloys. The interaction of Cu with Al can be interpreted within the framework of a Fano-like interaction between highly localized states and extended states16. The fact that here, the Al partial spectral curves both display a marked depletion exactly at the energy of the maximum of the localized states points out that Al states still retain an extended-like character in these compounds. Using the same methodology as for the HR alloys, we have analyzed several CMAs of much larger unit cells than HR compounds, including icosahedral (i-) and approximants crystals (especially from Al-Cu-Fe system). Again, like in βAl55Cu33Fe12, it was observed that the valence band of Al-Cu-Fe samples shows that Fe 3d states overlap the Al sub-bands edges nearby EF, Cu 3d states are found about 4 eV below EF, whereas the Al sub-bands overlap each other over the whole extent of the VB, namely over about 12 eV. In all these samples, it was observed that the intensity of the Al subbands at EF departs from its value in fcc Al. It is lower compared to the pure metal, so that the corresponding valence edges have no longer their half-maximum intensity set at EF. This more or less pronounced depletion that appears in the Al DOS at EF, points out the formation of a pseudo-gap. We refer now mainly to Al 3p states because (a) in pure fcc aluminum, these states are originally of extended-like character and therefore are more sensitive to changes of the electronic interactions than d states, (b) they are obtained alone by XES whereas this technique gives always d and s states together. To quantify the pseudo-gap of the Al 3p partial DOS, we shall restrict ourselves to using only the intensity at EF, labelled n(EF) hereafter. It is expressed in arbitrary units, with a value n(EF) = 0.5 in the pure metal, since the inflexion point of the DOS is by

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nEF (un. arb.)

definition located at EF and at half-maximum in a free-electron system. A summary of a large number of numerical data for n(EF) is given in Fig. 4.

e/a Fig. 4. Variation of the partial Al3p DOS at the Fermi energy n(EF) as a function of the electron-to-atom (e/a) ratio in Al-Cu-Fe intermetallics1. The DOS is expressed in arbitrary units: 0.5 stands for pure, fcc Al (cross at the right upper corner of the figure). The diamonds in the middle of the figure are for Al-Cu compounds, open squares for B2, CsCl-type β-cubic phases, the open circle for tetragonal ω-Al70Cu20Fe10. The black symbols at the bottom of the curve (which serves only to guide the eye) are for the icosahedral Al62Cu25Fe13 compound and its approximants. Observe that there is no real difference of n(EF) for a quasicrystal and for its closely related crystalline approximants of large unit cell and nearly identical electron concentration. In contrast, the difference is much more marked with respect to smaller crystal unit cells, yet also at identical electron concentration.

Figure 4 demonstrates that the minimum of n(EF) is obtained for quasicrystals with i) a lattice of very high perfection and b) containing a transition metal (TM) of the mid-series, preferably a 5d TM alloyed to a TM of the right hand side of the series like Cu or Pd. However, clearly enough, the minimum of n(EF) cannot be taken as a unique property because CMAs of very large unit cell and nearly identical chemical composition exhibit almost identical values of n(EF). This conclusion holds true for all CMAs known so far: physical properties vary in inverse proportion to the size of the unit cell and undergo an extreme at the ultimate size of the unit cell, but no gap is observed when the size of the


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unit cell approaches its maximum, for instance infinite size like in an icosahedral quasicrystal. Actually, a pseudo-gap is expected in most CMAs due to the HumeRothery effect, i.e. the interaction of the Fermi Surface with the Jones zone, an effect which in turn stabilizes the crystal structure of the compound. Therefore, the more complex the compound, in other words the closer to a sphere the Jones zone, the more efficient the HR effect and henceforth the deeper the pseudo-gap. The formation of a pseudogap was very carefully studied by Mizutani in a large number of γ-brass compounds having 52 atoms in the unit cell3. This work clearly established the origin of the HR mechanism and assigned it to a resonance between Fermi electrons and certain Bragg planes spaced by about.4 nm. The number of such planes may vary, depending on the details of each crystal lattice. However, hybridization effects between sp and d states deepen and broaden the pseudo-gap more efficiently than the HR effect. This argument was demonstrated both by computations17 and experimental studies of compounds like Al2Ru18, which as a matter of facts shows the opening of a true, tiny gap of 0.17eV at Fermi energy. Much broader gaps are expected in CMAs containing 5d TM elements, but so far none was synthesized19. 3.2. Transport properties Space is too limited in this introductory chapter to comment all properties measured so far in CMAs. The reader should refer to reference 1 and to the present book for a state of the art regarding transport properties of CMAs. Experimental determinations of optical conductivity, thermal conductivity, thermo-electric parameters like the Seebek coefficient were achieved in a rather systematic way for most CMAs known so far and contrasted to theoretical analysis, especially by Macia et al.20. For the sake of illustration, a blend of several resistivity measurements between liquid helium and room temperature is presented in Fig. 5 for several Al-based CMAs21. Surprisingly enough, the low temperature resistivity of those samples varies inversely to the unit cell

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size and presents a change of the sign of the temperature coefficient of resistivity (TCR) when the unit cell becomes large, with remarkably a zero-TCR at a specific composition (and crystal structure).

Fig. 5. Electronic resistivity of a variety of CMAs with different unit cell sizes21. Icosahedral CMAs are located in the upper part of the figure, O1-orthorhombic compounds are labelled O-AlCrFe and O-AlCuCrFe whereas a large unit cell AlPdMn CMA and its superstructures22, noted Ψ and ξ’, respectively, fall in the middle of the figure with a zero TCR. The notation ω-AlCuFe is for the tetragonal ω-Al7Cu2Fe compound and AlCuB is for a superstructure of the AlCu β-phase8 doped with 1 at% of boron. Observe the TCR, which marks the transition from normal metallic behaviour (small unit cell size) to an unexpected behaviour for a system made of metals (large unit cells).


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This observation has triggered a large research effort to understand better the transition from usual metallic behavior (small unit cell size) to a behavior much more resembling that of a semi-conductor. In the first regime, the diffusion of charge carriers follows a ballistic law, i.e. is proportional to time t and Einstein’s conductivity applies in proportion to N(EF), the total DOS at EF. In the second regime, conductivity is instead proportional to [N(EF)]2 and diffusion follows a tβ regime, with 0≤ β ≤123. Essentially, it was concluded that the breakdown of Bloch's theorem at large to infinite unit cell size is accompanied by the formation of socalled critical states, neither extended, nor fully localized like in a totally disordered medium24. Further analysis by Mizutani3 and others25, 26 has concluded to a transport mechanism by hopping in highly complex CMAs such as quasicrystals and their approximants, in total contrast to the usual behavior of alloys. 3.3. Mechanical properties The study of large unit cell CMAs and of quasicrystals under compression stress at high temperature was for some time hesitating, concluding first that plasticity was carried by glide of dislocations and later by climb. Finally, climb assisted by phason jumps appeared to be the most important mechanism for icosahedral quasicrystals27, but a general view at all possible deformation mechanisms of CMAs remains still to be worked out. A mechanism specific to large unit cell CMAs, called metadislocation, was discovered to be able to generate plasticity on the basis of very local rearrangements of structural units28. The situation is quite similar as far as contact mechanical properties (at much lower temperatures) are concerned, although it was recognized soon after the discovery of quasicrystals that friction is significantly reduced against metallic antagonists like steel29. A fairly illustrative example of the contrast to be expected regarding friction between a normal metallic crystal and an aperiodic CMA riding against a steel antagonist is provided by Fig. 6. It shows the data recorded during a pinon-disk experiment performed at a low residual pressure (typically 10-5 mbar) and a velocity of the disk relative to the pin of 5.10-4 m s-1. These parameters were chosen in such a way that a full layer of oxide has no

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Fig. 6. Pin-on-disk experiment in vacuum (1.9 10-6 mbar), using a hard Cr-steel ball of 6 mm diameter riding under a normal load of 1N on a mono-domain Al-Ni-Co decagonal single crystal at a linear velocity of 5 10-4 ms-1. The upper curse (open dots) is for the friction coefficient μ and the lower curve for the vertical position of the pin (see text). Two successive maxima of this curve mark the length of a full circular trace of the indenter in contact with the surface of the single crystal (diameter of the trace 3 mm). At the beginning of the experiment, the decagonal sample is covered by a thin layer of its native oxide, namely amorphous alumina. Due to a slight misalignment of the specimen surface with respect to a perfectly horizontal position, the maxima of friction coincide with the middle of the ‘up-hill’ part of the trace (solid vertical bars). Therefore, right after the test has started, friction on the amorphous oxide is isotropic as expected in the sample surface plane. The oxide layer is removed after two turns of contact with the indenter and friction switches to another regime. New peaks appear, superimposed on the ones due to horizontal misalignment. More specifically, two maxima (respectively, minima) of friction show up during one single rotation (dashed vertical bars), which clearly fits with half the previous period. Careful examination of the angular position of the single crystal on the disk holder proves that the maxima of friction coincide with the pin riding along the periodic stacking direction in the crystal whereas the minima are observed when the rider moves perpendicular to this direction.

time to grow in the time interval left between two successive passages of the indenter on the disk. Immediately after the beginning of the test, the native surface oxide inherited from exposure to ambient atmosphere of the sample is broken through by the pin. In Fig. 6, this operation takes two turns of the disk (about 0.012 m). After this point, the two naked pin and disk surfaces come into contact. The data shown in Fig. 6 was obtained using a single grain of the AlNi-Co decagonal phase, a material characterized by periodicity along one


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direction, and loss of periodicity perpendicularly to this direction. The stacking period is about 0.4 nm. Therefore, it is possible to probe the respective effects of periodicity and aperiodicity using the very same sample. The key point is that a large difference of the friction coefficient μ = FT/FN is observed depending on whether friction occurs along the periodic stacking direction of the single crystal or perpendicular to it (FT represents the tangential force that works against the movement of the pin, and FN is the normal load applied to the pin, here 1N). In other words, the measurement of the tangential force FT (Fig. 6) leads to a friction coefficient μ =FT/FN that returns to the same value with a period equal to the length of one turn (or a time periodicity equal to the duration of one rotation) when the test starts whereas it shows half that periodicity when pin and decagonal sample come in contact (beyond 0.012 m riding distance). This means that friction on the amorphous native oxide is isotropic (the slight variation of μ when one rotation proceeds is due to a misalignment of the sample surface against a perfectly horizontal plane, which causes friction forces when the pin goes ‘down-hill’ and ‘up-hill’ to be different (the vertical position of the pin is also shown in figure 6, which in turn supplies us with a reference for the position of the pin along the circular trace). After two turns, the oxide layers are broken and the pin comes into contact with the naked surface of the specimen. Then, it appears that the period of μ is no longer that of one full rotation, but only half a turn. Within one rotation, two peaks of μ are now visible, one of rather large amplitude, going from μ = 0.25-0.30 at its minimum up to μ = 0.5 at its maximum whereas the other peak exhibits intermediate values of μ. This situation remains unchanged for quite a significant number of pin-on-disk turns, until wear debris and severe plastic deformation of the surfaces in contact disturb the quality of the experiment. The simple example above illustrates the variety of friction conditions experienced on CMAs, but so far, the example of the decagonal phase is unique as was first pointed out by Park et al.30. Many more data was collected by the author, especially on Al-based binary CMAs 1, 31. In the coming section, the main contribution to friction in vacuum is assigned to the (reduced) adhesion properties of Al-based CMAs and therefore to the (lower) surface energy of these materials.

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These results have direct relevance to technological applications, especially hindrance of cold welding in aerospace mechanisms or cutting tools. 3.4. Chemical properties At least for Al-based CMAs, chemical properties are disappointingly determined by the constituents introduced in the complex alloys: the presence of elements like Cr or Mo enhances corrosion resistance, especially when pitting corrosion occurs1, 32, oxidation resistance is pretty good, including up to temperatures far above room temperature (e.g. 700-900 K for alloys that melt in the range 1200-1500 K) thanks to the passivating role of aluminum dioxide that is well documented in conventional alloys of this element1, etc. Two different chemical properties that benefit from complexity of the CMA lattice have been pointed however. The first is the catalytic efficiency of nano-domains of Cu or Pd prepared by etching a quasicrystal in an alkaline solution33. The beneficial effect results from a combination of properties. One the one hand, pure metal nanograins of the late transition metal (Cu or Pd) may be grown out the surface of the powdered CMA by etching and releasing it in a mixture of amorphous oxides of the other elements (i.e. Al and Fe or Mn, respectively). On the other hand, coarsening of the grains is prevented by the presence of the second TM, with which the enthalpy of mixing is negative. As a consequence, the catalytic activity of this type of material, used for instance for methanol reforming, is at least equal, if not superior to that of the pure, ultra-divided metal whereas savings are achieved on the quantity of catalyst and its preparation processing. Another challenging chemical property is hydrogen storage, which raises great interest in our community, see the program of the second CMA EuroSchool that is going to take place in May 200734. The hydrogen capacity of Ti-Zr-Ni icosahedral compounds, to a lesser extent their crystalline approximant, reaches a H/M atom ratio of 2 (H: number of stored hydrogen atoms; M: total number of Ti, Zr and Ni atoms)35. This represents a considerable amount of hydrogen that is comparable or even superior to the performance of more conventional metal hydrides.


17

The key point is that a very large number of tetrahedral sites form in the lattice, in relation to the complexity of the icosahedral structure, which are favorable sites to insert a H atom. Furthermore, the chemical composition yields TM atoms (e.g. Zr) that preferably bong to H atoms and are sitting on one or two vertices of these tetrahedral sites. However, technological difficulties are still there, such as an insufficiently long lifetime, or equivalently a too small number of load/unload cycles that are required for a commercial battery, which so far did not open the way to the usage of this type of CMAs in commercial devices. Furthermore, lighter materials such as Mg look more promising in respect of the H/M weight ratio. This is also a good reason to pursue a research on Mg-based CMAs, again trying to combine atom compositions with a large affinity for H and the presence of the largest possible number of tetrahedral traps for H atoms. 4. Surface energy 4.1. Surface energy in general As far as Al-based CMAs are concerned, surface energy (noted γS in the following) combines chemical physics with contact mechanical properties as we try to demonstrate in this section. It is a very important property which determines the cleavage energy of a solid, the equilibrium shape of a crystal, the wetting properties of both solids and liquids, the nucleation rate of a second phase (via the energy born at the interface), etc. It is however very difficult to assess experimentally, either by contact angle measurements or flow stress measurements at the approach of the melting point because of the need to employ single crystals with no defects emerging at the surface. Indirect assessment of the surface energy may be based on contact angle measurements of liquids wetting the surface of interest, but this also is a very difficult task for experimental reasons (contamination of the surfaces, high temperatures, etc.) and because the energy of the interface between solid and liquid is most often ignored36. In the case of CMAs, the number of different samples and the difficulty to grow them all as mono-domain samples simply rules out this route. Computer physics is more efficient

18

Jean-Marie Dubois

to calculate the surface energy of metals and simple alloys and was successfully applied to a large number of metals and pure elements37. Due to the limited power of computers, it is at present days limited to a small number of atoms per unit cell and cannot be applied to CMAs and even more so to quasicrystals. The importance of friction and wetting in potential applications of CMAs, and furthermore the need for a better understanding of the fundamentals that govern those properties, forced us to find a simpler method to assess approximately γS for a large number of CMAs of various compositions. This method, although far less accurate than the ones evocated above, is described in the following. 4.2. Experimental Placed in a vacuum chamber evacuated down to 10-6 mbar or less, a pinon-disk instrument allows us to measure the friction coefficient between a hard steel ball and the solid of interest without intervening artifacts like moisture or external contamination (refer to previous section). Tribooxidation however may play an important role as was noticed elsewhere for tests in ambient atmosphere38. This artifact is dramatically reduced if the relative velocity of the indenter to the disk is large enough to forbid the growth of a full oxide layer in the time interval elapsed between two successive passages of the indenter, whereas the native oxide always present at the surface of our samples is broken and disappears from the trace within very few passages at the beginning of the experiment. This experimental procedure was applied in a systematic way to a large number of samples prepared by sintering according to a standard procedure depicted in ref.1. After polishing these samples (diameter 20 mm, thickness 4-6 mm) down to mirror polish, the specimens were placed in a pin-on-disk set up housed in a vacuum chamber as explained already in the previous section. The friction coefficient μ was recorded continuously during each test, using the same parameters as given in the previous section. Careful examination of the contact trace was performed after each test, for both the steel ball indenter and the surface of interest, thus allowing an evaluation of the wear produced during the test. In most cases, but not all, wear appeared negligible.


19

Theoretical models were developed to understand better the origin of Admontons law FT = μFN39. Some of them take adhesion forces explicitly into account, but unfortunately require some information that is missing in our experiments (e.g. the actual contact area). As a much easier to handle test model, we shall assume that: μ= α/HV + β ΩSP

(1)

where the material (Vickers) hardness is noted HV, the work of adhesion (under these specific experimental conditions) of the steel pin P on the surface S is ΩSP and α and β represent calibration parameters which may be determined easily by producing the same experiment, but for a series a materials of known hardness and surface energy as explained in reference 40. Since wear is most often negligible, we furthermore assume that ΩSP is actually the reversible adhesion energy of P onto S: ΩSP = γS + γP - γSP with γP the surface energy of the pin and γSP the interfacial energy at equilibrium between S and P materials. This is obviously a very drastic assumption that can only be marginally valid for a pin-on-disk test, but it is strengthened by the very slow motion of the pin relative to the CMA solid and the (near) absence of wear. On top of this, we take γP-γSP =0, which means that we overestimate the value of γS that can be obtained after inverting Eq. 1, or: γS ≤ (μ−α/HV)/β

(2)

Calibration of Eq. 1 using known materials (metals, alloys, oxides) leads to a linear fit characterized by a regression coefficient very close to 140, which makes us very confident in the validity of the method. Yet, it must be insisted on the fact that instead of measuring γS, we just supply an estimate of its upper limit for a large number of the CMAs of interest here. 4.3. Data analysis and upper limit of γS Table 1 and Fig. 7 summarize our findings regarding the estimation of the surface energy of Al-TM CMAs (in this sub-section, TM represents one or two 3d metals).

20

Jean-Marie Dubois

Table 1. Pin on disk tests in vacuum on Al-based CMAs of varying composition and crystal structure: experimental results and estimated values of γS. Compound (at.% or atoms)

Crystal structure

Vickers Hardness (load 0.5 N) ± 8%

Friction coefficient ± 15%

Estimated γS (Jm-2) ± 25%

Al3Ti

tetragonal

604

0.6

1.72

AlTi

B2-cubic

277

0.7

1.86

Al3V

tetragonal

427

0.7

1.97

Al9Cr4

γ-brass

0.48

1.37

Al8Cr5 Al11Mn4

γ-brass triclinic

695

β-AlCo l-Al13Co4

720

0.49

1.40

B2-cubic

476 620

0.41 0.5

1.09 1.41

monoclinic

700-800

0.75-0.8

2.2 – 2.4

β-AlFe

B2-cubic

λ-Al13Fe4 Al3Ni

monoclinic

417 815

0.64 0.69

1.78 2.04

orthorhombic

662

0.45

1.27

Al2Cu γ-Al9Cu4

tetragonal

550 480

0.44 0.37

1.21 1.0

AlCu oF-Al3Cu4

hexagonal orthorhombic

806 710

0.3 0.24

0.82 0.62

φ-Al10Cu10Fe

cubic

650

0.32

0.9

cubic

β-Al55Cu30Fe15

B2-cubic

680

0.31

0.84

ω-Al70Cu20Fe10 Al62Cu25.5Fe12.5

tetragonal

640-640

0.6-0.75

1.7-2.0

icosahedral

780

0.26

0.55

Al59B3Cu25.5Fe12.5

icosahedral

790

0.21

0.54

Al70Pd20Mn10

icosahedral

750

0.3

0.82

First, one should notice that some CMAs, especially quasicrystals (large diamond symbols in Fig. 7) exhibit a particularly low surface energy compared to that of the constituent 3d metals (typically 2.2 J/m2 for Fe) and that of Al (1.15 J/m2) as well as to conventional binary intermetallics (for example, triangles in Fig. 7). Worth mentioning, AlCu binary compounds (squares in Fig. 7) are also characterised by a rather low surface energy. Second, the surface energy follows a smooth decrease with the filling-in of the valence band.


21

Fig. 7. Variation of γS with the total number of s, p and 3d electrons in the molecular composition of Al-3d TM CMA samples. Large open and grey diamonds stand for iAlCuFe and i-AlPdMn compounds, respectively. Grey squares represent Al-Cu, open triangles Al-Ni, grey triangles Al-V (large symbol) and Al-Ti (small symbol), open squares Al-Cr, grey dots Al-Mn, small solid diamond Al-Fe CMAs, respectively. Black solid triangles and black solid dots are for Al13Co4 and ω-Al7Cu2Fe compounds (two separate measurements on two distinct samples each), respectively. The top grey triangle is for Al8V5, but this estimate is rather uncertain due to the contribution of wear to friction.

Scrutinizing Fig. 7 shows that γS is smaller for TM alloying elements that belong to the right hand side of the 3d series whereas elements like V, Ti, Mn, Co or Cr are associated with much larger values of γS (grey and black symbols in Fig. 7). Nevertheless, quasicrystals are located significantly below average at a given electron concentration, an effect most presumably related to the formation of critical states and their unusual contribution to the valence band at the Fermi energy. Similarly, large differences in γS are observed for compounds of comparable chemical composition, but different TM content. This is the case for instance for β- and γ- compounds of the Al-Co and Al-Fe systems (Table 1). The two types of compounds are characterized by

22

Jean-Marie Dubois

values of γS that differ by a factor of nearly 2, a difference that must be related to the HR pseudo-gap existing at EF and to the respective abundance of s, p and d states in the surface electronic structure (see also Fig. 4). More work is in progress to understand better the possible role of the pseudo-gap in determining γS. 5. Inverse Nano-Structuration 5.1. Comparison to Conventional Nano-Structuration To sum up at this stage of the chapter, I shall remind the Reader that both the Hume-Rothery scattering of Fermi electrons and sp-d hybridization contribute to the formation of a deep pseudo-gap at Fermi energy in large unit cell CMAs. In few compounds, a true gap is open. The width and depth of the pseudo-gap are in proportion to the size of the unit cell (although no analytical exact expression that would account for the coupling has been worked out till now). A change of behavior of the transport properties is observed from normal metallic transport to a hopping mechanism when the size of the unit cell becomes comparable to the Fermi wavelength, i.e. when the lattice parameters reach approximately 1 nm or more. Associated with this fundamental characteristic of CMAs are numerous other effects that could not be introduced in the present chapter for the sake of brevity, but are illustrated elsewhere in this book and in review documents quoted in1. For the same range of lattice parameters, plasticity becomes one more essential property of CMAs and is carried by defects intrinsic to the lattice complexity of CMAs27. Specific chemical reactivity is not associated with the size of the unit cell of CMAs, although after etching it becomes quite clear that catalytic performance is only observed with Al-Cu-Fe CMAs of large unit cell, and not with more simple materials like the ω-Al7Cu2Fe phase. Altogether, such effects are considered a specific behavior of nanostructured materials, in association with the reduction of the size of constitutive (isolated) objects, or with the relatively higher importance of interfaces and surfaces compared to the volume of a bulk specimen synthesized from nanograined material. The comparison is made for


23

illustration in Fig. 8 between plasticity under compressive stress of nanograined ultra-pure copper41 and an Al-Cu-Fe-Cr orthorhombic of approximant of the decagonal quasicrystal42. 400

True stress (MPa)

b 300

c

a

200

100

0

0

2

4

6

8

10

12

True strain (%) Fig. 8. True stress-true strain curves recorded during compression testing at room temperature of microcrystalline copper (a) and nanocrystalline copper (b) of average grain size 50 nm41 compared to orthorhombic Al-Cu-Fe-Cr CMA (c) at 650°C42. Observe the absence of grain coarsening on curves b) and c).

In the former sample, the average size of the Cu grains is about 50 nm whereas in the latter, the grain size is a fraction of a millimeter. The stress-strain curves are however nearly identical because plastic deformation is due to the collective movement under stress of localized atomic defects, which go with the disorder installed in the respective materials. The occurrence of localized-like (critical) states, the change. transport regime of conductivity, the plasticity associated with atom jumps, the surface chemical reactivity that raises with increasing the size of the unit cell beyond a significant fraction of a nanometer (typically 1 nm) is coined Inverse Nanostructuration (INS) by the author. It is directly associated with the potential of CMAs for technology, see next section An oversimplified picture of the main effects encountered in conventional nanostructuration, when the size of the individual objects or the grain size in a composite material becomes small (typically below few nanometers), and INS is given in Fig. 9 and its caption.

24

Jean-Marie Dubois

Fig. 9. Simplified comparison between conventional nanostruturation (CNS, top part of the figure) and inverse nanostructuration (INS, bottom part of the figure). In CNS, the parameters that matter are the size A of the objects (large circles) and their respective interspacing distance a compared to the wavelength λ of the excitation. The presence or not of disorder (featured by small dots) between the separate objects, for instance contamination, dust, ill-grown particles, etc. also determines the response of the system to the excitation. The size of the system (e.g. a terrace on a single-grain wafer, light grey area) is supposed infinite, or much larger than the individual size of the objects. Effects of relevance to the small size of the system components manifest themselves when their size becomes small, i.e. when A ≈ λ whereas the separation distance a cannot be very large in order to observe a coupling between the objects, typically a ≥ λ. In INS, the situation is reversed. The individual objects are the atom clusters (large circles of diameter A) embedded in a periodic crystal having a unit cell of lattice parameter a, which is approximately the separation distance between individual clusters. Here also, there is some disorder between the clusters, often called ‘glue atoms’ (small dots). Effects due to INS manifest themselves when A ≈ λ, like in CNS, and when a ≥ λ, preferably when a >> λ. This supposes that the more complex the compound, equivalently the larger the unit cell, the more enhanced the effects of INS.

5.2. A great potential for future research Accessible due to very recent progress in materials science, CMAs offer great potential for innovation. Examples of this potential are heat insulation at low temperature (using e.g. Al-Cu-Fe compounds), hydrogen storage, thermoelectricity, enhanced catalytic efficiency at


25

lower cost, reduced friction, optimised composites, nanostructuration of metallic aggregates or thin films, development of innovative coating processes adapted to complex surface shapes, etc. Thermoelectricity is of special relevance nowadays that green energies are foreseen. Most presumably, clathrates and skutterudites offer the best CMA candidates for this purpose with quite respectable figures of merit achieved so far. Already mentioned, there are some doubts about the actual usefulness of CMAs regarding hydrogen storage in competition with light materials and especially nanotubes, but the challenge is worth a serious research effort. To end with, composite materials containing CMAs are the subject of various attempts for application in view of enhanced mechanical performance, whether they are prepared by blending with polymers or light metals or by in-situ reinforcement produced by nanoprecipitation of particles. Thanks to their low surface energy, grain coarsening is prevented, but direct usage of the specific surface energy is also foreseen in order to prevent cold-welding of mechanical parts kept in contact under severe load, for instance in satellites or vacuum technology applications. 6. Goals and organisation of the NoE CMA On this basis, a European network uniting 20 high-level core institutions in Europe, with a staff of more than 300 scientists and 60 PhD students, was designed to strengthen the competitiveness of European industries wherever materials need to offer hybrid properties, being both structural and functional, or embody an extraordinary combination of properties that are mutually excluding in conventional materials. Innovative management procedures for knowledge handling and networking, grant administration, organisation of conferences, exhibitions, industrial open days and specific measures for personnel exchange, access to platforms and durable integration of women in science are being taken, together with an ambitious program of summer schools and personnel training. The main purpose of a European Network of Excellence is to counterbalance the fragmentation of research in the continent. The essential tool used by CMA to achieve integration is the creation and functioning of so-called VIls, or Virtual Integrated Laboratories, in

26

Jean-Marie Dubois

conjunction with VIUs, or Virtual Integrated Units. These latter VIUs are service units that assist the VILs along their scientific strategy and needs (executive board and network office, gender mainstreaming, publications, mission service, transfer of knowledge and innovation, legal expertise). A very important VIU in this respect is the EuroSchool that has produced the present book. Six VILs were created, which assemble the expertise found in various European countries regarding metallurgy (VIL A), crystallography (VIL B), physical properties (VIL C), surface physics, chemistry and nanosciences (VIL D), surface technologies (VIL E) and finally applied physics of CMAs (VIL E). The degree of integration through the creation and functioning of the VILs can be appreciated from Fig. 10, which for the sake of simplicity presents only partial information on the exchange of data and deliverables between VILs and the external scientific community.

Fig. 10. Virtual Integrated Units of the CMA network of excellence, their relationships to each other and to the external world (simplified).


27

As an example, VIL A will supply the other VILs with wellcharacterized samples, for structure determination, property measurements or assessment of potential applications (shown by single-line arrows). On the other hand, output of VIL A relevant for the external world such as the discovery of new compounds, phase-diagram data and measured or computed thermodynamic properties, will be distributed via scientific publication (double-line arrows). There is therefore a clear will of CMA to establish integration both within a VIL, binding together various topically related laboratories, and between different VILs in order to forward co-operation between different communities. Acknowledgments Thanks are due to the Commission of the European Communities for partial support of my work over the years (Grants BRE 2 CT 92 0171, G5RD-CT-2001-00584 & NMP3-2005-CT-500145). The author also gratefully acknowledges the long-lasting support of his research by the local authorities in Nancy (Communauté Urbaine du Grand Nancy, Conseil Régional de Lorraine and Préfecture de la Région Lorraine). Thanks also go to Dr M. Feuerbacher, Forschungszentrum Juelich, Germany, for the provision of the X-ray data shown in Figure 1 and Prof. P. A. Thiel and her project team at Iowa State University, USA for the provision of the single grain decagonal sample used in part of this work. I am also deeply grateful to M. Sales, J. Brenner and A. Merstallinger, Austrian Research Centres, Seibersdorf, for providing access to the pinon-disk facility and experimental help. References 1. J.M. Dubois, Useful Quasicrystals (World Scientific, Singapore, 2005). 2. E. Belin-Ferré, J. Phys. Cond. Matter 14, R789 (2002). 3. U. Mizutani, The Theory of Electrons in Metals, (Cambridge University Press, Cambridge, 2001); U. Mizutani, in The Science of Complex Alloy Phases. Ed. T.B. Massalski and E.A. Turchi (The Minerals, Metals and Materials Society, Warrendale, 2005). 4. J.M. Dubois et al., Phil. Mag. 86-6-8, 797 (2006).

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Jean-Marie Dubois

5. J.M. Dubois and A. Pianelli, French Patent n° 2671808 (17-06-1994); US Patent n° 5432011 (11-07-1995). 6. J.M. Dubois, Robert F. Mehl Lecture 2007, (TMS Conf. Proceedings, Warrendale), to be published (2007). 7. For more details, see the website of CMA at: www.cma-ecnoe.org. 8. C. Dong, Q.H. Zhang, D.H. Wang and Y.M. Wang, Euro Phys. J. B 6, 25 (1998). 9. P. Häussler, J. Barzola-Quiquia, D. Hauschild, J. Rauchhaupt, M. Stiehler and M. Hackert, in The Science of Complex Alloy Phases, Ed.. T.B. Massalski and P.E.A. Turchi (The Minerals, Metals and Materials Society, Warrendale), pp.43-86 (2005). 10. L. Pauling, The Nature of the Chemical Bond, 3rd Edition, Chap. 11, (Cornell University Press, Cornell, 1960). 11. C. Dong and J.M. Dubois, J. Mat. Science, 26, 1647-1654 (1991). 12. V. Demange, J.S. Wu, V. Brien, F. Machizaud and J.M. Dubois, Mat. Sc. Eng. 294296, 79-81 (2000). 13. X. Z. Li, C. Dong and J.M. Dubois, J. Appl. Cryst. 28, 96-104 (1995). 14. A.P. Blandin, in Phase stability in metals and alloys, Ed. P.S. Rudman, J. Stringer, R.I. Jaffee (McGraw Hill, New York), 115 (1965). 15. V. Fournée, E. Belin-Ferré and J.M. Dubois, J. Phys.: Cond. Matter 10, 4231 (1998). 16. K. Terakura, J. Phys F: Met. Phys. 3, 1773 (1977). 17. G. Trambly de Laissardière, D. Nguyen Manh, L. Magaud, J.P. Julien, F. CyrotLackmann and D. Mayon, Phys. Rev. B 52, 7920 (1995). 18. D. Nguyen Manh, G. Trambly de Laissardière, J.P. Julien, D. Mayou and F. CyrotLackmann, Solid State Comm. 82, 329 (1992). 19. M. Krajci and J. Hafner, Mat. Res. Symp. Proc. 805, 121 (2004). 20. E. Macia, Phys. Rev. B 66, 174203 (2002); C.V. Landauro, E. Macia and H. Solbrig, Phys. Rev. B 67, 184206 (2003). 21. E. Belin-Ferré, M. Klansek, Z. Jaclic, J. Dolinsek, J. M. Dubois. J. Phys.: Condens. Matter 17, 6911 (2005). 22. L. Behara, M. Duneau, H. Klein and M. Audier, Philos. Mag. A 76, 587 (1997). 23. D. Mayou, in Quasicrystals, Current Topics, Ed. E. Belin-Ferré et al. (World Scientific, Singapore, 2000). 24. C. Sire, in Lectures on Quasicrystals,. Ed. F. Hippert and D. Gratias (Les Editions de Physique, Les Ulis, 1994). 25. C. Janot, Quasicrystals, a Primer, 2nd Edition (Clarendon Press, Oxford, 1994). 26. V. Demange, A. Milandri, M.C. de Weerd, F. Machizaud, G. Jeandel, J.M. Dubois, Phys. Rev. B 65, 144205 (2002). 27. F. Monpiou, D. Caillard and M. Feuerbacher, Philos. Mag., 84, 2777 (2004). 28. H. Klein, M. Feuerbacher, P. Shall and K. Urban, Phys. Rev. Lett. 82, 3468 (1999). 29. J.M. Dubois, S.S. Kang, J. von Stebut, J. Mat. Sc. Lett., 10, 537 (1991). 30. Jeong Young Park, D.F. Ogletree, M. Salmeron, R.A. Ribeiro, P.C. Canfield, C.J. Jenks and P.A. Thiel, Science 309, 1354 (2005).


29

31. E. Belin-Ferré and J.M. Dubois, Int. J. Mat. Res. 97, 7 (2006). 32. D. Veys, C. Rapin, X. Li, L. Aranda, V. Fournée, J.M. Dubois, J. Non Cryst. Sol. 347/1-3, 1 (2004). 33. A.P. Tsai and M. Yoshimura, Mat. Res. Soc. Symp. Proc. 643, K16.4.1 (2001). 34. See the website of the Euroschool at: http://euroschool-cma.ijs.si 35. A.M. Viano, R.M. Stroud, P.C. Gibbons, A.F. McDowell, M.S. Conradi and K.F. Kelton, Phys. Rev. B 51-17, 12026 (1995). 36. N. Eustatopoulos, M.-G. Nicholas and D. Drevet, Wettability at High Temperatures, Elsevier, Amsterdam (1999). 37. L. Vitos, A.V. Ruban, H.L. Skriver and J. Kollar, Surf. Science 411, 186 (1998). 38. I.L. Singer, J.M. Dubois, J.M. Soro, D. Rouxel and J. Von Stebut, in Quasicrystals, Ed.S. Takeuchi and T. Fujiwara, World Scientific Singapore 769 (1998). 39. Jianping Gao, D.W. Luedtke, D. Gourdon, M. Ruth, J.N. Israelachvili and Uzi Landman, J. Phys. Chem. B, 108, 3410 (2004) and references therein.D. R. Bates, Phys. Rev. , 492 (1950). 40. J.M. Dubois, M.C. de Weerd and J. Brenner, Ferroelectrics 305, 159 (2004). 41. Y. Champion, C. Langlois, S. Guérin-Mailly, P. Langlois, J.L. Bonnentien and M.J. Hÿtch, Science, 300, 310 (2003). 42. S.S. Kang and J.M. Dubois, Philos. Mag. A 66-1, 151 (1992).


CHAPTER 2

THERMODYNAMICS AND PHASE DIAGRAMS Livio Battezzati Dipartimento di Chimica IFMe Centro di Eccellenza NIS, Università di Torino, Via Pietro Giuria 7, 10125 Torino, Italy E-mail: [email protected] Phase diagrams are an extremely important tool for assessing alloy constitution, phase stability and for materials processing. In this tutorial chapter the strategy and practice of the construction of phase diagram are provided. The use of modern computer software for phase diagram calculation is outlined. Starting from basic thermodynamics, the free energy functions are derived for phases appearing in a given system: elements, solutions, compounds. The equilibrium conditions are defined for both metastable and stable states as a function of composition and temperature. The phase diagram is derived as an ensemble of equilibrium states with examples taken from binary systems. Examples of the use of thermodynamics for phase transformations are finally given.

1. Introduction The basic thermodynamics needed for the understanding and appropriate use of phase diagrams is given concisely in this chapter. The number of formulae has been kept as low as possible since extended treatises on the topic are readily available1-4. The equilibrium is defined for unary, binary and multicomponent systems composed of a single or multiple phases as function of temperature, pressure and composition. Throughout the chapter use is made of current computational tools for thermodynamic quantities and for assessment of phase diagrams5. Also the use of thermodynamics and phase diagrams is underlined for metastable equilibria often encountered in materials processing and phase transformations. Illustrations and examples are provided with numerical details so that they can be reproduced. 31

32

Livio Battezzati

2. Basic relationships and unary phase diagrams For a given system a reversible infinitesimal change of the Gibbs free energy, G, as a function of temperature, T, pressure, P, mole fraction, ni, is given by

dG = − SdT + VdP +

∑ μ dn + ..... i

i

(1)

i

with S, the entropy, V, the volume and μi the chemical potential of the i component, being the summation extended to all components of the system. The equilibrium condition is expressed by dG = 0........

(2)

d 2G ≥ 0...........

(3)

G being at a minimum or

and can be verified at constant T and/or P and/or ni,. The integral quantity, G = H - TS, can be made explicit when a reference state is defined. A standard element reference state (SER) is usually employed by posing H298 = 0 and S = S298 for the phase stable at 298 K. Therefore the enthalpy and entropy as a function of temperature are given by

H = H 298 +

S = S298 +

∫

∫

T

C p dT.....

(4)

C p d ln T .....

(5)

298

T 298

The specific heat is expressed through a series of empirical coefficients, ci multiplying various powers of temperature. Further possible contributions, such as magnetic, are accounted for by additional terms: C p = c1 + c2T + c3T 2 + c4 / T 2 + ... + C p ,mag + .......

(6)

Thermodynamics and Phase Diagrams

33

An example is now provided for a pure element: Fe, i.e. a unary system. The specific heat for the condensed phases having different structure, i.e. face centred cubic, fcc, body centred cubic, bcc, hexagonal closed packed, hcp, and liquid is computed using assessed coefficients. The specific heat curves are shown in Fig. 1. Note the cusp in the specific heat of the bcc structure due to the ferromagnetic to paramagnetic transformation. All curves have a small cusp at the melting point of the element. This is not an experimental fact but a feature introduced by the assessment of thermodynamic properties. Since the specific heat of a liquid (solid) element cannot be measured at temperatures below (above) the melting point, the quantity is estimated outside the field of existence of the phase from the ensemble of properties and from its behaviour in alloys. The specific heat of the liquid is brought towards that of the solid in the undercooling regime.

-1

Cp /J mol K

-1

60 bcc

50

liq

40 fcc, hcp 30 20

500

1000

1500

2000

Temperature /K Fig. 1. The specific heat of phases of Fe according to the compilation of the Scientific Group Thermodata Europe (SGTE)6.

The corresponding free energies of fcc, bcc, hcp and liquid phases of Fe at p = 1 bar are given in Figs. 1 and 2 using the free energy of the bcc phase as reference state for all temperatures and, therefore, set to zero. The equilibrium condition imposes that the stable phase at every temperature has the minimum free energy. This corresponds to the bcc phase from 298 K to 1184 K, the fcc phase from 1184 K to 1668 K (see insert in Fig. 2), the bcc phase again from 1668 K to 1809 K and the

34

Livio Battezzati

15000

Liquid

10000

Gibbs energy /J mol

Gibbs energy /J mol

-1

-1

liquid above 1809 K. At the temperatures just mentioned, two phases have the same free energy, i.e. the difference in their free energy, ΔG is nil. Therefore, the two phases coexist at equilibrium. 100 50 bcc

0 -50 -100 1000

fcc 1200

1400

1600

Temperature /K

5000

hcp fcc

0

bcc 0

500

1000

1500

2000

Temperature /K Fig. 2. The free energy of phases of Fe. The reference state at all temperatures is the free energy of the bcc phase stable in a wide temperature range. The insert is an enlargement of the plot in the temperature range of stability of the fcc phase.

Performing the same calculation at different pressures, the loci of equilibrium points in the p-T phase diagram are obtained as plotted in Fig. 3.

Temperature /K

3000 liq

2500 2000

fcc bcc

1500 1000

hcp bcc

0.0

10

10

2.0x10 4.0x10 Pressure /Pa

Fig. 3. The p-T of Fe. The phase stable in each field is marked in the figure.


35

It is useful to consider areas, lines, and triple points where lines join according to the Gibbs phase rule: v = c – f + 2, where v is the variance, the number of degrees of freedom of the system, c the number of components, f the number of phases in which components are distributed and the number 2 counts the physical variables defining such state, here pressure and temperature. Areas contain bivariant states, lines monovariant states and triple points invariant states. When a system is invariant, three phases are at equilibrium and no physical variable can be modified if equilibrium has to be maintained. When a system is monovariant, only one of the physical variables can be modified. The other one must follow its change to keep equilibrium. In bivariant states the physical variables can be modified independently within certain limits without changing the state of the system. Other variables, of magnetic, electrical, etc. origine, can be used to define a given system. In the following the pressure will be kept constant at p = 1 bar to consider systems containing two components and the Gibbs phase rule will be used as v = c – f + 1. 3. Simple binary phase diagrams

The free energy for each homogeneous phase, termed solution, is written as Gϕ =

ref

Gϕ + id Gϕ + exGϕ

(7)

where refGϕ is the free energy contribution of the pure elements, Gi, in their respective mole fractions, idGϕ is the ideal contribution to the free energy when a solution is formed. This is an entropic term stemming from the random occupancy of the sites in the structure of the system. ex ϕ G is called excess free energy and contains all non ideal contributions to the free energy of the phase. The excess term represents specific properties of the phase and, therefore, is model dependent. It is now common practice to express it via polynomial functions of composition containing empirical interaction parameters, vLi,jϕ ex

G ϕ = x A xB

ν L ∑ ν

A, B

ϕ

( x A − xB )ν

(8)

36

Livio Battezzati

Simple approaches correspond to all nLA,B j = 0 implying exGϕ = 0 (ideal solution); 0LA,Bj = const and nLA,B j = 0 for n > 0 (regular solution).The sum of idGϕ and exGϕ is termed free energy of mixing, ΔGmix, in that it expresses the variation of free energy when the solution is formed from pure components in their respective mole fractions. If the solution is ideal there is no enthalpy contribution to the free energy but only the ideal entropic term. If the solution is regular, the 0LA,B j interaction parameter can be either negative or positive expressing the favourable or unfavourable attitude of components to mix in a homogeneous phase. Considering a binary system A-B, the free energy of mixing is written as

ΔGmix = xA μA + xB μB

(9)

where μA and μB are the partial molar free energies of component A and B relative to the appropriate reference state or chemical potentials of A and B in the solution phase. The chemical potential is defined as the derivative of ΔGmix with respect to mole fraction of a given component taken at constant composition: ⎛ ∂ΔGmix ⎞ ⎟ ⎝ ∂x A ⎠T , P , xB

μA = ⎜

(10)

It expresses the behaviour of the component in the solution. When the chemical potential of A and B are equal in two phases, ϕ and ζ,

μ Aϕ = μ Aζ

(11)

μ Bϕ = μ B ζ

(12)

the condition for phase equilibrium is reached and the two phases coexist in the system. As an example, a set of free energy curves is shown in Fig. 4 for a regular solution having a positive interaction parameter at various


37

temperatures. At the highest temperature the free energy of mixing is negative and has positive curvature for all compositions: the solution is always stable in its homogeneous state. For all other temperatures, the free energy of mixing displays both positive and negative curvature. Taking its first derivative it can easily be shown that the chemical potential of both components is equal at the compositions of the two minima of the curve implying that two phases having such compositions must coexist in equilibrium.

Free energy /Jmol

-1

4000 2000 0 -2000 -4000 0.0

0.2

0.4

0.6

0.8

1.0

Mole fraction B Fig. 4. Free energy curves for a solution phase at various temperatures from 500 K to 1700 K at 200 K interval using the interaction parameter for a regular solution 0LA,Bj = 27000 Jmol-1.

Geometrically, the equality of chemical potential is represented by the construction of a common tangent to the two minima. The phases and the components have the same structure since their free energy is expressed with the same curve. Therefore, a single phase stable at high temperature, de-mixes on cooling forming two phases. The composition corresponding to the tangent points at every temperature are collected in a temperature-composition diagram containing a miscibility gap between components. The procedure followed up to now allows for the synthesis of a phase diagram from known free energy curves and reference states. For most practical cases, the free energy curves are only partially known from

38

Livio Battezzati

Free energy /Jmol-1

experiments. In addition points of the phase diagram are known where phase equilibria are established. The complete phase diagram is then optimised by fitting the set of relevant expression of free energy to the available data to obtain a reasonable number of interaction parameters. In the case of components having different structure, two free curves are needed. An example is shown in Fig. 5 for solution phases having negative interaction parameters. 0,β

GA

0,α

GB

5000

0 α

β

μA = μA

β

α

0.0

β

xB

α

xB

-5000 0.2

0.4

0.6

0.8

α

β

μB = μB

1.0

Mole fraction B Fig. 5. Free energy curves for two solution phases having different structure. GA0, β is the free energy of A in the β structure. GB0, α is the free energy of B in the α structure. The melting point of A, TmA, is 1200 K and its heat of fusion is ΔHm = 14 kJmol-1; The melting point of B, TmB, is 1000 K and its heat of fusion is ΔHm = 12 kJmol-1. Curves were computed at 600 K for two regular solutions with 0LA,Ba = -10000 Jmol-1 and 0LA,B b = -12000 Jmol-1.

The reference states are the pure A and B components in their stable state at the given temperature. Each curve extends from the stable state of one component (the zero of free energy) to the value of the free energy of the other component in the hypothetical state where the component has the structure of the other one. Such states are called lattice stabilities and need to be evaluated to draw a free energy curve for each phase. The lattice stabilities for elements have been compiled by the SGTE6. The equilibrium condition that the chemical potential be equal in the two phases is represented by the common tangent to the curves and, as above,


39

the corresponding compositions are collected in the binary phase diagram. When two phases of a binary system are in equilibrium at constant pressure the Gibbs phase rule implies v = 1, i.e. a single degree of freedom in order the system remains in the same state. If the temperature is modified, the composition of phases will follow accordingly. 4. Three phase equilibria: invariant reactions

Let us consider a system made of components A and B having the same structure and admitting two solution phases, having positive enthalpy of mixing, α, and ideal behaviour, β, respectively. A possible free energy scheme is shown in Fig. 6 at a temperature where the stability is given by the common tangent to the two minima of the first curve.

Free energy /Jmol

-1

8 0 00

T = 6 00 K

6 0 00 liqu id 4 0 00 2 0 00 so lid so lu tio n 0 0 .0

0 .2

0 .4

0 .6

0 .8

1.0

M o l fra ctio n B Fig. 6. Free energy curves computed for a hypothetical A-B system with: TmA = 1000 K and ΔHm = 12 kJmol-1; TmB = 1200 K and ΔHm = 14 kJmol-1. Ideal solution for liquid; regular solution for crystal (0LA,Bj = 30 kJmol-1).

The second curve lies above the first one or above the common tangent for all compositions. When the temperature is increased, the two curves are displaced relative to one another and the second one crosses the common tangent to the first curve branches (Fig. 7). The chemical potential of components is now equal in three phases which coexist at equilibrium and the system becomes invariant. No change in physical or

40

Livio Battezzati

chemical variables is possible until the three-phases equilibrium is maintained. On increasing the temperature further, the β phase acquires a stability range and coexists with α in composition ranges defined by the common tangents to the free energy curves of the two phases. The phase diagram obtained by collecting all composition points at equilibrium at every temperature contains monovariant lines and a horizontal line joining the three compositions coexisting at equilibrium. If the β phase is liquid the phase diagram is called eutectic, if the β phase is solid it is called eutectoid. The temperatures marked by the horizontal lines take the same names.

Free energy /Jmol

-1

6000 Teut = 735.5 K

liquid

4000

solid solution

2000 0

0.0

x2

xeut

x1 0.2

0.4

0.6

0.8

1.0

Mol fraction B Fig. 7. Free energy curves computed for a hypothetical A-B system at the eutectic temperature. Parameters as in Fig. 6.

An analogous phase diagram is obtained when three phases of different structure compose the system, all of them displaying upward curvature in their free energy curves as those of Fig. 7. In all these cases a high temperature phase will decompose isothermally into two phases at the temperature where the equality of chemical potentials occur. For the eutectic l ⇔ α1 + α 2 , if the solid phases have the same structure, and otherwise l ⇔ α + β . For the eutectoid the l phase is replaced by a γ solid phase.


41

T = 603.5

6000

liquid

4000

Free energy /Jmol

Free energy /Jmol

-1

-1

Should the relative position of the free energy curves differ from those employed up to now due to the values of the interaction parameters and/or of the lattice stabilities, other invariant reactions will appear. Fig. 8 shows a free energy scheme in which the liquid phase is placed at the extreme of a common tangent to three phases, two solid and a liquid one. 100 T = 603.5

0 -100 x2

xperi -200

0.96

0.98

Mol fraction B

2000 0

solid solution

xperi x2

x1

0.0

0.2

0.4

0.6

0.8

1.0

Mol fraction B Fig. 8. Free energy curves computed for a hypothetical A-B system with: TmA = 1000 K and ΔHm = 12 kJmol-1; TmB = 600 K and ΔHm = 6 kJmol-1. Regular solution assumed for both crystal and liquid (0LA,Bj = 20 kJmol-1 and 15 kJmol-1 respectively).

The resulting phase diagram has a reaction, named peritectic, in which on cooling a liquid reacts at a given temperature with a solid phase to form a single solid phase l + α1 ⇔ α 2 . As above, the structure of solid phases is often different and the reaction is l + α ⇔ β . If the l phase is replaced by a γ solid phase, the reaction is called peritectoid as is the corresponding temperature. In a binary system the occurrence of a miscibility gap in the liquid phase or of allotropic transformations of solid phases will cause modifications in the shape and relative position of free energy curves with the consequence of producing more invariant reactions. The local shape of the phase diagram, i.e. around the temperature where the reaction occurs, will be similar to those of the previous reactions,

42

Livio Battezzati

however the type and position of phases will be different and the reactions will deserve a new name as the corresponding temperature. A collection of all types of reaction is reported in Figs. 9a and b. liq 1

α

Temperature

liq 1 α

peritectic

liq

β

liq

α

β

eutectic

0.4 0.6 mol fraction B

α

α

α

0.2

0.8

liq

katatectic

α

Temperature

liq 2 monotectic

α

0.2

liq 2

syntectic

β

β1

monotectoid peritectoid β γ

eutectoid

0.4 0.6 mol fraction B

β2 γ

β

0.8

Fig. 9a and b. Upper panel, a: the position of monovariant lines at and close to eutectic, peritectic, monotectic, syntectic temperatures in hypothetical A-B systems. Lower panel, b: the position of monovariant lines at and close to eutectoid, peritectoid, monotectoid, katatectic temperatures in hypothetical A-B systems.

The monotectic implies a miscibility gap with two liquid phases in equilibrium with a solid phase. An analogous phase diagram can exist, made completely of solid phases, the monotectoid. The syntectic occurs


43

when two liquid phases solidify in a single solid phase. The analogous reaction in the solid state would be indistinguishable from a peritectoid. The katatectic is a case of inverse melting, although only of a portion of the system. In fact, here a solid phase decompose on cooling into a different solid phase and a liquid one. The analogous reaction in the solid state would be indistinguishable from a eutectoid. 5. Compounds

The free energy of formation of an ordered compound from the components is ΔGfor = ΔHfor -TΔSfor. The enthalpy of formation, ΔHfor, is usually negative and the entropy of formation, ΔSfor, is small since the compound is ordered at all temperatures. The free energy is, therefore, negative, centred at the stoichiometry of the compound, AnBm, and V-shaped since any compositional deviation from the exact stoichiometry would cause a sharp increase in free energy. In some cases the resulting field in the phase diagram is so narrow that it is drawn as a vertical line (line compound). The deviation from stoichiometry in ordered phases is accounted for by the sublattice model also called compound energy formalism5. From the knowledge of the structure of the phase, a minimum number of sublattices is defined which is needed to represent the order within the compound. Each sublattice is assigned a number of sites, Ns, which will host one of the components in the case of full order; otherwise, the other components or vacancies can mix on the same sites. The site occupation by component i is given by yi = ni/Ns with ni the number of atoms of the i component. The free energy of the compound is then written according to that of section 2. The ideal term stems from the entropy of mixing in the various sublattices is: ideal S mix = − kT

∑N ∑y s

s

s i

ln yis

(13)

i

The excess free energy is given as a function of the mole fraction of species in the sublattice, yi, via parameters, Li,j, expressing the interaction of components i and j inside a sublattice and between different sublattices, Li,j:k via component k. Taking a compound containing four

44

Livio Battezzati

components in two sublattices with n = m =1 and assuming a sublattice contains the A and B components and the other contains the C and D components, the simpler form for the excess terms is written as: G ex = y1A y1B L0A, B + yC2 y D2 L0C , D

(14)

This is equivalent to having a regular solution on both sublattices. A term for a regular solution between sublattices will be written as yA1yB1yC2 LA,B:C. More extended interactions will be given the form of RedlichKister polynomia. The reference state is the average of the free energy of binary compounds, weighted on the mole fraction of components in the respective sublattice, having the same structure as AnBm, the so called end members. For the case taken above (A, B)(C, D) it is given by ref

0 0 0 0 G 0 = y A yC G AC + y B yC GBC + y A y D G AD + y B y D GBD ..

(15)

As above, these free energies, as well as the interaction parameters, must be optimised by fitting to experimental data on thermodynamic quantities or phase diagram points. An example of such calculation is shown in Fig. 10. 1400 1400

T/K

T/K

1200 1200

liquid

1000 1000 800 800 600 600 400 400 0 0.0

Cu

Cu2Mg 0.2

0.4

CuMg2 0.6

x(Mg) x(Mg)

0.8

1 1.0

Mg

Fig. 10. The assessed Cu-Mg phase diagram. Parameters taken from7.


45

Free energy /J mol

-1

The Cu-Mg system is made of three eutectics involving the terminal solid solutions and two intermetallic compounds. CuMg2 is stoichiometric and described as a line compound. Cu2Mg can deviate from stoichiometry at high temperature and is described with two sublattices. The resulting free energy curve extends from the free energy of the end members to the minimum (Fig. 11). Common tangents to the neighbouring phases will touch the curve close to but not on the minimum. -20.0k -25.0k -30.0k

Mg hcp fcc

liquid

-35.0k -40.0k 0.0

Cu2Mg

0.2

0.4 0.6 XMg

CuMg2

0.8

1.0

Fig. 11. Free energy curves and values for phases in the Cu-Mg system at 700 K. Parameters taken from7.

6. Metastable phases and phase transformations

Thermodynamics deals with system at equilibrium. However, in various instances metastable phases and microstructures can be discussed in thermodynamic terms as long as they remain in the same state long enough to perform measurements of specific heat capacity or they transform to another well defined state with a measurable enthalpy change. In same cases thermodynamics is applied locally to a part of the system where a change in free energy occurs. A few examples of such applications will be provided in the following. The extension of solid solubility, frequent in processes involving quenching, can be explained

46

Livio Battezzati

by making use of the T0 concept8. The T0 line which can be superimposed to a phase diagram, is the locus of temperaturecomposition points at which the free energy of two phases is equal. With reference to the free energy scheme of Fig. 5, the relevant composition is recognized at the crossing of the two curves. This is not an equilibrium state since the minimum of free energy for this composition lies on the common tangent. Examples of T0 curves referring to a liquid and two crystalline phases are reported in Fig. 12. For every composition they represent the temperatures at which the liquid phase could transform to a solid without partition of composition. The T0 temperature can be reached if the liquid is undercooled bypassing the equilibrium solidification.

Fig. 12. A eutectic phase diagram (full lines) with superposition of T0 curves (dashed lines), i.e. the loci of composition-temperature points at which either the α or the β free energy equal that of the liquid.

Quenching can be performed not only from the liquid but, perhaps more commonly, from the solid state. In such processes, a high temperature phase is frozen at low temperature where it should not exist. The free energy scheme of Fig. 13 illustrates this event. At high temperature a homogeneous solution phase is stable at all compositions. After quenching it remains homogeneous and its free energy can lie in between the minima marking the condition for phase separation and also


47

within the spinodal points where the instability condition for the solution d2G < 0 is met. Any fluctuation in composition inside the solution would produce a decrease in free energy with the consequence that a spatially continuous decomposition of the solution can occur. The dashed line in the figure shows that the formation of a mixture of two arbitrary solutions having composition on both sides of the homogeneous one implies lowering of the free energy of the system. This is the starting point for describing the precipitation mechanism known as spinodal decomposition. 2000 Free energy /Jmol

-1

low T

0

spinodal points

-2000

-4000 0.0

high T 0.2

0.4

0.6

0.8

1.0

Mole fraction B Fig. 13. Free energy curves for a solid solution at high temperature (full miscibility) and low temperature (inside the miscibility gap). The curves are taken from Fig. 4. The dashed line indicates the free energy of the solution which has decomposed to some extent with a composition fluctuation.

Further, an example of more complex system will be provided with the aim of showing the procedure for drawing metastable phase diagrams. Fig. 14a reports the latest version of the Al-rich corner of the Al-Mn phase diagram9: the phases are solid and liquid solutions, 4 intermetallic compounds. All of the intermetallics, represented as line compounds, decompose peritectically on heating. In processes such as rapid solidification the compounds may not form and the system would be described by a metastable phase diagram. Ion terms of the calculation calculation of phase diagrams this means suspending a phase from the calculation and derive the equilibria as detailed above. Suspending the λ

48

Livio Battezzati

phase, the phase diagram of Fig. 14b results. The range of existence of Most phase transformations start with a nucleation event. The classical theory of nucleation evaluates the stability of a system containing a cluster of the new phase via the change in free energy, ΔGn, as a function of cluster volume, Vn and interfacial area, An: 1300 1300

T/K

T/K

1200 1200

liquid

μ

1100 1100

Al6Mn

1000 1000 900 900 800 800 700 700 0 0.00

Al

Al12Mn 0.06 0.06

0.12 0.12

0.18 0.18

0.24 0.24

x(Mn) x(Mn)

0.3 0.30

1300 1300 1200 1200

liquid λ

TT // K K

1100 1100

Al6Mn

1000 1000 900 900 800 800 700 700 0 0.00

Al

μ

Al12Mn 0.06

0.12

0.18

x(Mn) x(Mn)

0.24

0.3 0.30

Fig. 14a, upper panel: the assessed Al-Mn phase diagram. Parameters taken from9. Fig. 14b, lower panel: the metastable Al-Mn phase diagram where the λ phase has been suspended from calculation.

Al6Mn is extended although it still melts with a peritectic reaction.

ΔGn = Vn ΔGv + Anσ

(16)


49

The driving force for the formation of a nucleus of critical size which will eventually grow, is the difference in free energy between the nucleus and matrix phases, ΔGv whereas the process is adversely affected by the interfacial energy term, σ. In a transformation involving compositional partition, such as precipitation or primary solidification, the composition of the most probable nucleus will correspond to the maximum free energy gain in the process. This is determined by means of the parallel tangent construction as shown in Fig. 15. for solidification.

Free energy

crystal at liquidus temperature

crystal at nucleation temperature liquid at liquidus temperature

Alloy composition

mol fraction B Fig. 15. Free energy scheme for solidification of a crystal phase derived from curves in Fig. 6, illustrating the parallel tangent construction and the composition of the more likely nucleus differing from that of the equilibrium crystal (arrows).

At the equilibrium liquidus temperature the chemical potential of elements is provided by the common tangent construction. However, at this temperature there is no free energy available for the system to be spent as work of formation of the interface between new and old phases. Such free energy becomes available on undercooling the matrix. At the temperature were nucleation occurs, the relative position of the liquid and crystal phases will be modified as drawn in Fig. 15 with the liquid being metastable. The maximum change in chemical potential of elements is obtained by drawing the tangent to the free energy curve of the crystal phase parallel to the tangent to that of the liquid phase. The composition of the nucleus will differ from that expected by merely

50

Livio Battezzati

considering the liquid-crystal equilibrium. When the small nucleus of radius r forms, the overall composition of the matrix will not change appreciably; they will coexist through an unstable equilibrium established locally at the crystal liquid interface and defined by dGn =0 dr

(17)

The quantity ΔGv can be derived from the optimisation of the phase diagram for all phases allowing the prediction of phase selection by nucleation10 whereas the common tangent construction will show the driving force available for growth of the crystal after nucleation. References 1. D. A. Porter and K. E. Easterling, chap. 1 in Phase Transformations in Metals and Alloys, Chapman & Hall, London (1992). 2. D. R. Gaskell, Introduction to Metallurgical Thermodynamics, McGraw & Hill, New York (1973). 3. M. C. H. P. Lupis, Chemical Thermodynamics of Materials, PTR Prentice Hall, Englewood Cliffs, New Jersey (1983). 4. M. Hillert, Phase Equilibria, Phase Diagrams and Phase Transformations, their Thermodynamic Basis, Cambridge University Press, Cambridge (1998). 5. N. Saunders and A. P. Miodownik, CALPHAD Calculation of Phase Diagrams, A comprehensive Guide, Pergamon, Oxford (1998). 6. A. T. Dinsdale, in Calphad.Computer Coupling of Phase Diagrams and Thermochemistry, 15, 317 (1991). 7. P. Jiang et al., in Calphad.Computer Coupling of Phase Diagrams and Thermochemistry, 22, 527 (1998). 8. W. J. Boettinger, J. H. Perepezko, chap. 2 in Rapidly Solidified Alloys, Eds. H. H. Liebermann. and M. Dekker, New York (1993). 9. COST 507, Definition of thermochemical and thermophysical properties to provide a database for the development of new light alloys, Thermochemical database for light metal alloys, I Ansara., A. T. Dinsdale, and M. H. Rand Eds., European Communities, vol. 2, Bruxelles (1998). 10. L. Battezzati and A. Castellero, in Materials Science Foundation, Eds M Magini. and F. H. Wöhlbier., Trans. Tech. Publications Inc., Zurich, vol. 15 (2002).

CHAPTER 3

PERMANENT MAGNETS AND MICROSTRUCTURE Paul McGuiness Jožef Stefan Institute, Ljubljana, Slovenia E-mail: [email protected] Permanent magnets are vital components in many types of technology, from PCs and iPods to electric motors and generators. The strength of a permanent magnet depends to a large extent on the elements that comprise the magnetic material, and on the microstructure – the grain size, the grain shape, the distribution of phases, the occurrence of precipitates, the intergranular phases, etc. – that results from the way the material is processed. The most powerful magnets available today are based on rare earths and transition metals. These materials are not only interesting because of their excellent magnetic properties; they are also very interesting from the processing point of view, because a wide range of different techniques can be used to produce them. In this chapter we will look mainly at Nd–Fe–B-type magnets and how processing them with various techniques, like powdering and sintering, meltspinning, and hydrogen disproportionation, can be used to produce microstructures leading to the optimum magnetic properties for a particular application.

1. Historical introduction to permanent magnets Permanent magnets have been known since the discovery of the lodestone by the ancient Greeks. The first application, the compass, was invented in China; the earliest recorded use of lodestone as a direction finder was in a 4th-century Chinese book: Book of the Devil Valley Master. However, a systematic study was not carried out until William Gilbert (1544-1603) wrote his famous book De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure (On the Magnet and Magnetic body, and on That Great Magnet the Earth), which was published in 1600. From his experiments, he concluded that the Earth was itself 51

52

Paul McGuiness

magnetic and that this was the reason compasses pointed north (previously, some believed that it was the pole star (Polaris) or a large magnetic island on the north pole that attracted the compass). Before the invention of the electromagnet by Sturgeon in 1925 the only permanent-magnet materials were the naturally occurring lodestone, a form of magnetitie Fe3O4, and various forms of iron-carbon alloys. Many of these early magnets were built up from wires or strips, since these were easily magnetised by stroking them with another magnet. The development of modern permanent magnets can be said to have begun around the end of the 19th century, with the introduction of the steel magnet. This was improved upon in about 1900 by using tungsten steel as a starting material. The first substantial improvement came with the appearance of Honda steel, in which about 35% of the iron in the FeW-C steel was replaced by cobalt. However, owing to the high price of cobalt compared to iron there were very few applications for this material. The introduction of MK steel (an alloy of Fe, Ni, Co and Al) by Mishima was of considerable significance: not only did it have much better magnetic properties than Honda steel, it was also considerably cheaper. MK steel can be considered as the forerunner of Ticonal II, which was developed in 1936. This led to the development of an anisotropic form of the same material, called Triconal III, produced by annealing in a magnetic field. The next step forward in the story of permanent magnets was made when it was realised that larger values of magnetic anisotropy were needed to produce higher coercivities. The coercivity of a permanent magnet is its ability to withstand the effects of an opposing magnetic field. High anisotropies where found in materials that had highly anisotropic crystal structures and hexagonal or tetragonal symmetries. A good example of a material with this type of structure is the common, household ferrite magnet. This is the type of magnet you will often find stuck to the door of a fridge, but is also the workhorse magnet for thousands of industrial, automotive and domestic applications. Ferrites are oxide materials with the general formula M(Fe2O3)6, where M is one or more of the divalent metals barium, strontium or lead. Ferrites have a relatively low magnetisation, but their high coercivity and low

Permanent Magnets and Microstructure

53

price mean that they dominate the market for permanent magnets, at least in terms of tonnage, even today. There were few developments in permanent magnets until the 1960s. The second world war had seen advances in the separation and purification of rare earths, and with these metals now available researchers began to look at combinations of rare earths and transition metals. 1967 saw the first reports of RCo5 (R = rare earth) materials with CaCu5-type structures, which soon led to the commercial availablilty of SmCo5 magnets with properties that literally dwarfed those of ferrites, AlNiCos and the steel magnets that had come before – albeit at a price. Within a short time it was realised that the magnetic properties of these SmCo5 magnets were being limited by the magnetisation of the cobalt sub-lattice, and so a new type of magnet, based on R2Co17, quickly followed. These Sm–Co magnets made possible a wide range of new applications and presented tremendous possibilities for miniaturisation because of their enormous energy densities, but the high – and perhaps more importantly, the variable – price of cobalt was a problem. This situation became even worse with the cobalt crisis of 1979–81. The crisis was due to a rebellion in Zaire, source of about half of the world’s cobalt, when many of the mines were flooded. The price of cobalt increased six fold as a result, and this intensified the search for high-energy cobalt-free permanent magnet materials. The first announcements of the successful production of magnets based on neodymium, iron and boron were made at a meeting in Pittsburgh, PA, in 1983. At the same meeting there were reports from Sumitomo Special Metals of Japan and General Motors of the US of a new generation of permanent magnets based on a material with the chemical formula Nd2Fe14B. This Nd–Fe–B-type was an improvement in many ways over the existing Sm–Co materials, and could be produced using a number of different techniques. The Japanese had produced their Nd–Fe–B magnets via a relatively conventional powder-metallurgy sintering route, whereas the Americans had used a novel method called melt spinning. Within a few years there would more reports of good magnetic properties from groups working with techniques such as mechanical alloying, screen printing, sputtering, ablation and techniques based on hydriding.

54

Paul McGuiness

One more major discovery has been made since the arrival of Nd–Fe– B, and that occurred in 1990, when a group in Ireland reported hard magnetic properties in nitrided Sm–Fe-based materials. The Sm2Fe17N3 magnet had excellent properties, comparable in many ways to the market-leading Nd–Fe–B-based magnets, but difficulties associated with nitriding bulk samples has kept them from anything other than a niche market. A summary of the improvements made in permanent-magnet materials over the past 100 years can be seen in Fig. 1.

Fig. 1. Progress in permanent magnets, in terms of energy product.

In this chapter we will look only at the example of Nd–Fe–B permanent magnets, showing how it is possible to prepare magnets using a number of different techniques, so producing a wide range of microstructures that result in magnets with a variety of magnetic properties. 2. Permanent-magnet properties In order to describe a permanent magnet quantitatively we need to measure its magnetic properties. To do this we subject the magnet to a large positive magnetic field, to saturate the magnet, then we apply a large negative field in order to assess it ability to withstand a reverse magnetic field. This form of measurement is shown schematically in Fig. 2 and described in more detail below.


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Fig. 2. Hysteresis loop of a permanent magnet.

The measurement of the permanent magnet’s properties begins with a completely demagnetised magnet in a zero magnetic field at the crossing point of the x axis (the applied field) and the y axis (the magnetisation of the sample). The state of the magnet is illustrated by the empty rectangle, i.e., the magnet is unmagnetised. The first part of the measurement involves applying a large positive magnet field (+H). At this point the magnet becomes fully saturated (the red arrow) while it exists in a large positive field. The next stage is to remove the applied field and look at the magnetisation state of the magnet while there is no external field. With a good-quality magnet the internal magnetisation (the red arrow) will remain, even in the absence of the applied field. This point on the y axis, Br, is referred to as the remanence.. This is followed by a demagnetisation stage, where a negative field (-H) is applied to the sample. With a sufficiently high field the magnetisation of the sample will be reduced to nothing (no red arrow), and this field is referred to as the intrinsic coercivity, Hci, of the magnet. For most permanent-magnet applications we are looking for magnets with a high remanence and a high coercivity, although there are some applications when a very high coercivity would be disadvantageous. There is one other point that is important from the applications point of view, the normal coercivity, Hcb. This is the point where the external demagnetising field is equal

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and opposite to the internal magnetisation; the system, in effect, is equal to zero. This measurement is of more interest to electrical engineers than to material scientists, who are more interested in the material’s performance than the performance of the system. In order that we can have a single quantity to describe the quality of a magnet, the term energy product has been introduced. To calculate the energy product of a magnet we draw a straight line from the remanence point to the normal coercivity and then measure the area of the largest rectangle that we can fit under this line. The best Nd–Fe–B magnets have energy products in excess of 50 MGOe (mega gauss oersted), typical Sm–Co magnets are in the range 25–30 MGOe, and ferrites are about 4 MGOe. 3. Some applications of permanent magnets Permanent-magnet applications can be divided into four distinct groups: • Applications that make use of the attractive force that a magnet can have on a soft magnetic material, like magnetic particles in a slurry, or the repulsion between to permanent magnets, like in the case of magnetic bearings. • Applications that use the magnet’s magnetic field to convert mechanical energy into electrical energy. Examples include generators and alternators. • Applications that use the magnet’s magnetic field to convert electrical energy into mechanical energy. This is by far the largest category of applications, and includes all kinds of motors, meters, actuators and loudspeakers. • Applications that use the magnet’s magnetic field to control electron beams. The most obvious application here is the cathode-ray tube, but there are also a lot of magnets used in wave tubes, wigglers and cyclotrons. A particularly good example of a product that uses a lot of magnets is the car. Apart from the obvious applications like the starter motor, the alternator, the windscreen wipers and the loudspeakers for the radio, there is an enormous number of magnets in applications like door locks, ABS systems, fuel and water pumps, electric windows, seat-position actuators (as many as 50 magnets per seat) aerial motors, ignition


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systems, crankshaft-positioning, air-conditioning, cruise control, CD player, speedometer/tachometer, and tens if not hundreds of sensors. There are many cars on the road today that contain more than 1000 permanent magnets. 4. The crystal structure of Nd2Fe14B The first crystal-structure determination of Nd2Fe14B was reported in 1984. The structure is relatively complex, and there are 68 atoms in the unit cell. The tetragonal structure belongs to the space group P42/mnm; it comprises six crystallographically inequivalent Fe sites and two crystallographically inequivalnet Nd sites. The homogeneity range of Nd2Fe14B is very small, or even absent; it is effectively a line compound. A schematic representation of the crystal structure of Nd2Fe14B is shown in Fig. 3.

Fig. 3. The crystal structure of Nd2Fe14B.

5. Phase relationships in the Nd–Fe–B system The Nd–Fe–B system is characterised by three ternary compounds: Nd2Fe14B (sometimes called the φ phase), Nd1+εFe4B4 (sometimes called the η phase or the boride pahse), and Nd2FeB3; however the last of these

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does not figure in the compositions from which permanent magnets are fabricated. Fig. 4 shows a pseudobinary-type vertical section of the Nd– Fe–B phase diagram, with 100% Fe on the left-hand side and Nd/B = 2 on the other.

Fig. 4. Pseudobinary-type vertical section through the Nd–Fe–B phase diagram.

This section shows clearly the peritectic reaction L + Fe --> φ at 1180ºC. From the metallurgical point of view this is a critical reaction. In effect it is saying that you cannot melt an alloy with the composition Nd2Fe14B and hope to get a single-phase solid. Unless you go to extraordinary lengths, such a sample will always consist of islands of peritectically formed Fe surrounded by Nd2Fe14B and a liquid relatively rich in Nd. In order to avoid the formation of any peritectic iron it is necessary to be cooling down on the right-hand side of the diagram so that the reaction sequence is L --> L + φ. For this reason the permanent magnets based on the Nd–Fe–B system usually have compositions richer in Nd than Nd2Fe14B, close to Nd2.6Fe13B1.4, or Nd15Fe77B8 as it is more commonly expressed, and the microstructures of solidified samples are normally composed of three phases: Nd2Fe14B (typically around 85%), Nd1+εFe4B4 (typically around 2–3%) and a phase that is very rich in neodymium, called the Nd-rich phase, which normally constitutes about 12–13% of the material.


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6. Coercivity and microstructure in Nd–Fe–B permanent magnets The origin of the coercivity in all rare-earth–transition-metal permanent magnets is their high easy-axis magnetocrystalline anisotropy. However, since the coercivity remains well below the value of the anisotropy field, by a factor of about four, the coercivity is clearly also very dependent on microstructure, with magnetisation reversal being the result of the nucleation and growth of reverse magnetic domains. Sintered Nd–Fe–B magnets show two main coercivity characteristics that suggest the coercivity mechanism is one of nucleation and growth rather than a pinning type mechanism: unmagnetised magnets have many domains per grain, and the coercivity increases with the size of the field used to magnetise the magnet. In order to generate high coercivities under such conditions it is important to produce materials with a small grain size, thereby limiting the surface areas of individual grains. This is in stark contrast to permanent magnet materials like the Sm2Co17-type magnets, where large coercivities can be generated in cast-and-annealed ingots with very large grains, and where the mechanism of the coercivity is related to the pinning of domains within the volume of the grains by precipitates. 7. Processing Nd–Fe–B permanent magnets Nd–Fe–B permanent magnets are multiphase metallic structures. Irrespective of the processing route employed to produce them, the starting point is nearly always an as-cast alloy with a composition in the region of Nd15Fe77B8. Depending on the casting conditions, in the as-cast state this material will exhibit the three phases mentioned above as well as, possibly, a small amount of dendritic iron. Although this ferromagnetically soft form of iron is potentially very detrimental to the permanent magnetic properties of the material, in particular the coercivity, the processing of the alloy into a permanent magnet has the effect of removing this iron, providing the dendrites are not too coarse. The aim when processing Nd–Fe–B magnets is twofold: • First, we need to reduce the grain size down to micron or submicron sizes, thereby maximizing the potential coercivity of the sample.

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• Second, we need to orient the grains as much as possible so that the c axes (which are the hard-magnetic axes) of the grains are pointing in the same direction, thereby maximizing the potential remanence of the sample. This is shown in Fig. 5, where Fig. 5a shows schematically a typical as-cast microstructure for a Nd–Fe–B alloy with a composition of Nd15Fe77B8, cast into a mould containing about 10–15 kg of material, and Fig. 5b shows schematically an idealised magnet microstructure with grains of less than 10 micrometers.

Fig. 5a (left panel): Schematic microstructure of Nd15Fe77B8 cast alloy. Fig. 5b, (right panel): Schematic microstructure of idealized permanent-magnet microstructure.

7.1. Processing Nd–Fe–B magnets via the sintering route The powder-metallurgy sintering route has for a long time been the main processing route for metals and alloys when shape and uniformity of properties are important. Sintering has also played a major role in permanent magnets, dating back to the earliest sintered AlNiCo and ferrite magnets of the mid-20th century. Sintering was also the processing route of choice for Sm–Co-type magnets too, because it made it possible to produce the extremely brittle Sm–Co magnets at close to net shape, and, more importantly, it made it possible to align the powder particles with the crystal c axes all in the same direction prior to sintering in order that the material can exhibit a high degree of remanence. The differences


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in the magnetic properties of an aligned sample (the anisotropic case) and an unaligned sample (the isotropic case) are shown in Figs. 6a and 6b.

Fig. 6a. Schematic diagram of an unaligned (isotropic) permanent magnet and the associated magnetic properties.

Fig. 6b. Schematic diagram of an aligned (anisotropic) permanent magnet and the associated magnetic properties.

Under a scanning electron microscope or an optical microscope the microstructures of these two types of magnets would look to a large extent the same, because it is not easy to tell the orientation of a grain of Nd2Fe14B simply from looking at a polished surface. However, a measurement of the magnetic properties would quickly reveal which sample was well aligned, because a well-oriented sample would exhibit a

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remanence not far short of the saturation magnetisation (Ms), whereas the non-aligned (isotropic) sample would have a remanence close to about half of the value of Ms. But before we look at aligning Nd–Fe–B powders, let us have a look at the first stage of the powder-metallurgy sintering process, casting. 7.1.1. Casting Nd–Fe–B-type sintered magnets usually begin as cast ingots produced from appropriate amounts of neodymium, iron and ferroboron. These materials are then induction melted at temperatures in the range 1400– 1500ºC in an inert atmosphere before being poured into thin, bookshaped moulds to provide fast cooling rates. Industrial castings range in size from about 10 to 100 kgs. 7.1.2. Powdering The next, and arguably the most critical, stage is producing the powder from the ingot. The target size for the powder is 2–5 microns, but this is very difficult to achieve in a single step. The usual procedure is to crush the ingots to centimetre-sized lumps, then hammer-mill these lumps to pieces a few millimetres in size, and then finally attritor-mill or jet-mill these pieces to the final size of a few microns. Of course due to the high reactivity of rare-earth-based powders all of these procedures must be carried out in a protective atmosphere. In the cases of crushing and hammering this atmosphere is usually nitrogen, but for jet-milling the gas used is normally argon. 7.1.3. Pressing and aligning The next stage after powdering is to press the powders into compacted pellets, usually referred to as green compacts, so that we can achieve the right shape of magnet, while at the same time aligning the individual powder particles so that the magnetic c axes are all in the same direction. Since the initial grain size of the cast alloy is in the range of tens or hundreds of microns, we can be reasonably sure that most of the powder


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particles consist of a single grain with a single c axis. The fields required depend on the size and shape of the magnet, but they are usually in the range of about 1 Tesla. Like with the powder production, the pressing and aligning procedures must be carried out in such a way as to avoid oxidation as much as possible. 7.1.4. Sintering Once enough green compacts have been produced they can be loaded into the sintering furnace. The individual green compacts are normally arranged on stainless-steel trays. In order to maximise the density of the resulting sintered magnets the green compacts are sintered in a vacuum. The sintering-temperature varies considerably depending on composition, particle size, required properties, but generally involves slow heating to the sintering temperature in the range 1050–1100ºC, followed by a hold at this temperature for 1–2 hours, with subsequent holds during cooling at 900–950ºC and 600–650ºC. The complete sintering and heat-treatment cycle could last 12 or more hours. 7.1.5. Machining, coating and magnetising The sintered samples usually need machining to meet dimensional requirements. Because of the very brittle nature and high reactivity of Nd–Fe–B magnets this normally involves centreless grinding using nonreactive fluids. Nd–Fe–B magnets also require coating, because of their sensitivity to the atmosphere, and the coatings are usually based on nickel. The final magnetisation sometimes takes place as part of the production process of the magnets; however, increasingly magnets are being magnetised after they are fixed to the assembly in the final product. Fig. 7 summarises the key steps in the production of sintered Nd–Fe– B permanent magnets.

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Fig. 7. The main stages in the processing of sintered Nd–Fe–B permanent magnets.

Microstructures of the as-cast Nd15Fe77B8 alloy and a sintered magnet are shown in Fig. 8. Fig. 8a is an optical micrograph; Fig. 8b was produced from a scanning electron microscope. Note the dramatic reduction in the grain size and the even distribution of phases, both of which are very important for the development of a high intrinsic coercivity. The sintered magnet is also highly oriented, although this is not visible in this type of micrograph.

Fig. 8a (left panel): Microstructure of as-cast Nd15Fe77B8 alloy. Fig. 8b, (right panel): Microstructure of sintered Nd15Fe77B8 magnet.


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7.2. Processing Nd–Fe–B magnets via the melt-spinning route The melt-spinning production route, unlike the powder-metallurgy sintering route, was not used for previous generations of permanent magnets. The combination of melt-spinning and Nd–Fe–B-type magnets was pioneered by General Motors in the US, a development that ran in parallel with Sumitomo Special Metals’ research on the sintered route for Nd–Fe–B. In the melt-spinning process a jet of molten alloy comes from material in an induction-melting crucible and hits a rapidly rotating water-cooled copper wheel. Under such conditions cooling rates can be as high as 106Ks-1. The Nd–Fe–B alloy tends to form in the shape of ribbons, a few centimetres long and about 30 microns thick, which are then thrown from the copper wheel and collected a metre or so away in a hopper. Like with the sintering process, everything is carried out in a protective, inert atmosphere. A schematic diagram of the melt-spinner and some crushed Nd–Fe–B ribbons are shown in Figs. 9a and 9b.

Fig. 9a, left panel: Schematic diagram of melt-spinner. Fig. 9b, right panel: Crushed melt-spun ribbons of Nd–Fe–B.

The usual procedure involves over-quenching the ribbons to produce a largely amorphous structure, and then heat treating them at 600–700ºC to produce Nd2Fe14B grains that are 0.5–1.0 microns in size. The grain size of melt-spun magnets tends to be smaller than with sintered magnets, but this is a consequence of many factors, not least of which is the radically different processing route. The most important feature of

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Nd–Fe–B melt-spun ribbons is that they are completely isotropic, i.e., there is no preferred orientation of the c axes in the material leading to relatively low values for the remanence. Nevertheless, the simplicity of the ribbon-production process, the intrinsic stability of the ribbons in the atmosphere and the ease with which these materials can be mixed with polymers and other binders and moulded into intricate shapes makes them a very attractive material. 7.2.1. Hot pressing melt-spun ribbons These Nd–Fe–B melt-spun ribbons are also suitable for hot pressing. In such a process the ribbon pieces are placed in a die and compacted under high loads and temperatures in the range 700–800ºC to form 100%-dense solids. Of course, since there is no possibility of aligning the grains in a magnetic field these materials are still isotropic, but their high density gives them an advantage over the polymer-bonded variants. 7.2.2. Die-upset forging of melt-spun ribbons A third possibility is to die-upset forge the compacted melt-spin ribbons. This rather expensive process involves first of all producing a straightforward 100%-dense solid from the ribbons as described above, and then repressing the dense compact in an over-sized die, so causing the material to flow in a direction perpendicular to the direction of pressing. The two hot pressing techniques are illustrated in Figs. 10a and 10b.

Fig. 10a (left panel). Hot pressing of melt-spun powder to produce a fully dense isotropic magnet and Fig. 10b (right panel): Die-upset forging of a fully dense isotropic magnet to produce an anisotropic magnet.

This flow under pressure at high temperatures causes the material to become highly oriented along the pressing direction. This reorientation is


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the result of the growth of favourably oriented grains in combination with grain-boundary sliding, boundary diffusion and diffusion slip. The process can be enhanced by small additions of elements such as gallium, although die-upsetting ratios of about four, i.e., the compact must be reduced to a quarter of its original height, are required. Microstructures of the hot-pressed melt-spun ribbon and the subsequently die-upset variant are shown in Figs. 11a and 11b.

Fig. 11a (left panel): Microstructure of hot-pressed Nd–Fe–B melt-spun ribbon. Fig. 11b (right panel): Microstructure of subsequently die-upset forged Nd–Fe–B material.

7.2.3. Processing Nd–Fe–B magnets via the hydrogenationdisproportionation-desorption-recombination route A third method for producing permanent magnets from a starting material of as-cast Nd15Fe77B8 alloy is called the hydrogenationdisproportionation-desorption-recombination process, or HDDR process, for short. The process involves heating the as-cast Nd15Fe77B8 alloy in an atmosphere of hydrogen to about 700–750ºC, holding for a period of minutes to hours, and then cooling the material to room temperature in a vacuum. During the first stage of the process the material reacts with the hydrogen to form interstial hydrides and a lot of cracks form in the material due to the expansion of the crystal lattice with the formation of the hydrides. However, as the temperature increases these hydrides become unstable and the material disproportionates to form iron,

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ferroboron and neodymium hydride. The disproportionation reaction can be represented as: Nd2Fe14BHx 2HdHx/2 + 12Fe + Fe2B + ΔH

(1)

where ΔH is the heat of reaction. The value of x is dependent on the hydrogen pressure used and the exact composition of the starting alloy. The disproportionated mixture consists of a very finely divided mixture of iron, neodymium hydride and ferroboron. During the second stage of the process, when the material is subjected to vacuum conditions at high temperature the neodymium hydride desorbs to form neodymium metal, which then leads to the neodymium, iron and ferroboron recombining to form large amounts of Nd2Fe14B phase, together with some Nd–Fe intergranular material, only now the grain size of the Nd2Fe14B phase is in the 0.1–1.0-micron range. In simple terms the HDDR process converts a coarse-grained as-cast material into a very fine-grained powdered material via a reversible chemical reaction involving hydrogen. The process is shown schematically in Fig. 12.

Fig. 12. Schematic representation of the hydrogenation-disproportionation-desorptionrecombination (HDDR) process.

Like with melt-spinning the process is intrinsically isotropic, and the resulting HDDR powders tend to have remanences close to about half of


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the saturation magnetisation; however, in the case of the HDDR process it is possible to produce anisotropic material with the use of additives like zirconium, hafnium and gallium, and closely controlled processing conditions. HDDR-processed powder is also very suitable for hot pressing fully dense magnets in a similar to the procedure used for the melt-spun ribbons. 7.2.4. Other processing techniques Nd–Fe–B materials are remarkable in many ways, but perhaps their most remarkable characteristic is the number of different methods that can be used to produce high-coercivity permanent magnets from basically the same starting material. From the commercial point of view the three techniques already discussed – sintering, melt-spinning and HDDR – are the most important, but from the research perspective techniques like rapid casting, hot working, mechanical alloying, laser ablation, pulsedlaser deposition, rotary forging, gas atomisation and explosive compaction have provided valuable insights into the capabilities and limitations of the Nd–Fe–B system. 8. Magnetic properties The magnetic properties obtainable with Nd–Fe–B materials are in general higher than those available with other magnetic materials. In some situations Sm–Co materials might be a better choice because of a need to operate at high temperatures, or AlNiCo magnets, when temperature stability is required, but for most modern high-technology applications where cost is not the only factor, Nd–Fe–B magnets are the magnets of choice. Fig. 13 shows typical properties obtainable for Nd– Fe–B magnets produced by the three techniques described here. The highest remanences and energy products are the result of processing the material with a powder-metallurgy sintering route. Over the past 20 years improvements in laboratory and industrial processing have resulted in remanence increases from about 1.2 Tesla to more than 1.3 Tesla; this means that magnets with energy products in excess of 50 MGOe are available. Another magnet with a high remanence and energy

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product is the anisotropic die-upset-forged melt-spun magnet. Although the remanences are not quite as high as for sintered material, the magnets can be made net shape and can be considered as more than niche products. Anisotropic HDDR bonded magnets are still increasing their market share and are the only option when a 1 Tesla+ bonded magnet is required for an application. The processing of HDDR material is relatively high, comparable to the various processes associated with the melt-spinning option, and so HDDR magnets will never be in a position to displace sintered Nd–Fe–B from most applications.

Fig. 13. The magnetic properties of Nd–Fe–B magnets processed using a variety of techniques.

Lower down the energy-product scale we have isotropic hot-pressed melt-spun materials, which can be produced with very high coercivities, and isotropic HDDR and melt-spun bonded magnets which can be easily produced in large numbers through processes like injection moulding and extrusion.


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9. Summary This has been a brief overview of the processing of Nd–Fe–B magnets and the relationships between processing, microstructure and magnetic properties. We have looked at just one composition – the Nd15Fe77B8 ternary compound – and three types of processing – sintering, meltspinning and the HDDR process – and found that each of these techniques is able to produce high-quality magnets with useful properties for many applications, providing the right kind of microstructure is obtained. There are, of course, many other magnetic materials, some of which were mentioned in the text, that have other interesting processing– microstructure–properties relationships; however, that would require space than a single chapter. Further reading There is an enormous body of literature on permanent magnets, properties, and microstructures. Below are listed some widely available books that will provide you with plenty of information on all aspects of magnetism and magnetic materials. Permanent-magnet Materials and Their Applications. K.H.J. Buschow. Trans Tech Publications (ISBN: 087849796X) Permanent Magnet Materials and Their Application. Peter Campbell. Cambridge University Press. (SBN: 0521566886) Introduction to Magnetism and Magnetic Materials. David C. Jiles. CRC Press Inc. (ISBN: 0412798603) Rare-Earth Iron Permanent Magnets. Ed. J.M.D. Coey. Clarendon Press. (ISBN 0198517920) Hidden Attraction. Gerrit L. Verschuur. Oxford University Press. (ISBN 0195106555) Driving Force. James D. Livingston. Harvard University Press. (ISBN 0674216458) Modern Magnetic Materials: Principles and Applications. Robert C. O’Handley. John Wiley & Sons. (ISBN 0471155667)


CHAPTER 4

SOLIDIFICATION Peter Gille Crystallography Section, Department of Earth and Environmental Sciences, Ludwig-Maximilians-Universität München, Theresienstrasse 41, D-80333 München, Germany E-mail: [email protected] Solidification of a metallic melt is basic to various technological processes like ingot casting, directional freezing of composite alloys, single-crystal growth and rapid solidification of metallic glasses. Apart from varying scientific or industrial goals and significant technical differences in these areas of application, most of the fundamental problems are common to these fields. Starting from slight deviations from equilibrium thermodynamics, various aspects of the transformation process of a melt to the solid state are treated in this tutorial chapter: homogeneous and heterogeneous nucleation, kinetic aspects of crystal growth, segregation phenomena, and interface instability caused by constitutional supercooling. An understanding of the mechanisms of solidification and how they influence practical processes and alloy properties are the main objectives rather than a complete treatment of all solidification techniques. A short overview of the most important methods of bulk crystal growth from the melt is given.

1. Introduction Solidification means any process transferring a fluid phase into the solid state. In a narrower sense it is understood as crystallizing a liquid caused by lowering its temperature below the melting point or its liquidus temperature. In this chapter, basic principles of solidification are treated that should be considered when crystallizing a binary or highercomponent melt. Solidification of well-defined samples or even single crystals may be regarded as the goal of these processes. The same

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principles, problems and equations are also fundamental to several other technical disciplines of solidification, like casting or welding. Often it is only the quantity of some parameter that makes the main difference between these applications: the extent of the deviation from equilibrium, the solidification rate or simply the amount of the melt to be frozen. 2. Thermodynamics and nucleation Thermodynamics gives the expressions how Gibbs free energy G of all phases that may occur changes with temperature. At each temperature T, the phase with the lowest content of Gibbs free energy is expected to exist. In Fig. 1 the principle course of Gibbs free energy curves of a solid and its melt in a one-component system is pictured.

Fig. 1. Volume free energy for a pure component as a function of temperature for solid and liquid phases. Dashed lines represent branches of the curves that are unfavourable with respect to energy minimization.

From equilibrium thermodynamics, lowering the temperature of a melt below its melting point should immediately start the process of solidification with a driving force ΔG that is proportional to the undercooling Tm − T :

Solidification

ΔG =

ΔH m (Tm − T ) Tm

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(1)

with ΔHm being the latent heat of crystallization. But, this is only true for a specific volume of the melt that can reach a lower content of Gibbs free energy by becoming solidified. If the influence of the interface that separates the solid phase to be formed from the mother liquid is regarded as well, the total amount of Gibbs free energy might actually be even higher. This would prevent the occurrence of the new phase even at temperatures lower than the melting point. Since the influence of the surface strongly depends on the size of such a particle, these effects are only restricted to very small scales, typically in the nm-region. But, it is just this tiny size a new phase has to start with before it may form a large-scale crystal.

Fig. 2. Free energy change associated with homogeneous nucleation of a sphere of radius r.

In the nucleation theory usually spherical particles of the new phase, i.e. the crystalline phase of a radius r, are assumed to form in the fluid phase with a specific solid/liquid interface energy γ. The change in the total Gibbs free energy of the system is then composed of the volume-

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depending part being proportional to r3 and the surface-depending term with a r2-function:

4 ΔH m ΔG = − π r 3 ΔT + 4π r 2γ 3 Tm

(2)

that is pictured in Fig. 2. Once a particle as large as r* that is called nucleus has formed by some fluctuation, crystallization will spontaneously proceed accompanied by a lowering of the total Gibbs free energy of the system. The radius of such a nucleus can be derived from Eq. 2 to be: r* =

2Tmγ ΔH m ΔT

(3)

with an activation energy for nucleation that amounts to: ΔG* = ΔG (r = r*) =

16 Tm 2γ 3 π 3 ΔH m 2 Δ T 2

(4)

What is most important is the reverse proportionality to the degree of undercooling ΔT in these two equations. Thus, with a very low deviation from equilibrium, nucleation has almost no chance to occur and the nucleation rate is practically zero. If nucleation does not happen in the above mentioned homogeneous way but in contact with a third phase (ampoule wall, foreign particle etc.) that may act as a substrate, the process is called heterogeneous nucleation and the leading equations stay formally almost unchanged. It is a geometrical factor that includes the interface energies between the three different phases under consideration and will change the onset of nucleation to smaller nuclei and lower nucleation energies at the same amount of undercooling. Therefore, whenever foreign phases exist that are wetted by the melt, heterogeneous instead of homogeneous nucleation will occur because of its exponentially higher nucleation rate. This is schematically pictured in Fig. 3 assuming some detection limit for the nucleation rate, expressed by the number of nuclei N that may have formed within some time and in a given volume. The negligible chance of nucleus formation with small deviations from equilibrium is

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sometimes a problem during first solidification of a new phase. But on the other hand, it is of great help in single crystal preparation when the growth of the solid phase without the risk of a parasitic grain formation is the intention.

Fig. 3. Variation of the free energy of nucleation ΔG* with undercooling ΔT for homogeneous and heterogeneous nucleation (solid lines) together with the corresponding nucleation rates (dashed lines).

3. Growth kinetics

Once the solid phase exists, either formed by spontaneous nucleation or given by seeding, solidification may be regarded simply as the growth of the solid phase by the movement of the solid/liquid interface. Kinetically, this can be described as a 3-dimensional periodic attachment of specific building units. A very simple but powerful model has been suggested by Kossel1 describing the crystal as being constituted of cubic units and crystal growth as a process of the periodic arrangement of these cubes (see Fig. 4). For the sake of this chapter I will stay with these simple building units, but do emphasize that nothing is said about the nature of

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these cubes, and even their shape is not really important for this basic introduction. The elementary units may be single atoms or ions, molecules or even large clusters of atoms that have already formed in the melt prior to the interface attachment. According to Kossel’s model, the attachment energy of a new cube that enters the crystal will be different for the various sites pictured in Fig. 4. While for a single building unit attached not to a step but to the extended surface the gain in the free energy is the lowest, the kink position with three of six cubic faces to be attached to the already existing crystal is by far the best position. Apart from the highest gain in free energy that is obtained from the attachment at a kink position, it is the repeatability of this step that makes the kink position deciding for crystal growth kinetics. All other alternative sites pictured in Fig. 4 would qualitatively change the surface. But, after having occupied the kink position, a new kink has been formed for further growth.

Fig. 4. Surface of a simple cubic crystal (“Kossel’s crystal”) where the surface atoms or building units have various numbers of nearest neighbours in the crystal depending on the site occupied: (1) vacancy, (2) jag, (3) kink, (4) step site, (5) adsorbed unit.

Since the number of kink sites at a crystalline surface is so important for growth kinetics, the question arises which parameters do determine the atomistic state of a crystal surface. This problem has been first

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treated by Jackson2 for a simple one-layer boundary between the solid and the fluid phase similar to Kossel’s model. An atomically smooth interface with no empty site or additionally attached building blocks is the optimum with respect to the total enthalpy of the surface. But, at temperatures T > 0 K the entropic contribution to the free energy gain has to be regarded as well and the question arises whether or not an atomically rough interface is preferred that consists of many empty positions. The various states of surface roughness can be described by the occupation factor c of surface sites being the ratio of the number of occupied positions to the total number N of sites at the surface. The calculation of the free energy ΔF of the contribution of a specific surface (hkl) can be simply obtained by counting the number of dangling bonds within the boundary layer and using the statistic formula for the surface entropy

ϕ ⎡ ⎤ ΔF = NkT ⎢c (1 − c ) ξ hkl + c ln c + (1 − c ) ln (1 − c ) ⎥ kT ⎣ ⎦

(5)

with k being Boltzmann’s constant, ϕ being the latent heat of crystallization per building unit and ξhkl the anisotropic factor giving the ratio of bonds within the surface (hkl) compared to all bonds of a unit. The normalized plot of Eq. 5 is given in Fig. 5. The parameter α of the various curves stands for:

α=

ϕ kT

ξ hkl

(6)

From Fig. 5 it is clearly seen that the parameter α well divides the free energy curves into those with a local minimum at c = 0.5, i.e. with a preference to atomically rough surfaces (α < 2), and those (α > 2) having the local minimum of the symmetric curves next to c = 0 or c = 1 which means almost atomically smooth surfaces. Therefore, Jackson’s factor α is well suited to describe the tendency of a crystalline surface to become atomically smooth or rough. A high latent heat of crystallization, a low phase transition temperature and/or a dense-packed crystalline interface increase the tendency to form an atomically smooth surface. On the other hand, with

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a low Jackson factor the crystalline interface may be assumed as consisting of a huge number of kink sites that allow the easy attachment of the crystallizing building units. In these cases, the rate of crystallization will be proportional to the deviation from equilibrium, i.e. to the undercooling ΔT. Therefore, a linear dependence of the growth rate v will be observed.

Fig. 5. Plot of the relative free energy ΔF NkT as a function of surface occupation for various values of Jackson’s factor α.

With an atomically smooth surface, the question arises how crystal growth can kinetically take place. It is obvious that a new layer of elementary building blocks has to start with a very first block, that will

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create quite a large new interface but result only in a little gain in binding energy. This first step is similar to the above mentioned nucleation problem but reduced to a 2-dimensional nucleus of only elementary step height. Consequently, the conditions to form a stable 2-dimensional nucleus at the top layer of the crystal can be expressed by similar formulae like Eqs. 2 – 4. Once the nucleus of the new layer has been formed, it can be assumed that lateral growth of the new layer will immediately occur and soon be completed since the undercooling that has been accumulated for nucleation is high enough for step growth. Therefore, 2-dimensional nucleation is the limiting step and the growth rate of the crystal can be calculated from Boltzmann statistics using the activation energy that is required to start a new layer. It results in an exponential growth law: ⎛ A ⎞ v ∝ exp ⎜ − ⎟ ⎝ ΔT ⎠

(7)

with A being a constant factor coming from the specific surface. Burton et al.3 have suggested that a screw dislocation intersecting a growth interface can provide a continuous step on a surface for growth. Such a surface step winds up into an Archimedean spiral that leads to never ending steps. High dislocation densities are very common defects of crystals in almost all solidification processes, and spiral growth is therefore very likely to occur at smooth interfaces. The step edges overcome the problem of the 2-dimensional nucleation, can easily laterally proceed and result in a square law for the growth rate, v ∝ ΔT 2 . The three different kinds of how the growth rate of a crystal may depend on the degree of undercooling are summarized in Fig. 6. Of course, even with a parabolic or an exponential dependence in the case of spiral growth or 2-dimensional nucleation, respectively, the growth rate can never exceed that one of the continuous growth. With a very high supercooling, the density of steps and kinks at a formerly smooth interface will become as high as in the case of an atomically rough one, and growth will proceed in a continuous way.

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Fig. 6. Influence of interface undercooling ΔT on the growth rate for atomically rough and smooth interfaces.

4. Phase diagrams

Phase diagrams are very fundamental to the understanding of many effects and problems that occur in solidification. Whenever a melt is to be solidified that consists of more than one component some phase diagram knowledge is required. Regarding the preparation of multicomponent alloys, the growth of doped crystals or simply the casting of melts containing some impurity, there is nearly no technically interesting crystallization process in a true single-component system. Since basic knowledge on the thermodynamic background of phase diagrams is given in Chapter 2, here it is sufficient to state, that in each type of a binary phase diagram the liquidus and solidus lines may be approximated by linear slopes, at least in the narrow region under consideration. For not too large concentration changes, a constant segregation coefficient :

k0 = CS/CL

(8)

can be derived from the phase diagram with CS and CL being the equilibrium concentrations of the treated component in the solid (S) or in the liquid phase (L). For the scope of this chapter it is only necessary to know whether the equilibrium segregation coefficient k0 is lower or larger than unity.

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With k0 < 1, only a part of the regarded component of the melt can be incorporated into the solid phase. As a consequence, there should be expected an internal accumulation in the melt next to the phase boundary that depends on the rate of the interface movement. Assuming no other stirring mechanism in the vicinity of the phase boundary than diffusion, a steady-state concentration profile parallel to the growth direction will be formed that can be calculated from the 1-dimensional diffusion equation. This characteristic profile is pictured in Fig. 7 for a phase diagram region with k0 < 1.

Fig. 7. Solute concentration near an advancing solid/liquid interface.

From this, it seems to be reasonable to define an effective segregation coefficient keff that is again the ratio of the concentration in the solid to the one in the liquid. Contrary to the equilibrium segregation coefficient, now the melt composition at some distance to the phase boundary is regarded that is different from that one at the interface where equilibrium conditions are assumed. It is the problem of matter transport that makes the difference between keff and k0. Therefore the growth rate of the interface v, the diffusion coefficient D, and the diffusion boundary layer thickness δ enter the formula derived by Burton et al.4 : (9)

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The diffusion boundary layer that has first been introduced by Nernst5 should not be understood as a really stagnant layer, instead, there is a smooth transition between the dominant diffusive or convective transport and δ gives an implicit measure of the more or less strong convective mixing of the melt. 5. Interface stability

A stability analysis of the solid/liquid interface during growth of a solid phase has to answer the question of what happens if a disturbance suddenly occurs. Since temperature and concentration fluctuations are always present, stability or instability will depend only on the reaction of the specific system, whether it will decrease or increase a sudden fluctuation. This problem could have already been addressed to singlecomponent materials where temperature gradients have to ensure heat transport away from the growing solid phase. With a temperature maximum at the phase boundary position, i.e. with a decreasing temperature towards the melt, a sudden fluctuation of the interface position would be amplified. The resulting, locally enhanced, growth rate would produce even more latent heat that increases the original problem again. The result of such a thermally driven interface instability is a dendritic structure of the solid. In multi-component systems, the same phenomenon may occur even if the solid/liquid interface is not affected by a reverse temperature field but by a constitutional supercooling. This was first mentioned by Rutter and Chalmers6 and mathematically treated by Tiller et al.7. Since diffusive matter transport is orders of magnitude slower than heat dissipation, the stability limit that results from constitutional supercooling is much narrower than stability criteria originating from thermal problems. Starting with a concentration profile in a melt in front of a growth interface like that one in Figure 7, each position-dependent concentration has its own liquidus temperature. With the liquidus temperature being the lower limit of the single-phase melt stability, the question arises whether or not the position-depending specific liquidus temperature may have fallen below the actual temperature at some place. Therefore, the actual

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temperature gradient in the melt next to the interface gradTexp has to be compared to the fictive profile of the according liquidus temperatures TL(z). Assuming diffusion to be the only transport mechanism, the 1dimensional concentration profile CL(z) in the melt adjacent to an interface moving with the rate v amounts to: (10)

with C0 being the melt composition in some distance z to the interface, D the diffusion coefficient of the component under consideration and k0 the equilibrium segregation coefficient. According to the liquidus curve of the phase diagram with a linear slope, m = -dTL/dCL, the position-depending liquidus temperature of the melt can be calculated. The graphic construction is shown in Fig. 8.

Fig. 8. Schematic liquidus curve T(CL) of a binary phase diagram that projects a given exp solute concentration in the melt, CL(z), on the resulting temperature profile T(z). T1 exp and T2 represent different experimental temperature gradients that obey or violate the constitutional supercooling criterion.

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Since the highest gradient of the liquidus temperature occurs at the phase boundary ( z = 0 ), the toughest stability criterion may be expressed as:

(11) where Texp is the actual temperature profile in front of the growth interface. With Eq. 9 and the slope of the liquidus line, Eq. 11 turns to: (12) This expression derived by Tiller et al.7 has been known as constitutional supercooling criterion and may serve as a rough limit for the highest possible growth rate of a solid to be solidified from a multi-component melt of composition C0. Although, this expression has been derived for a diffusion-controlled regime, the problem of constitutional supercooling persists even with more or less intensive mixing by convection. One should always keep in mind that in the vicinity of a solid phase, convective motion becomes zero and diffusion is the only remaining process of matter transport. In the case of a too high solidification rate, the conditions in front of the phase boundary are unstable as mentioned above. In the melt, in some distance to the interface, there is a higher deviation from the equilibrium than at the interface itself. It is not just a real supercooling since the actual temperature may be well above that one at the phase boundary, but with respect to the actual melt composition at the specific position, it is constitutionally supercooled. Therefore, some interface fluctuation may reach a position where solidification can easier be achieved than at the solid/liquid interface. As a result of constitutional supercooling, typical dendritic structures may be formed that are morphologically not to be distinguished from those originated from thermally unstable conditions. Because of the relatively narrow limit of the constitutional supercooling criterion with respect to the applied temperature profile and the solidification rate used, these types of instability frequently occur in almost every casting process. On the other

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hand, they have strictly to be avoided in single crystal growth, independent of the technique that is used. Experimental conditions how to create steep temperature profiles next to the phase boundary as well as good mixing conditions in front of the growing crystal will be important arguments to judge about specific crystal growth methods. And, narrow growth rate limits have to be accepted not only with respect to an average rate, but also in the time intervals relevant for the ongoing solidification processes, i.e. within parts of seconds. 6. Segregation

From the phase diagram discussion it is obvious, excluding only a few very special cases, that a solid phase growing from a melt of composition CL will not solidify congruently but with a composition CS = kCL , with k being the segregation coefficient. Depending on the growth rate and the matter transport conditions, k may be taken from the effective segregation coefficient keff -formula (Eq. 8) or simply from the equilibrium phase diagram (k0). In any case, the incongruent solidification changes the composition of the remaining melt and the next infinitesimal layer to be crystallized starts from a changed situation. For a quantitative description of the resulting local distribution of some component in the solid phase (solid solution component, doping element or impurity), the mass balance of the liquid has to be calculated. This can easily be done for a 1-dimensional problem by assuming a complete mixing of the melt and no diffusion in the solid phase. For an ingot to be solidified from a finite amount of material that was completely molten at the very beginning, the problem of the component distribution has been solved by Scheil8. From an initial melt composition C0 one obtains: (13) with z being the axial position and L the total length of the ingot. The axial distribution obtained from the normal freezing process is pictured in Fig. 9 for various values of the segregation coefficient k.

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It is obvious from the mass balance that in systems with a segregation coefficient k < 1 , the distribution function is homogeneously increasing, while with k > 1 it results in a decreasing concentration curve. In the derivation of this normal freezing function (Eq. 13), the volumes of the melt and the solid have been considered instead of the axial position. Therefore, solidification problems in geometries that deviate from a constant cross section, the relative axial position z/L can be substituted for the volume portion of the already crystallized melt V/V0, and this formula may well be taken to explain component segregation in various geometrical configurations of casting processes and in crystal growth.

Fig. 9. Distribution curves for normal freezing, showing solute concentration in the solid versus distance in crystallized fraction, for various values of the distribution coefficient, k.

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Fig. 10. Schematic drawing of the geometry of zone melting.

There has been a second type of macroscopic segregation function that is fundamentally different from the normal freezing case. It has been derived by Pfann9 when he invented the principle of zone melting. Having an ingot with a constant cross section and an initial composition C0, a narrow zone is molten and made to pass along the ingot (see Fig. 10). If we assume the same rates of solidification and melting at the two solid/liquid interfaces, the molten zone has a fixed volume that can be expressed by a constant zone length l. The resulting distribution function amounts to: ⎡ ⎛ CS ( z ) = C0 ⎢1 − (1 − k ) exp ⎜ − k ⎝ ⎣

z ⎞⎤ ⎟ l ⎠ ⎥⎦

(14)

Again, k is the segregation coefficient and z the axial position within the ingot. From the plotted curves in Fig. 11, it can be seen that the zone melting distribution function CS(z) asymptotically reaches the initial composition C0 with the zone length being the characteristic length and depending on the segregation coefficient. With k-values next to unity, after a few zone lengths one gets the starting composition again, i.e. the molten zone has reached steady state conditions. On the other hand, with a very low segregation coefficient, the component under consideration is permanently accumulated in the liquid zone without any practical chance of the crystallized ingot’s composition to equal the initial one. This may be regarded as a powerful advantage if zone melting is carried out as a procedure to refine a material by passing a molten zone along the ingot. But, if a more or less macroscopically homogeneous solid is the aim, distribution coefficients much less than unity may be a severe problem.

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Fig. 11. Distribution curves for zone melting, showing solute concentration in the solid versus distance in zone lengths from beginning of charge, for various values of the distribution coefficient, k.

7. Techniques of single crystal growth

Some of the most important solidification methods for single crystal growth shall be briefly presented. According to the component distribution problems discussed in the previous paragraph, the techniques will be treated in two groups depending on their segregation characteristics. There are methods with totally molten material at the very beginning which leads to the normal freezing type of the component distribution (Bridgman, Czochralski, and Kyropoulos method) and those with zone melting characteristics (floating zone and Verneuil method).

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7.1. Bridgman method

The Bridgman method10 is regarded as the easiest to do, but nevertheless a very powerful crystal growth method from the melt. In this technique, an ampoule containing a melt is slowly lowered through a temperature field of a vertical tube furnace (Fig. 12).

Fig. 12. Bridgman crystal growth method.

Often a crucible with a pointed bottom is used to support nucleation, but seeded growth may be applied as well. Since a not too low temperature gradient at the solid/liquid phase boundary has been found necessary, e.g. to avoid constitutional supercooling, usually a twosegment furnace is used to produce a steep axial temperature profile at the interface with quite low gradients in the upper and lower parts of the tube furnace. This modification was suggested by Stockbarger11 and is nowadays included in almost all Bridgman-type experiments. For the process of solidification, it is not the decisive factor whether the crucible is mechanically passed through the temperature field of the furnace or the change of temperature is done by an electronically based temperature versus time program. Therefore, recent modifications of the Bridgman

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method may use multi-segment tube furnaces that are dynamically controlled to move a temperature field along the ampoule without any mechanical motion. 7.2. Czochralski method

Czochralski’s original idea12 was to measure the crystallization rate of metals by pulling a metallic wire from its native melt. Exceeding some upper limit of the pulling rate would separate the crystalline material from the liquid. This has been developed to the most important growth method for single crystals from the melt (Fig. 13) and is used for the large-scale production of electronic materials. Starting from a totally molten source material, a single-crystalline seed is brought in contact with the melt surface and wetted by the melt. After having reached equilibrium conditions, the seed is slowly pulled upwards which transfers the original interface to a lower-temperature position. Thus, the driving force for solidification is simultaneously created by the interface shift and the crystal growths downwards with approximately the pulling rate.

Fig. 13. Czochralski crystal growth method.

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It is really difficult to perfectly control the diameter of the growing crystal because it is influenced by the contact angle of the melt at the three-phase line (vapour/liquid/solid) where the meniscus of the melt touches the crystal interface. An increase of the crystal’s diameter, that is e.g. necessary at the end of the seeding procedure, is obtained either by lowering the pulling rate or by a slight reduction of the heating power. Conversely, a decreasing diameter is the result of an opposite change of one of these parameters. Counter rotation of the crystal and the melt is regarded as a decisive feature of the Czochralski method because it not only gives a better rotational symmetry but also ensures a good mixing of the melt in front of the growing interface. Thus, a more or less good homogeneity of the melt as assumed in the segregation analysis (Eq. 12) can much better be achieved than with the Bridgman method. The missing contact between the crystal and the crucible is one of the main advantages of this technique. 7.3. Kyropoulos method

The Kyropoulos method that is pictured in Fig. 14 looks technically very similar like the latter one. Now, the crystal grows into the melt instead of being pulled upwards. Therefore wetting does not play a dominant role, but solidification is exclusively driven by the temperature field. A slowly decreasing temperature, as well as the intensive cooling via the seed holder, make the crystal growing into the bulk melt. Crystal growth has to be completed by pulling the crystal rapidly out of the melt before the crystal might touch the crucible wall. Since the crystal is totally embedded during the solidification process, there are low thermal gradients in the solid. This is usually an advantage with respect to the mechanical stress that may be caused by inhomogeneous temperature fields. On the other hand, a low temperature gradient gives a rather narrow limit for the possible growth rate obtained from the constitutional supercooling criterion. With metallic melts, there is nearly no chance to in-situ observe the proceeding crystallization. Therefore, the Kyropoulos method is mainly used for transparent melts, i.e. for the growth of insulators and crystals for optical applications where large-diameter crystals are needed.

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Fig. 14. Kyropoulos crystal growth method.

7.4. Floating zone method

The floating zone method (Fig. 15) is the crucible-free modification of the vertical zone melting technique. Since it does not require any crucible, it can be used for crystals where a suitable container can hardly be found, either for reasons of its chemical reaction with the melt or because of the risk of impurity contamination at very high temperatures. The liquid is held by the surface tension that restricts the possible zone length to a height that depends on the liquid/vapour interface energy and the density of the melt. Growth proceeds by moving the heater that passes the liquid zone along the feed rod. Most frequently, a RF coil or an optical heating facility is used to create a steep axial temperature profile with a maximum in the floating zone. The feed material and the growing crystal are separately fixed at their ends and counter-rotated to enhance convective mixing as well as radial heat transfer. There are modifications with a much smaller diameter of the feed rod compared to the crystal to be solidified. In these cases, the feed rod must be moved relative to the growing crystal at a different rate.

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The separate preparation of the feed rod prior to the growth process makes the floating zone technique more complicated compared to other methods and therefore restricts it to the growth of ultra-pure materials and really high melting point substances for which suitable crucibles are missing.

Fig. 15. Floating zone crystal growth method.

7.5. Verneuil method

Verneuil was the first to develop a growth method for single crystals from the melt for commercial purposes15. His flame fusion method is especially dedicated to high melting point oxides like sapphire and spinel. Because of the heating by an oxyhydrogen burner, it can not be

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used for metals and other oxidizing substances, at least in the original version. In Fig. 16, the experimental set-up is shown that has almost not been changed during the last hundred years. The growing crystal is mounted on a ceramic rod that is rotated and slowly moved in vertical direction. Fine-grained powder is used as source material that trickles through a sieve driven by some vibration mechanism. Falling through the oxyhydrogen flame the powder grains become molten and enter the thin melt layer on top of the growing crystal. It is this thin layer that corresponds to the liquid zone in zone melting techniques. Since no other heating facility than the burner is used, the temperature field is simply the result of the flame and the ceramic insulating chamber and very steep temperature gradients occur in the crystal that produce considerable mechanical stress. Nevertheless, Verneuil’s method, which is a cruciblefree technique as well, has demonstrated its feasibility and high output, e.g. for the single crystal growth of gemstones. Before growing single crystals from the melt, one should thoroughly analyse all available properties of the substance to be solidified, including the melting behaviour, the vapour pressure of its components and the needs the crystal has to fulfil for application purposes. And the full range of crystal growth methods developed over the decades should be considered for growing this special crystal. There is no method that is suited for the whole spectrum of single crystals of interest. Only a short overview on the most prominent techniques of crystal growth from the melt has been given in this chapter. Detailed information on very special methods and a profound treatment of the phenomena that will influence the more or less successful growth experiments can be obtained from modern textbooks16, 17. References 1. W. Kossel., Zur Theorie des Kristallwachstums, Nachr. Akad. Wiss. Göttingen, Math.-phys. Kl., 135 (1927) (in German). 2. K. A. Jackson, Amer. Soc. Metals Cleveland, 174 (1959). 3. W. K. Burton, N. Cabrera and F. C. Frank, Philos. Trans. Roy. Soc., A 243, 299 (1951). 4. J. A. Burton, R. C. Prim and W. P. Slichter, J. Chem. Phys., 21, 1987 (1953). 5. W. Nernst, Z. phys. Chem., 47, 52 (1904) (in German).

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6. J. W. Rutter, and B. Chalmers, Can. J. Phys., 31, 15 (1953). 7. W. A. Tiller., K. A. Jackson, R. W. Rutter and B. Chalmers, Acta Met., 1, 428 (1953). 8. E. Scheil, Z. Metallkd., 34, 70 (1942) (in German). 9. W. G. Pfann, Trans. AIME, 194, 747 (1952). 10. P. W. Bridgman, Proc. Amer. Acad., 60, 303 (1925). 11. D. C. Stockbarger, Rev. Sci. Instr., 7, 133 (1936). 12. J. Czochralski, Z. phys. Chem., 92, 219 (1918) (in German). 13. S. Kyropoulos, Z. anorg. allg. Chem., 154, 308 (1926) (in German). 14. P. H. Keck, Phys. Rev., 89, 1297 (1953). 15. V. Verneuil, C. R. Acad. Sci. Paris, C 135, 791 (1902) (in French). 16. K. T. Wilke, and J. Bohm, Kristallzüchtung, Deutscher Verlag der Wissenschaften, Berlin, 1988 (in German). 17. D. T. J. Hurle, Ed., Handbook of Crystal Growth, 1-3, Elsevier, Amsterdam, (1993).


CHAPTER 5

DIFFUSIVE PHASE TRANSFORMATIONS Yves Bréchet SIMAP, Institut National Polytechnique de Grenoble, Domaine Universitaire, PB75, 38402 Saint Martin d’Heres cedex, France E-mail: [email protected] This chapter outlines the main analytical tools available to treat diffusion controlled phase transformations. These tools will be applied on one side to precipitation reactions, and on the other side to interface migrations. Special emphasis will be laid upon the necessary improvement to go from binary to multicomponent systems, and to integrate the models of phase transformation into a global alloy and process optimisation. Examples will be taken from aluminium alloys and steels.

1. Introduction and motivation In recent years, in addition to the pure knowledge driven research, the need to optimize alloy and process design, together with the enhanced possibilities of computers has motivated the development of a physically based modeling of phase transformations. The aim being to obtain the best compromise between properties, and the tools available for that being either the composition, or the the parameters of thermomechanical treatments, the understanding of the relations between process parameters and generated microstructures, and between microstructures and the resulting properties has become a necessary step for metallic alloys improvements. The microstructural features under consideration are the granular microstructures (grains size, texture), and the phase microstructures (nature, size and morphology). These microstructures are strongly influenced by the treatments materials undergo in the solid state. We focus in this course on the phases transformations leading to a multiphase structure. These multiphase structures control mechanical 99

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properties such as yield stress, work hardening and ductility. They may also govern functional properties such as critical currents in super conductors or coercivity in magnets. They certainly influence damage properties such as toughness or fatigue or corrosion, though the relation microstructure / properties is in these situations far less understood. Depending on the property under consideration, the microstructural features of interest are different, but in most situations, the nature, the volume fraction, the shape and the scale of the phases present have to be considered. The yield stress of an alloy with spherical precipitates depends both on the volume fraction f of precipitates, and on their size R. If the precipitates are bypassed, the yield stress scales as f1/2/R, if they are sheared, as (fR)1/2. The way dislocation interacts with them depends on their chemical nature, and on their size. Most phenomenological models would provide , at best, a description of the volume transformed after a given heat treatment. This illustrates the need to develop a more sophisticated approach for microstructure development which would incorporate the transformation mechanisms. The study of solid state phase transformations in metals and alloys is traditionnally divided in two main streams: the displacive transformations in which atoms move in a cooperative manner at a velocity approaching the one of sound waves, the diffusive transformations in which atoms move in a non cooperative manner, by diffusion processes. The emphasis laid upon various aspects of phase transition is different in the two cases. The distinction is certainly not as clear cut as it sounds (see for instance years of controversy concerning Bainite) but it remains useful at least to set limiting cases.For the martensitic transformation, treated in another chapter of this book, crystallography and back stresses generated during the transformation are a central issue, the thermodynamics of the problem enters mainly into the conditions for nucleation of the new phase, and since the propagation of the transformation is very rapid, the kinetics of invasion during continuous cooling is controlled by the possibility of repeated nucleation. The patterns emerging from these transformations reflect both the crystallographic constraints and the elastic interactions between different variants.

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For the diffusive transformation, the central role in modelling is given to mass transport via diffusion processes. The scale of the microstructure results from the competition between available free energy (driving force) and interfacial energy. The kinetics results from diffusive transport, and limited mobility of the interfaces. Although crystallography and elastic stresses may play a role they are not central to the main issue: transformation kinetics and morphology of the reaction products. The present contribution is limited to diffusive phase transformations. Table 1. Overview of the key differences between displacive and diffusive phase transformations. Displacive transformations Atoms move on interatomic distances a, in a cooperative manner Transformations occur below a critical temperature, at a rate independent of temperature The volume transformed depends on temperature only The chemical composition of the parent and daughter phases are identical There are strict crystallographic relations between the two phases

Diffusive transformations Atoms can move on distances 106a, in stochastic manner Transformation rate is highly dependant on temperature The volume transformed depends on time and temperature The chemical composition of the parent and daughter phases may differ There may be crystallographic relations between the two phases

2. Variety of situations in diffusive phase transformations Within the class of diffusive phase transformations, three other classifications can be proposed, in relation with the possible role of structural defects (homogeneous vs. heterogeneous), with the type of chemical evolution of the mother phase (continuous vs. discontinuous), and with the kinetics of transformation (linear or parabolic). In some situations, the transformation is homogeneous: it takes place in the grain interior, and as long as there is a thermodynamic driving force, it progressively transforms the mother phase. This process is typified by the example of fine precipitation in aluminium alloys obtained after quench and annealing below the solubility limit. But the

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transformation may also be heterogeneous, in the sense that it requires structural defects such as dislocations and grain boundaries to operate. Heterogeneous precipitation very often leads to coarser structures, with less interesting properties. Heterogeneous precipitation (similar to heterogeneous nucleation from the liquid, treated elsewhere in this book) occurs generally when the driving force is not sufficient, and the help of a structural defect decreases the nucleation barrier, but the scale remains large. In some situations, the transformation leads to a continuous evolution of the chemical composition of the mother phase. This is the case for the traditional precipitation reaction, where the precipitates progressively deplete the matrix supersaturated in solute. By contrast, in other situations, the decomposition process takes place by the migration of an interface through the mother phase. Across this moving interface, the composition, and even the crystallography may change in a discontinuous manner. These types of reactions can be found very often in steels (transformation from austenite to ferrite, or eutectoid reaction known as pearlite) but also in other systems. For instance, the same phase leading in sole circumstances to continuous precipitation, can lead to discontinous precipitation where a moving grain boundary sweeps a supersaturated solid solution to leave as a daughter phase a solid solution with less supersaturation, and regularly spaced lamellar or rod structures as precipitates. The third classification is according to the reaction rates. Depending on the morphology, the linear dimensions of the product phase may have different time dependence. In the case of non conserved shapes such as spherical precipitates growing from a matix, mass transport by diffusion, in absence of any interface limiting reaction, would lead to a parabolic growth rate. In the case of conserved shapes, such as needle growth or planar fronts, the situation is more complex. For planar fronts, the only way to obtain a constant velocity is when the daughter phase and the mother phase have the same average composition: this is the case for pearlite or for discontinuous precipitation. If the daughter phase has a composition different from the mother phase (such as for austenite to ferrite allotriomorphic transformation), the need for mass transport by diffusion imposes a parabolic growth. In the case of needles (such as


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Widmanstatten ferrite), the “volume of the mother phase explored” by the growing needle increasing while transformation proceeds, a constant velocity is possible even in the case of a non conserved composition. This description of the kinetics is highly idealized, one should rather speak of “constant velocity” vs. “decreasing velocity”. However, it sets the scene for the basic models of diffusive growth that are currently used in microstructural modeling. 3. Diffusion and diffusion equations 3.1. Basics of diffusion Diffusion processes in crystalline solids are closely related to point defects (vacancies and intersticials) and the activation energy for this thermally activated process is the sum of the defect formation energy, the defect migration energy and the binding energy between the defects and the solute atoms diffusing. This activation energy is lowered when diffusion takes places at dislocations (by a factor 0.6), at grain boundaries (0.4) and at surfaces (0.2). The details of the diffusion processes are beyond the scope of this paper, but it is worth keeping in mind that diffusion via intersticials is far more rapid than the one taking place by vacancies. This fact is the reason for some rich features in ternary Fe-X-C alloys. The phenomenological description of diffusion relies on the thermodynamics of irreversible processes. A simple way to derive it for perfect solid solutions is the following: the driving force F for diffusion is the opposite of gradient in chemical potential. The velocities of atoms under this driving force is v=MF (assumption of linearity), and the mobility M is related to diffusion via Einstein relation M=D/kT. The flux is J=cV. For perfect solutions, the chemical potential is kTLn(c). The resulting flux is therefore J=-D grad (c) , known as the first Ficks law. The second Ficks law is simply the mass balance, and, in three dimensions, the governing partial differential equation for diffusion is:

∂c = D.Δc ∂t

(1)

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Modeling diffusive phase transformations relies entirely on the solution of Fick’s diffusion equations with the appropriate boundary conditions which reflect the thermodynamics of the system. 3.2. Classical exact solutions

In diffusion controlled phase transformation, an essential tool for modelling is the classical solutions of Fick's equation in various geometries. We will successively investigate planar and spherical geometries leading to the so called "Parabolic solutions", and the needle like geometries leading to shapes propagating at constant velocities. 3.2.1. Parabolic solutions The family of parabolic solutions relies on the classical solution of Fick's equation:

C = A + Berf ( erf (u ) =

2

π

x ) 4 Dt

(2)

u

∫

. exp( −λ )d λ 2

0

In such solutions it is possible to impose a constant concentration at a position defined by:

ξ = K 4 Dt

(3)

K = 0 corresponds to a static interface, whereas a non zero K corresponds to an interface propagating with a decreasing velocity. Depending on the boundary conditions, A, B and K can be determined, leading to a prediction of the diffusion field, and of the propagation rate. For instance, the case of precipitation of a β phase of composition Cβ from a solid solution of composition Cα larger than the equilibrium concentration Cαβ, leads to the following results for the diffusion field and the parabolic constant K:


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⎡1 − erf ( x / 4 Dt ⎤ C α ( x) = C α∞ + (C αβ − C α∞ ) ⎢ ⎥ ⎣ 1 − erf ( K ) ⎦

(4)

( −1/ π ) (C K≈

αβ

− C α∞ )

(C β − C αβ )

(5)

This can be readily extended with the same method to situations where the composition of the precipitate is allowed to vary, where diffusion is allowed in the product phase, where an interface separates a one phase region from a two phase region, etc. For a better grasp of the physical meaning of the method, it is worth going into detail through the derivation of the growth rate of a spherical precipitate from a supersaturated solid solution. In spherical coordinates, Fick’s equation can be rewritten under the assumption of an invariant diffusion field: d 2 dc (r . ) = 0 dr dr

(6)

With appropriate boundary conditions at the interface and at infinite distance (i.e. assuming a diluted precipitation), one gets the expression for the diffusion field: C (r ) − C α∞ = ( C αβ − C α∞ ) .

R r

(7)

The mass balance at the interface writes then: 4π R 2 .dR.(C β − C αβ ) = 4π R 2 .Jdt

(8)

The expression of J can be then readily obtained from the expression of the diffusion field, from which the rate of growth of precipitate radius can be derived and by direct integration one finds the expression of the precipitate size as function of its initial size, of the diffusion coefficient, and of the supersaturation: R 2 (t ) − R02 = 2 D.

C α∞ − C αβ .t C β − C αβ

(9)

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3.2.2. Self preserving shapes These solutions correspond to C(x-Vt). As can be readily seen, a planar solution is possible (with an exponential variation of the concentration field) only if the product phase has the same overall composition as the parent phase (as is the case with massive transformation, pearlitic reaction or discontinuous precipitation). But there is another family of solution propagating at constant velocity and preserving its shape: the cases where the surface of the reaction front is a quartic, and the most relevant situation in phase transformations, when it is a paraboloïd. In order to prove this, the corresponding total differential equation for the function of x-Vt is rewritten in a system of confocal parabolic coordinates. In such a system, the parabolae can be lines of constant concentration. Solving the equation in this system of coordinates leads to an expression of the diffusion field, and the mass balance produces a relation between the dimensionless supersaturation S and the Peclet number P defined by:

S =

C αβ − C α ∞ C αβ − C β

P=

Vρ 2D

(10)

This relation can be derived for all paraboloïds. The most frequently used are the cylindrical paraboloïd and the circular paraboloïd. For each of these geometries the relations between S and P are respectively: 2

∞

exp(−u π∫

S = π P .exp( P ).

2

)du

(11)

P

∞

exp(−u ) du u P

∫

S = P.exp( P ).

(12)

These exact solutions do not survive to the introduction of capillary effects, which introduces higer order terms due to curvature, leading to similar mathematical difficulties to the ones found in the theory of dendritic solidification.


R 2 (t ) − R02 = 2 D.


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(13)

3.3. Classical approximate solutions

The examples given above clearly indicate both the mathematical complexity, and the limitations of exact analytical solutions of diffusion equations. One can bypass these difficulties using numerical solutions, but it is however often enlightening, for the sake of a physical understanding of the observed transformations, and even for a relatively accurate quantitative description within the limits of experimental accuracy and of the quantitative knowledge of thermodynamic and diffusion data, to obtain approximate solutions. These “classical approximate solutions” are still of great interest. 3.3.1. Growth of a sphere: the constant field approximation The constant field approximation amounts to solve Fick’s equation assuming the diffusion field is stationary in the frame of the growing precipitate. The concentration field in spherical coordinates decays as 1/r (r being the distance to the origin), and the growth rate for a precipitate of β growing from a supersaturated solution α is given by: R 2 (t ) − R02 = 2 D.


(14)

3.3.2. Diffusional growth of a planar front: Zener approximation Zener approximation is a very simple approach which considers that the diffusion field facing a moving planar front is linear and extends on a distance ΔX. Mass conservation provides a relation between the layer thickness X, the diffusion extension ΔX, and the various concentrations indicated on Fig. 1. ΔC.Δx = 2 (-Cα+C°).X

(15)

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Cαγ

ΔC

C° ΔX

Cα

X Fig. 1. Diffusion profile in Zener’s approximation.

The next step is the solution of Ficks first equation, which, via a mass balance, gives an expression of the velocity: (Cα-Cαγ).V=D.(dC/dx)

(16)

The final integration provides an expression for the layer thickness at a function of time:

X2 =

(Cαγ

D(ΔC ) 2 .t − Cα )(C ° − Cα )

(17)

3.3.3. Diffusionnal growth of a needle: Hillert approximation For the growth of a needle of tip radius r, with a similar approximation of a linear diffusion gradient, Hillert proposed an expression for the constant growth rate V: V=

2 D Cαγ − C ° . r Cαγ − Cα

(18)


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4. Precipitation

We will first outline the most frequently used experimental methods, then summarize the classical understanding of the phenomena, and give indications on recent developments.

Fig. 2. Precipitation of Copper into Iron observed by 3D Atom probe (D.Blavette).

4.1. Experimental methods

Informations on precipitation can be obtained by a variety of manners, both by direct and indirect methods. Their relevance to get reliable informations on the size, shape, volume fraction and chemistry of precipitation are summarized in the following table, italics indicate some information which may be attaignable, with difficulty, in certain cases. Table 2. Experimental methods for the study of precipitation. Direct Methods

Transmission Electron Microscopy

Size, shape, Volume fraction, chem

Neutron and X Rays Scattering

Size, volume fraction, shape

3D atom probe

Size, Chemistry

Resistivity, Thermoelectric power

Evolution of the solid solution

Indirect

Calorimetry

Nature and volume fraction

Methods

Hardness

Combination of size and volume fraction

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4.2. Classical picture

Precipitation from a supersaturated solid solution starts usually by a nucleation step where fluctuations of compositions providing a sufficient gain in bulk energy to afford the cost in surface energy can grow. This defines a critical radius which is essentially the ratio of the surface energy to the bulk energy, and a nucleation barrier which scales as the inverse cube of the available driving force: the larger the available free energy , the finer will be the structure, the smaller the energy barrier and the more abundant the nucleation rate. The classical nucleation theory presented in the chapter on solidification can be translated, with some minor modifications, for the solid state. In recent years, numerical simulations, namely Kinetic Monte Carlo and cluster dynamics, have shed a new light on this nucleation process. When temperature is changing, both the transport efficiency and the driving force for precipitation are modified. At temperature close to the solubility limit, the driving force for precipitation is low, and, in spite of a fast transport, the transformation kinetic is slow. At low temperatures, the driving force is large but the transport by diffusion is limited: again the transformation is slow. The transformation rate will be maximum at an intermediate temperature where both the driving force is important, and the diffusion coefficient is large enough. This is the origin of the shape so called C curves plotting the time necessary at a given temperature to perform a prescribed percentage of the transformation. While precipitation proceeds, the solid solution is progressively depleted, the available driving force for precipitation decreases and the nucleation of new germs of the precipitate phase becomes less favourable. In the first stages new germs appear: it is the nucleation stage, then only germs already present can grow: it is the growth stage. Finally only the largest germs which have a lower surface energy cost can grow: it is the coarsening stage. In these three stages, the scales emerging are given, in the nucleation stage by the critical radius, in the growth stage by the parabolic law: R 2 (t ) − R02 = 2 D.


(19)


111

and in the coarsening stage, the growth follows the so called “Lifschitz Slyozov Wagner equation”, where the surface energy γ enters directly: 8 DγΩC αβ 2 R 3 − R(0)3 = . t kT 9

(20)

4.3. Recent developments

This classical distinction between the different stages in the precipitation sequence is somewhat artificial. In recent years, the so called “class models” which are describing a population of precipitate, and not simple an average, have allowed to account for progressive transitions between the different regimes. They have also allowed to deal with non isothermal treatments, and with phenomena such as reversion of precipitation when precipitates become unstable due to an increase in temperature resulting in a rapid increase in critical radius. This possibility to deal with non isothermal treatments has opened the path for the modelling of microstructure gradients in heat affected zones in the proximity of welds, an example of “integrated modelling”. The classical models in the literature are derived for binary alloys. Most of the situations encountered in real systems are multicomponents. The adaptation of the models to these situations is not necessarily straightforward. A simple hypothesis is to deal with a “quasi binary system”, where the alloying elements are not distinguished. This is by no way satisfactory, even if the inaccuracy of available data may end to make this drastic simplification admissible. To deal rigourously with multicomponent systems is nowadays understood only when the precipitates are stoechiometric. In ternary alloys for instance the equilibrium conditions at the interface can be given by any of the equilibrium conodes of the phase diagram. For each of these possibilities, one can compute the flux of the two elements, and only one among these possibilities will fulfill also the ratio on the fluxes imposed by the stoechiometry conditions. This selects uniquely the operating conode, and sets the evolution of the boundary conditions when the solid solution gets depleted.

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5. Interface migration

Phase transformations associated with interface migration are very important in systems presenting several allotropic phases such as steels (as shown in Fig. 3). But they can also occur without cristallographic changes as is the case in discontinuous precipitation. For the sake of simplicity in this section, we will focus on the austenite to ferrite transformation, where partitioning of C is thermodynamically favoured. • traditional

• decarburization

200 μm

Fe-0.1C-0.1Mo; 800C 1min

~ 4 mm

Fe-0.54C-0.51Mo; 825C 128min

Fig. 3. Examples of interface migration in the austenite to ferrite transformation (H.Zurob, C.Hutchinson) a) during an isothermal treatment, b) during a decarburization treatment leading.

The main features of these types of transformations is to start at the boundaries of the mother phase (austenite) and to generate microstructure whose length scale if of the order of microns. The modeling of these reaction rates remains at the continuous level, and the key issue is to understand the interplay between thermodynamics and kinetics to define the interfacial conditions governing the transformation velocity. 5.1. Experimental methods

While precipitation requires often characterization methods at very small scales, interface mediated transformations are more often observed at the micron scale. As a consequence, optical metallography and Scanning Electron Microscopy remain the major tools for their investigation (Table 3).


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Table 3. Experimental methods for the study of precipitation.

Direct Methods

Indirect Methods

Scanning Electron Microscopy Microprobe

Size, shape, Volume fraction, chem

Optical microscopy Neutron and X Rays Scattering

Volume fraction

3D atom probe

Interfacial Chemistry

Dilatometry

Evolution of the volume fraction

Calorimetry

Nature

Hardness

Combination of size and volume fraction

5.2. Classical picture

For binary alloys, a Zener type of description, assuming equilibrium conditions at the interface for the local carbon composition, describes accurately the kinetics. When one considers the ternary alloys (such as Fe-Ni-C), an extra difficulty appears, to decide whether or not the substitutional element partitions between the two phases according to the phase diagram. If this is the case, due to the slow diffusion of Ni compared to C, the transformation would be always very sluggish. As it is observed not to be so in many cases, the possibility of transformation without long range partitioning of Ni had to be considered. Carbon diffusion is controlled by the interfacial concentration which depends on the concentration in alloying element X. The best solution from a thermodynamics viewpoint, and always possible, is “Full local equilibrium with partitioning” (Fig. 4.a) but since both X and C have to be partitioned, the predicted kinetics is very slow (being controlled by the diffusion of X). Less favourable from the thermodynamics view point, the “local equilibrium with negligible partitioning, or LENP (Fig. 4.b) is still a local thermodynamic condition at the interface. There is no long range partitionning of X, but a short range spike. The kinetics is controlled by C diffusion, the presence of the X spike modifies the carbon concentration at the the interface. This gives a fast reaction. This condition is possible only for compositions below dashed line in Fig. 4.b

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Fig. 4. The classical interfacial conditions : a) Local equilibrium, b) local equilibrium with negligible partitioning, c) paraequilibrium.

(no partition limit): above, the C concentration would be such that C would go toward α, instead of being rejected from α. In the “Paraequilibrium hypothesis” (Fig. 4.c), the interfacial conditions are given by a constrained equilibrium: the composition in X is forced to be the same in both a and b. The dashed lines are obtained by a double tangent construction between the curves G(C, X=cte) for a and g. There is no long range partition of X, no X spike. This is a fast


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reaction, only possible between the dashed lines on Fig. 4c indicating the «paraequilibrium limit». As is shown in Fig. 5, the choice between the different interfacial conditions has significant consequences on the predicted kinetics. 450

Fe-0.51C 775°C

450

Layer thickness (microns)

400 350

LE

Fe-0.50C-0.97Ni 775°C

400


500

350

PE LENP

300

300

250

250

200

200

150

150

100

100

50

50

0

0 0

50

100

150

200

250

0

300

50

100

150

200

250

300

time(min)

Time(min) 350

400

PE


200 150

LENP

100 50

PE

Fe-0.38C-1.03Ni 800°C

350

Layer Thickness (m icron

Fe-0.50C-1.66Ni 775°C 250 300

300 250

LENP

200 150 100 50

0

0

0

50

100

150

200

250

300

0

time(min)

Fig. 5. Modelling of decarburization kinetics temperature (C.Hutchinson, H.Zurob).

50

100

150

Time (min)

200

250

for various alloy compositions and

5.3. Recent developments

The selection criteria leading to a given interfacial condition is still a matter of active research. In recent years, some evidence has been given of the fact that the interfacial conditions are not constant when transformation proceeds. It seems that the interfacial condition at nucleation is close to paraequilibrium, whereas it progressively evolves building up a spike at the interface, toward LENP in the last stages of the

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reaction. It seems also that this transition depends itself on the interface velocity. The relative importance of solute drag (diffusive dissipation in the interface) and of cross interface diffusion is still under investigation. 6. Conclusions

This brief overview of diffusive phase transformations is far from being exhaustive. We have not dealt with lamellar structure such as discontinuous precipitation, or eutectoid transformations, neither with the puzzling aspects of the massive transformation, nor of the controversial nature of Bainite. The reader is referred to extensive treaties for these issues. We have presented an overview of the commonly admitted concepts and methods used in the understanding of diffusive phase transformations. More details on their application can be found in the lecture notes which were presented at the school and are on the website. To conclude, in recent years substantial progress has been made in the field due to better thermodynamic databases, better understanding of the coupling between kinetics and thermodynamics. Reliable systematic experimental investigations of model systems, and comparison between analytical modeling and atomistic simulations have clarified a lot the issue of nucleation in precipitation. For the case of interfacial migration, the evidence of the key role of the interfacial conditions is now well established, but the issue of nucleation is basically untouched, and the question of morphological instabilities remains open. For both precipitation and interface mediated transformation, the question of the interplay and the relative importance of thermodynamics, kinetics and crystallography is totally open. Acknowledgments

The author wants to thank his colleagues Prof A.Deschamps (Grenoble), Prof G.Purdy and H.Zurob (McMaster) and Dr C.Hutchinson (Monash) for invaluable discussions and collaborations over the years in the field of diffusive phase transformations. Constant support of Arcelor research is gratefully acknowledged.


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References In a chapter of a course, a list of specialized references would give an excessive weight to the topics selected which somewhat reflect the interest of the author. The field of phase transformations in metals and alloys has a long history, which has also led to a jargon which somewhat makes it less accessible to specialists of other fields. As a result, the general knowledge on precipitation for instance, is much more developped than the one on phase transformations controlled by interface migration. The general references listed here are, from the view point of the author, good overviews for a non specialist to enter the field. G1. H.Aaronson "Lectures in the theory of phase transformations",TMS (1999), recently reedited and updated is a must. The articles by M.Hillert on "thermodynamics" (Chapter 1) and by R.Sekerka on "Moving boundary problems" (Chapter 5) are masterpieces. G2. G.Kostorz "Phase transformations in Materials", Wiley VCH, (2002), is an updated version of the Treatise on Materials Science and Technology , volume 5 edited by R.Cahn, P.Haasen, E.Kramer. Of special relevance to the present topics are the review papers by R.Wagner et al. on "Homogeneous second phase precipitation" (Chapter 5) and G.Purdy et al. on "Transformations involving interfacial diffusion" (Chapter 7). G3. J.Philibert, "Atom movements, Diffusion and Mass transport in solids", Editions de Physique (1991) provides an excellent textbook on diffusion and its applications. G4. M.Hillert "Phase diagrams and Phase Transformations", Cambridge University Press (1999) is the key reference on the application of thermodynamics to phase transformations. G5. J.W.Martin, R.D.Doherty, B.Cantor "Stability of micostructures in metallic systems", Cambridge University Press (1997) is an encyclopedic visit of the world of microstructural evolution, and an invaluable source of references to original papers.


CHAPTER 6

DIFFUSIONLESS TRANSFORMATIONS C. Duhamel1,2, S. Venkataraman1,2, S. Scudino1,2 J. Eckert2 1

FG Physikalische Metallkunde, FB 11 Material- und Geowissenschaften, Technische Universität Darmstadt, Petersenstraße 23, D-64287 Darmstadt 2 IFW Dresden, Institut fürKomplexe Materialien, Postfach 270116, D-01171 Dresden E-mail: [email protected] Diffusionless transformations, also called displacive transformations, are solid state transformations that do not require diffusion, i.e. long range movements of atoms, for a change in the crystal structure to occur. They result from correlated atomic displacements where the atoms move less than one interatomic spacing and retain their relationship with their neighbours. The classical example is the martensitic transformation occurring in steels. The word “martensite” was originally used to name the hard and fine constituent formed in quenched steels. Later, other materials, such as non-ferrous alloys1 or ceramics2, were found to exhibit diffusionless transformations. The name “martensite” now refers more generally to the resulting product of such a transformation. The purpose of this chapter is to describe the basic features regarding the processes occurring during diffusionless transformations. First, the crystallographic theory of martensitic transformation will be presented. The crystallographic theory provides a description of the overall displacements involved in the transformation, consistent with the observed geometrical features. Then, the nucleation and growth of the martensitic phase will be described. In the last part, the mechanical effects in martensitic transformations will be discussed.

1. General characteristics of diffusionless transformations Although many of them occur at high temperature, diffusionless transformations are athermal. The transformation occurs at a very high velocity, independent of the temperature, which can be reached because

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of the absence of long-range atomic movements. The overall kinetics depends on the nucleation and growth steps and is limited by the slower of these two stages. The interface between the martensite and the parent phase is glissile and thermal activation is not required for its movement. Depending on the material, this interface can be fully coherent or semicoherent. For example, in ferrous alloys, the martensite / austenite boundary is semi-coherent. Only localized regions of the interface are coherently accommodated. On the contrary, the fcc → hcp transformation in pure Co leads to a fully coherent interface. The martensitic transformation implies coordinated structural changes with lattice correspondence and a planar parent / martensite interface, which gives an invariant-plane strain shape deformation. At a finer scale, inhomogeneities such as slip, twinning or faulting can be observed in the martensite, suggesting that a second deformation process occurs. This process is part of the overall diffusionless transformation. It gives the invariant plane conditions at a macroscopic scale and provides the semicoherent interface between martensite and the parent phase. 2. Martensite crystallography 2.1. General features Fig. 1 shows optical micrographs of typical martensite morphologies in iron-base alloys. The martensitic phase can have a lath, a lenticular or a

Fig. 1. Example of martensite morphologies in iron-base alloys: (a) lath in a Fe-9% Ni – 0.15% C alloy, (b) lenticular in a Fe – 29% Ni – 0.26% C alloy, (c) thin plates in a Fe – 31% Ni – 0.23% C alloy (From www.msm.cam.ac.uk/phase-trans/2005/Maki.ppt).

Diffusionless Transformations

121

thin plate shape. In steels, the orientation variants and the morphology of the plates depend on the alloy composition. The plates extend over the whole diameter of the grains and their growth occurs in a limited number of orientations. The contrast observed in the micrographs results from different lattice orientations with the initial parent grain. Different regions of martensite have undergone distortion in different ways with respect to the initial surface. This macroscopic distortion is known as shape deformation. The shape deformation exhibits features that are similar to simple shear, but is, in fact, associated with an invariant-plane strain where the plane of reference is the undistorted and unrotated habit plane. The invariant-plane strain is a homogeneous distortion in such a way that the displacement of any point is in a common direction and proportional to the distance from a plane of reference, the invariant plane, which is not influenced by the strain. It consists of an expansion (or contraction) normal to the invariant plane and a shear in a direction lying in the invariant plane. However, this invariant-plane strain is not sufficient to describe the martensitic transformation. 2.2. The Bain model This model was proposed by Bain3 in 1924 to explain how martensite in steel can be obtained from austenite with a minimum of atomic movement and a minimum of strain. 2.2.1. The solid solution of carbon in iron In this case, the parent phase is the austenite phase with fcc structure. In a fcc (or hcp) structure, two different types of sites can accommodate interstitial atoms: the tetrahedral sites with 4 near-neighbour atoms and the octahedral sites with 6 near-neighbour atoms. Assuming that the atoms are close-packed spheres, the maximum size of an interstitial atom that can be accommodated into these sites is given by: dt = 0.225×D = 0.568 Å do = 0.414×D = 1.044 Å

tetrahedral sites octahedral sites

with the diameter D of the lattice atoms. The numerical values are given for an iron structure.

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The diameter of a carbon atom is 1.54 Å. In the case of austenite, the two types of sites are too small to accommodate the carbon atoms. A distortion of the lattice is thus necessary and the carbon atoms will preferentially fill the octahedral sites. In a bcc structure, the maximum sizes of interstitial atoms are given by: dt = 0.291×D = 0.733 Å tetrahedral sites octahedral sites do = 0.155×D = 0.391 Å There is less space available for the interstitials in this structure. Although, the octahedral sites are smaller, the carbon atoms will still occupy these positions. The distortion induced is then huge and asymmetric. The lattice expands along the z direction leading to a bodycentered tetragonal (bct) structure. 2.2.2. The Bain model Bain3 revealed a particular correspondence between austenite and martensite. He has shown that a bcc cell can be drawn within two fcc cells (Fig. 2).

(a)

(b)

Fig. 2. The Bain’s model: (a) conventional unit cell, (b) relationship between the fcc and the bct cells. (From www.msm.cam.ac.uk/phase-trans/2002/encyclop.martensite.html).

There are other ways to generate a bcc cell from a fcc structure but the one shown in Fig. 2 involves the smallest strain. This strain is called the Bain strain. It involves a contraction of about 20% along the z-axis and an expansion of about 12% along the x- and y-axis. However, such


123

distortion will leave no plane invariant (undistorted and unrotated) and is inconsistent with the experimental proof of invariant-plane strain shape deformation. A lattice-invariant distortion should be added in the form of slip or stacks of twins. This additional deformation has to be latticeinvariant insofar as the structure change was already done with the Bain distortion. The purpose of the lattice-invariant distortion is to shear the cell resulting from the Bain distortion in order to obtain an undistorted interface between the martensite and the parent phase. This additional deformation is known as inhomogeneous shear or complementary shear. It leads to the formation of a substructure inside the martensite phase. However, even after this second deformation, the plane of contact between the martensite and the parent phase is still rotated and an third deformation, a rigid body rotation, should be added to maintain the habit plane unrotated. Microscopically, the habit plane of the martensite plates results from a succession of slipped planes or thin twins (Fig. 3). By averaging the distortion over many individual slipped planes or twins, the “net” distortion is found equal to zero. By adjusting the width and the angle of the individual features, the habit plane can adopt a large variety of orientations.

Fig. 3. Simplified mechanism for the martensitic transformation. From left to right: original austenite; effect of Bain strain; additional deformation by slip, additional deformation by twinning.

2.2.3. Comparison with experimental results A wide scatter in experimental measurements exists for the habit planes of a given type of steel. Additional elements also modify the habit planes. The reason why the martensitic transformation can lead to different habit planes is still unclear. Most probably, it results from

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different operative modes of the inhomogeneous shear (i.e. different planes or directions) or activation of multiple modes. However, a trend can be extracted. When the carbon contents increases, the transition between the habit planes obeys the following scheme:

{111}γ → {225}γ → {259}γ For low-carbon steels (up to 0.4% C), the habit plane is {111}γ and the martensite phase has a lath morphology or consists of bundles of needles lying on the {111}γ planes. The typical dimensions of a lath are 0.3 × 4 × 200 µm. The laths are untwinned but contain a high density of dislocations suggesting that the inhomogeneous shear is slip. The movement of interface dislocations affects the lattice-invariant deformation. When the habit plane is {225}γ and {259}γ, the martensite consists mainly of twinned plates or has a lens morphology. The lenticular plates are promoted because of their low-energy shape. They usually contain a high density of twins revealing that the inhomogeneous shear occurs via twinning. However, an accurate description of the morphology of martensite is difficult as, after growth, the martensitic phase has generally an irregular shape. 2.2.4. Summary

The martensitic transformation involves three different steps: - the Bain strain, which converts the crystal structure of the parent phase to that of the martensite phase - an inhomogeneous shear, which gives the lattice-invariant distortion - a rigid body rotation, which leaves the habit plane unrotated. 3. Martensite nucleation

Like numerous other phase transformations, the formation of martensite also occurs by a nucleation and growth mechanism. However, the whole transformation occurs at a velocity close to the speed of sound. A martensite plate, once formed, grows to its full size in 10-5 to 10-7 s. Due to the extreme rapidity of this process, it is difficult to study the transformation experimentally. One way of measuring the growth


125

Fig. 4. Schematic representation of the resistivity change during the martensitic transformation of a Fe-Ni alloy. (adapted from reference 4).

velocity of a nucleated martensite is to measure the resistivity change caused upon transformation from the parent austenite (γ phase) to the martensite (α phase). After a small initial increase in resistivity due to the initial strain in the austenite lattice caused by formation of the martensite nucleus, a steep decrease is observed upon transformation from γ to α (Fig. 4). The initial increase suggests that the austenite and the newly formed martensite nucleus are coherent. 3.1. Free energy change of martensitic transformation

Fig. 5(a) shows the free energy – temperature diagram for a diffusionless transformation.

(a)

(b)

Fig. 5. (a) Free energy – temperature diagram for the diffusionless transformation from the γ-phase to the α-phase; (b) variation of the martensite fractions with temperature upon cooling and heating.

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T0 is the temperature at which the parent phase and the martensite are in thermodynamic equilibrium. The reaction begins at the martensite start temperature (Ms) and is completed at the martensite finish temperature (Mf) below which further cooling does not increase the amount of martensite anymore (Fig. 5(b)). Theoretically, all austenite should have transformed into martensite at Mf. However, in practice, a small amount of austenite remains. The reverse transformation from martensite to austenite begins at the austenite start temperature (As) and is completed at the austenite finish temperature (Af). The chemical driving force to initiate diffusionless transformations is larger than for diffusional processes and is given by:

⎛ T − Ms ⎞ ΔG γ→α = ΔH γ→α ⎜ 0 ⎟ ⎝ T0 ⎠

(1)

where ΔH γ−α is the enthalpy for the transformation from the parent phase (γ-phase) to the martensitic phase (α-phase). 3.2. Formation of martensite nuclei

The increase in Gibbs free energy due to the formation of a coherent inclusion of martensite nuclei in the parent phase is expressed as:

ΔG = Aγ + VΔG s − VΔG v

(2)

where ΔG is the free energy change, A and V are the surface area and the volume of the nucleus, respectively, ΔGs is the strain energy, ΔGv is the volume free energy and γ is the interfacial free energy. Eq. (2) does not consider additional energies that might be active during martensite nucleation coming from thermal stresses generated upon cooling, addition of external stresses and stresses produced ahead of growing martensitic plates. Similar to other nucleation events, the surface and elastic terms tend to increase ΔG, while the volume component tends to decrease it. It has to be noted that the strain energy of the coherent nucleus contributes significantly towards the total free energy while the contribution from the interfacial (surface) component is relatively small.


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Let us consider the nucleation of a thin ellipsoid shaped nucleus, having a radius a, a semi-thickness c and volume V. The free energy increase can be given as: 4 2 2 ⎛ 2(2 − ν ) ⎞ c ΔG = 2πa 2 γ + 2μV ( s / 2 ) ⎜ ⎟ π − πa c.ΔG v ⎝ 8(1 − ν ) ⎠ a 3

(3)

where γ is the interfacial energy, ν is the Poisson’s ratio of the austenite, μ is the shear modulus and ΔGV is the free energy difference at the Ms temperature. For ν = 1/3, Eq. (3) can be simplified to: ΔG = 2πa 2 γ (surface) +

16 4 2 ( s / 2 ) μac 2 (elastic) − πa 2c.ΔG v (volume) (4) 3 3

The assumptions made are that nucleation occurs homogeneously without the aid of lattice defects like grain boundaries and that shear is responsible for nucleus formation. Additionally, the interface is coherent. The critical free energy minimum is obtained by differentiation of Eq. (4) with respect to a and c and can be expressed as: 4

512 γ3 ⎛s⎞ 2 ΔG = μ π joules/nucleus . . 2 ⎜ ⎟ 3 ( ΔG v ) ⎝ 2 ⎠ *

(5)

The critical nucleus size (c* and a*) are given by:

2γ c = ΔG v *

and

16γμ( s ) 2 2 a = 2 Δ G ( v) *

(6)

Metallographic studies on Fe-Ni single crystal particles have shown that upon cooling from the Ms temperature, martensite nuclei form heterogeneously. Several reasons have been given to account for this observation: 1. Cooling from below Ms down to 4 K does not result in complete transformation. 2. The number of nuclei formed is of the order of 104 per mm3, which is less than what is expected for homogeneous nucleation. 3. The number of nuclei increases with increased supercooling. 4. Surfaces and grain boundaries do not seem to be a preferred site for nucleation.

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This suggests also that the transformation is initiated at other defects: dislocations. 3.3. Role of dislocations in martensite nucleation

Based on nucleus density, it has been predicted that dislocations are preferred sites for martensite nucleation. It was first demonstrated by Zener5 how the movement of dislocations during twinning can generate a thin bcc lattice region from an fcc one. Fig. 6 shows layers of a fcc close packed plane, numbered 1,2,3.

Fig. 6. Zener’s model for nucleation of martensite by half-twinning shear (based on6).

The normal twinning vector b1 is formed by the dissociation of a 110 dislocations into two partials. 2

i.e.:

b = b1 + b2 a a a [110] = ⎡⎣ 211⎤⎦ + ⎡⎣ 12 1 ⎤⎦ 2 6 6

(7) (8)

For the generation of a bcc structure, all the “full circle” atoms 1 a (Level 3) jump by: b1 = ⎡⎣ 211⎤⎦ . 2 12


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This results in a thin nucleus. A thicker nucleus can be formed at dislocation pile-ups due to reduced slip vectors. An alternative approach of martensite nucleation was proposed by Venables7, with respect to martensite nucleation in stainless steels. According to Venables, the martensite forms via an intermediate phase having an hcp structure, denoted as epsilon martensite (ε):

γ→ε→α

(9)

Thickening of this ε structure occurs by non-homogeneous halftwinning on every {111}γ plane. However, there has been no direct evidence for this and electron microscopy studies on martensitic stainless steels have shown that the γ → ε or γ → α transformations occur independently of each other8. In case of pure cobalt, it has been demonstrated that half-twinning shear can induce a martensitic phase transition. This fcc → cph transformation occurs at 390°C. Transmission electron microscopy studies indicate that this transformation is a result of the formation of large number of dislocations with stacking faults appearing at the grain boundaries. Other models describing the role of dislocations in martensite nucleation have been reported9. They won’t be detailed here. 3.4. Dislocation strain energy assisted transformation

Let us now consider that the nucleation barrier necessary for the formation of coherent nuclei is reduced by the help of the elastic strain field of a dislocation. In this case, the dilatation associated with the extra half plan of the dislocation contributes to the Bain strain. Hence, the total Gibbs free energy can be represented as:

ΔG =Aγ + V ΔGs − V ΔGv − ΔGd

(10)

where ΔGd is the dislocation interaction energy which reduces the nucleation energy barrier. ΔGd, in turn, can be represented as:

ΔGd = 2μ sπ .ac.b where s is the shear strain of the nucleus and b is the Burgers vector.

(11)

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Fig. 7. Schematic diagram showing the need for twin nucleation when the martensitic plate reaches a given size.

Thus, using Eq. (10) and recalling Eq. (3), the total energy can be written as:

ΔG = 2π a 2γ +

16π 4 2 ( s / 2 ) μ ac 2 − π a 2c.ΔGv − 2μ sπ ac.b 3 3

(12)

It is clear from the above expression that the energy depends on the diameter (a), the thickness (c), the degree of assistance from strain field and whether it is twinned or not, as it is shown in Fig. 6. A fully coherent nucleus can reach a size of about 20 nm diameter and two to three atoms in thickness. However, further growth or thickening occurs only upon formation of twins or slip, which tends to reduce the strain energy. This theory offers the advantage to combine the crystallographic features of the Bain strain and inhomogeneous shear. The Ms temperature depends on the orientation or configuration of the dislocations. This is because large undercooling below Ms is necessary for nucleation. It suggests that a large chemical driving force is needed for the transformation and the presence of ideally oriented dislocations remains a statistical probability. 4. Martensite growth

Martensite growth occurs once the nucleation barrier has been overcome. It involves the edgewise propagation of the plate across the parent crystal


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and the thickening of the plate. Growth occurs rapidly until obstacles such as another plate or high angle grain boundaries are hit. Initially, very thin plates (with large a/c ratio) are formed and subsequently they thicken. A typical feature in high carbon martensites is the “midrib” of fine twins. Low carbon martensites are usually lath-shaped with a high dislocation density. Since the growth is a very high speed process, the interface between the parent phase and the martensite is most often a glissile semicoherent one. It consists of coherent regions separated by matching dislocations or twin boundaries. The movement of the glissile interface induces homogeneous transformation of the coherent areas as well as gliding of the matching dislocations in their slip planes or extension of twin boundaries. When the interface is moving, all the atoms in the parent phase are integrated in the martensite structure. This process involves no diffusion, no thermal activation and atomic movements of less than one interatomic spacing. It is thought that the Ms temperature dictates the mode of the inhomogeneous shear. At lower temperatures, slip-twinning transition is associated with increasing difficulty for nucleation of the dislocations necessary for slip. However, the chemical energy for the transformation is independent of the Ms temperature. Another factor affecting the growth is the mode of formation of the nucleus, i.e. by homogeneous Bain deformation or inhomogeneous shear. In the following, some aspects of the martensite growth in steels will be described. 4.1. Growth of lath martensite

The morphology of a lath of dimensions a > b >>c growing on a {111}γ plane involves nucleation and glide of the transforming dislocations, which move on distinct ledges behind the growing front. Dislocations nucleate due to the large misfit between the bct and fcc lattices. The required condition is that the stress at the interface should exceed the theoretical strength of the material. Using Eshelby’s approach10, the maximum shear stress at the martensite/austenite interface is given by:

σ ≅ 2 μ sc / a

(13)

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Fig. 8. Eq. (13) plotted versus the a/c ratio (modified from reference 6).

where μ is the shear modulus of the austenite. According to Kelly11, a theoretical shear strength of 0.025 μ can be used for fcc materials as a minimum or threshold stress for nucleation of dislocations. Assuming s = 0.2 (typical value for bulk and lath martensite), Eq. (13) is plotted in terms of a/c ratio in Fig. 8. The Kelly threshold stress can be exceeded in the case of lath martensite, but it is unlikely in the case of thinner plates. When nucleation of dislocations occurs at a highly strained lath interface, the misfit energy gets reduced and the lath is able to grow. Using internal friction measurements, it has been shown that, in lath martensite, the carbon density is higher at cell walls than inside the cells. This suggests that a limited amount of carbon might diffuse during transformation. Conventionally, the lath morphology is associated with higher Ms temperatures. This also allows dislocation climb and cell formation after transformation. The amount of retained austenite between laths is small in lath martensite, which is important for the mechanical properties of low-carbon steels. 4.2. Plate martensite

Plate martensite is a more common feature in medium and high-carbon steels. It is associated with a low Ms temperature and more retained


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austenite. The martensite is usually completely twinned. The morphology of the plates is much thinner compared with lath martensite and bainite. Once nucleated, twinned martensite grows very rapidly. Hence, the mechanism is not yet clear. The transition from twinning to dislocations can result from changes in the growth rate. Martensite formed at higher temperatures or slower rates grows by a slip mechanism while martensite formed at lower temperature or higher rate grows by twinning. Frank12 proposed a model for the formation of a dislocation-generated martensite. He considered that by inserting a set of screw dislocations at the interface, the lattice misfit could be reduced to a minimum. This is illustrated in Fig. 9.

Fig. 9. The Frank model (from reference 6).

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The close-packed planes of the fcc and bcc structures, (111)γ and (101)α respectively, meet along the habit plane (Fig. 9(a)). In order to reduce the misfit and to equalize the atomic spacings of the (111)γ and (101)α planes at the interface, a rotation ψ is introduced but is not sufficient. Inserting an array of screw dislocations with a spacing of six atom planes removes the misfit (Fig. 9(b)). 4.3. Stabilization

Stabilization can be defined as a process that occurs when a sample is cooled to a temperature between Ms and Mf and held there for some period of time prior to cooling again. The transformation to martensite does not immediately continue and so the total amount of martensite is less than that obtained by continuous cooling. The existing martensite plates do not grow. Instead, new plates nucleate. The amount of stabilization is a function of the time held at particular temperature between Ms and Mf. This phenomenon is not fully understood, yet. It seems that carbon atoms have enough time to diffuse to the interface, since the plate growth is under high stress. Additionally, local atomic relaxation at the interface increases the nucleation barrier necessary to generate dislocations 4.4. Effect of external stresses

External stresses aid martensite nucleation if the elastic strain components of the stress contribute to the Bain strain and raise the Ms temperature. Upon plastic deformation, there is an upper limit of Ms temperature defined as Md. The Ms temperature can be suppressed upon hydrostatic compression, since the increasing pressure stabilizes the close-packed austenite and lowers the driving force for the transformation to martensite. However, in the presence of a magnetic field, the Ms temperature is raised, since it favours the formation of a ferromagnetic phase. Too much plastic deformation can also hinder martensitic transformation. Although it increases the dislocation density and hence the number of nucleation sites, it also introduces restraints to the growth of nuclei. This increases the number of nucleation sites, and


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hence, refines the plate size. This phenomenon is called the “ausforming” process. The high strength of ausformed steel is due to a combination of fine plate size, solution hardening and dislocation hardening. The mechanical effects on martensitic transformations will be further discussed in section 6. 4.5. Role of grain size

Even though the grain size does not affect the number of martensite nuclei in a given volume, the final plate size is a function of grain size. In a large grain size material, the effect of large residual stresses can cause quench cracking and leads to an increase of the dislocation density of martensite. In general, fine grain-sized alloys along with a smaller martensite plate size exhibit superior mechanical properties13. 5. Tempering of ferrous martensites

The formed martensite always requires some heat treatment in order to improve the toughness and in some cases the strength of the steel. This is achieved by tempering. It is an ageing process where the martensite transforms to a mixture of ferrite and ε-carbide or ferrite and cementite.

α ' → α + ε − carbide or α ' → α + Fe3C

(14)

In the presence of strong carbide forming elements like Mo, Ti, Nb or V, the stable precipitates can be an alloy carbide instead of cementite. The phases that form depend on the heat treatment practice adopted. It is also possible that carbon segregation occurs during tempering. Especially in low-carbon steels, martensite starts to form at relatively high temperatures and can have sufficient time during quenching to segregate or precipitate as ε-carbide or cementite. ε-carbide has a hexagonal crystal structure and precipitates in the form of laths. Cementite (Fe3C) forms in most steels upon tempering between 250°C and 700°C. It has an orthorhombic structure. In case of alloyed steels having sufficient carbide-forming elements, the cementite composition can be represented

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as (FeM)3C where M is the carbide forming element. The precipitates form in situ whereby the alloy carbide can nucleate at several points at the cementite/ferrite interfaces and grows until the cementite disappears. Alternatively, they can form by a separate nucleation and growth process whereby the alloy carbides form heterogeneously within the ferrite on dislocations, lath boundaries and prior austenite boundaries. The actual mechanism depends on the alloy composition. Alloy carbides induce strength to the material and are advantageously used in high-speed tool steels. Tempering of molybdenum steels results in the formation of alloy carbides by a process commonly termed as secondary hardening. The effectiveness of carbides as strengtheners depends on their fineness and volume fraction. Finest precipitates are obtained from VC, NbC, TiC, HfC. These compounds are all close-packed intermetallics. The fineness depends on the free energy of formation. The volume fraction depends on the solubility of the alloy carbide in the austenite prior to quenching, relative to the stability in ferrite at a given tempering temperature. In most steels, containing more than 0.4 % C, austenite is retained upon quenching. It subsequently decomposes to bainite in the temperature range of 250-300 °C. As-quenched lath martensite contains high-angle grain boundaries, low-angle cell boundaries and dislocation tangles. Recovery occurs above 400 °C and leads to elimination of tangles and cell walls. However, the lath like structure is retained. Though the aim of tempering is to improve the ductility, in some steels, tempering in the range 350-575 °C can lead to embrittlement and thus loss of ductility. This is due to the segregation of impurity atoms like P, Sb or Sn to the prior austenite boundaries. 6. Non-thermoelastic and thermoelastic martensitic transformations 6.1. Non-thermoelastic transformations

6.1.1. General features Let us consider the cooling of a non-thermoelastic alloy. During cooling, nucleation and growth of martensite occur. If the cooling is stopped, the


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growth ceases and subsequent cooling does not lead to further growth of the martensite phase. The interface between the martensite and the parent phase apparently becomes immobilized. When the martensite is heated again, the interface does not move backward. The reverse martensite → parent phase transformation takes place by nucleation of small platelets of the parent phase inside the martensite. In steels, the usual stages of martensite tempering occur. For the non-thermoelastic transformations, a large difference between MS and Mf is observed.

6.1.2. Mechanical effects The martensitic transformation can be seen as a “shear” process that implies a cooperative shear movement of atoms. This process can be aided by an applied elastic stress. The non-thermoelastic martensitic transformation is associated with an apparent immobilized interface. If martensite formation is strain- or stress-induced, no retransformation of martensite into austenite is supposed to happen after unloading. The martensite phase is stable. It has been shown that ferrous martensite formed during deformation leads to a considerable increase of the work-hardening rate and elongation. This phenomenon is known as transformation-induced plasticity (TRIP)14-17.

6.1.3. Mechanical driving force When a stress is applied to the austenite at T1 (Fig. 10), a mechanical driving force U is added to the chemical driving force, ΔG Tγ→α . 1 Stress-induced martensite formation occurs when the stress reaches a critical stress σc. For this stress, the contribution of the mechanical and γ→α the chemical driving force is equal to the total driving force ΔG M s necessary to induce the martensitic transformation: γ→α ΔG M =ΔG Tγ→α + U' s 1

(15)

The mechanical driving force depends on the stress and the orientation of the forming martensite. It can be expressed by:

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Fig. 10. Schematic diagram showing the contribution of the mechanical driving force γ→α . (U’) and the chemical driving force ΔG Tγ→α to the total driving force ΔG M 1

s

U = τγ 0 + σε n

(16)

where τ is the shear stress along the transformation shear direction in the martensite habit plate, γ0 is the transformation shear strain along the shear direction in the habit plane, σ is the normal stress perpendicular to the habit plane and εn is the dilational component of the transformation strain. The applied stress σa can be decomposed into a shear component τ and a normal component σ, which are given by:

τ = (1/ 2) σa (sin 2θ) cos α and

σ = ±(1/ 2) σa (1 + cos 2θ)

(17)

where θ is the angle between the applied stress and the normal to the habit plane and α is the angle between the transformation shear direction and the maximum shear direction of the applied stress in the habit plane (Fig. 11). In Eq. (17) the plus (+) and the minus (-) correspond to a loading in tension or in compression, respectively. From Eqs. (15-17) follows: U = (1/ 2) σa ⎡γ ⎣ 0 (sin 2θ) cos α ± ε n (1 + cos 2θ ) ⎤⎦

(18)


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Fig. 11. Schmid-factor diagram. S is the direction of the shape strain for the martensite, Sm is the maximum shape strain elongation parallel to the habit plane HP, N is the normal to the habit plane.

When the martensitic transformation begins, due to the applied stress, the martensite plates, which are oriented in such a way that the mechanical driving force is maximum, form first. The maximum mechanical driving force is obtained for α = 0 and dU/dθ = 0 and Eq. (18) becomes: U ' = (1/ 2) σ c ⎡γ ⎣ 0 (sin 2θ ') cos α ± ε n (1 + cos 2θ ') ⎤⎦

(19)

where σc is the critical applied stress for the martensitic transformation to begin. The chemical driving force decreases linearly when the temperature increases for T > Ms. The critical applied stress σc is thus expected to increase linearly with the temperature in the same temperature range. This is true up to a certain temperature M sσ (Fig. 12). For T> M sσ , another trend is observed. Let us consider the deformation of the austenite for the temperature T = T2. Under the applied stress σa, the austenite will first deform elastically. Then, when σa = σ1, the plastic

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deformation of the austenite begins. Strain-hardening occurs and the stress increases up to σ2 where the martensitic transformation begins. As it can be seen in Fig. 12, σ2 is lower than the critical applied stress σc predicted by the linear trend. This decrease of the onset stress for martensitic transformation is due to plastic deformation of the austenite. However, the origin of this phenomenon is still unclear. One hypothesis is that the martensite nucleation is strain-induced. Another one is that the stress is locally concentrated at obstacles, such as grain boundaries, because of plastic deformation of the austenite. The local stress reaches the critical stress level σc, promoting the formation of martensite.

Fig. 12. Critical stress to induce martensite formation as a function of the temperature.

6.2. Thermoelastic transformations

6.2.1. General features Let us consider now the cooling of a thermoelastic alloy. During cooling, nucleation and growth of martensite occur. If the cooling is stopped, the growth ceases. However, in this case, if cooling starts again, the growth of the existing martensite continues. When martensite is heated, the


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interface moves backwards and shrinkage of the martensite plates is observed. The parent phase elastically accommodates the martensite plates. No dislocations are observed at the interface and the interface remains glissile. The stored elastic energy constitutes the driving force for the reverse transformation.

6.2.2. Mechanical effects The thermoelastic transformations are associated with remarkable mechanical effects in both the martensite and the parent phases. The most famous one is the shape memory effect (SME). However, this general name covers a large variety of interesting and unusual mechanical behaviours18,19. Some of them will be briefly described in the following section. Several alloys exhibit the SME. Among them, the classical examples are Ni-Ti alloys and their ternary variations20. The SME has also been revealed in Cu-based alloys21, 22 and Fe-base alloys23. In these alloys, the martensite is internally twinned or faulted, which is a sign of the inhomogeneous shear of the crystallographic theory.

6.2.3. The shape memory effect (SME) The shape memory effect is the ability of an alloy to be severely deformed and to return to its original shape after heating. The process to regain the initial shape is closely associated with the reverse transformation of the deformed martensite. The processes involved in this mechanical behaviour can be roughly described as follows (Fig. 13). Upon cooling, the parent phase transforms into martensite plates with different orientations. Usually, a single crystal of parent phase gives 24 orientations of martensite. During deformation, the different orientations of martensite reorganise to give one single orientation, which can be achieved by twinning and movements of certain interfaces. The remaining orientation is the one that will lead to the maximal elongation of the specimen as a whole in the tensile direction. When the material is heated again, the single orientation of martensite gives a single orientation of the parent phase and the material recovers its initial shape.

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The reverse transformation from martensite to the parent phase can be compared to an “unshearing” process.

Fig. 13. Schematic description of the shape memory http://www.cs.ualberta.ca/~database/MEMS/sma_mems/sma.html).

effect.

(From

6.2.4. The two-way shape memory effect The two-way shape memory effect (TWSME) is characterized by a spontaneous deformation of the specimen upon cooling from Ms to Mf and an “undeformation” of the specimen upon heating from As to Af. In order to obtain such an effect, a “training” of the material is necessary. This training can be done following two ways: (i) SME cycling: the specimen is cooled below Mf, deformed and then heated again above Af. The processes described previously occur. This procedure is repeated several times. (ii) SIM cycling: the specimen is deformed above Ms to produce stress-induced martensite (SIM). Then, the specimen is unloaded and the reversal of the SIM occurs. This procedure is repeated several times. After sufficient training, the first step of the TWSME occurs upon cooling where a large proportion of the martensite adopts a preferred


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orientation, leading to a spontaneous strain. The SME cycling seems to be less efficient than the SIM cycling to obtain the TWSME.

6.2.5. Pseudo-elastic effects The pseudo-elastic effects refer to the ability of a material, upon unloading at a constant temperature, to completely recover large strains, well above the elastic limit. Two categories of pseudo-elastic behaviours can be distinguished, depending on the nature of the driving forces and the mechanisms involved: Superelasticity: under the effect of an applied stress above Ms, mechanically elastic martensite is stress-induced and will disappear if the stress is released. Schematic stress-strain curves characteristic of a superelastic behavior are shown in Fig. 14. When the applied stress reaches the critical stress σc, stress-induced martensite is formed (upper plateau). Upon unloading, reversal of the stress-induced martensite occurs. Plates of the parent phase nucleate and grow (lower plateau). The plateau stresses depend on the temperature. The stress-strain curves in Fig. 14 are known as superelastic loops.

Fig. 14. Schematic illustration of the superelastic behavior.

Rubber-like behavior: although the superelasticity leads to a rubberlike behavior, this terminology refers to a mechanical effect that does not involve a phase transformation and is related to the reversible movement of twin boundaries or martensite boundaries.

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7. Summary

One of the purposes of this chapter was to give an overview on diffusionless transformations. A diffusionless transformation (or martensitic transformation) is a solid state transformation which is athermal, diffusionless and involves cooperative movements of the atoms less than one interatomic spacing. The phenomenological theory describing the crystallography of diffusionless transformations is based on the assumption that the macroscopic strain (or shape deformation strain) is associated with an invariant-plane strain. The invariant plane is the undistorted and unrotated habit plane. This macroscopic strain can be divided into three different components: – a Bain strain, which gives the lattice correspondence between the parent and the product phases – a inhomogeneous shear strain which leaves the invariant (or habit) plane undistorted – a rigid body rotation which leaves the invariant plane unrotated The nucleation and growth of martensite occur at a high velocity. The nucleation is heterogeneous and the transformation is mainly initiated at dislocations. Depending on the alloy composition, the temperature and the strain rate, the martensitic phase grows by dislocation slip or twinning. From a technological point of view, diffusionless transformations are of great interest because they are associated with improved mechanical properties such as transformation-induced plasticity or the shape memory effect. References 1. 2. 3. 4. 5. 6.

K. Otsuka, and X Ren,, Mater. Sci. Eng., 275, 89 (1999). P.M. Kelly, and L.R., Francis Rose, Prog. Mater. Sci., 47, 463 (2002). E.C., Bain, Trans AIME, 70, 25 (1924). R. Bunshah, and R.F., Mehl, Transactions AIME, 197 1251 (1953). C.Zener, Elasticity and Anelasticity of Metals, Univ. of Chicago Press, 1948. D.A. Porter and K.E. Easterling, Phase transformations in Metals and Alloys, 2nd Ed., Chapman & Hall, 1992. 7. J.A., Venables, Phil. Mag., 7, 35 (1942).


8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

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J.W., Brooks, M. H. Loretto, and R.E., Smallman, Acta Metall., 27, 1839 (1979). G.B. Olson, and M. Cohen, Metall. Trans., 7A, 1897 and 1905 and 1915, (1976). J.W Eshelby, Proc. Roy. Soc. A, 241, 376 (1957). A. Kelly, Strong Solids, Clarendon, Oxford, 1966. F. C. Frank, Acta Metall., 1, 15 (1953). D. H. Shin, W. G. Kim, J. Y. Ahn, et al., Mater. Sci. Forum, 503-504, 447 (2006). J. R. Patel, and M. Cohen., Acta Metall., 1, 531 (1953). T. Angel., J. Iron Steel Inst., 177, 165 (1954). P. J. Jacques, Curr. Opin. Sol. State Mater. Sci., 8, 285 (2004). S. Turteltaub, and A. S. J. Suiker, J. Mech. Phys. Sol., 53, 1747 (2005). C. M. Wayman, Prog. Mater. Sci., 36, 203 (1992). R. James, and K. F. Hane, Acta Mater., 48, 197 (2000). K. Otsuka, and X. Ren., Prog. Mater. Sci., 50, 511 (2005). C. Y. Chung, and C. W. H. Lam, Mater. Sci. Eng., 275, 622 (1999). F. C. Lovey, and V. Torra, Prog. Mater. Sci., 44, 189 (1999). N. Stanford, and D. P. Dunne, Mater. Sci. Eng., 422, 352 (2006).

Further reading:

C.M. Wayman, Phase transformation, non diffusive, Chap. 15, in Physical Metallurgy, 3rd Ed., ed. R.W. Cahn and P. Haasen, (North-Holland, Amsterdam), 1983. C.M. Wayman, Introduction to the Crystallography of Martensitic Transformations, Macmillan, New York, 1964. J.W. Christian, Theory of Transformation in Metals and Alloys, Pergamon, Oxford, 1965. J. Perkins, The shape memory effect in Alloys, Plenum, New York, 1975.


CHAPTER 7

INTERMETALLICS: CHARACTERISTICS, PROBLEMS AND PROSPECTS Gerhard Sauthoff Max-Planck-Institut für Eisenforschung GmbH, 40074 Düsseldorf, Germany E-mail: [email protected] Intermetallics is the short and summarizing designation for intermetallic compounds (IMCs) and phases, which result from the combination of various metals and which form a tremendously numerous and manifold class of materials. Intermetallics with outstanding physical properties led to the development of functional materials in the past. During the last 20 years intermetallics have aroused enormous and still-increasing interest in materials science and technology with respect to applications at high temperatures. Various new structural materials are being developed around the world, in particular in the United States, Japan, and Germany. The present report overviews thevarious intermetallics which have already been selected for materials developments or which have been and still are regarded as promising for materials developments with emphasis on structural materials.

1. Introduction An overview is given on the vast and manifold field of intermetallics which results from a tutorial for students of the 1st European School in Materials Science of the EU Network of Excellence "Complex Metallic Alloys" CMA. The aim is to point out the characteristics, problems and prospects of intermetallic phases and compounds and of related more or less complex alloys by using appropriate examples. Intermetallics have been subject of a long series of excellent reviews which is not to be continued by the present overview. This overview is based on a recent compilation of the knowledge on intermetallics1 and shorter reviews2, 3. Data of intermetallic materials were collected in reference 4. 147

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Applications of intermetallics are handicapped by their brittleness which is the reverse of the high strength and which is not sufficiently understood. The present still continuing enhanced interest in intermetallics results from the need for stronger metallic materials for applications at higher service temperatures e.g. in energy conversion for increasing the thermal efficiency. Thus the following discussion is done from the position of a physicist who tries to understand what mechanisms control the behaviour of the intermetallics in order to explore the possibilities for developing novel structural materials based on intermetallics for high-temperature applications. 1.1. Definition of intermetallics Intermetallics is the short and summarizing designation for the intermetallic phases and compounds which result from the combination of various metals and which form a tremendously numerous and manifold class of materials as will become visible in the following sections. An example is given by the binary Ni-Al phase diagram (Fig. 1) which comprises the phase NiAl with compositions around 50 at.% among other Ni-rich and Al-rich phases. This NiAl phase has a significantly higher melting temperature than the constituent metals Ni and Al indicating a much stronger bonding between the unlike Ni and Al atoms than between the alike Al atoms and Ni atoms. Its crystal structure is known as B2 structure (Strukturbericht designation) which is a bcc structure with atomic ordering and which clearly differs from the fcc structure of the constituent metals. Accordingly there is a simple general definition5, 6: intermetallic phases and compounds are chemical compounds of metals the crystal structures of which are different from those of the constituent metals. Examples are given by the phase diagrams of metal systems. The composition of an intermetallic may vary within a restricted composition range known as homogeneity range. This homogeneity range may be narrow or vanishing as is the case for a line compound and such phases are usually addressed as intermetallic compounds. Phases with a wide homogeneity range are usually addressed as intermetallic phases. Phases may exist only in a restricted temperature range. There are phases which show an order-disorder transition when heated above a critical

Intermetallics: Characteristics, Problems and Prospects

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temperature as is exemplified by the Cu3Au phase which is fcc ordered (L12 structure) at lower temperatures and disorders above the critical disordering temperature to form the familiar fcc solid solution. The latter phases are known as Kurnakov phases7. The Russian mineralogist Nikolai Semyonovich Kurnakov (1860 - 1941) in Sankt-Peterburg/Russia and the German chemist Gustav Tammann (1861 - 1938) in Göttingen/Germany initiated intensive and broad-scale scientific work on intermetallics at the beginning of the 20th century from which a large number of directive papers resulted8-13.

Fig. 1. Ni-Al phase diagram2

1.2. Historical Intermetallics have been made use of since the beginning of metallurgy as is visible in Table 1.

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Gerhard Sauthoff Table 1. Some applications of intermetallics4.

Only the applications in the 20th century were based on an understanding of the underlying metallurgical principles. The early applications as coatings made use of the high corrosion resistance of the respective intermetallics. The use of amalgams as dental restoratives is an early example of applications as structural materials. The more recent applications as functional materials rely on advantageous physical properties of particular intermetallics. The relatively high strengths of intermetallics are attractive for applications as structural materials at high temperatures. However, these high strengths are usually accompanied by brittleness, and the respective materials developments in the second half of last century have not yet led to widespread structural applications. Topical intermetallics of present interest are listed in Table 2.


151

Table 2. Topical intermetallics of present interest4. Alloy

Majority phase

Application

Functional materials (physical) A15 compounds Cu-Al-Ni, Cu-Zn-Al Fe-Al-Si rare earth magnet materials GaAs

Nb3Sn β-Cu-Al-Ni,β-Cu-Zn-Al Fe2Si SmCo5, Sm2Co17 Sm2Fe17(C,N), Nd2Fe14B

Superconductor shape-memory alloy thermoelectric generator permanent magnet

GaAs

transistor, laser diode, optoelectronic device, solar cell, acoustoelectric device electric heating elements

Kanthal Super, Mosilit Ni-Ti Permalloy Permendur Sendust silicide thin film

MoSi2 NiTi Ni3Fe FeCo Fe3(Si,Al) transition metal silicides,

alloys Terfenol-D Thermoelectrics Functional materials (chemical) M-Cr-Al-Y Pt aluminide rare-earth hydrides

precious metal silicides (Tb,Dy)Fe2 Bi2Te3

silicides Structural materials (under development) Advanced nickel

transition metal silicides

aluminide alloys alpha-2 titanium aluminide alloys B2 nickel aluminide alloys gamma titanium aluminide alloys iron aluminide alloys molybdenum disilicide alloys

NiAl Pt3Al LaNi5

Ni3Al

shape-memory alloy high permeability magnetic alloy high permeability magnetic alloy magnetic head material microelectronics, silicon integrated circuits giant magnetostrictive actuator thermoelectric generator

coating coating hydrogen storage, rechargeable battery coating

high-temperature-wear-resistant components

Ti3Al

jet engine components etc.

NiAl

gas turbine & automotive components

TiAl

aircraft, jet engine, automotive components etc.

Fe3Al, FeAl MoSi2

chemical engineering aircraft, automotive components etc.

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Gerhard Sauthoff

2. Principles There are a huge number of intermetallic phases which differ by stability, crystal structure and by their physical, chemical and mechanical properties. Stability, crystal structure and properties are a function of the particular atomic bonding. However, the principles controlling this function are still unclear as is discussed in the following. If the relation between bonding, phase stability and properties were understood, then it would be possible to predict the properties of given intermetallic phases and alloys which is a prerequisite for specific materials developments. 2.1. Bonding – stability – structure The examples in the various regarded metal systems -e.g. Ni-Al, Cu-Zn, Cu, Au - show that the relation between crystal structure type and atomic properties of the constituent atoms is not a simple one. Thus various criteria have been deduced in the past for correlating structure type and phase type, i.e. for predicting the crystal structure for a given phase or phase group. Compounds of metals from the left-hand side of the periodic table of elements with metals from the right-hand side are known as Zintl phases (named after the German chemist Eduard Zintl (1898 - 1941)). They are characterized by completely filled electron orbitals, in particular by a full octet shell in the normal case14-16. Thus they may be regarded as valence compounds which satisfy the familiar chemical valency rules6, 17. The Zintl phases have crystal structures which are characteristic for typical salts - e.g. NaTl with cubic B32 structure or Mg2Si with cubic C1 structure18 - and therefore ionic bonding is expected. However, all types of bonding - ionic, metallic and covalent - and mixtures thereof have been observed. Accordingly the Zintl phases may be regarded as electron compounds, the crystal structures of which are related to particular valence electron concentrations (average number of valence electrons per atom)17. Electron compounds are best exemplified by the Hume-Rothery phases (named after the British metallurgist William Hume Rothery (1899 - 1968)), the crystal structures of which are related to the valence electron


153

Table 3. Some electron compounds (with data from6). VEC

Structure

Phase

3/2

cubic

B2

complex cubic

A13

Zn3Co , Cu5Si

7/4

close-packed hexagonal close-packed hexagonal

A3 A3

Cu3Ga

21/13

complex cubic

D82

FeAl,, CoAl , NiAl Ag3Al

CuZn3 Ag5Al3 Cu5Zn8 Fe5Zn2

concentrations (VEC) of the phases - see Table 3. However, it has been found also in these cases that the bonding is not purely metallic in spite of the metal-like band structure. The bonding in NiAl is primarily covalent with some metallic character and no ionic component19 which can be understood in view of the band structures of Ni, Al and NiAl20. Ab-initio calculations have shown that - strictly speaking - the B2 aluminides are not Hume-Rothery electron compounds21. Obviously the Hume-Rothery rules, which relate crystal structure to valence electron concentration, are very simplifying rules which describe reality surprisingly well. There are crystal structures which have been derived from the size ratios and packing schemes of the constituent atoms. The respective intermetallic phases are known as size-factor compounds or topologically close-packed intermetallics or Frank-Kaspar phases6,17,22,23. The best-known size-factor compounds are the Laves phases with composition AB2 (named after the German mineralogist and crystallographer Fritz Laves (1906 - 1978)). They represent the most numerous group of intermetallics, which crystallize in the closely related hexagonal C14, cubic C15 or hexagonal C36 structures22. These structures result from an ideal packing of spherical atoms with a size ratio of √(3/2) = 1.225. However, nature is more complex as is visualized by Fig. 2. Obviously the ideal size ratio is more the exception than the rule. In addition it is noted that the valence electron concentration VEC seems to control the crystal structure at least in the case of the binary compounds AB2 of transition metals as is indicated by the border lines in Fig. 2a. However, there are ternary Laves phases A(B1-xCx)2 of transition metals which do not adhere to these border lines (Fig. 2b) and in the case of ternary Laves phases A(B1-xCx)2 with Al these border lines are no longer valid (Fig. 2c).

154

Gerhard Sauthoff

Radius ratio

Radius ratio

C15 C14

C15 C14 Zr(Co1-xCr)2

Valence electron concentration Valence electron concentration

a

b Radius ratio

C15 C14 a

c

a

a c c b

c b

b

b

Valence electron concentration Fig. 2. Atomic radius ratio as a function of valence electron concentration for various binary Laves phases of transition metals (data from27) with C14 (o) and C15 ( ∆) structure and the phase TaNi2 with C11 structure (the filled symbols indicate a temperature-dependent phase transition whereas indicates conflicting information on the crystal structure) (a), ternary Laves phases of transition metals (the interconnecting continuous / interrupted lines indicate the reported / supposed miscibility ranges, respectively) (b), and ternary Laves phases of transition metals and Al (a: Ti(Fe1-xAlx)2 2 , b: Nb(Co1-x,Alx2 , c: Ta(Fe1-xAlx)2 ) – see text28.

Obviously atomic size ratio and valence concentration are not sufficient for characterizing the complex bonding behaviour of the Laves phases. However, it is noted that the Laves phases show primarily metallic bonding according to the scarce information available5, 17. The complex


155

behaviour of the Laves phases is already exemplified by the binary phase diagrams of the systems of Cr-Ti, Co-Nb and Fe-Zr, which all contain all three crystal variants C14, C15 and C36 of the respective Laves phases24. The case of the ternary Laves phase Zr(Fe1-xAlx)2 is particularly intriguing since x can be varied between 0 and 1 which produces a sequence of crystal structure transformations with C15 → C14 → C15 → C1425. This cannot be rationalized by any argument based on atomic size ratios and valence concentrations since these parameters vary monotonously with x. Cooperative work is in progress for improving the understanding of the Laves phase behaviour26. A more specific characterization of the type of bonding should take account of the particular electron distribution of the respective atoms which is reflected in a simple way by the positions of the atoms in the periodic table. Accordingly David Pettifor numbered the atoms in the periodic table following the increase in the number of electrons in the electron shells to receive the so-called Mendeleeff numbers of the atoms29. Using the Mendeleev numbers as coordinates of binary intermetallic phases, Pettifor constructed structure maps, in which the phases with common crystal structure were located in restricted areas. However, the areas were not always cohesive, the borders were not well defined, areas overlapped, and there were phases in “wrong” areas. Thus the characterization of electronic structure by Mendeleev numbers is not sufficient and the predictive power of such structure maps is restricted. It has to be concluded that the bonding character and crystal structure of an intermetallic phase can be predicted only on the basis of a quantum-mechanical ab-initio calculation17, 30, 31. Much progress has been made in this respect32, 34 and for various important phases with not too complicated structures the crystal energy has been calculated as a function of lattice structure in order to determine the phase stability35-46. However, these calculations are very time-consuming even in simple cases - i.e. for small unit cells with few atoms - and more progress is necessary in the future with respect to computing power and understanding in order to give guidance to practical materials developments on the basis of multinary phases with less simple crystal structures.

156

Gerhard Sauthoff

The challenge of this problem is best exemplified by the case of the three simple binary phases FeAl, CoAl and NiAl which all crystallize in the simple bcc-ordered B2 structure and show similar melting temperatures in similar phase diagrams. The behaviour of these phases is well known, and in particular it is known that the most pronounced behaviour difference is observed for the transition from FeAl to CoAl. However, corresponding ab initio calculations showed that this FeAl-CoAl transition does not much affect the character and directionality of bonding as indicated by the valence electron density difference plot, whereas a distinct directional bonding increase is produced by the transition from CoAl to the most similar NiAl47. A common crystal structure is obviously not a sufficient criterion for similar behaviour. The problem is still more aggravated by the fact that the stability of a crystal structure may sensitively be affected by the presence of interstitial impurities as is demonstrated by Table 4. The expected normal case of solution of impurities without affecting the crystal structure is shown by the fcc-ordered Ni3Al phase which dissolves appreciable amounts of carbon without change of crystal structure. In contrast to this, the dissolution of less than 1 C atom per unit cell in the bcc-ordered Fe3Al phase produces a change of crystal structure to the fcc-ordered L12 structure with the C atom in the unit cell centre which is designated as L1’2 structure, i.e. this phase is also regarded as complex carbide with the perovskite-type structure48, 49. The most intriguing case is the case of the M5Si3 silicides with M = V, N, Ta, the crystal structure of which is affected even by much less than 1 C or N atom per unit cell50. As to the relation between crystal structure and properties, it is well known that there is a correlation of crystal symmetry and unit cell size on the one side and plastic deformability - i.e. brittleness - on the other side: the larger the unit cell and the less symmetric the lattice is, the less deformable the phase is. However, the correlation is a very loose one, and the plot of brittle-to-ductile transition temperatures as a function of the number of atoms per unit cell shows a huge scatter51. From the present discussion follows that the intermetallics do not form a homogeneous group of materials at all. Instead the term intermetallics comprises a huge variety of phases which differ drastically with respect to bonding, crystal structure and properties. Thus intermetallics cannot be


157

discussed generally with respect to all intermetallics, but can be discussed only by referring to specific groups of intermetallics. In view of the complexity of the classification of intermetallics, intermetallic phases are often grouped according to more practical criteria which refer to similarities in behaviour. Table 4. Crystal structures of phases without and with interstitial impurities (data from18). Phase Pearson Symbol cP4 (filled) cP4

Structure StructureBericht Ll2 Ll2

Fe3Al Fe3AlC ξ. When Coulomb repulsion between charge carriers is considered, Efros-Shklovskii found that the temperature dependence of the conductivity should instead follow (37) with a different, generally smaller characteristic temperature To and a square root behavior in the exponent. In the VRH magnetoresistance there is one term at small B, which is negative and linear in B and is due to interference of the contributions to the amplitude of the hopping probability. Shrinking of electronic wave functions in magnetic field gives a positive contribution to the magnetoresistance, which increases as B2. At still larger fields the B dependence is weaker. Hence in moderate fields (38)

308

Östen Rapp

Bo is the field for which one flux quantum e/h passes through the interference area (r3ξ)1/2. r has a different temperature dependence for Mott and Efros-Shklovskii hopping; rM ~ ξ (T'o/T)1/4 and rES ~ ξ (T'o/T)1/2, with the characteristic exponents of 1/4 and 1/2 respectively. This gives different temperature dependencies in the B2 region of Eq. (38). (39)

(40) cM and cES are characteristic constants for each hopping mode. The behavior of Eqs. (39, 40) is limited to a moderately low magnetic field region where the magnetic length ℓB = √ħ/eB < ξ. This limit marks the cross-over from the B2 region to a B2/3 behavior at larger fields. 3.5. A metal-insulator transition (MIT) in i-AlPdRe The strong increase of ρ (T) with decreasing T for QC's with large R has led to the conjecture that such samples are insulators. However, there is no energy gap as in semiconductors, and no factor ~ exp(-ε/kBT) in σ (T). On the other hand, theories for the metallic magnetoresistance break down at large R-values. Attempts to describe σ (T) from Eqs. (36, 37) have led to seemingly acceptable descriptions in various regions of T, but often with so different parameters that firm conclusions have not been obtained. A main difficulty is the existence, apparently in all QC's, of a zero temperature conductivity σ (0) >0, experimentally revealed by a flattening out of σ vs T at low T, sometimes only at T

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