Dislocations in Solids Volume 10
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Dislocations in Solids Volume 10 L12 Ordered Alloys
Edited by
E R. N. N ABARRO Department of Physics University of the Witwatersrand Johannesburg, South Africa and
M . S. D U E S B E R Y Fairfax Materials Research Inc. Springfield, VA, USA
1996 ELSEVIER Amsterdam • Lausanne • New York • Oxford • Shannon • Tokyo
ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands
ISBN: 0-444-82370-0 (volume) ISBN: 0-444-85269-7 (set) (~) 1996 ELSEVIER SCIENCE B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A.- This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands
Preface Volume 10 of this series is the first to appear under the joint editorship of E R. N. Nabarro and M. S. Duesbery. It contains a symposium on the behaviour of dislocations in alloys with the L 12 structure typified by Ni3A1. These structures are of engineering importance because they form an essential constituent, and, in modem alloys, the major constituent, of superalloys. They also present scientific problems in the behaviour of dislocations which are still not fully resolved. Unlike most metals and alloys, they have a flow stress which increases with increasing temperature over a wide range. Unlike most metals and alloys, they behave very differently in tensile tests and in creep tests. The first paper in the L12 symposium is a historically-oriented article by Westbrook, who observed the anomalous temperature dependence in 1967, and remarked that "Although several possible sources of this anomaly were investigated no unequivocal explanation could be established. Further study is indicated". Then Sun and Hazzledine report studies by transmission electron microscopy of the behaviour of dislocations in these alloys, while Caillard goes further in relating the structure of dislocation cores in these and other alloys to the anomalous temperature dependence of the flow stress. Vitek, Pope and Bassani, from the group which first expanded the ideas of Takeuchi and Kuramato on the anomalous temperature dependence into a detailed theory, show how their theory is modified in the light of recent experimental evidence. This theory considers the way in which dislocations in L12 become pinned at certain points, and the way in which these pinning obstacles can be overcome. Chrzan and Mills carry the analysis further by considering not only the individual pinning and unpinning events, but also the way in which the overcoming of one obstacle can trigger the overcoming of neighbouring obstacles. This leads them to an important link with the modern theory of phase transitions. The symposium ends with a rather controversial article by Veyssi~re and Saada, claiming that the model which Vitek, Pope and their colleagues have developed so successfully puts too much emphasis on the anisotropy of the flow stress and too little on other observations. They concentrate on the importance of the viscous motion of dislocations on the cube planes. The first chapter outside the symposium is by Takeuchi and Maeda on the effects of electronic excitation on the plasticity of semiconducting crystals. The effects are well documented, but their mechanism is not clear. The authors support a "phonon kick" interpretation. Finally, Joos examines the role of dislocations in melting, exploiting the old idea that the presence of small dislocation loops in thermal equilibrium reduces the energy required to form more loops, so leading to a first-order phase transition.
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Contents Volume 10
Preface v Contents vii List of contents of Volumes 1-9
ix
48. J. H. Westbrook
Superalloys (Ni-base) and dislocations
1
49. Y. Q. Sun and P. M. Hazzledine
Geometry of dislocation glide in L12 7 I-phase
27
50. D. Caillard and A. Couret
Dislocation cores and yield stress anomalies
69
51. V. Vitek, D. P. Pope and J. L. Bassani
Anomalous yield behaviour of compounds with Lie structure
135
52. D . C . Chrzan and M. J. Mills
Dynamics of dislocation motion in L12 compounds
187
53. P. Veyssi~re and G. Saada
Microscopy and plasticity of the LI2 "~ phase
253
54. K. Maeda and S. Takeuchi
Enhancement of dislocation mobility in semiconducting crystals 55. B. Jo6s
The role of dislocations in melting
505
Author index 595 Subject index 613
vii
443
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CONTENTS OF VOLUMES 1-9 VOLUME 1. The Elastic Theory 1979, 1st repr. 1980; ISBN 0-7204-0756-7
1. 2. 3. 4.
J. Friedel, Dislocations - a n introduction 1 A.M. Kosevich, Crystal dislocations and the theory of elasticity 33 J.W. Steeds and J.R. Willis, Dislocations in anisotropic media 143 J.D. Eshelby, Boundary problems 167 B.K.D. Gairola, Nonlinear elastic problems 223
VOLUME 2. Dislocations in Crystals 1979, 1st repr. 1982; ISBN 0-444-85004-x 5. 6. 7.
R. Bullough and V.K. Tewary, Lattice theories of dislocations S. Amelinckx, Dislocations in particular structures 67 J.W. Matthews, Misfit dislocations 461
1
VOLUME 3. Moving Dislocations 1980; 2nd printing 1983; ISBN 0-444-85015-5 8. 9. 10. 11. 12.
J. Weertman and J.R. Weertman, Moving dislocations 1 Resistance to the motion of dislocations (to be included in a supplementary volume) G. SchSck, Thermodynamics and thermal activation of dislocations 63 J.W. Christian and A.G. Crocker, Dislocations and lattice transformations 165 J.C. Savage, Dislocations in seismology 251
VOLUME 4. Dislocations in Metallurgy 1979; 2nd printing 1983; ISBN 0-444-85025-2 13. 14. 15.
16. 17.
R.W. Balluffi and A.V. Granato, Dislocations, vacancies and interstitials F.C. Larch6, Nucleation and precipitation on dislocations 135 P. Haasen, Solution hardening in.fc.c, metals 155 H. Suzuki, Solid solution hardening in body-centred cubic alloys 191 V. Gerold, Precipitation hardening 219 S.J. Basinski and Z.S. Basinski, Plastic deformation and work hardening E. Smith, Dislocations and cracks 363
1
261
VOLUME 5. Other Effects of Dislocations: Disclinations 1980; 2nd printing 1983; ISBN 0-444-85050-3 18. 19. 20. 21. 22.
C.J. Humphreys, Imaging of dislocations 1 B. Mutaftschiev, Crystal growth and dislocations 57 R. Labusch and W. SchriSter, Electrical properties of dislocations in semiconductors F.R.N. Nabarro and A.T. Quintanilha, Dislocations in superconductors 193 M. K16man, The general theory of disclinations 243
127
x 23. 24.
CONTENTS OF VOLUMES 1-9 Y. Bouligand, Defects and textures in liquid crystals 299 M. Kl6man, Dislocations, disclinations and magnetism 349
VOLUME 6. Applications and Recent Advances 1983; ISBN 0-444-86490-3
25. 26. 27. 28. 29. 30. 31. 32.
J.E Hirth and D.A. Rigney, The application of dislocation concepts in friction and wear 1 C. Laird, The application of dislocation concepts in fatigue 55 C.A.B. Ball and J.H. van der Merwe, The growth of dislocation-free layers 121 V.I. Startsev, Dislocations and strength of metals at very low temperatures 143 A.C. Anderson, The scattering of phonons by dislocations 235 J.G. Byrne, Dislocation studies with positrons 263 H. Neuhfiuser, Slip-line formation and collective dislocation motion 319 J.Th.M. De Hosson, O. Kanert and A.W. Sleeswyk, Dislocations in solids investigated by means of nuclear magnetic resonance 441
VOLUME 7 1986; ISBN 0-444-87011-3
33. 34. 35. 36. 37.
G. Bertotti, A. Ferro, F. Fiorillo and P. Mazzetti, Electrical noise associated with dislocations and plastic flow in metals 1 V.I. Alshits and V.L. lndenbom, Mechanisms of dislocation drag 43 H. Alexander, Dislocations in covalent crystals 113 B.O. Hall, Formation and evolution of dislocation structures during irradiation 235 G.B. Olson and M. Cohen, Dislocation theory of martensitic transformations 295
VOLUME 8. Basic Problems and Applications 1989; ISBN 0-444-70515-5
38. 39. 40. 41. 42. 43.
R.C. Pond, Line defects in inte~. aces 1 M.S. Duesbery, The dislocation core and plasticity 67 B.R. Watts, Conduction electron scattering in dislocated metals 175 W.A. Jesser and J.H. van der Merwe, The prediction of critical misfit and thickness in epitaxy P.J. Jackson, Microstresses and the mechanical properties of crystals 461 H. Conrad and A.F. Sprecher, The electroplastic effect in metals 497
421
VOLUME 9. Dislocations and Disclinations 1992; ISBN 0-444-89560-4 44. 45.
46. 47.
G.R. Anstis and J.L. Hutchison, High-resolution imaging of dislocations 1 I.G. Ritchie and G. Fantozzi, Internal .friction due to the intrinsic properties of dislocations in metals: Kink relaxations 57 N. Narita and J.-I. Takamura, Defi)rmation twinning in.fc.c, and b.c.c, metals 135 A.E. Romanov and V.I. Vladimirov, Disclinations in crystalline solids 191
CHAPTER 48
Superalloys (Ni-base) and DislocationsAn Introduction J. H. W E S T B R O O K Brookline Technologies 5 Brookline Road Ballston Spa, N Y 12020 USA
Dislocations in Solids 9 1996 Elsevier Science B.V. All rights reserved
Edited by E R. N. Nabarro and M. S. Duesbeo,
Contents 3 Background Superalloys, their history and nature 3 Dislocations in intermetallics and superalloys 9 Some significant problems and observations 15 4.1. On dislocations in intermetallics 15 4.2. On dislocations in superalloys 21 Future developments and challenges 23 6. Concluding remarks 24 References 24
1. 2. 3. 4.
~
1. Background A superalloy has been defined as "an alloy developed for elevated temperature service, usually based on Group VIIIA elements, where relatively severe mechanical stressing is encountered and where high surface stability is frequently required" (Sims and Hagel [1 ], p. ix). Compositionally superalloys are perhaps the most complex alloys ever developed in that they typically contain eight to twenty controlled alloying elements, the proportions of which must be exquisitely balanced in order to optimize their properties. Structurally, they are equally complex in that, unlike such common alloys as brass or Monel which are single-phase in character, superalloys have multi-phase microstructures in which the identity, size, shape, and disposition of each of the secondary phases must also be carefully managed for the best performance. Still further complications arise from the fact that superalloys typically are required to operate at temperatures of the order of 0.8Tmp or higher in aggressive atmospheres. Under such conditions most microstructures are unstable; new modes of deformation appear; and constituent elements can react with the environment. Small wonder, then, that understanding of how these alloys respond to mechanical stressing at high temperatures has evolved slowly. Most deformation of crystalline solids is now known to involve dislocations; thus, their structure, number, disposition, mutual interaction, and interaction with solute atoms and microstructural features all become critical. A large proportion of the content of the volumes in this series over the past 16 years has been devoted to such topics but generally with reference to other classes of materials. In order to understand how we arrived at the present state of the science of dislocations in nickel-base superalloys, we need to look at the separate historical development of superalloys and of dislocations, and then turn to the interaction of these two themes. Finally, we will attempt to project likely directions of future development, both in understanding and in application.
2. Superalloys, their history and nature Superalloys were not invented de novo but came as the culmination of a series of developments over hundreds of years aimed at improved structural materials for use at high temperatures. Although the fact that metals soften with increasing temperature has been known for thousands of years, for most of this period there was no interest in materials with useful high temperature strength. Only two instances are known of very early attempts to actually use metals at high temperatures. The Chinese (475-221 BC) used cast-iron molds for casting tool shapes from non-ferrous metals (Lu Da [2]). Here it was the relatively high melting point of the mold material (and hence minimal distortion and interaction) that was exploited, rather than high temperature strength per se.
4
J.H. Westbrook
Ch. 48
The next relevant development was the introduction of novel tool steels. With the advent of powered machine tools in the 19th c. and the increased use of iron and steel in machine construction, cutting materials superior to high carbon steel were badly needed. Perhaps the first of these was Robert Mushet's Special, introduced in 1868 and containing 1 wt.% Mn and 8-10 wt.% W. Although, as described by Keown [3], this was not a true high-speed steel, it did permit a 50% increase in cutting speed. The first real break-through in this type of material came in 1898 when Taylor and White [4] developed a tungsten steel with a 3.8% Cr content and a special heat treatment. At its first public demonstration at the Paris World Exposition in 1900, it was observed that both the tool tip and the chips coming from the workpiece were "red hot" at a cutting speed of 150 ft/min. (about 10x that possible with Mushet's Special). It was thus clear that the high temperatures generated in high-speed cutting required a tool material which softened substantially less than did the workpiece. Taylor and White's alloy was the forerunner of the 18W-4Cr-IV tool steels we know today. While casting molds and cutting tools are early instances of the need for high temperature materials, a major demand arose only with the progressive development of heat engines (Burstall [5], Ch. VI and VII, pp. 201-365). Following the invention of the first practical steam engines by Savery and Newcomen in 1700, the further improvements by Watt and Trevithick in the late 18th and early 19th c. ushered in the industrial revolution and the "steam power age". It was appreciated that both the boiler and the engine itself required structural materials which would operate at elevated temperatures - the higher the temperature and pressure, the greater the efficiency. However, the early steam engines were materials-limited, not so much by inadequate high temperature strength as by poor materials quality control, poor construction techniques, and lack of pressure measurement and control. At a later point, Charles A. Parsons in 1884 patented his design for a steam turbine that was an immediate commercial success and a further driver for improved high temperature materials. Design improvements over the next 40 years resulted in steady increases in size and efficiency. Over the same period, internal combustion engines, both gasoline and Diesel, were developed. For both the steam turbine and the internal combustion engine, the increases in power and efficiency that were achieved were aided in no small way by concurrent, evolutionary improvements in alloy steels (Dulieu [6]) 1. A greater incentive to the development of high temperature alloys was provided by superchargers intended for both piston engines and gas turbines (Moss [7]). However, only with the advent of the turbojet aircraft engine in the late 1930s, as a result of independent work by Whittle in England and von Ohain in Germany, did the higher temperatures and stresses encountered demand a truly discontinuous change in materials (Somerscales [8], Sims [9]). In answer to this challenge, metallurgists turned to some specialty alloys, originally developed for quite different purposes as summarized in table 1 (Westbrook [10]). Fifty 1For example, Ni-Cr-Mo steels were introduced in 1924 for steam turbine rotors; about the same time certain stainless steels for blades, and low C, 0.5 wt.% Mo steel for steam tubing with capability to 500~ and after 1930, Cr-Mo-V steel for a variety of applications. It is nonetheless remarkable that neither the ASM Handbook of 1930 nor the SAE Handbook of 1933 contain any data on the temperature dependence of mechanical properties!
w
Superalloys (Ni-base) and dislocations Table 1 Origins of the superalloy families (after Westbrook [10]). Class
Progenitor system
Fe-base
Fe-Cr stainless steel with/without Ni* (1914)
Co-base
Ni-base
Original application cutlery steel
Early superalloy descendant 16-25-6 (1940s)
Co-Cr-W-C**
cutting tool and hard facing
Haynes 23 (1946)
Co-Cr-Mo** (1929)
surgical and dental alloy
Vitallium (1950's)
Ni-Cr*** (1906)
resistance heating element
Nimonic 80 (1940's)
* Truman [ 11] ** Fritzlen [12] *** Marsh [13] 1600
Modem ? 1200
~2
800 ....,
g~ 400 _ Steam engine / ~/ r 0 1900
~ ~
1930s Air-cooled aircraft engine
~" /
~
1
I
I
I
1920
1940 Year
1960
1980
Fig. 1.20th c. increases in engine operating temperature made possible by new materials [14].
years of largely empirical work resulted in the families of superalloys we know today. This has been a truly remarkable achievement! The Ni-base alloys in particular are able to sustain stresses of 170 MPa for thousands of hours, running at temperatures of the order of 0.8Tmp, sometimes even higher. Another measure of the advance in high temperature structural materials since the days of the steam engine is shown in fig. 1. In parallel with materials development, processing of these materials became extremely sophisticated. Scientific understanding of the superalloy family also improved markedly, not just with respect to their phase constituency and the roles of individual alloying
6
J.H. Westbrook
Ch. 48
elements, but also with respect to strengthening mechanisms, deformation mechanisms, and oxidation behavior. In this paper we will restrict our attention to the history of a single family of superalloys, the nickel bases hardened principally by Ni3A1 (3"), not only for reasons of space but also because that family is the focus of most of the other papers in this volume. Although many of these alloys contain very small volume fractions of carbides and borides that play a significant role with respect to mechanical properties, we will concentrate on the characteristics of the Ni3A1 phase and its relation to the solid solution matrix. Similarly, we will not treat oxide dispersion strengthened superalloys, a powder metallurgy product. The fact that aluminum conferred age hardening behavior to a Fe-Ni or Fe-Ni-Cr base composition was deduced very early by Chevenard [15] from dilatometric and hardness tests. These effects and similar ones due to titanium and silicon additions were confirmed by a number of different workers during the 1930s, but results were inconsistent and confusing. In the late 1930s, at the beginning of World War II, when the aircraft gas turbine established an unprecedented requirement for strong high-temperature alloys, a broad systematic attack to generate precipitation-hardenable alloys from a NiCr base was launched at the Mond Nickel laboratories in England (Pfeil et al. [16] and Betteridge and Bishop [17]). This research produced the famous Nimonic series of alloys of which Nimonic 80, introduced in 1941, and later Nimonic 80A, were the first high temperature alloys 2 to be intentionally hardened by precipitation of a Ni3(A1, Ti) compound. Nimonic 80 was thus the forefather of all the "Inconel", "Rend", "Udimet", and similar alloys of today. Yet, while precipitation hardening was undoubtedly occurring, was responsible for the outstanding high temperature properties, and was associated with the Ti and A1 additions, no precipitates could be seen. Ni3A1 or 3" was not identified as the precipitate until about 1950 (Hignett [21]) with the aid of careful X-ray diffraction and electron microscopy, and the phase diagrams of the pertinent regions of the Ni-CrA1-Ti system were not published until 1952 (Taylor and Floyd [22, 23]). The further development of this class of alloys, via variations in composition and heat treatment, has again been largely empirical. Improved processing techniques, most importantly vacuum melting (permitting higher Ti and A1 contents and Nb and Hf additions without formation of deleterious oxides and nitrides) and directional solidification, introduced by Ver Snyder [24] (eliminating grain boundaries transverse to the stress axis or altogether, and allowing higher contents of refractory metal solutes) have brought about a 200~ increase in the permissible operating temperature and greatly improved reliability (10,000 hrs time between overhauls of jet engines). However, the reasons for all of these improvements were often in doubt despite the opportunities offered for "cleaner" experiments by the availability of superalloy single crystals. It has been observed (usually after the development of superior alloys) that: the solution hardening of the matrix was increased, the volume fraction of 3" increased, and the presence of deleterious phases avoided; but the more subtle and perhaps more important a s p e c t - the development of an optimal compositional relationship between matrix and 3" remained poorly understood and hence not realized. 2ICI Metals Division in England, now IMI, developed age-hardenable copper-base alloys, named Kunial, in the early 1930s. The hardening compound was later shown to be Ni3AI [18-20].
w
Superalloys (Ni-base) and dislocations
7
A1
1 Fig. 2. The L12, cP4 structure of Ni3AI. Note that the 5(110) vector, which is the perfect dislocation in the disordered f.c.c. "7 phase, becomes a superpartial dislocation in the ordered 7' phase. Thus movement of a pair of such partials is required to restore order (after Sun [25]).
We must next digress briefly to consider what "),' is and why it is important to Ni-base superalloys. Ni3A1 ('7') possesses a f.c.c, structure (hence the 3' in reference to the usual nomenclature in Fe-based systems) and is ordered (indicated by the prime symbol) as shown in fig. 2. Intermetallic compounds have been recognized as such only since the work of Karsten [26]. The idea of atomic ordering was introduced by Tammann [27] and confirmed by the X-ray diffraction studies of Bain [28]. The first determination of the ordered f.c.c, arrangement of Ni3A1 (L12, cP4) was by Westgren and Almin [29]. In contrast to many other cases (e.g., Cu-Au) where this structure forms on cooling from a disordered solid solution at high temperatures, Ni3A1 remains ordered to its solution temperature or melting point, forming directly by peritectic reaction between the disordered Ni solid solution and the liquid, as shown in fig. 3. 3 This circumstance is itself indicative of the strong bonding between nickel and aluminum atoms at this composition. The surprisingly high hardness characteristic of compounds formed, even between two relatively soft metals, was recognized early on, e.g., Martens [31], but studies of the temperature dependence of mechanical properties of intermetallics were not undertaken until much later. Tammann and Dahl [32] studied over 30 intermetallics and found that while all were brittle in the macroscopic sense at room temperature (albeit, some showed slip lines in individual crystals) macroscale plasticity could always be observed at sufficiently elevated temperatures. In the 1930s Shishokin and co-workers [33-36] studied the temperature and compositional dependence of hardness and flow stress (extrusion pressure) of a number of intermetallics and found a minimum in the temperature dependence of these properties at a composition corresponding to the ideal stoichiometry of any intermetallic formed, thus foreshadowing their potential as high temperature 3The peritectic and eutectic reaction temperatures for the binary are very close as seen in fig. 3 as also are the compositions of 3/ and eutectic liquid. These relationships may well be altered with ternary and higher alloys. There is reason to believe that in practical multicomponent superalloys the eutectic composition lies between 3, and 7', rather than between NiA1 and "3,' as seen in fig. 3.
8
Ch. 48
J.H. Westbrook
Weight Percent Nickel 1450 v 87Hil A 88Bre 9 90Jia - 91Udol o 91Udo3
\ r,.)
L
1400
o
\
~
v
/
v
cD
A1Ni ~ , /
1350
^ • 1360~
'~,.---*'~ o /~
^ ~ 4^ 1362~
\~ ~
76
78
(Ni)
1300 68
70
72
74
80
Atomic percent nickel Fig. 3. Partial A1-Ni phase diagram for the 68-80 at.% Ni region between 1300 and 1450~ (after Okamoto [30]). structural materials. Other aspects of the history of intermetallic compounds have been previously reviewed by the author [10, 37-40]. In the early post-World-War-II period, research in this area took two general paths. On the one hand there was continued empirical development in alloying of superalloys and improvements in their processing, accompanied by increased effort to understand their behavior; and, on the other, some pioneering work was begun to see if intermetallics themselves might be exploited as a new class of high-temperature materials, rather than simply as constituents of more conventional alloys. In connection with the first theme, an appreciation of the uniqueness of Ni3A1 began to be realized. No aluminide with comparable structure occurs in Fe- or Co-based systems. While in the Ni-A1 binary the compositional range of Ni3A1 is quite restricted, it can accommodate considerable solid solution alloying by ternary additions on both the Ni and A1 sites, leading to both significant hardening and the ability to adjust the lattice mismatch between Ni3A1 and the matrix. Thus, the improved scientific understanding led to concentration on Ni-base superalloy development rather than on Fe- or Co-based alloys. With respect to the second theme, single phase intermetallics or alloys with very high volume fractions of intermetallic were increasingly seen as having attractive potential for high temperature applications from a fundamental point of view because of several factors:
w
Superalloys (Ni-base) and dislocations
9
Mechanical: 9 high intrinsic yield strength at high fractions of the melting point; 9 deformability at low homologous temperatures (at least for some compounds, especially as single crystals); 9 high strain hardening rate. Kinetic: 9 low diffusion rates (relative to metals of comparable melting point) contributing to microstructural stability; 9 low diffusion rates that slow degradation by oxidation or corrosion or by diffusive deterioration of coatings; 9 low diffusion rates retarding atomic deformation processes. Matrix compatibility: 9 equilibrium or quasi-equilibrium structures; 9 compositional similarity between compound and matrix; 9 adjustable coherency. In an early instance of this type of observation, Westbrook [41] found that two-phase Ni-Cr-A1 alloys, consisting of a high volume fraction of 3'1 in a 3' matrix, exhibited an unusual combination of low temperature toughness and high temperature strength. We conclude this summary of the history of superalloys by referring the reader to two sets of conference proceedings: the Seven Springs Superalloy Conferences (1968, 1972, 1976, 1980, 1984, 1988 and 1992)sponsored by TMS/ASM cover developments in superalloys, and the Materials Research Society Symposia (1984, 1986, 1988, 1990, 1992 and 1994) treat high temperature ordered intermetallic alloys. More focussed reviews of the mechanical properties of 3'~ are given by Pope and Ezz [42], Stoloff [43], Suzuki et al. [44], and Liu and Pope [45]. Nembach and Neite [46] and Anton [47] have supplied reviews of the role of 3'~ in superalloys. Later we will review some particular observations relevant to the interaction between superalloy structure and dislocations. Now we trace the history of our understanding of dislocations.
3. D i s l o c a t i o n s in intermetallics a n d superalloys There have been several recent histories of the development of dislocation concepts, most notably those of Hirth [48] and Schulze [49], and it would be pointless to recapitulate them here in any detail. A convenient summary is given by tables 2a, b and c, taken from Schulze. As is well known, the modern conception of the dislocation theory dates from the independent publications of Orowan, Polanyi and Taylor in 1934. Their model successfully accounted for the low shear strength of real crystals and for the strain hardening of crystalline materials. A flood of papers appeared over the next 20 years by authors from all over the world elaborating the basic concepts and interpreting a wide variety of experimental results in terms of the generation, motion, and interaction of dislocations, both with themselves and with other microstructural features and atomistic defects.
10
Ch. 48
J.H. Westbrook
Table 2 Key points in the evolution of dislocation theory: a) first period, 1892-1920; b) second period, 1921-1934; c) third period, 1934-1958 (after Schulze [49]). a) First period Event "strain-figure" in a granular medium as a modification in the structure of mechanical aether
Year 1892
Authors C.V. Burton
"intrinsic strain-form" in a medium showing strictly linear elasticity constituting an electron - moving in the stagnant aether
1897
J. Larmor
Observation of "slip steps" on Pb; PRIORITY?
1900
J.A. Ewing, W. Rosenhain
First observation of slip lines (NaC1; CaCO3); origin of terms "slip plane", "direction of easiest translation" and "weak positions" in crystals
1867
EE. Reusch (Poggendorfs An.) (once again brought to view by G.E.R. Schulze)
9 Really first observation of slip lines in metals (Au, Cu .... ); 9 Determination of slip systems; 9 Originator of term "translation"
1899
O. Miigge (once again brought to view by G.E.R. Schulze)
Theory of self-strain states in an elastic continuum ("distorsioni")
1901 1907 1912/15
G. Weingarten, V. Volterra, C. Somigliana
Theory of hysteresis in a lattice of rotating dipoles with slipable regions of abnormal alignment
1907
J.A. Ewing
"I have ventured to call them Dislocations"
1920
A.E.H. Love
b) Second period Event Proposed notch effect by internal cracks in order to overcome the contradiction between theoretical and measured critical shear stress
Year 1921
Authors A.A. Griffith
Systematic measurements of stress-strain curves with Polanyiapparatus using single and polycrystalline metals
twenties
M. Polanyi, E. Schmid, W. Boas, G. Sachs et al.
Shear stress law
1926?
E. Schmid
Model of flexural slip ("Biegegleitung")
1923
G. Masing, M. Polanyi
Interpretation of mechanical hysteresis on account of translation periodicity of crystal lattice
1928
L. Prandtl
Model of "Verhakung"
1929
U. Dehlinger
Observation of decorated dislocations; not interpreted as such
1905 1932 1932
H. Siedentopf, E. Rexer, A. Edner
"Lockerstellen"
about 1930
A. Smekal
Model of local slip
1934
E. Orowan
Superalloys (Ni-base) and dislocations
w
Table 2 (Continued) c) Third period Event Models of edge dislocation
Year 1934
Authors E. Orowan, M. Polanyi, G.I. Taylor
Model of screw dislocation
1939
J.M. Burgers
Idea of partial dislocations
1948
W. Shockley
Influence of screw dislocations on crystal growth
1949
W.K. Burton, N. Cabrera, F.C. Frank
Pictures and energy estimations of small angle boundaries
1949 1953
W. Shockley, W.T. Read
Model of dislocation multiplication (Frank-Read-source)
1950
EC. Frank, W.T. Read
Imaging of an edge dislocation in lattice fringes of platinum phthalocyanine
1956
J.W. Menter
Evidence of real existence of dislocations in metals by electron diffraction contrast
1956
P.B. Hirsch, R.W. Home, M.J. Whelan
1957/58
A. Seeger
1958/60 [ l 1960 1962
P.B. Hirsch Z.S. Basinski
1958
E. KrOner
First estimation of critical stress
Work hardening theories (I) Glide-zone model (II) Forest-theories (III) Jog-dragging theory (IV) Mushroom theory "The dislocation is the elemental residual-stress source"
P.B. Hirsch D. Kuhlmann-Wilsdorf
Embarrassingly, as Read [50] noted, not only could any particular result be given a reasonable dislocation interpretation, but often several radically different models appeared to serve equally well. Worse still was the fact that until the end of this period no one had observed a dislocation "in the flesh", although slip lines had been seen as early as 1867 by Reusch [51]. An enormous boost was given the field in the 1950's with the introduction of techniques which permitted the observation of dislocations: etch pitting (Vogel et al. [52]), decoration (Hedges and Mitchell [53]), and especially electron microscopy (Heidenreich [54], Hirsch et al. [55] and Bollman [56]) and field emission microscopy (Muller [57]). Not only were some dislocation configurations now seen to be exactly what had been imagined, but it was finally possible to perform critical experiments, obtain quantitative data such as dislocation mobilities (e.g., Johnston and Gilman [58]), make detailed observations of dislocation core structures (down to resolutions of 1-2 nm), and even, in the case of transmission electron microscopy, observe other dynamic effects by stressing samples in situ in the microscope (Suzuki et al. [59, 60]).
12
Ch. 48
J.H. Westbrook
APB
b
O@O@Q@O @0@0@0 @Q@O@O@ O@O@O@ O@O@Q@Q @O@Q@Q | @ 0 @0@ Q @ Q@ Q@ 0 @ O@O @ O @Q @0 @ 0 @O@ 0 @ O@ m
Superpartial Dislocations Fig. 4. Two superpartial dislocations coupled with a piece of antiphase boundary (after Sun [25]).
The improved atomistic level of understanding of plastic deformation in solids had some particular impacts on interpreting the behavior of intermetallic compounds and superalloys, and conversely experimental findings on the latter contributed to the former. An early problem was that what would have been a unit dislocation in the disordered lattice, in attempting to glide through an ordered lattice, should experience a large resistance due to the disordering effect brought about by its motion; yet some fully ordered alloys, e.g., Cu3Au, deformed quite readily. Koehler and Seitz [61] predicted that the ordered structure of most intermetallics would require a superdislocation, i.e., a pair of partial dislocations separated by a strip of material with the ordering of the atoms exactly out-of-phase with the normal structure, a so-called antiphase boundary (APB) as shown in fig. 4. They reasoned that, in motion of this complex, the order that is destroyed by the leading member of a pair would be immediately restored by the trailing dislocation and hence would encounter much lowered resistance. At that time, however, no one had knowingly seen a dislocation, let alone a closely spaced pair of dislocations. Only a few years later, however, such superdislocations were first revealed by Marcinkowski et al. [62] by electron microscopy. More recently weak beam electron microscopy was employed by Crawford and Ray [63] to demonstrate the expected four-fold dissociation of dislocations in Fe3A1 (D03 structure). Elastic anisotropies, energetic anisotropies of internal surfaces such as APBs and stacking faults, and the often complex core structures of the dislocations themselves lead to still further complications in the geometry and dynamics of dislocations in ordered alloys. One of the most famous of these is the so-called Kear-Wilsdorf lock (Kear and Wilsdorf [64]) shown in fig. 5, caused by a screw superdislocation dissociated into (111) APB and cross-slipped segments on (100) planes. Consideration of the behavior of dislocations in two-phase alloys, e.g., Ni3A1 particles in a Ni solid solution matrix, brought about more complications. For example, compar-
w
Superalloys (Ni-base) and dislocations
Stacking ~, fault ,,
Dissociated screw dislocation
Trace of[lO0]
S
Antiphase boundary
",,r
Trace of[lll]
Cross slip
"
", ",
(a)
13
',, (b)
,, (c)
Fig. 5. The Kear-Wilsdorf lock formed by cross-slip pinning (Kear and Wilsdorf [64]).
Fig. 6. Dislocation structure developed during primary creep at 850~ of a Ni-base single crystal superalloy, CMSX-3. Dislocations appear in the matrix channels between ~,~ particles (horizontal channels, upper part and vertical channels, lower part of the figure) (courtesy of T.M. Pollock). ing the creep behavior at 1000~ of "7, "7~, and a mixture of the two phases, Nathal et al. [65] found for the two-phase alloy a decrease in creep rate of about a factor of 1000 relative to "7' alone. Creep tests of Ni-base superalloy single crystals were coupled with stereo transmission electron microscopy by Pollock and Argon [66]. For their alloy
J.H. Westbrook
14
Ch. 48
10 000
1000
10o Ni-Cr-AI alloys 700~ - 21,200 psi
0.01
0.006
0.002 00.002
0.006
0.01
zXa, (a~,- a~), kX
Fig. 7. Effect of lattice mismatch
(Aa) on creep-rupture life of Ni-Cr-AI alloys (after Mirkin and Kancheev [67]).
with a 3" volume fraction ~ 67% and particle size ~ 0.5 gm (typical of present day superalloy microstructures), they found that for creep at 850~ or less the "T' was essentially dislocation-flee and undeformable. As a result, dislocations are forced to move through the relatively narrow channels between q" particles, ultimately forming complex 3D networks and thereby constituting the principal cause for the high creep resistance. The dislocation structure developed during early stages of primary creep at 850~ and 552 MPa is shown in fig. 6. No dislocations appear within the cuboidal '7~ particles that extend through the foil thickness. The dislocations visible occur in matrix channels between '7' precipitate particles: vertical channels (lower part of the figure) and horizontal channels (upper part of the figure) that are contained within the thickness of the foil. To be effective in practice, where superalloys are stressed for long times at high temperatures, the microstructure must be stable. The coherent interface between "7 and "7' contributes basically to a low interfacial energy, which is further minimized by appropriate alloying. The practical result of this is seen in the work of Mirkin and Kancheev [67] illustrated in fig. 7. Still further contribution to microstructural stability can be obtained by addition of elements such as Re, which dissolves almost exclusively in "7 rather than in "7' (Giamei and Anton [68]). One interpretation of this benefit is that any growing 3,~ particles must drive Re away from the "7/"7' interface, and thus Re diffusion becomes the limiting step. However, solutes not only partition between "7 and "7' but may also segregate to grain boundaries, APBs, or interfaces between phases and contribute to strengthening and microstructural stability through various mechanisms. Furthermore the relative degree of partitioning and segregation may vary with solute concentration and temperature, so a general picture of the contribution of alloying to microstructural stability and strengthening does not readily emerge.
w
Superalloys (Ni-base)and dislocations
15
Modern views of the dislocation behavior of superalloys may be found in Sims et al. [9], Anton [47], and Pollock and Argon [66]; Ardell [69] reviews dislocation/particle interactions in precipitation and dispersion strengthened systems at room temperature and low volume fraction of the dispersed phase. Another historical development that must be remarked on is the use of the computer to perform various simulation "experiments". These may be used either to generate possible structures or to make quantitative or semi-quantitative estimates of dislocation behavior given known or assumed values for various key parameters. Such studies began as early as that of Foreman and Makin [63] and continue to this day. A summary of early work with this approach is given by Puls [71]. The method may be expected to be even more fruitful in the future with better interatomic potential models and as better estimates or experimental measures of key modelling parameters come to hand, e.g., elastic constants, vacancy formation energy, stacking fault and antiphase boundary energies, etc. We conclude this section with a brief citation of some other key reviews. Duesbery and Richardson [72] discuss dislocation core structures in all crystalline materials. Veyssi~re and Douin [73] review dislocations in intermetallic compounds with particular emphasis on core structures and core-related mechanical properties. Sun [25] discusses how defects in intermetallics (including dislocations) differ from those in ordinary metals and disordered solid solutions and the implications these differences have for mechanical behavior at both room and elevated temperatures. Yoo et al. [74] provide a convenient summary of currently unsolved problems. It must finally be noted that there are several aspects of the relevance of dislocations to the formation and behavior of superalloys that are not considered by the other papers in this volume; among these are: low temperature toughness and fracture, solid solution hardening, dislocation interaction with solute atoms and point defects, microstructural stability, grain boundaries and crystal growth. These topics, which are summarized in detail in two reference works (Sims and Hagel [1] and Sims et al. [75]), therefore, receive only incidental mention in this Introduction.
4. Some significant problems and observations 4.1. On dislocations in intermetallics
Anomalous temperature dependence of flow stress. This phenomenon was first reported by Westbrook [76] from hot hardness measurements on Ni3A1 where it was found, as seen in fig. 8, that the hardness increased with increasing temperature in contrast to the behavior of all other intermetallic compounds he studied. 4 He also found that the effect was both intrinsic to the material and temperature-path history independent. This surprising result was soon confirmed (Flinn [79], Davies and Stoloff [80]), and 4It is recognized that more modest increases in strength with increasing temperature for intermetallics were previously reported for CuZn (Barrett [77]) and for Cu3Au (Ardley [78]). These results, however, are readily interpreted as due to change in domain size and/or degree of long range order. Ni3A1does not have a stable domain structure nor does its ordering degree decrease significantly with increasing temperature; hence its strength increase is truly anomalous.
16
Ch. 48
J.H. Westbrook
2000 oO:>.
~ 1000
|
9o,o~ 1 4 9 ooO_ _o
,_
w
Z~.~NX
9
"
ooo _~
OLeO OOoO0 o u
~ A..~#~....z~%~Y~z~ ~
~
~ , ~ ~ ~
r
~o 500 ~g
0
,'~
9
~~
o 999
A~j,A
o
o EttS~ oo
[] []
~
mm -mm"ram"
9
~-~
m',,'m. "
200
0
Cr7C3
,m
9 Cr23C6
A Co2Ti(MgCu2 type) 9 Fe2Ti(MgZn2 type) 100
,
0
I
200
9
Ni3Ti 9 Ni3AI
[]
,
I
,
I
400 600 Temperature, ~
,
I
800
,
1000
Fig. 8. Hardness as a function of temperature for several phases common in superaUoys. The vertical arrows indicate a Th --0.5 (after Westbrook [76]).
many other aspects of the phenomenon documented by Thornton et al. [81]. These are summarized by Vitek et al. in this volume to include: 1) 2) 3) 4) 5) 6)
the yield and/or flow stress increases with increasing temperature; flow stress is temperature-path history independent; failure to obey Schmid's law; different strengths obtain in tension and compression; flow stress is virtually strain rate independent in the anomalous region; slip occurs predominantly on { 111 }(101) system below peak temperature and on {001}(101) above it; 7) predominantly long (101) screw dislocations are seen in the anomalous region; 8) no anomaly occurs in the microstrain range, 10-6-10-5; 9) the temperature dependence of the (low) work hardening rate is unusual in the anomalous region.
Scores of papers over the nearly forty years past have shown that the phenomenon occurs in many other intermetallic structures, yet not all L12 structures exhibit it. Much effort has been expended over this period to develop a model that would satisfactorily include all of the above listed features. The five papers that comprise the bulk of this
w
Superalloys (Ni-base) and dislocations
17
volume present our current level of understanding, which in many respects appears to be very good. In essence, all the proposed models envision the thermally activated locking of superdislocations by cross-slip from the primary plane onto another plane, leading to an APB partly or completely out of the primary plane. They differ in the way they account for secondary characteristics of the phenomenon. Before the problem can be regarded as solved, however, we must have a generic mechanism which can both apply to structures other than L12 and account for the absence of anomalous flow stress behavior in certain L12 compounds. (See especially the chapter by Caillard and Couret.) Effects of stoichiometric deviation on flow stress. It was observed very early (Kurnakov and Zhemchuzhnii [82]) that deviations from ideal stoichiometry resulted in an increase in the strength properties of intermetallics. Later, when measurements could be extended to high homologous temperatures (Th) at least for model compounds, as seen in fig. 9 1000
I
I
I
I
I
I
Homologous temperature T[Tmp
100
/ O
O
off
r~
c~
0.7
-----.a. 0.8
35
I
I
I
40
45
50
I
55 A/o Mg
I
I
60
65
Fig. 9. Influence of defect concentration (deviation from stoichiometry) and homologous temperature on the hot hardness of single phase AgMg alloys (after Westbrook [83]).
18
Ch. 48
J. H. Westbrook
700 LX 23.5 at % A1 600
-
[]
24.1
9
o
/
9 25.3
/
,9
500
f.-.. Z
\ N-
\
/ ./
400
300
200
100
I
I
I
0
200
400
I
I
600 800 Temperature, K
I
I000
1200
Fig. 10. Effect of A1 concentration on yield strength of Ni3AI as a function of temperature (after Noguchi et al. [84]).
from hot hardness studies on B2 AgMg by Westbrook [83], it was found that the reverse situation obtained. Thus, the same defects that impeded slip at low Th enhanced diffusion and lowered diffusion-controlled deformation resistance at high Th. 5 The defect structure of off-stoichiometric AgMg is very simple: substitution of the excess species on atom sites of the other component for both sides of stoichiometry. Marked effects of stoichiometric deviation are also to be found in Ni3A1 as seen from the results of Noguchi et al. [84] in fig. 10. In this case the strength d e c r e a s e s rather than increases with departure from stoichiometry on the Ni-rich side. Whether this result occurs because of interaction of the anomalous yield phenomenon with the point defects or because of a more complex defect structure, the possibilities for which have been studied by Fleischer [85], is not yet clear. In any case, in superalloys the composition of the Ni3A1 particles in equilibrium with the nickel solid solution matrix is always Ni-rich, although its exact composition may vary with the bulk composition of the alloy as well as with temperature. 5Note that both species need to diffuse for dislocation climb to occur. A given type of defect arising from stoichiometric deviation will not necessarily affect the two diffusion rates equally, although both should increase with increasing defect concentration.
w
Superalloys (Ni-base) and dislocations
1200
~A
/ 1000
/
/
\
d
/o,~
/"
800
19
/
O
O
,
Hf
~g
/
~o r~
~: 600 o c5
O -/ 9
2/Hf O
400 -
-
/Q /. .,
o/ i/o
/
/ Ni3A,
9
_ -O1
/
200
I
0
200
I
I
I
400 600 800 Temperature, K
I
1000 1200
Fig. 11. Effect of a large misfit solute (Hf) on the yield strength-temperature characteristics of Ni3A1 (after Mishima et al. [86, 87]).
Anomalous solid solution hardening at high temperature. Mishima et al. [86, 87] have shown that at room temperature the solution hardening that is observed in Ni3A1 is consistent with the model that this is caused by atom size and modulus interactions with moving dislocations, just as in Ni or other metallic matrices. However, this model would predict a decrease in solid solution hardening at elevated temperatures in contrast to a persistance or even an increase that have been observed in intermetallic compounds. This anomaly obtains for both large misfit solutes such as Hf in Ni3A1 (see fig. 11) 6 or small misfit solutes such as Fe in CoAl (see fig. 12). Whether complications of the nature of defects introduced by alloying or changes in the character of dislocations 6Note that while the magnitude of the solid solution hardening increases at high temperatures, the relative increase is lessened.
20
Ch. 48
J.H. Westbrook
1000
~
)Al
i
E d 100 -
\
(Co,Fe)AI
10
I
I
-200
0
25 A/o Co 25 AJo Fe
i
i
\
i
200 400 600 Temperature, ~
\
1
800
1000
Fig. 12. Effect of a small misfit solute (Fe) on the hot hardness of CoAl (Westbrook, unpublished).
at high temperatures is responsible cannot presently be determined, as discussed by Fleischer [85]. Analogous effects are also seen in the creep resistance of ternary vs. binary stoichiometric B2 aluminides, as shown by Sauthoff [88] in fig. 13. The correlation with the effective diffusion coefficient suggests that this factor must also be taken into account, at least for long-time tests at elevated temperatures. This high temperature solid solution hardening anomaly is both something that must be understood and could well be exploited in practical materials.
Strength of ternary compounds. Many binary compounds have ternary counterparts, not simply solid solutions extending from the binary but true ternary compounds, usually possessing further degrees of order based on the same crystal structure motif as the binary. The few cases that have been examined- ternary sigma phase, Fe36CrlzMo10 vs. FeCr and CoCr (Westbrook [76, 83]), (Co, Ni)3V vs. Ni3V or Co3V (K6ster and Sperner [89]), Ni2TiA1 vs. Ni3A1 (Strutt and Polvani [90]), and Nb(Ni, A1)2 vs. NiA1 (Sauthoff [91-93]) - all show strength increases relative to the binary of as much as a factor of 3 or 4, especially at high temperatures. In the example shown in fig. 14, the ternary compound is a C14 Laves phase (hence better represented as Nb(Ni, A1)2) and is contrasted with binary B2 NiA1 and D022 A13Nb. Not only is this strength benefit desirable in and of itself, but diffusion rates in ternary compounds may be expected to be even lower than in
Superalloys (Ni-base) and dislocations
w
21
300 T = ~--
900~ 10-7
S-I
100 cq
2;
CoAl
~
50-
[] (Nio.sFeo.2)A1 ~n
20-
NiAI
o,,.~ tD
~.8)A1
10L) _
~ F e A 1
o\
10-19
I
10-17
I
10-15
Diffusion coefficient, m2/s Fig. 13. Effect of solid solution hardening of Fe in NiAI and Ni in FeAI on compressive creep resistance at 900~ and correlation with the diffusion coefficient (after Sauthoff [88]). their binary counterparts, thus improving strength in temperature regimes where diffusion control exists as well as contributing to microstructural stability.
Ductilization or embrittlement phenomena in intermetallics. Several different classes of these effects have been reviewed by Stoloff [43, 94] and Westbrook [ 10]: ductilization by macroalloying, ductilization by microalloying, ductilization by grain refinement, ductilization by increase in mobile dislocation density, and embrittlement by stoichiometric excess of the active metal component. Not every effect appears in every system, and dislocation-based models in this area are too poorly developed to permit useful exploitation of those effects beneficial in intermetallics or superalloys. They are not discussed in the other chapters in this volume.
4 . 2 . O n d i s l o c a t i o n s in s u p e r a l l o y s
Peak strengthening in two-phase alloys. While good success has been obtained in modelling the strength of under-aged and over-aged alloys at low temperatures and with low to moderate volume fractions of the ordered phase (Ardell [69]), the same model does not account well for the behavior of alloys aged to peak strength, especially where the intermetallic phase is very strong relative to the matrix. Strengthening by multi-modal particle size distributions. Empirical optimization of strength of practical superalloys by multi-stage heat treatments frequently results in bi- or
22
Ch. 48
J.H. Westbrook
2000 ~= 10-4S-1 A1-Nb-Ni-3 [] A1-Nb-Ni-~ 1500 -
[]
eq
AI-N
1000
nl r
A13Nb AINbNi
500 NiA1
\
'1~\~~~0
X.\o 0
!
0
I
,
200 400 600 800 1000 1200 1400 Temperature, ~
Fig. 14. 0.2% compressive proof stress for single phase binary compounds (NiAI and AI3Nb), a ternary compound (Nb(AI, Ni)2, shown as A1NbNi) and several three phase alloys (AI-Nb-Ni 1, 2 and 3) (after Sauthoff [88]).
even tri-modal distributions of particle sizes. In very crude terms this is understandable inasmuch as it is desired that the alloy be strong over an extended range of temperatures, and for short times as well as long times. Strength is favored by small particles at low temperatures and short times and by large particles at high temperatures and long times. More quantitative modelling of the situation has not been generally successful in two respects: different additivity rules must be invoked in different systems and the parameters introduced to weight the individual contributions of each size mode lack physical significance [69].
Directional coarsening.
Under creep conditions, the originally cuboidal Ni3A1 particles in nickel-base superalloys, change their morphology by coarsening into preferentially oriented plates or "rafts". This phenomenon was first studied by Tien and Copley [95] and more recently by Pollock and Argon [96]. The structural change is related to the misfit (both sign and degree) between the intermetallic phase and matrix and by the imposed stress. This rafting morphology is of practical significance because dislocations have a more difficult time passing the laterally elongated particles and hence the creep strength is improved. Quantitative understanding of the process and its contribution to high temperature strength is impeded by the complicated roles of stress, misfit dislocations, temperature and crystallographic orientation. Vall6s and Arrell [97] have had some recent success in computational modelling of rafting behavior.
w
SuperaUoys (Ni-base) and dislocations
23
5. Future d e v e l o p m e n t s a n d c h a l l e n g e s While we may be justifiably proud of the vastly improved understanding we have retrospectively of the high temperature behavior of the present generation of superalloys, our predictive capabilities for inventing new high temperature structural materials or improving existing ones are still inadequate. It is chagrining to note that "enlightened empiricists" (to use Bruce Chalmer's phrase) still continue to produce beneficial practical results which can only be understood after the fact. A few examples may illustrate the capabilities we wish we had.
A rationale for the selection of new compound bases.
As Fleischer [98] has reviewed, the primary selection criteria for an initial screening are now melting point, density, and elastic modulus, supplemented when possible by knowledge of crystal structures, phase equilibria, and environmental resistance. How helpful it would be, either for intermetallics in an appropriate matrix or for single phase intermetallics, if we could predict a priori from fundamental parameters, slip systems, dislocation structures and mobilities, and likely interactions with solutes and lattice defects.
A rationale for alloying to balance compound and matrix properties and stabilize microstructures. We are aware that for optimum behavior of two-phase alloys we need to balance not only their individual mechanical properties (how each responds to the generation, motion, and interaction of dislocations) but we must also maximize the microstructural stability of the system. Addition of certain alloying elements is effective in both respects, but the processes are not parallel and the results often not predictable.
A basis for specifying and achieving the optimum microstructure and dislocation structure of the alloy. Even when composition is fixed, the properties of a superalloy can still be varied substantially. Complex thermomechanical treatments are known to be capable of altering the size and disposition of the second phase particles as well as the dislocation structures, but we neither know what would be optimum nor specifically how to achieve it.
Basis and means for atomistic alloying.
Most alloying performed today is at the macro level of several atom percent and is understood largely in terms of how the bulk properties of matrix and dispersed phase are affected. 7 If we knew how to select solutes, perhaps even at the ppm level, that would preferentially segregate to APBs, stacking faults, particle interfaces, and grain boundaries, and if we further knew what the effects would be of such segregation on dislocation energy, dislocation mobility, stacking fault energy, diffusion suppression etc., we would be in a fair way to advance to a new level of sophistication in alloy design. Alas, this is not yet possible. 7It is recognized that compositional control in the p.p.m, range has been exercised in the past, both for deleterious elements (Se, Bi, Pb) and for beneficial elements (Zr, B, C). These well-known instances are of less consequence now in the era of single crystals, vacuum melting, and virgin melt stock, although B and C segregation to low angle grain boundaries in single crystals may still contribute significant strengthening.
24
J.H. Westbrook
Ch. 48
6. Concluding remarks This article has attempted to sketch briefly how far we have come and how far we have yet to go in understanding and exploiting the behavior of dislocations in Ni-base superalloys and related materials. The five other chapters in this volume concentrate on but a single aspect of that subject, the flow stress anomaly. In reflecting on where we are and what has been accomplished, the author is reminded of the remark by a famous scientist, "I have never encountered a problem, however complex, that after careful study did not turn out to be even more complex". In this sense metallurgists and physicists will find interesting employment with dislocations in superalloys for some time to come.
Acknowledgment Appreciation is gratefully extended to R.L. Fleischer, J.J. Gilman, T.M. Pollock, C.T. Sims and N.S. Stoloff for insightful reading of and critical comments on an early draft of this paper.
References [1] C.T. Sims and W.C. Hagel (eds), The Superalloys (Wiley, New York, 1970). [2] Lu Da, Acta Met. Siniatica (Peking) 9 (1966) 1; translation in: Durrer Festrschrift, Vita pro Ferro, ed. W.M. Guyan (Schaffhausen, 1965) p. 65. [3] S. Keown, Historical Met. 19(1) (1985) 97. [4] F.W. Taylor and M. White, Metal Cutting Tool and Method of Making Same, US Pat. 668,269 (19 Feb. 1901). [5] H.E Burstall, A History of Mechanical Engineering (MIT Press, Cambridge, MA, 1965). [6] D. Dulieu, Historical Met. 19(1) (I 985) 104. [7] S.A. Moss, Trans. ASME 66 (1944) 351. [8] E.EC. Somerscales, in: An Encyclopedia of the History of Technology, ed. I. McNeil (Rutledge, London, 1990) p. 272. [9] C.T. Sims, in: Superalloys II, eds C.T. Sims, N.S. Stoloff and W.C. Hagel (Wiley, New York, 1987) p. 3. [10] J.H. Westbrook, in: Structural Intermetallics, eds R. Darolia, J.J. Lewandowski, C.T. Liu, P.L. Martin, D.B. Miracle and M.V. Nathal (TMS, Warrendale, PA, 1993) p. 1. [ 11] J. Truman, Historical Met. 9(1) (1985) 116. [12] G.A. Fritzlen, in: High Temperature Materials, eds R.F. Hehemann and G.A. Ault (Wiley, 1959) p. 56. [13] J.H. Marsh, UK Patent 2129, US Patent 811,859 (1906). [14] National Research Council, Materials Science and Engineering for the 1990s (Nat. Acad. Press, Washington, DC, 1989) p. 21. [15] P. Chevenard, Compt. Rend. 189 (1929) 846. [16] L.B. Pfeil, N.P. Allen and C.G. Conway, Iron and Steel Inst. Special Report, No. 43 (1952) p. 37. [17] W. Betteridge and J. Bishop, The Nimonic Alloys (Arnold, London, 1974). [18] W.O. Alexander and D. Hanson, J. Inst. Met. 61 (1937) 83. [19] W.O. Alexander, J. Inst. Met. 63 (1938) 163. [20] W.O. Alexander, J. Inst. Met. 64 (1939) 499. [21] H.W.G. Hignett, High Temperature Alloys in British Jet Engines (Int. Nickel Co., New York, 1951). [22] A. Taylor and R.W. Floyd, J. Inst. Met. 81 (1952) 25. [23] A. Taylor and R.W. Floyd, J. Inst. Met. 81 (1952) 643. [24] EL. Ver Snyder, US Patent 3,260,505 (1966).
Superalloys (Ni-base) and dislocations [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
25
Y.Q. Sun, MRS Bull. 20(7) (1995) 29. K. Karsten, Pogg. Ann. Ser. 2 46 (1839) 160. G. Tammann, Z. Anorg. Chemie 107 (1919) 1. E.C. Bain, Chem. Metall. Eng. 28 (1923) 21. A. Westgren and A. Almin, Z. Phys. Chem. B5 (1929) 14. H. Okamoto, J. Phase Equil. 14 (1993) 25. A. Martens, Mitt. K. Tech. Versuchs-Anst. 8 (1890) 216. G. Tammann and K. Dahl, Z. Anorg. Chemie 126 (1923) 104. V.P. Shishokin, Tsvetnye Metalli (November 1930). V.P. Shishokin and V.A. Ageeva, Tsvetnye Metalli 2 (1932) 119. V.P. Shishokin, V.A. Ageeva and V.T. Mikhieva, Metallurg 10 (1935) 81. V.P. Shishokin, Bull. Acad. Sci. URSS, CI. Sci. Math. Natur. (1937) p. 341. J.H. Westbrook, in: Mechanical Properties of Intermetallic Compounds, ed. J.H. Westbrook (Wiley, New York, 1960)p. 1. [38] J.H. Westbrook, in: Ordered Alloys: Structural Applications and Physical Metallurgy, eds B.H. Kear, C.T. Sims, N.S. Stoloff and J.H. Westbrook (Claitor's, Baton Rouge, LA, 1970) p. 1. [39] J.H. Westbrook, Metall. Trans. 8A (1977) 1327. [40] J.H. Westbrook, in: Intermetallic Compounds: Principles and Practice, eds J.H. Westbrook and R.L. Fleischer (Wiley, Chichester, UK, 1994) p. 3. [41] J.H. Westbrook, General Electric Report 55RL1281 (1955). [42] D.P. Pope and S.S. Ezz, Int. Metall. Rev. 29 (1984) 136. [43] N.S. Stoloff, Int. Metall. Rev. 29 (1984) 123. [44] T. Suzuki, M. Ichihara and S. Miura, ISIJ Int. 29 (1989) 1. [45] C.T. Liu and D.P. Pope, in: Intermetallic Compounds: Principles and Practice, Vol. 2, eds J.H. Westbrook and R.L. Fleischer (Wiley, Chichester, UK, 1994) p. 17. [46] E. Nembach and G. Neite, Progr. Mater. Sci. 29 (1985) 177. [47] D.L. Anton, in: Intermetallic Compounds: Principles and Practice, Vol. 2, eds J.H. Westbrook and R.L. Fleischer (Wiley, Chichester, UK, 1994), p. 3. [48] J.P. Hirth, Metall. Trans. 16A (1985) 2085. [49] D. Schulze, Prakt. Metallogr. 26 (1989) 559, 604. [50] W.T. Read, Dislocations in Crystals (McGraw-Hill, New York, 1953). [51] E. Reusch, Ann. Phys. Chem. 132 (1867) 441. [52] EL. Vogel Jr., W.G. Pfann, C.L. Corey and G. Thomas, Phys. Rev. 90 (1953) 489. [53] J.N. Hedges and J.W. Mitchell, Philos. Mag. 44 (1953) 223. [54] R.D. Heidenreich, J. Appl. Phys. 20 (1949) 993. [55] P.B. Hirsch, R.W. Home and M.J. Whelan, Philos. Mag. 1 (1956) 677. [56] W. Bollman, Phys. Rev. 103 (1956) 1588. [57] E.W. Muller, Acta Metall. 6 (1958) 620. [58] W.G. Johnston and J.J. Gilman, J. Appl. Phys. 30 (1959) 129. [59] K. Suzuki, M. Ichihara and S. Takeuchi, in: Proc. 5th Int. Conf. HREM, eds T. Imura and H. Hashimoto (Jpn. Soc. of Electron Microscopy, 1977) p. 463. [60] K. Suzuki, E. Kuramoto, S. Takeuchi and M. Ichihara, Jpn. J. Appl. Phys. 16 (1977) 919. [61 ] J.S. Koehler and F. Seitz, J. Appl. Mech. 14 (1947) A217. [62] M.J. Marcinkowski, R.M. Fisher and N. Brown, J. Appl. Phys. 31 (1960) 1303. [63] R.C. Crawford and I.L.F. Ray, in: Proc. Int. Symp. on Electron Microscopy, Grenoble, 1970, p. 277. [64] B.H. Kear and H.G.F. Wilsdorf, Trans. Metall. Soc. AIME 224 (1962) 382. [65] M.V. Nathal, J.O. Diaz and R.V. Miner, in: High Temperature Ordered Intermetallic Alloys III, eds C.T. Liu, A.I. Taub, N.S. Stoloff and C.C. Koch (MRS, Pittsburgh, PA, 1989) p. 269. [66] T.M. Pollock and A.S. Argon, Acta Metall. Mater. 40 (1992) 1. [67] I.L. Mirkin and O.D. Kancheev, Met. Sci. Heat Treat. No. 1-2, (1967) 10. [68] A.F. Giamei and D.L. Anton, Metall. Trans. 16A (1985) 1997. [69] A.J. Ardell, in: Intermetallic Compounds: Principles and Practice, Vol. 2, eds J.H. Westbrook and R.L. Fleischer (Wiley, Chichester, UK, 1994), p. 257. [70] A.J.E. Foreman and M.J. Makin, Philos. Mag. 14 (1966) 131.
26
J. H. Westbrook
[71] M.E Puls, in: Dislocation Modelling of Physical Systems, eds M.E Ashby, R. Bullough, C.S. Hartley and J.P. Hirth (Pergamon, Oxford, 1981) p. 249. [72] M.S. Duesbery and G.Y. Richardson, CRC Critical Rev. in Solid State and Mater. Sci. 17 (1991) 1. [73] P. Veyssi~re and J. Douin, in: Intermetallic Compounds: Principles and Practice, Vol. 1, eds J.H. Westbrook and R.L. Fleischer (Wiley, Chichester, UK, 1994) p. 519. [74] M.H. Yoo, S.L. Sass, C.L. Fu, M.J. Mills, D.M. Dimiduk and E.P. George, Acta Metall. Mater. 41 (1993) 987. [75] C.T. Sims, N.S. Stoloff and W.C. Hagel (eds), Superalloys II (Wiley, New York, 1987). [76] J.H. Westbrook, Trans. Metall. Soc. AIME 209 (1975) 898. [77] C.S. Barrett, Trans. Metall. Soc. AIME 200 (1954) 1003. [78] G.W. Ardley, Acta Metall. 3 (1955) 525. [79] P.A. Flinn, Trans. Metall. Soc. AIME 218 (1960) 145. [80] R.G. Davies and N.S. Stoloff, Trans. Metall. Soc. AIME 233 (1965) 714. [81] P.H. Thornton, P.G. Davies and T.L. Johnston, Metall. Trans. 1 (1970) 207. [82] N.S. Kurnakov and S.F. Zhemchuzhnii, Z. Anorg. Chemie 60 (1908) 1; translated from J. Russ. Phys. Chem. Soc. 32 (1907) 448. [83] J.H. Westbrook, J. Electrochem. Soc. 104 (1957) 369. [84] O. Noguchi, Y. Oya and T. Suzuki, Metall. Trans. 12A (1981) 1647. [85] R.L. Fleischer, in: Structural Intermetallic Compounds, eds R. Darolia, J.J. Lewandowski, C.T. Liu, P.L. Martin, D.B. Miracle and M.V. Nathal (TMS, Warrendale, PA, 1993) p. 691. [86] Y. Mishima, S. Ochiai and Y.M. Yodagawa, Trans. Jpn. Inst. Met. 27 (1986) 648. [87] Y. Mishima, S. Ochiai and Y.M. Yodagawa, Trans. Jpn. Inst. Met. 27 (1986) 656. [88] G. Sauthoff, Intermetallics (VCH, Weinheim, Germany, 1995). [89] W. Kt~ster and E Sperner, Z. Metallkd. 48 (1959) 540. [90] P.R. Strutt and R.S. Polvani, Scr. Metall. 7 (1973) 1221. [91] G. Sauthoff, Z. Metallkd. 80 (1989) 337. [92] G. Sauthoff, Z. Metallkd. 81 (1990) 855. [93] G. Sauthoff, in: Proc. Int. Symp. on Intermetallic Compounds - Structure and Mechanical Properties, JIMIS-6, ed. O. Izumi (Jpn. Inst. of Metals, 1991) p. 371. [94] N.S. Stoloff, Metall. Trans. 24A (1993) 561. [95] J.K. Tien and S.M. Copley, Metall. Trans. 2 (1971) 219. [96] T.M. Pollock and A.S. Argon, Acta Metall. Mater. 42 (1994) 1859. [97] J.L. Vall6s and D.J. Arrell, Paper M10C6, in: Proc. 14th Int. CODATA Conf., Chamb6ry, France, 1994 (to be published). [98] R.L. Fleischer, in: Intermetallic Compounds: Principles and Practice, Vol. 2, eds J.H. Westbrook and R.L. Fleischer (Wiley, Chichester, UK, 1994) p. 237.
CHAPTER 49
Geometry of Dislocation Glide in L12 7~-phase: TEM Observations Y. Q. SUN Department of Materials University of O~.ord Parks Road, O~.ord, OX1 3PH UK and
R M. HAZZLEDINE UES Inc. 4401 Dayton-Xenia Road Dayton, OH 45432-1894 USA
Dislocations in Solids 9 1996 Elsevier Science B.V. All rights reserved
Edited by F. R. N. Nabarro and M. S. Duesberv
Contents 1. 2. 3. 4.
Introduction 29 Selection of foil orientations 32 Normal slip mode (100){001} 33 Normal slip mode (110){001} 37 4.1. The low mobility of edge and screw dislocations in (110){001} 38 4.2. Observation of dissociations 39 4.3. Slip mechanism 45 5. Dislocations in the anomalous (101) { 111 } slip 45 5.1. Distribution of dislocations 45 5.2. Observation of dissociations 47 5.3. Superkinks and switched superpartials 48 5.4. Increasing activity on the cube cross slip plane with temperature 50 5.5. APB tubes and edge dipoles 51 5.6. Dislocation propagation in (101){111} slip as suggested by TEM observations 6. (101){111} slip" SISF dissociation 55 7. A summary of observations relevant to yielding 58 8. Geometrical aspects of work hardening 65 References 66
53
1. Introduction The only active slip direction in the L12 ordered '7~-phase is the close-packed (110), except at very high temperatures (above ~ 800~ Yet depending on whether that slip occurs on {111} or {001}, the material's response to an external load and to changes in temperature is very different. When the slip plane is { 111 } the yield stress increases with temperature (fig. 1). Such an abnormal variation in the yield stress with temperature is also accompanied by a number of other distinguishing properties- violation of the Schmid law and asymmetry of yield stress in tension and compression, near-zero sensitivity of the flow stress to strain-rate changes, and a yield stress that is reversible with respect to temperature changes, etc. - properties which make the ( 101 ) { 111 } slip in "7' outstanding and anomalous. However, when slip in (110) changes to {001}, occurring when the temperature is above a certain value Tp (fig. 1), the "7~ phase is normal: its yield stress decreases with temperature and increases substantially with strain-rate, the Schmid law is obeyed, and the yield stress is the same in tension and compression. (For a full review of the mechanical properties of "7~ the reader is referred to several review articles, e.g., [1, 2].) The aim of this chapter is to present transmission electron microscopy (TEM) observations of dislocations and other related defects that characterise these two contrasting slip modes in "7~ and, on this basis, to explore the controlling dislocation mechanisms. When the temperature exceeds about 800~ the slip direction changes to (100), the shortest lattice translation vector in L12; the origin for this transition will also be investigated. It was using a transmission electron microscope that Marcinkowski, Brown and Fisher [3] obtained the first direct experimental evidence for an important hypothesis, proposed almost half a century ago by Koehler and Seitz [4], about dislocations in superlattice structures: the dislocations would propagate more readily if they existed in groups coupled by antiphase boundaries (APB). Marcinkowski et al observed in the TEM that dislocations in L12-ordered Cu3Au indeed contained two components or superpartials which were held together by a strip of APB. Today, as exemplified in this volume, understanding of the varied and sometimes intriguing plastic properties of intermetallic compounds, notably the L12 ordered "7~-phase, is based on dislocation structures with complexities that have gone far beyond that of the simple splitting into two superpartials. The further splitting of the superpartials (e.g., each 89 (110} dislocation splitting into two 1 (112) Shockley partials) is known to play an important role in influencing the motion of superdislocations and the properties of plastic flow. In some cases the APB does not represent the major mode of dissociation and faults of other types (e.g., superlattice intrinsic stacking fault SISF) dominate the coupling of the partials. Owing, to a large extent, to advances in TEM techniques, experimental examination of dislocations with complicated internal structures can be performed, often to reveal details approaching atomic resolution; this may be made in parallel with observations of dislocation substructure, from which evidence for the relative mobility of dislocations can be obtained.
30
Ch. 49
Y. Q. Sun and P. M. Haz~ztedine
(/) or) CIJ t..4--
. _..,,,
TEMPERATURE [10]'1(111)
--
-------
i []'10](001) ~ [010](001)... [001](010)
[
Fig. 1. A schematic illustration of the yield stress (YS) and work hardening rate (WHR) of 71 and their variations with temperature. For most crystal orientations the active slip system is (103){ 111 } below the peak temperature T o and <110){001} above Tp. The work hardening rate shows two maxima. The first occurs at a significantly lower temperature than the yield stress peak. The peak at the high temperature is found to be associated with the activation of a new slip system, (010){001 }.
(101>{ 111 } octahedral slip in L12 has been studied most extensively. Experimentally it has been established that, when the yield stress shows an anomalous increase with temperature, the (101) superdislocations are dissociated into two 89 superpartials bordering the APB on { 111 }, i.e., 1
1
~(10T> + APB{lll} + ~(101).
(1)
The 89 (101) superpartials correspond to the unit dislocations in the face-centred cubic 1 (f.c.c.) lattice and were postulated to split further into g(l12) Shockley partials on
{111} [3], 1
-
1
~(101> -~ g1(1- 1 2 ) + C S F + g (2ii),
(2)
where the CSF connecting the Shockleys stands for a complex stacking fault which involves not just a disruption in the stacking sequence but also changes in the bonding between the atoms across the fault. Although the above dissociation mode was initially proposed on the basis of a hard-sphere model, it has been substantiated by atomistic simulations (e.g., [5]) and has received direct experimental confirmation from lattice imaging TEM [6-8]. The above dissociation modes have formed the basis of all the current models for the yield stress anomaly in 7' [9-11 ].
w1
Geometry of dislocation glide in L12 "Y~-phase
31
Another possible mode of dissociation for (110){111} dislocations involves splitting 1 (112), i.e into two super-lattice Shockley partials .~ 1
(3)
(101)_+~-1 (211) + SISF + ~(112),
where the superlattice intrinsic stacking fault SISF contains a fault in the stacking sequence but maintains the bonding condition among the nearest neighbouring atoms. Although analysed theoretically in detail [12-14], this dissociation mode is not observed frequently in "7' and there is a lack of experimental characterisation of the associated deformation properties. This dissociation mode has been suggested to lead to normal temperature dependence of the yield stress, but under the condition of the { 111 } APB _ being fully unstable [13, 14]. In Pt3A1 for which the slip system is (101){111} with a normal temperature dependence of the yield stress, it has been suggested that the dissociation involves SISF [13, 14], but so far there has been no direct experimental evidence, e.g., with TEM, to support this suggestion. When the temperature exceeds Tp (fig. 1), slip is still in a (110) direction but the slip plane changes to {001 }. Experimental observations have shown that when the superdislocation dissociates into two superpartials, the APB is on the glide plane {001 }, i.e.,
1
1
(4)
(110) -+ ~(110) -+- APB{001} + ~(110).
Investigations of the further splitting of the 1(110){001} superpartials have centred on the screw orientation, but edge dislocations have also received some attention. On {001}, screw and edge dislocations are special because their lines are parallel to the intersection of {001} with two {111} planes on which further glide splitting can take place. For screws the further splitting into Shockleys is the same as in eq. (2) but on planes alternating between the two co-zonal {111} planes at distances of b/2, b being the magnitude of ~1 (110), the Burgers vector of the superpartial. This further dissociation renders the dislocation core non-planar. Along the edge orientation a ~1 (110) superpartial on {001} is the equivalent of a Lomer dislocation in f.c.c. [15] and is thus expected to dissociate into a Lomer-Cottrell lock [16, 17] involving two Shockleys on the two { 111 } planes and a stair-rod dislocation, i.e., -
~[1T0]I -+ 61[11~'] + C S F ( l l- l ) + ~I [ l l 0 ] + C S F ( l l l ) +
1
-
g [112].
(5)
A schematic illustration of this structure and its experimental observation will be shown later in fig. 6. The above are the basic dislocation dissociation modes in L12 '7~ which have been studied and observed most extensively; they have formed the basis for the investigation of dislocation behaviour and mechanical properties. This paper demonstrates the experimental verification of the above dissociation schemes and also several of the variations derived thereon. Some of the dissociation schemes, e.g., (1, 2), have been studied
32
Y. Q. Sun and P. M. Ha~ledine
Ch. 49
extensively with atomistic simulations, while others, e.g., (5), proposed on the basis of crystallography and also directly observed, have received little atomistic study. In general TEM observations in recent years have put more dislocation types under scrutiny than atomistic studies (e.g., Lomer-Cottrell locks and B5 locks on which there have been several experimental observations but little atomistic simulation so far). This paper also aims to relate the observed dislocation substructure and, to certain extent, mechanical properties, to the dislocation dissociations.
2. S e l e c t i o n o f f o i l o r i e n t a t i o n s A full account of the experimental details, related to TEM, in single crystal preparation, sample testing and foil thinning is beyond the scope of this chapter, but the selection of foil orientation from deformed single crystal samples for TEM observations deserves an appropriate exposition. TEM micrographs recorded from foil specimens, usually 1000 or less in thickness as needed for sufficient electron wave transmission, inevitably offer a very limited view of the overall dislocation distribution in the bulk, and it is thus imperative that the thin foils be prepared parallel to selected crystallographic planes, including the macroscopic slip plane. In single crystals deformed by a uniaxial load, very often only one slip system, i.e., the primary slip system, is activated and, to observe the arrangement of dislocations in that slip system, the foil plane is often selected to be parallel to the macroscopic slip plane which is identified by optical microscope observations of surface slip markings. Hence many of the foils selected for the study of {101){111} and (110){001} slips are parallel to the primary slip planes, i.e., (111)and (001), with the loading axis in the 001-011-111 unit triangle. Such samples enable the distribution of dislocations to be viewed over a field tens of microns in width. In the ~"-phase the extent of cross slip of screw dislocations is an important experimental feature to observe and this can be revealed either by tilting the sample around the screw dislocation line direction to detect any line curvature off the slip plane, or by imaging foils which are parallel to selected crystallographic planes co-zonal with the slip direction or Burgers vector, or a combination of both. In these samples, the relative mobilities of dislocations are deduced from the arrangement of dislocations and those which dominate the structure and for which the dominance is not caused by self energies are thought to be the least mobile. Samples for the above observations are in general deformed by a small amount, typically 1-3 %. The observation of the dissociation structures of dislocations can be made on similarly oriented samples using the weak-beam method which is capable of resolving partials that are 15 A apart. When the partial separation is close to the resolution limit, the experimental error makes it unreliable to determine the dissociation plane, which in many cases is not the same as the glide plane, by trace analysis. Performing lattice resolution HREM observations on the basis of the weak-beam result has been found to be useful in providing further details of small-scale dissociations. For HREM observations, the foil orientation is selected such that the dislocations to be observed are normal to the foil plane so that they can be examined end-on. Also, owing to the required large magnification (~ 106 times), a high dislocation density is needed, thus necessitating a o
w
Geometry of dislocation glide in L12 ")/'-phase
33
relatively large amount of plastic strain (up to 10%). In the "/phase, as established by weak-beam observations, the substructure of deformed samples is often dominated by one type of dislocation (screws) in the case of (101){ 111 } slip, or two types (screw and edge dislocations) in (110){001} slip; dislocations with mixed characters have a very low density. This property ensures that foils containing the majority dislocations end-on are not influenced much by other dislocations. Image simulations will not be addressed here and references will be given when they are of concern. It is generally found that the identification of the Burgers vector of dislocations in 3,~ can be performed quite straightforwardly from their visibility at different 9" b or 9" b x u values, and the effects of elastic anisotropy do not appear to introduce significant ambiguity in the determination of Burgers vector b (9 represents the diffraction vector used to form the image and u is the line direction). Image calculation has been found necessary when the exact partial separation is required for the accurate measurement of fault energies (e.g., [18-20]) or when the relative image intensity becomes a factor for differentiating two dissociation schemes [21]. For the lattice imaging work, comprehensive image matching has not been thought to be feasible because the experimental imaging condition apparently change significantly within short intervals; this was reflected by the change of image contrast owing perhaps to the buckling of foils or surface contamination when the sample is subjected to a highly condensed electron beam. The combination of two tools, i.e., of weak-beam and lattice imaging, is the authors' approach for an improved identification of the dissociation mode of dislocations.
3. N o r m a l slip m o d e ( 1 0 0 ) { 0 0 1 } (100} is the shortest lattice translation vector in the L12 structure. On the basis of the isotropic elasticity theory of dislocations, (100) dislocations have smaller self energies than (110) dislocations. Using anisotropic elasticity and taking the dissociation of (110) 's into account, Hazzledine et al. [22] made a comparison of the line energies of (100) and (110) dislocations for Ni3A1. They found that except for a small angular range in which (100) dislocations are elastically unstable, (100) dislocations have lower line energies than (110) dislocations. Although the self energy is the lowest among the known dislocation systems in 3/, (100){001 } slip does not operate until the temperature is very high, above about 800~ higher than the temperature range of (110){001 } slip [23, 24]. This suggests that (100){001 } dislocations probably have the least mobility and require the highest thermal activation to operate. TEM observations have demonstrated that the low mobility of (100){001} dislocations is associated with the non-planar dissociation structures on 45 ~ dislocations, or B5 locks. (100) dislocations were first noted in a polycrystalline Ni3A1 deformed at 600~ [25], and they were in the form of short segments linked to (110) dislocations; they were thus thought to be the result of dislocation reactions between simultaneously active (110){001 } dislocations, which are likely in a polycrystalline sample. Later it was found in both Ni3Ga and Ni3A1 deformed at around 850~ that substantial numbers of (100) dislocations existed in the deformed samples [2, 24]. At such temperatures, (100) dislocations are in the form of long segments or loops and thus clearly contributed to the
34
Y. Q. Sun and P. M. Hazzledine
Ch. 49
plastic strain. (110) dislocations that dominate the plastic strain at low temperatures are still present at the high temperatures but with a much lower density. Figure 2 shows a TEM micrograph of Ni3A1 deformed at 900~ showing dislocations with (100) Burgers vectors. The occurrence of (100) slip is accompanied by a rise in the work hardening rate (WHR, fig. 1). The high work hardening rate is consistent with multiple slip. There are always at least two (010){001 } slip systems that possess the same highest Schmid factor. This makes the work hardening associated with (100){001 } slip essentially the same as in stage-II deformation of f.c.c, metals where the high work hardening rate is caused by the activation of a secondary slip system. There is as yet no quantitative modelling for the work hardening rate of multiple (100){001 } slip. Further details will be given in section 8. In the L12 structure a {001} plane has a smaller interplanar spacing than {111} and thus on {001} a lower dislocation mobility is generally expected. The fact that (100){001) requires a higher temperature to operate than (110){001} suggests that (100) dislocations are less mobile than (110)'s, considering that the former are energetically more favourable. In experiment, (100) dislocations on {001) were found to have particularly low mobilities along (110) directions along which they are 45 ~ in character. The low mobility of (100){001} dislocations was found to lie in the non-planar core structures formed on dislocations along these line orientations. From the curvature of the dislocations in fig. 2, the active glide plane has been identified as {001 }. The dislocation labelled D in fig. 2 has the Burgers vector (111) and lies exactly along the cube edge (100). It is thought to be formed by the reaction (110){001} + (001){010) -+ (111), with line direction parallel to (100). This is a locked dislocation because its allowed glide plane is (111) x (100) - {011} which is a high friction plane. The most distinguishing feature, however, is that most of the (100) dislocations are preferentially aligned along (110) 45 ~ directions. These observations led to the suggestion [23] that, since (110) is the intersection line of two { 111) planes with the {001 ) glide plane, a 45 ~ dislocation could split on the close-packed planes to lower its self energy, rendering the core nonplanar. The resultant structure is in fact the equivalent of B5 locks in f.c.c. [27]; the difference is that in f.c.c. B5 locks, like Lomer-Cottrell locks, are formed as a result of a special reaction between (110){111} dislocations, whereas in 1,/ they arise from the operation of a single (100){001} slip system. As in f.c.c., 45~ } dislocations are expected to dissociate to form non-planar structures subtending either an acute or an obtuse angle, fig. 3(a). The lattice imaging observations have shown the existence of non-planar cores on 45 ~ (100){001 } dislocations and an example is given in fig. 3(b). This image was recorded in the lattice imaging mode from a Ni3A1 sample deformed at 850~ The foil is parallel to (110) in which 45 ~ dislocations in the [010](001) primary system lie end-on. From the closure failure of a Burgers circuit built around the entire dislocation, the component of the Burgers vector orthogonal to the electron beam is found to be 1 [~ 10], consistent with the observed dislocation being a 45 ~ [010]. The three-fold non-planar structure observed in fig. 3(b) is thought to represent the following dissociation scheme [010] ~ ~1 [112.] + C S F ( l l l ) +
1
-
1
[031](stair-rod) + CSF(111) + ~ [i21] .
M 'w
Fig. 2. A weak-beam image to show 45°(010){001} dislocations in a Nij(A1, Ti) sample deformed at 900OC. Dislocations marked A have [loo] Burgers vector and the dislocation marked B has [OIO]. A and B dislocations are preferentially oriented along 45' (1 10) directions. Dislocations marked C are [I101 edges. Two (010) dislocations with orthogonal Burgers vectors have reacted to form a (110) dislocation lying in the edge orientation. Dislocation labelled D lies along cube edge and has Burgers vector (1 1 I ) . (From P.M. Hazzledine, M.H. Yo0 and Y.Q. Sun [22], by courtesy of Pergamon Press PLC).
VI w
36
Ch. 49
Y. Q. Sun and P. M. Hazzledine
(a)
x"
(OOl)
b,
Fig. 3. (a) B5 locks on 45 ~ (010){001 } dislocations. (b) A lattice imaging TEM picture showing the internal structure of a 45 ~ (010){001} dislocation lying end-on in the thin foil. The net Burgers vector normal to the electron beam is given by the closure failure of the Burgers circuit. The non-planar core structure is responsible for the low mobility of 45 ~ (010){001 } dislocations.
w
Geometry of dislocation glide in L12 "y~-phase
37
The locking of 45~ dislocations is similar to that of (111) screw dislocations in some b.c.c, metals because, in both cases, the extended non-planar structure must constrict to make the dislocation movable, which requires the assistance of thermal fluctuations, and the mobile state is unstable to the locking transformation. However, unlike (111) slip in b.c.c., in a given {001} plane a (010) dislocation may be locked along two (orthogonal) (110) directions, causing the entire dislocation loop to be trapped along these two directions. In b.c.c, only the screw dislocations are locked and the plastic flow can still be carried by the non-screw dislocations which remain mobile (at least during the early stage of plastic deformation). We shall see in the next section that the same property (i.e., loop trapping)is also possessed by (110){001} slip.
4. Normal slip mode (110){001 } Except in samples oriented very close to [001] (within approximately 5 ~ around [001]), the active slip system above the peak temperature Tp is the primary (110){001} slip; below Tp the slip direction is also (110) but on a { 111 } plane. Tp has been found to vary substantially with sample orientation and strain rate [2]. Since the onset of (110){001} slip with rising temperature marks the termination of the preferred (110){111} slip mode (preferred technologically because it raises the strength with increasing temperature), pushing the peak strength of Ni3A1 to still higher temperatures would require the suppression of the (110){001} glide, dislocation mechanisms for which need to be identified. The behaviour of (110){001} dislocations is important also because it is a necessary element in the current models for the yield stress anomaly of (110){ 111 } slip [9-11 ]. A feature common in these models is that the pinning of the dislocations, initially gliding on {111}, starts with some screw segments cross slipping onto {010}. But the models differ in the extent to which such cross slip events need to occur in order to become effective barriers (see, e.g., Vitek, Pope and Bassani chapter in this volume for a full description of these models). When the APB is completely on the cube cross slip plane, the screw dislocation is known as a Kear-Wilsdorf lock [28]. An understanding of the dislocation processes controlling pure (110){001 } slip is thus also relevant to the interpretations of the anomalous regime associated with (101){111} slip. (For a single crystal sample the cube cross slip [10i](010) (associated with the primary octahedral system [101](111))is not the most highly stressed {110){001} system, thus pure cube slip is possible only with the primary system [110](001).) A pair of 89 screw superpartials coupled by the APB on {010} have a lower energy than if the APB is on { 111}, and this is the fundamental driving force for { 111 } -+ {010} cross slip, another key element common to the models for the yield stress anomaly. This energy anisotropy for the screws is caused by a combination of two effects: a lower APB energy on {010} than on {111} [29] and a torque force, arising from elastic anisotropy, which acts in the direction of forcing the dislocation pair onto {010} [30]. As a result, an APB-dissociated (110) screw is bi-stable with dissociations on {111} and {010} as the two stable states [30, 31] although {010} is usually the most stable dissociation plane (i.e., dissociation on { 111 ) is metastable) even if "~010/> 3'111 (3'010 and 3'111 are APB energies). Despite being favoured by lower
38
Y Q. Sun and P M. Hazzledine
Ch. 49
self energies, (110){001 } slip requires substantially higher temperatures to operate than (110){111}; this is usually attributed to the low mobility of these dislocations. This section is aimed at identifying the origin of the low mobility of (110){001 } dislocations.
4.1. The low mobility of edge and screw dislocations in (110){001} Slip-line observations [32] have shown that the primary (110){001} cube slip is planar with no indication of extensive cross slip. In keeping with this conclusion, TEM observations (fig. 4) also show that in (110){001 } slip dislocation propagation is confined to the {001} glide plane, which is in contrast to the anomalous (101){111} slip in which screw dislocations are seen to cross slip frequently off the macroscopic slip plane {111} onto {010} (see section 5). Glide on {001 } is, in general, expected to experience stronger frictional resistance than on { 111 } because of the smaller interplanar spacing (d001 < dill). Experiments have shown that the mobility is particularly low for screw and edge dislocations. As shown in fig. 4, a weak-beam image of dislocations in a Ni3Ga sample deformed at 700~ ( ~ 100 ~ above the yield stress peak Tp), the majority of the dislocations lie along screw and edge orientations with very short lengths of dislocations having mixed characters. In fig. 4 the specimen foil is parallel to the macroscopic slip
Fig. 4. A weak-beam image to show the distribution of dislocations in the primary (110){001 } slip in a sample deformed just above the yield stress peak. The plane of the foil sample is (001). The substructure is dominated by screw (S) and edge (E) dislocations, with very few dislocations having mixed characters. Such observations indicate that both screw and edge (110) dislocations are relatively immobile on {001}. (From Y.Q. Sun, EM. Hazzledine, M.A. Crimp and A. Couret [17], by courtesy of Taylor Francis Ltd.)
w
Geometry of dislocation glide in L12 .yt_phase
39
plane {001} and the dislocations can be traced over long distances, indicating that they are lying within the foil, and thus their propagation is largely planar. The edge dislocations exhibit a pronounced rectilinear morphology while the screw dislocations, which jointly dominate the substructure with the edges, are slightly curved. Using anisotropic elasticity Douin et al. [33] analyzed the effect of line energy on dislocation loop shape using the method of the inverse Wulff plot. The study showed that, without the energy reduction due to Lomer-Cottrell dissociation, on {001} near-screw (110) dislocations are favoured, but the pure edge orientation should be (elastically) unstable. The presence of rectilinear edge dislocations therefore suggests that they are less mobile than other dislocations, consistent with the locked structures observed. The fact that the screw dislocations are able to retain their curvature without an external load suggests that they also experience a high frictional force. The scarcity of mixed dislocations shows that dislocations with intermediate characters are mobile.
4.2. Observation of dissociations
The low mobilities of screw and edge (110){001} dislocations have been found to be caused by their non-planar dissociation structures and this is demonstrated by the TEM observations with a combination of weak-beam and lattice imaging methods. There is a general agreement on the screw dislocations being immobile as a result of the superpartials' further splitting into Shockleys on the intersecting { 111 } planes. This can be envisaged on the basis of a hard sphere model and has been substantiated by the atomistic simulations of Yamaguchi et al. [12] which, although concerned with {111} -+ {010} cross slip, can be applied to pure (110){001} cube slip. The dissociation of the (110) superdislocation on {001} is described by eq. (4)in which the (110) superdislocation 1 first splits into two ~(110) superpartials with the APB on the glide plane {001}. The further splitting of the screw superpartials into Shockleys, according to eq. (2), may occur on two {111 } planes which are co-zonal with {001 } and alternate between the two at lattice positions separated by b/2, as shown in fig. 5(a). Superpartial configurations at positions in between are not known and can be assumed to be fully constricted [10]. 1 However, C16ment et al. [34] hypothesized a planar dissociation of the ~(110) screw superpartials on {001} with a view to accounting for the in-situ observations and an anomaly in the flow stress for (110){001} slip in a Ni-based 3'/7' two-phase superalloy. Such a core has not been substantiated by atomistic simulations, nor was there any direct experimental verification. Direct experimental evidence for the dissociated structure of screw dislocations in (110){001} slip has been provided by lattice resolution TEM, shown in fig. 5(b)[6, 7]. In this picture a (110) screw dislocation lies normal to the thin foil and parallel to the electron beam, i.e., it is viewed end-on. The image is from a Ni3A1 sample deformed at 400~ at which, although the primary (110){001} slip has not yet fully started, the large extent of cross slip onto {010} permits the imaging of screw (110) dislocations with the APB on {010}. The image shows that the extended cores of the !(110) screw superpartials are aligned along {010} ' showing that the APB is on {010} " 2 The displacement of the APB is not visible since it is parallel to the direction of the
40
Ch. 49
Y. Q. Sun and P. M. Hazzledine
(a) ,,,, ~
APB
.
b/.,/ .
.
.
Fig. 5. (a) End-on schematic of the Kear-Wilsdorf lock on the screw (110){001} dislocation. For glide on {001}, the locking effect derives from the out-of-the-plane splitting into CSF-coupled Shockleys. (b) An HREM image showing a screw (110){001 } dislocation imaged end-on in a thin foil, showing the predicted Kear-Wilsdorf structure. Positions of the superpartials are indicated by the arrows.
Geometry of dislocation glide in L12 "y~-phase
w
41
electron beam. The possible supplementary displacement normal to the fault plane is not revealed in the HREM pictures. The two superpartials are spread on two parallel { 111 } planes. Splittings on different { 111 } planes have also been observed and they directly confirm the dissociation structure illustrated in fig. 5(a). Since the dissociation of the screw superpartials is off the glide plane, the screw dislocation is locked against motion on {001 }. The dislocation becomes mobile if the non-planar core is constricted and this requires thermal activation; this is the origin for the increased activity of (110) screws on {001} with increasing temperature. The screw dislocations in (110){001} therefore experience essentially the same type of lattice friction as 45~ } dislocations or as screw (111) dislocations in some b.c.c, alloys. Experimental observations have shown that the edge (110){001} dislocations should feel the same type of lattice friction since their core structures, though different from those of screws, are of the same nature, i.e., their mobile, constricted core structure is unstable and is driven to transform into a non-planar configuration. For the edge or near edge dislocations, several mechanisms have been proposed to explain their low mobility. Veyssi~re [35] noted that dislocation loops on {001} were segmented with screw dislocations showing smooth curvature and edge and near-edge dislocations having a rectilinear image. For the edge orientation Veyssi~re suggested a climb dissociation 1 mode in which the (110) superdislocation first splits into two 7(110) partials which then separate by a conservative climb dissociation on { 110} perpendicular to {001 }, i.e., [110]-+ ~1[110] + APB(il0) + 1 [~ 10] This is followed by a further decomposition of the superpartials by glide 1 ~[110]-+ ~1[111]+
1
[112]
Here between the Frank ( 89(111)) and Shockley partials is a strip of stacking fault which may be intrinsic or extrinsic depending on the direction of motion of the Shockleys. However, owing to the limitations of the weak-beam method, it was not possible to identify experimentally the dissociation mode in further detail. It was subsequently found that climb dissociation could also occur on {310} planes [36]. The pronounced lirrearity of edge dislocations was later confirmed by Sun and Hazzledine [23] in both Ni3A1 and Ni3Ga and by Korner [37] in Ni3(A1, Ti) and was also revealed later by in-situ experiments [38]. Different immobilization mechanisms were 1 proposed, however [22, 17, 24]. It was noted [23] that a 7(110) edge superpartial on {001 } was the equivalent of the Lomer dislocation in f.c.c., although in 7' such dislocations were a consequence of cube slip and not formed as a result of dislocation reaction as they are in the original models of Lomer [15] and Cottrell [16]. An edge 1 (110)(001} superdislocation dissociating into two 7(110) partials was called a double Lomer [17, 23], to which the dissociation mode proposed by Cottrell for f.c.c, can be extended to form a double Lomer-Cottrell lock, described by eq. (5) and illustrated in fig. 6(a).
42
Ch. 49
Y. Q. Sun and P. M. Hazzledine
(a)
B~
ar
~ J.
s
APB
~A
Bo~
~A
Fig. 6. (a). The double Lomer-Cottrell lock on the edge (110){001} dislocation. The dislocation is split into two edge superpartials each of which is the equivalent of the Lomer dislocation in f.c.c.. The entire lock consists of two ordinary Lomer-Cottrell barriers. (b) An HREM picture of a double Lomer-Cottrell lock in Ni3Ga deformed at 600~ (From Y.Q. Sun, P.M. Hazzledine, M.A. Crimp and A. Couret [17], by courtesy of Taylor and Francis Ltd). In the experiments on Ni3Ga, edge (110){001} dislocations in the double-Lomer configuration were observed by the weak-beam method in samples deformed at around 600~ Weak-beam observations showed that edge (110){001 } dislocations in the double Lomer configuration were less mobile than those of other angular characters. Limited by the resolution power of the weak-beam method, no further splitting can be resolved on the individual Lomer superpartials, although the Lomer-Cottrell dissociation provided the most straight-forward explanation for their low mobility [23]. Lattice resolution microscopy was used to reveal the further splittings by viewing the edge dislocations end-on in specially oriented foils. Figure 6(b) shows an example of a Ni3Ga sample
w
Geometry of dislocation glide in LI 2
,3[ ! -phase
43
parallel to (110) with the edge dislocations in the primary cube system [110](001) lying end-on. The dislocation is seen to consist of two A-shaped components lying on the trace of the (001) plane, consistent with their being two superpartials each of which has dissociated according to scheme (5). The dissociation into Lomer-Cottrell locks is allowed only at lattice positions separated by b. At intermediate positions there is no apparent crystallographic reason for the edge superpartials to split on {001} and it is most likely that they are effectively constricted, as a result of which there is no resistance to the locking dissociation. At higher temperatures, dissociation by climb was observed on edge (110){001} dislocations, but the mode of dissociation, as revealed by lattice imaging TEM, is different from that initially proposed by Veyssi~re et al. [35, 36]. With the weak-beam method, as shown in fig. 7(a) for a Ni3Ga sample deformed at 750~ the pure edge dislocations are seen to have split into three partials while the non-edge dislocations are split into two. The transition from pure edge to non-edge is sharp, as pointed to by arrows in fig. 7(a), and all the partials recombine at the junctions. An undissociated (110){001} edge dislocation is called a super-Lomer since its Burgers vector is twice as long as that of a Lomer in f.c.c. [ 17]. At the temperature where super-Lomers are observed, the substructure is still dominated by screw and edge dislocations, but with the edges having visibly larger proportions than the screws, as noted in a number of studies [24, 35-38]. The screw dislocations are curved smoothly and thought to be the major contributor to the plastic strain. Observations such as fig. 7(a) suggest that at this temperature the super-Lomers are probably effectively immobilised by their non-planar core structures. It proved to be impossible to index the Burgers vectors of the three partials involved with the weak-beam method, and lattice resolution TEM was thus used which showed that the dissociation has taken place by climb, as shown in fig. 7(b). Here the sample was deformed under conditions similar to those for fig. 7(a) except that a much higher dislocation density was introduced by straining to about 10%. The foil is parallel to (110) so that the edge dislocations in the primary [110](001) slip are lying normal to the foil plane. The dislocation in fig. 7(b) has been identified to be a (110){001 } edge dislocation by constructing a Burgers circuit around the entire dislocation and indexing its closure failure, giving b - [110] [17]. The dislocation is seen to split into three partials, two on the intersecting { 111 } planes ((1 i 1) and (111)) and the third, the stair-rod dislocation, at their junction. The two partial dislocations at the extremities of the V-shaped structure are connected to the stair-rod dislocation by stacking faults on { 111 }. By viewing along the traces of the two { 111 } planes, it is seen that the stacking faults on both { 111 } planes are characterised by the removal of a { 111 } layer of atoms and thus the stacking faults 1 are intrinsic and the two extreme partials are therefore Frank partials with b = .~ (111). This observation is explained by the following dissociation scheme, -
1
1
1
[110] -+ ~ [ l i l ] + S I ( E ) S F ( l i l ) + 5[li0] + SI(E)SF + g[liT], where the choice of SISF or SESF depends on the climb direction of the Frank partials. In fig. 7(b) SISFs are involved, but many of the dislocations observed contain an SISF on one arm and an SESF on the other [17]. In fig. 7(b) further dissociation on the intersecting (111 } planes can be seen on the two Frank partials and such dissociation is
Fig. 7. (a) A weak-beam image showing the three-fold splitting on edge (110){001} dislocations. The image plane is (001). The three images constrict at the sharp junctions with the screw orientation at which the ordinary two-fold splitting is observed. (b). An HREM picture showing a super-Lamer-Cottrell lock on an edge (1 10){001} dislocation in Ni3Ga. The dissociation involves climb and the structure is illustrated in (c). Here ‘in.’ stands for superlattice intrinsic stacking fault (SISF). (From Y.Q. Sun, P.M. Hazzledine, M.A. Crimp and A. Couret [17], by courtesy of Taylor and Francis Ltd.)
9 %
Geometry of dislocation glide in L12 "y~-phase
w
45
thought to be of the same type as the formation of a stacking fault tetrahedron from a prismatic Frank dislocation loop [39], i.e., -
1
-
1
-
1[111]--9 ~ [ l l 2 ] + C S F ( 1 T i ) + ~[110] A schematic illustration of the climb dissociation of the super-Lomer dislocation is given in fig. 7(c).
4.3.
Slip mechanism
(110){001} slip has the property that dislocations are locked along two directions, the Kear-Wilsdorf lock on screws and Lomer-Cottrell locks on edges. For a given slip system, if the dislocations are locked along only one orientation, slip can still proceed through the propagation of the mobile dislocations and the deformed structure consists of a high density of dislocations locked in the specific orientation. This is the case in some b.c.c, metals in which (111) screw dislocations are locked in the three-fold non-planar core structures, and it is also true of the anomalous (110){111) slip in L12 (see later) for which locking also occurs along the screw orientation. If locking occurs on two or more orientations, the entire dislocation loop may be blocked along these orientations, forming rhomboidal or polygonal loops. The locking of both screw and edge dislocations in (110){001} is essentially the same as that of the screws in b.c.c, in that the mobile state is unstable to the locking dissociation. Continuation of loop expansion requires its unlocking and thus at a given stress level (110){001} slip starts when the temperature is sufficiently high to assist the recombination of the partials involved in the non-planar dissociations.
5. Dislocations in the anomalous ( 1 0 1 ) { 111 } slip 5.1.
Distribution
of dislocations
A well established fact concerning dislocations in samples that have undergone (101){111} slip with the yield stress anomaly is that the substructure is dominated almost entirely by screw dislocations with very few edge segments present. This was noted in the early TEM observations of deformed Cu3Au and led to a mechanism, proposed by Kear and Wilsdorf [28], in which pinning was caused by the screw dislocation cross slipping onto a cube plane. When the screw dislocation is fully dissociated on the cube plane, the structure is known as a Kear-Wilsdorf lock. This microstructural feature was subsequently observed in Ni3A1 and Ni3Ga containing various alloying additions where the dominance of the screw dislocations was more pronounced and occurred in the entire temperature range of the yield stress anomaly [9, 40-45]. Figure 8 shows a typical example of dislocations in a Ni3Ga single crystal sample deformed at 200~ which is within the temperature region of the yield stress anomaly and at which the active slip system is (101){111}. The foil plane in fig. 8 is parallel to the macroscopic { 111 } slip plane and the 202. diffraction vector used to form the weak-beam images is m
46
Y. Q. Sun and P. M. Hazzledine
Ch. 49
Fig. 8. A dislocation loop in a Ni3Ga sample deformed at 200~ The loop on the whole is elongated along the screw orientation. A superkink is marked SK. The four pictures were taken while the specimen was tilted around the screw dislocation line. In (a) the electron beam direction (viewing direction) is close to [111] and in (d) the beam direction is very close to [010]; for (b) and (c) the beam is at intermediate orientations. The pictures show that the edge parts of the loop, marked E in (a), are on the (111) plane, but some of the screw parts, marked C, are curved on (010).
w
Geometry of dislocation glide in LI 2 "7~-phase
47
parallel to the Burgers vector [101]. This microstructural feature, showing dominating screw dislocations in the substructure, characterises the entire regime of the yield stress anomaly in many L 12 compounds. Although the macroscopic slip plane is { 111 }, as has been confirmed frequently by slip-line observations (e.g., [32]), the screw dislocations are dissociated with the APB on the cube cross slip plane {010}, i.e., they are in the Kear-Wilsdorf configuration. In this experiment the dissociation plane of the screws is determined by tilting the sample around the screw dislocation line direction and measuring the separation of the superpartials when they are imaged along different directions. In fig. 8 the four TEM images were taken while the sample was tilted around the screw dislocation line direction to four different orientations relative to the incident electron beam. The top picture (fig. 8(a)) views the sample near the [111] direction and the bottom picture (fig. 8(d)) views the sample along [010]; figs 8 (b) and (c) are in between. The predominance of screws in the deformed microstructure is usually taken to indicate that the edge dislocations are relatively more mobile than the screws. The slip system (101){111} is known to possess a yield stress which does not obey the Schmid law and differs in tension and compression [32, 46, 47]. These macroscopic properties have been interpreted on the basis of the effects of the uniaxial load on the further dissociation of the screw superpartials [10, 32]. 5.2. Observation of dissociations
Dissociation of (101) superdislocations into two APB-coupled superpartials on {111} was first observed in L12 ordered Cu3Au by Marcinkowski et al. [3] using the TEM bright-field imaging method. Development of the weak-beam method [48, 49] has greatly enhanced the resolution capability of the TEM for dislocations and partials separated by around 15 ,~ can be resolved. The method has been used extensively to identify dislocation dissociations in ordered alloys, including the -y'-phase. Edge dislocations on { 111 } are expected to exhibit wider dissociations owing to their stronger elastic interactions and resolving the dissociation for the (101){l 1 l) edge dislocations using the weak-beam method has been found to be relatively straight forward; although in practice their rarity in some alloys is more of a problem than resolving the partials. The full dissociation mode of edge (101){111} dislocations, including the dissociation into Shockleys, has been observed together with a systematic measurement of dissociation widths and fault energies [ 19, 20, 50, 51 ]. A certain degree of splitting of the superpartials has been reported frequently on edge dislocations with the weak-beam method, but this requires a very large deviation from the Bragg diffraction condition and as a result the exposure time can be very long. The most systematic study using the weak-beam method is perhaps that of Hemker and Mills [20] who observed, on near-edge dislocations, the further splitting of the superpartials and also performed corrections for diffraction effects to obtain the true dissociation widths. The width of dissociation into Shockley partials was found to be around 2 nm on the near-edge dislocations in a binary Ni3A1 compound, corresponding to a CSF energy of approximately 200 mJ/m 2. Of particular relevance to the understanding of this slip system and other microstructural features, to be outlined in the following sections, is the elucidation of the dissociation mode of the screw dislocations and the extent to which the superpartials are able to
Y Q. Sun and P M. Hazzledine
48
Ch. 49
split further into Shockley partials. Weak-beam experiments conducted on a number of alloys have confirmed that along the screw orientation the superpartials are on the cube cross slip plane; the dissociation of the cross slipped screws is therefore the same as that of the screws in the pure (110){001} slip. With the weak-beam method, resolving the further splitting into Shockley partials on screws has proved to be difficult. Along the screw orientation the expected separation between the Shockleys is around 0.7 nm based on the CSF energies measured on the edge dislocations, and this is beyond the resolution limit of the weak-beam technique. Using the lattice imaging method [6-8] the CSF-coupled splitting was resolved by imaging the screw dislocations end-on in thin foils, as shown in fig. 5(b). Owing to the surface effects, the observed core extension representing the dissociation into Shockley partials may not represent the true dissociation width in the bulk, but the observed small core extension is in keeping with the weak-beam experiments. The fact that the CSFs on the two superpartials of the same superdislocation are sometimes on different { 111 } planes suggests that the CSF dissociation plane is able to change under thermal activation, perhaps through the well-developed kink models [52, 53].
5.3. Superkinks and switched superpartials Whereas it was established that the screw dislocations dominated the substructure in samples deformed in the anomalous regime, TEM observations in recent years have shown a number of other important features that also characterise the anomalous (101){111} slip. Of particular importance are the superkinks, screws bowing-out on the cube plane, and APB tubes and short edge dipoles; these features are dealt with in the next three sections and their interpretations will be given in section 5.6. Weak-beam TEM observations have shown in a number of anomalous L12 alloys that although the screw dislocations dominating the substructure are dissociated on {010}, they are not perfectly straight but contain many steps, known as superkinks, which are lying on the {111} slip plane [43, 45, 54-56]. One of such superkinks can be seen in fig. 8 marked by SK. That the superkinks lie on, and are therefore dissociated in, the { 111 } slip plane, is determined by noting the change in the projected kink height while the sample is viewed along different crystallographic orientations. Most of the superkinks have heights many times the APB width on screw dislocations, but smaller kinks were also noticed in early studies [54, 56]. Couret et al. [57, 58] carried out a statistical measurement of kink distribution (i.e., kink population vs. kink height) in Ni3Ga samples deformed at various temperatures. They found that, except for the kinks with height corresponding approximately to the APB width in samples deformed at low temperatures, the kink number N(h) follows an exponential variation with kink height h. The results for 20, 200 and 400~ are shown in fig. 9. This shows that the superkink distribution can be written in the following form,
N(h)-
h N'exp ( - ~),
where the parameter N ' is related to the total number of kinks and 1/h is the slope in the In N - h plot, h being the statistical average of kink height. The mean kink height
Geometry of dislocation glide in L12 ,yt_phase
w
(a)
20 ~
(b_)
~
In
49
200=(3
114
In80
4
"i
4
_=
=,.
,
I
100
a
200
I
300 h(.~)
I
I
400
500
100
(c)
200
300 h(A)
400
500
400"(3 41
i.-=1
i.-,.=1
I
I
100
200
I
300 h(A)
I
400
_
,
500
Fig. 9. The logarithm of kink numbers (N) plotted against kink height (h) to show experimental measurements of the distribution of superkinks in Ni3Ga samples deformed at three different temperatures. The distribution follows an exponential relation except for the kinks with height scaling approximately with the APB width. The average kink height decreases with increasing temperature. (From A. Couret, Y.Q. Sun and EB. Hirsch [58], by courtesy of Taylor and Francis Ltd.)
50
Y Q. Sun and P. M. Hazzledine
Ch. 49
Fig. 10. A TEM weak-beam micrograph to show switched superpartials (indicated by an arrow). At some places, e.g., those marked C, the two partials are combined and full switching can not be identified. is found to decrease with temperature, with h - 170 A, h - 120 ,~, and h - 80 ,h,, at the above three temperatures. This study also shows that in samples deformed at 20 and 200~ there is an exceptionally large number of kinks which have heights approximately equal to the width of the APB (50-100 ,~); in fig. 9 for 20~ and 200~ such kinks are represented by data points significantly off the exponential distribution. Bontemps and Veyssi~re [56] have shown that there are also many kinks on which only one of the partials is stepped and the other remains straight. Another important feature characterizing the superkinks is that the two APB-coupled superpartials are frequently seen to have changed partners across the superkink. Figure 10 shows an example where the switched partials are indicated by an arrow. By tilting the sample inside the TEM over a wide angular range, it has been confirmed that the switching of the partials is real and is not a deception due to projection [24, 58]. There are also numerous cases where the full switching is not shown in the images, but instead the images of the superpartials are seen to have constricted; examples of such constrictions are marked C in fig. 10. It is possible that these images of constrictions also represent switching of the partials (fig. 10).
5.4. Increasing activity on the cube cross slip plane with temperature TEM evidence of significant activity on the cross slip cube plane below Tp was noted in early studies by Staton-Bevan and Rawlings [42] who found many curved dislocations on the cube cross slip plane in samples deformed at temperatures near but still below the yield stress peak. Later investigations showed that in some alloys dislocation activity on the cross slip cube plane existed at much lower temperatures, e.g., room temperature [43, 59]. While most of the screw dislocations locked in the Kear-Wilsdorf configuration are straight (apart from the superkinks shown above), some of the screw dislocations have been found to bow out on their dissociation plane, i.e., the cube cross slip plane which is not the dominant macroscopic slip plane. Such examples can be
w
Geometry of dislocation glide in L12 "g'-phase
51
seen in fig. 8 where the screw segments marked C are nearly straight in fig. 8(a) when the sample is viewed along [111], but are curved when viewed along [010], fig. 8(d). Such observations show that whereas the edge dislocations have propagated on { 111 }, numerous screw segments have propagated on {010} as well as being dissociated on that plane. The fact that the screw dislocations are able to maintain their curvature on {010} after the removal of the load indicates that the lattice frictional force on the screw dislocations is high. Recently, Karnthaler et al. [60] compared the degree of activity on the cube cross slip plane for two samples deformed along the <001) cube orientation and along (123). They found that bowing on {010} appeared only in the sample deformed along <123) but not in that deformed along <001), as expected from Schmid factor considerations.
5.5. APB tubes and edge dipoles In Ni3A1 deformed in the regime of the yield stress anomaly, the presence of numerous edge dipoles, in addition to the dominating screws, was noted by Staton-Bevan and Rawlings [42]. They also showed that the edge dipoles were in general quite short and tended to become even shorter at higher temperatures. The presence of edge dipoles has been confirmed in recent investigations [24, 60-63]. In samples subjected to fatigue tests the proportion of edge dipoles relative to screw dislocations becomes substantially larger and their accumulation was thought to be responsible for the eventual failure [64]. In general the presence of such dipoles is thought to result from the cross slip of screw dislocations. Recent research has shown that some of the short edge dipoles are connected to antiphase boundary tubes. An APB tube encloses a region in which the lattice is displaced by an APB displacement vector/~ with respect to the lattice outside the tube; the tube wall is therefore a continuous interface of APB. An APB tube produces contrast in the TEM for two reasons. One is associated with the phase shift of 27rg. R in the electron waves scattered from the inside and outside of the tube. The interference between the waves forms an image if g" R r integer which is possible only with 9 representing certain superlattice reflections. This contrast mechanism was used to observe APB tubes in FeA1 and Ni3A1 [67, 68]. Another reason is that, in addition to the rigid APB displacement, an APB tube also possesses a continuous strain field with large components orthogonal to the tube axis but no component parallel to the axis [62, 69]. Like the strain field of dislocations, this strain field makes tubes visible because it changes the local diffraction condition. This allows the tubes to be observed in reflections, superlattice or fundamental, with large components perpendicular to the tube axis, whereas the tube should be invisible in reflections parallel to the tube axis. The strain field was thought to originate from the surface tension of the APB which compresses the material inside the tube and shears the material outside [62]. At the atomic scale the origin lies in the relaxations of atoms from perfect lattice positions and particularly large distortions are expected near sharp corners [62, 69]. In Ni3Ga it has been shown [62, 65] that many APB tubes terminate at the small edge dipoles, an example being shown in fig. 11. In fig. 1 l(a) the image was formed
52
Y. Q. Sun and P. M. Hazzledine
Ch. 49
Fig. 11. TEM dark-field images to show APB tubes. The tubes are invisible in the reflection parallel to the tube axis (a) and are visible as straight lines in reflections containing large components normal to the tube axis, such as (b). (From Y.Q. Sun [62], by courtesy of Taylor and Francis Ltd.) m
with fundamental reflection 202 parallel to the tube axis and the tubes were out of contrast. In fig. 11 (b) where the reflection 010 is perpendicular to the tube axis, the APB tubes are imaged as faint straight lines parallel to the Burgers vector of the surrounding screw dislocations. It is seen that some tubes end at edge dipoles marked A and B. A scrutiny of the literature shows that image features characteristic of APB tubes exist in several publications, often with dipoles at the terminating ends [45, 70, 71]. These image features escaped the attention of the early investigators apparently owing to the fact that the contrast selection characteristics of APB tubes became known only recently. It can however be said now that the presence of APB tubes and the edge dipoles is also a characteristic feature of -y~ deformed in the temperature region of the yield stress anomaly.
w
Geometry of dislocation glide in L12 "t't-phase
53
5.6. Dislocation propagation in (101) { 111 } slip as suggested by TEM observations The TEM observations presented above have shown that while the macroscopic (101) slip is on { 111 }, significant activity of the screw segments occurs on the cube cross slip plane. This, together with the superkinks, switching of the partials, and the formation of APB tubes, are the most distinguishing features of the dislocation substructure in "),~ deformed by the anomalous (101){ 111 } slip. These characteristic features are interpreted as being formed through the dislocation processes illustrated in fig. 12. Figure 12(a) shows a dislocation segment that has cross slipped off the { 111 } macroscopic glide plane and has made a number of cross slip jumps on (010) and (111) before finally settling down in the Kear-Wilsdorf structure with the APB on (010). To explain the switching of the partials and APB tubes attached to dipoles, it is necessary to assume that on the screw segment both superpartials cross slip off the glide plane { 111 }; this is unlike the PPV model [10] in which only the leading partial has moved on {010} by a short distance. Assuming the cross slip occurs on a segment of a finite length, the adjoining mobile part of the dislocation continues propagation on the initial { 111 } plane ahead the cross slipped segment, fig. 12(a). When the mobile part also cross slips following a path similar to that shown in fig. 12(a) (this second cross slip event is not shown in fig. 12), the structure formed has two cross slipped segments linked by a step or superkink. The height of the kink depends on the probability of cross slipping and also on dislocation velocity.
5.6.1. The switching of the superpartials The continuous lateral propagation of the mobile kink in fig. 12(a) can be shown to lead to the switching of the two superpartials. In the local pinning models, such as that of PPV, breaking-down of the small pinning points should not invoke the switching of the partials, whereas in the recent model by Hirsch [11], unlocking takes place on fully cross slipped segments-segments which have made several jumps on {010) and { 111 } and with long lengths, as illustrated in fig. 12(a). In fig. 12 L, L ~ and T, T' are, respectively, the leading and trailing superpartials before cross slip. As the adjoining mobile dislocation continues to expand on the glide plane { 111 }, a short segment on the leading partial is created (indicated by an arrow in fig. 12(b)) which has a line direction opposite to that of the trailing partial. When this segment annihilates with a segment on the trailing partial, the two partials change partner, leading to the apparent switching of the superpartials, i.e., L joins T ~ and T joins L ~ (fig. 12(c)). 5.6.2. Edge dipoles and APB tubes connected to dipoles In the model by Hirsch [11], the above partial switching process is a necessary step in the unlocking or by-passing of the fully cross-slipped segments, but it is not thought to directly determine the critical stress for the complete freeing of the dislocations. Instead the mobile superkinks are pinned by the edge dipoles, marked ED in fig. 12(d), and it is the by-passing of the edge dipole which has been proposed to be rate-controlling. It is shown in the following that by-passing of the edge dipole leaves behind either a short edge dipole or an APB tube connected to a dipole; both features are characteristic of the dislocation structure observed in the TEM. Because the mobile dislocation is pinned at
54
Ch. 49
Y. Q. Sun and P. M. Hazzledine T
L
,~/N~...~(~~~.i.~........~.~~/~
(c)
.~
/4
T
(b)
..~ ""~~';~i'~"';a" ~
L
)__ -,
.-" .-
DIPOLE
\\\ // // .// //"
v__#j
)
,.
Fig. 12. Dislocation by-passing process in (110){ 111} slip as suggested by TEM observations. (a) The cross slip, over a finite segment, involves both partials moving off the (111) glide plane. LLI and TT / are the leading and trailing superpartials respectively before cross slip. (b) As the mobile non-screws (superkinks) continue to expand on (111), a short segment (indicated by an arrow) of the leading partial acquires a line direction opposite to that of the trailing partial. (c) When the opposite partials annihilate, the adjoining partials change partner, i.e., L ~joins Tt and T joins L t. (d) The pinning of the mobile part is caused by the edge dipole, marked ED, which does not annihilate without sufficient diffusion. The continuing expansion of the mobile part leads to the formation of the screw dipole which is marked SD. (e) If the screw dipole annihilates imperfectly - such annihilation occurs by cross slip - an APB tube is generated attached to the edge dipole. If the annihilation is complete, only the edge dipole is left behind. the edge dipole ED, the continuous lateral propagation of the superkink generates a new screw segment, with an opposite line direction, which is trapped by the cross slipped screw to form a screw dipole, marked SD in fig. 12(d). The edge dipole can annihilate if the temperature is high e n o u g h for diffusion to take place, whereas the screw dipole
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Geometry of dislocation glide in L12 ~/~-phase
55
can annihilate by cross slip even at low temperatures. If the screw dipole in fig. 12(d) annihilates completely, only a closed edge dipole loop is left. If the annihilation is imperfect as a result of the superpartials following different paths during cross slip [67, 72], the result is a dipole with a tube attached to it (fig. 12(e)).
5.6.3. Superkink distribution and probability of locking The observation that the distribution of superkinks follows an exponential distribution N(h) - N' exp(h/h), and that the average kink height h decreases with temperature are evidence that the kink height is determined by a thermally activated process. The superkink distribution is best explained by assuming that each kink results from a cross slip locking event. The kink population distribution can be shown to be related to the probability of locking via N - N' exp{-Ph}, where P is the probability of locking per unit distance and is given by p-
v~ exp ( G1 }
where u0 is the attempt frequency, v is the free-flight velocity, and G1 is the locking activation enthalpy; k and T have their usual meanings. A full discussion can be found in [58].
6. ( 1 0 1 ) { 1 11 } slip: SISF dissociation Based on the crystal symmetry of the L12 structure an SISF on { 111 } can be shown to 1 be a generally stable fault with its displacement (g(112) type) corresponding to a local minimum in the 3,I-surface, whereas the pure APB displacement 71 (110) (based on the hard-sphere model) in general does not correspond to a local minimum; true, metastable APB displacement is obtained by adding a supplementary vector orthogonal to the ideal APB vector [5]. With increases in the ordering energy, the magnitude of the supplementary displacement increases and { 111 } APB is predicted to become completely unstable eventually, leaving SISF as the only stable fault on { 111 }. In experiment the dissoci1 ation of a (101)superdislocation into SISF-coupled .~(112)super-Shockleys has been observed only in some cases in alloys deformed at low temperatures. In Ni3Ga [24, 73] weak-beam observations in samples deformed at 77 K showed that the superdislocations were dissociated into two partials with non-parallel Burgers vectors which were identi1 fled to be .~(112) super-Shockleys. (In the same material the superdislocation was found to be APB-dissociated at and above room temperature.) In Ni3A1 lattice-resolution TEM also showed dissociations consistent with SISF dissociation [73]. Nevertheless, such dissociation has not been widely reported [43]. Also, the reason for SISF dissociation being favoured at low temperatures is not entirely clear. It is generally observed that, with the decrease of testing temperature to below room temperature, ribbons of SISF enclosed
56
Y. Q. Sun and P. M. Hazzledine
Ch. 49
1 by .~(112) loops become very common, but it is not yet established fully if these SISF ribbons are formed while the superdislocations are in APB or SISF dissociation. Suzuki et al. [74] suggested that the SISF-ribbons were nucleated on APB-dissociated (110) superdislocations, but it is not evident what is the driving force for this transformation. On the contrary, Pak et al. [70] suggested that the SISF loops could be explained if the /101) superdislocations were dissociated into SISF-coupled super-Shockleys. The latter view was supported by Sun et al. [73] who performed 9 " b invisibility analysis for the individual partials. It was found that the two partials constituting the (101) superdislocation were not out of contrast in the same reflections and they therefore must possess non-parallel 89 (112) Burgers vectors. The dislocation structure in a Ni3Ga sample deformed at 77 K is shown in fig. 13 where the g1 i l l 2 ) loops enclosing SISF's are 1 seen to be pulled off the (101) superdislocation from only one side. The .~ (112) loops in the substructure possess the same Burgers vector and by cutting foils with different orientations they were identified to represent the trailing super-Shockley partial [73]. By tilting the sample around the screw line direction and recording the change in the projected partial separation, the two partials on the screws were shown to be lying on the (111) glide plane, definitely not in the Kear-Wilsdorf configuration as occurs in samples deformed at higher temperatures. Since the sample was inevitably held for significant lengths of time at room temperature before being examined in the TEM and it is known that at room temperature screw dislocations dissociated into APB coupled superpartials are able to transform into the Kear-Wilsdorf configuration, the fact that the screws in samples deformed at 77 K are not in the Kear-Wilsdorf configuration suggests that the real dissociation mode is at least different from that of APB-coupled superpartials. On the whole the circumstances in which SISF ribbons are formed vary from alloy to alloy but clearly fall into two distinct categories, i.e., when the surrounding superdislocations in the post-mortem samples are ABP-dissociated and when they are SISF-dissociated. It is anticipated that in-situ experiments may help to make further clarifications. A recognizable feature of the dislocations in samples deformed at low temperatures is that the distribution of the dislocations is not aligned only along the screw orientation as it is in samples deformed at high temperatures. A close examination shows that the preferred directions are (110), as evidenced by the 30 ~ Shockley partial loops in fig. 13. This feature was noted in early observations by Giamei et al. [75] and also by Pak et al. [70], and led to a suggestion by Giamei et al. of a locking mechanism operative only along 30 ~ g1 (112) super-Shockley partials. Similar to the formation of the Kear-Wilsdorf lock, it also involves the cross slip of 89 (1 TO) superpartials, but assumes the following dissociation of the super-Shockleys,
- -+ 1 [121] + APB(111) + 1[10~ ] 11112] or
~1,2]1 [1 --+ 61[2il] + APB(1 11) + 1 [01 i]
Fig. 13. A we&-beam micrograph to show the SISF dissociation and the formation of SISF dipoles (loops) from one side of the superdislocation. The dipoles form from the trailing super-Shockley and they are aligned predominantly along 30' directions. bl and b, are Burgers vectors of the leading and trailing super-Shockleys which are separated by about 60 A on the screw superdislocation. (Y.Q. Sun, M.A. Crimp and P.M. Hazzledine [73], by courtesy of Taylor and Francis Ltd.)
vI 4
Y. Q. Sun and P. M. Hazzledine
58
(a)
Ch. 49
s,s .......... i
APB~.J__L CSF
Fig. 14. (a) The Giamei lock on a 30~ 1 dislocation. The lock is formed by two outer Shockleys cross slipping to transfer the APB onto an {010} plane. (b) An HREM picture of a Giamei lock in Ni3A1, showing steps on (010). (Y.Q. Sun, M.A. Crimp and P.M. Hazzledine [73], by courtesy of Taylor and Francis Ltd). 1
--
in which the ~(110) superpartials are expected to exhibit further splitting as discussed 1 in the preceding sections. When the g[117.] super-Shockley is in 30 ~ orientations (along [10i] or [011]), the constituent super-partial is a screw which may cross slip to transfer the APB from (111) to {010}, making the dislocation core non-planar; this is known 1 as a Giamei lock [75] (fig. 14(a)). The core structure of 30 ~ .x(l12) super-Shockleys has been observed end-on using lattice imaging TEM [73], an example being shown in fig. 14(b) which shows a step on a cube plane. This is an experimental confirmation of the operation of a Giamei lock [73, 75].
7. A summary of observations relevant to yielding In this chapter we have emphasised the geometry of dislocation glide in L12 alloys on three scales: the shape of the spreading dislocations on a scale of microns or more, the
w
Geometry of dislocation glide in LI 2 ~/~-phase
59
gross dissociation into APB- or SISF-separated superpartials on a scale of nanometers, and the fine dissociation into partials of the parent f.c.c, lattice on a scale of angstroms. These observations are summarised as schematic illustrations in fig. 15. Figure 15(a) shows the representative temperature regimes for which the dislocation structures are illustrated in fig. 15(b)-(f). At temperatures below Tp slip occurs on the (111) plane (fig. 15 (b) and (c)) and at temperatures above Tp it occurs on {001} planes, fig. 15(e) and (f). Close to but below To, there is an intermediate form of slip in which the edges move on (111) and the screws bow out on the cross slip plane (010), fig. 15(d). Well below room temperature the [101] dislocations on (111) may split into two 1 7(112) super-Shockleys separated by SISF, fig. 15(b), and each super-Shockley may further dissociate into three Shockleys when the dislocation is mobile on (111), i.e., 1
1
1
1
-( 33,11~., -+ g (2ii) + CSF + g (121) + APB + g (11~.) The super-Shockleys may be locked along 30~ directions, fig. 15(b), because they form Giamei locks in which a pair of Shockleys constrict and cross slip on (010) a short distance before redissociating on (111). As noted before, this dissociation mode is exceptional since it is not present in every case. Around and above room temperature in the early part of the anomalous region, fig. 15(c), the major dislocation configuration is 71 (101) pairs coupled by an APB (which can easily be formed on { 111 } from an SISF by annihilation of a Shockley dipole). Screw dislocations are locked because their fault is on (010) whereas all other dislocations with the APB on (111) are mobile. The result is long Kear-Wilsdorf screw dislocations linked by mobile edge superkinks. This is the only slip system in which the slip zone is not fully enclosed by locked dislocations. When the temperature approaches To more and more of the Kear-Wilsdorfs begin to glide on (010), fig. 15(d). At temperatures above the peak in the yield stress, fig. 15(e), slip transfers to the plane (001). The gross dissociation forms APBs on (001) and the fine dissociation forms CSFs on inclined { 111 } planes to form the Kear-Wilsdorf lock on screws and Lomer-Cottrell locks on edges. At higher temperatures the double Lomers (DL) change their form into climb-dissociated super-Lomers (SL). At the highest temperatures, fig. 15(f), [001](010) and [010](001) slips occur simultaneously. There is no gross dissociation, but fine dissociation forms CSFs on inclined { 111 } planes in the configuration of B5 locks along two orthogonal directions. The TEM observations have shown that in the order of increasing temperature the following four slip systems or dissociation modes are activated in the 3" phase: (101){111} with SISF dissociation--+ (101){111} with APB dissociation-+ (110){001} with APB dissociation-+ (100){001 }. The above sequence is however just the reverse of that ordered in terms of dislocation line energies. (a) (110){001} vs. (100){001}. An undissociated (110) dislocation has a higher line energy than an undissociated (100) dislocation at all angular characters. A comparison of dislocation line energies between APB-dissociated (110){001} dislocations and (100){001} dislocations has been calculated for Ni3A1 by Hazzledine et al. [22] using
Y. Q. Sun and P. M. Hazzledine
60
Ch. 49
(a)
[b)
It}
(d)
Tp
(el
(f}
I
I
I
I
I
I
Temperature
(b)
t[ool] SISF
//~/
APe(010) " / \ /,CSF
SISF dissociation
Giamei lock
l CSF
j
-'-,, APB dissociation
K-W lock
Fig. 15. Schematic illustrations of the geometry of dislocation structure in the 7 ' phase. (a) The yield stress vs. temperature curve. The dislocation structures at temperatures near b, c, d, e, f are illustrated in the following l five figures. (b) The dislocation structure at low temperatures. The partials are .~(112) super-Shockleys and they can be locked in the Giamei structure at 0 ~ and -t-60 ~ ((110)) directions. (c) At this temperature only the screws are locked in the Kear-Wilsdorf structure. The locked screws are joined by superkinks (SK).
w
Geometry of dislocation glide in L12 3,~-phase
61
(d)
\
?PB (010}
K-W lock
(e)
APB(001)
SISF
DL
SL CSF
Z. /CSF
/
B5
CSFJXx/~
Fig. 15 (continued). Schematic illustrations of the geometry of dislocation structure in the 3/ phase. (d) At high temperatures but still below the peak, many of the screws bow out on the cube cross slip plane (010). The edges and superkinks are still on (111). (e) Above the yield stress peak dislocations in the primary cube system are locked by Kear-Wilsdorf (KW) structure on screws and by double-Lomer-Cottrell (DL) or superLomer-Cottrell (SL) locks on the edges. (f) At still higher temperatures, the work hardening rate is high, two (010){001} systems are equally active. (010){001} dislocations are locked at 45 ~ directions by B5 locks.
62
Y. Q. Sun and P M. Hazzledine
Ch. 49
anisotropic elasticity. The calculation shows that, except for a small angular range in which (100){001} dislocations are elastically unstable, the line energy of (010){001} dislocations is lower than that of APB-dissociated (110) {001 } dislocations. (b) APB-dissociated (101){ 111} vs. (110){001 }. A comprehensive anisotropic elasticity study has not been performed to compare these two cases for all dislocation characters. However the energy relation for the screw dislocations dissociated on { 111 } and {001 } is well known [30, 31]. Except for unrealistically large 3'001/7111 ratios, dissociation on {001 } gives a lower self energy than dissociation on { 111 }. (c) SISF-dissociated ( 1 0 i ) { l l l } vs. APB-dissociated / 1 0 1 ) { I l l ) . In terms of the self energy of the partials, dislocation line energy in the SISF dissociation is expected to be higher than in the APB-dissociation owing to the longer partial Burgers vectors in the SISF mode. The self energy in the SISF dissociation may be lower than the APB dissociation when the SISF energy is exceptionally smaller than "7111. Suzuki et al. [74] found, using isotropic elasticity and ignoring the further splitting of the super-Shockleys, I of the APB energy to make the SISF that the SISF energy must be lower than about .~ dissociation energetically favourable. An estimate of the relative values of the SISF and APB energies from weak-beam observations [76] showed that this condition is not fulfilled in Ni3A1. Further research is clearly needed to investigate the energetics of SISF vs. APB dissociation by incorporating elastic anisotropy and the further splitting of the super-Shockleys. The dislocation self energies calculated with elasticity are comparatively insensitive to temperature, so it is safe to conclude that the L 12 7' alloys do not respond to an imposed temperature and strain-rate by multiplying dislocations with the lowest line energies. Instead, the slip system which operates, and the mechanical properties which result, must be controlled by dislocation mobility. In any temperature interval, the slip system in operation is that which generates strain at the lowest stress. The fact that there can be as many as four distinct slip systems in four temperature intervals (note the absence of SISF dissociation in some alloys) is a reflection of the great variety of dissociations available to the dislocations. This in turn leads to a variety in the temperature dependence of the mobility of the dissociated dislocations. Above Tp the temperature dependence of the yield stress is normal and the two slip systems which operate ((110){001}, (010){001})have one property in common: in the state with the minimum energy the dislocations are immobile as a result of being dissociated out of the glide plane, while the mobile state is unstable to the locking transformation. All of the dislocations have small dissociations (a few atoms wide) out of their {001} glide planes. (We ignore the super-Lomer dislocations at this stage.) The core structures are analogous to those of the screw dislocations in b.c.c, metals except that there are two orientations, not just screws, along which the dislocations are dissociated out of their glide planes. In order for the dislocations to move, their cores must be constricted by thermal activation. The resulting compact core is in a mobile but high-energy state. As soon as it has moved by one lattice repeat distance, it may lower its energy by redissociation. In this way the dislocations may advance in a Peierlslike manner. The higher the temperature, the more frequently constrictions occur and the lower is the yield stress. The fact that (010){001} slip needs higher temperatures to activate than (110){001 } implies that the Kear-Wilsdorf and Lomer-Cottrell locks in
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Geometry of dislocation glide in L12 "Y'-phase
63
(110){001} are weaker locks than the B5's in (010){001). Further research is needed to identify reasons for this difference, presumably by carrying out atomistic simulations. Below Tp the temperature dependence of the yield stress is anomalous and the two dissociation modes, SISF and APB on (111), have one property in common" their dislocations, of all characters, have planar dissociations when they are mobile (into four Shockleys in the case of APB dissociation and into six Shockleys in the case of SISF dissociation); in the mobile state the dislocations are at a local minimum in energy. (For SISF dissociation the planar configuration represents a local minimum in energy if the APB is stable on {111} [13].) These dislocations are like dislocations in f.c.c, metals and they would have a yield stress which declines gradually with increasing temperature were it not for one complication: both SISF and APB dislocations can lower their energy by transferring their APBs to the cube cross slip plane. The process of transferring the APB requires a pair of Shockleys to constrict with the help of thermal activation and so the probability of locking increases as the temperature rises. The resulting locks, Kear-Wilsdorf and Giamei, have different properties; we discuss Kear-Wilsdorf first. The APB-dissociated [101](111) slip system is unique in that only the screw dislocations may lock. We can imagine a complete mobile loop of dislocation spreading on the (111) plane with planar dissociation. Where the dislocation is in screw orientation its leading and trailing superpartials may constrict, cross slip and redissociate either on a parallel (111) plane or on (111). This process requires an increase in energy at first, but is rewarded by an eventual reduction in energy. The screw dislocation becomes locked temporarily but the adjoining segments may continue to glide. There is general agreement that it is this process, the increased frequency of locking as the temperature rises, which is responsible for the yield stress anomaly [9-11]. The two theoretical models [10, 11] diverge at this point. In the Paidar-Pope-Vitek [10] model the pinning points remain small (of atomic dimensions) and are destroyed by thermal activation [77], whereas in the Hirsch model [ 11] the pinning segments spread into full Kear-Wilsdorf locks involving both superpartials cross slipping off the original glide plane. It is worth noting that the spread of a pinning point on the (010) plane involves the motion of a non-planar screw and a non-planar edge and is in fact the same as the motion of a superpartial in the [110](001) slip system. This motion is Peierls-like, becoming easier at high temperatures, and would require a stress equal to the yield stress at Tp but a higher stress at lower temperatures. The SISF dissociated [10i](lll) slip system forms Giamei locks wherever one of the super-Shockleys may lock; along the other (110) directions either the leading or the trailing super-Shockley may lock. The slip zones are elongated along [101], presumably because along this orientation both super-Shockleys are locked, and they are fully enclosed by locked dislocations. As in the case of the APB [10T] dislocations, thermal activation is required to form Giamei locks and from this point of view anomalous yield would be expected so long as some mechanism exists to break the locks. On the other hand, if the { 111 } APB is sufficiently unstable, the driving force for the transition into a core structure with APB on {010} may dwarf the constriction energy, as a result of which a normal temperature dependence of the yield stress could be predicted in alloys with high ordering energies [13, 14]. It is not known experimentally whether the yield is normal or anomalous in the temperature regime of SISF dissociation. The reason why
64
Y. Q. Sun and P. M. Hazzledine
Ch. 49
SISF dissociation is preferred at low temperatures is possibly that the dislocations remain planar for long enough periods that they move macroscopic distances before they are locked. In this case the yield stress would be essentially independent of temperature. Whichever model is correct, PPV or Hirsch, or some intermediate model, the general features of the dislocation glide loop are the same: edge dislocations are highly mobile because of their planar dissociation, the slip zones are consequently elongated along the Burgers vector direction, the screw dislocations are segmented, consisting of locked segments linked by near-edge superkinks. But the detailed structures observed in TEM, e.g., partial switching, APB tubes attached to dipoles and isolated short edge dipoles, support a dislocation locking and by-passing process. To date only the model developed by Hirsch [11] contains the detailed geometrical elements in the unlocking of the Kear-Wilsdorf locks, which are consistent with the above characteristics observed in the TEM. The two models of the yield stress (PPV and Hirsch) at, say, room temperature, both assume that small nuclei of Kear-Wilsdorf locks form through thermally activated constriction of screw superpartials. In the PPV model, the locks do not grow during deformation but are constantly formed and unformed. The electron microscope observations of fully formed Kear-Wilsdorf locks are explained on the basis that when the load is removed from the specimen and the TEM samples are prepared, the nuclei have time to grow into locks. The Hirsch model [ 11] takes the view that the electron microscope observations reflect the geometry of the dislocations during deformation and that any relaxations during specimen preparation are minor. In this model, Kear-Wilsdorf screws form in a segmented fashion with locked screws linked by mobile superkinks. The exponential distribution of superkink heights reflects the fact that as a mobile screw segment advances it has a certain probability per unit time of constricting and cross slipping. As the temperature rises the probability of cross slip increases and the average superkink height decreases. The Kear-Wilsdorfs are linked to the superkinks bowing out on (111), acting as single operation sources. As they bow out they annihilate the old Kear-Wilsdorf screws. The yield stress rises because, as the temperature rises, the "source length" of the superkinks decreases. These assumed processes have two consequences which are amenable to the TEM observations: (i) As the superkinks bow out, the superpartials should be swapped over, the leader becoming the trailing and vice versa; (ii) When the superkinks annihilate the old dislocations, the process is likely to be imperfect so edge dipoles and APB tubes are left behind as debris. Both of these features have been observed, as well as the exponential distribution of kink heights in the higher part of the anomalous temperature range and they provide strong evidence that Hirsch-type mechanisms operate and that electron microscope observations reflect the deformation processes. In the lower temperature range, say 4 K to 150 K, the evidence is not so clear. The dislocations are rounded rather than kinked; these could be the remnants of PPV locked dislocations. It would not be surprising if at low temperatures (and consequently low stresses) the locks remain small during deformation. These problems, and the role and behaviour of SISF dislocations at low temperatures, which are of scientific but not technological importance, remain to be solved.
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Geometry of dislocation glide in L12 "y~-phase
65
8. Geometrical aspects of work hardening Work hardening rates in L12 alloys are known to be anomalously high (>> 0.003#, the usual value for f.c.c, metals) and to display a complicated temperature dependence in some circumstances [78, 79]. The initial work hardening rate (0) (after 2% strain) and its variation with temperature has been illustrated in fig. 1. The work hardening rate peaks at a temperature which is lower than the peak in the yield stress curve, then it falls to almost zero before rising to high values at very high temperatures. The very high values of 0 in two different temperature ranges indicate that work hardening mechanisms characteristic of ordered alloys operate and that there is more than one mechanism. No systematic theory has been developed for work hardening in L12 alloys, but it is possible to rationalize the form of the temperature dependence and the large values of 0 from the geometry of the dislocations observed to be present shortly after yield. At temperatures above Tp normal slip operates, first the (110){001} system, then the (010){001} systems. In both of these systems the slip is planar, that is, although there is some small scale dissociation out of the glide plane, the spreading dislocation loops stay on a single cube plane. The initial work hardening rate in single crystals is that of easy glide. At higher temperatures the same argument could be applied to the (010){001} slip system except for one fact: the Schmid factor on the most highly stressed system is always exactly matched by the Schmid factor on a second system. Thus the systems [010] (001) and [001] (010) always operate simultaneously and act as forest barriers to each other. It is observed that on the cube planes there are also (110) dislocations mixed in with the (001) dislocations and these also act as forest obstacles to dislocations spreading on the other cube plane. The rise in the work hardening rate at the highest temperatures could therefore be caused by the gradual dominance of slip by [010] (001), [001] (010) slip as the temperature rises. There is as yet no quantitative model to account for the (high) level of work hardening rate in this regime. In the anomalous temperature range, below Tp, the situation is much more complicated because there are many possible obstacles to glide which increase in density as the strain increases. These obstacles are of two kinds: (1) APB tubes which can only form in ordered alloys and (2) special forest dislocation obstacles which are characteristic of the glide geometry in L12 alloys. APB tubes may form either by the Vidoz-Brown [66] mechanism, being pulled out from misaligned jogs on non-screw dislocations or else by the cross slip and imperfect annihilation of screw dislocations on close slip planes proposed by Chou and Hirsch [67]. In either of these cases tubes would be observed to be attached to single edge dislocations and to impede the motion of the edges by pulling on them with a force which is similar to the line tension of a dislocation. A variant of the Chou and Hirsch mechanism is described in section 5 and fig. 12(e) of this chapter. When a mobile superkink bows out to unlock a Kear-Wilsdorf dislocation, the annihilation of the Kear-Wilsdorf is often imperfect so that short lengths of APB tubes are left behind as debris. These APB tubes may be attached to edge dipoles and many observations have been made of such associated tubes and dipoles [60, 62, 65]. Primary APB tubes have very little effect on primary dislocations [80], but their stress fields do interact with co-planar dislocations and they act as forest obstacles to dislocations on secondary planes. It has been shown [80] that the effects from the APB tubes are insufficient to explain the observed high values of 0.
66
Y. Q. Sun and P. M. Hazzledine
Ch. 49
Perhaps more important than the APB tubes are the special forest dislocations generated by glide in L12 alloys. It has been known for some years [41, 81] that edge dislocations move at a much lower stress than screw dislocations in L12 alloys in the regime of the anomalous (101){111} slip. Consequently, at a stress which is sufficient for the primary screw dislocations to move (and to give macroscopic yield), edge dislocations on a number of secondary slip systems will already have moved. For this reason, when glide commences on the primary plane there will already be a multiplying forest of secondary dislocations present to act as obstacles and to give rise to work hardening. A recent observation has shown a substantial density of "forest" dislocations in Ni3A1 deformed at about 200 degrees below the peak temperature Tp [82]. A second source of forest obstacles to primary dislocations is generated by the non-planarity of slip in L12 alloys. At lower temperatures, well below Tp, primary screws dissociate not on their glide plane but on the (010) plane and with increasing temperature more of the screw segments bow out on that plane. The screw superpartials are joined to the edge superkinks by Lomer dislocations which thread the (111) slip planes and therefore form a forest of obstacles to dislocations moving on parallel (111) planes [24, 83]. A possible explanation for the variation of 0 with temperature is that at low temperatures the forest of cross slipped dislocations is small because the cross slip distances are small. For the same reason, screws are more likely to annihilate perfectly rather than forming tubes by imperfect annihilation [72]. As the temperature rises, cross slip distances increase, but the forest density and the tube density for a given strain increases and hence the work hardening rate increases. As the temperature Tp is approached, the value of 0 falls for two reasons. First, because APB tubes have a finite life since they can disappear as a result of short-range diffusion, consequently the density of APB tubes decreases as diffusion becomes more rapid, and second, (110){001 } slip, which is not subject to significant work hardening, becomes increasingly predominant. Quantification of a work hardening model for L 12 alloys remains an outstanding challenge in dislocation mechanics.
Acknowledgement
Y. Q. Sun wishes to thank the Royal Society for a Mr and Mrs John Jaffe Donation Research Fellowship, and Wolfson College for a research fellowship. E M. Hazzledine acknowledges support from United States Air Force contract number F 33615-91-C-5663. We would like to thank Dr D. M. Dimiduk for many suggestions and comments.
References
[1] [2] [3] [4] [5] [6]
D.E Pope and S.S. Ezz, Int. Metall. Rev. 29 (1984) 136. T. Suzuki, M. Ichihara and S. Miura, ISIJ Intern. 29 (1989) 1. M.J. Marcinkowski, N. Brown and R.M. Fisher, Acta Metall. 9 (1961) 129. J.S. Koehler and E Seitz, J. Appl. Mech. 14 (1947) A217. M. Yamaguchi, V. Paidar, D.P. Pope and V. Vitek, Philos. Mag. A 45 (1982) 867. M.A. Crimp and P.M. Hazzledine, MRS Syrup. Proc. 133 (1989) 131.
Geometry of dislocation glide in L12 "TI-phase [7] M.A. Crimp, Philos. Mag. Lett. 60 (1989) 45. [8] N. Baluc, H.P. Kamthaler and M.J. Mills, Inst. Phys. Conf. Set. No. 93 (1988) 463. [9] S. Takeuchi and E. Kuramoto, Acta Metall. 21 (1973) 415. [10] V. Paidar, D.E Pope and V. Vitek, Acta Metall. 32 (1984) 435. [11] EB. Hirsch, Philos. Mag. 65A (1992) 569. [12] M. Yamaguchi, V. Vitek and D.E Pope, Philos. Mag. A 43 (1981) 1027. [13] G. Tichy, V. Vitek and D.E Pope, Philos. Mag. A 53 (1985) 467. [14] G. Tichy, V. Vitek and D.E Pope, Philos. Mag. A 53 (1985) 485. [ 15] W.M. Lomer, Philos. Mag. 42 (195 l) 1327. [16] A.H. Cottrell, Philos. Mag. 43 (1952) 645. [17] Y.Q. Sun, EM. Hazzledine, M.A. Crimp and A. Couret, Philos. Mag. A 64 (1991) 311. [18] N. Baluc, R. Sch~iublin and K.J. Hemker, Philos. Mag. Lett. 64 (1991) 327. [19] A. Korner, D.J.H. Cockayne and Y.Q. Sun, Philos. Mag. A 68 (1993) 993. [20] K.J. Hemker and M.J. Mills, Philos. Mag. A 68 (1993) 305. [21] P. Veyssi~re and D.G. Morris, Philos. Mag. A 67 (1993) 491. [22] P.M. Hazzledine, M.H. Yoo and Y.Q. Sun, Acta metall. 37 (1989) 3235. [23] Y.Q. Sun and P.M. Hazzledine, MRS Proc. 133 (1989) 197. [24] Y.Q. Sun, D.Phil. Thesis (University of Oxford, 1990). [25] P. Veyssi~re and J. Douin, Philos. Mag. A 51 (1985) L 1. [26] P.B. Hirsch (ed.), The Properties of Metals, Pt. 2 (Cambridge Univ. Press, Cambridge, 1975). [27] J.P. Hirth and J. Lothe, Theory of Dislocations (Pergamon, Oxford, 1982). [28] B.H. Kear and H.G.E Wilsdorf, Trans. Metall. Soc. AIME 224 (1962) 382. [29] P.A. Flinn, Trans. Metall. Soc. AIME 218 (1960) 145. [30] M.H. Yoo, Scr. Metall. 20 (1986) 915. [31] G. Saada and E Veyssi6re, Philos. Mag. A 66 (1992) 1081. [32] C. Lall, S. Chin and D.P. Pope, Metall. Trans. A 10 (1979) 1323. [33] J. Douin, E Veyssi~re and E Beauchamp, Philos. Mag. A 54 (1986) 375. [34] N. Clement, A. Couret and D. Caillard, Philos. Mag. A 64 (1991) 669. [35] E Veyssi~re, Philos. Mag. A 50 (1984) 189. [36] J. Douin, E Beauchamp and P. Veyssi~re, Philos. Mag. A 58 (1988) 923. [37] A. Korner, Philos. Mag. Lett. 60 (1989) 103. [38] G. Molenat and D. Caillard, Philos. Mag. A 65 (1992) 1327. [39] J. Silcox and EB. Hirsch, Philos. Mag. 4 (1959) 72. [40] B.H. Kear and M.E Hornbecker, Trans. ASM 59 (1966) 155. [41] EH. Thornton, R.G. Davies and T.L. Johnston, Metall. Trans. 1 (1970) 207. [42] A.E. Staton-Bevan and R.D. Rawlings, Philos. Mag. 32 (1975) 787. [43] P. Veyssi~re, MRS Symp. Proc. 133 (1989) 175. [44] P.M. Hazzledine and Y.Q. Sun, MRS Symp. Proc. 213 (1991) 209. [45] D.M. Dimiduk, PhD Thesis (Carnegie-Mellon University, 1989). [46] S.S. Ezz, D.P. Pope and V. Paidar, Acta Metall. 30 (1982) 921. [47] Y. Umakoshi, D.P. Pope and V. Vitek, Acta Metall. 32 (1984) 449. [48] D.J.H. Cockayne, J. Microsc. 98 (1973) 116. [49] D.J.H. Cockayne, M.L. Jenkins and I.LF. Ray, Philos. Mag. 24 (1971) 1383. [50] N. Baluc, H.P. Karnthaler and M.J. Mills, Philos. Mag. A 64 (1991) 137. [51] D.M. Dimiduk, A.W. Thompson and J.C. Williams, Philos. Mag. A 67 (1993) 675. [52] B. Escaig, J. Phys. 29 (1968) 225. [53] J. Bonneville and B. Escaig, Acta Metall. 27 (1979) 1477. [54] Y.Q. Sun and EM. Hazzledine, Philos. Mag. A 58 (1988) 603. [55] M.J. Mills, N. Baluc and H.P. Karnthaler, MRS Symp. Proc. 133 (1989) 203. [56] C. Bontemps and P. Veyssi~re, Philos. Mag. Lett. 61 (1990) 256. [57] A. Couret, Y.Q. Sun and P.M. Hazzledine, MRS Symp. Proc. 213 (1991) 317. [58] A. Couret, Y.Q. Sun and P.B. Hirsch, Philos. Mag. A 67 (1993) 29. [59] A. Korner, Philos. Mag. A 63 (1991) 407. [60] H.P. Karnthaler, C. Rentengerger and E. Mtihlbacher, MRS Symp. Proc. 288 (1993) 293.
67
68 [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83]
Y. Q. Sun and P. M. Hazg.ledine
A. Korner, Philos. Mag. Lett. 59 (1989) I. Y.Q. Sun, Philos. Mag. 65 (1992) 287. P.B. Hirsch and Y.Q. Sun, Mat. Sci. Eng. A 164 (1993) 395. L.M. Hsiung and N.S. Stoloff, Acta Metall. 38 (1990) 1191. N. Jiang and Y.Q. Sun, Philos. Mag. Lett. 68 (1993) 107. A.E. Vidoz and L.M. Brown, Philos. Mag. 7 (1962) 1167. C.T. Chou and P.B. Hirsch, Proc. R. Soc. London Ser. A: 387 (1983) 91. C.T. Chou, P.B. Hirsch, M. McLean and E. Hondros, Nature 300 (1982) 621. A.H.W. Ngan, I.P. Jones and R.E. Smallman, Philos. Mag. A 66 (1992) 55. H. Pak, T. Saburi and S. Nenno, Scr. Metall. 10 (1976) 1081. A. Korner, Philos. Mag. A 58 (1988) 507. C.T. Chou, P.M. Hazzledine, P.B. Hirsch and G.R. Anstis, Philos. Mag. A 56 (1987) 799. Y.Q. Sun, M.A. Crimp and P.M. Hazzledine, Philos. Mag. A 64 (1991) 223. K. Suzuki, M. Ichihara and S. Takeuchi, Acta Metall. 27 (1979) 193. A.F. Giamei, J.M. Oblak, B.H. Kear and W.H. Rand, in: Proc. 19th Annual Meeting EMSA (Claitor's Pub. Div., Baton Rouge, 1971) p. 112. P. Veyssi~re, J. Douin and P. Beauchamp, Philos. Mag. A 51 (1985) 469. V. Vitek and Y. Sodani, Scr. Metall. 25 (1991) 939. N.S. Stoloff, Strengthening Methods in Crystals, eds A. Kelly and R.B. Nicholson (Elsevier, New York, 1971) p. 193. A.E. Staton-Bevan, Philos. Mag. A 47 (1983) 939. P.M. Hazzledine and Y.Q. Sun, Mat. Sci. Eng. A 152 (1992) 189. R.A. Mulford and D.P. Pope, Acta Metall. 21 (1973) 1375. S.S. Ezz, Y.Q. Sun and P.B. Hirsch, MRS Symp. Proc. 364 (1994) 695. S.S. Ezz and P.B. Hirsch, Philos. Mag. A 69 (1994) 105.
CHAPTER 50
Dislocation Cores and Yield Stress Anomalies D. CAILLARD
and
A. COURET
CEMES-LOE/CNRS 29 rue Jeanne Marvig, BP 4347 31055 Toulouse-Cedex France
Dislocations in Solids 9 1996 Elsevier Science B.V. All rights reserved
Edited by F. R. N. Nabarro and M. S. Duesbery
Contents 1. 2. 3. 4.
Introduction 71 Some remarks on the experimental techniques 72 Dilute alloys (f.c.c. steel and h.c.p, titanium) 73 Prismatic glide of h.c.p, beryllium 75 4.1. Experimental results 75 4.2. Models 76 5. Cube glide in L12 intermetallic alloys 80 5.1. Experimental results 80 5.2. Interpretations 83 6. Octahedral glide in L12 alloys 84 6.1. Macroscopic results in nickel-based alloys 85 6.2. Microscopic observations in nickel-based alloys 87 6.3. A discussion of microscopic dislocation processes 91 6.4. Discussion of models of yield stress anomaly in nickel-based alloys 93 6.5. Other L12 alloys 103 6.6. Discussion on cross slip in L12 alloys 107 7. TiA1 (L10 structure) 108 7.1. Mechanical properties 109 7.2. Microscopic observations 110 7.3. Discussion of microscopic dislocation processes 112 7.4. Models of the yield stress anomaly 113 8. /3-CuZn and other B2 alloys 114 8.1. Macroscopic results on/3-CuZn 114 8.2. Microscopic observations in/3-CuZn 116 8.3. Models of the yield stress anomaly (/3-CuZn) 117 8.4. Other intermetallics with b.c.c.-derivative ordered structures 119 8.5. Discussion of the yield stress anomalies in B2 compounds 123 9. General discussion 124 9.1. General considerations on yield stress anomalies [129, 166] 124 9.2. Comparison of experimental results 125 9.3. Stress anomalies in case of APB cross-slip 126 9.4. Stress anomalies in case of diffusion-controlled frictional forces (A13Ti, steel) 128 9.5. Yield stress anomalies in case of Peierls type frictional forces 129 9.6. Comparison of the mechanisms proposed 130 References 131
1. Introduction In addition to high yield stresses and good creep properties up to high temperatures, materials with attractive mechanical properties for high-temperature applications must have sufficiently good fracture toughness at both low and high temperatures. These properties are however a priori incompatible, simply because high flow stresses generally result from high frictional forces acting on mobile dislocations. When the frictional stress is high, the classical laws which describe the movement of dislocations and the plasticity of materials indicate that owing to the thermal activation the deformation stress steeply increases when the temperature is decreased. High yield stresses at high temperatures are thus coupled with such high yield stresses at low temperatures that little plastic deformation is allowed and brittle fracture occurs more easily (fig. 1). This is actually what is observed in materials with strong covalent bonding like ceramics. Materials with such a priori incompatible mechanical properties however do exist: for instance, superalloys have a high and constant yield stress from room temperature up to about 800~ which contributes to their good mechanical properties at all temperatures. As shown in fig. 2, this results from an appropriate combination of the mechanical properties of its two constituent phases: the disordered 7 phase (f.c.c. structure) with an almost "normal" negative temperature variation of the yield stress, and the ordered 7' phase (L12 structure) with an "anomalous" positive temperature variation of the yield stress [ 1]. Yield stress anomalies thus appear to be an essential physical element in the origin of high-temperature applications. They also constitute a challenge for the theory of dislocations, from which new dislocation properties and mechanisms may emerge. Stress anomalies are found in numerous intermetallic alloys with many different crystallographic structures as well as in some pure metals and disordered alloys. They occur with many different glide systems and their amplitude can for instance be small, forming a plateau-like stress temperature curve, or large, leading to an increase of yield stress by a factor of 10. For these reasons, an exhaustive review of all the situations investigated will not be made in this article. On the contrary, the main general features of yield stress anomalies will be highlighted tentatively in several different but well studied cases, and compared with current models. It will be shown that no satisfying model exists at present which is compatible with all the microscopic and macroscopic properties. After some remarks on the experimental techniques, experimental results are reviewed in several cases and compared with the models, first in the case of metals and conventional alloys, second in the case of Ni3A1 and other intermetallics. All the similarities and differences between the situations investigated are then discussed, and three groups of materials and glide systems with common properties and possibly common explanations for their yield stress anomalies are then defined.
D. Caillardand A. Couret
72
Ch. 50
N (7"
Fig. 1. Normal (N) and anomalous (A) temperature variation of the yield stress (schematic). ,oo I I
(a)
.o 20%7
~
i 40%y'
BO A
13
13
60%)"'
o .,... u'}
60
(.h hl l'U'}
8 0 % Z '~
o .J
40
o 2O
oy 1
0 0
1
200
1
1
,oo
_1
1
1
600
1
800
1
L__
iooo
TEMPERATURE ('C)
Fig. 2. Flow stress of polycrystalline Ni-AI-Cr alloys as a function of temperature for different volume fractions of the 7 t phase (L 12 structure). From Beardmore et al. [1].
2. S o m e r e m a r k s on the e x p e r i m e n t a l t e c h n i q u e s Results from macroscopic mechanical tests have to be compared systematically with microscopic observations, and vice-versa, in order to formulate realistic models. Macroscopic deformation experiments yield reliable and precise data relating to the average of all microscopic mechanisms. Except for slip line observations, samples are however like black boxes, and the macroscopic properties can often be explained by several microscopic processes. In the same way, microscopic observations are useful only if they are related to mechanical properties.
w
Dislocation cores and yield stress anomalies
73
Post-mortem observations in weak beam conditions are the classical complement of macroscopic deformation experiments, because they provide a correct three-dimensional view of the microstructure, at any stage of deformation, by a careful selection of the different sample planes. However, the structure is often partly relaxed, after the stress is released and the temperature is changed. A more serious disadvantage is that post-mortem observations provide no direct information on the kinetics of dislocations, and the mechanisms controlling dislocation movements. For instance, it is difficult to distinguish between those dislocations which are active during deformation, and the others which may be much less important. Moreover, there are usually several dislocation mechanisms which could give rise to the same dislocation structure. High resolution post-mortem observations provide the detailed structure of dislocation cores. However, the results have also to be interpreted carefully, since surface effects may be important. In-situ deformation experiments allow the observation of mobile dislocations under experimental conditions close to those in the bulk material, provided some precautions are taken. These precautions have been described in detail elsewhere [2]. It is sufficient to mention here that surface effects are unimportant on dislocations that are distant from the free surfaces by more than #b/cr, where # is the shear modulus and b the Burgers vector, which corresponds to their radius of curvature under the shear stress or. This condition is sometimes not satisfied. However, many in-situ observations are made on dislocations satisfying the above condition and this ensures the validity of such observations. It is also straightforward to make sure that possible irradiation effects are unimportant, by noting that the behaviour of dislocations does not change within the interval of an in-situ observation in one given area (typically less than a few minutes). Lastly, samples with a simple shape are used because the stress distribution is locally similar to the macroscopic one. Even when the stress tensor is obviously very complex, for instance in the case of cracked samples, local shear stress values can be directly obtained from the radius of curvature of mobile dislocations, at different temperatures. The accuracy of these measurements is rather poor, but sufficient to allow a comparison with macroscopic data. When the agreement between microscopic and macroscopic stress values is satisfactory, it indicates that the bulk behaviour is reproduced in the thin foil. All observation methods have thus their own advantages and disadvantages, and thus they should be used together as often as possible, and their results compared. In particular, in-situ experiments allow one to relate, in a more realistic way, the results of macroscopic deformation tests and microscopic post-mortem observations. When available, converging results from different methods provide a reliable basis for proposing models of mechanical properties.
3. Dilute alloys (fc.c. steel and h.c.p, titanium) Stress anomalies have been observed in stainless steel (f.c.c. structure) in the temperature range 200-600~ [3, 4], in association with a jerky flow or Portevin-Le Chatelier
D. Caillard and A. Couret
74
Ch. 50
T/T. 5"0 0-2 I t
0"3 "l
0-4
-I
l
'i
0-5 '"
I
I
0"6 i
-
w
0 -
i
'
I
_o
b
Jt~
X
blJ
I'0
0"51_ 0
1
I 200
1
I__ 400
t
J 600
l
!. . . . . A 800 900
T(*C) Fig. 3. Flow stress versus temperature curves of a 316 stainless steel, for different amounts of deformation (respectively 0.002, 0.02, 0.10 and 0.24). The range of temperatures corresponding to the jerky flow is shown by the bar. From Kashyap et al. [4].
(PLC) effect (fig. 3). A less pronounced anomalous behaviour corresponding to a plateau in the stress versus temperature curve has also been observed in impure titanium between 300~ and 400~ [5] and in the 7 phase of some superalloys (see fig. 2). In both cases, as proposed first by Cottrell, the jerky flow is attributed to a dynamic interaction between mobile dislocations and solute atoms [6-8]: dislocation glide is more difficult at a low speed, when solute atoms have time to segregate onto dislocations, than at a high speed, when solute atoms are effectively immobile with respect to fast dislocations. Under such conditions, the crystal deforms more easily by the fast movement of a very low density of dislocations than by the slow movement of all potentially mobile dislocations. The sample deforms too rapidly for the applied strain-rate, in such a way that the applied stress decreases, the velocity of dislocations decreases, and the deformation stops. The stress then increases again and activates new dislocation movements. This process induces jerky flow and a very small or negative stress-strain rate sensitivity. Since diffusion of solute atoms onto dislocations is thermally activated, it is natural that the same process also induces in some cases an increase of the deformation stress with increasing temperature.
w
Dislocation cores and yield stress anomalies
75
4. Prismatic glide of h.c.p, beryllium 4.1. Experimental results Beryllium is a metal with a hexagonal closed-packed (h.c.p.) structure, which can deform by the glide of a dislocations (Burgers vectors 89 in the basal and the prismatic planes. The basal glide is however easier than the prismatic glide, owing to the small core spreading of a dislocations in the basal plane [9, 10]. Pure prismatic glide can be activated for deformation axes parallel to the basal plane. Under such conditions, R6gnier and Dupouy [11] have shown that the variation of the corresponding critical resolved shear stress (CRSS) with temperature exhibits a complex pattern, with an anomalous behaviour between 160 K and 320 K (fig. 4). The activation area is small (a few tens of b2) except in the temperature range of the yield stress anomaly. Straight slip lines parallel to prismatic planes have been observed at low temperatures, but the slip traces are wavy at high temperatures [ 11 ]. The post-mortem dislocation substructure observed by Jonsson and Beuers [12] consists of straight screw dislocations connected by macrokinks. The first in-situ observations were made by Pollock and Wilsdorf [13] and by Beuers, Jonsson and Petzow [14]. These experiments indicate stable prismatic slip at all temperatures. The observations of Beuers et al. reveal complex behaviour of kinked edge dislocations at all temperatures. Screw dislocations cannot however be observed easily with the sample orientation of these experiments. More systematic in-situ experiments have been conducted later by the authors [15]. Local stress measurements have been made at different temperatures which reproduce satisfactorily the macroscopic curve (fig. 4), showing that the physical process at the origin of the stress anomaly was also present in the thin foils during the experiments. 9
-_
,
i
=
t
.
,
100 Q..t
cr
~
X
10
"-
0
, 0
160
260
360
TIK) 400 500
Fig. 4. Temperature variation of the critical resolved shear stress for prismatic slip in beryllium. Dotted line: macroscopic results of R6gnier and Dupouy [11]. Full line: in-situ measurements of Couret and Caillard [15].
76
D. Caillard and A. Couret
Ch. 50
Fig. 5. Jerky glide of screw dislocations in prismatic planes of beryllium, in the temperature range of the yield stress anomaly (T -- 240 K). From Couret and Caillard [15]. The results clearly show that prismatic slip is controlled by a Peierls-type frictional force acting on rectilinear screw dislocations at all temperatures, the origin of which is the spreading of screw dislocation cores in the basal plane. The motion of the screw dislocations was however not steady as expected from a Peierls mechanism but jerky, with each jump corresponding to the nucleation of a macrokink pair piling up at both dislocation extremities (fig. 5). Waiting time measurements allowed us to estimate with fairly good accuracy the probability of unlocking per unit time at different temperatures, for similar strain rates and dislocation velocities. The mean jump length was also measured in the same conditions (table 1).
4.2. Models
The first model was proposed by R6gnier and Dupouy [11]. According to these authors, the origin of the stress anomaly is that screw dislocations may have two possible configurations, corresponding to dissociations in the basal and the prismatic planes, and that the energy barrier to the prismatic-basal cross slip is lower than that to the reverse process. It is noted that this prediction is in agreement with calculations of Legrand [ 10] which indicate the possibility of dislocation spreading in prismatic planes with higher stacking-fault energy. Three temperature domains can thus be defined. (a) The low-temperature domain (T < 170 K), where screw dislocations are expected to glide and multiply freely in the prismatic planes, since cross slip onto the basal plane is not active due to the lack of thermal activation.
w
77
Dislocation cores and yield stress anomalies
Table l AN In situ measurements of the mean velocity v, probability of unlocking Pul (slope of In 6-Tff = f(tB)), and
kink height (mean jump length A = slope of In ~AN - __ f (A)) of screw dislocations gliding in prismatic planes of beryllium at various temperatures. From Couret and Caillard [ 15].
T
V
PuL~.~
In AN = f(,,.,~) and .~ (pro)
In &N = f(tB)and PUL(S-11
or
btB
(K) {~m s1) (HPo)
(pro s I} j'l['= O.010,um
PUL~ 21 s-1 t
] 363
0.14
53"9
\
5
\
0-20
I
I
125 i n
25 .In
r
-
h~(s)
I
9 Ji
i::).2 '
o
o~
~l~m2
0
1
0.1
0:2 m
~ : O.06gym
PUL = 3.5 s -1 ,,,
,
]
In 300
0.26
64•
5.
2.5, In
2-5 !
0
,
I
9
0.2
i
x
o
JL
0.23 56_.+6
x
.,t(~,) 0-1
0
i
PUL = 3"5s-1 2t,.0
_ 0~2
=0-056pm 0-20
i
i-
t
s .i~.,.L.,"
5
In
zs ~
2.5. In
J •
~
0
0.2Z,
,!
tB(S) ,
...,
9
0"2
tB(s) 0
~,
0
0.1
0-2
-
78
Ch. 50
D. Caillard and A. Couret
(b) The domain of yield stress anomaly (170 K < T < 330 K), where the cross slip from prismatic to basal planes would be activated more easily as the temperature increases, whereas the reverse cross slip (basal-prismatic) would not yet be activated. Since the basal plane is unstressed in the orientation used, this process corresponds to a thermally activated locking of mobile dislocations and is presumably responsible for the stress anomaly. The very strong initial hardening would be limited at the early stages of deformation as a result of annihilations by glide in basal planes, under internal stresses. (c) The high-temperature domain (T > 330 K), where the CRSS decreases again. Here, unlocking by cross slip from basal to prismatic glide is expected to become possible, resulting in a second decrease in the CRSS. This model is supported by the observation of straight slip lines parallel to prismatic planes at low temperatures, and wavy slip traces at higher temperatures. It does not however correspond to in situ observations of screw dislocation movements: the behaviour corresponding to the high temperature domain of R6gnier and Dupouy has indeed been observed in the whole temperature range investigated, including the domain of stress anomaly. The deformation mechanism corresponding to the jerky movement of dislocations observed in situ has been described in detail in several articles [15-17]. Similar behaviour has been proposed by Greenberg for L12 alloys (section 6.4) and by Hug for TiA1 [18]. It is a series of locking and unlocking processes by double cross slip between basal and prismatic planes, with activation energies of GI (prismatic-basal) and Gul (basal-prismatic) respectively (fig. 6). It is based on the assumption already made by Rdgnier that screw dislocations can have two different core structures, extended in the basal and the prismatic planes respectively. Under such conditions, the average velocity of screw dislocations can be expressed as:
P.: 13 = VF-~I ~
where Pul
-
Pulo exp
(-
Gul '~
and
Gl kT
PI - Pl0 e x p - ~
are respectively the probabilities of unlocking and locking per unit time, and VF is the free flight velocity of screw dislocations with their core extended in the prismatic plane. The average velocity of screw dislocations is thus"
ulo (
V ~---VF-~I~ exp
Gul - Gl /
kT
The mean jump length is: A - vF/Pl, leading to v = APul. It can be verified in table 1 that, for similar values of v, the mean jump length A varies as the inverse of the probability of unlocking Pul, and similar values of A at 240 K and 300 K correspond to similar values of Pul.
w
Dislocation cores and yield stress anomalies
79
(a) metastabte and gtissite
stable and sessite
(b)
~"
_LGL
stone and sessile
Cc) Fig. 6. Peierls type frictional forces in metals and alloy: (a) and (b) Peierls and locking-unlocking (double cross-slip) mechanisms; (c) energy diagram of the locking-unlocking mechanism. From Cailllard et al. [16].
Both in-situ observations and theoretical estimates lead to Gul >> G1, in such a way that the movement of screw dislocations is mainly controlled by the unlocking process, i.e., by cross slip from the basal plane onto the prismatic plane. The same result is obtained if the dislocation velocity is set as v = APul, considering that the mean jump length A varies only slowly with temperature. It has been shown recently [16, 17] that this locking-unlocking mechanism is not fundamentally different from the Peierls mechanism, since the Peierls mechanism is a limiting case of the locking-unlocking mechanism, when the jump distance between locked positions decreases and scales with interatomic distances. The locking-unlocking mechanism has been evidenced in several h.c.p, metals, with either normal or anomalous stress-temperature relationships [17, 19]. The transition between locking-unlocking and Peierls mechanisms has been studied in titanium [20]. Since the velocity of screw dislocations is controlled by a positive activation energy, Gul - G1, the anomalous behaviour of beryllium cannot result solely from the lockingunlocking mechanism. A model for the yield stress anomaly has thus been proposed by the present authors [15], on the basis of their in-situ measurements of screw dislocation velocities. The main assumption is that, since the deformation is controlled by the cross slip of screw dislocations from the basal plane onto the prismatic plane in the whole temperature range, and since the probability of this cross-slip process is shown to be constant at constant strain rate in spite of a simultaneous increase of stress and temperature in the
80
D. Caillardand A. Couret
Ch. 50
domain of yield stress anomaly (table 1), there is necessarily an increase in the difficulty of cross slip at a constant stress, as the temperature is increased. Such a behaviour corresponds to an evolution of the dislocation core structure with temperature in basal and/or prismatic planes. Assuming dislocations in basal planes are split into two Shockley partials separated by a stacking fault (in spite of their very narrow spreading), this evolution can be described by a decrease in the stacking fault energy with increasing temperature, which may be accounted for by the change in the relative stabilities of h.c.p, and f.c.c. phases with temperature [ 15, 21 ]. On the basis of their own in-situ observations, Beuers et al. [14] proposed another model where the CRSS is controlled by the movement of salient points on edge dislocations, resulting from double cross-slip processes between basal and prismatic planes. Under such a condition, an increasing density of salient points (which are in fact macrokinks) with increasing temperature may hinder dislocation movements and give rise to a flow stress anomaly. This model is incomplete due to lack of details, and the role of screw dislocations has probably been underestimated on account of the special orientation of the microsamples.
5. Cube glide in L12 intermetallic alloys 5.1. Experimental results
Alloys with the L12 structure are deformed by the glide of pairs of dislocations with 7(110) Burgers vectors (similar to those in f.c.c, metals) separated by an antiphase boundary (APB) ribbon which is most often on octahedral ({111}) and cube ({100}) 1 planes. Each 7(110) dislocation is called a superpartial dislocation, and one pair with a (110) Burgers vector is a superdislocation. Superdislocations can glide on octahedral and cube planes, however with very different characteristic features. The existence of an anomalous temperature dependence of the CRSS of cube glide was pointed out after an in-situ study of the glide mechanism in the "7' phase of a nickelbased superalloy (C16ment, Couret, Caillard [22] and C16ment, Mol6nat, Caillard [23]). Experiments with a (110) tensile axis for which octahedral and cube glides have the same Schmid factor (0.35-0.4) reveal that both cube and octahedral glides are activated extensively in a wide temperature range down to 140 K. Since the two glide systems have different Burgers vectors, it is clear that dislocations are able to multiply and glide over large distances in both slip systems. This behaviour is rather surprising when the flow stress anomaly in octahedral planes is compared with the normal stress increase which is expected to occur in cube planes with decreasing temperatures (fig. 7). On the basis of this comparison, cube glide should be inactive below about 500 K, because its CRSS should become much higher than the CRSS of octahedral glide. The CRSS of cube glide seems thus to be closer than expected to the CRSS of octahedral glide between 140 K and 500 K in this material. This hypothesis has been directly verified by measuring the local stress necessary to move screw dislocations in the cube plane of "7' single crystals, as a function of temperature (fig. 8). In spite of rather large uncertainties, the curve clearly exhibits a
w
81
Dislocation cores and yield stress anomalies
"%
O-
r-' K Fig. 7. Schematic description of the CRSS versus temperature curves of cube and octahedral glide in Ni3A1. Dotted lines correspond to the behaviour of cube glide expected at low temperature.
600
Latt f 400
in
situ(t')
\
oL.INi3( AL,Nb))
-. ~ ~ "'~ - + ~i.._'~" ' ~ Umokoshiet al.. ,''""~ ]---~',"d~" (Ni3(At,Ta))
1
~
"~'."%
~q,, q"q',~
%
to o~ L~ 200 !
anomalous behaviou r 9
200
9
9
400
Saburi et aL. ( Ni3 (At,W)) ,
9
9
600
,
800
~
9
r
T(K)
Fig. 8. Temperature variation of the flow stress of cube glide in several nickel-based intermetallic alloys (LI2 structure). Dotted lines refer to macroscopic deformation experiments, and the full line refers to in-situ measurements in the 3,t phase. From [23-26]. plateau or a stress anomaly between 140 K and 600 K, and the CRSS of cube glide remains close to the CRSS of octahedral glide at temperatures as low as 140 K. This unexpected result is confirmed by the close inspection of the macroscopic results of Lall, Chin and Pope [24] in Ni3(A1, Nb), Umakoshi, Pope and Vitek [25] in Ni3(A1, Ta), Saburi, Hamana, Nenno and Pak [26] in Ni3(A1, W). These authors reported that, with tensile axes close to a (111) direction, extensive slip occurs on cube planes above room temperature. The corresponding CRSS values are also plotted in fig. 8. They exhibit the same temperature variation as those measured in situ, with the same small stress anomaly. A similar positive temperature dependence of the CRSS of cube glide has been measured in Pt3A1, with the same L 12 structure, by Wee, Pope and Vitek [27]. Unlike Ni3A1,
82
D. Caillard and A. Couret
Ch. 50
80 0
0
70
x
60
o
50
.IC
1D >
40
0 v) t...
-0 U
30
4.-
20
"C] ~C].~.
(001)[~10]
( 010 ) [ ~011 10
l
I
200
400
1
.
600
Temperoture
I
800
1 1000
( K )
Fig. 9. Temperature dependence of the CRSS of octahedral and cube glide in Pt3A1 (filled symbols correspond to measurements on the same specimen). From Wee et al. [27].
the CRSS of octahedral glide is not anomalous and exhibits a rapid increase with decreasing temperature. Here, cube glide is the primary glide system at all temperatures for tensile axes far away from (100). Its CRSS decreases rapidly from 20 to 500 K, and then increases from 500 to 1100 K (fig. 9). Conversely, the temperature dependence of the flow stress is always negative in Fe3Ge which has been shown recently to deform by cube glide [28]. Several post-mortem observations in Ni3A1, Ni3(A1, Ti), and Ni3Ga [29-31] above 673 K (in the domain of negative stress-temperature dependence) indicate that dislocation loops in cube planes have straight portions parallel to edge and screw orientations. Screw dislocations are dissociated in two superpartials separated by an APB in the cube plane. Each superpartial is dissociated in one of the two intersecting octahedral planes, which leads to a sessile configuration. Edge segments are dissociated in "super Lomer-Cottrell" or "double Lomer-Cottrell" type non-planar configurations. In-situ observations in Ni3A1 and 7' show that edge and screw segments move steadily or jerkily, which indicates that the glide is controlled either by a Peierls mechanism or by a locking-unlocking mechanism [22, 32]. Most post-mortem observations have been made on alloys which do not exhibit slip traces on cube planes at low temperatures. Only Korner [33] mentioned that many dislocations have glided in the primary cube plane after deformation of a Ni3(A1, Ti) alloy at room temperature.
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Dislocation cores and yield stress anomalies
83
Fig. 10. Glide of rectilinear screw dislocations in a cube plane of a 7 t nickel based alloy (L12 structure). In-situ experiment at 140 K. From Cl6ment et al. [22]. In-situ observations in 3,t reveal that in the cube plane rectilinear screw dislocations move jerkily in the whole temperature range 120 K - 1000 K, as in beryllium, and edge dislocations are sometimes sessile (fig. 10) [23]. Dislocation movements are proceeded by avalanches in the temperature range 300-673 K (unpublished result). No microstructural observation is available on Pt3A1. Superdislocations gliding in cube planes are however likely to be dissociated into two superpartials separated by an APB, since SISF are restricted to octahedral planes.
5.2. Interpretations Considering that the glide of screw dislocations is controlled by a Peierls-type mechanism in the whole temperature range, the stress versus temperature curve associated with cube glide in -),t (which has been obtained at constant velocity of screw dislocations) and Ni3A1 is truly anomalous for the following reason: For all thermally activated controlling mechanisms, the gradient of the stress versus temperature curve is: 0~*] OT
-H v = bAT'
(1)
where or* = o" - o- i (o- i is the athermal internal stress), H is the activation enthalpy, A is the activation area, v is the velocity of superdislocations (screws in the present case), and b is their Burgers vector.
84
D. Caillard and A. Couret
Ch. 50
When only orders of magnitude are concerned, the following approximation can usually be made: H ~ G ~ c k T , where G is the activation energy, and
c =
In go ln~
(2)
is effectively constant (c ~ 25 usually). This leads to: ~)cr*], ~
ck
OT ,~
bA"
(3)
In the case of a Peierls type frictional force, A is low and decreases as cr increases. This indicates that cr should increase rapidly at low temperatures. The experimental curve is thus unambiguously anomalous. The same conclusion would be derived even in the case of a zero or small negative temperature variation of the flow stress below 600 K. Under such conditions, and since the core structures of superpartials are similar to those of dislocations in beryllium ({ 111 } planes correspond to the basal plane, and { 100} planes to prismatic planes), the stress anomaly has been interpreted in the same way, i.e., it may result from an increase in the energy necessary to recombine superpartials dissociated in octahedral planes, as the temperature is increased. Such an increase may be explained by a short-range diffusional lowering of the stacking fault energy, as suggested by Ahlers [34]. Short-range diffusional processes indeed take place in disordered alloys at similar homologous temperatures, leading to a Portevin-Le Chatelier effect (section 3). Regarding Pt3A1, it should be noted that the same explanation was proposed earlier, in 1984, by Wee, Pope and Vitek [27]. It was not however mentioned again in subsequent publications on the same material [35].
6. Octahedral glide in L12 alloys Glide in octahedral planes in alloys with the L12 structure exhibiting an anomalous yield stress-temperature dependence is clearly one of the best documented situations in the study of plastic properties of materials. Many intermetallic compounds have been found to possess such anomalous behaviour [36]. The most extensive studies, however, have been conducted on nickel-based alloys, especially Ni3A1 and Ni3(A1, X) alloys. Important work has also been done on Ni3Ga and Ni3Si and will be mentioned in what follows. Nickel-based alloys have very similar properties, which are also apparently representative of the properties of L12 alloys with anomalous behaviour. We shall thus focus in the following on the properties of nickel-based alloys. Other L12 alloys are treated in section 6.5.
w
85
Dislocation cores and yield stress anomalies
6.1. Macroscopic results in n i c k e l - b a s e d alloys The main macroscopic properties of Ni3(A1, X) alloys in the domain of stress anomaly where deformation is achieved by octahedral glide are summarized in several review articles (e.g., [37-41 ]). We only mention below the main characteristics which are thought to be of importance to the understanding of the microscopic mechanisms in the domain of yield stress anomaly. (i) There is a strong effect of orientation and sense of the applied stress on the CRSS of octahedral glide. These effects, known as the violation of the Schmid law (fig. 11) and the tension-compression asymmetry, are described in many articles [25, 42-45]. They indicate that the CRSS is related to cross-slip processes, in which the forces acting on the Shockley partials have to be taken into account, according to Escaig [46]. We can only remark that this effect is difficult to explain quantitatively, since its amplitude as a function of the orientation of the applied stress varies significantly from one material to another. (ii) Below the peak temperature the strain rate sensitivity 1 0
S-- TSln~
I
T
has a rather low value in Ni3A1, decreasing slowly as the temperature is increased (fig. 12, from Thornton, Davies and Johnston [48]). The corresponding activation area has been measured from relaxation experiments in Ni3(A1, 1 at.% Ta) and has a high but finite value (Bonneville, Baluc and Martin [49]; Baluc et al. [50]). It is however not yet clear 4o0
t Orientation10~ I"1 Orientation[ i23] A Orientation[ i 11] ~.,
300
A 200
V
lOO
0
200
400
600
800
I000
1200
1400
T e m p e r a t u r e (K)
Fig. 11. CRSS of octahedral glide in Ni3(AI, 0.25at.% Hf)as a function of temperature, for different orientations of the compression axis. From Bontemps [47].
D. Caillard and A. Couret
86
25
-
Ch. 50
t
STRAIN RATE SENSITIVITY S I I j Iog~ ~
A /92.5
Ni3AI
/ /
20
INCREASE --~--- DECREASE
/ /
%
v
I0
-
0
" ,,, -..
,,s
1. . . . . 0
200
1
..
.1
400 600 TEMPERATURE, "C
..
I 800
IOOO
Fig. 12. Strain-rate sensitivity of Ni3A1as a function of temperature. From Thornton et al. [48]. whether the activation areas vary monotonically or not with temperature, after being corrected for the effect of strain hardening [50]. Ni3(Si, 11 at.% Ti) is exceptional because S reaches values about 10 times larger than in Ni3A1 (Takasugi et al. [44]). All these results clearly indicate that thermal activation plays an important role in the mechanisms which control deformation in the domain of the stress anomaly. (iii) The variation of the CRSS is largely reversible upon temperature changes. This property was found for the first time by Davies and Stoloff [51] who showed that a sample prestrained at a high temperature (high CRSS value) and subsequently strained at a low temperature had the same yield stress as a virgin sample deformed only at low temperature. In fact, more recent experiments of Yoo and Liu [52], Dimiduk [39], Hemker [53], Couret, Sun and Hirsch [54] show that the process is a little bit more complex: the second flow stress is identified with the CRSS of a virgin sample at low temperature, plus the strain hardening of the high temperature predeformation. This indicates that if the motion of dislocations in the perfect crystal is fully reversible upon temperature changes, the strain hardening due, in particular, to the accumulation of sessile dislocations and debris is not eliminated by cooling the samples. (iv) The CRSS is rather ill defined. It has already been shown by Thornton, Davies and Johnston [48] that the yield stress anomaly appears for very small amounts of plastic deformation, as small as e - 10 -5. The deformation stress increases with increasing strain, in such a way that the amplitude of the stress anomaly also increases in the same proportion. Experiments of Baluc [55] show that this homothetic increase goes on up to a 25% shear strain. Creep experiments of Hemker et al. [56] also show that about 0.5% deformation can be achieved by glide in { 111 } planes at a stress much lower than that needed for the 0.2% strain which is usually taken as the CRSS. This stress has however the same
w
87
Dislocation cores and yield stress anomalies
10000
0
(MPa) P~
8000
----
1' = 7xlO-Ss "~
-
1' = 7 x l O ' ~ s "1
,,,
6000 4000
//
2000
'~
\o \~
I
!
, i
200
I
~"
/
o
/
~'~'~'~~
-2000 -4000
;
400
(K)
, i
600
9
1
800
-
i
i
1000
1200
-
i
1400
Fig. 13. Strain-hardening coefficients at 0.2% deformation in Ni3(A1, 1 at.% Ta) as a function of temperature. From Baluc [55].
anomalous temperature dependence as the CRSS. As pointed out by Hemker et al., all these observations (including those of Thornton et al.) show that { 111 } glide is not controlled by the movement of the edge portions of dislocation loops when it is activated at stresses lower than the 0.2% proof stress. Hemker concludes that there is no critical stress for the motion of screw dislocations which controls plastic deformation. This conclusion may however be modified by recent results of Sp~itig, Bonneville and Martin [57] who show that activation areas versus strain curves exhibit an inflexion point at e ~ 0.2%, typical of a macroscopic elastic limit. (v) The strain hardening coefficient is high, even in single slip, especially for load axes close to (111 ). It also exhibits an anomalous temperature dependence, which indicates that strain hardening is associated with the same cross-slip process that controls the movement of dislocations (fig. 13) [47, 55]. Sp~itig et al. [57] have shown that strain hardening during relaxation tests is even higher than strain hardening during constant strain rate deformation.
6.2. Microscopic observations in nickel-based alloys Extensive microscopic observations have been made, giving indications of many dislocation mechanisms. It is thus especially difficult to determine which are the most important in explaining the origin of the yield stress anomaly. A review of microscopic observations has been made by Veyssi~re [58]. Only those which are considered to be of some importance by most authors are briefly described below. (i) The most striking characteristics, reported after the first TEM observations of Kear and Hornbecker [59] and Thornton, Davies and Johnston [48] is the presence of long and straight screw superdislocations in deformed samples. This indicates that screws are much less mobile than non-screws, as confirmed later by several in-situ deformation experiments [60-62]. Evidence of locking along 60 ~ orientations at low temperatures (below the domain of strong positive temperature dependence of the flow stress) has also been given in [63, 64].
88
D. Caillard and A. Couret
(Ill)
1/2[i01] ~
~
Ch. 50
~
APB
(121) Fig. 14. Scheme of an incomplete Kear-Wilsdorf lock. The apparent APB plane is shown by the dotted line.
(ii) Superdislocations are mostly dissociated into two superpartial dislocations with the same 71 (110) Burgers vector, separated by an APB ribbon. According to several authors (see a discussion in [39]), dissociations into super Shockley dislocations separated by superlattice intrinsic stacking fault (SISF) are not frequently observed in the domain of strong positive temperature dependence of the yield stress. The first weak-beam observations showed that screw superdislocations were mostly in the Kear-Wilsdorf configuration, with the dissociation into the cube cross-slip plane. More recent in situ and post-mortem observations however reveal that straight screw superdislocations are sometimes dissociated in the octahedral glide plane at low temperatures and room temperature [47, 65]. Intermediate situations have also been found, the APB ribbon being partly in the octahedral and partly in the cube cross-slip plane [66-68] (fig. 14). Lastly, further in-situ investigations in the octahedral plane have shown that when rectilinear screws are seen to be dissociated in the octahedral plane the APB is also probably slightly extended in the cube plane [62]. (iii) Many macrokinks along straight screw dislocations can be observed post mortem. They are lying in the primary octahedral plane, and detailed analyses allow them to be classified into three categories [41]: regular kinks on superdislocations, switched over kinks reversing the order of superpartials, and simple kinks affecting one superpartial only. Statistical measurements of macrokink heights indicate a small decrease with increasing temperature and increasing stress. The mean height of macrokinks has been measured by Couret, Sun and Hirsch in Ni3Ga [69], and Bontemps in Ni3(A1, Hf) [47]. The results obtained vary from 18-20 nm at 300 K to 10-12 nm at 673 K. Similar measurements performed by Dimiduk [64] indicate that the height of the macrokinks varies as the inverse of the strength of the alloys (the corresponding data are however not given). The distribution of macrokink heights, measured at different temperatures by Couret et al., exhibits an exponential decrease (fig. 15) similar to that observed in beryllium (section 4, table 1). This has been interpreted in the same way, i.e., in terms of a thermally activated locking process. An abnormally large number of macrokinks with 5-10 nm height is however observed after deformation at room temperature and 200~ It is discussed in point (v) below. (iv) In the higher temperature range of the yield stress anomaly (and in some cases above room temperature [33]), dislocations cross slip and glide in the cube plane as
w
Dislocation cores and yield stress anomalies
80
log 9 N
400"C
4
70
89
60 5O 4O
h(~)
30 2O
100 2OO I
lo.
~
5
66o
h(~)
0
(a)
Z.00
, ~
~oo 2oo
,--,
4oo
log N ,.._,i"log 11/.
200"C
4 2
(b)
h(A}
\
0 lOO 26o
"
~o
9
6~
Fig. 15. Statistical measurements of macrokink heights in Ni3Ga deformed at 200~ and 400~ Note the abnormally large number of macrokinks of 5-10 nm height after deformation at 200~ From Couret et al. [69, 166]. shown in fig. 16. This result is deduced from both post-mortem [33, 58, 63, 70] and in-situ [71, 72] observations. Macrokinks in octahedral planes act as anchoring points, in such a way that dislocations appear as a series of adjacent segments bowing out in the cube plane. The amount of deviation in the cube plane is higher when the tensile axis is far from (100), and at high temperatures. Post-mortem observations in Ni3Ga [70] and in-situ observations in Ni3A1 [72] however indicate that the reverse crossslip process from cube onto octahedral planes is also operating at high temperatures (fig. 17). (v) In-situ observations in Ni3(A1, 0.25 at.% Hf) show that straight screw dislocations move jerkily. When they are dissociated in the octahedral plane in the lower temperature range of the stress anomaly (cf. point (ii)), they jump over distances often scaling with their dissociation width [62] (fig. 18), in accordance with the observation of an abnormally large number of kinks with height 5-10 nm in Ni3Ga (fig. 15) (cf. point (iii)). In the higher temperature range of the yield stress anomaly, octahedral glide proceeds by bursts [60, 61, 71-73] with indications of double cross-slip between octahedral and cube planes [72], in accordance with the post-mortem observations described above (point (iv)). All dislocations are dissociated in cube planes just after the bursts.
90
D. Caillard and A. Couret
Ch. 50
Fig. 17. Dislocation D curved in a cube plane and cross-slipping (segment S) onto an octahedral plane in Ni3Ga deformed at 400~ (a), (b) and (c) - Observations under different tilting conditions; (d) - Scheme of the dislocation structure. From Mol6nat et al. [70].
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Dislocation cores and yield stress anomalies
91
Fig. 18. Screw dislocation dissociated in a (111) plane and gliding by jumps over a distance equal to its dissociation width, in Ni3(A1, 0.25at.%Hf). In-situ experiment at room temperature. From Mol6nat and Caillard [62].
6.3. A discussion of microscopic dislocation processes Although microscopic observations reveal rather complex dislocation behaviour in the domain of the yield stress anomaly, post-mortem and in-situ observations provide several consistent results which form a suitable basis for discussing the different models of anomalous mechanical properties.
6.3.1. Sessile configurations The deformation is controlled by the movement of screw superdislocations, which tend to cross slip into the cube plane. The driving force for cross slip is a combination of Yoo's force couplet [75] and a lower APB energy in cube planes. Kear-Wilsdorf locks are formed at high temperatures. They tend to glide in the cube plane especially for loading
92
Ch. 50
D. Caillard and A. Couret
"~---
t CSF9 AP,
I
>.C~ (111),
_ ..~
_-__
9
P
Ca)
:
(b)
-
unlocking -- - " ~ ~ --
-',K P
(c)
~]
II
(~
i -
-
(d)
",!
II II
,~
(e)
Q
Q
R
R
Q
(0
R
't, (g)
(h)
I
Fig. 19. Schematic description of the glide mechanism deduced from in situ observations in Ni3A1 at 300 K. w is the width of the APB ribbon lying in the cube plane. Cycle B corresponds to jumps over a length scaling with the dissociation width. From [62]. axes far away from (100) directions. Cross slip however appears to be less pronounced than expected at lower temperatures, as indicated by the observation of incomplete KearWilsdorf locks, and screw dislocations dissociated in octahedral planes. A model has been set up with the aim of explaining jumps of rectilinear screw dislocations dissociated in octahedral planes over a length equal to their dissociation width, and the formation of macrokinks of height 5-10 nm [62]. It is based on the short range cross slip of the leading dislocation into the cube plane over its whole length (fig. 19). This process, which is different from the local pinning process described by Paidar, Pope and Vitek [76], locks the superdislocation until the trailing superpartial cross slips in turn and follows the same path as the leading one. Two situations may arise according to whether or not the leading superpartial cross slips again when the superdislocation is locked. Jumps over the width of the APB are obtained in the second case, in agreement with the observations. This observation provides evidence of unlocking by cross slip. These results show that screw dislocations can take many non-planar configurations of different energies, especially at low temperatures. Calculations of Saada and Veyssi~re [77] show that incomplete Kear-Wilsdorf locks cannot exist without a strong frictional force in the cube plane. Such a frictional force is obviously present as discussed in section 5. However, in order to explain observations of incomplete Kear-Wilsdorf locks, it is also necessary to assume that the ratio of APB energies in octahedral and cube planes, ")'lll/T10o, is closer to 1 than expected from weak beam measurements of dissociation widths. According to Saada and Veyssi~re, stable configurations with the APB lying either in the octahedral plane or partly in octahedral and cube plane have very high energies for 7111/3'100 much larger than 0.9 in Ni3A1. The stability of incomplete Kear-Wilsdorf locks has been calculated under dynamical conditions as a function of the applied stress and as a function of the frictional force in the cube plane by Paidar, Mol6nat and Caillard [78] and Chou and Hirsch [79]. The results show that incomplete Kear-Wilsdorf locks can be stable under stress. They may however be transformed into complete locks upon unloading.
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Dislocation cores and yield stress anomalies
93
6.3.2. Locking process and kink motion There is a general agreement on the origin of macrokinks, which result from thermally activated glissile-sessile transitions. This can be deduced from the exponential distribution of macrokink heights, as discussed in [69]. This is also in a good agreement with the observed decrease of macrokink heights with increasing temperature. However, it should be noticed that the height of macrokinks can be written as: h = vFt, where t is the life time in the glissile configuration before locking and is temperature dependent, and VF is the stress-dependent free dislocation velocity in the glissile configuration. Life time t is expected to decrease with increasing temperature, thus decreasing h. However, the CRSS increases at the same time, thus increasing VF. This means that the macrokink height results from an equilibrium between these two opposite contributions. There is little data concerning the mobility of kinks along screw dislocations. Insitu experiments indicate that this velocity is high. Most jumps indeed correspond to nucleations and propagations of macrokinks which are too fast to be recorded. Theoretical estimates of the kink velocity also yield high values [80]. The average kink velocity may however be controlled by interactions with obstacles which are not seen in thin foils. 6.3.3. Unlocking processes No definitive conclusions about the existence and the nature of unlocking processes can be deduced from the experimental observations. Unlocking may be impossible under normal deforming conditions. It could also be initiated either at the non-screw glissile portions of dislocation loops like macrokinks, or at screw sessile portions (a general discussion about these two mechanisms is given in section 9). Experimental results however show that initiation of unlocking by cross slip at sessile screw portions does exist at least in two situations: when dislocations dissociated in cube planes are seen to cross-slip back into the octahedral plane (section 6.2 point (iv)), and when short-range double cross slip is shown to occur, in the case of jumps over the dissociation width (section 6.3 point i).
6.4. Discussion of models of yield stress anomaly in nickel-based alloys The different models have been simplified and classified according to their main hypotheses. Some of them are compared on table 2.
6.4.1. Accumulation of locked dislocations This model has been developed mainly by Greenberg and co-workers [81]. It is very similar to the model developed earlier by R6gnier and Dupouy [11] for beryllium. A general expression for the flow stress is: O" -- k v / - N -[- o-f,
(4)
where N is the density of dislocations, and O'f is a frictional force acting on mobile dislocations.
Table 2 Principles of several models of the stress anomaly in L I Z alloys. Thick lines correspond to locked dislocations. Note that I, h, and G,I may have different meanings (see text).
I
TakcuchiKurmo~oI821
I
G
2 ,
kT
"..--wcp-'
I
I
SOdaniVitek [851
I
dc Bussac et al. [89]
P
v=hlP,l v = vocxp
-5 kT
3
v = voexp -
GUI-TGI kT
? 0-
a=@
VP
o-cxp-
G
,14 G xcxp-' kT
,
I.
0
P VI
0
w
Dislocation cores and yield stress anomalies
95
Three cases have been defined for the behaviour of screw dislocations: - When the glissile-sessile transition by the formation of Kear-Wilsdorf locks is thermally activated, but the temperature is too low to activate the reverse sessile-glissile transition, N increases rapidly with increasing plastic strain. The storage rate of sessile dislocations also increases with increasing temperature, leading to an increase of cr at constant strain, ~cr/~T > O. At higher temperatures, the glissile-sessile transition becomes athermal and the reverse sessile-glissile transitions become thermally activated. Then, O~r/OT < 0. As shown in [16, 17, 20], this situation corresponds to a Peierls mechanism. Intermediate situations may be found when both transitions are thermally activated. This situation which has been developed in section 4 corresponds to the lockingunlocking mechanism for which Ocr/OT < 0 (higher temperature range in the model of R6gnier). However, according to Greenberg, O~r/OT > 0 may also result from a very low unlocking rate leading to the first situation. -
-
This model agrees with microscopic observations only qualitatively. Variations in the dislocation density N as a function of deformation and temperature are indeed too low to explain both high values of strain-hardening and the positive temperature dependence of the yield stress. This model may explain the high strain-hardening rate and the tension-compression asymmetry. Thermal activation may result from the term O'f although there is no experimental evidence of such a frictional force. The model however is inconsistent with the reversibility of the flow stress after a temperature change.
6.4.2. Local pinning models A local pinning model was first proposed by Takeuchi and Kuramoto (TK) [82]. The movement of dislocations results from a balance between two processes acting in opposite directions. The first is the thermally activated locking by cross slip, which starts at some points, and tends to extend along dislocations. The second is the break-away from this local cross slip, through the rapid glide of dislocation segments which are not yet locked (table 2). In other words, it is a competition between the velocity of the extension of the locked configuration along the dislocation line, and the velocity of parts of the dislocations which are not yet locked. In this model, dislocations glide at the minimum velocity which prevents the extension of locking over their whole length. This velocity is proportional to the probability of locking for a length l, P~ o( l exp[-Gl/(kT)], where GI denotes the enthalpy for locking. Namely, v o( hlexp[-G1/(kT)]. Since l and h are shown to be inversely proportional to stress, and since Gl depends on the shear stress in the cube plane, the dislocation velocity is v o( (1/crZ)exp[-Gl(cr)/(kT)]. On the other hand, since v is a free glide dislocation velocity (v = VF), it determines the stress ~r through a relation of the type VF = K o typical of viscous glide. This leads to cr3 c~ exp[-Gl(O)/(kT)] and
~r o( exp
3kT
"
D. Caillard and A. Couret
96
Ch. 50
When there is no applied stress in the cube plane (for (100) tensile axes), Gl does not depend on tr, and v oc (1/cr2)exp[-Gl/(kT)], whence cr oc exp[-G/(3kT)]. This model has been improved by Paidar, Pope and Vitek (PPV) [76] in order to account for the orientation effects and the tension-compression asymmetry. These two models explain successfully the reversibility of the flow stress upon temperature changes. Moreover, since mobile dislocations slowing down just below their minimum stable velocity are immediately trapped into Peierls valleys, a high strain-hardening rate is expected. These models are rather difficult to prove by microscopic observations because the density of mobile dislocations would be very low, of the order of 0.04 cm -2 according to Hirsch [83, 84]. Post-mortem and in-situ observations may thus concern only immobile dislocations and dislocations with unimportant movements. The main objection against these models is however their stress-strain dependence. If the dislocation velocity is considered to be controlled by free flight glide, i.e., v = vF cx Kcr, it will lead to ~ = pmbKcr. Unless Pm varies inversely proportional to the stress, low values of the strain rate dependence of the flow stress (S') cannot be explained according to Hirsch [84]. It can however be considered that (at least when there is no stress in the cube plane) cr is determined only by the temperature in such a way that the dislocation velocity is a constant at a given temperature. Strain-rate changes are thus necessarily accommodated by changes in the mobile dislocation density Pm, and the strain-rate sensitivity must be zero. A new model has been formulated by Sodani and Vitek [85] to account for the low strain-rate sensitivity. According to this new model, the probability of locking no longer determines the velocity of dislocations, but the distance between pinning points: 1 cx exp[G1/(kT)]. This can be compared with the TK model, which yields: 1 ~x l/or cx exp[Gl/(3kT)]. The two models are thus different. Another illustration of this difference is that
v o ( 1 / o "2exp
( c,) -~
,
within the TK model, and
v = vo exp
_ Gui
where v0 is a constant and Gul denotes the activation energy of unpinning, according to Sodani and Vitek. v is lower than in the TK and PPV models since it includes waiting times to recombine pinning points. Since the latter relation implies: Gul = kT In vo v
w
Dislocation cores and yield stress anomalies
97
and Gul - G u l o - crbAul, where Aul denotes the activation area for unlocking, Aul = lb cx exp[Gl/(kT)], the stress is"
cr oc co(Gulo - kTln V~~) eXp ( - -~T ).
(6)
The increase in stress is due to a decrease in the activation area of the unlocking process with increasing temperature. This model gives higher values of the density of mobile dislocations (Pm - 1.6 • 106 cm-2). However, the distance between obstacles at 1000 K seems too large to be realistic [86]. It should also be noted that according to Hirsch [84] there is no physical basis for assuming l oc exp[Gl/(kT)], since Gl controls the rate at which cross-slip occurs, not the steady state concentration. Moreover, Gul is taken as 3 eV which is quite large. Lastly, the model assumes that Gul < Gl, which is inconsistent with in-situ observations indicating that locking is easier than unlocking, and with estimates based on a lower energy of locked dislocation segments. The model has been then further modified by Khantha, Cserti and Vitek [87, 88] in order to explain the observed discontinuities in activation area versus temperature curves [49, 50]. The modification relies on the assumption that dislocations move after release of single pinning points at low temperatures, while double or multiple release is necessary at higher temperatures. It is however not certain that corrected values of activation areas are really discontinuous (cf. section 6.1 point (ii)). Another model has been proposed by de Bussac, Webb and Antolovich [89]. In contrast with the above models, the density of pinning points in this model is assumed to depend only on an activation energy of formation G1 and an activation energy of vanishing Gul. With this assumption, it is possible to justify that an equilibrium density can be reached. Its value 1 is [1 + e x p [ ( G l - Gul)/(kT)]] -1, which is to be compared with 1-1 - exp[-G1/(kT)] in the model of Sodani and Vitek. The flow stress is then" ere( co[1 + e x p Gl - Gul] -1 k~ '
(7)
where co is a drag force per cross-slipped segment. It is concluded that cr increases with increasing temperature, provided Gl > Gul. The term co is however not discussed. It is clear that it should be related to Gul, and thus should decrease as the temperature is increased. In fact, the origin of the stress anomaly is the same as in the model of Sodani and Vitek, since it results from a decrease in the driving force per pinning point, i.e., a decrease in the activation area, as the temperature is increased. The main objections are the inequality Gul < GI (as in the previous models) and the stability of pinning points which must tend to expand or to shrink easily. l A modified expression is proposed in [90]" [1 + (1/n) exp(AHn/ (kT))]--1, where AHn is the "enthalpy change associated with the formation of a cross-slip segment of length nb". The main conclusions are the same.
98
D. Caillard and A. Couret
Ch. 50
Fig. 20. Simulation of the expansion of a dislocation loop in Ni3AI, for a temperature of 300~ and a stress of 240 MPa. The pinning points are represented by blackened points. From Mills and Chrzan [91]. Computer simulations of Mills and Chrzan [91, 92] allow the local pinning model to be tested under more realistic hypotheses, namely using a statistical distribution of pinning points and a free flight velocity of dislocations including line tension effects. Since unpinning is athermal, these simulations reflect dislocation behaviour in TK and PPV models (dynamic break away). In particular, the same very low density of mobile dislocations is obtained. It is interesting to note that owing to the line tension effects, more or less rectilinear screw dislocations connected by macrokinks are obtained, as observed in TEM (fig. 20). Such computations should be improved by taking into account the lateral extension of cross slipped segments (Hirsch [79]). The same stress-strain rate dependence as found in the TK and PPV models is obtained a priori. The authors propose that strain-rate changes are almost completely accommodated by changes of mobile dislocation density, whence a low stress-strain rate dependence.
6.4.3. Sources at macrokinks The basic idea of this type of model was proposed first by Mills, Baluc and Karnthaler [93]. Since the critical stress for multiplying dislocations from Orowan-type sources is inversely proportional to the separation between anchoring points, a simple explanation for the yield stress anomaly is obtained if two hypotheses are verified: - The junction between macrokinks and screw segments (points A and B in Table 2) is either pinned or difficult to move. - The height of macrokinks decreases as the temperature is increased. The model proposed by Hirsch [83] can be summarized as follows, with some simplifications. The height of macrokinks, h, is related to the probability of locking per unit time, Pl, by the relation: h - vF/PI, where VF is the free flight velocity of glissile dislocations, proportional to the stress or. Since the probability of locking is P~ cx l exp[-G~/(kT)], where 1 is the distance between macrokinks, and is assumed to vary as the reverse of the stress or, the kink height is: h ~ cr2 exp[-Gl/(kT)]. Since macrokinks are very mobile along screw dislocations [79], strong pinning points must be formed by complex cross-slip processes, the details of which are not reproduced here. In this model, the stress necessary to emit dislocation loops at macrokinks is cr -O'i -~-O'*, where O"i is close to the Orowan stress # b / h (internal stress) and or* is the effective stress necessary to cut pinning points with the help of thermal activation.
w
Dislocation cores and yield stress anomalies
99
The average dislocation velocity is thus: v o~ hPul. It can be written: Gul - G l ) v - vo exp
-
kT
'
where vo is a constant, Gul = Gulo -o*bAul, and Gul > Gl. This yields: O"-
O"i
nt- O'*
#b
- -h-- +
Gu2o - k T In ~V - G! haul
Since Aul o( hb and h o( a 2 exp[Gl/(kT)], the stress is:
1[
GI ---Gl]exp(-~--~),
cr o( ~-~ #b 3 + Gulo - k T In VOv whence
]1/3 o" o(
~ b 3 -+- Gulo - k T In --v~ _ G1 v
exp ( -
Gi
3---~)
(8)
This model has many common properties with the model of Sodani et al. The temperature variation of cr is however weaker, owing to the factor 3 in the exponential term (as in the TK model). In addition, the main contribution to stress is the Orowan stress (term #b 3) which is not included in the model of Sodani et al. The activation area decreases with increasing temperature. Assuming that two anchoring mechanisms operate at low and high temperatures, namely a short range cross-slip at low temperatures, and the formation of Kear-Wilsdorf locks at high temperatures, Hirsch obtains two domains in the activation area versus temperature curve, in agreement with the measurements of Bonneville [49, 50]. This model is in good agreement with experimental observations of macrokinks, incomplete Kear-Wilsdorf locks, rapid glissile-sessile transitions. The mechanism leading to the formation of pinning points has however to be confirmed experimentally. In addition, the average kink height is far from being inversely proportional to the stress: it is found to decrease by only 40% when the yield stress increases by a factor of 3 between 300 K and 673 K in Ni3A1 and Ni3Ga (section 6.2 point iii). 6.4.4. Models based on the activation o f cube glide
The early model of Thornton, Davies and Johnston [48] is based on the strong hardening due to the interaction between dislocations gliding in the octahedral plane and another dislocation family gliding in the cube plane, which is more and more easily activated as the temperature is increased. Microstructural observations do not fit with this hypothesis, since the density of dislocations in cube planes is far from being high enough to explain the stress increase, at least with a (100) deformation axis. Moreover, the reversibility of the CRSS upon temperature changes is at variance with this model, since a high density of dislocations in the cube plane cannot be eliminated by cooling the sample.
100
D. Caillard and A. Couret
Ch. 50
Saada and Veyssi~re [94] have proposed a model of yield stress anomaly based on the more and more pronounced curvature of screw dislocations in cube planes, which is observed as the temperature is increased. The movement of screws is assumed to be controlled by the lateral motion of macrokinks which are always produced at a sufficient rate. This process is quite easy at low temperatures, since screw superdislocations are straight. It may however be more difficult at higher temperatures, since the bowing of screw superdislocations in cube planes should inhibit the lateral movement of superkinks which allow screw superdislocations to glide in octahedral planes. The total frictional force on kinks has been computed by Saada and Veyssi~re and is given by: O'fc O'f -- O'fo -]
(q0c O" -- O'fc)
o*[ 1- (~,~a--O'f~)~],/2,
(9)
where O'fo and O'fc are the frictional forces in the octahedral and cube planes, respectively, cr is the applied stress, CPc is the Schmid factor on the cube plane, O'c and o-* are constants depending on the height and the separation of kinks. Provided macrokinks are mobile, it is, however, likely that the segments with larger bowing will grow at the expense of the shorter ones, leading to the coalescence of macrokinks, a favoured process since it decreases the total energy by increasing the total work done by the resolved shear stress in the cube plane. Dipole formation may also inhibit this process [70, 71]. The violation of the Schmid law may be explained by this model. However, according to Saada and Veyssi~re, the stress anomaly does not result straightforwardly from this mechanism, unless the height of kinks decreases, and their distance increases, as the temperature is increased. A small decrease of the kink height has been measured. However, no data is available on their distance.
6.4.5. Double cross-slip mechanisms Here, dislocation movements are assumed to be controlled by a cross-slip process leading to the nucleation and the rapid extension of a glissile loop on a rectilinear screw dislocation. This dislocation loop is rapidly locked by another cross-slip event, which leads to the formation of a macrokink pair of height h. Macrokinks are assumed to glide further rapidly along the screw dislocation and move it over the distance h. In this model, dislocations move by series of thermally activated locking and unlocking processes by double cross slip, similar to those described in the case of prismatic glide in beryllium, with Gul > Gl (section 4). Such a suggestion was also made by Sun and Hazzledine [95] for the 3" phase of superalloys. As in the case of beryllium, the dislocation velocity can be set equal to: v = hPul, where Pul = Pulo exp[-Gul/(kT)] is the probability of unlocking per unit time and unit length for each screw dislocation, and h -1 c~ ~ c~ exp[-Gl/(kT)], which yields v -- v0 e x p [ - ( G u l - Gl)/(kT)], where v0 is a constant. Since h does not vary very much with stress and temperature, the velocity of dislocations is controlled by the unlocking process. Nabarro has proposed a model of stress anomaly based on this double cross-slip process, however with some modifications [96]. The main assumption is that after the
w
101
Dislocation cores and yield stress anomalies
8.
0.6 l o~
l
i
"t"p
7a
04 90.2
0 2
/ MPo
lO0
i/lo 0 0" 9u
~
! 10
! 20
wtb
b.
Fig. 21. Stress (or) necessary to unlock incomplete Kear-Wilsdorf locks, for various values of the extent of the APB in the cube plane, w, and frictional stress in the cube plane, trfc. The dotted line refers to the situations where the driving stress trfc on the trailing superpartial is equal to the driving stress crfc t on the leading superpartial. From Mol6nat et al. [97]. jump the dislocation reaches an equilibrium position illustrated in table 2, which means that the macrokinks thus formed are not mobile 9 Accordingly, the area swept by each unlocking process is A oc (1/F!3/2), instead of A ~ h oc (1/Pl) in the above model. This yields a dislocation velocity 9 v = v 0 e x p [ - ( G u l - 3 G l ) / ( k T ) ] (instead of v v0 e x p [ - ( G u l - G l ) / ( k T ) ] in the above model). A yield stress anomaly is thus obtained if 3Gl > 2Gul. Since the kinetics of dislocation movements indicate that G1 < Gul, both conditions are satisfied if ~Gl > Gul > Gl. This model is however incompatible with observations of highly mobile macrokinks. Since the model developed for beryllium explains the experimental observations most satisfactorily, the yield stress anomaly can only arise from an increased difficulty of crossslipping from sessile to glissile configurations (as already concluded in sections 4 and 5). Sun and Hazzledine proposed that the formation of a pair of macrokinks is hindered by an increased tendency of superpartials to dissociate onto the cross-slip octahedral plane, as the temperature is increased [95]. Another possibility has been suggested by Mol6nat et al. [97]. Following the calculations of Paidar et al. [78], the stress cr necessary to unlock incomplete Kear-Wilsdorf locks (fig. 21 (b)) has been calculated as a function of the extension w of the APB ribbon in the cube plane, and as a function of the stress Crfc necessary to move the trailing
D. Caillard and A. Couret
102
Ch. 50
dislocation in the cube plane (the frictional stress in the cube plane) (fig. 21(a)). ~r is shown to increase with increasing w and Crfc. The dotted line is the critical stress crc for which the total driving force for glide in the cube cross-slip plane is the same for leading and trailing superpartials (tree - Cr~c). According to [78, 79, 97], this situation results in jumps over a distance equal to the dissociation width in the octahedral plane (see fig. 19). For low values of w, the critical stress crc has been calculated by Paidar et al. [78] and Hirsch [79, 83], and is given by Crc- o71( -7-- 1
70 1 +(2/A))31/2 "71
)
where A is the elastic anisotropy ratio, ~'0 is the APB energy in the cube plane and 71 is the APB energy in the octahedral plane. For the maximum value of w, the critical stress is
,,( 1 ,o3'1 31/2,)
Oc---b-
"
It corresponds to the stress level necessary to destroy a Kear-Wilsdorf lock under slightly different hypotheses, according to Saada and Veyssi~re [77]. The dynamic behaviour of incomplete Kear-Wilsdorf locks is considered next. Chou and Hirsch [79] have shown that incomplete Kear-Wilsdorf locks evolve rapidly towards complete Kear-Wilsdorf locks at stresses lower than crc. Stable movements of incomplete Kear-Wilsdorf locks by locking and unlocking processes are thus possible only for O ' ~ O " c.
Recently, Mol6nat et al. [97] have postulated that octahedral glide may be controlled by the repeated locking and unlocking of incomplete Kear-Wilsdorf locks, with values of w increasing with temperature. This model is supported by several microscopic observations (sections 6.2 and 6.3). At low temperatures, the yield stress should be of the order of or slightly larger than Crc(w-b)-b-(1
701+(2/A)) '71
31/2
"
At high temperatures, the yield stress should be of the order of: Crc(W = d0) = bl- (1
"Yl 3 1 / 2
'
where do is the dissociation width in the cube plane. A general expression is thus:
[
3'1 cr ~ -b- 1
( 2 w,T> 0)1 .
1 3'0 1 + - 31/2 "71 A
do
(10)
w
Dislocation cores and yield stress anomalies
103
Three parameters are important: "71 (or "70), "70/"71, and the CSF energy which determines the temperature variations of w (larger values of the CSF energy makes cross slip from {111} onto {100} easier, whence higher values of w at a given temperature). Orientation effects and tension-compression asymmetries could also be explained by variations of w. The stress can be shown to be very sensitive to the ratio "70/"71, with "70/"71 ~> 1, as already pointed out by Saada and Veyssi~re [76]. The yield stress can also be expressed as cr = o i a t- o * , where O'i is an internal stress, and or* is an effective stress associated with the frictional stress acting on superpartials in the cube plane (fig. 21b). For "70 = 150 mJ/m 2 and "70/"71 -- 0.9, estimations give a ~ 100 MPa for w = b, and cr ,-~ 250 MPa for w = do. These values fit with yield stresses at 300 K and 673 K in Ni3Ga, where clear indications of double cross slip with corresponding low and maximum values of w have been obtained. Further developments are however needed in order to test other properties, e.g., the stress-strain rate dependence and the reversibility of the flow stress upon temperature changes. Three-dimensional configurations including the formation of pairs of macrokinks also need to be considered.
6.5. Other L12 alloys Many other intermetallic alloys with the L 12 structure exhibit the yield stress anomaly to different degrees (Wee et al. [36]). Nevertheless only those on which substantial information is available on mechanical properties, glide systems and/or dislocation structures will be discussed here. 6.5.1. Co3Ti
Mechanical properties of Co3Ti have several common properties with those of nickelbased alloys (Takasugi et al. [98]). Deformation is produced by octahedral slip, strong orientation effects can be observed in the domain of positive temperature dependence of the yield stress (fig. 22), and the stress-strain rate sensitivity is low. Cube slip is however more difficult to activate than in nickel-based alloys, since it is observed only for a (111 ) tensile axis, above the peak temperature. TEM observations reveal rectilinear screw superdislocations dissociated into two superpartials separated by an APB ribbon probably in the cube plane, after deformation in the anomalous domain (873 K) (Liu et al. [99]). The origin of the stress anomaly is thus likely to be the same as in nickel-based alloys. In the low-temperature regime, the normal flow stress-temperature dependence has been attributed to a dissociation of superdislocations into two super Shockley dislocations, separated by a ribbon of SISF [99]. 6.5.2. Cu3Au
The mechanical properties of Cu3Au exhibit important differences with respect to those of nickel-based alloys (Kuramoto and Pope [100]). The yield stress indeed increases steeply up to the order-disorder transition temperature and then sharply decreases (fig. 23). In addition, orientation effects are much smaller than in nickel-based alloys in the whole temperature range investigated.
104
Ch. 50
D. Caillard and A. Couret I
250 -
w
I
r-"
22 at'/', Ti 3.Sx10-4s-1
1
,
', 9
d'~,a__ 1s 200 -
o A
150 "
v--
t"
o
o5 v5 to
100 -
50-
ff"E closed
0 0
2; 0
' 400
6;o
' BOO
marks 10011[il O] ' 1000
' 1200
Temperature I K Fig. 22. Temperature and orientation dependence of the CRSS for octahedral slip of Co3Ti. From Takasugi et al. [98].
TEM observations indicate that a high density of rectilinear screw dislocations are present in deformed samples (Kear and Wilsdorf [101], Sastry and Raswamany [102]). The origin of the stress anomaly seems to be strongly correlated with the orderdisorder transition, and it may thus be different from that in nickel-based alloys and Co3Ti. According to Pope [103], it could result from an interaction between dislocations and local regions of disorder. This hypothesis does not however fit with microscopic observations, since wavy dislocations with no specific orientation would be observed in this case. Yamaguchi and Umakoshi [40] suggested that Flinn's model may also apply. This model is based on the change of the APB plane from {111} onto {100} by diffusion (or climb dissociation). This hypothesis is also in conflict with microscopic observations, since frictional forces would act on all dislocation characters, not just screw. The same remark also holds true for Brown's model [104]. The principle of Brown's mechanism is schematized on fig. 24. The APB is assumed to lie in the dislocation glide plane. The leading dislocation creates a fresh APB of energy ")/1 which relaxes rapidly by diffusion and lowers its energy (~,[). Then, the trailing dislocation does not restores the perfect crystal, but creates a fault of energy ~,{', which subsequently vanishes also by a diffusional process. The stress required for moving the dislocation is then: (7" -- ("Yl -t- "Yl' -- 7 [ ) / ( 2 b ) .
w
105
Dislocation cores and yield stress anomalies
Cu3Au o v
u') bJ
/:
5
l-V'}
rr L.
I
f
w
o "'
I
,
IAA
3
0 w nr _j
9 e,/
2.
O---A e--- B A---C X--- D
u
l
.t
o0
.
100
t
I
200
300
.I
400
. z
-, L J. ,
500
600
'700
800
TEMPERATURE (K) Fig. 23. Temperature dependence of the CRSS for octahedral slip of Cu3Au. From Kuramoto and Pope [100].
I
1. w z
7 "_I 7? t~
I
. . . . . . . . . .
%1 ,
I
I
,
==>
Fig. 24. Schematic description of the Brown mechanism.
This model may explain the yield stress anomaly close to the order-disorder transition temperature Tc. Diffusional effects could indeed strongly modify the structure and the energy of APB's above 0.75Tc in L12 alloys, according to Sanchez, Eng, Wu, and Tien [ 105]. Models based on the formation of Kear-Wilsdorf locks would explain microscopic observations better. It is not clear, however, why orientation effects should be so small if cross slip is controlling the strain-rate. Combining cross-slip and Brown's mechanisms may explain both the observation of rectilinear screw dislocations and the relation between the yield stress anomaly and the order-disorder transition. The APB of immobile Kear-Wilsdorf locks may indeed relax substantially even at intermediate temperatures. As pointed out by Morris [106], during the cross-slip process of Kear-Wilsdorf dislocations onto octahedral planes, a fresh APB is thus created in the octahedral plane whereas a relaxed APB is erased in the cube plane,
106
Ch. 50
D. Caillard and A. Couret
300 R
-2 k
S
~260 (? ,~
,o
I
220
/ t
.O
1 -0.5 -0
CH
c5
1.5
180
t:) 140
I
100
i / d
A
"13 05 m
--1
t~ (0.2%)
--1.5
. . 2 6 d ' :46d' i3bd'i36d "l'0bb 2
Temperature (~ Fig. 25. Temperature dependence of the yield stress tr and the strain-rate sensitivity S in a AI62,5Ti24,5Cu13 alloy. From Potez et al. [108].
resulting in a more difficult unlocking process of screw dislocations as the temperature is increased. The yield stress anomaly cannot however be discussed in more detail without further microscopic observations.
6.5.3. Al3Ti Titanium trialuminides may be changed into the Lie structure by adding some ternary elements. The corresponding yield stress versus temperature curves exhibit in several cases a weak anomaly between 300~ and 500-700~ (Morris, Gunther and Lerf [ 107], Potez, Lapasset and Kubin [108]) (fig. 25). In both studies, strain instabilities such as the Portevin-Le Chatelier effect have been clearly identified in the domain of the flow stress anomaly (fig. 26). In one study, these instabilities are shown to be the cause for a dip in the strain rate sensitivity versus temperature curve, which becomes negative (fig. 25). Activation areas are small at room temperature (,-~ 25b a) and large at 600~ (~ 300b 2) [1091. Microscopic observations show that deformation is due to (110) superdislocations gliding in octahedral planes. These dislocations are seen to be dissociated (within the resolution of weak beam) only above 500~ and they do not exhibit any strong directionality. Intensive cross-slip from octahedral onto cube planes is observed at 700~ A Peierls mechanism may explain the small values of activation areas at 20~ although no strong directionality is seen at this temperature [ 109]. The yield stress anomaly was first explained on the basis of the formation of KearWilsdorf locks [ 109]. The more recent studies however indicate that the stress anomaly may be coupled to the Portevin-Le Chatelier effect, owing to dynamic strain ageing processes as in dilute alloys (section 3) [107, 108]. Static age-hardening due to the precipitation of the A12Ti phase may also contribute to the increase of yield stress with increasing temperature [107]. Precipitates of A12Ti phase have been observed in dislocation cores in [ 110].
w
Dislocation cores and yield stress anomalies
107
I ioo M~d 300 oC
jr
40ON::
Strain Fig. 26. Serrations on stress-strain curves of a AI-Ti-Fe alloy. From Morris et al. [ 107].
6.6. Discussion on cross slip in L12 alloys
Except for A13Ti, and to a less extent Cu3Au, most experimental results show that the yield stress anomalies are related to an increased propensity of superdislocations to cross slip from octahedral onto cube planes, as the temperature is increased. As already discussed in many review articles, cross slip can be primarily induced by a lower APB energy on cube planes. Lower APB energies in cube planes are related to a lower stability of the L12 phase with respect to D022 or D019 phases (Wee et al. [36]). The lower the stability of the L12 phase, the larger is the tendency for a yield stress anomaly (fig. 27). A good example of this correlation is the progressive vanishing of the stress anomaly of (Ni, Fe)3Ge alloys, as the amount of nickel decreases, and as the phase stability of the L12 structure with respect to the D022 structure simultaneously increases (Suzuki, Oya, Wee [ 111 ]) (fig. 28). The driving force due to the difference in APB energies is however (at least in most alloys) not high enough to promote cross slip, whence the important role of the torque introduced by Yoo [75]. This torque depends on the elastic anisotropy factor A, which varies from 3.33 in Ni3A1, to 1.30 in Pt3A1 and 0.90 in A13Sc which do not exhibit stress anomalies (Yoo [112]). Saada and Veyssi~re [41 ] pointed out that the very small splitting width of superpartials is inconsistent with cross-slip mechanisms which remain active over several hundreds of degrees. As mentioned in section 4, cube glide, which is also controlled by the splitting of superpartials, is also thermally activated over a large temperature range (900~ in -y', 700~ in Ni3(A1, Ta) and Ni3 (A1, Nb) - cf. fig. 8). An explanation for this inconsistency may lie in the temperature dependence of the core of superpartials, as discussed in section 4. However, all the models presented in this section either suffer from serious discrepancies from experimental results, or are not developed systematically. Further developments are thus needed, especially on the kinetics of macrokinks along screw superdislocations.
108
Ch. 50
D. CaiUard and A. Couret
A1 S1 Gc] fie In Sn Sb Pb
Nls Pt,
Fig. 27. Part of the periodic table, showing the increasing trend of the magnitude of the anomalous strength behaviour when changing the combination of elements in A3B alloys. From Wee et al. [36].
(a)
I
(b)
I
1200
--"-v ---~ . . . . . . .~.
,
v
I 1000
lOOO
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Q.
~
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,'~
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I / / / / o ' , Fe25,'~ /v /
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9
.
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400
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300
200 0
200
400
600
Temperature [ K )
800
1000
0
200
400
600
800
1000
Temperature (K)
Fig. 28. Yield stress versus temperature curves for various (Ni, Fe)3Ge L12 alloys. From Suzuki et al. [111].
7. TiAl (Llo structure) Crystals with the L10 structure can be regarded as f.c.c, crystals with alternating layers of each element parallel to one face of the cube. In this crystal structure there are two types of perfect dislocations, with Burgers vectors 89 (110] and (011] respectively (following 1 the notation introduced by Hug et al. [18]). g( 110] ordinary dislocations are comparable to g1 (110) dislocations in the f.c.c, structure, while (011] superdislocations are similar to (011 ) superdislocations in the L 12 structure, although the detailed core structures are different.
w
109
Dislocation cores and yield stress anomalies
7.1.
Mechanical
properties
Many deformation experiments have been made on single phase 3' (L10) polycrystals and two-phase "7 + c~2 polycrystals and single crystals. The most significant results on deformation mechanisms involving the yield stress anomaly have however been obtained on single crystals of several A1 rich single-phase TiA1 alloys by Kawabata, Kanai and Izumi [113] and more recently by Stuke, Dimiduk and Hazzledine [114]. The following conclusions have been given by Kawabata et al.: - Yield stress versus temperature curves exhibit pronounced anomalies and similar stress levels when either ordinary 89 (110] or super (011] dislocations have the highest Schmid factor (fig. 29(a)). - In both cases the flow stress-strain rate sensitivity is low in the anomalous domain and goes through a minimum close to zero in the middle of this domain (fig. 29(b)). - The strain hardening coefficient increases with increasing temperature when ordinary dislocations are thought to be activated. It decreases normally when superdislocations are thought to be activated. - Twinning is observed in a few orientations in connection with either ordinary or superdislocations. In addition, experiments of Stuke et al. revealed that when ordinary dislocations are likely to be activated, the flow stress is not reversible when samples are predeformed at a high temperature and restrained at room temperature (fig. 30). This behaviour is (a)
(b) 400 %
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9
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500 Temperoture
1000 (K)
1500
0 o
500 Temperoture
1000
1500
(K)
Fig. 29. Yield stress (a) and strain-rate sensitivity (b) versus temperature curves in single phase TiA1. Solid and open marks refer to orientations with high Schmid factors on superdislocations and on ordinary dislocations, respectively. From Kawabata et al. [113].
110
Ch. 50
D. Caillard and A. Couret
500
400 296K ~"
=_..
300
20o
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2.5
(%)
Fig. 30. Schematic description of the stress-strain curve for a single room-temperature test and a twotemperature compression test in TiA1. From Stucke et al. [114]. at variance from that observed in Ni3A1 (section 6.1 point (iii)). The strain hardening coefficient (which is found to increase with increasing temperature, in agreement with Kawabata's results) does not depend on the prestraining conditions, contrary to Ni3A1.
7.2. Microscopic observations Microscopic observations reveal substantial differences in the dislocation microstructure of different deformed single-phase and two-phase alloys [115-124]. Ordinary dislocations, super-dislocations and twinning are however always observed with the following common properties: (i) Ordinary dislocations are observed most frequently after deformation at all temperatures. They are rectilinear along the screw orientation in the anomalous domain, and contain some jogs which act as anchoring points (fig. 31). In-situ straining experiments [125, 126] indicate that rectilinear screws move more slowly than edges, by a series of jumps over short distances often clearly related to nucleations and propagations of macrokink pairs (fig. 32). Jogs are seen to originate from double cross slip. They significantly slow down the screw dislocations. Individual dislocation movements are observed at room temperature, whereas deformation is produced by bursts of dislocation movements at 200~ and 400~ In the latter case, dislocations are immobilized in the screw orientation and cease to move upon subsequent deformation, in which new dislocations bursts are activated. Intensive double cross slip has been observed between octahedral and cube planes at all temperatures [125, 126]. (ii) Superdislocations are observed less frequently than ordinary dislocations. They can be absent either at low or high temperatures, depending on the alloys investigated.
w
Dislocation cores and yield stress anomalies
111
Fig. 32. Glide of ordinary dislocations in TiAI. Note the jerky movement of rectilinear screw segments. In situ experiment at 20~ From Farenc and Couret [125].
112
Ch. 50
D. Caillard and A. Couret
Direction of projection:
[100l [100]
~10] ~ 0 ]
[010] ~[010]
f Fig. 33. Scheme of screw superdislocations viewed along several directions in TiAI deformed at high temperature. From Hug [18]. They are very rectilinear along the screw orientation. Rectilinear screws have complex dissociations, which are shown to lie in the octahedral glide plane at low temperatures and in the cube cross-slip plane at high temperatures [18, 121, 122] (fig. 33). No cube glide seems to be possible for superdislocations. In situ observations have shown that rectilinear screw superdislocations move jerkily with an average velocity much smaller than that of edge segments. As in the case of macroscopic deformation tests, these dislocations are dissociated in the { 111 } glide plane at room temperature, whereas the APB straddles {111} and {100} planes at higher temperatures [125, 126]. (iii) Twinning has been observed extensively both post mortem and in situ (for a review, see for instance [127]). No significant change in the twinning processes has been detected between low and high temperature deformation experiments. Twinning proceeds by steady movements of Shockley partial dislocations on adjacent { 1 ! 1} planes, which leads to formation of almost perfect twins. Several twin-twin and twin-dislocation interactions have been reported.
7.3. Discussion of microscopic dislocation processes
The experimental results on TiA1 are more fragmentary than those on Ni3A1 alloys. Dislocation mechanisms can however be proposed on the basis of consistent results obtained post mortem and in situ by different authors. Glide of ordinary dislocations is possible at low and high temperatures. It is probably controlled by a Peierls-type frictional force on screw segments. Jerky movements however indicate that a double cross-slip (locking-unlocking) mechanism may operate instead of a pure Peierls mechanism, as in beryllium (section 4). In such cases, pinning points may not control directly the velocity of dislocations. The origin and the exact role of pinning points has however to be investigated more carefully.
w
Dislocation cores and yieM stress anomalies
113
Two distinct origins for the Peierls-type frictional forces have been proposed: a nonplanar extended core as in prismatic glide of beryllium (section 4), cube glide of Ni3A1 (section 5), and b.c.c, metals, or covalent bonding as in semiconducting materials [ 116, 128]. The second hypothesis has been formulated after calculations of charge densities around Ti atoms, showing that directional bonding develops between Ti atoms. However a detailed analysis of the kinetics of screw dislocations indicates that the Peierls forces are better explained by an extended core [129], although an extended core has not been predicted by atomistic calculations [130] and edge dislocations observed in high resolution TEM have a compact core [ 131 ]. - Glide of superdislocations is also possible, although not observed in all cases. Several observations report planar dissociations at room temperature and non-planar dissociations (more or less amount of cross slip into the cube plane) at higher temperatures. This behaviour is similar to that observed in Ni3A1, although with more complex dissociations, and no cube glide has been observed in TiAI. - Twinning is frequently observed at all temperatures. The exact role of twinning in the anomalous variation of yield stress with temperature is however unclear.
7.4. Models of the yield stress anomaly When superdislocations are activated, it is generally assumed that the yield stress anomalies in TiA1 and L12 alloys arise from a common origin. This hypothesis is based on measurements of similar orientation effects and values of stress-strain rate sensitivity and observations of cross slip into the cube plane. Greenberg et al. [132] considered that many locking processes can occur on superdislocations, e.g., local pinning, formations of roof type or Kear-Wilsdorf type barriers (see also [133]) which are all thermally activated with higher frequencies as the temperature is increased. The stress anomaly is thus interpreted within the frame of the models detailed section 6.4. Hug came to the same conclusions on the basis of his own observations [18]. It is much more difficult to account for the yield stress anomaly when ordinary dislocations are activated. Greenberg has proposed a model based on the existence of strong covalent frictional forces (deep Peierls valleys) along Ti rows of { 111 } planes [ 128]. For ordinary dislocations, this corresponds to the screw orientation. Ordinary dislocations are thus assumed to be more likely to be locked in the screw orientation as the temperature is increased. The thermally activated process leading to dislocation locking is however not described in detail. In addition, unlocking must be impossible, or at least very difficult in the domain of the anomaly. This model agrees with post-mortem observations. However, it conflicts with results of in-situ experiments which show that rectilinear screw dislocations have a jerky movement, for which unlocking is always possible and controls deformation. In addition, the kinetics of glide of ordinary dislocations does not correspond to the occurrence of covalent friction forces, as mentioned in section 7.3. Since the movement of dislocations is similar to those observed on prismatic planes of beryllium and cube planes of the -,/' phase (L12 structure), the origin of the yield stress anomaly could be the same, namely an increase in the width of non-planar screw dislocation cores as the temperature is increased. It would however be surprising if the yield stress
114
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D. Caillard and A. Couret
anomalies associated with ordinary and superdislocations should have different origins, as they have so similar properties (the same stress level, the same values of strain-strain rate sensitivity).
8. /3-CuZn and other B2 alloys The B2 structure is based on the body-centred cubic (b.c.c.) lattice. Accordingly, superdislocations with Burgers vectors (111) made of two superpartials with Burgers vec1 tors 7(111) (as in b.c.c, metals) separated by an APB ribbon have been found in some materials. However, ordinary dislocations with (100) Burgers vectors have also been found in some B2 alloys. Among b.c.c, type intermetallic alloys,/3 brass with the B2 structure has been studied most extensively. Materials with other superlattice structures based on the b.c.c, lattice are also treated in section 8 (subsections 8.4 and 8.5).
8.1. M a c r o s c o p i c
results on fl-CuZn
The flow stress in fl-CuZn increases with increasing temperature up to a peak temperature of about 200~ which is far below the order-disorder transition temperature (Tc = 460~ (fig. 34). 20--
1
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1
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I
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1
300
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Fig. 34. Yield stress of fl-CuZn as a function of temperature. From Umakoshi et al. [134].
w
115
Dislocation cores and yield stress anomalies
A c c o r d i n g to U m a k o s h i et al. [134] and N o h a r a et al. [135], slip lines are parallel to
{110} planes in the domain of the yield stress anomaly. {112} slip is also observed at low temperature ( - 196~ and above the peak [ 135]. In addition to the yield stress anomaly, the yield stress-temperature curves exhibit strong dependence on orientation, as first shown by Umakoshi et al. using compression tests (fig. 34). Later, Nohara et al. showed that there is also a tension-compression asymmetry. When samples are prestrained at room temperature in tension and further deformed at different temperatures in compression, the peak temperature is the same as if the crystal was only deformed in tension, and vice-versa [136]. These orientation effects cannot be explained as in L12 crystals, since <111) dislocations are not dissociated into Shockley partials and the Escaig theory does not apply (a) 15o
(b) 150
1 O0
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Fa .J
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TEMPERATURE,
300 "C
Fig. 35. Critical shear stresses of fl-CuZn deformed in pure shear, as a function of temperature (from [138]). (a) Along (112) [lli] (easy sense, open symbols) and (112) [iil] (hard sense, filled symbols). [liT] superdislocations are activated below the peak. (b) Along (101) [lli] (open symbols) and (101) [ill]) (filled symbols). [lli] dislocations are activated below the peak. (c) Along (010) [100] ([100] ordinary dislocations are activated).
116
D. Caillard and A. Couret
Ch. 50
directly. They may be explained as in b.c.c, metals by glide in { 112} planes either in the twinning or in the antitwinning sense (see for instance Duesbery [ 137]). To test this hypothesis, macroscopic deformation experiments have been made in direct shear along a (111) direction, in {110} and {112} planes (fig. 35(a))(Matsumoto and Saka [138]). The flow stress-temperature curves do not exhibit any asymmetry when the sense of the shear is reversed, which shows that orientation effects arise from a different origin. They may be due to normal stresses which are not present in pure shear tests. Duesbery discussed the possible effect of different components of the stress tensor on the core structure of 89 dislocations in b.c.c, metals [139]. These effects, similar to the Escaig effect for f.c.c, crystals, may play a role in the orientation effects in CuZn [ 138]. More surprising is the resemblance between the yield stress versus temperature curves obtained in pure shear along (111) {110} and (111) {112} glide systems [138] (fig. 35 (a) and (b)). Both curves exhibit an anomalous behaviour with a peak at about 230~ It is worth noting that a weak anomaly is also found in the (100) {010} pure shear system, with a peak at about 200~ (fig. 35(c)).
8.2. Microscopic observations in/~-CuZn From several post-mortem observations of Saka et al. (see for instance [138]), it is evident that dislocations with {111 ) Burgers vectors are activated in the domain of the yield stress anomaly, whereas dislocations with other Burgers vectors are found above the peak temperature. According to Zhu and Saka [140], and Dirras, Beauchamp and Veyssibre [141, 142], ( 111 ) dislocation loops are elongated in the screw orientation (fig. 36). However, they do not seem to be as straight as in Ni3A1. They are dissociated into two 89 (111) partial dislocations, separated by an APB ribbon. The APB tends to extend out of the glide plane as the temperature is increased, on dislocations of all
Fig. 36. (111) dislocation loop elongated parallel to its screw orientation, in j3-CuZn deformed at 180~ From Dirras [ 141].
w
Dislocation cores and yield stress anomalies
117
characters. This occurs by cross slip along the screw orientation, leading to dissociations in { 112} planes, and by climb dissociation along the non-screw orientations. The problem of the possible rearrangements of the dislocation substructure during the preparation of the microsamples has been raised by both authors. According to Veyssi~re, samples have to be quenched rapidly in order to freeze the substructure. However, Saka showed that climb dissociation is less well defined in samples deformed at - 7 3 ~ than in samples deformed at higher temperatures, which indicates that climb dissociation does occur during bulk deformation. In-situ experiments have also been made on/3-CuZn by Nohara [143] who found rapid and jerky movements of (111) dislocations in {110} planes below the peak, in contrast to steady movements of non (111) dislocations above the peak. The velocity of the screws is similar to the velocity of the non-screws at 20~ The velocity of the screws is however slightly smaller than the velocity of the non-screws at 150~ in the anomalous domain, leading to dislocation loops elongated parallel to the screw orientation.
8.3. Models of the yield stress anomaly (/~-CuZn) According to Umakoshi et al. [ 134], orientation effects can be interpreted by the ease of cross slip from {110} to {I12} planes in the twinning sense. Therefore, the yield stress anomaly may result from the thermally activated locking of (111) screw dislocations by cross slip from {110} onto {112} planes, and several models developed for L12 crystals may apply. No interaction torque acts on (111) dislocations lying in {110} planes in order to help a change of APB plane, as for (110) dislocations in the L12 structure [112]. Accordingly, the APB is expected to lie in the glide plane, either {110} or { 112}. A torque is however acting at intermediate angular positions, which may destabilize {110} dissociations and stabilize {112} dissociations, even for APB energy values ")'112 > ")'110 (Saada and Veyssi~re [144], Sun [145]). Elastic calculations indicate that both dissociations could be stable, although experimental data are not sufficient to derive clear conclusions. Stable non-planar configurations may also be found [144, 145]. On the basis of their experimental results, Nohara et al. [135] concluded that Umakoshi's hypothesis is inconsistent with the amplitude of the tension-compression asymmetry. Moreover, there is no twinning-antitwinning effect in direct shear parallel to {112} planes, and stress anomalies cannot be explained in both (111) {110} and (111) {112} glide systems. On the basis of this experiment and the subsequent TEM postmortem observations, Zhu and Saka [140] concluded that the yield stress anomaly arises from increased climb dissociation of mixed dislocations with increasing temperature. The climb dissociation of mixed dislocations is assumed to induce the cross slip of the screw parts, in such a way that all components of dislocation loops are locked (fig. 37). Dislocation movements controlled by climb dissociation of edge segments would however produce dislocation loops elongated in the edge orientation, at variance with TEM observations. In addition, the cross-slip mechanism proposed by Zhu and Saka is likely to take place only for large dissociation widths (i.e., not much smaller than the size of dislocation loops), in order to obtain substantial line tension effects. Dirras et al. [141, 142] and Nohara [143] interpret their results by two effects: climb dissociation, as above, and the Brown mechanism (section 6.5).
118
Ch. 50
D. Caillard and A. Couret
plan view
Fig. 37. Mechanism by which screw segments of (111) dislocations in fl-CuZn may dissociate off the original plane. The pure screw superpartials are rotated by line tension effects induced by climb dissociated non-screw segments. From Zhu and Saka [ 140]. 2O0 0 theoretical A exp.(A) a exp. (D)
13_ v co tj~ (i)
too
0 0
2O0
4( )0
600
800
Temperature (K)
Fig. 38. Predicted flow stress dependence upon temperature in Cu-Zn using Brown's model, and comparison with experimental results. From Beauchamp et al. [146]. Using the approximation of the cluster variation method, Beauchamp et al. have estimated the temperature dependence of the thickness and the energy of an APB in a { 110} plane [146]. With Tc being the order--disorder transition temperature (Tc ~ 460~ there is a significant increase in the thickness of the APB, and a decrease in its energy, above 0.35Tc. The calculations give a correct value of the peak temperature (fig. 38). Kinetic processes may however have to be included in order to obtain the correct stress level. These kinematical processes have been studied by Nohara [143]. A comparison between the velocity of diffusion and the velocity of dislocations, as observed in situ, shows that
w
Dislocation cores and yield stress anomalies
119
the Brown mechanism may take place, at least when dislocations are temporarily slowed down. This model however cannot explain orientation effects. Additional effects may also arise from the proximity of a martensitic transformation, according to Ahlers [34]: the core of screw dislocations may relax via a local martensiticlike structure when the temperature is changed, leading to changes in the Peierls stress. Such a process, similar to those proposed in beryllium (section 4) and 3" (section 5) would explain stress anomalies in both { 110} and {112} planes, as well as the anomalous behaviour of dislocations with (100) Burgers vector gliding in {001} planes. As a conclusion, it is clear that more experimental results such as measurements of stress-strain rate sensitivities and in-situ experiments are needed.
8.4. Other intermetallics with b.c.c.-derivative ordered structures 8.4.1. F e C o
The mechanical properties of Fe(Co, 2 at.% V) with the B2 structure have been studied by Stoloff and Davies [147]. The CRSS is almost independent of temperature when the temperature is increased to 670~ It increases markedly between 670~ and 700~ and decreases steeply above 700~ As in Cu3Au (L12 structure), the peak in the yield stress is strongly correlated with the order-disorder transition at 720~ Deformation tests at room temperature in specimens annealed at different temperatures and subsequently quenched indicate that dislocations glide easily in the fully ordered material whereas they are more difficult to move when the degree of long-range order is lower, i.e., after annealing just below the transition temperature. A model for the yield stress anomaly was proposed by Stoloff and Davis [147] on the basis of this observation. At low temperatures, deformation is achieved by the glide of superdislocations, as in most intermetallic alloys. However, as the temperature is increased, the degree of order decreases, and superdislocations tend to decompose into their two constituent superpartial dislocations, which however have to trail an APB fault behind them. This requires a higher stress, whence the stress anomaly. Above Tc, the APB vanishes, and the stress decreases normally. No microstructural observation is however available to support this model. 8.4.2. A g 2 M g Z n
This alloy has the D03 structure at low temperatures and the B2 structure above 250~ The D03 structure corresponds to a higher degree of order than the B2 structure, in such a way that four 89 (111 ) dislocations are necessary to restore the perfect crystal. These quadruplets are in fact loosely correlated double pairs, which decompose into two pairs when the D03 structure is transformed into the B2 structure. Ag2MgZn has anomalous behaviour with a peak temperature just below the D03-B2 transition temperature, and no orientation effect is observed [148]. (111) {110} glide or (111) { 112} glide is observed as a function of the orientation of the compression axis. It is noticeable that the stress versus temperature curves for these two glide systems are similar and exhibit the same anomalous behaviour as in/3-CuZn. According to Yamaguchi and Umakoshi [ 148] the decomposition of double pairs into single pairs may be at the origin of the stress anomaly, following a process similar to
120
Ch. 50
D. Caillard and A. Couret
that proposed in FeCo. Here again, no microstructural observations exist to confirm this hypothesis.
8.4.3. Fe3Al The plastic properties of Fe3A1 polycrystals were investigated first by Stoloff and Davies [147]. Fe3A1 and closely stoichiometric Fe3A1 single crystals have been subsequently studied by Hanada et al. [149], and more recently by Schr/3er et al. [150, 151]. Macroscopic results are in good agreement. The stress-temperature curves exhibit an anomalous behaviour above 600 K, with a peak temperature close to the temperature of the D03-B2 transition (825 K), as in the case of Ag2MgZn. However, the peak is found either below or above the transition temperature, depending on to the crystal composition [ 149] (fig. 39). There are strong orientation effects, and slip line observations show that dislocations glide in { 110} planes below the peak temperature, and either in { 110} or in { 112} planes above, depending on the ratio of the Schmid factors in these two 9 o
Fe-24.Sat %At Fe-25.8at %AI
200 Tc
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er U
loo
_.
50
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" 600
' 700
~ 800
Temperature
. . 900 /
.
.
30*
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Fig. 39. Temperature dependence of the critical resolved shear stress and corresponding operative slip plane ({ 110} for qo = 0) of Fe - 24.8 at.% A1 (A) and Fe - 25.8 at.% A1 (A I) crystals. Tc is the D03-B2 transition temperature. From Hanada et al. [149].
w
Dislocation cores and yield stress anomalies
121
planes. Glide in { 112) is also sometimes reported below the peak [151]. A minimum value of the stress-strain rate sensitivity has been measured by Schr6er et al. in the anomalous domain (fig. 40) [150]. Deformation tests have been performed at room temperature after annealing at different temperatures and quenching, in order to test the influence of the degree of long-range order. The results are inconsistent, since Stoloff and Davies [147] found a strong dependence of room temperature yield stress on the degree of long-range order (deformation stress is higher when the degree of long range order is lower) whereas Schr6er et al. [ 151 ] only found a very weak dependence. Post-mortem observations reveal a high density of rectilinear screw dislocations after deformation at room temperature, below the anomalous regime [150-152]. Dislocation pairs are mostly present in crystals deformed below the transition temperature instead of the expected double pairs [153, 154] (cf. section 8.4.2). In fact, according to Morris et al. [154], double pairs are already activated at yield, but they are rapidly uncoupled upon subsequent deformation. This can be explained in most cases by the low values of the APB energy corresponding to the D03 order (Crawford and Ray [155]). Double pairs have been however reported by Schr~3er after deformation just below the temperature range of yield stress anomaly. The first in-situ experiments have been performed at room temperature by Kubin et al. [156]. They indicate that the deformation is controlled by the glide of pairs of rectilinear screw dislocations in { 110} planes. More recent in-situ experiments provide further information on glide processes in FeToA130 in the temperature range of the stress anomaly [157]: groups of paired dislocations glide very rapidly over large distances in (110} planes until they lock themselves along the screw orientation and sometimes cross-slip and glide steadily in a { 112} plane. For the lower temperature 300 w
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_
0.075
c ffj
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IT" ~
-
o I--
100 --
O
m
.m
-- O.025
~
m
0'
I 2O0
300
'
I
400
'
I
500
"
I
600
Temperature
'--i-
700
'
I
800
'
I
900
0,000
'
1000
[K]
Fig. 40. Critical resolved shear stress (open symbols) and strain-rate sensitivity of Fe - 30 at.% AI as a function of temperature. SR is the ratio of the Schmid factors for slip in {112} and { 110} planes. From SchrOer et al. [150].
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range of negative temperature-stress dependence the deformation is also controlled by the movement of rectilinear screw dislocations, however with a more steady motion [157]. These observations indicate that deformation is controlled by the movement of screw dislocations subjected to Peierls stresses. Several models have been proposed for the yield stress anomaly. The first is the model of Stoloff and Davies [ 147], already put forward for explaining mechanical properties of FeCo and Ag2MgZn. This model is consistent with yield stress measurements in quenched samples only when there is a strong influence of the quenching temperature on the low temperature yield stress. A modified model has been proposed by Schr6er et al. [ 150, 151 ], based on the observation of different dislocation structures at different temperatures. The yield stress anomaly no longer arises from a variation of the degree of long range order with temperature. The normal negative temperature-stress dependence at low temperatures is interpreted in terms of a Peierls mechanism acting on pairs. The minimum stress corresponds to both the vanishing of the Peierls stress and the presence of highly mobile double pairs. The yield stress anomaly then results from the uncoupling of double pairs by thermally activated cross-slip, in such a manner that an increasing density of dislocation pairs glide in a D03 crystal (the authors assume here that the movement of dislocations does not depend on the degree of long-range order, in agreement with their experimental results). This explanation is clearly at variance from the explanation of Stoloff and Davies which relies on the glide process of double pairs in a more and more disordered D03 (i.e., closer to B2) crystal. These two models however do not explain the peak temperatures above the transition temperature and the orientation effects observed by Hanada et al. [149]. In addition, in-situ experiments indicate that Peierls stresses are important at all temperatures. A model similar to that proposed by Umakoshi et al. in/3-CuZn may thus also be considered [149], although this model encounters serious difficulties.
8.4.4. CoTi, CoZn, CoHf These alloys are different from the preceding because they maintain their B2 structure up to the melting temperature. They exhibit a strong anomalous behaviour (Nakamura and Sakka [158], Takasugi and Izumi [159]). Deformation experiments on single crystals reveal orientation effects in the temperature domain of the yield stress anomaly and at high temperatures above the peak, and no orientation effect at low temperatures (fig. 41). The flow stress-strain rate sensitivity is low at low temperatures and high at high temperatures. It is however either low or high in the temperature domain of the anomaly, depending on the crystal orientation (Takasugi et al. [160]). Strain hardening exhibits the same anomalous behaviour as the yield stress, and jerky flow is observed in the anomalous temperature range in Co(Ti, Zr) and (Co, Ni)Ti compounds [161]. Slip line and microstructural observations all reveal that (100) {110} glide systems are activated in the whole temperature range, with some indication of glide of (110) dislocations at high temperature for a (001) straining axis. These dislocations are not seen to be dissociated. They exhibit a strong directionality, sometimes with a rectangular shape, especially below the peak. These directions, which are often different from the screw and edge orientations, are however not mentioned clearly (they may be (111) and /112)). Local pinning on screw dislocations is also reported in CoTi deformed in the anomalous temperature range [ 162, 163].
w
123
Dislocation cores and yield stress anomalies i
I
i
i
8O0
_
I
i
I
I
I
"
I
I
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i
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1
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*,
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.
0
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.
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I I i. t ~ 0 200 400 600 800 1000 1200 0
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200
,
i
i
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200 400 600 800 1000 1200 Temperature / K
0
I 0
o,.
200 400 600 800 1000 1200
(b)
(c)
Fig. 41. Temperature dependence of the yield stress of a CoTi alloy, for two strain rates and three orientations. K denotes deformation by kinks. From Takasugi et al. [160].
It is concluded that Peierls forces acting on different crystallographic directions of dislocations in { 110} planes play an important role in the plastic properties of these alloys. These Peierls forces may be connected to the non-planar core configurations computed by Farkas et al. [164]. Models for the yield stress anomaly which are specific to superdislocations, such as the dragging of faults in a more or less disordered material or cross-slip pinning as in L12 alloys obviously do not apply, since (100) dislocations are ordinary dislocations similar to those which could be found in disordered crystals. On the basis of the relationship between the anomalous behaviour and the stability of the B2 phase in several alloys belonging to the same category, Takasugi et al. [165] suggested that the stress anomaly may originate from glissile-sessile (planar-non-planar) transitions associated with a lower stability of the B2 phase.
8.5. Discussion of the yield stress anomalies in B2 compounds A comparison between experimental results and interpretations in b.c.c, based intermetallic compounds is somewhat confusing. As a matter of fact, it is quite impossible to find a single correlation between the different situations treated above. It can be seen that the anomalies are not limited to a particular glide system. They are found either in the cases of (111} superdislocations or in the case of (100) ordinary dislocations (cases of CoTi and CuZn). In addition, in the case of (111) dislocations, anomalies are found either in { 110} or {112} glide planes (CuZn and Ag2MgZn). There is no common property with respect to orientation effects (strong/weak), the flow stressstrain rate dependence (high/low). Instabilities are sometimes reported in the anomalous domain. Yield stress anomalies are in some cases correlated with B2-disordered and D03-B2 transitions. However, the peak temperature may be higher than the corresponding critical
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temperatures (Fe3A1). On the other hand, there is no phase transition in CoTi which could explain the anomalous behaviour. Lastly, microstructural observations are not as systematic as in L12 alloys. The directionality of dislocations is sometimes weak, indicating that Peierls stresses may be lower than in TiA1 and Ni3A1.
9. General discussion 9.1. General considerations on yield stress anomalies [129, 166] The elastic limit of crystals is bound to both the nature of the obstacles opposing dislocation movements and the way in which these obstacles are crossed in order to propagate mobile dislocations over long distances. Since the crossing mechanisms can only be either athermal or thermally activated, that is, they cannot become more difficult at increasing temperatures, yield stress anomalies necessarily arise from an increase in obstacle efficiency. Obviously, this increase must overcome the normal effect of thermal activation on the crossing process, if any. Increasing the frequency of the successive dislocation-obstacle interactions with increasing temperature is almost inefficient, since it only requires shorter waiting times for unlocking, which is equivalent to an increase of the strain rate for a constant density of obstacles. The only aspect to consider is thus the single dislocation-obstacle interaction. To discuss models for the yield stress anomaly, it is sometimes helpful to consider the two topologically different situations, illustrated schematically in fig. 42. If the obstacles are localized or discontinuous (fig. 42(a)), they can be by-passed by mobile dislocation segments bowing in the free areas. The efficiency of the obstacles is increased either by increasing their strength or by decreasing their separation, the latter situation being considered more often. On the contrary, if the obstacles constitute a continuous front
(a)
Cb) Fig. 42. Two possible topologies of dislocation-obstacle interactions: (a) By-passing and cutting; (b) Cutting. From Caillard et al. [166].
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Dislocation cores and yield stress anomalies
125
opposing dislocation movements (fig. 42(b)), they are necessarily cut and yield stress anomalies can only arise from an increase in their strength. The experimental results are now compared, and the possible origins of stress anomalies are then compared with reference to the above discussion of cutting and by-passing mechanisms.
9.2.
Comparison
of experimental
results
From the analysis of yield stress anomalies in different pure metals, alloys and intermetallic alloys, it is evident that many different situations can be found, which leads to the following conclusions: - Yield stress anomalies are not specific to superdislocations in intermetallics. They are observed in pure metals (Be), disordered alloys (steel), and in several intermetallics deforming by glide of ordinary dislocations (TiAI,/3-CuZn, CoTi). - Yield stress anomalies can be observed in several slip planes for one given Burgers vector in a given material ((110) dislocations in { 111 } and { 100} planes of -y', (111) dislocations in {110} and {112) planes of/3-CuZn). Anomalies can also be observed for different Burgers vectors in one material ((111) and (100) in/3-CuZn). - Anomalies are sometimes correlated with orientation effects and/or tension-compression asymmetries (Ni3A1, C03Ti, TiA1,/3-CuZn, Fe3A1, CoTi). Orientation effects are however not observed in Cu3Au and Ag2MgZn. - Anomalies are correlated with many different dislocation microstructures including planar and non-planar dislocations, with no directionality or strong directionalities. - Yield stress anomalies are often observed in correlation with a low flow stress-strain rate dependence (Be, Ni3A1, C03Ti, TiAI, Fe3A1, CoTi along some orientations) or a negative stress-strain rate dependence (A13Ti). Exceptions to this rule are however Ni3(Si, 11 at.% Ti) [44] and CoTi (several orientations). In several cases, yield stress anomalies and low values of the stress-strain rate dependence are correlated with jerky flow and/or glide of dislocations by avalanches (cube glide in -),', octahedral glide in Ni3A1, TiA1, A13Ti, { 110} glide in Fe3A1 and/3-CuZn). - Anomalies are sometimes correlated with an increase in the strain-hardening coefficient (Ni3A1, TiA1 for high Schmid factors on ordinary dislocations). No anomalous strain hardening is observed in TiA1 when superdislocations have the highest Schmid factor. - Y i e l d stress anomalies in intermetallics are in some cases correlated with orderdisorder transitions (Cu3Au, FeCo, Ag2MgZn, Fe3A1). This behaviour is however at variance with that of/3-CuZn, where the order-disorder temperature transition is much higher than the peak temperature. In addition, the peak temperature is found either below or above the transition temperature in Fe3A1. Except for the low values of the flow stress-strain rate sensitivity, the jerky flow and the dislocation movements by avalanches which yield indication of dynamic dislocation processes in almost all cases, experimental observations are thought to give no guiding idea about a possible common interpretation of the anomalies. It is however possible to
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classify tentatively the different situations investigated into three categories, with several important common features and possible interpretations. - Ni3A1, Co3Ti, TiA1 (superdislocations): anomalies in these materials are clearly related to APB cross-slip, i.e., to the cross slip of superdislocations from the primary { 111 } glide plane onto the { 100} plane (these dislocations have been referred as OA type in [ 129]). The frequency or the extent of the cross slip process accounts for the increase in the yield stress as the temperature is increased, via several detailed models. This explanation may hold true for/3-CuZn since cross slip from {110} onto {112} may also be possible even if the torque force is less efficient than in L12 crystals. This conclusion disagrees, however, with the observation of anomalous behaviour in both {110} and {112} planes. The same remark may also be valid for Fe3A1 which has many common features with ~3-CuZn. Since superdislocations are expected to have their lowest energy in the cube plane, this explanation does not hold for cube slip in "1t' and Pt3A1. - A13Ti, steel: anomalies in these materials are clearly related to stress instabilities and the Portevin-Le Chatelier effect. No APB cross slip takes place and no Peierls stress has been observed in this case (which is natural in steel where ordinary 89 dislocations glide freely in { 111 } planes). - Others: anomalies in these cases are more difficult to interpret. APB cross slip is either irrelevant or unlikely to be important. However, evidence of Peierls stresses and/or orientation effects is always found, which indicates that dislocations may have a nonplanar core structure. Dislocations in this case have been referred to as PM type (pure metal type) in [129]. This category includes: Dislocations in pure metals (Be); Ordinary dislocations in intermetallics ( 89(110] dislocations in TiA1, (100) dislocations in fl-CuZn and CoTi); Superdislocations with their APB in the glide plane, such as (110) dislocations in cube planes of 7' and (probably) Pt3A1. (111) screw dislocations gliding in {110} and {112} planes of/3-CuZn 2, Ag2MgZn, and Fe3A1 may also have a planar APB, since the torque which tends to induce cross slip into another plane is less efficient than in L 12 crystals [ 112, 145, 164]. This hypothesis is in accordance with the similarity between CRSS values in {110} and {112} planes of Ag2MgZn and /3-CuZn, since in that case the mobility of superdislocations would be controlled by the mobility of individual superpartials (Yoo, [112]).
9.3. S t r e s s a n o m a l i e s in c a s e o f A P B c r o s s - s l i p
(Octahedral glide in Ni3Al, other related nickel-based alloys and C03Ti, glide of superdislocations in TiAl, possibly (111) { 110} glide in fl-CuZn and Fe3AI.) All the models developed for Ni3A1 can apply in this case. They are based on the thermally activated locking of superdislocations by cross slip from the primary plane 2provided climb dissociation is not rate controlling. (111) glide in /3-CuZn could thus be out of this classification.
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Dislocation cores and yieM stress anomalies
(octahedral or {110)) onto another plane (cube or {112}), leading to an APB partly or completely out of the primary plane. The models which are based on the hardening process due to an increase of the total dislocation density (Greenberg, Thornton) can be ruled out on the basis of room temperature deformation experiments after predeformation at high temperature. The model of dislocation bending in cube planes of NiaA1 is now considered by its authors to be a contribution to strain-hardening rather than a model for the yield stress anomaly [94]. In addition, this model cannot be transposed to TiA1 where long-range cube glide is inhibited. The other models all assume that locked screw dislocations can be released. They can be classified into two groups according to the topology of the obstacles discussed in section 9.1.
extension of locking
vF
---.-i,
l'v, intrinsic unlocking
~s
by-passing
initiation of locking
-
(local pinning models)
~v ,
~' extension of locking
~
"~'.~ movement of a macrokink
1 "~
~
intrinsic unlocking (double cross-slip models)
I A by-passing (kink models)
Fig. 43. By-passing and intrinsic unlocking (cutting) mechanisms leading to several types of models for the yield stress anomaly. From Caillard et al. [129].
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- The cross-slip pinning models [81, 82, 85, 87-89, 91, 92] and the kink models [83, 93] are by-passing mechanisms in nature. In both cases, the yield stress is related directly to the length of the free dislocation segments which are limited by locked screw segments. Only the character of the free dislocation segments (screw and nonscrew respectively) differ in these two models. Accordingly, the yield stress anomaly originates from a decrease in the separation between obstacles as the temperature is increased. - The double cross-slip mechanisms is on the contrary a cutting mechanism, for which the yield stress anomaly is a consequence of an increase in the obstacle strength as the temperature is increased [15, 95, 97]. As discussed by Hirsch [79] and Caillard et al. [129] the type of dislocation-obstacle interaction in Ni3A1 (by-passing/cutting) depends in reality only on the respective velocities of free dislocation segments and glissile-sessile transitions along screw dislocations (fig. 43). The extension of locking along a screw dislocation is thought to be a very fast process, and by-passing may not take place unless kinks meet extrinsic obstacles so that v' < VF (fig. 43). This is the reason why Hirsch introduces pinning points formed by cross-slip. Whether large amounts of pinning points are really present or not could not be checked unambiguously, however. This discussion also shows that the calculations of Chzran and Mills [91, 92] must take into account the extension of locking along screw dislocations, with reasonable velocities, in order to be more realistic. Many points remain however unclear regarding hardening effects, and the origin of the low flow stress-strain rate sensitivity (although it may be correlated to jerky flow).
9.4. Stress a n o m a l i e s in case of diffusion-controlled frictional forces
(A13Ti, steel) Dislocations are here assumed to be slowed down by diffusion controlled mechanisms, such as due to relaxation of APBs, climb dissociation, and core interaction with solute atoms. An equilibrium configuration is progressively attained after some temperaturedependent relaxation time. It is important to note that many dislocation characters are subject to these mechanisms, which results in an isotropic dislocation substructure different from that caused by a Peierls mechanism. Three domains can be defined [6, 107, 108]: dislocation velocities, low temperatures: diffusion is too slow to modify the structure of mobile dislocations, and the temperature dependence of the flow stress is negative. - Low dislocation velocities, high temperatures: diffusion is so fast that even mobile dislocations have reached their equilibrium configurations. The temperature dependence of the flow stress is also negative. - Intermediate dislocation velocities and temperatures: the speed of diffusion is comparable to the velocity of dislocations, in such a way that dislocations move easier at
-High
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Dish)cation cores and yield stress anomalies
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a higher velocity, whence negative stress-velocity dependence. At a given strain rate, only a very low density of dislocations is gliding very rapidly at the same time, which leads in some cases to jerky flow and Portevin-Le Chatelier effects. These last two properties account for low or negative values of the flow stress-strain rate dependence. The state of mobile dislocations nevertheless varies with temperature which results in a yield stress anomaly. Such an increase in the obstacle strength is consistent with a cutting mechanism, according to section 9.1.
9.5. Yield stress anomalies in case of Peieris type frictional forces
(Dislocations in pure metals: Be; ordinary dislocations." 89 (110) in TiAl, (100) in/3-CuZn and CoTi; superdislocations with planar APBs: (110) in cube planes of .),1 and Pt3AI, possibly (111) superdislocations in {110} and {112} planes of t3-CuZn and ag2MgZn, possibly (111) superdislocations in (110} planes of Fe3al.) Since dislocation substructures have a tendency to lie along crystallographic directions (most often the screw orientation) a, it can be concluded that dislocations have a minimum velocity along these directions, i.e., the strain rate is controlled by the mobility of those dislocation segments which are subjected to Peierls frictional stresses. For that reason, no mechanism based on the trailing of order faults can be rate-controlling. For dislocations in pure metals and ordinary dislocations in intermetallics, Peierls forces probably originate from a non-planar core structure, as in b.c.c, metals (this conclusion has been shown to be the most probable in TiA1). In the same way, since APBs are expected to be planar 4, Peierls forces on superdislocations are expected to originate from a non-planar core structure of superpartials. As in cross-slip models (section 9.3), several explanations can be proposed for the origin of stress anomalies. By-passing mechanisms may be activated, although they have never been proposed for Peierls-type mechanisms. For instance, the kink model may be transposed to this situation provided metastable glissile configurations exist along the screw orientation (as also assumed in the locking-unlocking model). Cutting mechanisms correspond either to the more classical Peierls mechanism, i.e., the nucleation and the propagation of kink pairs, or to the locking-unlocking mechanism, described for beryllium. In these latter cases, however, it is necessary that the strength of the Peierls forces, i.e., the depth of the Peierls valleys, increases with increasing temperature. Such a variation of the Peierls stress has been measured by an indirect method in beryllium. An explanation for the increase in the Peierls stress with increasing temperature has been proposed for beryllium. Another mechanism may apply to all the glide systems considered here. It is known that dislocation core structures responsible for Peierls stresses are very sensitive to material purity. For instance, it was shown that the Peierls stresses which control prismatic slip in titanium are much higher when the oxygen content is higher [126, 167, 168]. It is thus possible that the Peierls stresses are increased by the chemical interaction between dislocation cores and diffusing solute atoms or impurities, 3fl-CuZn is an exception since dislocations exhibit no directionality. Orientation effects however indicate that frictional stresses originating from non-planar dislocation cores could be important. 4Except may be for non-screw (111) dislocations in 3-CuZn for which climb dissociation also takes place.
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at increasing temperatures. Even if all dislocation characters are concerned with such a diffusion process, the hardening effect could be larger on dislocations subjected to Peierls stresses. This hardening effect is thus expected to be higher than the classical hardening effect due to the direct interaction between all dislocation segments and solute atoms, which is the origin of the yield stress anomalies in steel and A13Ti (section 9.4.). This mechanism may induce processes similar to those described section 9.4, however with dislocations lying along crystallographic directions. It may thus explain substructure observations, jerky flow and avalanches of dislocation movements.
9.6. Comparison of the mechanisms proposed Three types of explanation are thus proposed for explaining yield stress anomalies in these different cases, on the basis of cross-slip leading to non-planar APBs, diffusion controlled frictional forces, and Peierls-type frictional forces, respectively. One common feature of yield stress anomalies in all the situations investigated is their correlation with jerky flow and/or avalanches of dislocation movements. Jerky flow and avalanches are macroscopic and microscopic aspects of the same physical processes, although microscopic instabilities (avalanches of dislocation movements) may not always be strong enough to give rise to macroscopic instabilities (jerky flow). Plastic instabilities can explain easily low values of flow stress-strain rate sensitivities, since strain-rate changes can be accommodated by a change in the density of mobile dislocations instead of a change in the dislocation velocity. Stress instabilities have the same origin in mechanisms based on diffusion-controlled frictional forces, and diffusion-enhanced Peierls forces. The origin of flow stress instabilities is however still unclear in cross-slip mechanisms leading to non-planar APBs. Analogies can also be found between screw dislocations blocked by cross-slip leading to a non-planar APB and dislocations blocked in Peierls valleys. In both cases, dislocations have a non-planar structure, and superdislocations with their APB out of the glide plane can be considered as lying in a peculiar type of Peierls valley. Accordingly, glide mechanisms may have the same origin, based either on by-passing or cutting processes. In case of a cutting mechanism, however, it is important to note that the origins of the increase in the locking strength could be different, controlled by cross-slip and diffusion mechanisms, respectively. In the case of locking by cross-slip leading to a non-planar APB, the increase in the locking strength may indeed be due to an increase in the extent of cross-slip into the cube plane leading to an APB partly in the octahedral plane and partly in the cube plane, whereas in the case of locking by a Peierls mechanism, the increase in locking strength may result from a chemical interaction between diffusing solute atoms or impurities and dislocations. Lastly, different dislocation mechanisms may take place simultaneously. For instance, diffusional effects onto APBs may be important in the cross-slip processes leading to non-planar APBs at high temperatures.
Dish)cation cores and yield stress anomalies
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CHAPTER 51
Anomalous Yield Behaviour of Compounds with L 12 Structure V. VITEK,
D. R POPE
Department of Materials Science and Engineering University of Pennsylvania Philadelphia, PA 19104-6272 USA and
J. L. BASSANI Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania Philadelphia, PA 19104-6272 USA
Dislocations in Solids 0 1996 Elsevier Science B. V All rights reserved
Edited by E R. N. Nabarro and M. S. Duesbery
Contents 1. 2. 3.
Introduction 137 Characteristics of the anomalous regime 139 Dislocation dissociation and dislocation core structure 144 3.1. Anti-phase domain boundaries and stacking faults 144 3.2. Dislocation dissociation 146 3.3. Core structure of (101 } screw dislocations 147 4. Theory of the yield anomaly 150 4.1. The Takeuchi-Kuramoto model 150 4.2. The PPV model 154 4.3. Strain-rate effects 163 5. Continuum theory of crystal plasticity of L12 compounds in the anomalous regime 170 5.1. Flow behaviour and the Schmid stress 170 5.2. Yield criterion with non-Schmid stresses and hardening 172 5.3. Tension/compression asymmetry and its orientation dependence 173 5.4. Restricted slip and polycrystalline response 175 5.5. Shear band formation 176 6. Conclusions 178 References 182
1. Introduction Ordered alloys and compounds with A3B compositions crystallize in a number of different structures (Westbrook [1]) but the mechanical behaviour of alloys possessing the L12 (Cu3Au-type) structure which has a cubic symmetry, has been studied most extensively. This is an f.c.c.-derivative structure, the unit cell of which contains four atoms, with A atoms occupying face-centered positions and B atoms corner positions. It can be viewed as resulting from the stacking of three close-packed { 111 } planes in the usual f.c.c, stacking sequence. Even though the slip directions ((110)) as well as principal slip planes ({111}) are the same as in f.c.c, metals, significant differences between the mechanical behaviour of L12 alloys and f.c.c, metals arise due to chemical ordering. In this chapter we concentrate on the unusual temperature dependence of the yield strength observed in a number of L12 compounds. Of the many L I2 compounds, Ni3A1 (3't) has been investigated most frequently because it is of both technological and scientific interest. This alloy is the most important strengthening constituent in commercial nickel-base superalloys and is responsible for the high-temperature strength and creep resistance of those alloys. At the same time it exhibits a number of surprising mechanical properties, very different from disordered alloys. The most distinguished and well-known is the increase of the yield strength (rather than the more common decrease) with increasing temperature (Westbrook [2], Flinn [3], Davies and Stoloff [4], Copley and Kear [5]; for recent reviews see Pope and Ezz [6], Pope and Vitek [7], Pope [8, 9], Dimiduk [ 10]). An important accompanying phenomenon is a strong orientation dependence of the yield stress and tension-compression asymmetry (Takeuchi and Kuramoto [11, 12], Lall et al. [13], Ezz et al. [14], Umakoshi et al. [15]) so that the Schmid law (Schmid [16], Schmid and Boas [17]) is not obeyed (Qin and Bassani [18]). The strength reaches a maximum around 500 to 900~ above which Ni3A1 shows a change in slip systems from {111}(110) to {001}(110), except for special orientations, and the strength then rapidly decreases. This phenomenon has come to be known in the literature as the "yield anomaly". This anomalous yield behaviour has been observed in a number of L12 compounds (Suzuki et al. [19], Pope [8, 9], Dimiduk [10]), but Ni3A1 is the prototypical material and thus in this chapter we concentrate on this alloy. However, the yield anomaly is not a ubiquitous feature of the mechanical behaviour of compounds with L12 structure. A number of such compounds display the "normal" behaviour, i.e. the yield stress increases with decreasing temperature and either no increase or only a very small increase of the yield stress occurs at high temperatures. Examples of such materials are Pt3A1 (Wee et al. [20, 21], Heredia et al. [22]), A13Sc (Schneibel and Hazzledine [23]) and alloyed A13Ti in the L12 form (Wu et al. [24-26], Yamaguchi et al. [27]). This suggests that the reasons for the unusual yielding properties of alloys such as Ni3A1 cannot be sought in the crystal structure alone and it has, indeed, been proposed that the ordering energy
138
V. Vitek et aL
Ch. 51
plays the major role (Vitek et al. [28, 29]). This point will also be discussed briefly in this chapter. Since the discovery of the anomalous temperature dependence of the yield stress in Ni3A1, a number of reasons for this phenomenon have been put forward. Some of the first explanations were based on changes in long range order, following Brown's [30] explanation of the anomalous yield behaviour in/3-brass as being related to such changes. This can, indeed, explain the anomalous increase of the yield stress observed in L12 Cu3Au where the temperature at the peak of the yield stress is very close to the order-disorder transition temperature (Pope [31]). However, Ni3A1 remains ordered up to melting and therefore changes in order cannot explain the phenomenon. An additional important feature of the yield anomaly, which needs to be explained at the same time, is the orientation dependence of the yield stress - something that is absent in the case of Cu3Au (Kuramoto and Pope [32]). The models which most successfully explain these phenomena are based on the assumption that individual dislocations carrying the plastic deformation change their splitting and/or core structure in response to changes in temperature. The first such suggestion was made by Flinn [3] who hypothesized that at high temperatures dislocations, originally gliding on {111} planes, climb into {001} planes and thus become immobile. The driving force for this transition is that the energy of the anti-phase boundary (APB)is lower on {001 } planes than on {111} planes. While control by diffusive mechanisms is not plausible, since the strain rate sensitivity of the flow stress in the anomalous regime is very small (Davies and Stoloff [4], Thornton et al. [33], Umakoshi et al. [15], Miura et al. [34], Bonneville and Martin [35], Sp~itig et al. [36]), the idea that immobilization of dislocations is associated with different energies of APB's on {111} and {001 } planes remains important. It was, in fact, the underlying motive for the suggestion that the increase of the yield stress with increasing temperature occurs owing to the cross slip of screw dislocations from { 111 } planes, where they are mobile, into {001} planes where they become immobilized (Kear and Wilsdorf [37], Thornton et al. [33]). Such a cross slip is, indeed, commonly observed and leads to the formation of Kear-Wilsdorf locks (see the chapter by Sun and Hazzledine in this volume). The cross-slip induced immobilization of screw dislocations is then the underlying concept which led to the development of more recent quantitative models of the yield anomaly in L12 compounds. The first such model was proposed by Takeuchi and Kuramoto [12] who suggested that the increase of the yield stress is proportional to the density of the dislocation segments that have cross-slipped into the {001 } plane and which then provide local pinning points that hinder the motion of the rest of the dislocation. The production of such pinning points is thermally activated. This model was then significantly advanced by Paidar, Pope and Vitek [38] who demonstrated that the observed orientation dependence of the yield stress is a natural consequence of the pinning transformations of screw dislocations. This model, commonly referred to as the PPV model, is discussed in detail later in this chapter. An important feature of this model is that while it is based on the above-mentioned concept of the cross slip of screw dislocations, it does not involve a true transition of the dislocation into the cross slip plane. Rather, the transition involved is a transformation of the core of screw dislocations from a glissile form into a sessile form which is akin to cross slip, analogously as the transformation of sessile screw dislocations in b.c.c, metals into glissile ones (Vitek [39, 40],
w
Anomalous yield behaviour of compounds with L12 structure
139
Duesbery [41 ]). The corresponding core structures have been revealed by atomistic calculations (Paidar et al. [42], Yamaguchi et al. [43], Vitek et al. [28, 29, 44]) the results of which are also reviewed in this chapter. More recently the PPV model was extended to include strain rate effects (Khantha et al. [45-48]) and employed in the development of constitutive relations needed in continuum crystal plasticity which include the corresponding non-Schmid characteristics of the slip in L12 alloys in the anomalous regime (Qin and Bassani [18, 49], Bassani [50]). The continuum theory predicts that, due to non-Schmid effects, polycrystalline ordered alloys are harder than corresponding disordered alloys and tendency to strain localization is increased. These latest developments are also discussed in this chapter.
2. Characteristics o f the anomalous regime In this section we summarize briefly the important attributes of the deformation behaviour of L 12 alloys in the regime of the yield anomaly. The experimental work devoted to this phenomenon in the last two decades is very extensive and we do not attempt to review it here in detail. Rather, we outline the distinguishing features which have to be elucidated by any comprehensive theoretical description of the yield anomaly. The following are the most important characteristics of the anomalous yield behaviour found in a wide variety of experimental studies" (1) The yield and/or flow stress increases with increasing temperature (anomalous behaviour) until a peak is reached ("peak temperature"), after which the flow stress steadily decreases. In Ni3A1 the anomalous regime extends from room temperature up to a peak temperature, To ~ 800-1200 K. This phenomenon was first observed in hardness measurements by Westbrook (Westbrook [2]) and in yield stress measurements by Flinn (Flinn [3]) and since then studied by many researchers. Flinn's results are shown in fig. 1. (2) The flow stress depends only on temperature and not on the sequence of test temperatures employed. In other words, if a sample is first deformed at a high temperature where the yield stress is high and then deformed again at a lower temperature, its yield stress is practically the same as that of a virgin sample deformed at this lower temperature. A small difference in the yield stresses of these two samples is, of course, observed but this is due to the usual work hardening occurring during deformation (Davies and Stoloff [4], Dimiduk [10]). (3) Slip occurs on the { 111 }(101) system below the peak temperature (Staton-Bevan and Rawlings [51, 52]). (4) Above the peak temperature slip occurs predominantly on the {001}/101 ) system [51] except for samples with the tensile/compressive axis oriented near [001 ], which continue to slip on (111)[101] (Umakoshi et al. [15]). (5) A remarkable feature of the dislocation substructure observed in the anomalous region is the predominance of long (101) screw dislocations (Kear and Hornbecker [53], Staton-Bevan and Rawlings [51, 52]). These dislocations are usually immobile KearWilsdorf locks [37] resulting from the cross slip from { 111 } onto {001 ) planes (see also the chapter by Sun and Hazzledine in this volume). E
140
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(6) In the anomalous regime the critical resolved shear stress (CRSS) depends strongly on the orientation of the tensile (or compression) axis (Takeuchi and Kuramoto [ 11, 12]) (see fig. 2) and on the sense of the applied stress. The latter is manifested by a remarkable tension-compression asymmetry (Lall et al. [13], Ezz et al. [14], Umakoshi et al. [15], Heredia and Pope [54-56]) (see fig. 3). This means that the non-glide components of the stress tensor play an important role, i.e. the yield stress in the anomalous regime does not obey the Schmid law (Schmid [16]; Schmid and Boas [17]) and exhibits so called "non-Schmid" behaviour. (7) The strain-rate sensitivity,/3 - ( ~ ) (~ is the strain rate and a the flow stress), is positive but very small below the peak temperature, commonly less than 1% for an order of magnitude change in the strain-rate (Thornton et al. [33], Takeuchi and Kuramoto [ 12], Miura et al. [34], Bonneville and Martin [35], Sp~itig et al. [36]) (see fig. 4). However, a strong positive strain-rate sensitivity is observed at temperatures higher than the peak temperature (Umakoshi et al. [15]). In the anomalous regime the activation volume V * = lr measured in Ni3(A1, Ta) using stress relaxation experiments exhibits a discontinuity at a temperature, Tc, well below the peak temperature (Baluc et al. [57], Bonneville et al. [58], Bonneville and Martin [35], Spfitig et al. [36, 59]) (see also
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Anomalous yield behaviour of compounds with L12 structure
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fig. 20). However, the same discontinuity has not been observed in Ni3(A1, Hf) alloys (Sptitig et al. [60], Ezz and Hirsch [61], Bonneville et al. [62], Ezz et al. [63]). (8) No anomaly occurs in the microstrain range, at offset strains of 10-6-10 -5, but the anomaly is fully developed for the offset strain ..~ 10 -3 (see fig. 5) (Thornton et al. [33], Mulford and Pope [64], Ezz and Hirsch [65]). (9) While the contribution of the work-hardening to the increase of the yield stress is small compared to the anomaly, the work hardening rate also exhibits an unusual behaviour in the anomalous regime. At strains ~ 1.5%, the work-hardening rate first increases with increasing temperature, reaches a peak at temperatures close to half of Tp (500 K for Ni3A1 for which Tp ,.~ 1000 K) and then decreases with increasing temperature (Staton-Bevan [66]). However, at low strains (less than 0.2%) the work-hardening rate increases steadily with increasing temperature up to Tp (Ezz and Hirsch [65]). (10) Most of these properties are also seen in Ni-base superalloys (Heredia and Pope [54]). The conclusion which can be drawn from (2) is that the anomaly does not result from work hardening since if the opposite were true, a sample work-hardened at a high temperature would exhibit the yield stress of the same magnitude after cooling. Furthermore, it follows from (3) that diffusion-controlled processes do not play any role in the yield anomaly since otherwise the strain rate dependence would have to be much higher. Hence the yield anomaly is intrinsic to materials such as Ni3A1 and an explanation must be sought in those properties of individual dislocations carrying the plastic deformation which are strongly affected by the type of the crystal structure. It is now generally accepted that the core structure of the dislocations controls those deformation phenomena that cannot be ascribed to long-range interactions between dislocations and/or between
144
Ch. 51
V. Vitek et al. I00
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dislocations and other crystal defects (Vitek [40, 67], Veyssiere [68], Duesbery [41], Duesbery and Richardson [69]). Hence, in the following section we summarize our present understanding of the dislocation cores in L12 alloys. This includes the usual dislocation splitting into partial dislocations which we can regard as a special form of the dislocation core.
3. Dislocation dissociation and dislocation core structure 3.1. Anti-phase domain boundaries and stacking faults The distribution of atoms in the (111) planes is shown schematically in fig. 6. Three layers, marked successively A, B and C as in the f.c.c, case, are shown here. Small circles represent the majority atoms (e.g. Ni) and large circles the minority atoms (e.g. A1). Generally, it is assumed that three distinct metastable planar faults, whose displacement vectors are shown in fig. 6, can be formed on (111) planes. They are the APB with the displacement 71[]-01] , the complex stacking fault (CSF) with the displacement ~ []-]-2] and 1 [~11]. While the the superlattice intrinsic stacking fault (SISF) with the displacement _~ metastability of such faults has often been assumed a priori, it does not follow from the symmetry of the L12 lattice and needs to be assessed carefully. Stacking fault-type defects can, in general, be studied using the concept of the 7surface (Vitek [70]). We cut the crystal along the given crystal plane and rigidly displace
w
145
Anomalous yield behaviour of compounds with L12 structure
|
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Fig. 6. The distribution of atoms in the three adjacent (111) planes. the upper part with respect to the lower part by a vector t, parallel to the plane of the cut. The energy of such a fault, "/(t), can be evaluated, at least in principle, using the chosen description of atomic interactions. Repeating this calculation for various vectors t within the repeat cell of the crystal plane, we obtain an energy-displacement surface which is called the -,/-surface. The local minima on this surface determine possible metastable stacking faults on the crystal plane considered and certain faults can be anticipated using the crystal symmetry. In particular, the -)'-surface possesses extrema (minima, maxima or saddle points) for such displacements for which there are at least two non-parallel mirror planes in the ideal lattice; the first derivative of the "/-surface is then zero in two different directions (Yamaguchi et al. [71]). Whether any of these extrema correspond to minima, and thus metastable faults, can often be decided by considering the first nearest neighbour interactions, in particular possible overlaps of the neighbours. Hence, by analysing the symmetry properties of a ")'-surface one can assess, in advance, the possible existence of metastable stacking faults. Such stacking faults are then common to all materials crystallizing in a given structure. However, for a particular material other minima than those associated with symmetry-dictated extrema may exist, but these cannot be anticipated on crystallographic grounds. The existence of these extrema depends on the details of the atomic interactions. As seen in fig. 6, there is only one mirror plane for the displacement corresponding to the APB. It is parallel to the [121] direction and its trace is marked m l . Hence, the 89
146
V. Vitek et al.
Ch. 51
APB on the (111) plane may be, but need not be, a metastable fault depending on the details of atomic bonding. Furthermore, even when it is metastable, the corresponding displacement vector may not be exactly 89 []-01] but may possess a component parallel to the [121] direction. The same applies in the case of the 1 []--f2] CSF as there is also only one mirror plane for this displacement. It is parallel to the [112] direction and marked m3 in fig. 6. Thus, in addition to APB, the CSF may be, but need not be, metastable on (111) planes, depending on the details of atomic bonding. Indeed, both pair potentials and many body potentials were constructed for which the CSF and APB are unstable (Tichy et al. [72], Vitek et al. [44]). On the other hand for the displacement !3 [211] corresponding to the SISF, there are three mirror planes perpendicular to the (111) plane. Their traces, parallel to the directions [121], [211] and [112], respectively, are marked ml, m2 and m3 in fig. 6. The ")'-surface, therefore, must have an extremum for this displacement and since the separations and stoichiometry of the first and second nearest neighbours remain the same as in the perfect lattice, it is likely to be a minimum. Hence, the SISF is a metastable fault which may exist in any material crystallizing in the L 12 structure. For the {010} planes in the L12 structure the situation is very simple. There are two mirror planes perpendicular to the (010) plane and parallel to the [101] and [101] directions, respectively, for the displacement 89 [TO1]. This implies that the 7-surface must have an extremum for this displacement, and since the separations and stoichiometry of the first nearest neighbours remain the same as in the perfect lattice (Flinn [3]), it is likely to be a minimum. Hence, the l I-f01] APB on (010) planes is a metastable fault in any L 12 alloy.
3.2. D i s l o c a t i o n
dissociation !
Generally, it has been assumed that in L 12 alloys the [101] superdislocations may dissociate on both (111) and (010) planes according to the reaction [-1 0 1 ] - ~1[101] +
1
(3.1)
[101]
with an APB between the l [TO1] superpartials. However, as explained above, a metastable !2 []-01] APB can always be formed on {010} planes but not necessarily on {111} planes Hence, the above dissociation (3.1) may, but need not, be possible on (111) planes. Another common assumption is that 89 superpartials may further dissociate into []-]-2] and g1 [~11] Shockley partials separated by CSF on (111) planes. Again, the CSF may be, but need not be, metastable and, therefore, the dissociation into Shockley partials may, but need not, occur, depending on the material. On the other hand the SISF is always a possible metastable fault on (111) planes, and thus [101] superdislocation may always dissociate on these planes according to the reaction m
1
[-1 0 1 ] - ~1 [211] + ~ IT1-2].
(3.2)
w
147
Anomalous yield behaviour of compounds with L12 structure
This splitting will occur either when the APB on (111) planes is unstable or its energy is very high (Yamaguchi et al. [73]).
3.3. Core structure of (101) screw dislocations
Screw dislocations are frequently the most important dislocations when considering core effects. The reason is that they usually lie parallel to low index directions in which several low index crystallographic planes intersect and thus cross slip type spreading of the core into these planes can occur (Vitek [40]). Such cores are sessile and hinder the movement of screw dislocations which then control the deformation process owing to their low mobility. The most familiar examples are 89 (111) screw dislocations in b.c.c. materials, the core of which spreads into several {110} and {112} planes (Vitek [39], Duesbery [41]). In the L12 structure two {111} planes and one {010} plane intersect along any {110) direction, providing an opportunity for the screw dislocations to spread into several non-parallel crystallographic planes. Atomistic studies of the screw dislocation cores in L12 alloys were originally made using pair-potentials (Paidar et al. [42], Yamaguchi et al. [43]) and more recently employing central force many body potentials of the Finnis-Sinclair type (Finnis and Sinclair [74], Vitek et al. [28, 44], Vitek [40]) as well as the embedded atom method (Farkas and Savino [75], Yoo et al. [76], Pasianot et al. [77, 78]). The results of these calculations are very similar and here we summarize the findings of these studies by presenting the core structures calculated using the many-body potentials. Details of these potentials and corresponding atomistic studies can be found in [28, 40, 44]. The first set of many-body potentials, constructed to reproduce a number of equilibrium properties of Ni3A1, leads to metastable APB and CSF on the { 111 } plane with respective energy 189 and 226 m J m -2 and to metastable APB on {010} planes and SISF on {111} planes with energy 53 and 11.4 mJ m -2, respectively. The atomistic studies were carried out for [101] superdislocations dissociated according to (3.1), either on the (111) or on the (010) plane. Two alternate core configurations of the 1[]-01] superpartial, found when the APB is on the (111) plane, are shown in figs 7 (a) and (b). In this and the following e. - o. -. e. - .o .- e. -.o.- e. -. o. - e. .. o. - .e -. o. -.e .. o. -.o. .
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148
Ch. 51
V. Vitek et al.
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111) Fig. 8. The core configurations of the 1[]-01] superpartial found using the potentials for Ni3A1when the APB is on the (010) plane; the core is spread onto (111) plane. figures depicting the dislocation cores the atomic arrangement is shown in the projection onto the (101) plane. Small circles represent the Ni atoms, and large circles the A1 atoms. Two consecutive (101) planes are always shown and distinguished as darkly and lightly shaded circles. The [101] (screw) component of the relative displacement of the neighbouring atoms are represented by arrows, the length of which is proportional to the 1 []-01]1. The rows of arrows of magnitude of the displacement and normalized modulo 1~ constant length represent APBs and/or SISFs. The core shown in fig. 7(a) is planar, spread in the plane of the APB, and in this form the dislocation is glissile, while the core shown in fig. 7(b), is non-planar and extends not only in the (111) but also onto the (111) plane. The dislocation possessing the latter core configuration is sessile. Two symmetry related configurations were found when the APB is on the (010) plane with the core spread onto (111) and (111) plane, respectively. The latter configuration is shown in fig. 8. The core spreads onto the (111) plane and it is thus sessile. An equivalent core spread onto the (111) plane also exists and is sessile but no glissile configuration with the core spread onto the (010) plane was found. The common feature of all the four configurations is the spreading of the dislocation core into { 111 } planes. This core spreading can be depicted as dissociation into ] (112) type Shockley partials, but it must not be considered as the usual splitting, since the width of the core is of the order of three lattice spacings, too small to be regarded as well defined separate partial dislocations. When the sessile core structure shown in fig. 7(b) is compared with that displayed in fig. 8, a close resemblance of the core displacements is apparent. Therefore, this structure can be interpreted as consisting of a narrow strip of (010) APB, about 1~[101]1 wide (a is the lattice parameter), which connects the APB on the (111) plane with the dislocation spread onto the (111) plane. The assumption that such a sessile core configuration exists and is energetically favoured over the glissile configuration, is the basic assumption of m
Anomalous yieM behaviour of compounds with L12 structure
w
149
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Fig. 9. The core structure of the 1 [i01] superpartial terminating the APB on the (010) plane found using the model potentials for which the CSF on { 111 } planes is unstable.
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the model for the anomalous temperature dependence of the yield stress, in particular the PPV model, described below. The second set of many-body potentials was not constructed to describe any one specific material but rather a class of L 12 alloys with an unstable CSF. The APB on { 111 } planes is still metastable but its energy is so high that on these planes the super-dislocation 1 dissociates into .~(112)superpartials (3.2)rather than into 89 (3 1). On the (010) plane only dissociation (3.1) exists, as before, and the core structure of the ![T01] superpartial terminating an APB on this plane is shown in fig. 9. Its core is not 2 confined to the (010) plane but spreads simultaneously into both the (111) and (111) 1[~11] superpartial bounding planes and thus its core is sessile. The core structure of the .~ the SISF on the (111) plane is shown in fig. 10. Since this superpartial possesses an edge m
150
v. Viteket al.
Ch. 51 u
component, significant displacements perpendicular to the [101] direction exist in its core but only the screw component of the displacements is shown here. These displacements are spread simultaneously into the (111) plane, one layer above the plane of the SISE and to the (111) plane. The edge component of the displacements remains in the plane of the SISE Thus the core of this superpartial is again sessile and its screw component has practically the same form as the core of the 89 superpartial shown in fig. 9. These results suggest that there are L12 compounds in which no mobile screw dislocations exist. In these compounds APB on { 111 } plane is either unstable or possesses such a high energy that dissociation (3.2) is favoured over dissociation (3.1). The screw 1 superdislocations dissociated into g(l12) superpartials and gliding in such compounds on { 111 } planes will be equally difficult to move as the superdislocations dissociated into 89 superpartials on {001 } planes. The Peierls stress of such dislocations is very high but at non-zero temperatures the motion of sessile screw dislocations can be aided by thermal activation. This results in a strong temperature dependence of the yield stress at low temperatures. Such a temperature dependence of the yield stress has, indeed, been observed in Pt3A1 (Wee et al. [20, 21], Heredia et al. [22]), and recently in the L12modified A13Ti (Kumar and Pickens [79], Wu et al. [24-26], Kumar et al. [80], Kumar and Brown [81 ]). The plastic behaviour of L12 alloys of this type is not described in this chapter since we concentrate here only on those L 12 compounds which display the yield anomaly and, as explained below, this cannot occur in the alloys in which no glissile form of screw dislocations exists.
4. Theory o f the yield anomaly In the L 12 alloys displaying the yield anomaly the yield (flow stress) at low temperatures is much lower than at high temperatures and is practically temperature independent below room temperature (see figs 1-3). This implies that there must be fully mobile dislocations available in these materials at low temperatures. Since the slip system operating at these temperatures is { 111 }(101) the dislocations carrying the slip on the (111) plane are the [-101 ] superdislocations which dissociate according to (3.1) into two 1 [T01] superpartials separated by the APB as first proposed by Koehler and Seitz [82] and observed in the field ion microscope by Taunt and Ralph [83]. Each of the 1 IT01] superpartials either splits into two Shockley partials or, at least, their cores are planar, i.e. of the type shown in fig. 9(a). Hence, in these materials both the APB and CSF on { 111 } planes must be metastable. Otherwise no mobile screw dislocations would exist and the yield stress would be rapidly increasing with decreasing temperature.
4.1. The T a k e u c h i - K u r a m o t o m o d e l m
The first model employing a mechanism of immobilization of [101] screw dislocations at high temperatures was proposed by Takeuchi and Kuramoto [11, 12]. It is based on the original suggestion of Thornton et al. [33] that immobilization involves cross slip of screw dislocations from (111) to (010) planes, which is a variant of the work-hardening
w
Anomalous yield behaviour of compounds with L12 structure
151
Fig. 11. Schematic pictures showing the formation of the pinning points on the [101] dislocation moving on the (111) plane and its bowing-out around them.
theory of Kear and Wilsdorf [37]. The screw dislocations on (010) planes are, indeed, sessile owing to the spreading of their cores into (111) or (111) plane as demonstrated in the atomistic studies (see fig. 8). This cross slip process can be thermally activated and is therefore more frequent at higher temperatures. The driving force is provided by the anisotropy of the APB energy (Flinn [3]) and by the resolved shear stress (RSS) on (010)[ 101 ] system; using linear elasticity the splitting (3.1) is favoured on (010) planes, APB , where 7 APB and _ APB are the energies of APBs on {111} and provided 7~1]B > v Jfbl0 "~'010 {010 } planes, respectively. However, a permanent immobilization of dislocations would lead to a work-hardening type increase of the yield stress which would not be reversible upon decrease of temperature (see section 2, characteristic (2)). For this reason Takeuchi and Kuramoto [12] assumed that the cross slip occurs only locally along short segments of the dislocations, leading to the formation of pinning points. This is a reasonable assumption since these cross slips (in fact core transformations) occur on dislocations gliding on { 111 } planes and, as the transformed immobile part of the dislocation is being formed, the dislocation in the glissile form bows out around it away from the screw character. This prevents the transformation into the sessile form from spreading along the entire length of the dislocation and, therefore, only a small segment of the sessile dislocation, i.e. a pinning point, is formed. These pinning points then act as obstacles to the dislocation motion. This is shown schematically in fig. 11. An athermal release of the dislocations from the pinning points is then assumed to take place so that a steady state, corresponding to the creation of a constant average density of pinning points, is attained at a given temperature. Since the density of the pinning points increases with increasing temperature, a positive temperature dependence of the yield stress ensues. At the same time when the temperature is decreased a new steady state develops and the yield stress decreases accordingly. If the activation enthalpy for the formation of the pinning points is Hp, and their extent along the screw dislocation is go, the frequency of formation of the pinning points per unit length of the dislocation is (Friedel [84]) n
fp--~exp
-~
,
(4.1)
where Vd is the Debye frequency, b the Burgers vector of the dislocation, k the Boltzmann constant and T the temperature. At steady state one new pinning point is created for
152
V. Vitek et al.
Ch. 51
every pinning point annihilated by the release of the dislocation. Hence, if tu is the time the dislocation is, on average, trapped at a pinning point and L is the average separation between the pinning points, then
tuLfp = 1.
(4.2)
This is then the equation determining L. Similarly as in the case of solid solution hardening (see, e.g., Friedel [84]), we assume that the dislocation is bowing out between the pinning points, and unpinning occurs when the force acting on the pinning point reaches a critical value, which can be written as 2T sin 0c, where r is the line tension and 0c is the critical angle at which the unpinning occurs. The RSS in the (111) glide plane, apb (p implies primary slip plane, b implies in the direction of the Burgers vector), can be written as --
(4.3)
O'pb,
where the superscripts 0 and T distinguish the athermal and thermal components of the stress, respectively. O'pb 0 corresponds to the RSS at low temperatures and no motion of the dislocation occurs for smaller stresses. Hence, the bowing out of a dislocation is determined by O'pb x and, therefore, the radius of curvature of the bowed out dislocation R - 7/(aTpbb). It follows from obvious geometrical considerations that sin 0c - L / ( 2 R ) , and the critical stress in the primary plane needed to tear the dislocation away from the pinning point is then apTb = 2T sin 0c bL "
(4.4)
At this point the dislocation has bowed out to the distance d
-
-
Oc)
-
-
cos
Oc)
apTb
9
(4.5)
Since unpinning is athermal, the time tu is the time needed to move the dislocation to this distance by gliding in the (111) plane, i.e. tu = d/v, where v is the dislocation velocity. If a power law dependence of the velocity on the RSS is assumed, v - v0(apb) rn, where m is a positive exponent, eqs (4.1), (4.2) and (4.5) yield L-
(g0)2V0kvpb] exp "r(1 - cOS0c)Ud
( ) Hp
~
9
(4.6)
Inserting this value of L into (4.4) we obtain
O'pb = O"0 exp
- (m + 2 ) k T
,
(4.7)
w
Anomalous yield behaviour of compounds with L12 structure
153
where ] 1/(m+2) 2r 2 sin 0c ( 1 - cos 0c) Ud "0
--
be2vo
Takeuchi and Kuramoto [12] assumed m - 1, i.e. a Newtonian viscous motion of glissile dislocations in the (111) plane. Furthermore, the only stress component aiding the formation of pinning points was assumed to be the RSS on the (010) plane in the direction of the Burgers vector, Crcb, since glissile-sessile transformation is considered as a cross slip into this plane (c implies cross slip plane, b implies in the direction of the Burgers vector). Hence, they wrote the activation enthalpy for the formation of the pinning points as np
-
n ~ -
(4.8)
Vpl~cbl
where H ~ is a constant and Vo is the stress independent activation volume. Note that Hp does not depend only on the thermal components of the stress but on the total applied stress since no dislocation motion takes place during the glissile-sessile transformation. Furthermore, Hp does not depend on the sign of Crcb since both positive and negative shear stress in the (010) plane aids the cross slip into this plane. For a given applied uniaxial stress (tension or compression) Crcb and O'pb are related geometrically and a geometrical parameter N, depending on the orientation of the sample with respect to the tensile/compressive axis, can be introduced (Lall et al. [13]) such that
N-
O'pb
I
The implicit relation
Crpb -- o'0 exp
[ --
3kT
,410
then determines both the temperature and orientation dependence of the yield stress in the anomalous regime for temperatures lower than the peak temperature. Owing to the dependence on the orientation factor N eq. (4.10) predicts a breakdown in the Schmid law since O'pb, the CRSS for the (111)[101] slip system, will be different for different orientations of the tensile/compressive axis. The results of the early experimental studies of the orientation dependence of the CRSS, measured in compression for Ni3Ga (Takeuchi and Kuramoto [12]), Ni3Ge (Pak et al. [85]) and Ni3A1 (Saburi et al. [86], Aoki and Izumi [87], Kuramoto and Pope [88]) were in agreement with this model. However, these tests were flawed by the fact that only very limited number of sample orientations were tested. When tests were carried out for more orientations (Lall et al. [13], Ezz et al. [14], Umakoshi et al. [15]) significant discrepancies with the model were found. For example, the relative strength of samples having nearby orientations is, in some cases, opposite to the prediction of the model. However, most
154
V Vitek et al.
Ch. 51
significant is the observed tension/compression asymmetry (see section 2) which cannot be described in the framework of the Takeuchi-Kuramoto model because the activation enthalpy, Hp, does not depend on the sign of the applied stress. A comprehensive description of the yield anomaly requires, therefore, a detailed understanding of the transformation from the glissile to sessile form which takes fully into account the core structures of the corresponding dislocations. Such an analysis was carried out by Paidar et al. [38].
4.2. The PPV model
The starting points of this model are the same as in the case of Takeuchi-Kuramoto model. First it is assumed that there are two possible configurations of the 1 []-01] screw superpartials bounding an APB on the (111) plane" a glissile configuration with its core spread in the (111) plane and a sessile configuration with a non-planar core spread on the (111) plane. These two types of the cores are shown in figs 7(a) and (b), respectively. The second assumption is that the glissile screw dislocations transform with the aid of thermal activations into sessile ones since the sessile core is energetically favoured over the glissile core. Furthermore, as in the Takeuchi-Kuramoto model, it is assumed that these transformations occur locally on dislocations gliding on { 111} planes and this leads to the formation of the pinning points which hinder the dislocation motion (see fig. 11). As discussed in section 3.3, in the glissile core (fig. 7(a)) the Burgers vector is distributed in the similar way as if the dislocation were dissociated into two 1 (112) type Shockley partials on the (111) plane. The sessile core (fig. 7(b)) can be regarded as a narrow strip of the (010) APB, I~[101]1 wide, terminated by the dislocation spread onto the (111) plane. The Burgers vector is distributed in this plane again as if the dislocation dissociated into two ~(112 / type Shockley partials. Hence, in both cases the core is spread in the planes of the { 111 } type and only a narrow ribbon of the APB on the (010) plane occurs in the sessile form. For this reason, the core transformation is best described in three steps" (i) Constriction of the glissile core on the (111) plane, (ii) movement of the constricted dislocation along the (010) plane by the distance I~[101]1, and (iii) spreading of the core onto the (111) plane. A schematic picture showing the core of the 1 [101] screw superpartial before and after the glissile-sessile transformation is shown in fig. 12. This process is significantly different when compared with the model of Takeuchi and Kuramoto because this model only includes step (ii) since it is assumed that the transformation corresponds to the cross slip onto the (010) plane. The activation enthalpy, Hp, for the transformation process defined above is then composed of two parts (Paidar et al. [38]). The part associated with the steps (i) and (iii) consists of the energy of constrictions, Wc, formed on the dislocation cores spread into the (111) and (111) planes, respectively. The part associated with the step (ii) has been evaluated by regarding this step as the formation of a pair of kinks of the height I~[101]1 on the (010) plane. After reaching a critical separation,/c, corresponding to the maximum of the enthalpy when considered as a function of kink separation, the kinks m
w
Anomalous yield behaviour of compounds with L12 structure
w
155
)- . . . . . . . . . . . . . . . . . . . . r
1 [101"] SUPERPARTIALS 2
(olo)
"~
Fig. 12. Schematic picture showing the core of the l [T01] screw superpartial before and after the glissile-sessile transformation.
move apart and the dislocation has been displaced by the distance corresponding to the kink height. This part of the enthalpy is (Paidar et al. [38])
.o,0
1/2
27r
2 -
G
/
'
(4.11)
where c is the normalized self-energy of the kink, approximately equal to 0.5, and AE is the energy difference per unit length of the dislocation, between the sessile and glissile core configuration; this energy gain is the driving force of the transformation. The critical length of the transformed segment is ] ~/2 gc = b
G 167r - ~ +
(4.12)
We assume here that as the sessile segment of the screw dislocation is being formed the dislocation bows out around it away from the screw character which prevents the transformation into the sessile form from spreading along the entire length of the dislocation and only a pinning point is formed. The extent of this sessile segment will then be close to gc so that in (4.6), go can be identified with gc. In the original PPV model step (ii) was treated analogously to cross slip onto the (010) plane, as originally proposed by Kear and Wilsdorf [37]. This process is driven (a) by lowering of the APB energy owing to the decrease of the APB on the (111) plane and formation of a new segment of the APB on the (010) plane and (b) by the torque arising due to the elastic anisotropy, as proposed by Yoo [89-91]. The saddle point corresponds
156
Ch. 51
V. Vitek et al.
then to the formation of the pair of kinks of height b/2 having a critical width gc. AE was then evaluated using the standard elastic theory of dislocations. In this approximation
m E --
b
(1 + f X/2)
where f = v ~ ( A -
_ APB] - "/010 J
(4.13)
1)/(A + 2) arises from the anisotropic torque term (Yoo [89, 901);
A = 2 C 4 4 / ( C l l - C12) is the Zener anisotropy ratio for cubic structures.
However, it should be emphasized that while this analysis relates AE to the energies of APBs and the elastic anisotropy of the material, it must still be regarded as a model calculation of the core transformation. Indeed, the width of the ribbon of the APB on the (010) plane is only b/2 and the decrease of the extent of the APB on the (111) plane is of the same order, i.e. both are of atomic dimensions. These transformations are, therefore, not equivalent to the formation of Kear-Wilsdorf locks. Hence, either AE or "/1A1PBand "Y010 - APB have to be regarded as disposable parameters and the measured values of the energy of APBs (see e.g. Douin et al. [92], Baluc et al. [93], Dimiduk et al. [94], Hemker and Mills [95]) can only serve as guidelines for a judicious choice of these parameters. Assuming that b2lCrcbl << AN and expanding the square root in (4.11) into a Taylor series while neglecting terms of quadratic and higher order in IO-cbl, we obtain for H m~ an expression of the same type as in eq. (4.8) where
27r
2-
~
and
Vp-~
~
.
Thus the Takeuchi-Kuramoto model is a special case of the PPV model when only the step (ii) is considered and the work done by the applied stress is much smaller than the energy gain due to the transformation into the sessile form. The energies of constrictions can be determined following Escaig's analysis of cross slip in f.c.c, materials (Escaig [96-98]) since the spreading of the cores into the (111) and/or (111) plane can be regarded as splittings into Shockley partials. A very important aspect of this analysis is the realization that, unless the dislocation is pressed against an obstacle, the formation of the constrictions is not affected by the RSS in either the primary or cross slip plane but only by the shear stresses acting in these planes perpendicular to the Burgers vector. The reason is as follows: while the total dislocation is screw, the partials possess edge components of opposite sign and the shear stress component, O'pe, in the primary plane acting in the direction of those edge components, exerts equal and opposite forces on the partials. These equal and opposite forces either increase or decrease the width of splitting, dp, in the primary plane. The effect of the corresponding shear stress component, O'se, on the width of splitting in the cross slip plane, ds, is similar. Figure 13 shows schematically the dislocation splittings in the primary, (111), and cross slip, (111), planes in an f.c.c, lattice, together with the corresponding shear stress components affecting their widths.
Anomalous yield behaviour of compounds with L12 structure
w
157
Fig. 13. Schematic picture showing screw dislocations split in (111) and (1T 1) planes of an f.c.c, lattice together with the shear stress components playing role in the cross slip process, trpe is in the direction perpendicular to the total Burgers vector, in the primary (111) plane; Crse is in the direction perpendicular to the total Burgers vector in the cross slip (111) plane. m
As shown by Escaig ([96, 97], Bonneville and Escaig [99]) dp-
ao 1-
v"3o~
and
ds =
2"7
ao 1- v~'
(4.14)
2"7
where do is the width of splitting in an unstressed solid and 3' is the stacking fault energy. Both O'pe and O'se are taken as positive when they extend the corresponding width of splitting. Employing Stroh's analysis [ 100] of constrictions on dissociated dislocations in f.c.c, crystals, Escaig derived the following expression for the energy of constrictions w h e n O'pe and Crse << "y/b, i.e. when the splittings are narrow:
We-
Gb3 h +
27r
(O'pe -/'~O'se)
,
(4.15)
where h and A are dimensionless constants depending on do and t~ is a constant which describes the relative importance of O'pe and Crse; according to Escaig [97] 0.7 ~< t~ ~< 1. Equation (4.15) also determines the energies of constriction in the L12 case since the cores of the 1 []-01] superpartials undergo transitions entirely analogous to cross slip in the f.c.c, lattice. The constants h, a and n could be, in principle, evaluated as functions of the elastic moduli and ")'csv, using the elastic theory of dislocations (Escaig [96, 97], Bonneville and Escaig [99], Paidar et al. [38]). However, as in step (ii), the splitting from which the constrictions are being formed are only two to three lattice spacings wide and hence the process must again be regarded as a model for the core transformation. Thus, it is more appropriate to regard constants h, a and t~ as disposable parameters somewhat different from those used in the f.c.c, case (Paidar et al. [38]).
v. Vitek et al.
158
Ch. 51
The activation enthalpy for the glissile-sessile transformation leading to the formation of the pinning points is then 1/2
Hp - Gb327r h + ~ (ape - aa~) + 2 -
G
o
(4.16)
If we assume, as in the Takeuchi-Kuramoto model, that the unpinning is an athermal process and a steady state develops at every temperature, the temperature dependence of the RSS on the (111) plane is again determined by (4.7) with Hp given by (4.16). Furthermore, a Newtonian viscous motion of glissile dislocations in the (111) plane has been assumed in [38] and thus m = 1.
4.2.1. Orientation dependence and tension~compression asymmetry of the yield stress It is seen from eq. (4.16) that when ape > 0, which increases the width of the core on the (111) plane, the activation enthalpy increases, and when ase > 0, which increases the width of the core on the (111) plane, there is an opposite effect. Note that O'pe and O'se change sign when going from tension to compression and thus Hp, in general, will be different in tension and compression for the same magnitude of the applied uniaxial stress. This is in contrast with Takeuchi and Kuramoto expression (4.8) for Hp which does not depend on the sign of the applied stress. Clearly, it is the effect of O'pe and ase on the widths of the cores in (111) and (111) planes, respectively, which gives rise to the tension/compression asymmetry of the flow stress 1. Similarly to the geometrical factor N (4.9), which relates acb to Crpb, a geometrical factor Q relating ape to Crpb under uniaxial stress was defined via the relation (Paidar et al. [38])" m
Q = ape. O'pb
(4.17)
The range of values of N and Q for the tension/compression axes in the standard [001], [011] and [111] unit triangle is seen in fig. 14. It can be shown that
0"se ~-
Q ( N + v / 3 ) ( 2 N - v/3) ( 3 N - v/-3) v/3 apb.
Hence in (4.15) we can write O'pe
K-Q
1-~
--
NO'se
( N + v / - 3 ) ( 2 N - v/3) (3N-v~)v~
(4.18) --
KO'pb, where
(4.19)
IEscaig showed that in f.c.c, metals the dominant part of the activation enthalpy for the cross slip is the energy of constrictions and it is, therefore, practically independent of the shear stress acting in the direction of the total Burgers vector of the dislocation in the cross slip plane. This immediately suggests that a tension/compression asymmetry should exist in the case of the cross slip in f.c.c, materials. This prediction was, indeed, confirmed by experimental studies of cross slip in copper (Bonneville and Escaig [99]).
Anomalous yield behaviour of compounds with L12 structure
w
Q
[111]
[011]
1
159
0.4 -
o_ o
1.o
N
45
-0.4
[001] Fig. 14. A plot of Q vs. N showing the boundaries of the [001], [011], [111] standard unit triangle.
and it is this quantity that depends only on the sample orientation and the parameter ~;, which determines the extent of the tension/compression asymmetry. In particular, if K > 0 the CRSS in tension (apb > 0) is smaller than in compression since Hp(tension) > Hp(compression) and thus fewer pinning points are formed in tension. On the other hand the CRSS in tension is larger than in compression when K < 0 since Hp(tension) < Hp(compression). When K = 0 the term responsible for the tension/compression asymmetry vanishes and thus there is always a value of N, dependent on the value of ~;, for which there is no tension/compression asymmetry. It should be noted that Q -4 0 does not necessarily imply that K (and also ase) converge to zero. The reason is that Q and N are related: for example, on the [001]-[011] boundary of the unit triangle Q
__
3N-
x/~
x/'3(x/~- N) and on the [011 ]-[111] boundary 3NQ=
x/~
x/~(v/-~ + N)
Thus when Q --4 0, N ~ 1/x/~, and appropriate limits have to be taken in (4.18) and (4.19). The predictions of the tension/compression asymmetry for different orientations of the tensile/compressive axis, based on the calculated dependence of the parameter K on the orientation factor N, are shown in fig. 15. Independent of the value of ~;, K is negative near the [001] corner and it reaches a maximum near the [111] corner. The orientations for which there is no asymmetry, i.e. K - 0, depend on the value of ~; and, in general, they lie on the line which deviates significantly towards the [001 ] corner from the [012]-[113] line for which Q - 0. In this calculation n was set equal to 0.3 [38]. These predictions are in an excellent agreement with observations of the orientation
160
Ch. 51
V. Vitek et al.
111
TENSION
x" \ 001
/
~
~
\ 012
011
/
TENSION = COMPRESSION K=0 Fig. 15. Summary of the predictions of the PPV model for tension/compression asymmetry for various orientations of the tensile/compressive axis inside the standard triangle. m
dependence of the CRSS for the (111)[101] slip in Ni3A1 doped with Nb (Ezz et al. [14], Umakoshi et al. [15]) the results of which are shown in fig. 16. As seen from fig. 16(a), the CRSS in tension is, indeed, larger than in compression near [001], while the opposite is true for orientations near [110] (fig. 16(e)) and [111] (fig. 16(d)). The asymmetry is the largest near [111] as predicted by the largest value of K for this orientation. For the orientations corresponding to the line K - 0 the asymmetry vanishes as seen in fig. 16(b); this line is displaced away from the [012]-[113] line towards the [001] corner. This possibility was, in fact, first predicted theoretically by Paidar et al. [38] and later confirmed in the studies of Umakoshi et al. [15]. However, recent observations on binary Ni3A1 (Heredia and Pope [55]) and Ni3Ga (Ezz et al. [101 ]) show that in these alloys the orientations for which the tension/compression asymmetry vanishes lie very close to the [012]-[113] line (Q - 0) and the asymmetry is approximately the same for orientations near [001] and [110] though still in the opposite sense. For this reason a modification of the PPV model which reproduces the observed features of the tension/compression asymmetry in binary L 12 compounds, and suggests an additional alloying effect to explain the behaviour of ternary compounds, has been proposed by (Khantha et al. [45-47]). The magnitude of the deviation of the line of the zero tension/compression asymmetry from the Q - 0 line is a measure of the relative significance of O-pe and Crse and it was proposed, therefore, that ~rse does not play any role, i.e. ~; - 0 in (4.14). In this case Hp is the same in tension and compression only when ape vanishes, i.e. when Q - 0. At the same time the tension/compression asymmetry depends on IQI and is, therefore, the largest for orientations near [001] and [011] where IQI ~ 1/v~. However, unlike in the original PPV model where constrictions on both the (111) and (111) planes are considered, the asymmetry is virtually of the same magnitude for these two orientations. These features are in good agreement with observations in stoichiometric binary compounds [101, 55]. The following may be the physical reason for a negligible role of Crs~" while constriction of the core on the (111) plane is clearly necessary, the extent of the core re-dissociation on the (111) plane at the saddle point and thus the need for constrictions in this plane is not, a priori, clear. The saddle point, configuration m
Anomalous yield behaviour of compounds with L12 structure
w
161
(a) 500
I~.
-
J
-
400-
i/
~ ,oo Ten,i,. ,,,,,,[/~.~//i
!
Oc U
200
(b)
~
Compressive
v.,,;[io,]
If) if)
300
U
200
i,
Tensile lriiii i [TOiI
I00
qoo,,t~,o]
I00
- ; 2 o
4 'oo
' 600
' IZO0 ' IO00
' 8oo
o
Temperature (K)
[
I
o 400 n .....
2o0
\
l
J
1000 1200
(d}
r
200
r~oo, [i,o] Tensile rti.l [iol']
I O0, I 200
l
800
"o,
/-/ comp/e,s,v,,;KT=o //.~ *,oo,,v,oj h.. ~6/ ,;n,,,, *,,,,,C~o,3 t
I00 I
0
I
600
400
o n
-,El
400
Temperature (K)
50O
u') 300 v)
I
200
500 ~ (c)
(..)
t ~k
Tensile ,, I
i 1 l I 9 400 600 800 iOOOI200 Temperature (K }
I
J
200
I
I
I
l
400 6oo eoo I o o o i z o o Temperoture (K)
.500
IC (e) " 9 400
g.
300 o3 rr L~ 200
/
~
~l Tensile r, ooi,[i,o]
// "',. -",] "/Compressive.._ o,.~esslve ,,..,~-~ " % l"too,j[;,o]
Tens~e r.,lj[To, ]
~.~
"i
I00
'
0
200
I
l
I
I
I
400 6oo 800 ,000 i200 Temperature [ K)
Fig. 16. The temperature dependence of the CRSS for the (111)[101] and (010)[101] slip in Ni3A1 doped with Nb measured in tension and compression for different orientations of the tensile/compressive axis at a strain rate of 1.3 x 10 -3 s -1 (Umakoshi et al. [15]).
162
V. Vitek et al.
Ch. 51
involves only a single atomic step of the dislocation along the (010) plane, attained when forming the pair of kinks, and the critical length, gc, of the transformed segment is also only of the order of a lattice spacing (Paidar et al. [38]). Hence, while core spreading into the (111) plane must ultimately occur, since this is the final low energy sessile core configuration, it has not necessarily already taken place at the saddle point. Nevertheless, the observations in ternary compounds (Ezz et al. [14], Umakoshi et al. [15], Heredia and Pope [56]) definitely indicate that the tension- compression line is shifted to the left of the Q - 0 line and it is not clear why Crse should play a more significant role in this case. Hence, it was proposed that internal stresses induced by the ternary additions are responsible for the shift of the line of the zero tension/compression asymmetry from the Q -- 0 line in ternary compounds (Khantha et al. [45]). It is well known that ternary additions provoke dramatic strengthening in Ni3A1 (Pope and Ezz [6], Suzuki et al. [19], Heredia and Pope [56]) which is believed to be related to both the size and modulus effect as well as to the enhancement of the frequency of the cross slip by impurities (Pope [8]). The latter occurs because impurities may generate suitable sites for formation of constrictions on dissociated dislocations. For example, if the dislocation moves on the (111) plane and encounters an impurity which blocks its motion, it will be pressed towards the obstacle by the applied stress Opb, and the constriction will be easier to form. This effect of the impurities can be taken into account by replacing Crpein the activation enthalpy, Hp, by Crpeff- (Q +/3c)Crpb, where/3c is a constant that depends only on the concentration of impurities. This formulation simply reflects the fact that the effect of impurities always decreases the width of the core in the (111) plane. (A similar approach was adopted by Bonneville and Escaig [99] in their studies of cross slip in copper in order to include the effect of internal stresses arising due to forest dislocations introduced by prior deformation.) The value of/3c appropriate for a given concentration of an impurity cannot be quantified theoretically but it can be estimated from the orientation Q0 for which the tene f t _ 0): tic ~ _ Q 0 . In ternary compounds sion/compression asymmetry vanishes (i.e. rrpe of Ni3A1, Q0 is positive, and hence, tic is generally negative. For a given orientation and magnitude of the applied stress the width of the core is larger when O'pe is positive and vice versa and thus a negative value of tic decreases the core width everywhere in the standard triangle in both tension and compression, in agreement with the above interpretation of this quantity. A decrease in the core width in ternary compounds implies a lower constriction energy and hence a greater propensity for glissile-sessile transformations. For Ni3(A1, Ta), tic has been estimated to be -0.09 (Khantha et al. [48]) on the basis of the orientation dependence determined in (Heredia and Pope [55]). As described in the following section the temperature dependence of the yield stress in this ternary alloy is then very well described (Khantha et al. [47, 48]). A further discussion of the tension/compression asymmetry and, in particular, the influence of materials parameters that enter multiple-slip yield criteria that are applicable for a general state of stress can be found in (Qin and Bassani [18]) and sections 5.2 and 5.3.
4.2.2. The peak temperature In the framework of the PPV model the yield stress determined by eq. (4.7) always increases with temperature. However, the validity of this equation hinges on the assumption that glissile-sessile transformations do not extend along the entire length of
Anomalousyieldbehaviourof compoundswithL12 structure
w
163
the dislocation but rather that well separated pinning points are formed. From (4.6) and (4.7), the separation between the pinning points can be written as Hp L = Lo exp 3kT'
(4.20)
where
Lo - (s )2/3 (4v~ Vd
"
Hence, the separation of the pinning points decreases with increasing temperature until L becomes comparable with the size of the pinning points, ec, and the screw dislocations become completely transformed into the sessile form. At this point the mechanism assumed in the PPV model no longer applies. We identify the temperature at which this occurs as the peak temperature Tp. Khantha et al. [48] (see also section 4.3) showed that very reasonable values of Tp are obtained on the basis of this assumption. For T > To the movement of the sessile screw dislocations can occur via a thermally activated process of formation of pairs of kinks as, for example, in the case of screw dislocations in b.c.c, metals (Duesbery and Foxall [102], Duesbery [41 ]). The yield stress then decreases in the usual way with increasing temperature. Since in this regime the screw dislocations are sessile on both {111} and {010} planes, the slip on the latter planes will be preferred if possible, because the dislocations gliding on {010} planes possess a lower energy owing to the lower energy of the APB on these planes.
4.3. Strain-rate effects
As mentioned in section 2, one of the remarkable characteristics of the anomalous regime is a very small strain-rate sensitivity,
below the peak temperature (Thornton et al. [33], Takeuchi and Kuramoto [12], Miura et al. [34], Bonneville and Martin [35], Sp~itig et al. [36]). Nevertheless, the strain-rate sensitivity and/or the activation volume V* = kT/3 have recently been measured by several authors using stress relaxation techniques (Baluc et al. [57], Bonneville et al. [58, 62], Bonneville and Martin [35], Spatig et al. [36, 59, 60]) and in strain-rate jump tests (Ezz and Hirsch [61], Ezz et al. [63]). While the strain-rate sensitivity can always be measured, the activation volume has a good physical meaning only if the motion of dislocations is associated with a thermally activated process aiding the dislocation glide. In this case V* = -OHg/OCrapp,where Hg is the activation enthalpy of this process and Crapp is the applied stress. The origin of such a thermally activated process in the anomalous regime has not yet been established unequivocally, but two alternatives have been proposed: the first
164
V. Vitek et al.
Ch. 51
possibility is that the thermally activated process is not directly related to the mechanism of the anomalous yielding. This has been recently proposed by Hirsch and co-workers [61, 63] who concluded that, at least in Ni3(A1, Hf), the strain rate sensitivity of the flow is associated with work-hardening and converges to zero as the strain, and thus the work-hardening related part of the yield stress, approaches zero. The second possibility is that a thermally activated process is aiding the release of the screw dislocations which have been pinned by the transformation into the sessile form. Several such mechanisms have been proposed recently. In the model of Hirsch [103-107], it is reasoned that the jogs created by the PPV cross-slip mechanism are actually highly glissile and can therefore lead to the formation of long cross slipped segments rather than to the formation of pinning points. These long segments are then stabilized either by a second cross slip step of the leading superpartials resulting in the formation of strong edge dipole barriers or the cross slip continues until Kear-Wilsdorf locks are formed. In the former case, the barriers are unlocked by the movement of the superkinks of edge character which link the adjacent cross slipped segments. This motion is a thermally activated process with a large athermal component. The small thermally activated contribution is introduced in this model to give rise to a small strain-rate sensitivity. A different thermally activated process of release is proposed for the Kear-Wilsdorf locks that also comprises a significant athermal contribution but gives rise to a larger activation volume than the previous process. This mechanism is suggested to explain the observed jump in the activation volume vs. temperature dependence (Bonneville and Martin [35], Sp~itig et al. [36]). A dynamical simulation of the motion of [101] superdislocations was carried out by Mills and Chrzan [108, 109] and is described by these authors in detail in this Volume. In this study the evolution of a dislocation discretized into segments was studied by imposing a certain probability of the formation of PPV-type pinning points at each discrete unit, depending on its orientation. The localized barriers can dissolve athermally when the adjacent segment is bowed-out to a critical angle. A large propensity for the formation of long cross slipped segments terminated by superkinks was found. The motion of the sessile dislocations then occurs by the glide of unpinned superkinks of edge character. Fluctuations in the population of superkinks along a given superdislocation can lead to their exhaustion and thus immobilization of the dislocation. It is then suggested that the small strain-rate sensitivity may be due to the strong dependence of the density of mobile superkinks on the applied stress. However, the unpinning process may be aided by thermal agitations directly, as is the pinning process. A model which considers thermally assisted unpinning from individual pinning points has been suggested in [29, 110, 111]. The unpinning is assumed to trigger a major breakaway, i.e. release from many pinning points at the same time, and the unpinned dislocation then glides freely before being pinned again. Yet, it was recognized by Khantha et al. [45, 46] that release from a single pin is rate controlling only at low temperatures, when the mean separation between the pinning points is large. However, as the temperature and the stress increase and the mean separation between the pinning points (also formed with the aid of thermal activations) decreases, major breakaway becomes unlikely after the release from a single pin since re-pinning becomes more probable than further unpinning. The release is then controlled by more difficult, m
165
Anomalous yield behaviour of compounds with L12 structure
w
(11) .
.
.
.
.
.
liB.
.
.
.
.
(21)
(31)
(33)
(22)
.
(32)
Fig. 17. Equilibrium configurations of a dislocation with two pinning points. but alternate, reaction paths that involve multi-pin activations. A similar situation arises when considering release of dislocations pinned at impurities, and it was investigated in detail in conjunction with studies of internal friction (Teutonico et al. [112, 113], Blair et al. [ 114], Granato and Lticke [ 115]). Combining these results with the analysis of the pinning process via the PPV mechanism, Khantha et al. [45-48] developed a strain-rate dependent model of the anomalous yielding which is summarised below. As the first step in the development of this model we consider the thermally assisted release of a dislocation from two pinning points separated by a distance L. There are six non-equivalent equilibrium configurations of such a dislocation which are shown in fig. 17. The notation (ij) ( i , j - 1,2,3) refers to the displacement states at the two pins with zero displacement at the ends: the pinned, saddle-point and unpinned states are denoted as 1, 2 and 3, respectively. The configurations (11), (31) and (33) correspond to energy minima, (21) and (32) to saddle-points and (22) to an energy maximum. The saddle-point (21) involves activation at one pin while the saddle-point (32) involves simultaneous activation at both pins. Major breakaway can result from unpinnings which occur either via the path (11)-(21)-(31)-(32)-(33) or the path (11)(32)-(33) (Blair et al. [114]). The (21) saddle-point is rate-controlling for transitions via the first path (since the subsequent passage over the state (32) is easier) while the (32) saddle-point controls the activation energy for the second path. For large values of L, the release via the second path is unlikely since simultaneous activation over two pins is difficult to achieve. The transition via the second path becomes likely only when L is small and it will be favoured if the transition via the (easier) first path is inhibited owing to the backward jumps. However, when the backward jumps become easier than the forward jumps, the controlling path becomes the simultaneous unpinning from three or more pins. The path (21) is, therefore, rate controlling provided the activation enthalpy for unpinning, Hu, is smaller than the activation enthalpy, Hb, for the backward movement which leads to re-pinning. However, as the value of L decreases, a situation when Hu ~> Hb will ensue at a certain stress. When this happens, the release of the dislocation via the (21) path is unlikely if the attempt frequencies for unpinning and re-pinning are equal. Major breakaway now becomes possible only via the second path that takes the dislocation directly from (11) to (32), bypassing (31) [114]. Thus a transition from one reaction path to another occurs at those values of L, for which Hu ~ Hb. General expressions for the activation enthalpies Hu and Hb as functions of the applied stress, L, and displacements at the pins in the saddle point and unpinned states, respectively, have been derived in [46, 114] assuming a reasonable dependence of the dislocation-pin interaction energy on the distance from the pin.
v. Viteket al.
166
Ch. 51
After the major breakaway from the pinning points, the dislocation is assumed to glide freely on the (111) plane with a velocity v, traveling a distance d until enough pinning points have nucleated on the unpinned segment that it again becomes sessile. To determine a steady state we consider a random distribution of pinning points separated by a distance, L, on average, along a segment of length Ls of a screw dislocation. Let Np(t) be the number of pinned segments at time t. Np(t) changes as follows: (i) it decreases when the dislocation is released from a certain number of pinning points resulting in the major breakaway and (ii) it increases when new pinning points are formed on the unpinned gliding segments. Clearly, the pinning process must dominate over the unpinning process to provoke the anomalous temperature dependence of the yield stress. In order to attain a steady state in which both pinned and unpinned segments are present, a gain-loss rate equation must couple the dominant (pinning) and weak (unpinning) processes. Otherwise, the dominant process wins which in the present case would imply a complete transformation of screw dislocations into the sessile form. The rate equations of predator-prey models, logistic equation and autocatalytic reactions (Haberman [116], Reichl [117]) are examples in which strong and weak processes are coupled in a non-linear manner. A similar but simpler approach was adopted by Khantha et al. [46]. The gain term is considered to be proportional to the product of the number of unpinned segments that can glide freely at time t, Np(t) e x p ( - H u / ( k T ) ) , and the number of pinning points formed during the free flight, equal to
v gc exp
( Ho
Here udb/g c is the attempt frequency for pinning, as in (4.1) and it can be approximated as Ud since gc ~ b. The rate equation for Np(t) can then be written as
_
where Ls/s c is the number of nucleation sites for pinning and Ls/L the number of nucleation sites for unpinning on the dislocation length Ls and Hp, the activation enthalpy for the nucleation of pinning points, is again given by (4.16). In the steady state when dNp/dt = 0, it is reasonable to take d ~ L for a random distribution of pinning points [117]. It follows then from (4.21) that in the steady state
/ gCbv L-
Hp
3V---~d exp 3kT
which is analogous to (4.20).
(4.22)
w
Anomalous yield behaviour of compounds with L12
structure
167
The rate of thermally assisted unpinning of screw dislocations moving on the (111) plane determines the strain rate which can be written as = e0exp
- ~
(4.23)
with e0 = p b A N ( v d b / L ) , where p is the density of mobile dislocations, A the area swept by the unpinned dislocation, and N the number of nucleation sites for unpinning per unit length of the dislocation. Since A is proportional to L 2 and N ~ 1/L, ~o can be treated as a constant. The RSS on the (111) plane is given by (4.3) and the activation enthalpy Hu is a complicated function of (rpTb and L which is given by (4.22) (see [46]). The activation volume related to the strain-rate sensitivity is in this case V* - --0Hu/0CrpTb. A discontinuity in V* will occur whenever there is a change in the dominant unpinning path; V* is, in general bigger for simultaneous release from (m + 1) pins than from m pins. T can be obtained by solving For a given strain rate the temperature dependence of (rpb eq. (4.23) for Crpb. T This can only be done numerically. Such calculations have been made by Khantha et al. [46] for binary Ni3A1 and ternary Ni3(A1, Ta) assuming that the principal difference between these two alloys is a different core width of the screw superpartials, i.e. in terms of dissociations, different energies of APBs and CSF. Furthermore, the effect of the ternary addition (Ta) was also taken into account via the parameter/3c, as described in section 4.2.1. In these calculations Hp and t?c were calculated using (4.16) and (4.12), respectively. In these equations the constriction energy, We, has been determined employing the complete formulas derived by Escaig [96, 97] and using the anisotropic elasticity to estimate the widths of the cores [45]. Following the recent measurements (Baluc et al. [93, 118]), the values of the APB energy on the (010) and (111) planes were taken as 100 and 120 m J m -2 in binary Ni3A1 and 200 and 237 mJm -2 in Ni3(A1, Ta), respectively. The CSF energy on the (111) planes was taken in these two materials as 200 and 290 mJ m -2, respectively; this choice leads to reasonable values of We for which the observed temperature dependence of the CRSS is well reproduced. The energy gain, AE, has not been evaluated explicitly but an order of magnitude estimate, 0.01 J m -1, was used in both Ni3A1 and Ni3(A1, Ta). The factor, 13c, is zero in the binary Ni3AI and in Ni3(A1, Ta) it was estimated to be -0.09 on the basis of the orientation for which no tension/compression asymmetry was observed (Heredia and Pope [56]). For the free flight velocity, v, the Leibfried's expression, v - lOcr~bb4Vd/(3kT), was used (Leibfried [119]). Finally, the maximum value of the dislocation-pin interaction energy, U0, was taken as 3.2 eV and 2.8 eV in Ni3A1 and Ni3(A1, Ta), respectively and eo - 1012 s -1. The sensitivity of the results to changes in U0, e0, AE and/3c have been investigated. Even a 50% change in the values of U0, AE and/3c and three to four orders of magnitude change in eo does not produce any significant difference in crpTb. The results are most sensitive to the value of the CSF energy which is to be expected since small changes in the constriction energy and hence, Hp, produce large changes in L. The calculations have always been made for unpinning from one, two or three pinning points concomitantly (m - 1,2, 3), and in each case the activation enthalpy for the backward movement,
168
Ch. 51
V. Vitek et al.
400
9
I
0
9
9
I
9
I
Compression(Exp.)
D
3OO
I
Tension (Exp.)
400
9
9
0
.. ;O
] ] ~,
Compression(Exp.)
D
300 /
TensionS:" / 9
~'200
Compression
E?
"
9
...., 200
/.~fO
~-./ 100
100
~ 0
,,
200
I
400
9
I
9
I
9
I
600 800 1000 Temperature T (K)
(a)
9
0
1200
200
~ m=l 9
!
9
400
m=~ '~> !
600
9
1
800
j
,
|
.
1000
1200
Temperature T (K)
(b)
Fig. 18. The calculated and measured (Heredia and Pope [55]) CRSS vs. T dependence in the binary Ni3A1 at the strain rate of g = 1.3 • 10 -3 s -l" (a) [3220] orientation, (b) [1922] orientations. In the case (b) the squares correspond to values calculated for the strain rate g = 1.3 • 10 -2 s -1
Hb, has also been calculated. The transition from one reaction path to another was then determined according to the criterion Hu ~ Hb. Figure 18(a) shows the calculated temperature dependence of the RSS in binary Ni3A1 for the [12 20] orientation of the tensile/compressive axis at g - 1.3 x 10 -3 s -1 along with the experimental values (Heredia and Pope [55]). Following the experimental data 0 was chosen as 59 MPa. For this orientation the RSS in tension is higher than in Crpb compression, which is borne out by the calculations. There are three distinct temperature regimes with different rate controlling paths. The path corresponding to the release from one pinning point (m - 1) is controlling from 300 K to ~., 600 K. Between 600 K and 650 K, when L ..~ 120b, Hb becomes smaller than Hu signaling a change in the reaction path to simultaneous activation over two pinning points (m = 2). When this path starts to dominate, the activation volume exhibits a jump from ~ 400b 3 to 2000b 3. The transition temperature is very close to 670 K, the temperature at which an anomaly was observed in strain-rate jump experiments performed on polycrystalline binary Ni3A1 (Thornton et al. [33], Bonneville et al. [58]). From 650-850 K, the m - 2 path is controlling. Between 850 K and 900 K, when L ~ 50b, there is another transition to the path corresponding to the concomitant release from three pinning points (m = 3). However, the jump in the activation volume is now much smaller. The peak temperature, Tp, determined from the condition that L ~ gc, formulated in section (4.2.2), is approximately 1150 K which agrees very well with observations [55]. The calculated temperature dependence of the RSS in binary Ni3A1 for the [19 22] orientation of the tensile/compressive axis at g - 1.3 x 10 -3 s -1 (Crp ~ - 69 MPa) along with the measured values [55] are presented in fig. 18(b). For this orientation the tension/compression asymmetry is almost absent and thus only the compression results are shown. The RSS calculated for the strain rate 1.3 x 10 -2 s -1 is also presented and this demonstrates the very low strain-rate sensitivity below Tp. Again different paths D
w
Anomalous yield behaviour of compounds with L12 structure 300
9
I
"
I
"
I
"
I
"
I
"
/I
169
"
#
0
/O
Compression (Exp.).. l
200 /
;b
/o
/ ~
(~"
100
0
9
I
,
I
,
I
.
m=3
I
i
I
,
I
.
200 300 400 500 600 700 800 900
Temperature T (K) Fig. 19. The calculated and measured (Heredia and Pope [56]) CRSS in compression vs. T dependence in Ni3(A1, Ta) at the strain rate of g = 1.3 x 10 -3 s -1 for the orientation [123].
3000
9
I
O
2500
"
I
"
I
"
I
Volume(Exp.)
0
v 2000 E
0
"6 1500 ~9
1000
>
m=2\O
m=I"~k.~~,,, ~~.......,..(~Om, :~
500 9
I
~
I
200 300 400
9
I
9
I
.
I
i
I
,
500 600 700 800 900
Temperature T (K) Fig. 20. The calculated and measured (Bonneville et al. [58]) activation volume vs. T dependence in Ni3(A1, Ta).
control the unpinning process in different ranges of the temperature, leading to jumps in the activation volume. Figure 19 shows the calculated temperature dependence of the RSS (in compression) in the ternary Ni3(A1, Ta) for the [123] orientation of the tensile/compressive axis at - 1.3 x 10 -5 s -1 (o-p ~ - 40 MPa) along with the experimental values (Bonneville et al. [58]). There are again three distinct rate-controlling paths and discontinuities in V* occur between 450-500 K and at 650 K. The calculated peak temperature Tp ~ 800 K. The variation of the activation volume with temperature is shown for this case in fig. 20 along with the values of the apparent activation volume measured in stress relaxation
170
V. Viteket al.
Ch. 51
experiments [58] 2. The jump in the V* observed at 470 K agrees very well with the present calculation, but the second jump, around 600 K, suggested by the calculations, is too small to be observable.
o
Continuum theory of crystal plasticity of L12 compounds in the anomalous regime
From the point of view of the continuum theory of crystal plasticity the most interesting deformation properties, observed in the L12 compounds with anomalous temperature dependence of the yield stress, are the orientation dependence of the CRSS and the tension/compression asymmetry. These represent so-called non-Schmid behaviour, i.e. the Schmid law (Schmid [16], Schmid and Boas [17]) which states that only the RSS on the slip plane in the direction of the Burgers vector governs slip on that system, does not apply. The continuum theory of plastically deforming crystals, with its origins dating back to the work of G.I. Taylor [120, 121], was originally based on the Schmid law and the recent monograph by Havner [122] provides a historical perspective and develops the mathematical theory of finitely-strained crystals that display Schmid-type behaviour. Recently this theory has been extended to include non-Schmid effects (Qin and Bassani [18, 49], Bassani [50]), and certain aspects of the theory are reviewed in this section, particularly those that relate to the anomalous behaviour of L 1z alloys. Relatively simple multi-slip yield criteria are formulated on the basis of the microscopic analysis of the anomalous yielding (PPV model), and it is shown that the observed tension/compression asymmetry at initial yield in single crystals and its orientation dependence are well characterized using these criteria. Furthermore, strain localization in single crystals and overall hardening of polycrystals are predicted to be significantly influenced by non-Schmid effects of the type observed in L12 alloys in the anomalous regime. Since non-Schmid stresses are introduced into the yield criteria, this theory is of a non-normality or non-associated-flow type. In contrast, the constitutive model for L12 crystals recently proposed by Cuitifio and Ortiz [123] takes the Schmid stress to be the driving force for slip (a normality-type relation) but accounts for non-Schmid effects through hardening alone. The latter is not in accord, for example, with characteristic (2) of section 2.
5.1. Flow behaviour and the Schmid stress
In the framework of the continuum crystal plasticity the plastic flow in the single crystal is assumed to result from continuous shearing or slip on well defined lattice planes in well defined directions, and the underlying lattice is assumed to be unaffected by these plastic slips. With superscript c~ denoting a particular slip system, where cx ranges from 2Recently Spiitig et al. [36, 59, 60] introduced the effective activation volume which excludes the effect of changes in the mobile dislocation density and in the internal stresses. The values of the effective activation volume are lower than those of the apparent activation volume but the functional dependence on the applied stress and/or temperature remains the same.
w
Anomalous yield behaviour of compounds with L12 structure
171
1 to 12 for {111}(110) slip systems in f.c.c.-based lattices, the rates of slip .~c, on well defined crystallographic planes with normals n a in directions m '~ in the deformed configuration ( m '~ is in the direction of the Burgers vector and m c' 9n '~ = 0) give rise to the plastic part of the rate of stretching (see Rice [124], Hill and Rice [125], Hill and Havner [ 126])
Dip = y ~ ~;y~(m~n; + m;n~).
(5.1)
OL
In the small-strain version, Dij = Di~ + Dip corresponds to a strain rate (see, e.g. Hill [127]); at finite strains there is a subtle difference between the rate of stretching, Dij, and strain rate, but for the present treatment this difference is not important. Since the lattice is assumed to be unaffected by slip, the slip vectors are taken to convect with the elastic (lattice) deformation which also determines the stress via an elastic strain-energy function. In order to determine the slip rates, "~'~, a slip-system flow rule is required that, in general, depends on the stress in the crystal, the hardness of each system, and possibly the stress rate. In a time-independent theory this involves a yield criterion and hardening rule for each slip system. For a slip system to be potentially active the stress must be at yield. For a potentially-active system to be active, the rate of stressing must keep up with the rate of hardening. In a time-dependent theory typically the same dependence on stress appearing in the yield criterion is assumed to determine the slip rate, and in this case there is no stress-rate dependence (Asaro [128]). In conventional crystal plasticity the Schmid stress on a particular system determines the corresponding slip-rate. The Schmid stress apb '~ is defined such that ap'~-~'~ is the rate of plastic working (per unit reference volume) due to slip on system a; under multiple slip, the rate of plastic working, with (5.1), is
SijDip = ~
l ~ s i j (m~'n~ + mj'~n'~), -~
(5.2)
ot
where Sq is the (Kirchhoff) stress acting on the crystal. Consequently, the Schmid stress for slip system a is defined as 1
(5.3a)
or since Sij = Sji
apb~ -- Sijm ianj.a
(5.3b)
In the setting of a time-independent flow theory of plasticity, Schmid law states that a slip system is potentially active when the RSS on the slip plane attains the value of the critical resolved shear stress (CRSS). When the rate of shear stressing equals the rate of hardening of the CRSS then the slip system is said to be active. Conversely, if ~c~ # 0,
172
Ch. 51
V. Vitek et al.
then the Schmid stress equals the CRSS. Two important consequences of Schmid law for well-annealed cubic crystals are that the CRSS is independent of orientation of the uniaxial loading axis with respect to the lattice and there is no tension/compression asymmetry.
5.2. Yield criterion with non-Schmid stresses and hardening To include non-Schmid behaviours, Qin and Bassani [18] proposed that slip system c~ is potentially active when a yield function that equals the Schmid stress plus a linear combination of other components of stress attains a critical value (which in general is different from the CRSS). For cross slip phenomena in f.c.c, materials (Escaig [96, 97], Bonneville and Escaig [99]) and the anomalous yielding of L 12 intermetallic compounds (see section 4.2) these additional non-Schmid stresses are also shear stresses. Therefore, the yield function for a given slip system depends on the stress in the crystal and a set of pairs of lattice vectors denoting plane normals and directions for each shear stress just as r~~ and rn ~ determine the Schmid (shear) stress. The additional shear stresses entering the PPV theory of the anomalous yielding in L12 compounds (see (4.15)) include the shear stress on the primary slip plane, (111), in the direction perpendicular to the Burgers vector of the superpartial (ape), the shear stress on the secondary slip plane, (111), in the direction perpendicular to the Burgers vector of the corresponding superpartial (Crse), and the shear stress on the (010) plane in the direction of Burgers vector (Crcb). Because of the core configuration of the superpartials, the effects of O'pe and Crse on dislocation motion depend on their sense as described in detail in section 4.2. These stresses are also shear stresses and, therefore, are given in terms of the stress Sij acting on the crystal by equations analogous to (5.3). A table of the planes and directions for each slip system that define these shear stresses which enter the yield criterion given below for L12 crystals is found in [18] and [50]. Slip system yield criteria for L12 intermetallic compounds that involve a linear combination of ffpb, O'pe, O'se and Crcb can be deduced from the knowledge of the activation enthalpy, Hp, for the glissile-sessile transformation given by eq. (4.15). As explained in section 4.2, assuming that b2[Crcb[ << AE, Hp depends linearly on (O'pe - t~Crse) and Crcb while the stress component Opb is driving the slip on the slip plane. For this reason Qin and Bassani [18]) proposed the yield criterion
I
ffpb -+- A O'pe -- NO'se -~- /30"g --~ -I-O'er
(O~ = 1 , . . . ,
12),
(5.4)
where crgr is the critical hardness on system c~ and A, B, and t~ are material parameters that depend on temperature and, in principle, can evolve with the deformation although here they are taken as the same constant for each slip system. Activation of a slip system in the negative sense is implied when equality is met with the right-hand-side negative. These criteria clearly indicate the dependence on the sense of the stress. The magnitudes of the non-Schmid factors A,/3, and ec are typically less than unity. The absolute value of the term involving O'pe and ~rse reflects their effects on the core of the superpartials.
w
Anomalous yield behaviour of compounds with L12 structure
173
The evolution of hardening of each slip system under multiple slip deformation is described in the usual manner (see, e.g. Hill [127]): ~r~r -- ~
h , ~ ~,
(5.5)
where the instantaneous hardening moduli h,~ are, in general, functionals of the entire deformation history (see Wu et al. [ 129], Bassani [50]). The diagonal components of h~z account for self hardening while the off-diagonal components can be included to account for latent hardening. A simple form of the hardening matrix where the instantaneous hardening moduli monotonically decrease with increasing slip is h~=
h0
1 + I~1
+hs
(c~ = fl)
"},o
h~z- q(h~ + hz~) (a ~ /~),
(5.6)
where ho, hs, 70, and q are taken to be positive constants. Specifically, h0 and hs characterize the initial and the saturated values of the incremental hardening moduli, respectively, and q is the latent hardening parameter. This is a strain-dependent variant of the Hutchinson hardening law [130]. When h0 = 0 and q = 1, (5.6) reduces to the isotropic Taylor hardening law. The form of the hardening matrix in (5.6) will be used below for predictions of polycrystalline response.
5.3. Tension/compression asymmetry and its orientation dependence Uniaxial stressing for orientations of the loading axes in the standard triangle can readily be investigated using the yield criteria (5.4). For initial yielding of well-annealed crystals, the critical values are assumed to be the same for each slip systems, i.e. O'er - a~, which characterizes a well-annealed crystal. Initial yield is reached when one of the 12 slip systems is activated, i.e., at least one equation in (5.4) is satisfied for a given stress, e.g. uniaxial, applied to the crystal. The CRSS is plotted in fig. 21 for all orientations of the uniaxial stressing axis over the standard unit triangle (see fig. 15) for values of the non-Schmid parameters A - -0.3, B - -0.1, and ~ - 0.4 chosen to qualitatively agree with the experimentally observed tension/compression asymmetry for Ni3A1 shown in figs 2, 15 and 16. As shown in [18], the degree of the tension/compression asymmetry is controlled by the factor A, and increasing its magnitude produces a larger separation of the tension and compression surfaces. Changing the sign of A exchanges the two surfaces. The factor ~ determines the orientations where the CRSS at initial yield in tension is the same as in compression. The cross-slip factor B influences those crystals oriented near the [011 ]-[111] boundary. High-symmetry orientations of the uniaxial loading axis at the corners of the standard triangle can be examined analytically, and explicit expressions are obtained for the CRSS and its dependence on the sense of the applied uniaxial stress [18]. For the particular
174
Ch. 51
V. Vitek et al.
"CO / "l;cr Ni3AL: A = -0.3 B = -0.1 k--- 0.4
2.0
C
T
1.0
0.0 ~
~
[-111]
[ooI]
~ [oI]]
~
Fig. 21. Predicted RSS on the primary slip system at yield for both tension (T) and compression (C) for different orientations of the loading axis within the standard triangle for Ni3AI.
choice of the parameters A, B, and ~ noted above, that describe Ni3A1, the most highly stressed slip systems which correspond to those activated at initial yield for each of these special orientations are the same as for crystals obeying Schmid law, i.e. A - B - 0 in (5.4). One such system is (111)[110]. For values of the non-Schmid parameters that differ significantly from these, some other system(s) may become the critical one(s). The ratio of CRSS in tension and in compression for uniaxial stressing at the corners of the standard triangle can be easily found as a function of A, B, and t~ from (5.4) with table of the lattice vectors given in [18]" u
l-A(1-
~O'pbIi0Oll- =
;lI0lll%u --
t~)/V~
1 + A(1 - n)/v/3 '
1 + A/x/~ + ~/3B/2 1 - A/v/3 + v ~ B / 2 '
(5.7)
I + A ( 1 - ~)/v/3+ v/-3B 'r; lET1 = 1 - A ( 1 - ro)/v~ + x/3B" From these expressions, the three non-Schmid factors A, B, and t~ can be readily determined by the measurements of the tension/compression asymmetry at the corners of the spherical triangle. Other aspects of the yield behaviour including yield surfaces and restricted multiple slip are discussed in [18, 50].
w
Anomalous yield behaviour of compounds with L12 structure
175
5.4. Restricted slip and polycrystalline response One issue concerning restricted slip which is rather important for the behaviour of polycrystals is briefly discussed here. Geometrically, the slip systems associated with lattices based on the f.c.c, structure possess a high degree of symmetry. As observed by Taylor [120, 121], Bishop and Hill [131] and Bishop [132], due to the symmetry of the f.c.c, lattice there are many special stress states that can accommodate an arbitrary plastic strain by simultaneously activating five or more linearly-independent slip systems. These correspond to vertices on the single crystal yield surface. In particular, there are 28 such stress states for f.c.c, crystals obeying Schmid law with ac~ = tr~ on each slip system, and each state activates either six or eight slip systems. Each of these stress states are of four types (see Bishop [132] for f.c.c, crystals obeying Schmid law and Qin and Bassani [ 18] when non-Schmid effects are included). On the other hand, yield criteria containing non-Schmid terms such as, e.g. (5.4), do not possess the high degree of symmetry of those based on Schmid law. As a consequence, when non-Schmid stresses enter the yield criteria not only are yield surfaces distorted, but it also is more difficult to activate many systems simultaneously as generally required in large plastic deformations of polycrystals. For example, this has been demonstrated utilizing an extension of the elastic-plastic self-consistent approximation for polycrystals introduced by Hill [133] and applied by Hutchinson [130]. Figure 22 is a plot of the overall stress-strain response for a random polycrystal of NiaA1 with individual grains governed by the yield criteria (5.4) and hardening rule (5.5) with (5.6) (and q = 0) under uniaxial tension and compression. Recall that the corresponding single crystal initial yield behaviour in uniaxial stress is plotted in fig. 21. A comparison
0"1%* 8.0
_
c
6.0
4.0
2.0
0.0
,,i
0.0
i
i
i
I
0.05
i
l
i
i
I
0.1
Fig. 22. Comparison between predicted uniaxial stress-strain curves of polycrystals comprised of L12 (solid curves) and f.c.c, single crystals obeying the Schmid law (dashed curve).
176
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Ch. 51
with an f.c.c, polycrystal obeying Schmid law, i.e. A = B = n = 0 in (5.4), which does not display a tension compression asymmetry is also shown in fig. 22. Note, for example, that at an overall strain of 0.05, in tension the predicted flow stress for the Ni3A1 polycrystal is nearly 10% higher than that for the f.c.c, polycrystal obeying Schmid law while in compression it is over 25% higher. This stiffer response arises, in general, from the compatibility of deformation between individual grains in the polycrystal which requires extensive multiple slip once the overall plastic strains exceed the elastic ones. This stiffer response, i.e. higher stresses to generate a given level of plastic deformation, may contribute to the brittle behaviour of certain intermetallic polycrystals such as Ni3A1 when compared with the single crystal response.
5.5. Shear band formation
After some more or less uniform straining, plastic deformation often concentrates within narrow bands. Upon further straining, the deformation may become diffuse again or it may continue to concentrate leading to fracture in the vicinity of the bands. Strain localization has been analyzed in the form of a shear band bifurcation, i.e. a discontinuity in tangential components of velocity across the band, that results from a constitutive and geometric instability (see Bassani [50] for a review of related studies). With coarse slip band formation in mind, Asaro and Rice [134] analyzed shear bifurcations under single slip conditions. For Schmid-type behaviour they found that bifurcations are precluded at positive values of the (instantaneous) slip-system hardening modulus. On the other hand, under truly single-slip flow, non-Schmid effects (e.g., non-zero values of A or B in (5.4)) trigger shear bifurcations at positive hardening (Asaro and Rice [134], Qin and Bassani [49]). These effects are also seen under multiple slip conditions; an example is given below for the case of planar double slip. Asaro [135] developed a symmetric double-slip model in two dimensions and demonstrated that with Schmid-type behaviour the critical hardening modulus at which localization occurs is positive although rather small. Qin and Bassani [49] extended Asaro's [135] symmetric double-slip model to include non-Schmid effects. As expected, and consistent with the single slip results, non-Schmid effects tend to enhance localization as compared with Schmid-type behaviour for the case of symmetric double slip. The geometry of these models is shown in fig. 23(a) where the loading stress is or. The two slip systems are taken to be symmetrically disposed about the uniaxial loading axis, which can represent the behaviour in uniaxial stressing of a single crystal where lattice rotations tend to bring the primary and secondary slip systems into such a configuration. In fig. 23(a), the angle ~b gives the orientation of the two slip systems (and, therefore, the Schmid stress) and 0 represents the direction of localization band. A non-Schmid shear stress is introduced for each slip system, whose orientation is characterized by the angle ~b (dashed lines in fig. 23(a)). In the double-slip model, the yield criterion for the two slip systems that includes non-Schmid effects is -- % +
O'ns- O'gr
l, 2),
(5.8)
w
177
Anomalous yield behaviour of compounds with L12 structure
(a)
2
I T t
,
,
"~,,0
=
\
/ \\
I
iI
\
......-.-
J
F ' " \ ""
/
\
/
\
/
(b)
\
0.1
q =0.0
q=0.3 q
t~
0.05
-
0.0
~
;
0.0
0.6
q=0.9
1
0.2
,
I
0.4
!
]
0.6
Fig. 23. Predicted influence of the non-Schmid factor on the critical slope of the localization surfaces.
178
v Vitek et al.
Ch. 51
where cr~ are the Schmid stress components on the two systems at angles of +~b to the loading axis, Cr~s are the non-Schmid stress components oriented at -t-~ and al ~ 0 is the non-Schmid factor which characterizes the deviation from Schmid-type behaviour when al = 0. For an incompressible single crystal with two symmetrically-oriented systems undergoing equal slip analytical solutions for the bifurcation condition including non-Schmid effects have been obtained by Qin and Bassani [49]. A plot of the critical level of the instantaneous hardening, hcr, normalized by the applied tensile stress versus the magnitude of the non-Schmid parameter, a, is plotted in fig. 23(b) for the case where q~ = 35 ~ and ~b = 10 ~ There is an approximately linear increase in (h/O')cr with increasing al due to non-Schmid effects. Increasing the latent hardening ratio q in (5.6) tends to decrease the critical ratio. Examples of these and other results that emphasize such non-Schmid effects are given in [49].
6.
Conclusions
It is generally accepted that the anomalous increase of the yield and flow stresses with increasing temperature, observed in a number of L 12 intermetallic compounds, is related to the thermally assisted immobilization of [101] superdislocations gliding on (111) planes when they are in the screw orientation. This immediately suggests that the reason why the anomalous yield behaviour almost disappears for very small offset strains (characteristic (8) in section 2) is that at this stage screw dislocations practically do not participate in the yielding process which is controlled by the motion of mobile non-screw dislocations. This is analogous to the situation observed during the micro-deformation of b.c.c, metals (Brown and Ekwall [ 136]) when the temperature dependence of the flow stress is much weaker than when large deformation develops since the sessile screw dislocations barely move in this regime. The PPV model (Paidar et al. [38]), described in detail in this chapter, successfully explains a number of important features characterizing the anomalous regime. This model, which builds up on the original suggestion of Takeuchi and Kuramoto [12], is founded on the assumption, born out by atomistic studies (Vitek et al. [28, 29, 40]), that in the compounds with metastable APBs and CSFs on (111) planes the 1 [T01] superpartials bounding APBs on these planes may possess two alternate core configurations: one glissile, spread in the primary (111) plane, and the other sessile, spread in the cross slip (111) plane. Additionally, in the latter case a narrow ribbon of the APB on the (010) plane is another intrinsic feature of the core (see fig. 7). The core spreading into the { 111 } planes is akin to the splitting into the Shockley partials separated by the CSF. However, this splitting is only two to three lattice spacings wide and thus it represents a model for the dislocation core. The immobilization of the dislocations is associated with the glissile-sessile transformations occurring with the help of thermal activation. The driving force for these transformations is the lower energy of the sessile core and, possibly, effects of elastic anisotropy (Yoo [89, 90]). These transformations can be identified with the first elementary step of cross slip onto the (010) plane consisting of the formation of the u
w
Anomalous yield behaviour of compounds with L12 structure
179
ribbon of the APB of length I~[101]l on this plane and spreading of the core onto the (111) plane. In the PPV model it is then assumed that such transformations do not spread along the entire length of the dislocation but are blocked due to bowing out of the gliding dislocations around the transformed segments. This contradicts the assumption that such transformations automatically lead to a complete cross slip onto the (010) plane and thus formation of Kear-Wilsdorf locks [37]. These segments then represent pinning points which hinder dislocation motion. Their density increases with increasing temperature inducing the increase of the yield stress. At each temperature a steady state is achieved in which the average number of pinning points per unit length of dislocation is constant. When the temperature is altered a new steady state is attained and the yield/flow stress decreases or increases accordingly. This is the reason why the major contribution to the flow stress depends only on temperature and not on the sequence of test temperatures employed (and the associated workhardening). However, the thermally activated process leading to the formation of the pinning points may also sometimes result in the development of Kear-Wilsdorf locks. This occurs if the glissile-sessile transformation is not blocked by the bowing of the dislocation but extends along a substantial length of the screw dislocation. Such sessile dislocation may continue to move with the aid of thermal activation along the (010) plane since energy is gained by replacing the high energy APB on the (111) plane by a lower energy APB on the (010) plane. Such a dislocation is always sessile since its core never spreads into the (010) plane but either into the (111) or (1]1) plane (see fig. 8), forming a Kear-Wilsdorf lock. Notwithstanding, such complete transformations into the sessile form will always occur at a temperature at which the separation of the pinning points becomes comparable with their extent. This temperature can then be identified with the peak temperature, To. While below To the dislocations move on the (111) plane, where they are glissile, above To the dominant slip plane will be of the {010} type since the APB energy is lower on this plane and the dislocations are now sessile on both {111} and {010} planes (see section 4.2.2). At this stage their movement is thermally activated and the yield/flow stress decreases with increasing temperature. The occurrence of thermally assisted glissile-sessile transformations leading to the development of pinning points, which is the basis of the PPV model, explains characteristics (1)-(5) of the slip behaviour in the anomalous regime, summarized in section 2. However, the model also can explain in detail the unusual orientation dependence and the tension/compression asymmetry of the yield stress observed in the anomalous regime (characteristics (6)). Significant orientation dependence of slip is typical when the core structure of dislocations controls the slip mechanism (Duesbery and Richardson [69], Vitek [40]), and it is, indeed, the special structure of the cores of 1 []-01] superpartials which is responsible for these phenomena in L12 compounds. The most important characteristics of dislocation cores which leads to the explanation of orientation dependence is the fact that they always spread into { 111 } type planes and this spreading can be considered as dissociation into the corresponding Shockley partials. Thus, the sessile-glissile transformation involves formation of a constriction on the glissile dislocation in the (111) plane, movement of the constricted dislocation along the (010) plane by the distance I~[101]1, and spreading of the core onto the (1]-1) plane.
180
Ch. 51
v. Vitek et al.
The formation of the constrictions on { 111 } planes can be treated in the same manner as cross slip in f.c.c, metals and the most important realization, originally due to Escaig [96, 97], is that these are aided by shear stress components in these {111} planes which are perpendicular to the total Burgers vector, in the present case stresses Crpe and Crse. The opposite effect of these stresses for tension and compression causes the tension/compression asymmetry. Furthermore, dislocation movement along the (010) plane is aided by the shear stress Crcb in this plane in the direction of the Burgers vector. Consequently, the activation enthalpy for the glissile-sessile transformation (eq. (4.16)) depends in a complex way on all these shear stress components. The predicted orientation dependence of the CRSS successfully explains the observed behaviour as shown in section 4.2.1. The only remaining uncertainty is the effect of ternary impurities, though it can also be accounted for, at least phenomenologically, by assuming that the formation of constrictions in the (111) plane is aided by pinning of dislocations at impurities (Khantha et al. [45]) (see also section 4.2.1). Nevertheless, further investigation of alloying effects is needed for a more quantitative understanding of this phenomenon. The orientation dependence and the tension/compression asymmetry in the anomalous regime lead to the break down of the Schmid law. This is again common in materials where the dislocation core effects play an important role (Duesbery and Richardson [69], Vitek [40]). Hence, the L12 compounds exhibit so-called non-Schmid behaviour in the anomalous regime, i.e. slip is not controlled only by the RSS on the slip plane in the direction of the Burgers vector but other stress components play a significant role. This has important consequences for the macroscopic plastic behaviour of these materials, even in polycrystalline form. This was recently investigated by Qin and Bassani [18, 49] and it is discussed in section 5. A rigorous three-dimensional, finite strain, continuum theory was developed for elastic-plastic deformation with non-Schmid effects. This phenomenological theory not only captures the orientation dependence and tension/compression asymmetry in anomalous regime, but also predicts a greater tendency for localization of strain in shear bands and a stiffer polycrystalline response when non-Schmid effects are present compared to the behaviour of similar materials obeying Schmid law. In fact, certain features associated with the anomalous behaviour of single crystals are also predicted for polycrystals. The modification of the PPV model that includes a thermally assisted unpinning [46] also explains certain salient features of the strain-rate sensitivity (see section 4.3), in particular the very weak dependence of the yield stress in the anomalous regime on the strain rate and unusual discontinuities in the activation volume observed in Ni3(A1, Ta) (Baluc et al. [57], Bonneville et al. [58], Bonneville and Martin [35], Sp~itig et al. [36]). However, this aspect of the anomalous yielding is still somewhat controversial. For example, in Ni3(A1, Hf) alloys (Sp~itig et al. [60], Ezz and Hirsch [61], Ezz et al. [63]) no unusual jumps in the activation volume have been found, and recent strain-rate jump experiments of Hirsch and co-workers [61, 63] indicate that the strain rate sensitivity converges to zero as the strain approaches zero. They argue that the process controlling the anomalous behaviour, i.e. the immobilization of the [101] superdislocations, is entirely strain-rate independent and any observed strain-rate sensitivity arises due to work-hardening. The contribution of the work-hardening to the strain-rate sensitivity is D
w
Anomalous yield behaviour of compounds with L12 structure
181
certainly indisputable and it may dominate over other contributions. It is possible, for example, that while it dominates in some materials (e.g. Ni3(A1, Hf)) it may not be of principal importance in others where thermally activated unpinning may dominate (e.g. Ni3(A1, Ta)). On the other hand, Sp~itig et al. [36, 59, 60] maintain that while the apparent activation volume measured in both strain-rate jump and stress relaxation experiments includes effects of the changes of the mobile dislocation density and internal stresses, an effective activation volume can be determined which is related only to the activation enthalpy which controls the dislocation velocity. The property of the anomalous regime which cannot be readily related to the immobilization of screw dislocations, and thus to a model of the PPV type, is the temperature dependence of the work-hardening rate (characteristic (9)). Nevertheless, an indirect relationship to the increase of the density of immobile Kear-Wilsdorf locks can be envisaged. For example, it has been proposed by Ezz et al. [63] that the increase of the work-hardening rate at temperatures up to 500 K (in Ni3(A1, Hf)) correlates with the density of [101] dislocations on (010) planes (Kear-Wilsdorf locks) which gradually increases in this temperature regime. The decrease in the work-hardening rate at temperatures higher than 500 K can then be attributed to the annihilation of these screw dislocations by cross slip which is becoming easier as the temperature increases. Finally, it should be noted that the anomalous increase of the flow stress with increasing temperature is not a generic feature of the L12 compounds. In a number of cases the yield stress exhibits a "normal" behaviour, i.e. it increases with decreasing temperature while at high temperatures it remains almost constant. Such a behaviour was observed, for example, in Pt3A1 (Wee et al. [20, 21], Heredia et al. [22]), A13Sc (Schneibel and Hazzledine [23]) and alloyed A13Ti in the L12 form (Wu et al. [24-26], Yamaguchi et al. [27]). This temperature dependence of the yield stress suggests an intrinsic mechanism analogous to that in b.c.c, metals: the screw dislocations always possess a sessile core and their motion can be aided by thermal activation via a mechanism of the formation of pairs of kinks (Vitek [39, 40], Christian [137], Duesbery [41]). In L12 alloys such a situation will arise if the CSF, and possibly also APB, on { 111 } planes are not stable. This is seen in figs 9 and 10 which demonstrate that in this case the screw dislocations are sessile on both {111} and {010} planes. A theory of the temperature dependence of the yield stress in L12 compounds with sessile screw dislocations was developed by Tichy, Vitek and Pope [138] and it reproduces the observed temperature and orientation dependence of the yield stress in Pt3A1 at low temperatures [22]. Thus, there are at least two classes of L12 compounds which principally differ according to the stability of the CSF (and possibly APB) on { 111 } plane. When the CSF is stable both glissile and sessile core configurations of the screw 89 superpartials exist, and such alloys display the anomalous yield behaviour. When the CSF is unstable only sessile core configuration of the screw 1 []-01] superpartials exists and normal yield behaviour ensues. The latter class of the alloys generally possesses a higher ordering energy than the former though in neither case disordering occurs prior to the melting. In the alloys which undergo order-disorder transition well below melting, such as Cu3Au, the temperature dependence of the yield stress correlates with the amount of disorder (Pope [311).
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Acknowledgements The authors would like to thank Dr M. Khantha for many valuable discussions and comments. This research was supported by the National Science Foundation, grant no. DMR-92-19089 and the MRSEC Program DMR-91-20668.
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J.H. Westbrook, Intermetallic Compounds (John Wiley, New York, 1967). J.H. Westbrook, Trans. TMS-AIME 209 (1959) 898. P. Flinn, Trans. TMS-AIME 218 (1960) 145. R.G. Davies and N.S. Stoloff, Trans. TMS-AIME 233 (1965) 714. S.M. Copley and B.H. Kear, Trans. TMS-AIME 239 (1967) 977. D.P. Pope and S.S. Ezz, Int. Metall. Rev. 25 (1984) 233. D.P. Pope and V. Vitek, in: High-Temperature Ordered Intermetallic Alloys, MRS Symp. Proc., Vol. 39 (Materials Res. Soc., Pittsburgh, 1985) p. 183. [8] D.P. Pope, in: High Temperature Aluminides and Intermetallics, eds S.H. Whang, C.T. Liu, D.P. Pope and J.O. Stiegler (TMS, Warrendale, PA, 1990) p. 51. [9] D.P. Pope, in: Ordered Intermetallics - Physical Metallurgy and Mechanical Behaviour, eds C.T. Liu, R.W. Cahn and G. Sauthoff (Kluwer, Dordrecht, The Netherlands, 1992) p. 143. [ 10] D.M. Dimiduk, J. Phys. (Paris) III 1 (1991) 1025. [11] S. Takeuchi and E. Kuramoto, J. Phys. Soc. Jpn 31 (1971) 1282. [12] S. Takeuchi and E. Kuramoto, Acta Metall. 21 (1973) 415. [13] C. Lall, S. Chin and D.P. Pope, Metall. Trans. A 10 (1979) 1323. [14] S.S. Ezz, D.P. Pope and V. Paidar, Acta Metall. 30 (1982) 921. [15] Y. Umakoshi, D.P. Pope and V. Vitek, Acta Metall. 32 (1984) 449. [16] E. Schmid, in: Proc. Int. Congr. Appl. Mech., Delft, 1924, p. 342. [17] E. Schmid and W. Boas, Kristallplastizit~it (Springer, Berlin, 1928). [18] Q. Qin and J.L. Bassani, J. Mech. Phys. Sol. 40 (1992) 813. [19] T. Suzuki, Y. Mishima and S. Miura, J. Iron Steel Inst. Jpn 29 (1989) 1. [20] D.M. Wee, O. Noguchi, Y. Oya and T. Suzuki, Trans. Jpn Inst. Met. 21 (1980) 237. [21] D.M. Wee, D.P. Pope and V. Vitek, Acta Metall. 32 (1984), 829. [22] EE. Heredia, G. Tichy, D.P. Pope and V. Vitek, Acta Metall. 37 (1989) 2755. [23] J.H. Schneibel and P.M. Hazzledine, J. Mater. Res. 7 (1992) 868. [24] Z.L. Wu, D.P. Pope and V. Vitek, Scr. Metall. 24 (1990) 2187. [25] Z.L. Wu, D.P. Pope and V. Vitek, Scr. Metall. 24 (1990) 2191. [26] Z.L. Wu, D.P. Pope and V. Vitek, Philos. Mag. A 70 (1994) 159. [27] M. Yamaguchi, S.R. Nishitani and Y. Shirai, in: High Temperature Aluminides and Intermetallics, eds S.H. Whang, C.T. Liu, D.P. Pope and J.O. Stiegler (TMS, Warrendale, PA, 1990) p. 63. [28] V. Vitek, M. Khantha, J. Cserti and Y. Sodani, in: Int. Symp. on Intermetallic Compounds, ed. O. Izumi (Japan Inst. of Metals, Sendai, 1991) p. 3. [29] V. Vitek, Y. Sodani and J. Cserti, in: High-Temperature Ordered Intermetallic Alloys IV, MRS Symp. Proc., Vol. 213, eds L.A. Johnson, D.P. Pope and J.O. Stiegler (Materials Res. Soc., Pittsburgh, 1991) p. 195. [30] N. Brown, Philos. Mag. A 4 (1959) 185. [31] D.P. Pope, Philos. Mag. 25 (1972) 917. [32] E. Kuramoto and D.P. Pope, Philos. Mag. 33 (1976) 675. [33] P.H. Thornton, R.G. Davies and T.L. Johnston, Metall. Trans. A 1 (1970) 207. [34] S. Miura, S. Ochiai, Y. Oya, Y. Mishima and T. Suzuki, in: High Temperature Ordered Intermetallic Alloys III, MRS Symp. Proc., Vol. 133, eds C.T. Liu, A.I. Taub, N.S. Stoloff and C.C. Koch (Materials Res. Soc., Pittsburgh, 1989) p. 341. [35] J. Bonneville and J.L. Martin, in: High-Temperature Ordered Intermetallic Alloys IV, MRS Symp. Proc., Vol. 213, eds L.A. Johnson, D.P. Pope and J.O. Stiegler (Materials Res. Soc., Pittsburgh, 1991) p. 629.
Anomalous yield behaviour of compounds with L12 structure [36] [37] [38] [39] [40] [41]
183
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185
CHAPTER 52
Dynamics of Dislocation Motion in L12 Compounds D. C. CHRZAN*
and
M.J. MILLS**
Sandia National Laboratories Livermore, CA 94551 USA
* Present address: Department of Materials Science and Mineral Engineering, University of California, Berkeley, CA 94720, and Materials Science Division, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA. ** Present address: Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43220, USA Dislocations in Solids 9 1996 Elsevier Science B. V All rights reserved
Edited by E R. N. Nabarro and M. S. Duesbery
Contents 1. 2.
Introduction 189 Dynamical simulations of dislocation motion 193 2.1. Description of the simulations 193 2.2. Results of continuum-based simulations 195 2.3. Implications of the simulations 205 3. Quantitative analysis of mechanical properties 211 3.1. Introduction to quantitative analysis 211 3.2. Pinning/depinning transition, scaling and creep 214 3.3. Future 235 4. Discussion and conclusions 239 5. Appendix A. Continuum-based dynamical simulations 243 6. Appendix B 245 6.1. Discrete model for dislocation motion 245 6.2. Discrete model for dislocation motion including thermal activation 250 References 251
1. Introduction The increase of the yield strength with temperature observed in the L12 intermetallic compounds has been the focus of intensive research efforts over the last several decades. The principle motivation for these studies has been to understand the fundamental cause for the inverse or "anomalous" temperature dependence of the yield strength. A characteristic of deformation in the anomalous regime is that the dislocation microstructure is remarkably homogeneous, with little apparent tendency to form the subgrains or cell structures found in pure metals. Therefore, it is plausible that this regime is amenable to dislocation-based modeling. In fact, central to most modeling efforts, including that presented below, is the assumption that the macroscopic mechanical properties of these compounds are determined largely by the behavior of individual dislocations, while the interaction between dislocations is far less important. In fact, it is generally accepted that the structure of dislocation cores in L 12 compounds has a profound influence on the macroscopic mechanical behavior. The first formal link between the mechanical properties and dislocation core structure in these compounds is provided by Takeuchi and Kuramoto [1]. This correlation is established even more clearly by the work of Paidar, Pope and Vitek (PPV) [2] who are able to predict the non-Schmid effects observed in these compounds, including the dependence of the yield strength on crystal orientation [1] and the asymmetry of the yield strength for tension and compression [3]. The key to this success is their careful analysis of the cross slip of a (101) screw dislocations from { 111 } planes, on which glide is relatively easy, to {010} planes [here, as throughout this chapter, the word dislocation refers to the a(101) superdislocation]. Cross slip to the {010} planes is thought to be driven by both the lowering of the antiphase boundary (APB) energy [4] and non-radial elastic interactions [5]. An estimate of the energy cost to form a small cross-slip segment from the (111) to the (010) plane leads to the calculation of the activation enthalpy for pinning. This enthalpy depends on temperature, stress and crystal orientation. In the PPV proposal, the critical resolved shear stress (CRSS) was defined as the stress at which screw dislocations achieve a mobile steady state. The dislocation-level model chosen to calculate this stress was the geometrically simple one originally suggested by Takeuchi and Kuramoto [1]. In this configuration, the pinning points were assumed to be distributed periodically along the dislocation line (see fig. 1). The yield strength was calculated as the applied stress at which the free dislocation bowing between the pinning points on the { 111 } plane achieves a critical bowing angle at which the dislocation can "break away" from the cross-slip pinning point. In spite of the simplified nature of the model, many important trends displayed by the yield strength in the anomalous regime were reproduced by this cross-slip pinning (CSP) model. The original CSP model can been called into question on the basis of several objections. First, transmission electron microscopy (TEM) observations [6-13] indicate a microstructure different from that anticipated from the CSP model: Most of the dislocation
190
Ch. 52
D. C. Chrzan and M. J. Mills
on (111) plane 9
(a )
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~
~-
-
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line length lies on the (111) plane with only localized points which have cross-slipped onto the (010) plane. Instead, the observed dislocation microstructure is dominated by Kear-Wilsdorf (KW) locks [6] - long, rectilinear (at low temperature) sections of the dislocation, aligned along screw orientation, with their their APB widely spread on the (010) plane. These KW locks are connected by segments of mixed character (mostly edge) lying on the { 111 } plane. As a second objection, it is noted that the CSP model exhibits a large strain-rate sensitivity which is not observed experimentally [1, 14]. Third, the CSP model predicts abrupt yielding, with no incorporation of strain hardening. However, high rates of strain hardening are an important feature of deformation in the anomalous regime (below the peak temperature) [14, 15]. The first two inconsistencies have been addressed through modifications of the CSP model [ 16, 17], and through the development of new models [ 18, 19]. Inconsistency between the observed microstructure and that proposed in the CSP model has led to an alternate picture of flow in which near-screw character dislocations are assumed to propagate by the lateral, kink-like motion of the mixed segments between the KW locks [6, 8, 9, 19, 20]. Since these mixed segments are not constrained to the periodicity of the lattice and can be quite long, these segments have been termed "superkinks." The assumption that superkink motion leads to propagation of the dislocation is consistent with the fact that only these segments lie on the macroscopic { 111 } glide planes [14, 21]. As proposed originally by Mills et al. [8], a dislocation structure containing numerous superkinks could be formed during initial expansion of a dislocation loop. An alternate method for generating superkinks, proposed by Molenat and Caillard [22] involves a sessile-to-glissile transformation of the dislocation in which the KW locks are not fully formed. Instead, the rearward superpartial of the partially formed lock structure is freed through thermal activation, and the dislocation glides forward for a variable distance before locking occurs once again. The primary evidence for this mechanism arises from in-situ TEM straining experiments.
w1
Dynamics of dislocation motion in L12 compounds
191
The early qualitative, superkink models [6, 8, 9, 11] share the same difficulty concerning the strain-rate sensitivity as the CSP model. In order to alleviate this difficulty, the most recent models [16--19] incorporate a stress dependence on the rate of unlocking from the pinning points or KW locks. If the unlocking process has a large activation volume, then a small strain-rate sensitivity can be derived from these models. This assumption eliminates the discrepancy between experiment and theory concerning the small strain-rate sensitivity, but the analysis of the simulated dislocation dynamics presented below suggests that the assumption may not be necessary. These latest models, particularly that of Hirsch [19] which incorporates a superkink mechanism, can explain most features of the yield strength in a manner consistent with the microstructure- including the temperature dependence, the variation with crystal orientation, the tension/compression asymmetry and the small strain-rate sensitivity. It is important to recognize, however, that this agreement for yielding behavior is obtained using models which are explicitly steady-state descriptions. Examples of dislocation propagation via the CSP and Hirsch-superkink models are shown in fig. 1. Of particular note is that in both models (a) the obstacles are distributed periodically and (b) no change in the obstacle density for a given dislocation is allowed. While condition (a) has been relaxed in a recent treatment by de Bussac et al. [18], condition (b) is still enforced. Hence, these models should be valid strictly only for steady-state, constant-structure conditions. An important aspect of deformation in the L12 compounds, however, is the absence of steady-state behavior under both constant strain-rate and creep conditions (below the peak temperature). For constant strain-rate tests, extraordinary rates of strain hardening are observed until just below the peak temperature [14, 15]. In contrast, the proposed models predict the immediate saturation of the flow stress, as shown in fig. 2(a). It is noted that incorporation of a thermally-activated bypassing of obstacles would round the point of yielding. However, since large activation volumes are required to explain the small strain-rate sensitivity, such models would still be characterized by a sharp point of yielding. Under creep conditions (fig. 2(b)), exhaustion behavior is observed at lower temperatures [ 14]. Slip traces and TEM investigation of the dislocation structures indicate the prevalence of octahedral glide after creep at intermediate temperatures [23], and it is expected that this behavior should extend to lower temperatures as well. Yet the existing models would predict that constant creep-rates are achieved immediately, with the creep-rate determined by the frequency of unpinning from obstacles and the frequency with which they reform (which determines the distance for free-flight motion). It should be noted that high rates of strain hardening are found even for single crystals deformed in a single slip orientation [23, 24]. The strain hardening appears to be an intrinsic property of the dislocation dynamics in these compounds. In all of the above models, it is assumed implicitly that there are two steady-state dynamical "phases" of dislocation motion: an immobile "pinned phase" and a mobile "unpinned phase." Note that the "pinned" and "unpinned phases" do not refer to the state of a local segment of the dislocation, but instead refer to whether or not the dislocation remains mobile for all time. The CRSS is, then, the stress demarcating the boundary between the pinned and unpinned dynamical phases. It is assumed further that the mobile unpinned phase is the appropriate choice for modeling of mechanical properties. However, the observed high rates of strain hardening, apparently intrinsic to the motion of
192
D. C. Chrzan and M. J. Mills
Ch. 52
Observed
Zcrss
(a)
Strain Observed
~
(b) Time Fig. 2. (a) Schematic comparison of the constant strain-rate test predicted from the constant-density-of-obstacle models with the typical experimental result. (b) Schematic comparison of the primary creep transients predicted by the constant-density-of-obstacles models and the observed primary creep transient.
individual dislocations, force one to reconsider and dismiss this latter assumption. Under experimentally observable conditions, mobile dislocations simply do not remain mobile for all time. One concludes that in order to understand fully the mechanical properties of these compounds one must understand both the pinned and unpinned "phases" of dislocation motion, as well as the dynamics governing the transition between them (i.e., the pinning/depinning transition). This realization allows several important observations to be made. First, the experimentally measured yield strength, arbitrarily defined as the 0.2% offset stress, is not obviously equal to the transition stress. Second, the pinning/depinning transition is the manifestation of the collective behavior of the pinning points. (A single pinning point cannot pin completely the dislocation, it takes many pinning points to do so.) It follows that the dynamics of the transition are related intimately to the spatial and temporal correlations between the pinning points (or KW locks). Therefore, any reliable theory of dislocation dynamics in these compounds must clearly justify the imposed correlations, or, better yet, let the correlations arise from the dynamics as they do in nature. Third, the dynamic, nonequilibrium nature of the transition implies that fluctuations play an important role and cannot be neglected in any complete formulation of the mechanical properties. Thus the "steady-state" phases are steady-state only in a statistical sense. Fluctuations about the mean values must be understood and included in the analysis. In the next section of this chapter, a dynamical simulation of dislocation motion in the L12 compounds displaying the yield strength anomaly is developed and studied. All assumptions concerning both the spatial and temporal correlations between the obstacles
w
Dynamics of dislocation motion in L12 compounds
193
are eliminated. Instead, simple, physically motivated rules are adopted to govern the kinetics of dislocation motion, pinning and unpinning. Then, the resulting simulated dislocation structures and dynamics are evaluated. The dislocation structures are found to be in qualitative and quantitative agreement with TEM observations. The most important result of the simulations is that the dislocations display the proposed pinning/depinning transition and allow, for the first time, study of the dynamics of dislocation motion in the pinned phase. It is discovered that many of the experimental observations can be understood qualitatively as arising from the pinned phase of dislocation dynamics. The successful qualitative understanding of the pinned phase suggests that further work toward developing a quantitative theory of the mechanical properties is warranted. As a first step towards this theory, the third section of this chapter relates directly the collective dynamics of the pinning/depinning transition to the experimental measurement of the primary creep transient. The algebraic expression derived to describe the primary creep transient is found to be in quantitative agreement with recent experiments. This agreement provides striking evidence that the understanding of the pinning/depinning transition can be exploited to span the length- and time-scales to produce a microstructure-based theory for the mechanical properties of these compounds. Finally, the fourth section of this chapter presents a brief discussion of the relationship between the results presented here and other models of flow. In addition, some general conclusions are drawn.
2. Dynamical simulations of dislocation motion A computer simulation of dislocation motion has been constructed in order to study the dislocation structures which develop, as well as the dynamics by which dislocations move [25]. The simulations are described briefly in the next section, and more completely in Appendix A. The structural and dynamical features of the simulated dislocations are then presented. These features are discussed in terms of their implications for macroscopic mechanical properties.
2.1. Description of the simulations In these simulations, the dislocation is modeled as a discrete array of points. The stress on each point is the sum of the applied stress and the bowing stress due to dislocation line tension (assumed isotropic). The simulation of dislocation motion in this case requires three components: (1) a kinetic law for the motion of the free segments; (2) a mechanism for the formation of cross-slipped points or segments; and (3) a prescription for defeating the pinning points. The particular rules chosen for these three key components are now addressed. A simple, damped equation of motion for all dislocation segments (independent of line orientation) is used to describe the motion of the free segments. Dislocation motion on the { 111 } planes in these compounds is inherently "easy", as indicated by the low flow stresses observed at low temperatures (i.e., in the absence of abundant cross-slip pinning).
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Ch. 52
In-situ TEM observations [22] and optical microscopy of slip-band traces [26] suggest rapid dislocation motion. In addition, extraordinarily rapid plastic strain upon loading of single-crystal creep specimens at low temperatures is observed [27]. Consequently, under conditions where cross-slip pinning is influencing strongly the flow stress (anomalous regime), motion of free segments may be assumed to approach free-flight conditions [28]. The possible temperature dependence of the free-flight velocity has been ignored in the present simulations. For simplicity, a damped equation of motion linear in stress has been chosen. The general features of the simulations are not believed to be sensitive to the precise stress dependence chosen here. Cross-slip pinning is assumed initiated in a manner similar to that described by PPV [2]. This allows for the same temperature (and stress-orientation) dependence for the pinning frequency as given in PPV. However, in contrast to CSP models where a particular arrangement of pinning points is assumed, here a given dislocation orientation has a well defined probability for pinning, and the pinning points form randomly, governed only by the pinning probability. The probability for cross slip pinning is largest for pure screw orientation, and decreases with increasing deviation from screw character. The functions assumed to accomplish this are given in the Appendix A. Several dynamical paths for the formation of complete KW locks [extended on the (010) cross slip plane] have been offered recently by Hirsch [19]. However, formation of the KW locks is initiated through the PPV-like state, and so this state has been assumed here. It should be noted that an explicit description for the possible expansion of the pinning "point" into a KW lock extended laterally along screw orientation is not considered in the simulations. Whether or not a pinning point remains spatially localized depends on the relative mobility of the short edge jog (J) on the (010) plane compared with the mobility of the adjacent dislocation on the (111) plane (SD), as shown in fig. 3. If the dislocation on the (111) plane can sweep past the point at which cross slip initially occurs (at "A" in the figure), then bowing of SD will oppose the motion of J to the right. Since the relative mobility of SD and J can not be determined directly, it is assumed for simplicity in the present simulations that the cross slipped segments remain localized. A key result of the simulations, however, is that in spite of this assumption one obtains features which are dynamically equivalent to KW locks - obstacles which immobilize extended segments near screw character. Pinning points are assumed to dissolve when an adjoining free dislocation segment achieves a critical bowing angle (0c in fig. 3) with respect to screw orientation (0c corresponds to a critical force exerted on the pinning point by the adjacent dislocation segment). When this occurs, the previously pinned point is unpinned and allowed to move as a free segment. Theoretical estimates for 0c range from 10 ~ [2] to about 50 ~ [19], depending primarily on the extent of cross slip which has occurred on the (010) plane. An angle of 30 ~ has been chosen for the present simulations, although other values of 0c are found to give qualitative similar results. Note that the unpinning is assumed to be athermal. As is discussed below, the overall time-scales for the simulated dislocations are much too short to account for many of the experimental observations. This result suggests that the unpinning may, in fact, be a thermally-activated process as argued by other authors. This point is addressed further in section 3 of this chapter.
Dynamics of dislocation motion in L12 compounds
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(b) Fig. 3. Depiction of the formation of a pinning point and the subsequent evolution of the dislocation structure. (a) The cross-slipped "point" is most likely to form along the screw-oriented portion of the dislocation SD (dissociated into two superpartial dislocations SP1 and SP2). (b) The advancement of the jog J to the right in the figure begins immediately but, the bowing adjacent dislocation segment rapidly bows about the pinning point exerting a force that opposes the advancement of the jog J. When the adjacent segment bows to an angle 0c the force exerted on J overwhelms the force driving expansion of the pinning point and forces the jog to move to the left in the figure, thus dissolving the pinning point. Two types of simulations are conducted. In the first, the motion of a complete loop is considered. The second type of simulation propagates a finite length of near-screwcharacter dislocation and is used to study these slowest-moving components of the loops.
2.2. Results of c o n t i n u u m - b a s e d simulations
2.2.1. General structural and dynamical features Simulation results for the expansion of an initially circular dislocation loop at two different stresses are shown in fig. 4. Five configurations of the loops after equal time increments are shown. For both stresses, the near-edge portions of the loops move more rapidly than the near-screw portions. This is expected since pinning takes place only near screw orientation. The effect of the pinning on the overall shape of the loops is clearly stress dependent: the loop aspect ratio is roughly 5:1 at the lower stress and about 4"3 at the higher stress. The distribution of pinning points (indicated as heavy black dots) along the loops is significantly different at the two stresses. At the lower stress, the pinned points clump together into segments of near-screw orientation. These highly pinned segments are connected by unpinned segments of mixed character. As a result, the near-screw portion of the loop takes on a "stepped" appearance. In contrast, at the high stress, the pinning points are more evenly distributed, resulting in the much flatter appearance of the near-screw portions of the loop. The clumping tendency of the pinning points, which is apparent particularly at the lower stress, is a key result of the simulations. The origins of pinning correlations are
196
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D. C. Chrzan and M. J. Mills
1 ~tm
b
m
(a)
(b)
Fig. 4. Simulations of the expansions of dislocation loops at a temperature of 300~ and a stress of (a) 240 MPa and (b) 350 MPa. Configurations after identical time steps of 2 • 10 - 7 and 1.5 x 10 - 7 seconds, respectively, are shown. The initial configuration, not shown, is a circular loop 1 ~m in diameter. shown schematically in fig. 5. Let us consider the motion of a free dislocation segment of mixed character which is adjacent to a pinned segment. This situation exists, for example, near the leading edge of an expanding loop. The shaded lens-shaped region in the figure corresponds to the region in which an additional pinning point can form and create an immobile segment of the dislocation. The total probability for a pinning point to form in the lens-shaped region is related to time that the dislocation spends in that "pinning zone." At low stresses, the pinning zone increases in size (since the radius of curvature required to achieve 0c is larger) and the dislocation velocity decreases. Thus, there is a high probability of forming a new immobile segment. At high stresses, the lens-shaped region becomes smaller, and the dislocation moves more rapidly, dramatically reducing the probability of creating an additional immobile segment. This is the physical basis for the change in dislocation structure apparent in fig. 4, and also lies at the heart of the pinning/depinning transition. The correlations in the formation of pinning points determine not only the simulated structures of the loops, but also the dynamics which govern the motion of the near-screw portions of the dislocation. At the lower stress, the highly-pinned segments near screw character are static, and the overall propagation of these portions of the loop occurs by the l a t e r a l motion of the unpinned segments of mixed character. In the course of their motion, these mobile, mixed segments annihilate the highly-pinned segments of nearscrew orientation in their path. New, highly-pinned segments are formed in the wake of this motion. This type of lateral motion is similar to that envisioned in the "superkink"
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Dynamics of dislocation motion in L12 compounds
197
Fig. 6. TEM image of a typical superdislocation. The segments labelled KW are the completely cross slipped Kear-Wilsdorf locks, and the segments labelled SK are the superkinks which lie on the (111) plane. models. It should be noted that only a small fraction of the total dislocation line length is mobile at a given time. At the high stress, the near-screw portion of the loop no longer proceeds by lateral motion of superkinks, but rather by forward bowing (in the direction perpendicular to the Burgers vector) between the pinning points. This mode of motion is closer to that envisaged in the CSP model, although the distance between pinning points is clearly not periodic. Two points are important to emphasize: (1) Loop expansion and motion of the near-screw portion of the loop clearly can proceed at stresses much lower than that required for motion in the CSP mode; (2) Superkink-type motion is exhibited by this model, even though the pinning points are assumed spatially localized. The highly-pinned segments resemble the KW locks observed using TEM. In fact, the overall "stepped" structure of the dislocation is in good qualitative agreement with previous microstructural analysis [6-13]. An example of the dislocation structures which are found after creep deformation at room temperature [29] is shown in fig. 6. It should be noted that while the pinned, static segments in the simulated dislocation structures resemble KW locks, they are not structurally identical since the KW locks are a contin-
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D. C. Chrzan and M. J. Mills
Ch. 52
uous, rectilinear screw segment dissociated on the (010) plane. However, the fact that extended, static segments near screw orientation arise naturally in the simulations suggests a mechanism by which KW locks might develop after being nucleated by localized cross slipped points (section 2.3.7). An important feature of the dynamics of dislocation motion is that propagation is governed by the lateral motion of superkink-like segments (in the low stress regime). This regime, the pinned phase, is thought to be the most significant for understanding macroscopic flow, particularly under constant strain-rate conditions, since loop expansion and forward dislocation motion can proceed readily in this phase (for a finite time). There is no need to apply the stresses necessary to reach the high stress regime since dislocations exhibit ample mobility at lower stresses. Before presenting a detailed analysis of the dynamical features of the model, a more quantitative characterization of the dislocation structures, including the average population of superkinks, is pursued. 2.2.2. Distribution of superkink lengths
An apparent feature of the simulated dislocation configurations at the lower stress shown in fig. 4 is that at any given time, a distribution of superkink lengths is observed. This distribution is an important to the dynamics of dislocation motion, since these are the only potentially mobile dislocation segments. The actual superkink distribution is determined by the correlations between pinning points described above. An example of the calculated distribution of superkinks based on simulations performed in the low stress regime is shown in fig. 7. The superkink density p(l) is defined as the probability of finding a free segment whose length lies between l and l + dl when selecting a free segment at random. In practice, this probability distribution is calculated by simulating the movement of a near-screw portion of dislocation and periodically counting and binning the number of segments of length l. The most notable feature of the distribution is that it contains two different branches. For small segment lengths, the distribution is well characterized by an exponential form: p(l) ,,~ e -t/t~
(1)
Another branch of the distribution is seen for larger segments, where the distribution also appears nearly exponential (with a larger value of l0 than for the small-length branch). The transition between the two branches of the distribution occurs near a critical length, lc: Ic =
2/zb sin(0c), "ra
(2)
where # is the shear modulus, b is the Burgers vector of the dislocation and 7"a is the applied stress. Unpinned segments aligned along pure-screw orientation that are shorter than lc are immobile; segments longer than Ic are mobile. Because of the critical angle condition for depinning, it is possible to have a kink shorter than lc that is still mobile, but these kinks are relatively few in number and move fairly slowly. For practical purposes, the length lc defines the distinction between mobile and immobile kinks. The conclusion
Dynamics of dislocation motion in L12 compounds
w
199
10 o | |
1 0 "1 -
10 .2
I D I I | II I |
I | |
Cx.
10 .3 _
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| | | | I | | | |
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0% 0 mo<
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,
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00 I ....
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4000
Fig. 7. Probability that a superkink selected at random in the crystal will be of length l as calculated from the simulations for a stress of 220 MPa and a temperature of 300~ The steep regime at small lengths follows the form of eq. (1). The critical length, lc, is 110 nm, and closely corresponds to the transition to the second branch of the distribution.
one draws from fig. 7 is that the distribution of mobile kinks differs substantially from the distribution of immobile kinks. An exponential dependence for the distribution of superkink lengths may be qualitatively understood based on the correlated pinning process illustrated in fig. 5. As the dislocation segment sweeps around the pinning point there is a probability per unit time p that the segment will pin. Assume that in a time step At, the segment adjacent to the pinning point moves a distance vat, where v is the local average velocity for the moving segment. Then, the probability for the segment to advance a distance l -- rwAt (implying that the segment has not pinned after n time steps) is (1 - p)n = (1 - p)l/(vAt), which is of the form p(1) ~ e -t/t~ with l0 - - l n ( 1 - p)/(vAt). These arguments explain the exponential form for the "immobile" branch of the simulated distribution. The "mobile" branch of the distribution has a more complicated structure that cannot be understood as simply. This "mobile" branch in composed of very large superkinks much larger than could be generated by the mechanism outlined in the previous paragraph. The physical mechanism by which these kinks evolve is illustrated in fig. 8. As the immobile dislocation structure is laid out, through the correlated pinning, the superdislocation assumes a gradual slope relative to screw orientation. After the formation of each individual pinning point there is a finite probability that the adjacent segment will move far enough past the screw character to generate a mobile superkink moving in the direction opposite to that of the initial superkink. This newly spawned
200
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D. C. Chrzan and M. J. Mills
J
I ! I ! I
b
Fig. 8. Origins of the large superkinks. As the superkink labelled (1) moves to the left in the figure, it leaves behind the slightly inclined, pinned structure shown. A mobile superkink labelled (2) is spawned at A in the figure, and immediately advances to the right in the figure. The slope of the pinned structure insures that this newly spawned superkink increases in length and remains mobile for some time. This mechanism allows for the production of superkinks much longer than Ic.
mobile superkink sweeps back along the superdislocation and increases in length. It is by superkink multiplication processes such as these, as well as superkink interactions described in more detail below, that the mobile branch of the distribution is established. Thus, the two branches of the superkink population distribution are not independent. The immobile branch is established during the motion of the mobile superkinks. Conversely, the area swept out by a mobile superkink depends on the spatial configuration of the immobile segments. Experimental confirmation of an exponential superkink distribution has been provided recently by Couret et al. [31] who used weak-beam TEM to measure superkink heights in Ni3Ga after deformation under constant-strain-rate conditions. This distribution was found to follow an exponential form similar to eq. (1) p(h) ~ e -h/h~
(3)
with h0 = 13 nm at 473 K and h0 = 9 nm at 673 K. The resolved shear strength at yielding in these tests were 150 MPa and 260 MPa at 200~ and 400~ respectively. (At 200~ the distributions revealed an anomalously large number of superkinks with heights corresponding to the (111) APB width [31 ]. This is commented on further below.) The distribution of heights obtained from the simulations can be estimated from the distribution of superkink lengths by accounting for the average angle of inclination of the superkinks [25]. The values so obtained from the immobile branch of the distributions range from h0 = 3 nm for an applied shear stress of 220 MPa and a temperature of 573 K to h0 = 13 nm for an applied stress of 350 MPa and a temperature of 573 K, which compare favorably with the values measured in the experiment. The appearance after room temperature deformation of an excess number of superkinks corresponding approximately to the APB width is discussed in section 4. The successful comparison of the experimental results with those predicted from the simulation is encouraging. However, this result begs the question as to why the mobile
$2.2
Dynamics
of
dislocution motion in LI2 compounds
20 1
roo 4 @
Fig. 9. Superkink distribution as a function of stress for simulations performed at 300°C
branch does not appear in the experimental measurements. It is speculated that the experimental determination will be strongly skewed toward the immobile branch due to (a) the thin-foil geometry used for TEM and (b) inadequate statistics for determining properties of the mobile branch. It should also be noted that the comparison between simulated and experimentally measured distributions is not straightforward. The experimentally observed distributions are determined after constant strain-rate tests in which the stress varies continually. The simulations are all performed at a constant stress. As is discussed in section 2.2.3, a principle result from the simulations is that exhaustion of dislocation motion occurs frequently, especially at the low stresses initially applied in a constant strain-rate test. Since an exhausted dislocation has no mobile superkinks along its length, the dislocation substructure should be comprised primarily of static dislocations and, therefore, the experimental measurement of the superkink distribution will be dominated by the immobile branch. The form of the superkink distribution raises doubts about trying to understand the mobility of the dislocations based on an average superkink with an average length. From the distributions it is apparent that the average superkink is in fact immobile since the distribution is skewed heavily towards the shorter superkink lengths. Consequently, attempts to understand the plastic deformation of the LIZ compounds based on this average superkink are inappropriate. Rather, the simulations indicate that flow in these compounds is determined largely by the population of mobile superkinks which is actually a small subset of the total superhnk population. These simulations offer the opportunity to study the properties and dynamics of these mobile superkinks. The dynamic aspects of the mobile superkinks are discussed in the next section. The effect of stress on the simulated superkink distributions can be seen in fig. 9. As stress increases, the value of I , (eq. (2)) decreases, resulting in a decrease in the extent
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D. C. Chrzan and M. J. Mills
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of the static branch. Meanwhile, both the extent and slope (on the log-normal plot) of the mobile branch increase with stress. These trends are physically reasonable since it is expected that the total mobile line length of the dislocation increases with stress. This interpretation is also consistent with the trends seen in the simulated dislocation configurations of fig. 4. At low stresses (in the pinned phase), most of the dislocation is static, while at the high stress (in the unpinned phase) most of the line length is mobile. The simulations therefore demonstrate a tendency toward a behavior similar to that assumed in the CSP model. However, this behavior is found in the simulations only at stresses so high that pinning has little effect on the dislocation mobility (as evidenced by the nearly circular loop configuration of fig. 4(b)). Dislocation motion can instead proceed at much lower stresses via superkink motion while most of the line length is in fact pinned and immobile. As will be discussed in the next section, in this low stress "pinned phase" the overall motion of the dislocation is rendered extremely sensitive to fluctuations in the superkink population along each individual dislocation. The origins and consequences of these fluctuations are now addressed. 2.2.3. Fluctuations in dislocation velocity and exhaustion o f dislocation motion
The superkink distributions presented above reflect time-averaged quantities determined from a large number of dislocation configurations. However, in the low stress regime the velocity of a particular, near-screw character dislocation is determined by the number and length of superkinks present on that dislocation. The population of superkinks along a given dislocation varies considerably as a function of time, and, consequently, the velocity of the dislocation fluctuates. In fig. 10, the fractional velocity of a near-screw portion of dislocation (averaged over its length) is shown as a function of time for four different stress levels. This average fractional velocity is defined as
1 dA Vavg-- v f L d t '
(4)
where A is the area swept out by the superdislocation, "Of is the free flight velocity of the superdislocation, and L is the width of the crystal (the nominal length of the dislocation). At the highest stress, the value of "Oavgshows small fluctuations, which is consistent with the fact that most of the dislocation is moving at nearly the free flight velocity. As the stress decreases, the amplitude of the fluctuations in the value of "Oavg increases. At the lowest stress shown, the root-mean-square fluctuation of the average velocity is approximately 25% of "OavgFigure 11 shows four sequential configurations of the same near-screw portion of dislocation moving in the low stress regime. As a given superkink moves laterally along the dislocation, its length can either increase or decrease, depending on the local inclination of the pinned structure in its path. This results in a change in the overall dislocation velocity. More dramatic changes in velocity can arise due to (a) the spawning of a new superkink during the motion of an existing one (see also fig. 8 above), (b) interaction of two moving superkinks which, in general, leads to the annihilation of one of them, and (c) superkink annihilation at the ends of a finite length of dislocation. If two superkinks
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Dynamics of dislocation motion in LI 2 compounds
203
1.0
0.8
(a) 0.6
(b)
(c)
t~
0.4
(d)
0.2
0.0
9
0.0
'
'
I
0.2
I
0.4
'
"
I
"
'
0.6
'
!
0.8
9
'
1.0
time [10 .6 sec] Fig. 10. A plot of Vavg vs. time at 300~ for four stresses: (a) 400 MPa, (b) 350 MPa, (c) 300 MPa and (d) 250 MPa. The most notable feature of this plot is the fact that the velocities are not constant but display large fluctuations. The magnitude of the fluctuations increases relative to the magnitude of the velocity as the stress decreases. These fluctuations can ultimately arrest the dislocation's motion.
of the same sense (i.e., moving in the same direction) interact, then the net length of superkink on the whole dislocation is conserved more or less by the formation of a single, long superkink. Conversely, if two superkinks of opposite sense interact, then annihilation occurs, resulting in a dramatic reduction in the net length of superkink for that dislocation. It is the complex interplay between these processes of multiplication and annihilation which lead to fluctuations in the superkink population along a given dislocation, and to the time-dependence of the average fractional velocity. The influence of cross slip pinning does result in the retardation of the dislocation velocity relative to the free flight velocity. However, the effect on the dislocation mobility in the unpinned phase is not too dramatic since even at the lowest stress shown in fig. 10 the mean value of the velocity is still a significant fraction of the free flight velocity. This result indicates that the superkinks, while present, efficiently mediate plastic flow. A far more dramatic and important effect of the cross slip pinning is related to the velocity fluctuations described above. At lower stresses in which a relatively small number of superkinks are responsible for propagating the near-screw portion of the dislocation, the various superkink annihilation processes can result in a dislocation configuration which lacks mobile superkinks: the dislocation is rendered immobile. It is the fluctuation to this exhausted configuration that is believed to be an important, intrinsic source of strain hardening in the L12 compounds. Dislocation exhaustion is a prominent feature of the simulated dislocation dynamics at lower stresses. Exhaustion occurs due to statistical fluctuations in the superkink population along each dislocation. Consequently, given an initial population of mobile
204
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D. C. Chrzan and M. J. Milb
1 ~tm |
(4)
(3)
(2)
(1) Fig. 11. A time sequence of the motion of a near-screw dislocation segment advancing under an applied stress of 390 MPa and a temperature of 500~ The scale has been magnified by a factor of ten in the vertical direction for clarity. In addition, the dislocations have been offset in the vertical direction so that the time steps can be easily distinguished. (1) The dislocation as it appeared 4 x 10 - 9 seconds after the edge segments of the loop exited the crystal. (2) and (3) are the configurations at 8 x 10 - 9 and 12 x 10 - 9 seconds, respectively, after the edges of the loop exited the crystal. Configuration (4) is the completely exhausted state occurring 15 • 10 - 9 seconds after the edge segments of the loop exited the crystal. dislocations of finite length, a distribution in the time required for them to exhaust is expected. Such exhaustion distributions have been calculated using the simulations, and they are found to obey approximately an exponential cut-off at longer times: N(t)
~ e -t/t~
(:5)
Since strain hardening in these materials should be related to intimately the exhaustion distribution, it is crucial to characterize completely the features of the distributions, and the dependence of the distribution on model parameters. For example, the value of to is found to depend on the stress relative to the pinning frequency (i.e., temperature), as well as the length of the dislocation [25]. Deviations from the exponential form are also observed at shorter times. Unfortunately the simulations are computationally intensive, making a complete, statistical analysis of the exhaustion distribution intractable. In section 3, a discrete model describing dislocation motion is presented which yields resuits similar to the continuum-based model, but enables quantification of the exhaustion behavior. The stress dependence of the exhaustion behavior is noteworthy. At lower stresses where dislocations propagate via superkink motion, exhaustion is always observed (for a dislocation of finite length). For sufficiently high stresses at a given temperature, the strong correlations between pinning points no longer exist and dislocation motion
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Dynamics of dislocation motion in L12 compounds
205
is never observed to exhaust (over time-scales amenable to computer modeling). It is argued in section 3 that this change in exhaustion behavior is direct evidence of a non-equilibrium dynamical-phase transition between the low-stress pinned phase and the high-stress unpinned phase of dislocation motion. The characterization of this transition, and in particular the distribution of exhaustion events, allows the prediction of the form of the primary creep transient in the compounds under consideration.
2.3. Implications of the simulations The results of the simulations allow several general expectations to be stated concerning important aspects of the observed, macroscopic behavior. However, it is noted that the quantitative prediction of the mechanical properties, particularly under constant strainrate conditions, requires: (1) the assumption of a "reasonable" distribution of stresses for the activation of intrinsic sources, (2) a complete understanding of the dislocation exhaustion and reinitiation behavior with varying stress, and (3) incorporation of interdislocation forces for the estimate of internal stresses. These three aspects of the problem have not been addressed within the current simulations. Nevertheless, the simulations lend insight into the expected behavior, as summarized below.
2.3.1. Lack of anomaly at small strains As the stress increases in a constant-strain-rate test and available sources begin to operate, the near-edge portions move first and extend outward, leaving behind long near-screw dislocations. Since the motion of the edges is not hindered by the effects of cross slip pinning (provided the stress is high enough to enable loop expansion), then no yield strength anomaly is expected for very small strains, as is found experimentally [14]. 2.3.2. Lack of a sharp yield point In order to achieve larger strains (e.g., the macroscopic 0.2% offset), motion of all dislocation components is required (so that sources remain operative). The near-screw portions of the dislocations begin to propagate by the motion of the largest superkinks generated during initial loop expansion. Since there is naturally a distribution in the sizes of superkinks along each dislocation, and from dislocation to dislocation, a spectrum of stresses will be required to cause propagation of the near-screw dislocations. This will round the flow curve at the yield point, in agreement with experimental observations. In addition, strain hardening due to exhaustion (see section 2.3.5) is expected even at small strains. This anticipated behavior is in contrast to that predicted by the previous models of flow which assume a periodic array of cross slip obstacles [1, 2, 16, 17, 19]. 2.3.3. Anomalous temperature dependence of flow Since the pinning event assumed in the present simulations is that analyzed by PPV, the anomalous temperature and stress-orientation dependence of flow can be incorporated naturally through the value of the activation enthalpy, AH in eq. (A.3). Making the simplifying assumption that the yield strength corresponds to the stress level at which near-screw dislocations begin to move, then the temperature dependence inherent in
206
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D. C. Chrzan and M. J. Mills
400 350
s ss S
300
sS
250 200
150 100
50
300
I
I
I
I
400
500
600
700
temperature [K] Fig. 12. Applied stress needed to achieve a velocity of V a v g --" 0.25 as a function of temperature assuming the form for the activation enthalpy given in eq. (A2). The solid circles refer to points calculated using the simulations. The solid line is a guide for the eye. The dashed line is the prediction of the CSP models using the same activation enthalpy.
the simulations can be explored. In fig. 12, the applied stress required to achieve a mean dislocation velocity of approximately Vavg = 0.25 is plotted as a function of temperature. At this velocity, dislocations are moving by the lateral motion of superkinks. The exponential form resulting from the assumption of a periodic array of pinning points generated by an Arrhenius relation such as eq. (A.3) is also plotted for reference (with AH = 0.32 eV). For both plots, the stress-dependence of the activation enthalpy has been ignored. These results demonstrate that the temperature dependence resulting from a periodic array of pinning points is similar to that exhibited by the simulations. While the simulated temperature dependence is in reasonable agreement with the experimental observations, a more quantitative comparison with experiment is not appropriate since many important effects have been ignored in these calculations.
2.3.4. Small strain-rate sensitivity A remarkable feature of deformation in the anomalous regime is the reported very small strain-rate sensitivity [ 1, 14]. This characteristic implies that both the dislocation velocity and mobile density adjust to the new conditions. If only the dislocation velocity were to change, a strong strain-rate sensitivity is expected, based on the present simulations, since the velocity is assumed to vary linearly with stress. However, let us consider the change in mobile superkink density which occurs upon a stress increase. This change corresponds to a decrease in lc (eq. (2)). Because the superkink number density decreases exponentially with length [see eq. (1) and fig. 7], a small increase in stress results in a large increase in the number of mobile superkinks. Since the superkink distribution is averaged over many dislocation configurations, a large change in the mobile dislocation density should also result. In other words, since the superkink distribution depends strongly on length (i.e., stress), a weak strain-rate sensitivity should result. It is also noted that the determination
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Dynamics of dislocation motion in L12 compounds
207
of the strain-rate sensitivity is performed generally using separate samples such that the transient behavior between strain-rates is not observed. In fact, an instantaneous strainrate sensitivity has been reported by Thornton, Davies and Johnston (TDJ) [14], although this effect has not received much attention in the subsequent literature. It is suggested that the transients associated with strain-rate (and stress) changes may be helpful in exploring the dynamics of superkink motion, multiplication and annihilation.
2.3.5. Origins of strain hardening The exhaustion of dislocation motion provides a natural explanation for the extremely high rates of strain hardening observed within the anomalous regime under both creep and constant strain-rate conditions. An intrinsic source of hardening such as dislocation exhaustion (rather than a forest hardening mechanism) is necessary to explain the high hardening rates observed at lower temperatures and small strains. In contrast to prior theories, the simulations show that yielding and hardening are reflections of the same dislocation dynamics; attempts to distinguish these as separate mechanisms are artificial. Under constant strain-rate conditions, dislocation exhaustion contributes to strain hardening in several ways: (1) Reinitiation of dislocation motion may require a significant increase in the applied stress, depending upon how much shorter than Ic is the longest immobile superkink; (2) Blockage of operating sources thus requiring increased stress to activate new sources; (3) Storage of dislocations and increased interdislocation forces leading to back stresses. Although not treated in the present simulations, KW locks on exhausted dislocations will be permitted sufficient time under stress at higher temperatures to bow out on the cross slip cube plane leading to forest hardening of subsequent octahedral glide. This could provide an additional hardening effect for deformation orientations with a significant applied shear stress on the cross slip (010) plane [7]. Conversely, this bowing on the (010) plane produces strain which could lead to a reduction in the rate of strain hardening at higher temperatures. The quantitative prediction of strain hardening rates requires consideration of all these contributions. The first step in modeling these effects is a complete understanding of the exhaustion process itself.
2.3.6. Thermal reversibility of flow and the Cottrell-Stokes experiment The thermally-reversible and irreversible contributions to the flow stress have recently been explored [10, 32-35] using Cottrell-Stokes-type experiments [36]. In these tests (see fig. 13), a specimen is deformed at constant strain-rate and high temperature, Thigh, to some point on the flow curve. The test is then interrupted and the load removed while the specimen is cooled to a temperature, Tlow. Deformation at the initial strain-rate is then resumed at Tlow, and the subsequent flow behavior is compared with that which would have been obtained for monotonic tests at both Thigh and Tlow. If flow is perfectly thermally reversible, then the flow curve following the temperature change should follow the monotonic one at Tlow. This is the behavior expected from the CSP models since the frequency of pinning (and hence the dislocation mobility) would change instantaneously with temperature. At the other extreme, if the flow stress is determined by substructure formation and obstacle hardening, then upon deforming once again at Tlow, the stress should return to its previous high value at Thigh. The observed behavior in Ni3A1 lies between these two extremes. A large fraction of the stress is recovered on changing temperature, but a hardening increment remains.
208
D. C. Chrzan and M. J. Mills
Ch. 52
Thigh Reversible r~ r~
Irreversible T]ow
Strain Fig. 13. Diagram depicting the typical results of Cotrell-Stokes-type experiments. The sample is deformed at Thigh to a predetermined strain, unloaded, cooled and restrained at the same strain-rate at Tlow. See the text for details. This non-reversible hardening corresponds to approximately the strain hardening (crflowO'0.2%yield) imparted to the sample during deformation at Thigh. An additional feature of these tests is that the rate of strain hardening also appears thermally reversible. Thus, in order to reconcile these tests, a deformation model must allow for an immediate response (dislocation mobility change) to a temperature change, as well as provision for permanent strain hardening. The results of the present simulations offer a simple, consistent explanation for the Cottrell-Stokes results. The irreversible contribution arises from strain hardening due to dislocation exhaustion and storage (see section 2.3.5) during deformation at Thigh. Since dislocations exhausted at Thigh are not remobilized at the lower stress and temperature, strain softening should not occur and a permanent hardening increment is expected. Now consider the thermally-reversible contribution to the flow stress. Since the pinned dislocation line length is not altered immediately by the temperature change, one might not expect the same strain-rate to be achieved at a lower stress at TZow. The answer lies, once again, in the existence of a distribution of superkink lengths. Dislocations which were mobile at the higher stress and Thigh can contain superkinks significantly longer than lc. These large superkinks can, in principle, move at stresses lower than the final stress at Thigh. At a lower stress and Thigh, these superkinks would be quickly immobilized since the high rate of pinning would overwhelm their lateral motion. However, at Tlow the same large superkinks can move and quickly spawn new superkinks due to the lower pinning frequency. Concomitant with superkink motion is the conversion of pinned structure formed at Thigh to structure consistent with Tlow. Thus, in the course of superkink motion, an existing m o b i l e dislocation quickly loses its "memory" of the refined structure at the higher temperature. The results of dynamical simulations intended to explore the thermal reversibility of the dislocation mobility confirm the above interpretation. One such calculation is shown in fig. 14 in which the average velocity, Vavg (see eq. (4)), of a single, nearscrew-character dislocation is plotted as a function of time. For t < tchange, the pinning frequency is consistent with that at 700 K, and an applied stress of 360 MPa is imposed
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Dynamics of dislocation motion in LI 2 compounds
209
i t i t
0.4a
0.3
0.2
0.1 700 K %100 M e a
5 x 1 0 -8
t [seconds] Fig. 14. The average velocity of a single, near-screw oriented dislocation plotted as a function of time. The early time portion of the plot displays the velocity at 700 K for a dislocation moving under an applied stress of 360 MPa. At the dashed line (corresponding to tchange described in the text) two curves are shown: (1) the dislocation continues to advance under an applied stress of 100 MPa at a temperature of 400 K, and (2) the same dislocation exhausts under an applied stress of 100 MPa at 700 K. See the text for details.
such that the mean value of '0avg ~ 0.25 (see fig. 10). These conditions correspond to dislocation propagation via lateral superkink motion, and the large fluctuations in the velocity typical of this state are apparent. For t > tchange, the pinning frequency is that consistent with 400 K, and the applied stress has been set at 100 MPa (the value of stress which should yield a mean value of ~0avg --- 0.25 at 400 K). Two aspects of these velocity-time plots are worthy of note and discussion: (1) the dislocation does not exhaust immediately after the change in deformation conditions, and (2) the dislocation rapidly attains the expected velocity at the lower stress and temperature. Feature (1) clearly occurs only because there exists one or more superkinks longer than lc (corresponding to 100 MPa) along the length of the dislocation. The response of the same dislocation to a change in stress only (from 360 MPa to 100 MPa) without changing the temperature has also been shown in fig. 14 for reference. It is clear that the high pinning frequency immediately overwhelms lateral superkink motion, leading to exhaustion of motion. Feature (b), the brief duration of the transient, is due to the fact that the rate at which new superkinks are spawned from pre-existing ones changes immediately with the temperature change. In addition, the entire pinned structure of the dislocation is converted from that consistent with Thigh to that for Tlow as soon as superkinks have passed over its entire length. A rudimentary and extremely conservative estimate for the strain necessary for this transformation of the pinned structure can be made in the following way. Making the assumptions that the mobile superkinks
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are extremely long (1 ~m) and that most dislocations in the substructure are mobile (suggesting that the mobile dislocation density is Nm = 1012 m-2), then the total strain necessary to erase the structure formed at Thigh is C - - NmbA x - - 0 . 0 0 0 5 , which is far less than the strain required for macroscopic yielding. In fact, due to the effects of exhaustion, the mobile density should be much smaller than the total density, indicating that the erasure of the previous structure formed at Thigh would easily occur in the microstrain regime. If the dynamical recovery of the dislocation structures were not a rapid process, then one might expect to see flow softening at smaller strains.
2.3.7. Pinning points and Kear-Wilsdorf locks In the CSP model, the pinning points are assumed to be localized spatially on any mobile dislocation. The argument for this claim is that because the pinning points are formed along screw-oriented dislocation segments, the dislocation segments adjacent to the pinned point, which are moving very rapidly, immediately bow about the pinning point. The force due to this bowing, then, prevents the spreading of the pinning point on the (010) plane. However, TEM images of post-deformation dislocation structures show very clearly the presence of KW locks. In fact, at high temperatures the locks are observed to be bowed on the (010) plane, suggesting that they form while the sample is under load. The question remains: are KW locks relevant to the motion of a dislocation, or do they form after a dislocation's motion has been completely arrested? In the superkink models, it is proposed that KW locks are relevant to the motion of the near-screw-oriented segments of the dislocations. In this view, the spreading rate of the jogs on the (010) plane is comparable to, but still less than, the rate of advancement of the glissile segments on the (111) plane. Hence, once a pinning point forms, it has sufficient time to cross slip further onto the (010) plane before the bowing of the adjacent segments halts this progress. This process is envisioned to create a structure like that observed in the TEM studies. Forming KW locks in this manner implies that the ratio of the velocities of the (010) jogs versus the adjacent (111) dislocations must fall within a fairly narrow range. If this ratio were too small, the cross slipped segments would remain localized. If this ratio were too large, then bowing past the cross slipped segment (and the creation of superkinks) would not be possible. The dynamical simulations presented here shine a clarifying light on this issue. It is certainly true that an isolated pinning point would be rapidly annihilated by the adjacent bowing dislocations making the pinning point's lifetime too short to allow for the formation of KW locks. However, as shown above, in the low stress regime, pinning points do not form in isolation. The correlated pinning displayed in fig. 4 suggests that pinning points beget pinning points. Whereas one pinning point in isolation has a very short lifetime, a pinning point stabilized by adjacent pinning points may have sufficient time to further cross slip and spread onto the cube plane. This is one possible mechanism by which the KW locks may come into existence. It is interesting to note that the overall structure of the near-screw portions of the dislocation loops in fig. 4(a) is reminiscent of the TEM images (e.g. fig. 6). Even though the dynamical simulations are based on the hypothesis that pinning points remain pinning points, it is found that those pinning points are arranged, by the dynamics inherent to dislocation motion, in a fashion that resembles KW locks. This suggests that
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the locks are an integral part of the dynamics of a moving dislocation. It is therefore worthwhile to comment on the differences in dynamics expected from the replacement of densely pinned regions of pinning points with bona fide KW locks. At first glance, one would not expect the dynamics of the superkinks (in an athermal model) to be significantly altered by the replacement of densely pinned regions with KW locks. In fact one would expect a superkink/KW lock structure very similar to the superkink/pinning point structures observed in the present simulations. Because the dynamics of dislocation motion are governed by this structure, one would still expect the various superkink spawning, annihilation and scattering processes to operate, and hence their collective behavior should be very similar. However, if the further cross slip of the pinning point into the KW lock configuration is not rapid, it is possible that the strength of a particular lock will increase with the lifetime of the lock. This possibility is made even more likely given the observation of bowing of the KW locks on the cube plane, which has been considered in some detail by Saada and Veyssi~re [20]. It is not known presently whether or not this time dependence will produce significant changes in the observed dynamics, and this is likely to be a fruitful direction for future research. An additional point concerns the question of thermally-activated bypassing of the locks. As mentioned above, the time-scales in the simulation (and the discrete model discussed below) are too short to be consistent with experimental observations. Preliminary study of a discrete model of dislocation dynamics including a thermally-activated step in the lock-bypassing mechanism (see section 3.3) indicates that the time-scales can be extended into the appropriate range. It therefore seems reasonable that future work should incorporate a thermally-activated KW lock bypass mechanism(s), possibly that suggested by Hirsch [19].
3. Quantitative analysis o f m e c h a n i c a l properties 3.1. Introduction to quantitative analysis The simulations provide qualitative insight into many of the mechanical properties of the L 12 compounds displaying the yield strength anomaly. However, connecting directly dislocation core structures with mechanical properties requires a quantitative theory. This theory must involve time-dependent, structural evolution since flow in the anomalous regime is not under steady-state conditions. As a first step, a rudimentary microstructurebased model of the primary creep transient is proposed and studied [37]. The difficulties inherent in the attempt to develop a microstructure-based quantitative theory of mechanical properties must not be understated. Typical times for atomic-scale phenomena are measured in picoseconds, while creep transients are observed to vary over times of the order of hours. Therefore, the desired theory must span fifteen orders of magnitude in time! For this reason, the direct simulation of creep curves using the continuum-based simulation is impractical. To further complicate matters, one is attempting to understand the collective behavior of particles (dislocations) which interact through long-range forces and, in addition, have internal degrees of freedom! At first
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glance it seems that even the most ambitious attempts to develop the theory are doomed to failure. At the very least, any theory must include simplifying assumptions. Fortunately, features of the observed post-deformation microstructure in the L 12 compounds in the anomalous regime allow reasonable assumptions to be made and a rudimentary theory to be developed. The fact that dislocations are spread nearly uniformly throughout the crystal, with no evidence for the formation of dislocation cell structures, suggests that direct dislocation--dislocation interactions may be negligible. Also, the internal stress, assumed to be uniform in space, may be included in the definition of the net applied stress. Due to the apparent difficulty of dislocation multiplication (see section 3.2.2), the total dislocation density (mobile and immobile) is assumed constant, and it is assumed further that the increase in immobile dislocation density does not significantly alter the internal stress, which is valid only for relatively small strains. These simplifying assumptions reduce the problem to the study of the dynamic evolution of a population of noninteracting dislocations. Even within this highly-idealized picture, one gains significant insight into the mechanical properties of these compounds. The analysis, then, requires the development of a quantitative, statistical theory for the dynamic properties of individual dislocations. As demonstrated by the simulations, single-dislocation dynamics are governed by the collective properties of the pinning points (or KW locks). Therefore, the analysis must include these collective effects. In general, the collective behavior of a group of objects is much more complicated than the behavior of the individual objects. In addition, this complexity manifests itself on large length- and time-scales. Thermodynamic phase transitions are examples of this complexity, and provide a useful analogue to the dynamical "phase" transition which has been discussed above. Consider the order-disorder transition of CuZn (/3-brass). At low temperatures, the Cu and Zn atoms each preferentially occupy sites on their individual sublattices - the ordered phase. At high temperatures, the individual sublattices are occupied equally by both types of atoms - the disordered phase. The temperature demarcating the boundary between the two regimes is called the critical temperature, denoted Tc. The equilibrium state of the alloy at Tc is known as the critical state. The existence of the two alloy phases stems from two competing tendencies: (1) the tendency to disorder in order to create a large configurational entropy, and (2) the tendency to order to reduce the internal energy. At temperatures below Tc, the ordering tendency is strongest, whereas above Tc, the disordering tendency dominates. Precisely at Tc, no single tendency dominates the free energy, resulting in the critical state [38]. The features of this state have been well studied and documented [38, 39]. For the purposes of the current discussion, the most significant property is that the state displays scale invariance. At a temperatures slightly above Tc, the disordering tendency is still dominant. However, the reduction of free energy possible through the creation of some degree of shortrange order is not negligible. If one denotes the two simple-cubic sublattices by A and B, the observed state of the system will consist of patches of the crystal in which the A sublattice is occupied preferentially by Cu atoms, and other patches in which the B sublattice is occupied preferentially by Cu atoms. These patches assume a distribution of sizes, with the size of the largest patches determined by the temperature. Precisely at the critical temperature, the largest connected patch spans an infinite length. Also, the
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distribution of patch sizes becomes scale invariant. Practically, this means that if one is given an image of the patches of order in the alloy in its critical state, there is no feature in the image which allows one to determine the magnification. Dynamical systems can display behavior similar to that found in continuous thermodynamic-phase transitions [40]. As a simplified example pertinent to the current problem, let us consider a crystal which has an initial number of mobile dislocations, No. Let us also suppose that a constant stress is applied and that internal stresses can be neglected. Further let us suppose that mobile dislocations can spawn new dislocations at a rate a per dislocation, where a increases with the net applied stress, and that the dislocations leave the crystal (through its surface) at a fixed rate,/3, per mobile dislocation. Finally, let us assume that mobile dislocations annihilate with other mobile dislocations at a rate of con per dislocation, where N is the number of mobile dislocations at time t. The rate equation governing the number of mobile dislocations becomes: dN dt = N ( # - coN)
(6)
with # = a - / 3 . This equation is identical in form to that proposed by Li for the description of transient creep in metals [41 ]. The solution to eq. (6) is N -
#N~ # - coNo(eUt - 1)"
(7)
Let us consider the long-time behavior of this solution. One can identify two regimes: (i) # < 0 and (ii) # > 0. In regime (i), the rate at which dislocations exit the crystal is higher than the rate at which new mobile dislocations are spawned (a /3). In the long time limit, the number of mobile dislocations reaches a steady state in which N = # / w . In this regime, the crystal deforms plastically at a constant rate. The point separating the two regimes is # = 0. At this point, the rate of spawning of dislocations exactly balances the rate at which mobile dislocations leave the crystal. The solution to eq. (6) becomes (assuming No --+ oe) 1 N
~
--
(8)
cot" Note that this simple power-law decay possesses no intrinsic time-scale. The curve resulting from plotting N as a function of the number of hours is a trivially scaled version of the curve resulting from plotting N as a function of the number of seconds. In this sense, the function describing the number of mobile dislocations as a function of time becomes scale-invariant at the point # --- 0. The analogy with the order-disorder transition of/3-brass is now clear. The two regimes (phases) arise from the competition between two competing dynamic processes: 1) the multiplication of mobile dislocations, and 2) the elimination of dislocations through the
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surface of the crystal. Regime (i) is a dynamic phase in which all mobile dislocations eventually become immobile, leading to exhaustion of plastic deformation. Regime (ii) is a dynamic phase in which the number of mobile dislocations approaches a constant value and plastic deformation proceeds at a constant rate for all time. At the point demarcating the boundary of the two regimes (# = 0), the critical point, the function describing the time dependence of the number of mobile dislocations becomes time-scale invariant. Examples of dynamical systems displaying transitions similar to continuous thermodynamic phase transitions abound [40]. Perhaps the best studied case is the electric-field driven pinning/depinning transition studied in charge-density wave systems [42]. Recent studies of the dynamics of earthquake faults suggest that they, too, display such transitions [43]. It is argued below that the stress-driven pinning/depinning transition which is observed in the continuum-based simulations is similar to a continuous thermodynamic-phase transition. It is argued further that the mechanical properties of the compounds are linked directly to the scale invariance associated with the critical point of the transition. This point is crucial to the development of the quantitative theory that describes the microstructural evolution (i.e., the change in mobile dislocation density) during extended periods of transient flow. If the dislocation dynamics become truly scale invariant at a critical stress, one can study the properties of small systems for brief periods of time and use the scaling behavior to extrapolate across the time-scales to the regime of relevance for the creep experiments.
3.2. Pinning/depinning transition, scaling and creep In this section, the postulate that the pinning/depinning transition observed in the simulations displays critical behavior is explored fully. The experimental evidence for this hypothesis is reviewed briefly. Scaling arguments are then employed to relate the properties of the transition to the primary creep and stress relaxation transients, and algebraic expressions for the time dependence of both transients are derived. Results from a discrete version of dislocation dynamics are used to establish the existence of the transition. The cellular automata displays the pinning/depinning transition, complete with scaling behavior. In addition, results for the discrete model reflect the influence of finite size. The discrete model is used to deduce values of parameters necessary for the prediction of the creep and stress relaxation transients, and the mechanical properties so deduced are compared with the available experimental data. It is useful to define precisely the characteristics of the proposed transition. The simulations show two distinct regimes of dislocation motion. The most distinguishing characteristic of these two regimes, as discussed above, is related to the long-time behavior of the dislocations. In the high-stress, "unpinned" phase, once a dislocation is mobilized, it remains mobile for all time. In contrast, in the low-stress, "pinned" phase, all mobile dislocations eventually become immobile. The stress demarcating this pinning/depinning transition is the critical stress, Tc. As in the examples of the transitions described above, the existence of these two dynamical phases arises from the competition between two processes which dominate
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on either side of the transition: (1) the thermally-activated pinning point formation, and (2) the athermal bypassing of these obstacles at a critical angle. For applied stresses below the critical stress, the pinning points form at a rate more rapid than that at which they are dynamically dissolved, and the entire dislocation becomes pinned. At stresses above the critical stress, the pinning point formation rate is exactly balanced, on average, by the pinning point dissolution rate leading to the mobile state. 3.2.1. Evidence for the pinning/depinning transition
The experimental evidence for the existence of the low-stress phase comes directly from the high rates of strain hardening which are observed in the anomalous regime (below the peak temperature). Using single crystals oriented for single-system slip, both creep [23, 27] and constant strain-rate [24] tests show a large amount of strain hardening, indicating that this hardening is intrinsic to the dynamics of the individual dislocations, and does not originate from dislocation-dislocation interactions. Therefore, one is forced to assume that in these experiments, mobile dislocations eventually become immobile, which is the distinguishing characteristic of the low-stress, pinned phase. Also, postdeformation microstructures are consistent with those produced in the low-stress phase by the simulations. To date, there is no definitive evidence for the high-stress, unpinned phase. This is particularly surprising, given that much of the modeling of the yield strength anomaly has centered on the existence of this phase. Creep experiments give very large strains upon loading at higher stresses [14, 27], a behavior which may be indicative of the unpinned phase. It is physically reasonable to expect that there is a stress at which the cross slip pinning process cannot pin the entire dislocation. It has been implicitly assumed that the experimentally measured CRSS represents this critical stress and that dislocations move only under an applied stress larger than CRSS. However, the dynamical simulations show that a dislocation can move a substantial distance before coming to rest in the pinned "phase," i.e. at a stress below the critical stress of the transition. Therefore, dislocation mobility for some time is not an indicator of the unpinned "phase," and there is no a priori reason to associate CRSS with the critical stress of the proposed pinning/depinning transition. However, a relationship between CRSS and Tc probably does exist. Let us consider a constant strain-rate test performed on a single crystal oriented for single system slip. At low values of the strain (and stress) the dislocations must be advancing at a stress below ~'c. This implies that the dislocations are exhausting at some, stress dependent rate (i.e. the characteristic dislocation exhaustion time increases with applied stress). In order to maintain the fixed strain-rate, the decrease in mobile dislocation density (and, additionally, the increase in the internal stress) must be accompanied by an increase in the applied stress, which activates new dislocations and sources. It seems, then, that the net applied stress will increase monotonically, and be driven toward the critical stress. It is not obvious, however, that the net applied stress is driven to the critical stress. All that is required is that a sufficient number of dislocations move to accommodate the imposed strain-rate. Since dislocations can move at stresses below the critical stress, it is not obvious that the applied stress will have to equal To. It is likely, however, that
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the applied stress will approach "re from below, which will increase the characteristic time for exhaustion and thereby reduce the rate of strain-hardening. Thus, the continual decrease the in rate of strain hardening noted near the CRSS probably stems from the fact that the applied stress is near to, but not necessarily at, "re. The observation of a smooth constant strain-rate curve is a reflection of the continuous nature of the transition. There is additional evidence suggesting that the dislocations undergo a continuous pinning/depinning transition. Thornton et al. [ 14] reported results from creep experiments on polycrystalline samples of Ni3A1. These authors identified two regimes for the primary creep transient: (1) a regime at lower stresses in which the creep-rate behaved as t -1, with t the time (yielding logarithmic creep), and (2) a regime at higher stresses in which the creep-rate scaled as t -2/3 (yielding t 1/3 creep). It is intriguing to note that the creep response has been modeled by simple power-law functions of the time. Recall that in the simple example of a dynamic-phase transition presented above (section 3.2) the number of mobile dislocations at the critical point was also described by a simple power-law dependence on the time. Thornton et al. interpret these power laws along classical lines of dislocation mobility-controlled flow versus recovery effects. Instead, it is argued below that a more complete expression for the creep-rate, which could be misconstrued as these simple power-law forms, stem from the scale-invariance associated with the critical point of the pinning/depinning transition. 3.2.2. Scaling theory of the primary creep transient Having postulated the existence of the pinning/depinning transition, the relationship of the transition to the mechanical properties must be made formal. In order to do so, it is necessary to model an appropriate experiment. The most straightforward experiment to model is the measurement of the primary creep transient of a single crystal oriented for single-system slip because: (1) the external applied stress remains constant, and (2) no activation of additional intrinsic sources should occur beyond those operating after initial loading of the sample. Once the behavior of the compound under creep conditions is understood, it is hoped that studying the response to other, more complex experimental conditions (e.g. a constant strain-rate test) will be possible. An additional assumption made for the case of the L12 compounds is that soon after loading the sample, no dislocation multiplication occurs and the total dislocation density remains constant. This assumption is based on the fact that cross slip to (010) planes is strongly favored over double cross slip between { 111 } planes. As evidenced by the coplanar arrangement of non-screw portions of dislocations [44], and the rectilinear nature of slip traces [ 14, 21], this normally important multiplication mechanism appears to be inhibited. At higher stresses this assumption may no longer be valid as KW locks and bowed segments on the (010) may be able cross slip back to the { 111 } planes [11, 19] and act as Frank-Read sources. The reduction in the rate of strain hardening which might result from such a multiplication process is not treated here. Let us consider, then, the following experimental situation. Suppose that a constant load is placed on a single crystal oriented for single slip, such that the applied stress is less than "re. The sample creeps until all dislocations become immobile and the strain-rate goes to zero. The stress is then incremented a small amount. In response to this change in stress, some formerly immobile dislocations begin to move, and the sample begins to
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creep. The strain as a function of time, measured from the time the load is increased, is recorded. With the assumptions outlined above, calculation of the evolution of the mobile dislocation density and velocity allows the prediction of the creep transient. Let n(a, t)dadt be the number of dislocations initially mobile which exhaust at a time between t and t + dt after sweeping out an area between a and a + da. The total number of dislocations which exhaust between t and t + dt is then given by ~(t)dt with
~(t) -
f0 ~
dan(a,t).
(9)
The average area swept out by a dislocation which exhausts at a time between t and t + dt, defined to be g(t), is:
-6(t) -
f o da an(a, t) -Ted . . . . . . fo da n(a, t)
9
(lo)
Finally, define v(t, t ~) to be the average areal velocity at time t of the dislocations destined to exhaust at time t ~. The definition of v(t, t I) requires
v(t, t') = o
vt >1 t',
(11)
dt'v(t',t).
(12)
and
g(t) -
f0 ~
The functional dependence for v(t, t ~) has been kept as general as possible for the following reasons. First, over its lifetime, a dislocation does not maintain necessarily a constant velocity. For example, the velocity is expected to vary with the time to exhaustion (e.g. dislocations which exhaust after a short time never acquire a high population of mobile superkinks, and therefore do not move as fast as a dislocation which exhausts after much longer times). Also, it should be emphasized that the important fluctuations in the superkink population are incorporated in this theory directly through the dependence of v(t, t ~) on exhaustion time, t ~. However, this quantity represents an average over the velocity fluctuations which occur for an individual dislocation (e.g. see fig. 10). The shear strain, 3'(t), is proportional to the total area swept out by all the dislocations. The shear strain-rate, dT/dt is given by
d__j_7= _b dt V
f oodt'v(t
t')~(t'),
(13)
where b is the Burgers vector and V is the crystal volume. This expression follows directly from the definition of ~(t) and v(t, t').
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The scale invariance which is postulated to describe the pinning/depinning transition at the critical point is now invoked to derive an expression for ~(t). The physical picture for the dislocations based on this hypothesis is the following. As the critical stress is approached from below, the characteristic time for dislocation exhaustion increases. At "rc, the characteristic time becomes infinite and the distribution of exhaustion times becomes scale invariant. Practically, this means that there is no typical time to exhaustion. Mathematically, this statement is equivalent to assuming that the distribution n(a, t) is a homogenous function of its arguments: (14) where c~ and 3 are the scaling exponents of the transition. Rescaling the area by a factor As and time by a factor A6 gives the original distribution scaled by a factor of A. In order to obtain an expression for ~,(t), multiply both sides of eq. (9) by A, and redefine the variable of integration to give:
)~(t)-- ~ foo~176
- ~---g~()~6t).
(15)
Since eq. (15) holds for all values of A, it will hold for the particular value of A chosen such that A6t -- 1. This choice of A implies that ~(t)-~(1)t
-~ ,
~ = - - -l-+~c. ~
(16)
Hence the assumption of scale invariance leads to directly a simple power-law time dependence for the number of dislocations which exhaust at a time between t and t + dr. In order to arrive at an expression for v(t, t'), one refers to the physical basis of the scaling hypothesis. Equation (14) states that there is no intrinsic time-scale in the system. However, consider a particular dislocation destined to exhaust at time t 1. The exhaustion time for this dislocation does set a time-scale for this particular event. Since there are no other relevant time-scales in the problem, this can be the only time-scale governing that particular dislocation's motion. Therefore, at the critical point in an infinite system one can write [recalling that v(t, t I) is the average areal velocity time dependence of a dislocation destined to exhaust at time t']
v(t, t') = t'o9(t/t'),
(17)
where 9(x) is defined to be the areal-velocity scaling function which has the property that 9(x) = 0 for all x >~ 1. It is shown in section 3.2.3 that this scaling relation holds for the cellular automata model described below. Using eqs (10), (12) and (14), one can show that r / - c~/~;- 1. The physics succinctly stated in eq. (17) is remarkable. Consider two dislocations mobilized at the beginning of the creep test. Suppose that one of these dislocations is destined to exhaust at late times, and one is destined to exhaust almost immediately.
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Equation (17) states that, on average, the areal-velocity profiles of the two dislocations are related by a trivial scaling transformation. It is this fact that allows the spanning of the time-scales to produce a quantitative theory of the mechanical properties of these compounds. One calculates the velocity scaling function for the dislocation events that exhaust rapidly, and hence are amenable to computer modeling. The scaling transformation is then used to produce v(t, t ~) for the much larger values of t ~ relevant to macroscopic strain measurements. Given this understanding of the scaling properties of ~(t) and v(t, t'), and eq. (13), one can show that the creep-rate at the critical point becomes d3' = b~,(1)tl+n_ ( dt V
s
9(x) dz. z2+'7-~
(18)
Thus the creep-rate time dependence reduces to a simple power-law, provided the integral on the right-hand side of eq. (18) is finite. The assumption ofscale invariance at a critical stress leads directly to the conclusion that the creep-rate shows a simple power-law dependence on the time. It is important to note that the above analysis holds strictly only at the critical stress, Tc. However, as reasoned above in section 3.2.1, mechanical tests are not always performed at the critical stress. It is, therefore, important to analyze the behavior of the system for stresses 7- -r 7-c. For 7- < 7-c, n(a, t) is no longer scale invariant. Instead, there is a characteristic time beyond which the number of mobile dislocations decays exponentially. It is reasonable to assume that most mechanical tests are performed at stresses near the critical stress. For stresses much less than "re, the characteristic time is too short to allow for measurable plastic strain. In many systems displaying scale invariance at a critical point, the function which becomes scale-invariant at "re assumes the Ornstein-Zernike form [39] near the critical stress. The following, therefore, is conjectured to hold near to and below the critical stress:
~(t) = ~tt-~e -t/t~
(19)
with ~t a normalization constant and to, the characteristic exhaustion time, diverging at Tc according to
to - Alrc - 7.1- ~ ,
(20)
with g' a critical exponent of the transition and A a constant. The value of ~t is determined by requiring that the integral from some initial time (nonzero, to avoid the singularity at t = 0) to infinity of ~(t)dt be the total number of dislocations mobile at the initial time. Note that no analytical theory has been developed to support the conjecture in eq. (19). However, it has been demonstrated to hold analytically in related models [43]. Furthermore, the data generated from the cellular automata studied below is consistent with this expression, as shown in the next section.
D. C Chrzan and M. J. Mills
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Ch. 52
Assuming that the scaling form for v(t, t ~) holds near the critical stress, and also assuming ~(t) is of the form of eq. (19) the creep-rate becomes
d__~_~= bfi,t tl+n_ ~ dt V
L1
9(x)e
-t/(xto)
X 2+rl-~
dx
(21) "
Note that the integral in this equation is finite as along as 9(0) is not infinite. The finiteness of 9(0) is insured because the physics of motion of a finite length dislocation does not allow an infinite areal velocity. It should also be noted that the above expression does not reduce to a simple power-law time dependence. Finally, it is noted that eq. (21) is based on the assumption that v(t, t ~) is described well by the scaling form of eq. (17) even for stresses less than "rc. Equation (17) is likely to be accurate for t ~ < to, but certainly cannot be applied to times t > to. Thus application of eq. (21) to experimental data must be restricted to times less than to. Equation (21) is of central importance to the theory. First, it is the predicted constitutive law for the creep transient based on the postulated pinning/depinning transition. In addition, it embodies the direct connection between the areal-velocity scaling function and the creep-rate during the primary creep transient. The interpretation of an experimental creep curve using eq. (21) allows one to extract directly information about the dynamics of dislocation motion.
3.2.3. Discrete model results and test of the scaling hypothesis In this section the scaling properties of the proposed transition are explored fully. The continuum-based simulations presented above are too numerically intensive to allow for detailed numerical study. Therefore, a discrete version of the simulations has been developed. A detailed description of the discrete model is presented in Appendix B. The discrete model is designed to mimic as closely as possible, the physical processes thought to be important for a proper description of dislocation dynamics. These processes are concerned mostly with the superkink spawning, annihilation and scattering processes described in section 2 of this chapter. The correlations between pinning points discussed in connection with fig. 5 are determined by both the pinning frequency and the dislocation velocity. It is reasonable to expect that the probability for creating an additional immobile dislocation segment increases with the temperature and decreases with the net stress (including the bowing stress) on the free dislocation segment. Therefore, the probability of creating an immobile dislocation segment in the discrete model depends on the following ratio @ Tn~
(22)
where O is a parameter which increases with increasing in temperature to reflect a higher probability of pinning, -r is a parameter which represents the applied stress (including the internal stress) and t~ is a parameter reflecting the bowing stress and related to the local configuration of the dislocation. See Appendix B for more details. There are two goals for the following calculations. Most importantly, because the possibility that the dislocation pinning/depinning transition may display critical behavior
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Dynamics of dislocation motion in L12 compounds
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has not been appreciated until recently [37], the physical phenomena associated with the transition is unknown. It is hoped that study of a simplified version of dislocation dynamics will allow a detailed understanding of the general physics expected from such a transition. The second goal of these calculations is to make quantitative predictions (if possible) for the critical exponents of the transition, enabling calculation of the form of the velocity scaling function and the creep transient through eq. (21). In the next subsection, the best estimates for the critical exponents are given, and a brief discussion of finite-size effects is provided. For the interested reader, a detailed test of the proposed scaling hypothesis for the pinning/depinning transition is discussed in subsection 3.3.3.2. Others may wish to skip subsection 3.3.3.2. 3.2.3.1. Summary of the results from the discrete model. Figure 15 displays the dislocation configurations resulting from the discrete model described in Appendix B. The black lines represent the position of the dislocation upon exhaustion of its motion. The white patches represent the area swept out by the dislocation before exhaustion. Panel (a) contains the successive configurations generated at a stress parameter "r well below the critical stress parameter rc. Note that the white areas are all relatively small, suggesting that there is some upper limit to the area swept out by a single dislocation. Panel (b) contains the successive configurations of a dislocation moving under an applied stress near T = Tc. Note that the white areas are distributed over a broad range of sizes, and one cannot identify easily a "typical" area. This panel displays directly the scale invariance of the exhaustion event distribution. Panel (c) is the result for a net applied stress above the transition stress. An interesting feature of this panel is that dislocations do exhaust, but only after sweeping out very large areas. The exhaustion of dislocations at an applied stress above the critical stress is a finite size effect, which is expected in the experiments as well. The discrete model is used to determine the scaling exponents appearing in eqs (14) through (21). The results are best described using the exponents c~ = - 1 / 2 and 5 - 1 / 3 . This choice of c~ and 5 imply that ~ = 3/2 and r / = 1,/2. These represent a selfconsistent set of values for the exponents that are deduced from analyzing the results of the discrete model in several different ways. It should be noted that in contrast to thermodynamic phase transitions, there is no theory for dynamic systems stating that the critical exponents are "universal" [38]. In this context, "universal" means that the exponents are determined by overall symmetries of the system rather than the detailed physical mechanisms. Thus there is no guarantee that the simple model described in Appendix B is governed by the same set of scaling exponents as the actual dislocations in the solid. Nevertheless, it is hoped that the simple model studied can provide accurate estimates of the exponents. Most importantly, however, the discrete model establishes that the pinning/depinning transition does occur for this model system, that the transition displays critical behavior, and that much can be learned about the transition through the study of this example. One important result from the study the model is the observation of the effect of the finite size of the simulated system. These finite-size effects manifest themselves, for example, in the deviation of the simulated results for ~(t) from the Ornstein-Zernike form of eq. (19). The deviation occurs for times of the order of those required for a
D. C. Chrzan and M. J. Mills
222
Ch. 52
(a)
(o)
(c)
Fig. 15. The exhausted configurations of a single dislocation as calculated from the discrete model described in section 6.1 (Appendix B). The dislocations are composed of 100 segments. The temperature parameter is fixed to be 69 = 6.5. (a) The stress, 7" = 10, is well below the critical stress. (b) The stress, 7" -- 12, is near the critical stress. (c) The stress, 7" -- 15, is well above the critical stress. At low stresses, the events are all small in duration and area (the white patches correspond to the areas of the events). At high stresses, all events are large. At stresses near the critical stress, it is difficult to assign a characteristic size to the events. This is a manifestation of the scale invariance associated with the critical point.
Dynamics of dislocation motion in L12 compounds
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223
superkink to pass over the entire length of the dislocation, and are thought to arise from the interaction with and loss of mobile superkinks to the ends of the dislocation. Evidence is also presented in the next section suggesting that the collective dynamics of these "large events" are different from the smaller events. These large events are expected in the real world, dictated by the size of the crystal or grain being deformed. Therefore, the presence of the large events may be important to consider when interpreting the experimental creep curves. Unfortunately, as of this writing, characterization of the collective dynamics of these large events is incomplete. This means that the comparison of the experimental creep transient with that predicted using eq. (21) should strictly be valid only for stresses near to the critical stress, but not so near as to include many events which interact directly with the edge of the crystal. As it will be shown in section 3.2.4, using eq. (19) and the values of the critical exponents listed above, good agreement is obtained for these stresses. It is hoped that accurate prediction of the entire creep transient will be possible when the dynamics of the large events are fully understood.
3.2.3.2. Test of the scaling hypothesis. The starting point for the analysis is evidence that the pinning/depinning transition exists in the discrete model. Equation (20) is one of the fundamental assumptions of the theories treating such transitions. The creep curves calculated from the automata are fit to eq. (21) [using a = - 1 / 2 and ~ = - 1 / 3 ] and a value of to is determined. (The quality of those fits is addressed below.) This value of to is plotted as a function of the applied stress (at fixed temperature) in fig. 16. The solid line is a fit to the form of eq. (20) with ~b = 2.14 and "rc = 97.3 This result is consistent with the divergence of the characteristic dislocation exhaustion time at a critical stress. 6000 5000 4000
t.__.a
3000 2000 I000
80
85
90
I
!
95
100
Fig. 16. The characteristic time, to, as a function of the applied stress for a dislocation containing 10000 segments and a temperature parameter 6) -- 50. The circles are the values of to measured directly from the simulations (as described in the text) and the solid line is a fitted curve of the form of eq. (20). The critical stress is ~-c -- 97.3 and the exponent characterizing the divergence is ~ --- 2.14. These values must be viewed only as approximate as placing error bars on the measurementis difficult.
D. C. Chrzan and M. J. Mills
224 1.0
Ch. 52
-
0.8
,~ ~l/,~" ~ ~[~
~
0.6
~ ~ %
e
0.4
9 o [] + <>
t=30 t = 60 t = 90 t = 120 t = 150
<>r
I
0.2
0.0
t1.5
1.0
1.5
2.1)
a/t ~5 Fig. 17. Scaling analysis showing a plot of t-1/6n(a, t) as a function of a/t ~/6 with c~ = - 0 . 5 1 and -- - 0 . 3 5 . The data was obtained from a dislocation with 1000 segments, a temperature parameter O - 50, and a net applied stress of "r - 95. The times for each of the curves are presented in the legend.
The value of the exponent if' should not be taken seriously, as placing error bars on its value is difficult. To establish whether scale invariance occurs at the critical point in this model, one must investigate directly the scaling assumption contained in eq. (14). This "scaling analysis" is performed in the following manner. Since eq. (14) must hold for any value of A, it must hold for the value chosen so that A6t = 1. This choice of A implies that
t - 1 / ~ n ( a , t) - n
t--dT-g, 1
- f
(o)
t--gT-g ,
(23)
with f ( x ) defined to be the distribution scaling function. Equation (23) implies that a plot of t - 1 / 6 n ( a , t) as a function of a / t ~/6 for different values of t should produce one curve: f ( x ) . This curve is constructed for the choice of a -- -0.51 and ~ = - 0 . 3 5 in fig. 17. This choice of a and ~ produce the "best" data collapse, where "best" is judged by eye. Thus the distribution of exhaustion events does display scale invariance at the critical stress. Figure 18 shows the histograms for ~,(t). The curves are the results from the discrete model for four different applied stresses, where the applied stress increases monotonically from (a) to (d). The dashed line displays a t -3/2 dependence which is predicted from the above scaling analysis. Note that the short-time behavior for each of the curves follows closely the t -3/2 dependence, providing confirmation of the scaling analysis. It can also be shown that the long time dependence of the curves labelled (a) and (b) is well
w
225
Dynamics of dislocation motion in LI 2 compounds
10 8 10 7
%% %%
10 6 10 5 10 4 t~
10 3 10 2 101 10 0
(a)
m,
"
(b)
10 "1 10
0
10
1
10
2
10
3
10
4
time [steps] Fig. 18. A plot of n(t)dt (with dt -- 1) for a dislocation containing 1000 segments, a temperature parameter of O = 50, and a net applied stress of (a) 80, (b) 90, (c) 95 and (d) 100. The critical stress for these conditions is estimated to be "re = 97.4. The dashed line displays the t -3/2 dependence expected from the analysis presented in the text. The peak in curve (d) is a finite size effect. described by eq. (19) - the function decays exponentially at long time. The appearance of the peak for long times in curve (d) is a finite size effect. This finite size effect is expected to be present in experiments as well, and is discussed below in more detail. Since the exhaustion distribution, n(a, t), displays scale invariance at the critical stress, eq. (10) implies that the average area swept out with time, ~(t), will become a simple power law:
-d(t) ~ t ~/5.
(24)
This behavior has been observed in the discrete model. Figure 19 contains a plot of ~(t) as a function of t near the critical stress. The dashed line represents the t 3/2 dependence expected from the values of a and 6 deduced from the scaling analysis. Finally, scaling of the areal-velocity profile is also observed. Figure 20 is a plot of t-~v(t, t') as a function of t/t' for 77 = 1/2 for various values of t'. Note that the function resulting from this procedure is the areal-velocity scaling function, g(x) in eq. (17). The data collapse for different times is adequate. At stresses nearer to the critical stress, in which the finite size of the system plays more of a role, the data from different times collapse less convincingly onto one curve. This is a finite size effect. The physical interpretation of fig. 20 is worth comment. The areal-velocity scaling function describes the velocity profile of a typical dislocation from the time that motion
D. C. Chrzan and M. J. Mills
226
10 lO o.
10
Ch. 52
5 4 sS
3
sS
2
SS
S
SIS SSS
lO lO
1 sS
0
10 0
101
10 2
10 3
t [steps] Fig. 19. A plot of ~(t) as a function of t for a dislocation composed of 1000 segments, O = 50 and a net applied stress of r --- 100. The dashed line displays the t 3/2 dependence expected from the analysis in the text. (Note the unit of area in the automata is steps 2 because in each step, a mobile segment moves one unit of distance.)
0.8 0.7
0.6 ~.. ,~
i~. ~..
0.5 0.4 0.3 0.2 i I
0.1
~
0.0
, 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
t/t' Fig. 20. Scaling analysis showing a plot of t~-nv(t, t ~) as a function of t/t' for a dislocation composed of 1000 segments, O = 50 and r = 90 (77 = 1/2). The value of t ~ ranges from 30 to 300 steps in increments of 30. This plot is the areal-velocity scaling function for the cellular automata.
w
Dynamics of dislocation motion in L12 compounds
227
is initiated to its exhaustion time. In particular, for the dislocations described by the discrete model, the velocity starts at some small value, the dislocation accelerates for roughly half its lifetime, and then finally decelerates to zero at its exhaustion time. This description applies to the average dislocation m o t i o n - a particular dislocation's motion may be substantially different. With the exception of finite-size effects, the results from the discrete model presented thus far are consistent with the scaling predictions founded on eq. (14). Any critical state must display nontrivial finite size effects and in this sense, the observation of finite size effects in the model reinforces the notion that the pinning/depinning transition should be viewed as a critical phenomenon. Attention is now focused on these finite-size effects. Let us consider the curve labeled (d) in fig. 18. This curve appears to behave according to a simple power law for times less than 1000 steps. It should be noted that this data was obtained from a dislocation containing 1000 segments, and that a mobile superkink in the automata moves at a rate of one site per step. Thus the time 1000 steps is precisely the time it takes for a superkink to traverse the length of the dislocation. Events lasting for times larger than 1000 steps are, thus, very likely to "feel" the edges of the dislocation, whereas smaller events are less likely, on average, to interact with the edges. Given this fact, the following picture for the origins of the peak in the histogram is proposed. Let us consider a dislocation moving under an applied stress well below the critical stress. The characteristic time for exhaustion of this dislocation, to, is small, and the events are less likely to "feel" the edges of the dislocation. As the stress is increased toward the critical stress, the characteristic time for the events becomes larger and more events interact with the edges of the dislocations. Very near to the critical stress, the characteristic time for the events is much larger than the time it takes for a superkink to traverse the dislocation. At these stresses, a dislocation which would have moved for a very long time will be cut short arbitrarily by the loss of one or more mobile superkinks to the edges of the dislocation. Consequently, this event will be included in the histogram for some time less than its "destined" time. Most often, the times into which the events are folded are larger than L steps. The peak arises, then, as a compression of the majority of the events destined to last for times larger than L steps into a smaller time range near L steps. There is evidence, however, that the large events which are folded into the histogram are dynamically different from the small events. Figure 21 is a plot of ~(t) for a system of size L = 1000 at a stress near %. The dashed line displays the t 3/2 dependence expected from the scaling analysis. It is clear the small events (those with times less than 1000 steps) display this dependence. In contrast, the large events (those with times greater than 1000 steps) display a nearly linear dependence on the time (the dotted line). This indicates that the collective dynamics of the large events differ in some important respects from those of the small events. At present, the explanation for these different collective dynamics remains an open question. If finite size is playing a role in the histograms calculated from the discrete model, it may also influence the results of experiments. In order to determine whether or not this is the case, one must consider the system size which corresponds to an actual dislocation. If each site in the model represents a region in which a pinning point can form, then each site is roughly 1 nm in length. Using a generous estimate of the "typical" dislocation length of 10 -3 meters, then the calculation for such a case would involve 106 sites.
D . C . Chrzan and M. J. Mills
228
10 10 e,l
10 10
Ch. 52
7
I
6 5 4
10 3
10 10
2 1
10 ~ 9
10 0
i
iiiiiii
,
101
,
iiiiiii
,
10 2
i
iiiiiii
. . . .
10 3
iiiii
i
1111
10 4
t [steps] Fig. 21. A plot of ~(t) as a function of t for a dislocation composed of 1000 segments, 69 = 50 and 7- = 95. Events lasting for times less than 1000 steps are not as likely to be influenced by finite size, and reflect the t 3/2 dependence (indicated by the dashed line) expected from the scaling analysis. The events for longer times seem to be well characterized by a linear dependence on t, as indicated by the dotted line. This different time dependence suggests that the collective dynamics of the large events differ from those of the smaller events.
Although this is a large number, it certainly does not represent the thermodynamic limit, suggesting that finite size will play an important role in the experiments, and that some effort should be made to understand its importance. It should also be noted that given this same system size, and considering the velocity of mobile superkinks (approximately vf/10 = 10 m/s based on the continuum simulations), this analysis suggests that the scaling behavior of the system will extend only for the initial 10 -4 seconds. This indicates that the theory must be extended to span an additional 8 orders of magnitude. This failure to model the proper range of time-scales will be discussed below in section 3.3, where remedies for this flaw are proposed. In summary, the results from the discrete model indicate that dislocations in the L12 intermetallic compounds to undergo a pinning/depinning transition which displays scale invariance at a critical stress. Below the critical stress, a moving dislocation ultimately comes to rest. The characteristic time for exhaustion is designated to. This characteristic time is observed to diverge at a critical value of the stress, 7c, with and exponent of g' = 2.1 4. At 7-c, the distribution of exhaustion events becomes scale invariant, with the scaling exponents [see eq. (14)] c~ - - 1 / 2 and 6 -- - 1 / 3 . These exponents obtained from the scaling analysis imply that the other critical exponents of the transition are -- 3/2 and 77 = 1/2. Direct measurements of the exponents using the automata agrees well with these estimates, indicating that this scaling forms used are self-consistent. It is also demonstrated that the effects of finite size are nontrivial, and are likely to be present in experiments. The effects of finite size suggest that the scaling theory, as presented here, is only capable of modeling the early time (tinit < t < tO) behavior of the primary creep transient and stresses for which to is much less than the typical time
Dynamics of dislocation motion in L12 compounds
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229
for a superkink to traverse the dislocation. In the next section, the implication of these results with respect to the primary creep transient are discussed.
3.2.4. Analysis of the primary creep transient and comparison with experiment Given the values for the scaling exponents deduced from the discrete model, eq. (21) for the creep-rate during the primary creep transient can be recast as
d"/ dt
=
b~t f l
g(x)e-t/(xto)
V J0
x
dx.
(25)
Note that the sum 1 + 77- ~ = 0, so that at the critical point [eq. (18)] the creep-rate reduces to a constant (provided the integral is well defined). Thus at the critical point in a an infinite system, one should observe steady-state creep. This is not surprising given that at stresses above the critical stress (i.e. the "unpinned phase") exhaustion will not occur in an infinite system. It is also important to note that below the critical stress, the time dependence of the primary creep transient is entirely determined by the form for ~(t) given in eq. (19). In order to make further progress, one must assume a form for the areal-velocity scaling function. An example of the behavior of this function based on the model results is shown in fig. 20. However, when comparing with experiment, this function may differ in form, and its shape is suggested to contain important physical insight into the statistically-averaged movement of dislocations. Consequently, a generalized form for the scaling function is desirable. Noting that one can define a complete set of orthogonal polynomials on the interval [0,1], 9(x) can be expanded as a simple polynomial: O(3
9(x) =
(26)
9rex m. m=O
Substitution of eq. (26) into eq. (25) allows one to rewrite eq. (25) as
(t)
dTbnt ~ d---( = V ~ ffm Em + l -~0
(27)
m-0
with En (x) defined to be the exponential integral: En (x) - f l ~ y-------~. dy e -vx
(28)
Equation (27) can be integrated to give as the final expression for the primary creep transient:
~(~) -- ")/@init)-~
V
~ m--O
[ (init
gm Em+2
230
Ch. 52
D. C. Chrzan and M. J. Mills
1.0 0.8 0.6 0.4
0.2 0.0
I '
0.01
.......
'
0'.1
.......
r
i
,
,
i
i
iii
10
o
Fig. 22. The creep curve predicted from eq. (29) using the areal-velocity scaling function of fig. 20. This unusual sigmoidal shape has not been noted in the literature. See the text for details. where tinit is the earliest time at which the ~(t) distribution is well described by eq. (19). Note that eq. (19) cannot possibly be correct down to t = 0: the total number of mobile dislocations must be finite. In practice, the areal-velocity scaling function is found to be smooth, so that the series can be truncated at some m = mmax. The results from the automata suggest that mmax = 4 produces sufficient accuracy. Figure 22 is the creep curve predicted from the scaling function of fig. 20 (fit to a fourth-order polynomial) and eq. (29). Note the logarithmic z-axis. This unusual sigmoidal shape has n o t been noted in the literature, though the TDJ data does display some of the features observed in this curve [14]. In particular, the region of fig. 22 near t / t o = 1 displays logarithmic behavior. In addition, especially given the historical bias of the interpretation of these measurements, the early time portion of this curve may be mistaken for a t 1/3 dependence. These are the two time-dependencies which were identified as indicative of physically distinct regimes by TDJ [14]. Also, several of the creep curves in reference [14] display a saturation of strain at long times. Before proceeding, it is worthwhile investigating how well eq. (29) describes the data obtained from the discrete model. Figure 23 is a comparison of the data obtained from the model and the fit to eq. (29) holding the scaling function fixed to be that measured at shorter times. The fitting procedure includes the entire range of data shown. The agreement between the simulation data and the fit is reasonable, but not perfect. The value of to deduced from the fit to the entire range of data is to = 323 steps. The origin of the disagreement between the fitted curve and the data stems from the application of the scaling law (14) beyond the value of to. (Recall that the scaling form for the velocity is based on the idea that the only time-scale in a particular event is the ultimate exhaustion time of that event. If the dislocation exhausts at a time larger than to, however, this argument is not correct and one expects the scaling form to break down.) In addition, at this value of the stress the presence of the large events cannot be neglected. Both of
w
Dynamics of dislocation motion in L12 compounds
231
1.0 0.8 0.6
0.4 0.2
...••/-"
- ....
fit
0.0
lo
.....
i00
......
ii/00
t [steps] Fig. 23. A comparison of the creep curve resulting directly from the model with the fitted curve resulting from holding the areal-velocity scaling function fixed to the form indicated in fig. 20. (The model data was collected under the same conditions as the data in fig. 20.) The fit resulted in a value of to = 323 steps whereas the curve was fit over the entire range of data shown. The disagreement between these two curves stems from application of eq. (29) to times beyond to. The fitted curve resulting from limiting the range of data to t < 323 steps is indistinguishable from the model data. these effects contribute to the disagreement between the fit and the measured expression. When one restricts the range of the fit to coincide with range of times over which the scaling function was determined, the agreement between eq. (29) and the model data is excellent. Given that the form of eq. (29) is adequate for describing the results of the discrete model, it is reasonable to investigate to what extent data for the primary creep-transient can be understood within the framework proposed here. Figure 24 is an example of a fit to the data reported by TDJ. The paucity of data points makes performing the fitting procedure for the scaling function difficult. Therefore, only the m = 0 term of eq. (27) is included in the scaling function. Though the number of parameters (three) is too large (compared with the number of data points) for this to constitute a true test of the theory, it is clear that the eq. (27) is able to fit the data adequately. It should be noted that this data is from the temperature regime in which TDJ claim to observe a t 1/3 time-dependence for creep. The apparent success in fitting the creep results of the TDJ suggests that a further, more stringent test of the theory is warranted. Figure 25, panel (a) contains an experimental measurement of the primary creep transient in Ni3A1 [27] (the solid line). The data was obtained at room temperature from an annealed, single crystal, oriented sample. Immediately upon application of the load, a strain of about 10% was recorded. Recording of the transient began at 33 seconds into the test, and it is this transient data which is reported here. Note that the large amount of strain observed in the initial few seconds of the test is consistent with the loop expansion model proposed above, and indicates the extremely rapid dislocation motion which is possible in these compounds. This
D. C. Chrzan and M. J. Mills
232
Ch. 52
1.0 0.9
,--,
0.8
N
0.7
oml
1.1 o
.
0.6
0.5 0.4 0.3
.
0.2
0.1 !
IO0
9
9
,
|
| | | 1 1
,
I000
,
w
i i i 1 !
I0OO0
t i m e [seconds] Fig. 24. A test of the fitting procedure on the TDJ data for 363 MPa and a temperature of 399 ~ C. Only the m - 0 term of eq. (27) is included in the fit. Equation (29) provides an adequate fit to the data, though the paucity of data points in comparison with the number of fitting parameters (three) makes accurate assessment of the fit difficult.
loop expansion ultimately results in the linear segments which then begin to advance and exhaust in the manner which has been modeled using the automata. It should be noted that during this loading strain, some rotation of the crystal toward a multipleslip orientation is also expected. The influence of this rotation on the operative slip systems and the effect on the subsequent strain hardening behavior is presently under investigation. The dashed line in panel (a) is the fit resulting from analyzing the data up to 2000 seconds. The value of to deduced from this fit is 2787 seconds, suggesting that the range of data fit is reasonable. The fitted curve provides a very accurate description of the initial stages of the primary creep transient. Panel (b) of fig. 25 contains the scaling function resulting from the fitting procedure. Note that this scaling function differs significantly from that resulting from discrete model (fig. 20). This difference is physically significant. In the model, dislocation motion is initiated through the creation of a superkink at some random location along a completely pinned dislocation. This superkink begins to propagate, spawn new superkinks, scatter, etc. resulting in the acceleration of the dislocation. Eventually, the dislocation begins to decelerate until exhaustion. The scaling function of fig. 20 displays this behavior. In contrast, the data contained in fig. 25 is obtained on a sample for which there is no prestrain. In this test, initial loop expansion creates near-screw dislocations which have been created during the motion of the edge portions of the loops - effectively two very large superkinks. These near-screw dislocations, therefore, contain a larger number of mobile superkinks, and have a higher velocity on average, than if they had been mobilized through the creation of a single mobile superkink. Thus the initial velocity of these dislocations is high. The scaling function in panel (b) of fig. 25 hence displays this different behavior. The typical dislocation decelerates for roughly the first third of
Dynamics of dislocation motion in L12 compounds
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233
1.0
(a)
0.9 .~
0.8
9-.
0.7
0.6 o =
0.5
~-. 0.4 0.3
0.2 .....
i oo
time [seconds] 3.0 2.5 2.0 ~"
1.5 1.0 0.5
(t,) 0.0 0.0
0.2
0.4
0.6
0.8
1.0
X Fig. 25. (a) Comparison of experimental primary creep transient and the fitted curve arising from eq. (29). The solid line is the experimental data obtained from a single crystal, [123]-oriented crystal (single-system slip), at room temperature. The dashed line is the fitted curve. (b) The areal-velocity scaling function arising from the fitting procedure. The initial deceleration is believed to result from the excess number of superkinks spawned during the initial loop expansion. See the text for details.
234
D. C. Chrzan and M. J. Mills
Ch. 52
its lifetime, settles into a near constant velocity for the second third, and decelerates to zero velocity over the final third. The above analysis points to the strength of this formulation for the primary creep transient. The fact that there are reasonable differences between the scaling functions for these different situations suggests that 9(x) is indeed a physically significant quantity. Measurement of the primary creep transient thus may provide a detailed description of the motion of a typical dislocation. 3.2.5. Formulation of the stress-relaxation tests The agreement between the predictions of the scaling theory for the primary creep transient and the experiments supports the postulated form for the exhaustion of dislocations under constant stress conditions. A measurement closely related to primary creep is the stress relaxation transient. In this test, a constant strain-rate test is performed until the sample reaches a predetermined strain. The machine crossheads are fixed, and the elastic strain is converted, through dislocation motion, to plastic strain. During this conversion, the applied stress (and presumably the total effective stress as well) decreases. The time dependence of the stress is recorded as the stress relaxation transient. For the case of Ni3A1, the applied stress changes by only 0.5-5% during the stress relaxation within the lower temperature regime of the flow strength anomaly [45, 46]. Consequently, it would seem reasonable to approach the analysis of the stress relaxation transient in a manner similar to that used above for the primary creep transient. Specifically, if the exhaustion of dislocations is responsible, in large measure, for the form primary creep transient, then it is essential that this process is also included in the interpretation of stress relaxation data. Since the stress is decreasing, it is also probably reasonable to assume that no new dislocation sources become operative during the the transient. In addition, since relatively small plastic strains are recovered during relaxations in Ni3A1 within the anomalous regime, it is perhaps reasonable to assume that the internal stresses are not changing dramatically during the transient. The greatest difficulty in modeling the stress-relaxation transient is determining a unique time at which the mobile dislocations became mobile. In order to make further progress toward modeling this transient, assumptions must be made. The assumptions described below are less satisfying than those used in modeling the primary creep transient, and can certainly be improved upon. In this regard, the predictions for the stress-relaxation transient presented here should be viewed only as evidence that further development of the scaling theory of the plastic deformation of the L12 compounds is warranted, and not as the final description of the transient. The approximate physical picture is the following. It is assumed that at the beginning of the stress-relaxation measurement that the dislocations have been mobile for a time short in comparison to the time span of the measurement. It is also assumed, that the dislocations are moving under an applied stress near the critical stress of the pinning/depinning transition. (This is consistent with the observation that the net applied stress must be driven towards the critical stress during a constant strain-rate test.) The equation describing the relaxation is
d~- = _ M d5 dt dr'
(30)
Dynamics of dislocation motion in L12 compounds
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235
where M is the constant describing the machine stiffness. Substitution of eq. (20) into eq. (27) gives the following equation for the stress relaxation transient. -dr ~ -- - M - -b~t ~ - E~ gmEm+l [A(Tc - T) ~pt].
(31)
m----O
Note that this transient reflects the hardening apparent in the measured transients. Equation (31) can be used to fit experimental data. Figure 26, panel (a) is a comparison of experimental data (solid line) and the fitted curve (dashed line) for a relaxation transient obtained from a Ni3A1 sample held at a temperature of 545 K [47]. The initial stress for the transient is 141.5 MPa, and the data is fit over the time range from 20 to 300 seconds. (The initial seconds of the test are omitted for two reasons - eq. (27) does not apply to short times, and the errors made in estimation of the mobilization time are most significant for the early time regime.) The fit is indistinguishable from the data for the time range fit. Panel (b) depicts the time dependence of the characteristic exhaustion time, to. Panel (c) of fig. 26 is the scaling function resulting from the fitting procedure. Note that this scaling function does not display the initial deceleration present in the scaling function of fig. 25. This is consistent with the fact that the stress relaxation test is performed after a significant strain. Thus the dislocations do not contain an overabundance of mobile superkinks, as is likely after initial loading and loop expansion. The agreement between the scaling theory of the stress-relaxation transient and the experimental result is also satisfying. It is important to note that this analysis is markedly different from the conventional interpretation of the stress-relaxation transients [48] which have been applied recently to measurements in Ni3A1 [45, 46]. Specifically, these interpretations are based on the premise that the mobile dislocation density is constant during the relaxation. The form of the transient is then dictated by the reduction in the dislocation velocity as the stress decreases, and is therefore thought to indicate the value of the activation volume in the framework of thermally-activated flow. This conventional approach results in the prediction of a "logarithmic" relaxation which is in excellent agreement with the transients measured for most temperatures within the anomalous regime [45-47]. However, the fact that dislocation exhaustion is clearly central to understanding the primary creep transient suggests that dislocation exhaustion should be at least as important in determining the form of the stress-relaxation transient. The above analysis indicates that such a view may also yield satisfactory fits to the relaxation transients. This represents only a first step toward a more refined theory which, for example, would properly account for a distribution of initiation times. Nevertheless, the value of this approach is that it provides a self-consistent explanation for both the stress relaxation and primary creep transients - the latter being inconsistent with previous models of anomalous flow.
3.3. Future
The rudimentary theory of the pinning/depinning transition presented here provides a satisfying explanation for much of the experimentally observed phenomena. In particular,
D. C. Chrzan and M. J. Mills
236
Ch. 52
140.2 140.1 140.0 e,-,,,,,t 139.9 139.8 '--' 139.7 r/j 139.6 ',,-* 139.5 139.4 139.3 139.2 0
50
100 150 200 250 300
t [seconds] Ill
(b)
650
,--,
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~
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Dynamics of dislocation motion in L12 compounds
237
it has been shown that one can exploit the properties of the pinning/depinning transition to connect directly dislocation dynamics and macroscopic mechanical properties. The theoretical identification and the subsequent experimental measurement of the arealvelocity scaling function is the most striking success of the theory. In spite of the current successes, the connection to macroscopic properties is far from complete. In particular, it is demonstrated above that the overall time-scale of the current formulation is roughly eight orders of magnitude too small to be consistent with experiments. In addition, even if the time-scale problem is addressed, the formulation leading up to eq. (27) is certainly applicable only to a restricted time range (due to finite size). Both of these problems need to be addressed in order to advance the theory. Understanding of the finite size effects awaits further study, and will not be commented on further in this chapter. The question of time-scales is addressed briefly below. The successful analysis of the primary creep transient within the scaling theory of the dislocation pinning/depinning transition suggests that the ideas behind the transition are sound, but that some fundamental time-scale has been underestimated. This forces one to reconsider the physics entering the formulation of the dynamical simulations as well as the discrete model of dislocation dynamics. A fundamental parameter which determines the overall duration of the creep transient is the free-flight velocity, vf (determined by the drag coefficient, B, in eq. A1). It as been reasoned above that 'Of should be large (see section 2.1). While the free-flight velocity can only be estimated based on experimental evidence, uncertainty in its value is not solely responsible for the time-scale discrepancy. It is the opinion of the current authors, that the most likely origin of the time-scale discrepancy stems from the conditions under which a pinning point (or KW lock) unbinds. In the dynamical simulations, it is assumed that the pinning points dissolve whenever the adjacent dislocation segment bows beyond a critical angle, 0c. However, several authors have proposed that this picture is too simplistic [16-18]. A more realistic picture is to assume that the dissolution of a pinning point can be achieved over a range of bowing angles through thermal activation. It is reasonable to assume that the critical angle for athermal dissolution is still well defined, but in addition, for bowing angles less than 0c, the energy barrier is an increasing function of (0c - 0 ) . The existence of a thermally-activated step in the dislocation dynamics may well resolve the discrepancy in time-scales between the pinning/depinning transition analysis and the experimental data. A question which then must be resolved is: If there is a thermally-activated step in the motion of dislocations, does this completely alter the analysis of the pinning/depinning transition? In order to test the influence of a thermally-activated step on the dynamics governing dislocation motion, the discrete model described in Appendix B was modified as detailed in the final section of that appendix, 6.2. The discrete model described above allowed only one value of edge character (i.e. slope = 2) for a mobile superkink, and a lesser value (slope = 1) for the immobile superkinks. In contrast, the modified model allows two different values of edge character for the mobile superkinks and one value of edge character for the immobile superkinks. In the modified model, the superkinks with the largest value of edge character (slope = 3) are able to overcome pinning points athermally. The superkinks with a slope = 2 overcome pinning points through a stochastic, thermally-activated step. A dislocation is deemed immobile when all the
238
D. C. Chrzan and M. J. Mills
Ch. 52
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t [steps] Fig. 27. Comparison of g(t) curves for the original automata and the automata modified to include thermallyactivated unpinning. The dashed line displays a t3/2dependence, the dotted line a t 5/3 dependence. Note that the overall time-scale for the thermally-activated automata is extended by two orders of magnitude. Note also that the scaling exponents of the thermally-activated model appear to be different from those describing the original automata (as indicated by the different value of the ratio a/b).
superkinks have so little edge character (slope - 1) that they cannot overcome the pinning points even if assisted by thermal fluctuations. The physical basis for these modifications to the original discrete model is the belief that one can separate the thermally-activated steps into those which are relevant for the time-scales over which mechanical properties are modeled and those for which the activation barrier is too large for the process to influence the final outcome of the experiments. In the real world, the activation barriers are distributed over a continuum of values. The modified model divides this continuous distribution into three categories: 1) zero energy barrier, 2) finite energy barrier, and 3) infinite energy barrier. Whether or not this division alters in a fundamental way the dynamics governing dislocation motion is not known. Nevertheless, the modified discrete model provides a means to study the effects of including a thermally-activated step in the dislocation dynamics. While the study of the discrete model is not complete at the time of this writing, some significant results are available. It is clear that the modified model displays a pinning/depinning transition. A precise estimate for all the scaling exponents is not yet available, but an estimate for the ratio a/~ can be obtained. Figure 27 is a plot of ~(t) obtained from the modified model near its critical stress compared with a similar result from the original discrete model. The dashed line displays the t 3/2 dependence expected from the original model (c~/3 = 3/2). The dotted line displays a t 5/3 dependence. It is clear from the plot that c~/~ = 5/3 is the better of the two estimates for the modified model. The introduction of a thermally-activated step has changed scaling exponents of the transition. Note that the short time behavior no longer reflects the behavior described by eq. (24), but that the long time behavior agrees well with this form.
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Dynamics of dislocation motion in L12 compounds
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The energy barrier for the thermally-activated step was chosen such that the probability for dissolution of a pinning point was 1/100 per time step. The plot in fig. 27 shows that the scaling behavior begins at roughly 100 time steps. Thus, the scaling behavior of the transition has been shifted to longer times. Therefore, it is plausible that the introduction of a thermally-activated step in the dislocation dynamics eliminates the time-scale discrepancy, but clearly, more research is needed. As a final note, other authors have argued that the form of the stress-relaxation transient measured in the L 12 compounds is indicative of the importance of a thermally-activated step, and indeed, a detailed theory for the temperature dependence of the activation volume has been developed by Kantha et al. [17] and an alternative theory has been developed by Hirsch [19]. The fundamental tenet of these theories is that the shape of the stress-relaxation transient is the manifestation of a thermally-activated process. In contrast, the analysis of the pinning/depinning transition presented above indicates that the algebraic form of the stress-relaxation transient stems from the continuous nature of the pinning/depinning transition, and that dislocation exhaustion plays an important role in determining the shape of the transient. The resolution of the differences between this and competing theories awaits further study. The preliminary analysis of the discrete model modified to include a thermallyactivated step suggests that the scaling exponents of the transition are influenced by this change in dynamics. If so, it is not clear that eq. (29) is a precisely correct interpretation of the experimental data. Determination of whether or not this is so remains for further research. However, eq. (21) is thought to hold regardless of the values of the scaling exponents, and the general formulation is not expected to be altered radically. Thus the quantitative results presented here must be viewed as preliminary. The study of the dynamics relevant to the pinning/depinning transition is still in its preliminary stages.
4. D i s c u s s i o n a n d c o n c l u s i o n s
In its original formulation, the CSP model is developed strictly to predict the temperature dependence of CRSS, and is not intended as a constitutive flow law for these compounds. Nevertheless, the assumptions of the model can be incorporated into a description of flow, and this route has been pursued by several authors [16-18]. Most modifications center on an explanation of the lack of strain-rate sensitivity in the yield strength. The starting point for these modifications is the Orowan equation, = pmbv.
(32)
In eq. (32), ~ is the strain-rate, Pm is the mobile dislocation density, b is the Burgers vector of the dislocation and v is the dislocation velocity. This equation states suscinctly that the strain-rate depends on both the mobile dislocation density and on the dislocation velocity. In principle, both the dislocation velocity and the mobile density depend on the applied stress and the time. The models for the yield strength [16-19], however, have assumed that the mobile density varies only weakly as a function of the stress, and not
D. C. Chrzan and M. J. Mills
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at all as a function of time (constant-structure conditions). Conversely, the dislocation velocity is dependent very strongly on the applied stress. The strong stress dependence, introduced through an activation volume, implies a weak strain-rate sensitivity. These theories go on to make quantitative predictions for the measured activation volumes that are in good agreement with the values deduced from experiment. It should be noted that for these predictions, the yield strength models are being used to predict the time dependence of the strain-rate under conditions of changing applied stress, even though these models are fundamentally steady-state formulations. Consequently, it is supposed that Pm is constant, presumably throughout the lifetime of the transient. In practice, however, Pm is known to change with time. The observation of strain hardening in single crystal samples oriented for single slip indicates that dislocation exhaustion is an intrinsic feature of the dynamics of dislocation motion, even under constant stress conditions. It should also be noted that measurement of the activation volume from the stress-relaxation transient requires a significant effort to "correct" for strain hardening [46]. Thus experimental evidence suggests that the mobile dislocation density is not constant over the life of the transient. In contrast to prior yield strength models, the current work centers on a more general formulation of the Orowan equation: a formulation in which the mobile dislocation density and the dislocation velocity are tied into one distribution, n(a, t). A more general form of eq. (32) can be obtained by replacing pm'O by a sum, over every dislocation in the solid, of the dislocations' velocities. This procedure results in eq. (13). It is possible to write the integral on the right-hand side of eq. (13) in the form of a simple product:
dt'v(t, t')~(t') - g(t)
dt' ~l,(t t) -- VDm
(33)
but this procedure does not simplify obviously the formulation: In order to define ~(t), one must know the value of the integral on the leftmost side of eq. (33). The average velocity for the dislocations that enters eq. (32) thus depends intimately on the mobile dislocation density. It has been mentioned above that strain hardening, as reflected in a changing mobile dislocation density, cannot be separated from yielding, as reflected in a change in dislocation velocity. Equation (33) is the mathematical equivalent of this statement. The successes of the more general approach described in the previous paragraph are detailed in the preceding sections of this chapter. In particular it is noted that the analysis presented here yields a quantitative description of both the primary creep transient and the stress relaxation transient. The algebraic forms of these laws are not dictated by the existence of a thermally-activated step, but rather stem from the structure of the integral on the left hand side of eq. (33). However, this work does not rule out the existence of a thermally-activated step in the dynamics of dislocation motion. In fact, the existence of such a step is required to correct the overall time-scale. But the work does cast some doubt on the common interpretation of the stress relaxation transient as reflecting the value of the activation volume. At this point, it is worthwhile to comment, where appropriate, on the relationship of this work to that of other authors. In particular, this work has some implications for the
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Dynamics of dislocation motion in L12 compounds
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theories of Molenat and Caillard [22], models derived from the CSP model [16-18], and the work of Hirsch [19]. These implications are described briefly below. However, one general statement about all of these works can be made: they do not reflect the strain hardening that is experimentally observed in these compounds, and given that this hardening is not separable from the dislocation velocity, they must be questioned on this basis. The Molenat and Caillard mechanism of the yield strength anomaly relies on the thermally-activated nucleation of two superkinks (referred to as macrokinks by these authors) and the subsequent lateral motion of these superkinks to advance the dislocation. The supporting evidence for this mechanism stems primarily from in-situ TEM studies. While in-situ observations of this phenomena are suggestive, they are not positive proof that this mechanism is controlling deformation. The simulations presented in this work show that superkinks can be created during the motion of existing, mobile superkinks even in the absence of thermal activation. The simulations also indicate that the distribution of mobile superkinks is significantly different from that of the immobile superkinks. In particular, fig. 7 shows that, even for mobile dislocations, less than one in every 1000 superkinks is mobile. In addition, the length of these mobile superkinks makes TEM observation difficult. The current in-situ TEM studies, therefore, may not be representative of dislocations containing these large superkinks, but instead, may reflect an alternate, and possibly more difficult, path by which a dislocation may move. A supporting argument for the locking/unlocking mechanism of Molenat and Calllard [22] is the observation of an exponentially-decaying distribution of superkink heights. However, the form of this distribution is also predicted by the current simulations, so this feature does not distinguish between the two mechanisms. At 200~ in Ni3Ga, in addition to the exponential distribution of superkink heights, there is an excess of superkinks with heights corresponding approximately to the width of the APB on the (111) plane. Molenat and Caillard argue that the locking/unlocking mechanism would lead to an abundance of superkinks with lengths corresponding to the APB width. While this is certainly the case, there is an alternate explanation for this "deviant" point which might also be consistent with the superkink dynamics found in the simulations. Hirsch [19] has postulated that the formation of KW locks may occur by several, successive cross slip events between (111) and (010) planes. This suggests that if, on the same initial cross slipped segment, the formation of a final KW lock configuration does not occur simultaneously along the entire length of the segment, then KW locks displaced by the width of one APB could be formed. This situation is more likely at low stresses (and temperatures) based on calculations by Hirsch [19], which explains why the "deviant" point on the distribution has only been observed after lower temperature deformation. Since this level of detail concerning the cross slip process is absent from the present simulations, this feature is not observed in computed distributions. When discussing the models derived from the PPV theory for the formation of the pinning point, one must be careful to distinguish between those results arising from the physics assumed for the model, and those results that arise from the use of the Takeuchi and Kuramoto approximation [10] for the distribution of pinning points. The simulations discussed in section 2 of this chapter are based on the assumed physics of the original PPV model for dislocation motion. The only real difference between the two approaches
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is that instead of assuming a fixed, periodic arrangement for the pinning points, the current simulations let the physics of dislocation glide and cross slip determine the distribution. The results of eliminating the periodicity are dramatic. For example, the pinning points form naturally into groups which resemble KW locks - a configuration very different from that envisioned in the Takeuchi and Kuramoto scheme. Also, the dislocation dynamics lead to the formation of mobile superkinks - a feature present in the microstructure, but absent in the CSP models. In addition, the structures that develop allow for natural explanations of the strain-rate sensitivity. Finally, by allowing the fluctuations and correlations to be unrestricted, the simulations lead directly to the origin of the experimentally observed strain hardening, a result not available from the analysis using the assumed distribution of pinning points. In the work of de Bussac et al. [18], the restriction of a periodic arrangement of pinning points is relaxed. Instead, the pinning points are thought to form at random along a dislocation which retains its near-screw orientation. This physical picture is then analyzed within in a mean-field statistical model to predict the mechanical properties of the compounds. This treatment probably provides a good description of flow in the very high-stress regime, where correlations in the pinning point formation process are not as important. However, the mean-field treatment is unable to address the low-stress, pinned phase in which correlations in the pinning process dominate the dislocation dynamics. Hirsch has provided plausible, detailed mechanisms by which the initial PPV pinning point may convert into the experimentally observed KW locks [19]. The simulations presented in this work do not reflect this process. Nevertheless, the assumptions of the simulations lead to a structure that is the dynamical equivalent of the KW locks in that it consists of long portions of dislocation, oriented along screw orientation, that are immobile, connected by mobile segments of mixed character. Since reasonable assumptions lead to the formation of KW locks in mobile dislocations, it seems plausible that these locks play a substantial role in the dynamics. However, within the analysis presented in this chapter, the precise mechanism by which the KW locks are formed does not seem to be the most important question. Many of the details may not propagate to the longer length- and time-scales relevant to the macroscopic behavior. In this regard, the simulations allow the identification of the important details. For example, it is clear that one must include some type of thermallyactivated process in order to produce reasonable overall time-scales. Also, the Hirsch description of KW lock formation suggests that the rate at which the ends of the locks are freed may depend on the lifetime of the particular lock. In addition, Saada and Veyssi~re [20] have considered the bowing of the locks on the cube plane, which may also change the strength of these obstacles with time. It is not clear a priori that these effects are important. In such a situation, the simulations provide a convenient means of conducting "numerical experiments" to address these issues. As a final note, the dynamical simulations and discrete model results presented here point to the most important philosophical conclusion of this work: The dislocation pinning/depinning transition reflects the collective behavior of obstacles formed through a thermally-activated process. Development of a quantitative theory of flow in these compounds requires a complete understanding of the critical dynamics of the transition. The current work does not present this complete theory, but rather presents a first attempt at developing this understanding.
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Acknowledgements The authors wish to thank J. Bonneville, M.S. Daw, S.M. Foiles, S.H. Goods, K.J. Hemker, M. Khantha, L. Kubin, J.-L. Martin, W.D. Nix, G. Saada, and W.G. Wolfer for stimulating discussions. Support for this work has been provided by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences under Contract No. DE-AC04-94AL85000.
5. Appendix A. Continuum-based dynamical simulations In these simulations, a simple equation of motion has been coupled with the concepts of cross slip pinning and athermal depinning. The dislocation is represented as a set of points. The kinetics for the motion of each point is governed by [49]:
v -
(Tab- A~;) B '
(A.1)
where v is the velocity normal to the dislocation segment, % is the resolved applied shear stress, b is the magnitude of the SD Burgers vector, A is the line tension (assumed to be isotropic), t~ is the local curvature, and B is the drag coefficient. (Note that the curvature can either impede or assist dislocation motion.) The parameter B is not easily obtained experimentally, so it has been chosen to give a straight, free moving dislocation a velocity of 100 m/sec while under an applied stress of 400 MPa, independent of temperature. The other parameters chosen for the simulations are contained in table A.1. Each point is moved based on eq. (A.1) and using a fourth-order RungeKutta integration [50] with a fixed time step. The time step was chosen to insure the stability of the integration, and depends inversely on the applied stress. For example, at 250 MPa, the time step is about 10 -1~ seconds. The spacing of the points was also chosen to maintain stability of the integration, and is dependent on both the applied stress and the temperature (which determines the pinning frequency). The parameters chosen correspond to a point separation of roughly 10 -8 m. As the dislocation moves, the interpoint spacing changes. Points are added and removed from the dislocation line to keep the spacing near the stable value. A point is added to the dislocation using an interpolation scheme based on a parabolic fit. Virtually identical results are obtained using both a linear interpolation scheme and an interpolation scheme based on a circular fit. The probability of pinning a particular point depends on the local line orientation of the dislocation. Strictly, cross slip can occur only along a dislocation of screw character. For small deviations from screw orientation, a dislocation can be assumed to be decomposed into pure screw segments connected by short mixed segments as shown in fig. A. 1. With increasing deviation from screw character (i.e. increasing values of a), the probability for cross slip pinning should decrease because the total available length of pure screw character decreases. In addition, since it is not possible for an arbitrarily short segment length to cross slip, the individual length of each pure screw segment (indicated as ls
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Table A. 1 Simulation parameters Parameter magnitude of the Burger's vector shear modulus SD line tension drag pinning attempt frequency per unit length activation enthalpy for cross slip
Symbol b /z A B f
Value 5.0 x 10 - l ~ m 5.0 x 10 l~ Pa #b 2 = 1.25 x 10-1~ N 2.0 x 10 -3 Pas 3.26 x 1019 m - l s - 1
AH
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~
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,
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in fig. A.1) must also be considered. To incorporate both the fact that the probability of pinning is related to the screw character of the SD at that point, and that there is a small segment cutoff to the pinning, the following geometrical factor, Pgeo, has been included in the pinning frequency:
Pgeo -" 1 c o s o~ 1 -
tan c~] tan 4)
0(/3
- c~).
(A.2)
Here, 1 is the length of dislocation that a particular point represents, c~ is the angle between the overall line direction and the pure screw orientation, q5 is the angle between the short mixed segments and the pure screw orientation, and /3 is a "cut-off" angle which reflects the minimum length of screw-oriented dislocation necessary for pinning to occur. O(x) is the theta function which is given by O(x) = 1 for x > 0, and O(x) = 0 for x ~< 0. The value of Pgeo decreases nearly linearly with ~, with Pgeo = 1 for c~ = 0, and Pgeo = 0 for c~ - / 3 . Strictly speaking, r should be chosen based on the minimum energy orientations for the SD, but for simplicity,/3 -- r = 20 ~ which yields a "critical" screw segment length ls ~ 0.5 nm. Other values of r and/3 and functional forms for Pgeo have also been investigated with no change in the general behavior. The other parameters used in the simulations are shown in table A.1.
Dynamics of dislocation motion in L12 compounds
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245
The total pinning probability also depends on temperature. As assumed in previous modeling [1, 2], the temperature-dependent probability for pinning, PT, is given by an Arrhenius relation: PT--fexp
( A,,) ---~
,
(A.3)
where f is the attempt frequency for cross slip per unit length, AH represents the barrier to cross slip, k is Boltzmann's constant, and T is the temperature. The values of f and AH shown in table A.1 are similar to those used previously [ 1, 2]. The total probability for the formation of a pinning point is given by .Ptot =
PgeoPTAt,
(A.4)
where At is the integration time step. The loop expansion simulations are initiated with a circular loop with a diameter of 1 ~tm. The calculation of the superkink distributions as well as the calculation of the dynamical properties of the dislocations (average velocity and exhaustion spectra) were performed by considering the motion of a near-screw dislocation of finite length. The starting configuration for these calculations was generated by advancing a loop under an applied stress for some time, then extracting the portions of the loop aligned roughly along screw character. This simulates, for example, the effect of losing the long edge portions of the loop to the sample surface. The free ends of the dislocation were assumed to be of pure-screw character. The distributions of segment lengths were calculated by periodically tabulating the number of kinks of each length in a single, mobile dislocation. It is assumed that the time averaging of this quantity is equivalent to averaging over many different configurations, though this has not been explicitly verified. The distribution of exhaustion times has been calculated by advancing a finite-length, near-screw segment until exhaustion. The times required for these near-screw segments to exhaust are then catalogued. By repeating this calculation many times, a histogram of exhaustion times can be compiled.
6. Appendix B 6.1. Discrete model for dislocation motion
The continuum simulations described in section 2 indicate that the dislocations in these compounds undergo a stress-driven pinning/depinning transition. Unfortunately, the simulations are too numerically intensive to allow a complete numerical study of the properties of the transition, and progress can be made through a substantial simplification of the simulations. The simplified version of the simulations, cast in the language of a cellular automata, is described in detail in this appendix. A remarkable feature of thermodynamic phase transitions is that critical exponents are universal [38]. The values for those exponents depend only on "symmetries", and not on
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the details of the microscopic interactions. Therefore, if one constructs a simple model with the proper symmetries, one can make a quantitative prediction for the values of the critical exponents. Universality has n o t been established in the context of nonequilibrium dynamic critical phenomena. Nevertheless, it is hoped that the exponents describing such nonequilibrium behavior also depend only on symmetries. This possibility has been exploited modeling crystal growth [51 ]. In addition, recent results on a simplified version of the Burridge-Knopoffmodel have demonstrated universality for that system [43]. One must use caution. In the problem considered here, it is not clear which "symmetries" determine the exponents. Therefore, the approach taken here is the construction of a discrete model describing dislocation motion which retains as many features from the original simulations as possible, but remains computationally tractable. Since the major goal of this research is to develop a general understanding of this type of transition, much can be learned from the study of a system displaying a similar transition, even if quantitative estimates for the exponents are incorrect. In constructing a model for dislocation motion, it is deemed important to retain the physical properties of the superkinks. In particular, it is important to allow superkinks to spawn superkinks, single superkink annihilation, and superkink-superkink scattering leading to annihilation of superkinks. The model described below is not obtained as the discrete limit of the continuum description, but is constructed to mimic as many properties of the superkinks as is possible. The rules governing the discrete model are described in the following. The discrete model is constructed to be similar to those used to model crystal growth. The dislocation is assumed to be composed of L segments, labelled by the index i. Each segment i rests at a position denoted Yi, restricted to be an integer. At each step of the model dynamics, the segment at site i advances with an advancement probability Pi. The advancement probabilities depend on the relative configuration of that segment's nearest neighbors. At each step of the simulation, L random numbers, Pio, uniformly distributed between zero and one, are generated and compared with the Pi. The advancement of the sites proceeds according to the rule:
{y~+
l,
Yi -
Yi ,
P~>Pio,
(B.1)
Pi < Pio.
The physics of the model is contained in the specification of the advancement probabilities. The advancement probabilities are designed so that lYi+l- Yil < 2 for all i. Figure B.1 depicts the allowed three-segment configurations which enter into the model. The black dots in the figure indicate segments, n o t pinning points. In the model, a mobile superkink is composed of one segment drawn from configurations (1)-(o), any number of segments (g), and a final segment drawn from the configurations (h)-(k). Figure B.2 depicts the structure of a typical mobile superkink in the model. Note that there are many more possible superkinks. The structure of this superkink is denoted (amgja) because it contains one segment of type (a) adjacent to a segment of type (m) adjacent to a segment of type (g) etc.
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Dynamics of dislocation motion in L12 compounds "-
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y
i
Fig. B. 1. The three dislocation segment arrangements allowed in the cellular automata version of dislocation motion. See Appendix B for details. The continuum simulations show that superkinks of sufficient edge character will advance in a lateral fashion. For the purposes of the model, a segments edge character is determined by the relative positions of its nearest neighbors. If both lYi+l - Yil and lYi-1 - Yil are less than two, site i is deemed to have insufficient edge character and is defined to be immobile: pi = 0. The central segment in the configurations labelled (a)-(f) is, therefore, immobile. The central segment in the configurations labelled (g) is mobile because its relationship to its neighbors gives it the largest degree of edge character of any of the segments. Its advancement probability is pi - 1. In the continuum simulations, the force exerted on a pinning point by an adjacent segment depends on its edge character. In the discrete model, the dissolution force on a site is deemed critical whenever (yi+l - Yi) and/or (Yi-1 - yi) equals two. Therefore, Pi = 1 for all configurations labelled (h)-(k). The central dislocation segment in each of the configurations (1)-(o) represents the upper portion of a superkink. Examination of the continuum simulations reveals that spawning of new superkinks by a mobile superkink occurs in this upper portion of the superkink. Alternatively, each one of these configurations can create a new pinning point: if any of the segments (1)-(o) fails to advance, it becomes a pinned segment. As the stress is increased, the probability for spawning a mobile superkink increases. Conversely, as the temperature increases, the probability of creating an additional pinned point also increases. The advancement probabilities of these segments, therefore, depend on the temperature and the applied stress.
248
D. C. Chrzan and M. J. Mills
Ch. 52
As described in section 2.2.1, the bowing stress slows segments and holds them in the pinning region for a longer period of time, thus increasing the probability of forming a new pinning point. The advancement probability depends, therefore, on the curvature of the dislocation segment under consideration, and decreases as this bowing increases. Also, an increase in the applied stress will reduce the probability for formation of a pinning point. Finally, the probability for pinning point formation increases with temperature. Therefore, the probability for the segments of type (1)-(o) to pin is 0 Ppin(i)
"" T -
(B.2)
[Nil'
where 0 increases with temperature reflecting the temperature dependence of pinning point formation, T is the net applied stress, and It~il is the bowing stress which slows the dislocation segment. The advancement probability for segment (1)-(o) is given, therefore, by 0, Pi -
max
O
1-
(B.3)
where
e c i - ~x2(i) 1 +
~xx(i)
,
(B.4)
and the derivatives are replaced by their discrete counterparts:
0x ~J
(B.5)
2
and i~2y (i)
-
+
-
2y
].
(B.6)
The advancement probabilities are thus determined by two quantities: the net applied stress "r, and the temperature dependent pinning frequency, 69. Let us consider the dynamics of a typical superkink of type (amgja) shown at the far left of fig. B.3. The figure displays three of the possible paths for the superkink under the dynamics dictated above. The path labelled by the arrow (A) converts the initial configuration into (abnkc). The path labelled by the arrows (A) and (C) is the path by which the superkink travels one unit and returns to its original shape. The path denoted by the arrows (B) to (F) to (G) is a path by which the mobile superkink spawns another superkink. The path (A) to (D) to (E) results in annihilation of a mobile superkink. Thus all of the processes deemed important to the dynamics of dislocation motion based on the continuum simulations are contained within the dynamics of the discrete model.
w
Dynamics of dislocation motion in L12 compounds
249
(a)(m) A
A
w
w
A
w
A
A
W
W
Fig. B.2. A typical superkink in the discrete model.
_ww
(A
(C)
Fig. B.3. Three possible dynamical paths for the superkink of fig. B.2. The path labelled (A)-(C) translates the superkink by one unit towards the right. The path labelled (A)-(B)-(F)-(G) results in the spawning of an additional mobile superkink. The path labelled (A)-(D)-(E) leads to exhaustion of the mobile superkink. The dynamics of the discrete model are explored as follows. A random, pinned, dislocation configuration [i.e. built only from the segment configurations labelled (a) through (e) in fig. B.1] is generated. The boundaries of the dislocation are considered to be a free crystal surface, and the dislocation is restricted to approach the boundary with zero slope. The practical consequence of this statement is that only the configurations labelled (e), (f), (h), or (o) can be the endpoints of the dislocation. A random point within the dislocation is selected and advanced by one unit: yi --+ yi + 1. This initiation mobilizes the dislocation and allows it to advance under the applied stress. At stresses below the critical stress, the dislocation advances for a finite time before once again arriving in an exhausted configuration. The initiation process is begun again starting from the exhausted configuration. This process is repeated until a statistically steadystate is obtained, and the histograms described in the main text are compiled.
250
D. C Chrzan and M. J. Mills
Ch. 52
6.2. Discrete model for dislocation motion including thermal activation The discussion in the main sections of this chapter have pointed to an overall discrepancy in time-scale between results of the discrete model described above and the experimental measurements. The discrepancy most certainly arises from an underestimate of a fundamental time-scale in the problem. Hirsch has proposed that superkinks do not bypass obstacles athermally, but rather require thermal activation to advance [19]. This proposal is based on a careful analysis of the likely origins of the KW locks and the likely paths for their destruction. The following modifications to the original discrete model were implemented to test the effect of including a thermally-activated step in the unpinning of the KW locks. The modifications rest on the idea that a large superkink will still be able to overcome a KW lock through an athermal process, but that smaller superkinks may overcome the locks through a thermally-activated process. In order to be consistent with the experimentally observed strain hardening, it must still be assumed that some superkinks are so small as to be incapable of overcoming the KW locks, even through thermal activation. The modified model can be obtained from the model described in section 6.1 through the following modifications. Since the intent is to have superkinks with three different mobilities, the number of allowed height differences must be increased. For this reason, the dynamics allow all configurations for which ]yi-Yi+ll - - 3. Thus, the total number of nearest-neighbor configurations increases from the 25 shown in fig. B.1 to a total of 49. The rules for the advancement of the segments are then formulated as above, subject to the modifications imposed by the existence of both an athermal means of pinning point dissolution and a thermally-activated pathway. In the new model, the configurations (a)-(f) shown in fig. B.1 remain immobile. The advancement probabilities of all three segment configurations with negative curvatures [evaluated using eq. (B.6)] are given by equation (B.3). This rule applies to the configurations (1)-(o) of fig. B.1, as well as similar configurations in which (y~ - Yi+l) = 3 etc. The configurations labelled (k) through (i) are those which can overcome the pinning points through a thermallyactivated step. Their advancement probabilities are deemed to be Pi = r , where r is the probability per unit time that a pinning point unpins. The configuration labelled (h) can advance through thermal activation, but since there are two segments which can lead to dissolution of the pinning point, its advancement probability is fixed to be Pi = 2r. The advancement probability of configurations (g) in fig. B.1 remains one, because this segment has sufficient edge character for mobility. All segments in which ( y i + l - y~) = 3 and/or (yi-1 - yi) = 3 advance with a probability Pi = 1. The dynamics of this modified discrete model are explored in the same fashion as the original model. A dislocation is deemed exhausted when all of its segments are drawn from the configurations labelled (a)-(f) in fig. B.1. As discussed in the text, this modification increases the overall time-scale of the problem, but does not eliminate the critical nature of the dynamics associated with the transition. However, there is some evidence, also discussed in the text, that the inclusion of a thermally-activated unpinning does influence the scaling exponents of the transition.
Dynamics of dislocation motion in L12 compounds
251
References [1] [2] [3] [4] [5] [6] [7] [8]
S. Takeuchi and E. Kuramoto, Acta Metall. 21 (1973) 415. V. Paidar, D.P. Pope and V. Vitek, Acta Metall. 32 (1984) 435. S.S. Ezz, D.P. Pope and V. Paidar, Acta Metall. 30 (1982) 921. P.A. Flinn, Trans. TMS-AIME 218 (1960) 145. M.H. Yoo, Scr. Metall. 20 (1986) 915. Y.Q. Sun and P.M. Hazzledine, Philos. Mag. A 58 (1988) 603. A. Korner, Philos. Mag. Lett. 58 (1988) 507. M.J. Mills, N. Baluc and H.P. Karnthaler, in: High Temperature Intermetallic Alloys III, MRS Symp. Proc., Vol. 133, eds C.T. Liu, A.I. Taub, N.S. Stoloff and C.C. Koch (1989) p. 203. [9] P. Veyssi~re, in: High Temperature Intermetallic Alloys III, MRS Symp. Proc., Vol. 133, eds C.T. Liu, A.I. Taub, N.S. Stoloff and C.C. Koch (1989) p. 175. [10] D.M. Dimiduk, PhD Thesis (Carnegie-Mellon University, 1989). [11] Y. Sun, PhD Thesis (University of Oxford, 1990). [12] N.L. Baluc, PhD Thesis (Ecole Polytechnique Federale Lausanne, 1990). [13] C. Bontemps and P. Veyssi~re, Philos. Mag. Lett. 61 (1990) 259. [14] P.H. Thornton, R.G. Davies and T.L. Johnston, Metall. Trans. 1 (1970) 207. [15] S.M. Copley and B.H. Kear, Trans. Metall. Soc. AIME 239 (1967) 977. [16] V. Vitek and Y. Sodani, Scr. Metall. Mater. 25 (1991) 939. [17] M. Khantha, J. Cserti and V. Vitek, Scr. Metall. Mater. 27 (1992) 481. [18] A. de Bussac, G. Webb and S.D. Antolovich, Metall. Trans. A 22 (1990) 125. [ 19] P.B. Hirsch, Philos. Mag. A 65 (1991) 569. [20] G. Saada and P. Veyssi~re, Philos. Mag. A 66 (1992) 1081. [21] A.E. Staton-Bevan and R.D. Rawlings, Phys. Status Solidi A: 29 (1975) 613. [22] G. Molenat and D. Caillard, Philos. Mag. A 64 (1991) 1291. [23] K.J. Hemker, M.J. Mills and W.D. Nix, Acta Metall. Mater. 39 (1991) 1901. [24] A.E. Steton-Bevan, Philos. Mag. A 47 (1983) 939. [25] M.J. Mills and D.C. Chrzan, Acta Metall. Mater. 40 (1992) 3051. [26] S. Takeuchi, K. Suzuki and M. Ichihara, Trans. Jpn Inst. Met. 20 (1979) 263. [27] S.H. Goods, unpublished research. [28] G. Leibfried, Z. Phys. 33 (1950) 127. [29] M.J. Mills and S.H. Goods, unpublished research. [30] A. Couret, Y. Sun and P.M. Hazzledine, High Temperature Intermetallic Alloys III, MRS Symp. Proc., Vol. 213, eds L.A. Johnson, D.P. Pope and J.O. Stiegler (1991) p. 317. [31] A. Couret, Y.Q. Sun and P.B. Hirsch, Philos. Mag. A 67 (1993) 29. [32] R.G. Davies and N.S. Stoloff, Trans. TMS-AIME 233 (1965) 714. [33] K.J. Hemker, M.J. Mills, K.R. Forbes, D.D. Sternbergh and W.D. Nix, Modeling the Deformation of Crystalline Solids, eds T.C. Lowe, A.D. Rollett, P.S. Follansbee and G.S. Daehn (The Minerals, Metals and Materials Soc., Pittsburgh, 1991) p. 395. [34] A. Couret, Y.Q. Sun and P.M. Hazzledine, in: Proc. Int. Symp. on Intermetallic Compounds: Structure and Mechanical Properties - JIMIS-6, ed. O. Izumi (Japan Inst. of Metals, Sendai, 1991) p. 397. [35] K.J. Hemker, M.J. Mills and W.D. Nix, J. Mater. Res. 7 (1992) 2059. [36] A.H. Cottrell and R.J. Stokes, Proc. R. Soc. London Ser. A: 233 (1955) 17. [37] D.C. Chrzan and M.J. Mills, Phys. Rev. Lett. 69 (1992) 2795; D.C. Chrzan and M.J. Mills, to be published; D.C. Chrzan and M.J. Mills, Mater. Sci. Eng. A 164 (1993) 82. [38] Shang-Keng Ma, Modern Theory of Critical Phenomena (Benjamin/Cummings, Reading, MA, 1976). [39] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford Univ. Press, New York, 1971). [40] H. Haken, Synergetics (Springer-Verlag, Berlin, 1983). [41] J.C.M. Li, Acta Metall. 11 (1963) 1269. [42] See, for example, D.S. Fisher, Phys. Rev. B: 31 (1985) 1396; O. Narayan and D.S. Fisher, Phys. Rev. B: 46 (1992) 11520. [43] E.J. Ding and Y.N. Lu, Phys. Rev. Lett. 70 (1993) 3627.
252
D. C. Chrzan and M. J. Mills
[44] G. Molenat, D. Caillard, Y. Sun and A. Couret, Mater. Sci. Eng. A 164 (1993) 407. [45] J. Stoiber, J. Bonneville, J.-L. Martin, in: Proc. ICSMA-8, Vol. 1, eds P.O. Kettunen, T.K. Lepisto and M.E. Lehtonen (Pergamon Press, London, 1988) p. 457. [46] J. Bonneville and J.-L. Martin, in: High Temperature Intermetallic Alloys III, MRS Symp. Proc., Vol. 213, eds L.A. Johnson, D.P. Pope and J.O. Stiegler (1991) p. 629. [47] J. Bonneville and J.-L. Martin, unpublished research. [48] E Guiu and P.L. Pratt, Phys. Status Solidi 6 (1964) 111. [49] H.J. Frost and M.E Ashby, J. Appl. Phys. 42 (1971) 5273. [50] W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes (Fortran) (Cambridge Univ. Press, Cambridge, 1990). [51] See J. Krug and H. Spohn, Solids far from Equilibrium, ed. C. Godreche (Cambridge Univ. Press, Cambridge, 1992).
CHAPTER 53
Microscopy and Plasticity of the L12 Patrick VEYSSII~RE
and
Phase
Georges SAADA
Laboratoire d'Etude des Microstructures CNRS-ONERA, BP 72 92322 Chatillon Cedex France
Dislocations in Solids 9 1996 Elsevier Science B.V. All rights reserved
Edited by E R. N. Nabarro and M. S. Duesbery
Contents 1. 2.
Introduction 255 Macroscopic properties 256 2.1. Domain A 257 2.2. Domain B 259 3. Properties of superdislocations in L12 intermetallics 290 3.1. Limitations of TEM investigations 290 3.2. Properties of planar defects 314 3.3. The deformation microstructure in domain A - rate-controlling mechanisms 317 3.4. The deformation microstructure in domain B 319 3.5. Dynamical behaviour 354 4. The implications of microstructural observations in domain B 359 4.1. Summary of facts 360 4.2. The contribution of individual dislocations 361 4.3. The collective behaviour of dislocations in domain B 389 4.4. Work hardening 397 4.5. Summary and concluding remarks 399 5. Analysis of theoretical contributions on the positive temperature dependence of the flow stress 401 5.1. Introduction 401 5.2. On the elementary cross-slip process 402 5.3. Miscellaneous models 405 5.4. Local pinning models 407 5.5. Locking/unlocking 413 5.6. Kink models 414 5.7. Modelling 425 6. General conclusion 429 6.1. Analogies and differences with previously analysed systems 429 6.2. The evolution of the microstructure 430 6.3. Consequences on mechanical properties 431 References 435
1. Introduction Single-phase ordered intermetallic alloys exhibit a variety of macroscopic mechanical properties that are dominated by the individual behaviour of dislocations and more precisely by transformations that occur within the core of superdislocations (Vitek [ 1], Duesbery [2], Veyssi6re [3]). In the following, the term core will refer to the fine structure of a dislocation regardless of its extent. Core-controlled behaviour is primarily manifested through pronounced temperature dependence of the flow stress (TDFS) and, in some cases, through some dependence of the flow stress upon the sense and orientation of the applied stress. Such properties are the object of particularly active investigations in L12, B2 and L lo intermetallics, some of which were initiated more than three decades ago. Positive temperature dependence of the flow stress has been reported in some metals and ceramics (see Caillard, this volume), but it is certainly in intermetallic systems that this behaviour is the most frequently encountered. In fact, a flow stress peak can be found in almost every "simple" ordered structure (L12, B2, L10, D019, D022.... ). However one should not assume that flow stress peaks all originate from the same dislocation mechanism (Veyssi~re [4, 5]). There is for instance consistent experimental evidence to suggest that the positive TDFS of/3-CuZn (B2) involves diffusion (Saka and Zhu [6], Zhu and Saka [7], Nohara [8], Dirras et al. [9]) whereas a non-conservative process appears inadequate in order to explain the flow stress peak in Ni3A1 (L 12) and in TiA1 (L 10). Even comparing the flow stress peaks of the two latter alloys is difficult since in Ni3A1, strain is ensured by dislocations with only one type of Burgers vector (e.g., b -- (110)), the anomaly being controlled by the locking of one family of dislocations, 1 whereas deformation in TiA1 is provided by three families of dislocations (b - ~(110], b - (101] and b = 1 (112]) whose locking properties and the dependence of these upon temperature have not yet been all elucidated. Moreover, the positive TDFS of some B2 alloys such as CoZr and CoHf is intriguing since deformation does not proceed by superdislocations of the B2 structure but by simple (100) dislocations (Yoshida and Takasugi [10], Francois and Veyssi~re [11 ]) which suggests that the locking mechanism is not a consequence of ordering on dislocation cores. The present contribution is limited to plasticity and core transformations in the family of L12 intermetallics since it is in this structure- in fact, mostly in Ni3Al-based alloys and, to a lesser extent, in Ni3Ga, in Ni3Fe, in (Ni, Fe)3Ge, in Ni3Si, Co3Ti and in some Pt3X compounds- that the temperature sensitivity of the flow stress and related properties are known in greatest detail and in the most consistent way. It should also be noted that these members of the family of L12 alloys happen to remain fully ordered up to their melting point. We shall restrict this review to properties intrinsically related to dislocation cores as opposed to extrinsic effects such as precipitation and static ageing. Extrinsic effects have been in fact identified as possible sources of strengthening and of mechanical
256
P. VeyssiOreand G. Saada
Ch. 53
instabilities at intermediate temperatures in L12 doped A13Ti alloys (Kumar [12], Morris et al. [13, 14], Wu and Pope [15]). Part 2 is devoted to a review of macroscopic mechanical properties including deformation under constant strain rate, creep and the analysis of transient tests. The results of microstructural analyses and especially of locking dislocation configurations are examined in part 3. In both parts 2 and 3 some emphasis is put on possible sources of experimental uncertainties. Part 4 discusses properties of organization of the deformation microstructure. Part 5 is aimed at reviewing the available theoretical models of the positive TDFS. Some of the arguments developed in part 5 are used in part 6 to discuss the consistency between the scenario developed in part 4 and the mechanical behaviour of L 12 alloys. Attesting to the level of interest still placed in this area in spite of more than 35 years of research, we would like to mention that since August 1995 where the present manuscript has been sent to the publisher, many new and original papers have been published. Recent results have little implication on the following account of experimental results since the latest mechanical tests and microstructural observations are mostly consistent with previously released works. This is not quite true with explanations and models and it is remarkable to realize how much points of view have changed over the last few months. Unfortunately, the most recent modelling papers could not be properly accounted for in the following unless significant changes had been made in too many places of the manuscript. We have therefore limited our latest additions to a few brief references to the contributions in question. For the sake of completeness, we thus advise the reader to refer in particular to the proceedings of the 3rd International Conference on High Temperature Intermetallics (ASM International, San Diego, spring 1994) and to the 6th volume on High Temperature Ordered Intermetallics Alloys (MRS, Boston, fall 1994, MRS Symposium Proceedings, Vol. 364, Parts 1 and 2).
2. M a c r o s c o p i c p r o p e r t i e s The most general flow-stress dependence upon temperature of a L12 alloy is represented in fig. 1, where three main domains can be distinguished (Suzuki et al. [16]). Some compounds such as Ni3A1- and Ni3Si-based alloys, but also some Pt-based alloys such as Pt3Ti, show only the flow-stress peak. A few others such as Pt3A1, Pt3(In, 0.1 at.% T1) and Fe3Ge exhibit only a strong negative TDFS at low temperature (Wee et al. [17], Suzuki and Oya [18]). Several systems such as Zr3A1 (Schulson and Roy [19]), Co3Ti, Pt3Cr, Pt3Ga, Pt3In, Pt3Sn, Pt4Sb (Wee et al. [17]), Fe3Ga (Suzuki et al. [20]), and Ni3Fe (Wee and Suzuki [21 ]) are known to exhibit both types of temperature dependence. There is some indication that even in Ni3Al-based alloys the flow stress increases slightly with decreasing temperature in the vicinity of 4 K (Mulford and Pope [22], BontempsNeveu [23]). Interesting enough is the system Ni3Ge-Fe3Ge (Suzuki et al. [24]), for it provides a spectacular example of a gradual transition from one type of dependence to another upon changes in composition. In this review, we shall concentrate on domains A and B, with major emphasis put on plasticity in domain B. Properties of domain C will be nevertheless addressed in a few particular situations (for instance in section 2.2.10)).
Microscopy and plasticity of the L12 ,)it phase
w
257
1;9
domain ~. B! ._/_.
domain A I 9
t
,
[,
.~
"-/doma~
I-" I ~
B2
domain B ,
,,,I
,
I
domain C I
,
I
Temperature Fig. 1. Generic temperature dependence of the flow stress in L I2 alloys (one can find some L I2 alloys that exhibit comparatively very little temperature dependence, see fig. 3) and the definition of the three domains of yield behaviour (Tp is the peak temperature, "rp the critical resolved shear stress at the peak, in the following ap will refer to the flow stress). Within domain B, some differences in mechanical properties on each side of the inflection point suggest a subdivision into two sub-domains, e.g., B 1 and B2. The peak sometimes exhibits a fine structure consisting in a succession of two maxima or more (not represented here).
2.1. Domain A
This domain is characterized by a more or less rapid decrease of the flow stress with increasing temperature, at low temperature. In domain A, increasing the test temperature by 100 K decreases the flow stress by 20 MPa in Co3Ti (Liu et al. [25]), by 50 MPa in Fe3Ga (Suzuki et al. [26]) and by 250 MPa in Pt3A1 (Wee et al. [27]). Suzuki et al. [24] have shown that a solid solution exists between Ni3Ge and Fe3Ge which provides a convenient system to study continuously the transition from a system which exhibits a flow stress peak only (Ni3Ge), to a system showing domain A only (Fe3Ge). As far as mechanical properties are concerned, Suzuki et al. [24] have found that the addition of Ni3Ge to Fe3Ge decreases gradually the positive TDFS up to an addition of Ni3Ge a little above 27.5 at.% (fig. 2(a)). For higher Ni3Ge concentrations, the flow stress decreases monotonously from 77 K to 900 K (fig. 2(b)). However, it should be noticed that between 0 at.% and 30 at.% of Ni3Ge, the level of the flow stress peak is stationary within 4-100 MPa and that the peak remains centered at about 700 K, while the flow stress at 77 K increases progressively from 320 MPa to 1200 MPa on increasing the Ni3Ge content from 0 at.% to 55 at.%. It follows that the gradual disappearance of the peak with increasing Fe3Ge content in the Ni3Ge-Fe3Ge system results essentially from an increase of the flow stress at low temperature (domain A), and not from a lowering of the flow stress in domain B. Surprisingly enough, above 55 at.% of Fe3Ge, the flow stress in domain A is shifted down quite abruptly by approximately 400 MPa and the most Fe3Ge-rich alloys are the weakest, which might be related to the fact that in Fe3Ge slip occurs most easily on the cube plane.
P. Veyssidre and G. Saada
258
(a) ~;;
1250 I
I l
(b) -~, 1250 I~
Fe0.4
F
IT 750
t-
250/Ni3Ge I 0 200
e,
%x~.%~
I Fe0_2~"~,l.1 F
I
~N_~\ I ~ ~:) I
I 400
I
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'
._*. ~-...
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a) 1000 -
6
Ch. 53
I
/
600 800 1000 Temperature (K)
o IT
lo0o
750
o~ O
250
0
I
I
I
I
200
400
600
800
1000
Temperature (K)
Fig. 2. Temperature dependence of the flow stress in a variety of Ni3Ge-Fe3Ge alloys, showing the gradual change of mechanical behaviour as the proportion of Ni relative to that of Fe is varied, that is, when the parameter z in (Nil_zFex)3Ge is varied from 0 to 1 (after Suzuki et al. [24]). (a) The flow stress peak disappears gradually up to z -- 0.4, as a result of the increase of the low temperature flow stress (the flow
stress at 600 K remains roughly unchanged, except for z = 0.1). (b) Temperature dependence of the flow stress for z/> 0.3. For the sake of comparison, Fe0.275and Feo4 are common to (a) and (b). Co3Ti and L12 trialuminides are among the few alloys in which the strain-rate sensitivity of the flow stress has been addressed in domain A. In Co3Ti, it has been determined that the flow stress is strain-rate independent from 77 K up to the peak (Liu et al. [25]). By contrast, in Mn-doped L12 A13Ti, activation volumes measured at 210 K are rather small (Morris et al. [13, 14]), indicative of significant strain-rate sensitivity. By analogy with similar behaviour encountered in a number of materials, such as b.c.c, crystals, some semiconductors and ceramics, the negative TDFS at low temperature of L12 systems should originate from lattice friction (section 3.3), which is thermally activated. However, there should be several microscopic origins for this behaviour since the operating slip system is not unique in domain A. It may differ from one L 12 crystal to another and one given crystal may exhibit two nonequivalent slip systems, both showing a negative TDFS: (i) Pt3A1 single crystals loaded near [001] deform by octahedral slip, whereas those oriented in the near-J111] and in the near-[ 123] orientations deform on the primary cube plane. The cube slip system exhibits the lowest CRSS (fig. 3) which, in addition, is independent of the orientation of the external stress (Wee et al. [27]). (ii) It seems that Fe3Ge deforms principally by cube slip (Ngan et al. [31, 32]). In Co3Ti on the other hand, slip occurs on octahedral planes - in both domains A and B - regardless of the orientation tested (Liu et al. [25]). In this latter alloy, the CRSS on (110){ 111 } is claimed to be orientation-dependent, but the reported variations are so modest that one may wonder whether they are actually meaningful (section 2.2.2). (iii) In L12 Fe- and Cr-doped A13Ti single crystals, the CRSS on (110){111} is independent of load orientation (Wu and Pope [15]).
Microscopy and plasticity of the LI 2 "7' phase
w tl:l
600 r-I Pt3A I (octa.)
s
o~
259
[] Ni3(AI,Nb )
9Pt3AI (cube)
O Ni3(AI,Ti )
500
O~ O~
,-r it) -o
400
300
>
o n-"
100 0
I 0
200
I 400
I
I
I
I
600
800
1000
1200
1400
Temperature (K) Fig. 3. Selected positive temperature dependencies of the shear stress resolved on (I01){111} in several L12 alloys deformed in compression in the neighbourhood of the [123] direction, except for Ni3(Si, Ti) and for Pt3AI in octahedral slip which were deformed near [111] and [001], respectively. Data are taken from the following studies: Ni3(AI, Nb): Lall et al. [28]; Ni3(Si, Ti): Takasugi and Yoshida [29]; Pt3AI: Wee et al. [27]" Ni3(AI, Hf)" Bontemps-Neveu [23]; Ni3(A1, Ti): Staton-Bevan and Rawlings [30]; Co3Ti (actual composition Co74Ni3Ti23): Liu et al. [25]. For the sake of clarity, the points plotted in this graph do not correspond to the temperatures at which the original measurements were conducted.
2.2. Domain B
Fu et al. [33] have recently pointed out that the first indication of a flow stress anomaly in L12 alloys was reported in Ni3Si by Lowrie [34]. It is nevertheless a study of the hardness dependence of Ni3A1 upon temperature (Westbrook [35]) that drew the attention of materials scientists to the anomalous plastic behaviour of L 12 alloys. However, because of the nature of a hardness test, it was not then possible to discriminate between the variations of the flow stress and those of work hardening. After 35 years of investigation, the ambiguity is still not fully elucidated in this category of alloys (see however [69, 71, 93, 280]). The domain of temperatures over which domain B takes place varies from one alloy to the other (fig. 3). This will be largely illustrated throughout this section and information on this subject can be found more consistently in the excellent review of Suzuki et al. [16], in the original works of Curwick [36], of Heredia [37] and in the paper of Heredia and Pope [38]. One considers in general the TDFS to be only slightly positive at 77 K. The case of Ni3(Si, Ti) should be mentioned since the flow stress for these materials increases continuously from 4.2 K to a temperature located between 600 and 800 K, depending upon sample orientation (Takasugi and Yoshida [39]). The mechanical behaviour is already anomalous at 4.2 K, as indicated by the fact that the flow stress is already orientation-dependent (see section 2.2.6). In this respect, the situation is in fact quite confusing in the low-temperature part of domain B, since all sorts of mechanical
260
P. VeyssiOreand G. Saada
Ch. 53
behavior can be encountered. Some alloys, such as Ni3(A1, Nb), that show quite a steep positive TDFS, have little orientation dependence (Ezz et al. [40]). On the other hand, amongst alloys that exhibit a small positive TDFS at 77 K, some, such as Ni3(A1, Ta), show pronounced orientation dependence all over domain B from 77 K to the peak, while others, such as Ni3(A1, Zr) and Ni3(A1, B), are orientation-insensitive at 77 K [37]. The most significant macroscopic features that characterize the positive TDFS (domain B) are analyzed in this section, but before addressing the possible origins of the positive TDFS, we find it important to shed some light on compositional effects. 2.2.1. Generalities on compositional effects
Our knowledge of the positive TDFS of L12 alloys relies essentially on measurements carried out on Ni3Al-based alloys. However, since binary Ni3A1 single crystals are difficult to grow near the stoichiometric composition, it should be kept in mind that most of the properties discussed below refer to off-stoichiometric and ternary compounds. One of the earliest indications of the role of composition seems to be due to Thornton et al. [41] 1 who pointed out niobium as a potent strengthener. The unpublished work of Curwick [36] is the first extensive study of compositional effects ever carried out on single crystals. In practice, composition adds to the overall complexity of core-controlled situations since modest changes, even at the level of one percent or less, may effect mechanical properties of Ni3Al-based alloys quite dramatically (Mishima et al. [42], Suzuki et al. [16], Khan et al. [43], Heredia and Pope [38], Dimiduk and Rao [44]). In binary alloys, it is known that above room temperature, the larger the atomic fraction of aluminium, the steeper the positive TDFS (Noguchi et al. [45]). According to Dimiduk [46], the composition dependence of the flow stress is even more pronounced in single crystals. On the other hand, Mishima et al. [47] have shown that for solute additions which substitute for Al-sites, transition metal elements induce extra hardening as compared to solute hardening obtained by additions of B-subgroup elements. Heredia [37] has in addition determined that composition strengthening differs according to whether ternary atoms are in substitutional or interstitial positions. The sensitivity of the flow stress to a deviation from stoichiometry, further complicated by the fact that site occupancy is not always identified in these alloys, is probably one of the reasons which make rationalizing compositional effects in ternary alloys so uncertain (Dimiduk et al. [48]). An illustration of these difficulties may be found, for instance, in the work of Heredia [37] in the case of Ta-doped Ni3A1 where the strengthening does not depend monotonically on the Ta content (70.2(2 at.% Ta) > 70.2(1 at.% Ta) at low temperatures, whereas above 800 K, T0.2(2 at.% Ta) > 7"0.2(5 at.% Ta), Tp = 1000 K). At the present time, it is not yet clearly elucidated whether chemistry acts intrinsically on core structure through modifications of the energy of planar defects, or else extrinsically, by the usual solution hardening process. Heredia [37] has concluded to the predominance of a classical solution-hardening effect at low temperatures, though he 1We would like to draw the reader's attention to the experimental work of Thornton, Davies and Johnston [41], in which almost every basic property was addressed very early and sometimes in great detail. Since then, this pioneering work has appeared instrumental to the understanding of the flow stress anomaly of L12 alloys.
w
Microscopy and plasticity of the L12 .yt phase
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relates the mechanical behaviour at higher temperatures to dislocation core transformations. In view of several major contributions (Mishima et al. [42, 47], Dimiduk [46, 49], Heredia [37], Hemker and Mills [50]), it seems in fact reasonable to consider that the composition dependence of the flow stress stems from a combination of both solution hardening and core transformation. A discussion of these effects, about which still very little is known theoretically, is beyond the scope of the present contribution. Compositional effects will nevertheless be addressed occasionally in the following when they may explain apparently inconsistent findings. In conclusion for this question, it appears that detailed comparisons between the mechanical properties of alloys with different solute contents should be considered with some care since no systematic trend has been established. For further and extensive details on solute effects, the reader should consult the studies of Curwick [36], Mishima et al. [42, 47], Miura et al. [51-53], Suzuki et al. [16], Dimiduk et al. [46, 48], Heredia [37] and Heredia and Pope [38].
2.2.2. Experimental uncertainties Besides difficulties which arise from compositional changes, mechanical tests are not always as reliable as suggested in the literature. Experimental uncertainties are not limited to the accuracy in measuring the conventional flow stress at 0.2% from the engineering stress-strain curves, which one may estimate to about 1% to 2%. For instance, in view of the large work-hardening rate at 0.2% of permanent strain (section 2.2.3), the experimental determination of the flow stress is inherently inaccurate (fig. 4) and one may wonder whether a difference of less than say 5% between two flowstress measurements allows one to claim significant differences in mechanical behaviour. This is particularly important in the discussion of the tension-compression asymmetry of the flow stress (section 2.2.6). The adverse effects of large work-hardening rates are also encountered in stress-relaxation experiments where resulting corrections to derive true activation volumes may represent a considerable fraction of the uncorrected signal (section 2.2.8.2). Although the appropriate precautions are usually taken, a further source of uncertainty may arise from sample preparation, since surface damage introduced during sample machining can increase the flow stress by a factor as large as two (Bontemps et al. [55]). The tension-compression asymmetry (section 2.2.6) is however unaffected. Still regarding boundary conditions, Mulford and Pope [22] have pointed out that prestraining at 77 K does minimize end effects such as plastic flow due to stress concentrations at the loading faces. Recent work has shown the importance of sample end effect under compression [286]. It should be kept in mind that since most straining devices exhibit significant thermal inertia, information derived from temperature transients should be regarded with care. Thermal stability is of critical importance in stress relaxation tests. Measurements depend strongly upon single crystal perfection, upon the accuracy in orientating single crystals and upon maintaining axiality during deformation. For instance, in order to explain a difference of approximately 20 MPa in the level of the CRSS at 77 K of a sample under one particular orientation (sample #5 in Ezz et al. [56]), with respect to the value expected from the value determined in every other orientations tested (~ 75 MPa), Ezz et al. [56] were prompted to invoke some experimental
262
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P. Veyssidre and G. Saada
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Fig. 4. Typical engineering stress-strain curves in Ni3(AI, 0.25 at.% Hf) deformed in compression along [123] at varied temperatures, showing the magnitude of the work-hardening within the early stages of deformation and the resulting difficulties in determining accurately the flow stress at 0.2% of permanent deformation. (a) Domain B" (b) End of domain B and domain C (after Bontemps-Neveu [23]). As remarked by Hernker et al. [54], the absence of an abrupt change in the stress-strain curves in domain B, suggests that flow stress depends on the formation/recovery of the microstructure variation of the apparent elastic slope that can be noted over these curves stems from the softness of the straining device and from sample end-effects (Shi et al. [286]). The extension of the preyield regime is however genuine. error. To the authors' k n o w l e d g e , it is only in the works of M u l f o r d and P o p e [22] and of H e r e d i a [37] that s o m e indications on crystal imperfections are reported, t h o u g h the i m p l i c a t i o n s of these on e x p e r i m e n t a l results are not clearly assessed. Uncertainties in crystal orientation m a y be estimated from fig. 1 in the work of U m a k o s h i et al. [57] as being of several degrees in various cases; it cannot be ensured of course that this estimate applies to every other e x p e r i m e n t a l study. Finally, in e x p e r i m e n t s w h e r e one
Microscopy and plasticity of the L12 3"t phase
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263
Table 1 Work-hardening rates in selected LI2 alloys. The results are given in units of/z (after Staton-Bevan [58], Bontemps-Neveu [23]). X and S refer to polycrystals and single crystals, respectively. Alloy composition Ni3Fe Cu3Au Zr3A1 Ni3(A1, Ti) Ni3(A1, 0.25% Hf) Ni3(A1, 1.5% Hf) Ni3(A1, 3.3% Hf)
Temperature range (K) X S X S X X X
298 77-262 100--1175 173-1273 4-1273 4-1273 77-573
Minimum work-hardening rate a t e = 1% -0.007 0.0004 0 - 1.02 0.006 0.006 0.006
Maximum work-hardening rate a t e = 1% 0.011 0.006 0.03 0.64 0.24 0.155 0.025
Reference
Arko and Liu [59] Kuramoto and Pope [60] Schulson and Roy [19] Staton-Bevan [58] Bontemps-Neveu [23] Bontemps-Neveu [23] Sp~itig et al. [61]
or two samples of each orientation are tested over the entire range of temperature, even though the samples are not allowed to yield more than 0.2% per test, one may wonder how much the sample axis is rotated after multiple deformation under single or multiple slip (the cumulated strain amounts to between 10 and 20%). One may thus question the overall precision whereby one given orientation is represented within the unit triangle. Recent dedicated experiments [282] indicate in addition that significant changes in the level of flow stress may occur as a result of cumulated prestrains at varied temperatures. It is not sure then how much the stored in microstructure contributes to these changes. Information on sources of errors in experimental procedures is scarce, which makes it difficult to estimate uncertainties in mechanical data, but this does not imply that errors can be ignored.
2.2.3. Work-hardening rate The work-hardening rate (WHR), identified as the slope of the shear stress-shear strain curve 2 h - d'r/d'y, can reach astonishingly large values (Staton-Bevan [58], Sun [62], B o n t e m p s - N e v e u [23]). The W H R at the conventional strain of 0.2% is larger than at 1%. At a permanent strain of 1%, h is significantly larger than 0.01# (in Ni3Al-based alloys #111 amounts to approximately 50 GPa). Staton-Bevan [58] has reported 0.64# in Ni3(A1, Ti) deformed along [111], and very large levels of W H R are found even under single slip (0.2/z in the same alloy deformed along [123]). For comparison, the W H R in f.c.c, metals is about 0.5 x 1 0 - 3 # under single slip and approximately one order of magnitude larger under multiple slip. The magnitude of the W H R in domain B is large even at the foot of the curve "r(T) where h may already overpass 0.01#. Some W H R determinations are gathered in table 1. It should be recalled that large values of h are h o w e v e r not limited to L12 alloys deforming in domain B. In fact, it has been remarked very early that high WHRs, though with a lesser magnitude than in L12 alloys, are a w
2The WHR is defined as 0 = dot/de, and h = dr/d3,(= ~--20) in polycrystals and single crystals, respectively, where q9 is the Schmid factor on the operating slip plane. However, because of confusion in notations between the surface energy of stacking faults and antiphase boundaries - denoted 3' in the following and the shear strain, we shall tend to use e systematically for the strain, its meaning is changed according to whether one is dealing with polycrystals or single crystals. On the other hand, the usual distinction between 7and ~r will be respected. -
264
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P. VeyssiOre and G. Saada
general manifestation of ordering (see for instance Kear and Wilsdorf [63], Kear [64, 65], Marcinkowski [66]). Figure 5(a) shows that, provided the WHR is measured beyond 1% strain, the variation of h with temperature exhibits pronounced anomalous temperature dependence. In general, h exhibits one peak at intermediate temperatures (Copley and Kear [67], StatonBevan [58], Dimiduk [46], Sun [62], Bontemps-Neveu [23]). However, in Ni3(A1, Ti) and Ni3(A1, Ta) deformed along [123], two peaks have been reported (Staton-Bevan [58] and Baluc et al. [68], respectively). The existence of a peak of WHR is strain dependent: at 0.15% strain and below, h increases with temperature and this increase is steady up to near the temperature of the peak of yield stress (Ezz, Sun and Hirsch [69]). It has been remarked that the peak position corresponds roughly with the temperature at which "r(T) is inflected (Saada and Veyssi~re [70]), irrespective of orientation (fig. 5). In Ni3(A1, Hf)B the WHR peak coincides with the onset of serrated flow (Ezz and Hirsch [71], see however section 2.2.11). The fact that the peak of WHR disappears when measured below say 0.2% strain is indicative of the extent of the microplastic domain; on the other hand, the fact that in this domain h increases with temperature deserves special attention, although one might question the meaning of h is this transient domain. The analysis by Staton-Bevan [58] of the composition and orientation dependencies of the WHR in Ni3(A1, Ti) samples constitutes the most extensive work published so far on this question; it can be read with benefit, although it lacks information on the temperature and orientation dependencies of the flow stress. The main findings and conclusions of Staton-Bevan's work are the following: m
In domain B, the work-hardening peak is the largest in the [111] orientation, it decreases for the [123] and [144] orientations and almost disappears for the [001] orientation (10 times less than in the [111 ] orientation). This behaviour is consistent with the magnitude of the Schmid factor for cross slip onto the cube plane relative to that of slip on the primary and cross-slip octahedral planes. - A zero WHR near 500~ for all orientations tested, a property which has been interpreted as corresponding to the onset of significant climb-assisted recovery.
-
B
Couret et al. [72] have measured the WHR in Ni3Ga single crystals deformed successively at two temperatures (400 -+ 200~ and 200 -4 20~ They show that the WHR at the second temperature compares with that of a virgin sample deformed at the same temperature. It does not depend upon whether the WHR at the second temperature is larger or smaller than that of the first temperature (section 3.2.1.6). According to these experiments, the WHR is little affected by a prestrain. The properties of reversibility of the flow stress are treated in section 2.2.5. Experimental results on work hardening in domain B will be discussed further in sections 2.2.4 and 2.2.8.1. Though beyond the scope of this review, the existence of a second orientationdependent peak of WHR in domain C should be mentioned (Staton-Bevan [58], Sun [62], Bontemps-Neveu [23]). Again above the temperature of the peak for the flow stress, L12 alloys exhibit pronounced work softening (see, for instance, fig. 1 in [58] and fig. 1 in [53]) which may be ascribed to the fact that at these temperatures superdislocations tend to be locked by climb dissociation (section 3.1.2.6) and that, as in other systems where the initial density of mobile dislocation is too low, the applied stress has to be raised in order to activate the appropriate amount of fresh sources.
Microscopy and plasticity of the L12 "7' phase
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266
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P. VeyssiOreand G. Saada
2.2.4. Strain dependence of the positive TDFS The critical resolved shear stress at a permanent strain of 0.2%, 7"0.2, is large and of the order of 10-3#. The TDFS is currently of the order of + 1 MPa per degree but this is markedly composition-dependent (fig. 3). Thornton et al. [41] have established that, beyond a permanent strain of 10 -5, whereas the peak magnitude is actually strain-dependent, the occurrence of a peak is independent of the strain at which the offset flow stress of a polycrystalline Ni3A1 sample is determined (fig. 6). At 400~ for instance, the flow stress at 0.2% is about 4 times larger than that measured at c - 10 -3, indicating that cr0.2 involves very strong microhardening. On the other hand, for a permanent strain less than 10 -5, the flow stress becomes temperature-independent. The particular strain dependence of the flow stress in domain B has been later confirmed by Mulford and Pope [22] and by Jumojni et al. [73] (it should be noted that no group has ever succeeded in reproducing the strain-dependence experiments of Thornton et al. (fig. 3) to the accuracy claimed by these authors in the domain of very modest strains). The former authors have succeeded in reproducing the results of Thornton et al. after prestraining. They show that, when loaded to the macroscopic yield p o i n t - here taken at a permanent strain of 0.1% - at 190 K and above, Ni3(A1, W) single crystals show no microstrain activity upon subsequent reloading at the same temperature. However, when the same prestraining is conducted at 77 K, a finite microstrain is again measured in the microyield region. As indicated by Mulford and Pope, it is likely that their overall observation is related to the exhaustion of the non-screw mobile dislocations. The particular effect of prestraining at 77 K itself suggests that there are many mobile dislocations formed at low temperature or else that unlocking of screws dislocations is easy. This simple experiment implies in addition that the positive TDFS does not originate from an increasing intrinsic resistance of the 100 r} v
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w
Microscopy and plasticity of the L12 "y~ phase
267
lattice to dislocation motion as envisaged in early hypotheses on the flow stress anomaly (section 5.2.1) (Davies and Stoloff [74], Johnston et al. [75] and Copley and Kear [67]). The transition from the elastic to the plastic regime takes place gradually over an extended domain of strain which, in some cases, can be as wide as 0.5% (fig. 4). The microdeformation behaviour of L 12 alloys deformed in domain B is thus reminiscent of that of b.c.c, metals deformed at low temperature, although the extent of the apparent microdeformation stage is larger in L12 alloys. The flow stress is essentially ill-defined, so that the usual conventional stress at 0.2% of permanent strain determines the yield stress quite arbitrarily. From the analysis of stress relaxation tests, Sp~itig et al. [61, 76] have concluded that the choice of or0.2 (or r0.2) is reasonable (section 2.2.8.2). On the other hand, Ezz and Hirsch [71] have considered the flow stress as the superimposition of two contributions, the yield stress 70.01 and the work-hardening stress th T = 7o.ol + 7-5,
(1)
which are governed by distinct mechanisms (section 2.2.8.1). This strain of 0.01% is actually defined as that at which the stress-strain curve first deviates in a measurable way from linearity, it is independent of strain rate and it exhibits a flow-stress anomaly with temperature. Nevertheless, the macroscopic elastic limit may not represent a critical stress for dislocation motion (Hemker et al. [54]) as suggested by the absence of a distinct yielding point on the stress/strain curves and on properties of primary creep that will be reviewed later (section 2.2.9). In the following, we shall make use of the traditional definition of 0.2%, essentially because this is the only flow stress determination that is made available in most experimental studies. 2.2.5. Flow stress reversibility In L12 alloys deformed in domain B, the flow stress is partly to fully reversible. In materials whose flow is determined by interactions between mobile dislocations and the microstructure, prestraining may affect the flow stress: in the absence of recovery, when strength is the largest at the prestraining temperature, the flow stress measured in the straining experiment coincides with the ultimate prestraining stress. Conversely, when flow is only determined by the mobility of individual dislocations, the flow stress should be almost reversible. A preliminary indication of a flow stress reversibility in Ni3Al-containing alloys is due to Davies and Stoloff [74]. In fact, the temperature reversibility of the flow stress appears implicitly in subsequent studies of the temperature dependence of the flow stress of L 12 alloys. It is quite frequent, indeed, to count many more points on the experimental curves of fl0w stress versus temperature than samples tested. This is due to a specific experimental procedure (section 2.2.2) in which T0.z(T) is determined over the whole temperature range using a limited number of samples, sometimes only one [56], two in other cases [37], tested successively at decreasing and/or increasing temperatures (see fig. 7). The permanent strain per test is modest. It has been checked, by comparison with deformation tests carried out under a different testing sequence or on non-prestrained
268
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Fig. 7. Example of a Cottrell-Stokes experiment conducted on Ni3(A1, 0.25 at.% Hf) deformed in compression along [123], first at 600~ to about 2%, then at room temperature (Shi [283]). The thin line (RT) corresponds to a stress-strain curve entirely carried out at room temperature. The work-hardening rate at room temperature is approximately the same whether the sample was predeformed or not. Above 2%, the level of stress is higher in the predeformed sample than in the sample deformed directly at room temperature. The difference is of the order of magnitude of the total work-hardening after deformation at 600~ However, at variance with results published elsewhere (Dimiduk [49], Hemker et al. [54]), the flow stress after the temperature change is not equal to the yield stress at the lowest temperature (20~ corrected for the stress caused by work-hardening during the prestraining at the highest temperature (600~ The curve (h6(x)) represents the correction in the present case, which yields a final stress nearly twice as large as the flow stress of the prestrained sample. It should be realized that the accord between the conclusions of Dimiduk [49] and those of Hemker et al. [54] regarding corrections for work hardening is only apparent. In fact, applying the same method of correction as that employed by Hemker et al. [54] to Dimiduk's results yields a resulting stress that is significantly larger than the flow stress of the prestrained sample. samples at various temperatures [37, 40, 55, 56], that the flow stress is partly reversible but this conclusion has been recently questioned by the work of Sp~itig [282]. Dedicated studies of the influence of prestraining at a temperature different from that of the straining test itself, known as Cottrell-Stokes experiments (fig. 7), have been conducted in single-phased Ni3Al-based alloys by Thornton et al. [41 ], Yoo and Liu [77], D o w l i n g and Gibala [78], Dimiduk [46], H e m k e r et al. [54, 79], B o n t e m p s - N e v e u [23], Couret et al. [72], Dimiduk and Parthasarathy [80], Shi et al. [286]. It is found that - t h e deformation at low temperature of prestrained samples occurs at a flow stress definitely less than the flow stress at high temperature; - in the samples deformed at low temperature, the work-hardening rates are not significantly effected by the prestraining; - however, the flow stress at low temperature of non-prestrained samples is increased by a quantity A'r0.2. This increase has been reported by Dimiduk [46] and H e m k e r et al. [54] as being close to the total work hardening stored in the sample during the high-temperature prestraining test. As shown in fig. 7, this cannot be considered as established. In a recent work, Dimiduk and Parthasarathy [80] have shown in comple-
w
Microscopy and plasticity of the L12 ,.it phase
269
ment that A7"0.2 is a function of the total prestrain (i.e., of the total hardening) at high temperature, the smaller e, the smaller AT0.2. Finally, in addition to a prestraining done at a temperature higher than that of the straining test (820 K and 77 K, respectively), Dowling and Gibala [78] have carried out prestraining at 77 K over 5% and more of deformation. They have shown that under these circumstances they introduce a sufficiently large amount of mobile dislocations to reduce the flow stress considerably at 820 K at macrostrain levels, especially in the [123] orientation. The flow stress of the prestrained sample converges to the stress level of the non-prestrained one after a permanent deformation of a few percent at 820 K. The required strain depends upon the amount of prestrain. Thornton et al. [41] have conducted deformation tests on polycrystalline Ni3A1 prestrained to e ~ 20% at 300 K and reported that the flow stress then becomes almost independent of test temperature. Its magnitude is less than the peak stress of a nonpredeformed sample. This last result shows that, upon adequate thermo-mechanical treatment, a stable work-hardened microstructure can still be introduced in this alloy just as in other crystals. The results contained in this paragraph together with others reported in sections 2.2.3 and 2.2.4 indicate that whereas the flow stress is essentially determined by properties of individual dislocations, it may also comprehend an irreversible component which is associated with work hardening. This conclusion is supported by the strain-rate sensitivity tests of Ezz and Hirsch [71 ], which show that, in the flow stress (eq. (1)), the component at near-zero strain, 7"0.01, is fully reversible and that it depends only on temperature, while the work-hardening component 7"h carries the other dependencies (section 2.2.9). For additional information on that matter, the reader should consult recent papers by Ezz and Hirsch [92, 280, 281].
2.2.6. Violations of the Schmid law There is ample experimental evidence that, in domain B, not only the CRSS on the primary slip plane does depend upon load orientation but the CRSS in tension, 7"T, differs from that measured in compression, 7"c. These two violations of the Schmid law are interesting for they raise questions on the behaviour of individual superdislocations which in turn are at the origin of active theoretical discussions (sections 4.2.1 and 5.4). 2.2.6.1. Chronological viewpoint. In L12 alloys, the violation of the Schmid law for (110) { 111 } slip was first discovered by Takeuchi and Kuramoto [81 ] in Ni3Ga deformed in compression. The same study contained in addition some of the basic ingredients of the point-pinning models of the flow stress anomaly (section 5.4). More refined experiments followed and a wider variety of load orientations were explored on a choice of alloy compositions. Furthermore, the role of the sense of the external stress was determined in great detail. This property is illustrated in figs 8, 9 and 11. According to Takeuchi and Kuramoto [81 ], the orientation of the applied stress affects the locking process through the ratio N between the Schmid factors, ~, on the cross-slip cube plane and on the primary octahedral slip plane cross-slip N = ~(11o){ool} _ 7-c q0(110){111} 7"0
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Fig. 8. The orientation dependence of the flow stress and the TC asymmetry in binary Ni-rich Ni3A1 (after Heredia [37]). Note that I-rT -- "rc] is not a maximum near [011] (f), but about the same as in the orientations of (a) and (b). Note also that the relative magnitude of the TC asymmetry depends upon the temperature at which it is determined (vertical dashed lines are drawn at 600 and 900 K, for a better comparison between the TC asymmetry at these particular temperatures). As to experimental uncertainties, ascribing a physical meaning to such small differences between 7"r and TC as those appearing in this series of figures, would imply that for distinct orientations located on the left-hand side of the neutrality line (chosen as that corresponding to 8(d), "rT is at the same time less (c) and greater (a) and (b) than rc. It should be however acknowledged that some systems exhibit much more dramatic TC asymmetries that those shown in figs 8 and 9 [40, 56, 57].
Microscopy and plasticity of the LI2 "T' phase
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Fig. 9. The orientation dependence of the flow stress and the TC asymmetry in binary Ni3AI + 0.3 at.% Zr (after Heredia [37]). Note the absence of a TC asymmetry for both figs (b) and (c), that is, for orientations located on either side of the line where Q -- 0 (eq. (3), fig. 10). In the near-[001] orientation, the amplitude of the TC asymmetry I'rr - rcl, is larger than in the near-[011] orientation, irrespective of temperature ((a) and (d), respectively). However, the difference is much smaller at 900 K than at 600 K (dashed vertical lines).
Hence, given the temperature within domain B, all samples having the same N should exhibit the same CRSS. Takeuchi and Kuramoto's results and prediction were confirmed on Ni3(A1, Ti) (Aoki and Izumi [82]) and on Ni3(A1, W) (Saburi et al. [83], Kuramoto and Pope [84]), but Lall et al. [28] soon showed that the choice of load orientations explored in these studies was too limited. In fact, Lall et al. [28] pointed out that when considering a wider variety of load orientations, the above prediction based on the orientation factor N is no longer in agreement with experiments. They remarked that some consistency with experiments could nevertheless be achieved, by introducing the following orientation parameter Q_
~(112){111},
(3)
~(110){111}
that accounts for the subdissociation of the leading superpartial into Shockley partials. Cross-slip will be hindered or promoted by the applied stress, when cores are widened or
272
Ch. 53
P. VeyssiOre and G. Saada
[111]
[113]
/
9
,:~:,;i~+~;:4~T , < "[7C .,,. / . , ,
/ Z,,+:i,~ Q= 0 [001]/"~T > ~ 4 [012]
[o11]
Fig. 10. The two domains of TC asymmetry within the unit triangle. The line that joins [113] to [012] corresponds to the locus of orientations where Q = 0 (eq. (3)). In view of the variety of results published on this subject, the hachured band schematizes what could be the uncertainty on the experimental determination of the line where the flow stress in tension, ~, is equal to that in compression, rc (see text for details of experiments). narrowed, respectively. More precisely, the larger Q, the lower the cross-slip rate and the CRSS. Lall et al. [28] conceded, though, that whereas the flow stress is determined by Q, it should also be influenced by the parameter N. Consistently, Takasugi et al. [85] have found that the orientation dependence of the flow stress of Ni3(Si, Ti) in compression is correlated to the orientation parameters N and Q, and this result has been recently shown to remain valid down to 4.2 K (Takasugi and Yoshida [39]). Q is zero on the [012]-[113] portion of the (121) great circle and, by definition, Q is positive in compression on the [001] side of the unit triangle and negative towards the [011]-[111] boundary (fig. 10). Given the load orientation, since Q changes sign upon reversal of the load, the CRSS in tension should differ from that measured in compression. The so-called tension-compression (TC) asymmetry was first checked experimentally by Ezz et al. [40] on Ni3(A1, Nb). The asymmetry was found in accord with predictions based on the parameter Q, except for the position of the line where the asymmetry disappears, which was in fact determined to lie nearer to [001] than predicted (see also section 2.2.6.2). Similar results were obtained on Ni3(A1, Ta) by Umakoshi et
al.
[57].
The exact position of this experimental neutrality or inversion line became the central element of the study of the flow stress anomaly. It is the detailed position of this line together with some atomistic computer simulations of dislocation cores in the L 1e structure which served as a foundation for the so-called PPV model ([86], section 5.4.2). In this latter model, a better fit - which was considered at this time as the actual experimental neutrality line - is obtained by adding to the above effects that of the applied stress on the core width in the octahedral cross-slip plane. Accordingly, the magnitude of the TC asymmetry is governed by a factor K defined as K - Q 1 + t~(N + V ~ ) ( 2 N (3N - v ~ ) v ~
v/3),
(4)
w
Microscopy and plasticity of the L12 ~/i phase
273
where the adjustable parameter t~ is introduced in order to account for the relative importance of the core widths on the primary and cross-slip octahedral planes. When K is zero, then 7-i- -- 7c. It should be noted that in practice, because of the parametrization of eq. (4), an agreement between the PPV model and experiments can be found for any neutrality line located between [001] and the line Q = 0 (fig. 10). Ezz et al. [56] obtained excellent accord between the provision of predictions of the PPV model and their experiments carried out on Ni3Ga. These authors further noted the very good agreement between the PPV predictions and tests conducted previously by Ezz et al. [40] and by Umakoshi et al. [57] on Ni3(A1, Nb) and on Ni3(A1, Ta), respectively. The position of the TC neutrality line together with another key prediction of the PPV model in which samples oriented at or in the near vicinity of [011] should exhibit the largest TC asymmetry, are discussed in the following section, which is aimed to show that a refined comparison between a given model and mechanical tests is almost hopeless. 2.2.6.2. Comments on the violations of the Schmid law. Based on a number of experimental determinations of TT and "rc in binary and ternary Ni3Al-based alloys and in Ni3Ga, the TC asymmetry is known to differ according to the position of the load axis within the unit stereographic triangle in the following way (fig. 10): D
IT ~ TC in the vicinity of the great circle that joins [012] to [113], that is, approximately to the locus of Q = 0; IT > ~-c for loads oriented within the area of the unit triangle located between the above neutrality line and the [001] axis; - ~-c > TT opposite to the preceding domain.
-
-
We believe that in fact a critical importance should not be given to the exact position though not the existence - of the neutrality line in the research of the origins of the flow stress anomaly, for the following reasons. (i) Given an alloy composition, one may question the accuracy to which this line can be determined experimentally. In fact, the change in behaviour from "rc > TT to ~-C < TT is quite smooth, which, in view of the considerable WHR at e = 0.2% (fig. 4), makes the determination of the neutrality line rather uncertain. For instance, the positions of the neutrality line claimed by Umakoshi et al. [57] and by Ezz et al. [56] are only about 5 ~ or less off the line Q = 0. In fact, in order to investigate accurately the position of the neutrality line, several orientations distant within a few degrees from one another within the region of interest should be tested. For obvious practical reasons, such as limitations in the availability of deformation samples and experimental uncertainties, this is unfeasible. In the study by Ezz et al. [56], it would have been helpful if the magnitude of the asymmetry of sample #4 could have been compared to that corresponding to a load exactly oriented on the line Q = 0. This would have allowed to check whether there is actually a significant difference between these two orientations. (ii) The position of the neutrality line is composition-dependent (figs 8, 9 and 11). The deviations from the line Q = 0 observed by Heredia [37] and summarized in the papers by Heredia and Pope [38, 87] are listed in table 2 as a function of alloy composition. It is in some ternary alloys that the observed neutrality line is the most deviated from the
-
P. Veyssikre and G. Saada
274
Ch. 53
Table 2 Orientations for zero tension/compression asymmetry in a variety of L12 alloys, sorted by ascending deviation from the line Q = 0. 0 is the angle between [001] and the load orientation [hkl]. N and Q are given by eqs (2) and (3), respectively. The orientation for zero tension-compression asymmetry is taken from Heredia [37]. In many instances, such as the ternary alloy containing 0.3 at.% Zr (question mark), this orientation is in fact loosely defined (fig. 9). In particular, it depends upon the temperature at which it is measured, as shown in fig. 8 on binary Ni3A1; this can yield considerable uncertainty on the actual neutral orientation. Alloy composition
Orientation for zero asymmetry
1.0 at.% Ni-rich 0.7 at.% 1.0 at.% 3.3 at.% 0.2 at.% 0.3 at.% 1.0 at.% 2.5 at.% 1.0 at.%
- 1 - 1 -1 - 1 - 1 - 1 -1 - 1 - 1 - 1
h Zr B Ta Hf B Zr B Ta Hf
k 2.8 9 4.1 2 3 2.4 9.3 2.6 1.6 3
0
N
Q
22.16 22.37 21.52 20.76 19.36 18.89 18.53 16.67 13.94 13.67
0.533 0.520 0.515 0.502 0.472 0.462 0.445 0.413 0.338 0.346
0.049 0.075 0.074 0.075 0.115 0.121 0.166 0.174 0.228 0.247
l 7.3 22 10.7 5.9 9 7.6 27.9 ? 9.3 7.6 13
line Q - 0, thus the best accounted for by the PPV model. On the other hand, in Ni-rich (76.6 at.%) binary Ni3A1, an alloy which should be least affected by extrinsic effects such as solution hardening, the orientation for which the TC asymmetry vanishes is very close to the great circle Q - 0; the coincidence between the curve in tension with that in compression is almost perfect for this orientation (fig. 8(d)). However, in all other orientations of this alloy tested, the asymmetry measured near the inflection point of "r(T) remains very modest, maybe indistinguishable within experimental uncertainties (section 2.2.2). Note also that in table 2, the question mark for the determination of the neutral orientation for the 0.3 at.% Zr alloy results from the fact that in the work of Heredia [37] a very good superimposition of the curves 7-(T) in tension and compr,.,:~ion is also observed for a sample orientation 1~ off the line Q - 0 (compare figs 8(c) and 8(d)). Surprisingly enough this latter orientation is located on the other side of the line Q - 0, that is, on the [111]-[011] side of the unit triangle. For the same alloy composition, it should be noted that, with regards to the TC asymmetry, figs 8(c) and 8(e) are almost identical although their load orientations are far apart on each side of the line Q - 0. Similarly, still from Heredia's excellent and comprehensive work, it is quite unclear, in Ni3A1 single crystals containing 1 at.% of Ta and containing 1 at.% of Zr, where the position of neutrality is actually located. This is essentially due to the fact that more orientations have to be tested within the area of interest. (iii) It is not straightforward to assimilate the core extension of a 89 (110) superpartial to a regular dissociation into two Shockley partials (section 4.2.1.1) and then to make use of linear elasticity, as is done in the PPV model and others. This issue will be addressed in more detail in sections 5.2 and 5.4. The above-mentioned inconsistency between the orientation dependence predicted from the PPV model and its experimental determination in a variety of alloys, has
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Microscopy and plasticity of the L12 .[t phase
275
led Khantha et al. [88] to revisit the orientation part of the PPV model and to propose a simplified though parametrized version of the cross-slip event (section 5.4.4). This new version ignores the CRSS in the cross-slip octahedral plane. In its present form, the treatment of the orientation-dependence of the cross-slip controlling mechanism resembles strongly the previous work by Lall et al. [28]. We shall return to this question in section 5.4. We now address the prediction of the PPV model that the position of the maximum TC asymmetry will be seen near [011], which has been verified experimentally in Ni3(A1, Ta) [57] and in Ni3Ga [56]. It is true that when measured at 400 K in Ni3(A1, Ta) and at 600 K in Ni3Ga, the difference between "rc and "IT is the largest for a load oriented along [011]. It should be noticed, however, that the magnitude of this difference depends on the temperature chosen for its determination. If one chooses this temperature at 600 K in Ni3(A1, Ta) and at 400 K in Ni3Ga, the determination of ]TC -- TTI near the [001] orientation is of the same magnitude as that found in the [011] orientation, if not larger. In other words, ITC -- TTI depends critically on temperature and it is not at all clear that the magnitude of the asymmetry follows the prediction of the PPV model. The difficulty in rationalizing the orientation dependence of I r c - TTI has been made even clearer more recently by Heredia [37], who showed that both in binary (fig. 8) and in some ternary Ni3Al-based alloys (such as in Ni3(A1, Zr), fig. 9, and in Ni3(A1, Hf)), the asymmetry is the largest near [001 ], while it conforms to the PPV model in B-doped Ni3A1. Inspection of figs 8 and 9 shows quite clearly the temperature dependence of the TC asymmetry (compare for instance the asymmetries between figs 9(a) and 9(d) at 600 K and at 900 K. 2.2.6.3. Conclusion. From one alloy composition to another, the measured orientation dependence may vary in a significantly inconsistent manner as shown by uncertainties on the position of maximum asymmetry. It is fair to say that the actual orientation dependence appears in fact to be more complicated than a simple theoretical analysis could predict, unless of course this theory is appropriately parametrized (section 5.4.4). Though the observed violations of the Schmid law are somewhat consistent with the idea that slip is affected by stress-induced core deformations taking place off the slip plane, it does not imply that this must be the only possible cause. Regarding available models, it appears that a too perfect agreement between predictions and experiments has been looked for in the past and, for this reason, the modelization of cross-slip has been carried far beyond the limits of validity of linear elasticity. As a consequence, it can be considered that major issues in the research of the origins of the positive TDFS have remained fogged for several years. We shall come back to this question in sections 5.2 and 5.4. Since the orientation dependence appears to differ to varied extents from one system to the next, we believe that the critical experimental orientation-dependent properties to which theoretical predictions can be confronted should be limited to the existence of two regions with opposite TC asymmetry in the unit triangle. Neither the existence and position of maximum asymmetry nor the position of the neutrality line can be regarded as firm experimental criteria. Regarding the latter, the safest assertion is to consider that it is located within a domain of finite width in the neighborhood of the line defined by Q = 0, maybe a little off towards the [001] direction (fig. 10).
P. Veyssidre and G. Saada
276
Ch. 53
Finally, it would have been quite profitable that the composition dependence of the work-hardening rate had been measured and more data made available in the course of the experiments on the orientation dependence of the flow stress. Comparing their combined behaviour is one of possible ways to discriminate between these two intricate quantities. 2.2.7. The peak stress The flow-stress peak can be interpreted in two different ways. On the one hand, it might be regarded as the level of stress capable of destroying the strongest obstacles that the microstructure can oppose to deformation (section 4.2.3.4). On the other hand, it can be considered that the flow stress keeps increasing with temperature as long as the sample is not softened by the gradually increasing activity of primary cube slip (the occurrence of the latter is indeed suggested by slip lines (section 3.4.1) at temperatures near and above the peak). In both interpretations, the peak position is expected to be orientation-dependent. In the former explanation, which is obstacle-controlled, the peak, also named the saturation stress, would be determined by the ratio between the stress on the primary octahedral plane and that on the cube cross-slip plane (section 4.2.2.3). Unless obstacle destruction is thermally activated, the peak should appear as a more or less extended plateau. Experiments show that this is not true and that the temperature extent of the flow stress peak is in general rather limited (see fig. 3), though exceptions can be found for instance in Ni3(A1, Ti) and in Ni3(A1, 1 at.% Ta) both oriented along [132] (StatonBevan and Rawlings [89], Bonneville and Martin [90]) and in Ni3(A1, 5 at.% Ta) oriented along [001] (Heredia [37]). A theoretical analysis of the obstacle-controlled saturation stress, based on the response of a Kear-Wilsdorf lock to an applied stress, shows no convincing correlations with experimental observations (section 4.2.3.4). Figure 11 shows the absence of a correlation between the variations of the flow stress with N, in tension and compression, for various alloys. In the interpretation by which the peak corresponds simply to the onset of primary cube slip - except for the [001]-oriented samples - the orientation dependence of the peak temperature should roughly coincide with that of the Schmid factor for primary cube slip, through the orientation factor m
9 primary
Nr = ~v<110){0ol}
(5)
~(110){111}
The larger N t , the lower the peak temperature Tp. Although Ezz et al. [56] have reported a correlation between the peak temperature and N t , nothing systematic has been found [37]. In particular, the peak temperature is almost orientation-independent in binary Ni3A1 strained in tension. Furthermore, it appears that the role of N I differs with the nature of the ternary element. Still regarding the orientation dependence of the peak, there is a general tendency for the flow stress to be the highest in the near-[001] orientation, that is, for samples which have zero or modest shear stress on cube planes. This remark should however, be
Microscopy and plasticity of the L12 .yt phase
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regarded with some care since experimental violations again have been encountered as shown by the work of Heredia [37]. For six of a total of eight compositions tested- the exceptions are the binary and perhaps the (0.7 at.% B)-doped alloys- the highest peak in compression does correspond to the near-[001] orientation, but this property is never true in tension, except maybe for an addition of 1 at.% of tantalum. Finally, ternary additions seem to decrease the peak temperature in single crystals [37]. 2.2.8. Transients
As it has been already made clear in the preceding sections, the study of the strainrate sensitivity of L12 alloys brings out interesting information on the mechanisms that determine the flow stress anomaly. This is why, despite the fact that few experiments are available, the space devoted here to their analysis is large relative to that of other properties. The experimental analysis is based on the assumption that the strain rate is governed by a thermally-activated mechanism, whose free energy AG* and activation volumes V* depend on stress and temperature. Then the strain-rate sensitivity/3 expresses as 1 - -
/~
(3[ln(~)])
_
_ _
1 (,lAG*])
V*
_ _
67"*
T
~ _
kT
67"--
T
-
-
(6a)
.
kT
In this expression, 7-* is the stress effectively applied to the dislocation, that is, the external stress corrected for the athermal stress 7-i (often taken as the long-range internal
278
P. Veyssi~re and G. Saada
Ch. 53
stress). The strain-rate sensitivity has been measured in L12 alloys by two methods (i) strain-rate jumps (Thornton et al. [41], Ezz and Hirsch [71]) and (ii) stress relaxation (Bonneville and Martin [90], Sp~itig et al. [76]). Correcting 3~- (or &r) for work hardening as well as for the microstructural changes that occur in L12 alloys during the test, constitutes a serious difficulty in both methods. As illustrated below, it is not at all ascertained that one technique is significantly more straightforward or more reliable than the other. It should be kept in mind that some confusion in comparing the responses to strainrate changes between different alloys may arise from the fact that some groups employ a different definition of this parameter, as
S-
~
~0-~- ]
T
~
6[l~(g)]
T
TT'
(6b)
in which/3 is scaled both by the test temperature and by the nominal flow stress. 2.2.8.1. Strain-rate jumps. Since Davies and Stoloff[74] pointed out that the flow stress is little sensitive to the strain rate, the study of this property has been essential in the analysis of the flow stress anomaly. Typically, &r/~r amounts to l0 -2 upon a hundredfold strain-rate increment and it remains very small up to the peak. For a thorough overview of the effect of strain rate on the flow stress and on the work-hardening rate, in the instance of Ni3(A1, Ti) deformed in a wide range of temperatures, the reader is referred to fig. 1 in the work of Miura et al. [52]. As shown by expression (6a), small strain-rate sensitivity is equivalent to large activation volumes. Thornton et al. [41] were the first to suggest that as temperature is raised in domain B, the flow stress anomaly is successively controlled by two dislocation mechanisms. This hypothesis relies partly on the fact that strain-rate jump experiments on Ni3A1 and Ni3(A1, Cr) polycrystals show two sorts of response (fig. 12(a))" (i) Atypical behaviour below 670 K (referred to as domain B1 (see fig. 1) in the following3), where the stress increment &r decreases gradually with strain until it reaches the flow stress at the lower strain-rate. In domain B 1, &r decreases more rapidly as temperature is increased. (ii) In the second type of response, e.g., above approximately 670 K and up to the peak (domain B2), &r no longer decreases but remains constant with strain. This behaviour compares with that of f.c.c, metals. Somewhat similar, though not identical, transient behavior has been reported in Ni3(A1, Hf, B) single crystals (fig. 12(b)) by Ezz and Hirsch [71] who, in addition, have identified a range of intermediate temperatures characterized by the onset of serrated flow. This transition domain is centered approximately at the inflection point of or(T). Due to the difficulties in measuring &r [277], exploration of the strain-rate sensitivity in Ni3(A1, Hf, B) has been limited so far to an upper temperature of 550 K. This domain of temperature has been expanded recently to higher temperatures up to 800 K [69, 280]. The measurement of/3 raises some difficulties and questions in Ni3AI:
3Temperatures are indicated for information for a given alloy. They should not serve for a comparison between different alloys, because the positions of domain B1 and B2 are stronglycomposition- and orientationdependent.
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Microscopy and plasticity o[ the LI 2 7 ~ phase
(a)
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~
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Fig. 12. The various shapes of yield drops encountered during strain-rate jump experiments in Ni3Al-based alloys. (a) After Thornton et al. [41], note the difference in response between domain B1 (200~ and B2 (550~ (b) After Ezz and Hirsch [71], showing the multiple possibilities for the determination of 67-.
(i) For instance, Thornton et al. [41] have pointed out that the magnitude of ]&r] depends upon whether the strain-rate is increased or decreased (fig. 12(a)). It is less for an increase of the strain rate, which is the experimental procedure chosen by Ezz and Hirsch [71] (for the sake of comparison, we recall that during a stress relaxation experiment, the strain rate is continuously decreased). (ii) The work-hardening rate at small strains is steep (figs 4 and 5(b)), which causes severe inaccuracy in the near vicinity of the yielding region (in fact, d;o- could not been determined in practice below a permanent strain of 0.5%). (iii) It should also be noted that at larger strains and at temperatures up to approximately 800 K, &r includes a yield drop (fig. 12) whose magnitude increases with increasing deformation [71,280] and decreases with increasing temperature [41 ]. There is in fact a lasting debate on the analysis of such transient behaviour (Basinski and Basinski [91]) and on its physical interpretations. It is agreed that upon a strain-rate change, the yield drop stems from an inadequate multiplication rate of dislocations during the transient. In
P. VeyssiOre and G. Saada
280 20
-
9 294K
~,.Jll
9 [] 398K
tl:i
13..
Ch. 53
9
-+
-
v
"
.so2K
~ " ~ ~ -
~ o
10
0
50
150
200
250
250
1~ - ~y ( M P a )
Fig. 13. The strain rate sensitivity of the flow stress,/3 (eq. (6)), as a function of work-hardening, for various test temperatures (after Ezz and Hirsch [71]).
the analysis of Ezz and Hirsch [71], 5or is taken as the total variation of stress up to the yield point. In the same circumstances, Thornton et al. [41] have chosen to correct 5or for the yield drop and yet obtain activation volumes consistent with those determined by Ezz and Hirsch (section 2.2.8.3). Because of such an indication of a rapidly changing density of mobile dislocations during strain-rate jumps, it is in fact unclear how these tests relate to the more steady processes which take place during deformation at constant strain-rate. (iv) Finally, in the analysis by Ezz and Hirsch [71], the effective stress applied to a segment undergoing a thermally activated jump over the rate-controlling obstacle, 7 " * - - T - - T i is identified directly to 75 = 7- - T0.01, which in turn implies that 7-i = 7"0.01. What the physical implications of this assumption could be is quite unclear. In fact, 7-i and 7"0.01 may not play the same physical role during deformation since 7"i, usually taken as the internal long-range stress, does not correspond to obstacles that can be thermally overcome; on the other hand, the "true" yield stress, 7"0.01, exhibits a positive TDFS. In a recent contribution, Ezz and Hirsch [92] have proposed to identify 7"0.01 to a bowing stress. The most interesting finding in the work of Ezz and Hirsch [71] is that, for any strainrate jump and at any given temperature within domain B 1, the Cottrell-Stokes law is obeyed for 7"h (eq. (1)), that is, 57" (or equivalently fl) is a linear function of 7"(e) which vanishes at e - 0.01 (fig. 13). In other words, it is experimentally demonstrated that 67" and 7"(e) are proportional quantities or else that they are proportional to the same physical quantity. The strain-rate sensitivity of Ni3(A1, Hf)B is reminiscent of that of pure f.c.c, metals, which contrasts with the fact mentioned in section 2.2.4 that the strain dependence of the flow stress 7"(e) reminds one very much of that of b.c.c, metals. It has been determined, in addition, that the quantity d;7-/7-h is independent of orientation even though the work hardening rate itself is actually orientation dependent. As to the activation volume, it increases with e or 7-h. The quantity (7-by*), which increases with temperature, is remarkably independent of stress at any given temperature within domain B 1, except in the near vicinity of domain B2.
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Microscopy and plasticity of the L12 "y~phase
281
Ezz and Hirsch [71] have concluded that, in domain B1, the WHR is determined by forest dislocations, that is, basically the same intersection mechanism as for f.c.c. crystals. The increase of WHR with temperature is attributed to an increasing density of forest dislocations originating from both the primary and cross-slip cube systems. The temperature and orientation dependencies are claimed to originate from the varied activities of cube slip, following a mechanism specific to L12 alloys. This forest mechanism had been first suggested by Thornton et al. [41 ], though for domain B2, it had later been supported by the TEM observations of Korner (1991b). It will be further discussed in section 5.3. Regarding domain B2, where the WHR decreases with increasing temperature, Ezz and Hirsch [71] invoke the enhanced annihilation of screw dislocations on the cube plane, which reduces the forest density, and the increased amount of strain on the cube plane (increasing dctotal, in dT/d~total).
2.2.8.2. Stress relaxation tests. In doing a stress relaxation test, one has a direct access to an apparent activation volume, Va. Deriving an effective, sometimes named "true", activation volume that would be more typical of the rate-controlling mechanism requires some corrections to be made. One usual reason why the two activation volumes differ is the stress dependence of the pre-exponential factor in the rate equation. Other causes of discrepancy may stem from the stiffness of the straining device and from the workhardening of the sample, which is large in L12 alloys. In order to account for the two latter effects, a method of successive relaxations has been designed by Sp~itig et al. [76]. By means of this method, the volume Va corrected for work hardening by a term vh transforms into an effective activation volume Veff. The volume ratio Va/Veff expresses as (1 + H / M ) , where M is the equivalent elastic modulus of the testing device including that of the sample, and H an athermal work-hardening rate, (d'ri/dc), defined as the local sensitivity of the internal stress to the shear strain. It is equally important to remark that in "normal" materials, the apparent activation volume decreases as a function of plastic strain during the microplastic stage and that it remains constant after the sample has yielded (Meakin [93], Escaig [94], Groh and Conte [95]). This is currently interpreted (i) beyond the yield strain, as the signature of the particular rate-controlling mechanism within a relatively constant density of mobile dislocations, whereas (ii) the marked decrease of the activation volume, which is typical of the microyield regime, reflects the rapidly changing density of mobile dislocations. This two-stage behaviour then provides a neat means to determine, on a physical basis, the strain at which the sample actually yields. Finally, a necessary condition in order to obtain a sound result which would be interpretable within the frame of thermal activation is that the sample under investigation responds logarithmically during its relaxation. Unless explicitly mentioned, the following results were obtained from Ta-doped Ni3A1. In the domain of positive TDFS, the stress relaxation of L12 alloys raises a few difficulties: (i) The stress relaxation behaves normally at any temperature within domain B, except in a narrow temperature range where relaxation is not logarithmic (Baluc et al. [97], Bonneville and Martin [90], see however the following point (iv)). This temperature range includes the inflection point of T(T).
282
Ch. 53
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Microscopy and plasticity of the L12 9/~phase
283
(ii) Va exhibits a discontinuity across the above temperature range (fig. 14(a)). The discontinuity is consistent with the observation of a low and a high temperature regime of strain-rate sensitivity (section 2.2.8.2). In the same vein, Bonneville and Martin [90] (see also Sp~itig et al. [96]) have re-visited the early experimental analysis of strainrate sensitivity, S, conducted by Thornton et al. [41 ]. They have shown that it contains a similar discontinuity (fig. 14(b)) and pointed out, in addition, that the discontinuity of 5' is particularly pronounced when the strain rate is decreased during the jump, i.e., when the experimental procedure is the most comparable to a stress relaxation test. The reported discontinuity in activation volume has prompted Hirsch [98-100] and Khantha et al. [88, 101]) to design models of the flow stress anomaly in which two distinct rate-controlling mechanisms would operate, one at low (B1) and the other at high temperature (B2). (iii) When considering the effective activation volume Veff, the above discontinuity seems to be significantly diminished, if not cancelled (fig. 14(a)). (iv) On the other hand, stress-relaxation experiments carried out recently by Sp~itig [96, 282] on other alloy compositions, such as Ni3(A1, Hf) and Ni-rich binary Ni3A1, have shown that the existence of a discontinuity is in fact composition-dependent (so far, found only in Ni3(A1, Ta)). The absence of a discontinuity is corroborated by the strain-rate jump tests by Ezz and Hirsch [281]. (v) Instead of showing a plateau, Va decreases continuously and significantly with deformation (fig. 14(c)). This holds true for '/3eft [102]. Sp~itig et al. [61, 76] have discussed the validity of the choice of the conventional yield stress at a permanent strain of 0.2%. Their conclusion, based on the strain dependence of the effective activation volume, is that the choice of 70.2 is adequate. (vi) The values of the athermal work-hardening rate H are 4 to 7 times larger than those of h determined directly on the stress-strain curve. This difficulty can be partly justified considering that the microstructure evolves differently depending upon whether the sample responds under constant strain rate or under relaxation, with an exhaustion rate significantly larger in the latter case (Saada [103]). Nevertheless, since "Oa/Veff -(1 + H / M ) , the volume ratio should not exceed 2 (otherwise (dT"i/d~) would be larger than the equivalent elastic modulus), whereas values as large as 4 have been reported [76, 96]. This suggests some lack of consistency in the interpretation of the method of successive relaxations. The above difficulties in measuring activation volumes suggest the operation of some potent and rapid exhaustion process (section 4.3), as it had been already mentioned by Hemker et al. [54] in view of a relaxation test that showed the stress to cease relaxing rather quickly (for less than 6% of the initial stress, at room temperature). Ways of including dislocation exhaustion in the analysis of repeated relaxation tests have been proposed very recently, assuming an exponential decay of the density of mobile dislocations (Bonneville et al. [104]). 2.2.8.3. Discussion Strain-rate jumps and relaxation tests provide distinct estimates of the permanent strain that defines when L12 samples actually yield, e.g., 0.01% for Ezz and Hirsch [71] and 0.1 to 0.5% by Sp~itig et al. [61], depending upon the orientation of the load axis.
P. Veyssidre and G. Saada
284 o0
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Ch. 53
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Determinations of activation volumes are not all consistent. The activation volumes measured by strain-rate jumps at small strains by Ezz and Hirsch [71] on single crystals do agree with those determined by Thornton et al. [41] on polycrystals with different composition. They are also in accord with the apparent activation volumes derived from stress relaxation experiments by Bonneville and Martin [90], but they are much less consistent with the effective activation volumes, that is, those corrected for work-hardening [61, 76]. This is shown in fig. 15 in two Hf-containing Ni3A1 based alloys. At this stage, the assumptions under which the strain-rate jump and relaxation tests are actually analysed are worth reconsidering. It is usually postulated that either the initial density of mobile dislocations remains fairly constant over the transient (and from one transient to the other in the course of a repeated relaxation test) or that the system is not sensitive to initial conditions and that the internal stress is unchanged. Neither of these hypotheses can be checked directly; instead, they are regarded as satisfied when results are self-consistent, which is certainly not the case in L12 alloys. What the deviations from "usual" behaviour outlined in sections 2.2.8.1 and 2.2.8.2, indicate is that - the density of mobile dislocations is varying over significant amounts during the transients, - this originates from a very efficient exhaustion process, - which itself results in an overwhelming WHR, from which the parameters that would inform on the actual rate-controlling mechanism at yield are difficult to draw unambiguously.
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Microscopy and plasticity of the LI 2 "7~ phase
285
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Fig. 16. Temperature dependence of creep curves of Ni3(A1, Hf, B) under a load of 745 MPa oriented in the near [001] direction (more precisely along [ - 0 . 0 5 0.2 1]). In the early stages of deformation, the creep rate decreases (primary creep) and then increases (inverse creep). The inverse creep rate increases with increasing temperature (after Hernker and Nix [111]).
2.2.9. Creep By far, there are not as many studies on creep as on deformation at constant strain rate, especially at temperatures below the peak. Although creep properties appear composition dependent (Anton et al. [105]), some general trends have been identified (Hazzledine and Schneibel [ 106, 107]). The stress dependence of the steady state creep rate differs depending upon whether tests are conducted above or below the peak temperature (Nicholls and Rawlings [108]). In what follows, we shall focus on the creep behaviour in the range of temperature that corresponds to domain B. In this particular case, it has been found that a steadystate creep is rarely attained in Ni3Al-based alloys (Anton et al. [105], Schneibel and Horton [109], Hemker et al. [79]). At 630 K creep occurs at a stress equal to 75% of the 0.2% elastic limit (Hemker [110]) which, by contrast with conclusions drawn from strain-rate jumps and relaxation tests, suggests that there is no distinct critical stress for dislocation propagation. This property is ignored in most models of the positive TDFS (section 5). As shown in fig. 16, creep involves two regimes [79]: (i) An initial portion that looks at first like an incubation period, where in fact the creep rate decreases continuously. It is remarkable in addition that the total amount of creep strain (after 25 h) shows an anomalous decrease with increasing T (%{630 K} = 9 x 10 -3, Cp{823 K} = 4 x 10-3). Deformation is provided by primary octahedral slip suggesting that yielding and primary creep are related phenomena. Since the deformation microstructure in primary creep is in addition very similar to that at yield, Hemker et al. [79] inferred that primary creep is controlled by the same dislocation process as that which is responsible for the positive TDFS. The fact that in primary creep the creep rate is a decreasing function of strain indicates that deformation is controlled by the development of a dislocation substructure (Hemker et al. [54]).
286
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Ch. 53
(ii) A regime of continuously increasing creep rate, named inverse creep - with some exceptions however, such as in Ni-rich Ta-containing Ni3A1 [105]. Deformation is provided by cube slip and it is remarkable that the operating slip plane is the cross-slip cube plane instead of the cube plane with maximum Schmid factor. As shown in fig. 16, inverse creep behaves normally in the sense that the creep rate shows a positive temperature dependence in domain B (Nicholls and Rawlings [108], Hemker et al. [79]). This behaviour is fully consistent with the fact that cube slip is markedly thermally-activated (sections 2.2.10 and 4.2.1). The fact that inverse creep behaves normally indicates that the mechanisms for yielding and inverse creep are different. It should be noted that, after a few hours and a relatively modest permanent strain (about 5 x 10 -3, fig. 16), primary creep is followed by an exhaustion stage, consistent with the hypothesis that dislocations become gradually and irreversibly locked. The exhaustion stage, that is, the absence of thermally-activated recovery, attests to some irreversibility of locking in creep. The experimental values of the stress exponents and of the activation volumes confirm that glide predominates during both the exhaustion and the inverse creep regimes. In domain B, there is no evidence for diffusion-assisted creep. Hemker and Nix [111] have reported that upon cooling under load, crept samples undergo significant plastic strains (fig. 17(a)). Typically, a sample crept 5% at 913 K nearly along [001] further deforms over 20% during the cooling to room temperature when the applied load of 745 MPa is maintained. In a subsequent paper, Hemker et al. [54] have reported that the resulting change in creep rate occurs after some delay. Because of the positive TDFS, the applied load becomes larger than the 0.2% flow stress at some stage during the cooling. Accordingly, the strain then occurs on the primary octahedral plane even though the sample was previously deformed in inverted creep, that is, under the cube cross-slip system. The dislocation microstructure after the test is typical of that of yielding. Hence, when a Ni3A1 sample is deformed in inverse creep, the microstructure which has developed extensively on cube planes to a significant plastic strain (about 5%), does not block irreversibly the mechanism that controls the yielding. This result should be regarded in the light of the experiments on flow stress reversibility (section 2.2.5). Finally, in an additional temperature-drop experiment under a constant load of 770 MPa, Hemker and Nix [111] demonstrated that a prestraining carried out at 913 K results at 813 K and 713 K in extensive deformation in octahedral slip (fig. 17(b)). In this alloy, the yield stresses at 913 K, 813 K and 713 K are 810, 800 and 690 MPa, respectively. The prestraining gives more immediate (or primary) strain than one observes when the sample is crept at 813 K and 713 K without prestrain.
2.2.10. Properties of cube slip Cube slip is prominent in domain C (fig. 1). Slip lines indeed indicate that above the peak temperature slip on the primary cube plane takes over from octahedral slip, except in the near-[001] orientation where the flow stress decreases under primary octahedral slip. The high-temperature CRSS for cube slip, 7-001, is in general orientation-dependent (Curwick [36], Lall et al. [28], Umakoshi et al. [57], Heredia [37]) and it should be
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Microscopy and plasticity of the L12 ~ phase
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Fig. 17. Temperature drop tests on Ni3(AI, Hf, B) (after Hemker and Nix [111]). (a) Creep at 913 K under 743 MPa and subsequent plastic strain attained on cooling at the same stress (the additional strain is 20%). (b) Successive creep tests under 770 MPa, at 913 K, 813 K and 713 K; note that the plastic strain at reloading is larger than the measured primary creep strain for tests conducted at a constant temperature. noted that a TC asymmetry can also be found in cube slip. The asymmetry is pronounced for some compositions, in particular in Ni-rich alloys crystals [37]. The TC asymmetry is decreased by ternary additions but in an unpredictable manner. Miura et al. [52] have reviewed properties of T001 at elevated temperature and concluded that its thermal contribution was controlled by a Peierls-Nabarro mechanism. The study of the orientation dependence of ~-0ol is limited to high temperatures (domain C). This is because below the peak (domains A and B), octahedral slip is in general much more strongly favoured than cube slip, unless deformation is carried out in the near-[111] orientation. In this case, a mixture of both primary slip systems can be observed, sometimes at rather low deformation temperatures. For instance, according to D
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Umakoshi et al. [57], 7-0ol exhibits a strong positive strain-rate dependence at intermediate temperatures. Heredia [37] has further supported this result by plotting 7-o01down to 77 K, and suggested a marked CRSS peak which, depending of course upon alloy composition, culminates at levels of stress comparable to those of the peak of 7-111, that is, about 400 to 500 MPa. There are nevertheless some differences in the analyses of the temperature dependence of 7-0o1: (i) Saburi et al. [83] concluded to a positive temperature dependence of 7-o01in Ni3(AI, W), below the peak temperature of the (110){111} slip system (slip lines indicate that at 200~ and 400~ cube slip always coexist with octahedral slip). (ii) In Ni3(A1, Nb), 7-001 would exhibit a slight increase (by less than 5%) between 350 K and 650 K. Again primary octahedral slip and cube slip are seen to operate simultaneously (Lall et al. [28]). (iii) In Ni3(A1, Ta), the presence of slip traces attesting to some cube slip activity is reported below the peak. Contrary to the preceding works, the CRSS for this system has been estimated at temperatures only above the peak, where it shows a regular negative TDFS (Umakoshi et al. [57]). (iv) Finally, it should be noted that by local measurements of dislocation curvature under stress, C16ment et al. [112, 113] have determined a fiat peak of 7-0ol located between 400 K and 600 K in a multicomponent 7 t phase strained in tension along [011] in an electron microscope. In Hf-doped Ni3A1 again strained in situ along [011 ], Mol6nat and Caillard [114] have reported a marked decrease of 7-0ol with increasing temperature, but from their experiments the occurrence of a peak at lower temperatures cannot be excluded. The existence of a true peak of 7-o01(T) depends on the confidence one can place on data obtained from samples where octahedral slip and cube slip have operated simultaneously. There is in fact always the possibility that sources are heterogeneously activated in cube slip as a consequence of local stress concentrations, with the consequence that flow stress measurements cannot scale with 7-0ol. Such situations would nevertheless give rise to visible localized slip traces. On the other hand, the measurement of 7-ool relies on the assumption that observing slip trace evidence of cube slip implies that the stress was sufficient to activate extensively the appropriate dislocation sources during straining. Henceforth, 7-0ol is simply the projection of the applied stress in the slip direction of the cube plane. In view of the viscous nature of cube slip (section 4.2.1.2), this procedure is perhaps somewhat naive when slip traces indicate that plastic strain is provided mostly by octahedral slip, as is often the case. The reason is that shear strains do not necessarily proceed at the same rate on octahedral and cube slip systems 4. In the case of dual slip, the main difficulty is then to ascertain that slip lines are produced under the same conditions. Since, as we shall see later, the microstructure at intermediate temperatures contains ample provision of sources for cube slip under the form of long screw superdislocations dissociated in the cube plane (sections 3.4.3.5, 4For the sake of completeness, the case of Pt3A1 should however be mentioned, since if offers what can be regarded as true example of a - slightly - positive TDFS dominated by cube slip (fig. 2 in Wee et al. [27]).
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Microscopy and plasticity of the L12 "7t p h a s e
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4.2.3.1 and 4.2.3.3) and since, because of the sluggish nature of cube slip, there is no threshold stress for its operation (section 4.2.1.2), there is nothing to prevent slip system on a cube plane from contributing to the total strain. Hence, what the observation of - perhaps a few n a r r o w - slip traces of cube slip means is that at any level of stress and at intermediate temperatures, the activation of sources in cube slip is extensive enough in order to give rise to slip lines. Because of the viscous nature of cube slip, it is nevertheless probable that the shear strain provided by a cube slip remains moderate relative to that provided by octahedral slip. The few slip lines on the cube system could well have taken the whole duration of the deformation test in order to be formed to such an extent that they are visible on sample surfaces. It should be thus kept in mind that in extrapolating "r0.01 to temperatures below the peak where cube slip does not contribute significantly to the total strain, one cannot guarantee that the effective strain rate is kept constant in cube slip. In fact, the effective shear strain-rate on this system should be decreased as temperature is decreased and it is not surprising, under such circumstances, that slip traces are present below the peak temperature and that these traces vanish gradually and not abruptly as test temperature is decreased. 2.2.11. Serrated flow The following rapid and somewhat incomplete survey of plastic instabilities in L 12 alloys has been motivated by the fact that in Ni3(A1, Hf)B, Ezz and Hirsch [71] have observed that the occurrence of serrated flow seems to coincide with the inflection point of -r(T) which, as we have noted, may correspond to a critical range of temperature. In the experiment of Ezz and Hirsch [71 ], the magnitude of instabilities is in fact large enough to preclude the analysis of strain-rate jump tests. Several authors have reported that plastic instabilities may occur during the deformation of L12 alloys. There is some disagreement however as to the conditions of their occurrence. We have tried to review in this section the relevant information based on published stress-strain curves that have been reproduced in the literature (it cannot be excluded that some curves have been "cleaned from noise" before publication). Differences in the results between different groups are accentuated by the fact that the visibility of serration, depends markedly on the stiffness of the straining device. The data of Curwick [36] show no serrated flow with ternary additions of Nb, Ta, Ti, W and Mo. There is no instability either in Co3Ti (Takasugi et al. [115]), in Ni3(A1, Ti) (Staton-Bevan [30, 58], Miura et al. [53], Korner [116]) and in Ni3(A1, W) (Saburi et al. [83]). Within the vast variety of compositions tested, Dimiduk [46] has not detected plastic instabilities, except in the course of a Cottrell-Stokes experiment on Ni-rich binary Ni3A1 deformed at room temperature, after prestraining at 600~ A similar observation has been made in Ni3(A1, Hf) deformed at room temperature after a prestraining at 773 K [283, 286]. However, in the same alloy, serrated flow occurred also at 773 K after a prestraining at a lower temperature of 473 K. There is some indication that serrated flow tends to appear at intermediate temperatures as in Ni3Ge (Pak et al. [117]), where it is observed at 300 K for two load orientations near [111] (in Ni3Ge 300 K is very close to Tp at least for one orientation). Plastic u
290
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Ch. 53
instabilities seem to appear in Ni3Ga tested between 77 K and 770 K, irrespective of load orientation (Takeuchi and Kuramoto [81]). Ni3Fe shows one of the most striking examples of serrated flow (Wee and Suzuki [21]). Finally, Takasugi and Yoshida [29] have recently reported plastic instabilities in Ni3(Si, Ti) deformed at 4.2 K, which is of course the lowest temperature at which serrated flow has ever been found in L 12 alloys. In Fe-doped L12 stabilized A13Ti alloys, which show a slight positive TDFS, serrated flow is observed during high temperature continuous straining (300 ~ to 600~ In this family of L12 alloys, this effect, which is also influenced by strain ageing, might be attributable to the instability of a second phase. These aspects are discussed by Potez et al. [118] and in recent overviews (Morris et al. [13, 14], Wu and Pope [15]).
3. Properties o f superdislocations in L12 intermetallics This section, is aimed at covering data, acquired mostly by transmission electron microscopy (TEM), on dislocations and microstructural organization in deformed L12 intermetallics. It is divided into four main parts. The core structure of dislocations in intermetallics remained poorly documented until the early 80s, when dedicated studies of superdislocation cores were initiated by means of TEM. A significant amount of information on the fine structure of dislocations, complemented in some cases by atomistic simulations, is now available. TEM has also contributed to a better knowledge of the overall organization of the deformation microstructure. These studies have been extremely fruitful not only regarding the mechanical behaviour of L12 alloys but also for the general understanding of dislocations in materials. Not every observation of dislocations in intermetallics has been correctly interpreted. When confusion is due to specific contrasts, difficulties can be overcome by means of appropriate image simulations. Quite currently, however, uncertainties arise from overor misinterpretations of the features available in thin foils in relation with deformation mechanisms in the bulk. These uncertainties transform into serious problems when they become legitimated in subsequent papers. This is why we have found it important to start this section by assessing some limits of reliability of TEM analyses of dislocation cores (section 3.1). Section 3.2 will review some general results of measurements of planar defect energies and the following two parts will deal with the microstructural properties of L12 samples deformed in domains A and B (sections 3.3 and 3.4, respectively). In the last section (3.5), we shall review the information on dislocation dynamics that has been gathered directly by in-situ tensile experiments.
3.1. Limitations of TEM investigations Rather than addressing in general terms the limitations of TEM applied to the study of deformation processes, we shall examine critically examples selected for their implications in dislocation processes in L12 alloys. Additional information on difficulties in TEM studies of dislocations in intermetallics can be found in several contributions
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Microscopy and plasticity of the L12 "tt phase
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(Veyssi~re [3, 4], C16ment et al. [112], Couret et al. [72], Hemker and Mills [50], Korner et al. [119], Veyssi~re et Douin [120]). The most frequently encountered dissociation configuration in intermetallic alloys is that of a superdislocation dissociated into two superpartials with collinear Burgers vectors 1
1
(llO) -+ ~(110) + APB + ~(110),
(7)
hereafter referred to as mode I, where APB stands for AntiPhase boundary. In principle, this reaction may occur conservatively in the screw orientation in any plane that contains the Burgers vector. Other than that, a (110) superdislocation may dissociate into two 1 5(112) superpartials which border a superlattice intrinsic stacking fault (SISF), a defect with rather low surface energy, according to 1
(110)--+ ~ ( l l 2 ) + S I S F +
1
(112)
(8)
(mode II). Clearly, the dissociation plane is now restricted to that containing both su1 perpartial Burgers vectors. In addition, either a l(110) or a _g(112) superpartial is liable to further subdissociate into Shockley partials, provided the stacking fault thus formed, named CSF (C stands for complex), is of sufficiently low energy (for a review, see Kear and Oblak [121]). As we shall see, the subdissociation constitutes one of the critical issues in the debate on the flow stress anomaly of L12 alloys. 3.1.1. HREM As to the fine structure of dislocations and properties of planar defects, high-resolution electron microscopy (HREM) provides the best attainable resolution (better than 0.2 nm), but this requires that
- foils be thinner than 10 nm in order to be considered as phase objects; - the defect under study be rigorously aligned along the direction of the optical axis (for instance a deviation of less than 3 ~ from the exact screw orientation should be enough to give rise in projection to a further core spreading over about 2b); - the defect core structure must remain unchanged from top to bottom of the foil. These conditions are particularly stringent. For instance, they forbid a safe HREM analysis of kink-containing dislocation lines or of stepped planar defects whose steps are not aligned with the electron beam. Insofar as L 12 alloys are concerned, HREM has been used in order to study dislocation cores in Ni3A1 and Ni3Ga (Crimp [122], Sun et al. [123, 124], M.J. Mills: unpublished results) together with some solid state properties such as the early stages of disordering through the wetting of APBs in Co3Pt (for a review, see Ducastelle et al. [125]). 3.1. I.I. The dissociation of screw 1(110) superpartials. In addition to the abovementioned difficulties, cores of screw dislocations are particularly difficult to study under HREM because displacements occur essentially along the electron beam, with the origin
292
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Fig. 18. The Eshelby twist causes a rotation of the crystal about the line direction of a screw dislocation in a thin foil. When two screw dislocations are in close proximity, the twist due to each dislocation is either added or cancelled in between the dislocations, depending on the relative signs of the Burgers vectors (M.J. Mills, N. Baluc and H.P. Kamthaler: unpublished results). (a) HREM image of two parallel screw superdislocations (SD1 and SD2), bl = b2 = [101], in Ni3Al observed along [101] (defocus ~ 12.5 nm, foil thickness ~ 8 nm). The image in between SDI and SD2 is consistent with untwisted crystal. The contrast changes to the outside of the superdislocations, where the twist is additive.
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Microscopy and plasticity of the L12 ")/phase
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Fig. 18 (continued). (b) Under the same imaging conditions in (a), the contrast is reversed so that the contrast from the untwisted crystal is in the outside of SD1 and SD2, while the tilted crystal is seen between them. In this case, the two superdislocations are forming a screw dipole. Note the absence of significant core extension at t SD1 and SD2 and the impossibility to locate the core of 31101] superpartials directly (courtesy of M.J. Mills).
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of displacements limited to the edge component (be - ~(112), ~ 0.14 nm) of the constituting partials, if any. Core extension is so narrow in Ni3A1 that contrast-producing deformations originating from companion Shockley partials compensate nearly exactly, so that the distortions of atomic rows remain almost invisible in the vicinity of a screw superpartial. Nevertheless, as a result of a subtle contrast property that stems from the so-called Eshelby twist effect, it is possible to localize 1(110) superpartials quite neatly on HREM images and to discriminate between a dissociated superdislocation and a superpartial dipole (fig. 18). The magnitude of superpartial subdissociation into Shockley partials as observed under HREM is claimed to be about 3b wide (M.J. Mills: unpublished result) or a little more (Crimp [122], Crimp and Hazzledine [126]). In view of the expected magnitude of distortions, this is in the range of the experimental uncertainty of the method, see fig. 5 in [126]. In addition, as stated earlier, an undissociated superpartial line that would be slightly bent off the screw orientation in its slip plane or, equivalently, that would consist in two screw segments connected by a kink of atomic size, should give rise to an extension of the core image of the order of b. Finally, it cannot be excluded that, just as in f.c.c, alloys (Hazzledine et al. [127]), the core of a superpartial relaxes under image forces in the vicinity of a free surface. Again, this would result in some widening of its projected image [126]. Further comments on TEM observations of the subdissociation of 89 (110) superpartials will be forthcoming in section 3.1.2.2. With regards to mechanical properties, the important result which emerges from HREM studies is that the spreading of the core of a screw 1 (110) superpartial in Ni3A1 is remarkably narrow. Assuming that the role of the APB during cross-slip could be neglected, Ni3A1 compares thus very much to aluminium or to nickel where cross-slip is easy. The implications of this on cross-slip processes in the L12 structure are of course important, they will be discussed in more detail in section 4.2.1.1.
3.1.1.2. The locking of edge 1 (110) superpartials. An edge 89 (110) superpartial undergoing cube slip offers the same potential of subdissociation into Shockley partials off the glide plane as a Lomer dislocation in the f.c.c, structure. It is therefore a good candidate for a locking process. There is abundant evidence that edge superdislocations tend to be blocked after deformation at and above the peak temperature (Thornton et al. [41] 5, Veyssi~re [128], Douin et al. [129], Sun [62], Bonneville et al. [130], Takasugi and Yoshida [39]). An asymmetrical type of subdissociation of an edge segment has been directly evidenced in Ni-rich Ni3A1 (Veyssi~re [128]) (fig. 19). It has been claimed on the other hand that, depending upon test temperature, two distinct configurations may take place (Sun et al. [124]): - the double Lomer-Cottrell lock, which contains an APB that belongs to the slip plane, - the super-Lomer dislocation whose APB is formed non-conservatively. 5In fig. 13(c) of Thornton et al. [41], superdislocations, which can be reasonably thought of as expanding in the (001) plane of the toil, are in contrast under the g = 220 reflection, which then defines the direction of their Burgers vector b. Since 9 is orthogonal to the direction of the dislocation greatest elongation, the longest segments are edge in character, while the shorter orthogonal ones are the mobile screws, conforming to the general aspect of the deformation microstructure in cube slip (Veyssi~re [128]). m
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Fig. 19. The subdissociation of an edge superdislocation, in Ni3A1 deformed at 800~ and slowly cooled (2b = [110]). In its nearly screw part (s), the superdislocation is dissociated in its slip plane, (001), whereas the mixed (m) and edge (e) parts are dissociated by climb perpendicular to the slip plane. The subdissociation of the edge segment into a configuration containing a stacking fault (sf) is visible under the form of the fringe system originating from the fault, which is in contrast under the reflections presented here. In both micrographs the beam direction B D = [011], is inclined at 45 ~ from the (001) plane, which is the actual slip plane of the superdislocation. (a) 200 reflection, the companion superpartials are in contrast; the arrow indicates the transition between the glide and the climb dissociated parts of the superdislocation. (b) Same projection, g --- 111, the superdislocation is entirely out of contrast, the stacking fault remains in contrast. m
The H R E M image of the double Lomer-Cottrell suggests that the extent of the CSF is between 3 and 4.5 nm, but this is clearly inconsistent with the direct observation of core spreading of a single superpartial which is at most of 3b (section 3.1.1.1). Furthermore, the image shows more than the expected four stacking-fault planes, that is, two per 1[ 1 2 0]-] superpartial (examined at a glancing angle along the trace of the (1]-1) plane, the left-hand side CSF in fig. 4 by Sun et al. [ 124] is connected at its left-hand side to another two stacking faults on (111)). On the other hand, it is difficult to reconcile the weakbeam images of super-Lomer--Cottrell dislocations with their H R E M h o m o l o g u e s in the same work, since the width of the stacking fault ribbon, as identified from H R E M , should make it in principle easy to detect the stacking fault contrast in the w e a k - b e a m images taken under the 220 reflection parallel to the Burgers vector of the edge superdislocation. This is in fact never the case.
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Further work is clearly needed before definitive conclusions can be drawn as to the sessile origin of edge superdislocation cores in the L12 structure after deformation at elevated temperature. Nevertheless, based on the observation of sessile edge segments in Ni3A1 during in-situ straining experiments and on an analogy in shape thus found with Sun et al.'s observations in Ni3Ga, Mol6nat and Caillard [114] (see also Caillard et al. [131]) have recently claimed the existence of a transition from a double LomerCottrell into a super Lomer configuration. Unfortunately, the imaging method in this particular in-situ observation cannot provide enough resolution to conclude as to the actual fine structure of these dislocations. Therefore, the Lomer-Cottrell type configuration, through perfectly acceptable as a working hypothesis, cannot be considered as legitimated, since it is not adequately assessed. Hemker et al. [79] report that in Ni3(A1, Ta) dislocation characters are randomly oriented on the cube plane during inverse creep (section 2.2.9). Regarding edge segments, these authors have not found evidence for segmentation in the edge orientation, thus at variance with the results of Sun et al. [124]. It should be added that inspection of the whole TEM work done under WB conditions on superdislocation properties in L 12 alloys demonstrates that the locking of edge dislocation in cube slip is highly compositiondependent. Douin and Veyssi~re [132] have indeed shown on polycrystalline binary Ni3A1 alloys that the larger the atomic fraction of aluminium, the narrower the dissociation and the greater the tendency to form locked edge segments (fig. 20). In other words, differences in locking properties from one alloy to the other are not necessarily associated to the presence of a ternary element. Composition seems to be the reason why sessile edge segments showing multiple line contrast have been evidenced in Ni3Ga (Sun et al. [124]), in Ni3(A1, Ti) (Korner [133, 134]) and in Ni3(A1, Ta) (Baluc [135]), - Hemker et al. [79] claim not to have observed Lomer locks after creep of Ni3(A1, Hf, B), - these features have been reported neither in Ni3Ga by Takeuchi and Kuramoto [81] nor in Ni3Ge (Pak et al. [117]).
-
The present detailed discussion of Lomer-Cottrell configurations will take some importance in what follows because of the role that Hirsch [99] has ascribed to these locks in the high-temperature stage of his model (section 5.6.2) where it is assumed that the long closing jogs (CJ) in the cube plane which constitute the extremities of KW locks (section 3.4.3.1) are stabilized by means of a core transformation into a sessile LomerCottrell type configuration. This locking step contributes to the high temperature/high stress bypass mechanism (sections 3.4.3.3 and 5.6.2.2). 1 Burgers vector 3.1.1.3. The locking of 1 (112) superpartials. Superpartials with .~(112)
are preferentially aligned along (110) directions. It was first suggested by Giamei et 1 al. [136] that the core of 7(112) superpartials is stabilized along 30"(110) directions by 1 transformation into a non-planar structure (a leading .~(112) superpartial creates a SISF in its wake, formula 8). A HREM study by Sun et al. [123] on Ni3(A1, Ti) deformed at 77 K has supported the existence of the so-called Giamei locks. This is however difficult to confirm based on the examples of non-dipolar SISF-containing defects given in this work (i.e., figs 11 and 12 of Sun et al. [ 124]). In fact, the defect shown in their fig. 11 is
5
pm
Fig. 20. Composition dependence of the locking of edge segments in polycrystalline Ni3AI samples, both deformed at 80OOC [132]. (a) Superdislocations with [I101 Burgers vector expanding in the (001) cube plane in an Al-rich sample, BD = [OOl]. The microstructure shows essentially two preferred orientations, screw (s) and edge (e). The latter is strictly rectilinear and there is evidence of a further subdissociation of some of the edge segments in places (thicker contrast). The screw part is smoothly curved in the cube slip plane suggesting that this is now the most mobile of all dislocation characters. (b) Ni-rich sample, the magnification is the same as in (a), B D = [ O i l ] . Note the dramatic change in superpartial separation between the two compositions, as well as the absence of preferred orientations for dislocation locking in the Ni-rich sample, in particular in the edge orientation (courtesy of J. Douin).
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fully planar and it contains two strips of stacking fault together with a pronounced but unfaulted contrast on its left. In their fig. 12, on the other hand, the arrowed step in the cube plane is not bordered on its right by a fault which, in accord with the indication provided in the work on Lomer locks [ 124], should be wide enough to be clearly visible.
3.1.2. Conventional TEM (weak-beam Under diffraction contrast and because waves and matter, the image position geometrical projection onto the plane beam. As a consequence,
dark-field) of the nature of the interaction between electron of a source of lattice distortion differs from its of observation in the direction of the diffracted
- the image of a dislocation line is systematically shifted, with potential adverse consequences on surface energy determinations, - there is not necessarily a one-to-one correspondence between the structure of a dislocation line and of the number of its images. Selected difficulties which have arisen in the course of weak-beam studies of dislocations in L12 alloys are presented in the following.
3.1.2.1. Measuring the energy of planar defects from dissociation widths. The determination of stacking-fault energies is confronted with several sorts of limitations. Regarding SISFs, the major difficulty stems from the fact that 7SISF is small while lattice friction 1 (112) superpartials is significant. Hence, it is not ascertained that one has ever seen on .~ a configuration dissociated in mode II that could be unambiguously claimed as being under stable equilibrium (section 3.1.2.3). On the other hand, for the same reason as under HREM (section 3.1.1.1), the experimental determination of CSF energy under weak-beam poses great difficulties: 7CSF is so large that the resulting dissociation would be both at the limit o f - in fact even beyond in the screw orientation- the resolution of the weak-beam technique (section 3.1.2.2), and at the limit of applicability of linear elasticity (Saada and Douin [ 137]). It should be kept in mind that when dissociation is of the order of 2 to 3b, the distortions in between the Shockley partials are so l a r g e - beyond linear elasticity - that the notion of a stacking fault does not really hold, so that 7CSF becomes pointless and the use of the usual linear elasticity expressions for equilibrium irrelevant. Hence, it is only in the case of APB energies that some confidence can be reasonably placed on experimental measurements in the L12 alloys under consideration here. Data on SISF and CSF energies should be regarded as indicative of an order of magnitude with an error bar of +50% and maybe more. APB energy determinations are mostly conducted in the weak-beam imaging mode; this is because the weak-beam technique offers the best compromise between adequate resolution and comfortable working conditions. In the case of dissociation under mode I (formula 7), companion superpartials are viewed under identical 9" b conditions (9 is the diffraction vector). Hence, the correction for image shift is expected to be less critical than for the dissociation into unlike Burgers vectors where a widening or a narrowing of the inter-partial s p a c i n g - which arises from the dipolar component of the configuration - may take place. Accordingly, a large fraction of the published surface energies have been determined from direct observations of the separation between the image peaks of partial dislocations (section 3.2). We shall see below that there are problems however.
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M2
M1 11
1
2
Fig. 21. Origin of the contrast asymmetry of a superdislocation dissociated into two superpartials with collinear Burgers vectors. Ml and M2 are equally shifted with respect to superpartials 1 and 2, respectively ( r l l - " V22 ) . The strain field is the superimposition of the individual strain fields caused by each superpartial. The asymmetry originates simply from the fact that rl2 differs from r21. This implies in turn that image formation under weak-beam conditions cannot take place in geometrically equivalent positions at companion superpartials; in other words, the images are shifted differently with respect to the dislocation lines. Furthermore, since the volume of crystal that satisfies Bragg conditions locally is inversely proportional to the distance from the dislocation core, neither the width not the intensity of companion images should be in general the same, that
is, two companion superpartials with identical Burgers vectors may show significantly distinct contrast (this property can be checked in many of the micrographs of this review and most particularly when the same superdislocation is presented under 9 and - 9 conditions, such as in fig. 22). Since APB energy determinations have been the object of several studies or reviews (Veyssi~re [138], Dimiduk [49], Dimiduk et al. [48]), the following analysis of the reliability of APB energies may be restricted to a few remarks. (i) Douin et al. [ 139] were the first to point to the importance of anisotropic elasticity in deriving APB energies from equilibrium separations. In the same alloy, the APB energy corrected for elastic anisotropy is about 90 mJ m -2 in the cube plane and 110 mJ m -e on the octahedral plane, instead of 140 mJ m -2 and 180 mJ m -a, respectively, as had been previously determined by Veyssi~re et al. [140] in the frame of isotropic elasticity. (ii) Another source of errors originates from the fact that it is not ascertained that observed configurations are actually all under equilibrium, especially when they are dissociated in the cube plane where lattice friction is large (section 4.1.1.2). (iii) Regarding dislocation contrast itself, it should be kept in mind that, corrected for orientation with respect to the projection plane, the separation between the images of companion superpartials still does not coincide with their actual separation in the crystal. This is due to the fact that under diffraction contrast, the image of a dislocation is always shifted with respect to its line and that the shift is not the same for each superpartial (fig. 21) (Veyssi~re and Morris [141]). In addition to this effect, Baluc et al. [142] have demonstrated that this effect is complicated by elastic anisotropy. In the conditions simulated by Baluc et al. [142], systematic corrections on ")/111 and 7001 are modest and significant, respectively (approximately 10% and 25%, respectively). As a consequence, the APB energies published before Baluc et al.'s contribution have been systematically underestimated in these planes (Dimiduk et al. [48]), in such a way that the value of the APB energy ratio ( = 7o01/")/111 (section 4.2.2) published before this work should actually be a little smaller than initially thought. (iv) Some more complex corrections are needed when, as practiced by some authors (Baluc et al. [143], Hemker and Mills [50]), imaging is carried out under 9 " b = 4 conditions (i.e., when the operating reflection is 9 = 440 parallel to the Burgers vector b),
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in which case multiple images may arise which requires the use of image simulation techniques (section 3.1.2.2). (v) Finally, it should be pointed out that in the case of narrow separations, say of the order of 3 nm or less, which are encountered in some ternary Ni3A1X alloys, lattice distortions induced at the APB interface by the proximity of the bordering superpartials cause slight atomic deviations off the positions they would have adopted in the infinite APB (D. Dimiduk: private communication). This may result in an additional correction whose magnitude is unpredictable (in between the superpartials, the larger the fraction of crystal distorted beyond the limit of applicability of linear elasticity, the larger the correction). In this case and after corrections, the APB energy in a given plane determined from the separation in the screw orientation should not coincide with that determined from the edge separation. A similar effect had been evidenced for the first time in Ge by Chiang et al. [144], who showed that the stacking-fault energy determined directly under HREM from the shape of Z- and S-shaped faulted dipoles increases anomalously below a critical dipole dimension. Some authors such as Baluc et al. [143], Hemker and Mills [50] and Korner and Karnthaler [145] determine dissociation widths statistically over a very large number of measurements, which does reduce some of the above difficulties quite significantly, though not totally. It also allows comparison between the dissociation width in the screw and that in the edge (or near edge) orientations where accuracy is the greatest. Properties of planar defects in L12 alloys and some determinations of planar defect energies in L12 alloys are further addressed in sections 3.2 and 3.4.3.9.
3.1.2.2. The subdissociation of screw ~1 (11 O) superpartials.
A difficulty in interpreting dissociation contrast arises from the fact that, under certain circumstances, the image of a single dislocation line may show more than one peak of intensity in the images (for example see fig. 6 in Veyssi~re [4]). In view of the importance played by the subdissociation of a superpartial into Shockley partials in every model of the flow stress anomaly, this property of contrast in L12 alloys is critical and deserves special attention. Since as long as the product 9" b is less than or equal to 2, no additional peaks are expected to arise from a single line (Hirsch et al. [146], Korner et al. [119]), weakbeam observations are then in principle artefact-free. However, it has been shown both experimentally and with the help of multi-beam image simulations that supplementary peaks may still appear under 9" b = 2. This occurs under specific orientations of the foil at which the core of a superpartial diffracts both the primary and a once diffracted beam (Bontemps-Neveu [23], Oliver [147], see also Veyssi~re and Douin [120]). Additional peaks are rather easy to identify in this case since they are expected to appear and to disappear upon slight foil tilting (i.e., such that ~Sg ..~ sg/lO). The work of Liu et al. [148] seems to be interpretable by such a contrast artefact rather than by a true fourfold dissociation. These authors indeed suggest a separation between Shockley partials, in the near screw orientation of the superdislocation, which would be by far larger than any other subdissociation observed by others in the edge orientation (see below). If it were correct, this dissociation should have been repeatedly reported in other studies of superdislocations in Ni3(Al, Ti) under weak beam conditions (see Sun et al. [124], consult also most of Korner's papers in the following list of
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references). The subsplittings reported by Sun and Hazzledine [149] in the ,),! phase of a superalloy as well as those reported in Ni3Ga by Couret et al. [72] and in Ni3(Si, Ti) (Yoshida and Takasugi [150], Takasugi and Yoshida [39]) might as well correspond to the above process of extra peak production by core double diffraction. This is suggested by the fact that the subsplittings are observed for a given foil orientation that does not particularly coincide with the projection under which the dissociation should be the widest within the tilting sequences. For instance in the observation of Couret et al. [72], the subdissociation over 1.5 nm of their fig. 3(c) - this subsplitting is in fact not seen in their fig. 4(a) though taken under similar conditions of orientation and resolution implies that the dissociation should be 1.8-2 nm wide in the (111) plane (or 5.4 nm in the (111) cross-slip plane). Hence, it should be visible in a comfortable and reproducible manner under standard weak-beam conditions, in foil orientations near the octahedral dissociation plane be it of primary or of cross-slip nature. Since this is the case neither in the observations of Couret et al. [72] nor more generally in similar observations made in L 12 alloys with a positive TDFS, it can be quite reasonably inferred that the reported image splittings of screw superdislocations do not result from a true subdissociation, but from the above-mentioned critical double diffraction condition. In fact, we do not know of a direct observation under weak-beam conditions of the true subdissociation of a screw superdislocation into four Shockley partials. In particular, the TEM work of Sastry and Ramaswani [151] on Cu3Au does not have adequate resolution to yield undisputable evidence of the claimed fourfold dissociation. A recent study by Hemker and Mills [50] features the subdissociation beautifully, but this is achieved in the edge orientation. Note, however, that this result has been obtained for 9 " b - 4, that is, an imaging condition that causes inherent additional peaks at each superpartial (Hirsch et al. [146], Korner et al. [119]. The origin of the different peaks under 9" b - 4 imaging conditions together with the subdissociation into Shockley partials bordering a complex stacking fault (CSF) are consistently demonstrated in Hemker and Mills' study. It remains to understand why edge superdislocations imaged under 9" b - 2 conditions exhibit a twofold dissociation, when four peaks are evidenced under 91 9b - 4 (and 91 - 404) in similar foil orientations (compare figs 3 and 4 of the paper of Baluc et al. [142]). Hemker and Mills [50] indicate that the vast majority of their images only exhibited three visible peaks, which suggests that experimental parameters that may yield multiple image p e a k s - such as maybe dislocation depth in the f o i l - are not all under control yet. Clearly, additional observations and accompanying simulations are needed in order to elucidate the remaining unknowns in this problem. Finally the observation of a subdissociation of a near-edge superpartial in Ni3Fe by Korner et al. [ 152] deserves special attention since it appears to be the largest apparent dissociation width ever reported in L12 alloys. It should be incidentally noted that this alloy does not show a positive TDFS, which suggests that the flow stress anomaly is directly related to the cross-slip capabilities of superdislocations (Korner [153]). m
3.1.2.3. Weak-beam analysis of the dissociation under mode H in Ni3Al. In this section we wish to demonstrate that superdislocations are generally dissociated under mode I at any temperature of deformation.
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Based on a comparison between faulted defects in Ni3Ga and in Ni3A1, both deformed at 77 K, Sun and Hazzledine [154] have concluded that in these alloys mode II (formula 8) is the preferred dissociation mode during deformation at low temperatures. The splitting distances would be almost identical to those of APB-coupled superdislocations in the same alloy (see also section 3.1.1.3). Their statement relies (Sun et al. [123]) on - the existence of elongated SISF dipoles attached to a superdislocation (fig. 22(a)), the fact that there seems to exist continuity between the SISF within the dipole and the planar defect located within the narrowly split mixed superdislocation segments, to which the dipole is attached; in Ni3Ga the superdislocation dissociation is hardly resolved in the relevant weak-beam images, - apparent analogies between SISF dipoles observed under weak-beam conditions in Ni3Ga and faulted defects in Ni3A1 imaged end-on under HREM. It is true that trailing SISF dipoles similar to those reported in Ni3Ga are currently observed in Ni3A1 after deformation up to room temperature, so that a comparison between Ni3Ga and Ni3A1 is in principle valid. However, a Shockley partial, located at the boundary between the SISF and the APB, can be evidenced in Ni3A1 when the appropriate reflection is set up under adequate resolution (fig. 22(b)), implying that superdislocations are dissociated under mode I in the narrowest part of the superdislocation and that mode II remains local at least in Ni3A1. This is actually also true in Ni3Ga as shown in figs 22 (e) to (i) (see the insert in fig. 22(g)). Sun et al.'s identification of the dissociation under mode II of the narrowest portions of the superdislocations [123] should be compared to the analysis made by Couret, Sun and Hirsch [72] of their fig. 4 in which a S-shaped dipole is similarly attached to a narrow superdislocation. Clearly, this defect is a SISF-containing dipole which, from the published tilting sequence, does lie on the (111) plane; unfortunately, no reflection that would permit to visualize the stacking fault contrast unambiguously is available (the fact that the dipole exhibits S shape is common for this category of defects; although the details of their formation mechanism are still not elucidated, such S-shaped dipoles are 1 consistent with the property of .~ (112) superpartials to stabilize along two of the three (110 / close-packed directions of a given octahedral plane). Hence, despite a striking analogy between the configurations analyzed in the studies of Sun et al. [123] and of Couret et al. [72], it is remarkable that in the latter work the superdislocation dissociation mode is analyzed in its narrow segments as being under mode I, while mode II was concluded in the former study. --+ Fig. 22. Examples of SISF dipoles connected to an APB-coupled superdislocation. (a) to (d): Ni3(AI, Hf) deformed along [123] at room temperature showing an APB-coupled superdislocation (Bontemps-Neveu [23]). Note that in addition to the screw direction, the superdislocation is markedly aligned along the 60 ~ [011] direction. A SISF dipole originates from a near-edge segment. (a) g = 202, both superpartials are visible. (b) g = 220, the upper superpartial that borders the SISF is out of contrast (b = + g i l l 2 ] ) , shown by an arrowhead in the insert is the Shockley partial that separates the SISF from the APB" note also that the ~l[]01] superpartials are both visible. (c) g -- 022, the lower superpartial that borders the SISF is m
--
1
out of contrast (b -- -t-g11211]). (d) g = 111 the SISF is in contrast, while the narrowly dissociated segment is invisible, confirming that it consists in two superpartials with 1 []01] collinear Burgers vectors.
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This particular example also points to the difficulty in correlating defects identified under weak-beam conditions to features observed edge-on under HREM, especially when these features are taken from two different alloys such as Ni3A1 and Ni3Ga. It should finally be mentioned that the nature of the dissociation mode after deformation at the temperature of liquid nitrogen has been checked by a number of authors (Baluc et al. [142], Bontemps-Neveu [23], Korner [116, 134]) and that mode I was the only dissociation mode that has been found to occur intensively. The evidence that dissociation under mode I dominates within the low temperature deformation microstructure is clear enough; it has been demonstrated in Ni3(A1, Hf) deformed at 4 K (fig. 23) (Bontemps-Neveu [23], Korner and Veyssi~re [155]). In conclusion for this section, SISF-containing dipoles whose habit plane coincides with the primary slip plane are local deformation debris, not representative of the actual dissociation mode. Further comments on the place taken by SISF-containing dipoles in the deformation microstructure of L12 alloys will be forthcoming in section 3.4.3.8.
3.1.2.4. The dissociation under mode H in L12 stabilized Al3Ti ternary alloys. The sometimes pronounced negative TDFS found in L12 alloys, which undergo octahedral slip in domain A has been attributed to dissociation under mode II (see for instance Vitek [1], Pope [156]). Atomistic simulations have indeed suggested that in L12 alloys in which the APB on {111} is unstable, if it occurs at all, (110) superdislocations may still dissociate under mode I in the cube plane or under mode II in the octahedral plane; and that superpartials should be spread spatially off the slip plane in both modes. Hence, one expects them to experience a strong lattice friction in the appropriate slip planes (Yamaguchi et al. [157], Tichy et al. [158, 159]). This hypothesis has been tested under weak-beam conditions in L12-stabilized A13Ti alloys. Atomistic simulations predict that in the range of strong negative TDFS, superdislocations dissociated under mode II should sit more favourably along (110) Peierls valleys. This is apparently not observed in a number of L 1z-stabilized A13Ti alloys deformed in domain A. The dissociation mode in domain A has been studied directly in +.__ Fig. 22 (continued). (e) to (i): Similar observation in Ni3Ga. Again a SISF dipole originates from a narrow dissociated dipole, however, the connection occurs at a kink on a kinked screw segment. In addition to the SISF dipole, two features are worth mentioning. Firstly, just as any APB-coupled superdislocation would behave, it is markedly kinked in its screw part, including repeated APB jumps on the right side of the SISF dipole. Secondly, below the kink on the screw superdislocation and overlapping partly with the SISF loop, one can observe a succession of near-edge dipoles with the bottom extremity on a given dipole aligned with the top extremity of its right-hand side neighbour. (e) g = 202, the superpartials bordering the SISF are both visible, the SISF is invisible (the fact that the inner part of the dipole appears brighter stems precisely from a dipolar effect as shown by comparison with figure (f)). Both I []-01] superpartials are clearly resolved and show an asymmetrical contrast, with the upper superpartial being the brightest. (f) g = 202, the inner part of the SISF dipole is darker than the background. Note that in the APB-coupled parts, the bottom superpartial is now the brightest. The dipolar nature of the sequence of short near-edge dipoles is made clear by comparison of figures (e) and (f). (g) g = 022, the upper superpartial that borders the SISF is out of contrast (the closing Shockley partial is visible in the insert). (h) g = 220, the other super-Shockley is now out of contrast (as expected, in the APB-coupled part, both il [101] superpartials remain visible). (i) g -- 1]-1 the fault is in contrast while the superdislocation shows only a faint residual contrast in its APB-coupled part. Note that the study of the Shockley partials that separates the SISF from the APB is made difficult by the fact that, because of their reduced length, they are not systematically visible under g 9b ~ 0 conditions. D
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several of these alloys. It has been claimed to occur under either mode I (Lerf and Morris [160], Vasudevan et al. [161], George et al. [162], Morris [163]) or mode II (Hu et al. [164], Inui et al. [165], Wu and Pope [15]). This difference is the object of an ongoing controversy (Morris et al. [13, 14]). In Fe-doped A13Ti, Inui et al. [165] have concluded to a change in dissociation mode, from mode II to mode I according to whether deformation is conducted at room temperature or at 600~ respectively. The weak-beam analysis is based on the assumption that mode II should give rise to an asymmetrical pair of peaks, whereas mode I should not [165]. This assumption is incorrect: although it is true that under mode II superpartials should in general show an asymmetrical contrast, there is an intrinsic cause of contrast asymmetry under mode I as well (fig. 21) (Veyssib~re and Morris [141]). Further weak-beam observations have been conducted in L12-stabilized Cr-doped A13Ti alloy deformed at liquid-nitrogen temperature and above. In these experiments the above mentioned transition in dissociation m o d e - between mode II at low temperature and mode I at high temperature- has been claimed to be confirmed experimentally (Wu and Pope [15]). However, the dislocations said to be SISF-coupled are not particularly 1 aligned in dense atomic rows as .~(112) are usually, but smoothly curved. Furthermore, the contrast evidence for mode II is still not convincing 6. Finally for what concerns dissociation in L12-stabilized A13Ti, Gao et al. [166] have reported an unusual threefold dissociation in as-prepared Co-doped A13Ti that consists essentially in an APB-coupled superdislocation of which one superpartial is widely subdissociated into Shockley partials bordering a CSE In conclusion and following Morris et al. [13, 14], we believe that the fact that superdislocations are dissociated under mode II in this particular alloy is not relevant to the dissociation mode of the majority of mobile dislocations. However, the debate on the dissociation mode in L12-stabilized A13Ti remains open. 6At this stage the procedure to prove mode II directly is to set the SISF in contrast, which can be done even when the SISF is both parallel to the foil and narrow (5 nm or more), by using an appropriate 200 or I 11 reflection and tilting the foil slightly until the strip becomes bright. -+ Fig. 23. Examples of kinked superdislocations in Ni3(AI, Hf) deformed along [123] at 4.2 K (the foil was sectioned parallel to (101)). Tilting sequence showing the details of the dissociation mode along the dislocation length to illustrate the role played by the octahedral cross-slip plane in the low-temperature deformation mode. Note that in both cases the superdislocations exhibit all the features that are typical of an APB-coupled superdislocation after deformation at low or moderate temperatures (A. Korner and P. Veyssi~re: unpublished result). (a) to (c): Note the presence of two segments indicated by arrowheads, that remain rectilinear irrespective of tilting. (a) 13D -- [121]: the superdislocation is rectilinear and it shows a single line over nearly all its length; the contrast is weak in places corresponding to the parts where the superdislocation is kinked. (b) B D -- [ 111]" in this orientation, the segments that remain rectilinear upon tilting are seen under nearly edge-on conditions, similar to those of figure (a), indicating that they are actually incomplete KW locks whose average habit plane lies is slightly off (111), i.e., there is a limited cross slip on the cube plane. (c) B D -- [101]" in view of the variation of its projected length, and in particular, of the fact that it is not straightened for this orientation of the foil, the right-hand side kink that happens to lie off the primary slip plane in (a), should be lying in the cross-slip octahedral plane.
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3.1.2.5. Core effects in Co3Ti. Based on the observation of a large density of SISFs and on further contrast peculiarities, it has been argued that a stress-assisted transition from mode I to mode II may occur in Co3Ti (Liu et al. [25, 167, 168]) and it has been proposed that this transition is at the origin of the dual TDFS exhibited by this alloy (section 3.3). The available analyses of mode II at individual superdislocations are based upon contrast asymmetries (section 3.1.2.4) of images taken under insufficiently weak weakbeam conditions (see fig. 7 in Liu et al. [167]). As discussed in the previous section, this hypothesis makes image interpretation disputable. TEM experiments similar to those of Liu et al. [25, 167, 169] conducted after deformation between 4.2 K and 873 K (fig. 24), including in-situ straining experiments at room temperature, have indicated that the evidence for mode II had been incorrectly interpreted (Oliver [147]). In post mortem Co3Ti samples resulting from deformation at moderate and low temperatures, the SISF contrast is so overwhelming that it can be misleadingly taken as the dominant mode during deformation (fig. 25). However, as pointed out by Takeuchi and Kuramoto [81], even in foils containing large densities of SISFs, this density may account for but a small fraction of the total deformation. Moreover, it can be shown that in Co3Ti samples the habit plane of a significant fraction of SISF-containing features differs from the primary slip plane. As shown elsewhere in this review (sections 3.1.1.3 and 3.4.3.8), SISFs are in fact a rather common feature in L 12 alloys strained at moderate and low temperatures. In most cases they can be regarded as deformation debris that are irrelevant to the rate controlling mechanisms. This holds true in Co3Ti since SISFs are generally located in an octahedral plane distinct from the primary octahedral slip plane (Oliver [147]). Properties of superdislocations in Co3Ti in domain A are further discussed in section 3.3. 3.1.2.6. Climb dissociation. The fact that in L12 alloys non-screw superdislocation segments are dissociated by climb after deformation above the peak temperature and sometimes below this temperature is a common observation (Veyssi~re et al. [170, 128, 171], Douin et al. [139], Korner unpublished result, Bontemps-Neveu [23], Sun et al. [123, 124], Mol6nat and Caillard [114], Morris et al. [13, 14], Takasugi and Yoshida [39]). Climb dissociation also appears to occur in a number of ordered alloys provided that superdislocations dissociate into two superpartials with collinear Burgers vectors and that deformation is conducted at sufficiently high temperature for diffusion to operate (see Veyssi~re [3-5], Veyssi~re and Douin [120]). Based on the fact that climb-dissociated superdislocations start to appear in samples deformed far below the peak (i.e., 350~ in Ni3A1, Veyssi~re et al. [140]) and following an early suggestion of Flinn [172], this locking mechanism has been incorrectly associated with the flow stress anomaly by Veyssi~re [128]. +___ Fig. 23 (continued). (d) to (f): By comparison with figures (a) to (c), this superdislocation has achieved a completely different motion, being essentially contained in the cross-slip octahedral plane. (d) 13D -- [121]' the superdislocation appears to be dissociated in the primary octahedral slip plane only in a limited number of segments indicated by the label 'p' (b). There is indication that the segment is an incomplete KW at 'ikw' (upon tilting, the segment remains parallel to the screw orientation and it shows its narrowest projection in the (101) orientation when the (010) plane is set on its edge). At 'kw', the variation of the dissociation width with tilting suggests that most of the APB is lying in the cube plane forming thus a nearly complete KW.
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Fig. 24. The dissociation mode of superdislocations in Co3Ti. (a) to (f): deformation at 4.2 K along []23], (h) and (i): room temperature in-situ deformation (courtesy of J. Oliver). (a) General view showing the segmentation of a superdislocation with [101] Burgers vector, along the screw as well as along a 60 ~ direction.
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Fig. 24 (continued). (b) to (g): Contrast and tilting experiments to show that the dissociation of the superdislocations occurs under mode I (note that the micrographs are not all presented under the same magnification). (b) and (c): The contrast of the pair of superpartials is reversed upon reversal of g, whereas if the dissociation occurred under mode II, the superpartial with Burgers vector bs = +~[211] should remain invisible (the letter D indicates dipolar filaments similar to those studied in figs 42 and 43). (d) Under g = 111 the superpartials are simultaneously out of contrast over the entire length of the superdislocation, with the exception of a few faulted defects (f) which are inclined to the (11 l) slip plane of the superdislocation. (e) to (g): Dependence of the dissociation with upon tilting: the screw parts consists in incomplete and complete KW locks.
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Fig. 24 (continued). (h) and (i): Weak-beam observation of superdislocations that stopped and were maintained under stress in the course of an in-situ straining test (tensile axis near [100]). Regardless whether they are stopped or they glide, all the mobile superdislocations are dissociated under mode I. The faulted features (f) that can be seen in the vicinity of the slip bands are sessile debris that are formed certainly as the result of the interaction of the superdislocation with the free surface as shown by the pair of superdislocations labelled f in the top center of figure (h). These faulted defects are left behind when superdislocations start to glide during the test, giving rise to triangular fauts and, less frequently, to more complicated arrangements such as (f~).
The idea that the locking of superdistocations by climb dissociation could be rate controlling (Flinn [172]) is inconsistent with macroscopic mechanical data (section 5.3) such as the insensitivities of the flow stress both to the strain-rate (section 2.2.8.1) and to static ageing (Davies and Stoloff [74]), as well as the temperature reversibility of the flow stress (section 2.2.4). In fact, Tounsi et al. [85] have shown that in polycrystalline Ni3Si deformed slightly below the peak temperature (420~ the dissociation mode is clearly a function of the cooling rate after the test. In samples cooled at a rate determined by the furnace inertia, mixed climb-dissociated superdislocations dominate, whereas in samples cooled in a matter of seconds down to room temperature the microstructure consists in superdislocations still mixed in character, but dissociated in their slip plane (Veyssi~re [5], see also [182]). This example of climb dissociation illustrates the fact that deformation microstructures
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Fig. 25. The room temperature deformation microstructure in Co3Ti deformed at about 10~ from [001], shows an overwhelming density of faulted defects superimposed and connected to a distribution of APB-coupled superdislocations. The debris nature of these SISF-containing defects follows from the fact that their octahedral habit plane is generally distinct from the primary octahedral slip plane of the parent APB-coupled superdislocations. These features result probably from the intersection of the different slip systems that operate under this particular orientation of the load. In this TEM experiment, the slip direction has been arbitrarily chosen as [101] instead of the usual [101] direction" the same feature labelled 'a' is pointed for the sake of comparison between the micrographs (courtesy of J. Oliver). (a) g is parallel to b, the surrounding super-Shockley partial are visible, the faults are out of contrast. (b) g = 111, the fault and the APB-coupled superdislocations are simultaneously in contrast. (c) g = 020, the APB-coupled superdislocations with [101] Burgers vector are out of contrast.
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may evolve before TEM observations, to quite unpredictable extents. Such difficulties can be circumvented by making use of in-situ straining experiments, unfortunately this latter technique cannot be regarded as a panacea either (section 3.5). The present discussion shows in addition that non-conservative aspects of the deformation microstructure in domain B are not elucidated yet. As a general conclusion for section 3.1, misinterpretations and errors may result from the tendency to fit unsufficiently criticized observations to clever schemes. Additional illustrations of this will be presented in the following.
3.2. Properties of planar defects In L12 alloys, ordering occurs as a first-order transition. Systems such as Ni3Al-based alloys do not however disorder until the melting point while others, such as Cu3Au or Fe3Ge, can be disordered below their melting temperature (TM). Since planar defects are, by definition, regions where crystal symmetry is disturbed, it is not surprising that properties of ordering and/or atomic segregation at interfaces, though reminiscent of the host lattice, could differ from those of the bulk. Planar defect energies have been and will probably remain the object of a vast experimental activity in L12 alloys. The interest of such measurements is at least twofold: (i) Planar defect energies are of primary importance in the debate on the locking properties of screw superdislocations in L12 alloys, as a part of the driving forces for cross-slip (section 4.1.2). For instance, there is still a number of unanswered questions as to the driving forces for cross-slip in alloys showing no domain B but a domain A. (ii) Planar defect energies can be regarded as parameters that are intrinsic to the crystal. Hence they can be confronted to the results derived from first-principles calculations. This has obvious applications regarding - the validity of potentials used in calculations, - the methods of calculations themselves, - fundamental properties such as the phenomenon of APB wetting and effects of relaxation and of composition. We now examine a few properties of planar defects. Extrapolated from the edge to the screw character, the core spreading of 89 superpartials is quite consistent with that observed directly under HREM (section 3.1.1.1). Hemker and Mills [50] have shown in addition that, upon slight addition of boron, the CSF energy increases substantially from 206 + 30 to 335-t-60 mJm -2, whereas the APB energy on the octahedral plane is unchanged at about 180 + 20 mJm -2. They conclude that the core-controlled origin of the observed strengthening by boron addition is however difficult to assess. As to APB energies, it should be noted first that the shear APB on the cube plane plays a central role in the L12 structure since, unlike all other APB configurations, there is no nearest neighbour violations (Flinn [172]). Vitek [ 1] has addressed the issue of APB stability in L12 alloys considering both crystal symmetry, which determines stationary
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surfaces, and "the overlap of atoms [...] using a hard sphere model". He concluded that the particular shear APB on the cube plane is stable, irrespective of nature of interatomic interactions, by contrast with other APB configurations that may become unstable. One should notice, however, that symmetry considerations can only prescribe equilibrium, not stability (F.R.N. Nabarro (1994): private communication). Using empirical pair potentials, it has been shown that considerable position relaxations could occur at APBs - again except for the shear APB on the cube plane - resulting in sometimes dramatic decrease of APB energies (Beauchamp et al. [173]). Regarding relaxation effects, Sanchez et al. [174] have found by CVM (Cluster Variation Method) calculations that, at 0.74TM, the concentration profile of a conservative APB on a { 111 } plane is not affected beyond the first neighbour atomic plane. As the temperature increases, a thick layer rich in the majority component develops up to the 6th neighbour at 0.98TM. The degree of segregation to the APB is strongly dependent on the alloy stoichiometry (3rd neighbour at 0.52TM). In view of its modest magnitude at low temperatures, this disordering effect should be unimportant in the mechanical properties in domain B. The phenomenon of APB wetting (Kikuchi and Cahn [175]) should be also mentioned, but since it occurs within a few degrees from the order-disorder transition, Tc, it can be regarded as marginal with respect to the main scope of this review. In the process of APB wetting an APB is coated by a distinct disordered phase with finite thickness (for a review of theoretical and experimental aspects of wetting, see Ducastelle et al. [ 125]). In the L12 structure, an APB is entirely defined only once its habit plane (normal N = [h k l]) and its displacement vector R are known. Douin et al. [129] have shown that an APB is fully characterized by the knowledge of N and of the parameter p --- 2 N . R. A value of p ~ 0 generates a non conservative APB which, in addition, violates crystal composition only if p is odd. Paidar [ 176] has predicted that within APBs formed non-conservatively by net transport of atoms, those where crystal composition is violated should be energetically the most favourable. Based on a weak-beam analysis of configurations adopted by climb dissociated superdislocations, Douin et al. [129] have evidenced experimental configurations that oppose Paidar's predictions. Dissociation widths of superdislocations split under mode I are highly compositiondependent, so are APB energies. As remarked by Dimiduk et al. [48], it is however very complicated to determine the role of ternary additions because of the dramatic effects of a deviation from the stoichiometric Ni/AI ratio. A compilation of selected APB energy determinations in a variety of alloys is given in table 3. Most of them should however be considered with some care since they have received no correction other than for elastic anisotropy. In addition t o - and may be in connection with -composition, the question of the temperature dependence of planar defect energy has been repeatedly addressed in L12 alloys. Results by different groups agree on the fact that in alloys showing a flow stress peak, this dependence is almost undetectable within domain B (Crimp [122], Korner [184], Veyssi~re et al. [ 177]). It seems however that this relative insensitivity to temperature is no longer true above the peak where a drop of the APB energy on the cube plane of about 30% has been reported in B-doped slightly Ni-rich polycrystalline Ni3A1 [177]. Crimp and Hazzledine [126] have reported a change in APB width of about 50% in Ni-rich Ni3A1 in the near vicinity of the peak. In fact, there is some indication
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Table 3 Energy values of planar defects in L12 alloys. In the references marked with a star, the values have been corrected based on image simulations (t assuming isotropic elasticity, # deformed at high temperature, ~t with elastic constants of Ni3(Ti, Si) z is the ratio of the APB energies, z -- 7111/~'~X)l). Alloy Ni3AI Ni3A1 Ni3A1 Ni3A1 Ni3A1
Composition (~ stoi-X)t (,.~ stoi-X) (24A1--0.24B-X) (24A1-0.24B-X) # (Ni-rich-X) #
7~x)l 140 90 126 72 75 (25#)
Ni3A1
(Al-rich-X)#
120 (100#)
Ni3A1 (22A1-S) Ni3A1 (22.9A1-S) Ni3A1 annealed lh 700~ Ni3A1 (24A1, S) Ni3A1 (24.2A1-S) Ni3A1 (25.9A1-S) Ni3AI (24.1A1-0.9Sn-S) Ni3AI (21. IAI-3.7Sn-S) Ni3A1 (23.4Al-I.0V-S) Ni3A1 (20.8A1-4.0V-S) Ni3AI (24.7AI-1Ta-S) Ni3A1 (24.7Al-lTa-S) Ni3A1 (24.7Al-lTa-S) Ni3A1 (24.7Al-lTa-S) Ni3A1 (17.4AI-6.2Ti-S) Ni3A1 (24A1-0.25Hf, S) Ni3A1 (24AI-2Hf, S) Ni3AI (23.8A1-0.7B, S) Ni3Ga Ni3Ga Ni3Si (23.1Si-X) ~t Ni3Si (24.5Si-X) ~t Ni3Si (10.9Si-10.7Ti-S)
104 104 92 122 170 129 146 155 201 155 200 225 200 250 120 170 17# observed 220 250 124
3'111 180 110 -
z 1.03-1.56 0.9-1.47 -
"TcsF
175 170 123 180 163 190 166 174 192 198 165 237 250 237 250 160 190 173 1I0 observed 220 250
1.34-2.13 1.34-1.96 1.05-1.65
235 4- 40
Ni3Si
(10.9Si-10.7Ti-S) #
77
Ni3Fe
(S)
55
93
Cu3Au
(S)
-
39
Co3Ti Co3 Ti
(23Ti-3Ni-S) (22Ti-SX)
210 130
270 155
206 -1- 30 1.06-1.65 0.86--1.44 1.04-1.57 0.95-1.48 0.96-1.58 0.71-1.35 1.03-1.36 1-1.04 0.88-1.4 0.92-1.54 0.8-1.25 1-1.6 1-1.6
185 4- 20 300 4- 40
335 4- 60 0.75-1.33 0.8-1.25
1.7
1-1.7 1-1.8
that the effect o f t e m p e r a t u r e is c o m p o s i t i o n - d e p e n d e n t :
Reference Veyssi6re et al. [ 140] Douin et al. [139] Veyssi~re et al. [ 177] Veyssi~re et al. [ 177] Douin and Veyssi~re [132] Douin and Veyssi6re [132] Karnthaler et al. [278] Dimiduk [49] Dimiduk [49] Hemker and Mills [50]* Dimiduk [49] Dimiduk [49] Dimiduk [49] Dimiduk [49] Dimiduk [49] Dimiduk [49] Baluc et al. [ 179] Baluc [135] Baluc et al. [143] Baluc et al. [142]* Korner [ 180] Bontemps-Neveu [23] Bontemps-Neveu [23] Hemker and Mills [50]* Suzuki et al. [ 181 ] Sun [62] Tounsi [ 182] Tounsi [ 182] Yoshida and Takasugi [183] Yoshida and Takasugi [183] Korner and Kamthaler [145] Sastry and Ramaswami [151] Oliver [ 147] Oliver [ 147]
a m a r k e d d e c r e a s e in A P B
e n e r g y has b e e n f o u n d in N i - r i c h binary Ni3A1 alloys w h e r e , in addition, A P B s s h o w r e s i d u a l c o n t r a s t u n d e r f u n d a m e n t a l reflections, i n d i c a t i v e o f local r e l a x a t i o n s ( D o u i n and Veyssi~re [132]). In Ni3(Si, Ti), t e m p e r a t u r e affects A P B e n e r g y a b o v e the p e a k t e m p e r a t u r e ( Y o s h i d a and T a k a s u g i [ 183]).
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The influence of temperature on APB energy has also been studied in great detail in Ni3Fe which, unlike Ni3Al-based alloys, undergoes an order-disorder transition at a temperature Tc (776 K) that is lower than its melting point. In addition, Ni3Fe does not exhibit a flow stress anomaly (Korner [153]). It is found that the APB energy decreases dramatically within 100 K of Tc (Korner and Karnthaler [145], Korner et al. [152, 153, 185], Schoeck and Korner [186]. This has been attributed to preferential disordering at the APB. Atomic shuffling occurs very rapidly and superpartials are seen to move apart at a rate controlled by atom diffusion onto the APB [186]. Ni3Fe shows some similarities with L12-stabilized AI3Ti alloys. For a variety of stabilizing ternary additions in the latter alloy, significant structural modifications are observed to take place in a matter of minutes at dislocation cores, causing the superdislocations to widen dramatically (Morris et al. [13, 14]). Additional evidence of structural changes at APBs can be found in rapidly solidified Ni3Al-based alloys. The enhanced visibility of APBs after rapid quenching- in some systems, APBs remain under strong contrast even under fundamental reflections - has been discussed quantitatively by Lasalmonie et al. [187] and interpreted in terms of a chemical relaxation.
3.3. T h e d e f o r m a t i o n
microstructure
in d o m a i n A - r a t e - c o n t r o l l i n g m e c h a n i s m s
Only a very small fraction of the L 12 alloys that exhibit a domain A have been examined under TEM and this is why the discussion of the low-temperature negative TDFS is limited to this section. So far the only documented cases are Co3Ti, Fe3Ge together with members of the family of L12-stabilized A13Ti and, to a very limited extent, with Ni3A1. The negative TDFS of L12 alloys has been modelled by Tichy et al. [159], based upon atomistic simulations of dislocation cores [158]. The model predicts octahedral slip for compression axes located on the [001] side of the triangle, while {001} slip should occur on the other side of the triangle; the location of the boundary between these domains should be temperature-dependent. The model relies on the fact that, in L12 alloys where the APB on the octahedral plane is unstable, dislocations experience strong Peierls-Nabarro forces, regardless whether slip occurs on the octahedral or on 1 the cube plane. In the latter case, dissociation occurs into two ~(110) superpartials bordering an APB (mode I) and the Peierls forces originate from the core spreading of these in two octahedral planes. In the former case, (110) superdislocations split into two 1 7(112) superpartials with a SISF in between (mode II); the latter superpartials are also intrinsically sessile (section 3.1.13). By analogy with the work of Paidar et al. [86], it may be anticipated that since the rate-controlling core configurations are spread off the slip plane, be it cubic or octahedral, then the CRSS must be somewhat orientation-dependent. Some orientation dependence of cube slip has been indeed reported by Heredia [37] in Ni3Al-based alloys deformed at intermediate and elevated temperatures (see however section 2.2.10). Regarding deformation in domain A, it results from a set of experiments, however limited to two neighbouring orientations, that the CRSS on the cube plane is nearly orientation insensitive. Tichy et al. [ 158] have in addition predicted a tension-compression asymmetry for
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{ 111 } slip which has not been checked experimentally yet, since mechanical tests have been conducted only in compression. Electron microscopists have attempted to identify the dissociation mode II and some have claimed evidence for its occurrence. The situation is in fact much more confused than it appears from a first inspection of the published literature. Fe3Ge is the simplest case to start with since in this alloy octahedral slip is intrinsically inhibited, forcing cube slip to occur at temperatures where octahedral slip would normally operate in other L 12 alloys. The negative TDFS thus takes its origin in the marked thermal activation of cube slip. The reason for the difficulty of octahedral slip in this alloy is ascribed to the fact that an APB on { 111 } would be only "marginally stable", implying that (110} superdislocations slip as undissociated in this plane (Ngan et al. [31]). Two characteristics of the deformation microstructure are worth mentioning (i) Fe3Ge exhibits extensive grown-in stacking faults on { 111} habit planes with 1 .~(112} displacement vectors, suggesting relatively low SSF energy. The as-annealed microstructure also contains a small amount of thermal APBs lying on cube habit planes. The reason why deformation proceeds by cube slip by APB-coupled superdislocations rather than by octahedral slip in mode II, can be reasonably ascribed to the poor mobility 1 of .~(112} superpartials [31]. (ii) FeaGe contains "APB tubes" [188], that is, defects elongated in (110) direction, showing a faint contrast very similar to that previously reported by Sun [ 189] in a study of Ni3A1 (section 3.4.3.6). The apparent inconsistency in Fe3Ge between the "marginal stability" of APBs on octahedral planes and the presence of APB tubes, which has been pointed out by Ngan et al. [ 188], is a question that will not be discussed in the present review, but that certainly deserves further attention. By contrast with FeaGe, deformation in Co3Ti proceeds almost exclusively on octahedral planes over domains A and B. According to Liu et al. [25, 167], the glissile configuration in the octahedral plane would, in domain B, be the APB-coupled one (mode I), whilst the most energetically favourable though sessile configuration would be SISF-coupled (mode II). Assuming that the transition from the sessile mode II to the glissile mode I can be thermally activated, the negative TDFS would be due to the increasing difficulty to activate this transition from mode II to mode I under stress as temperature is decreased. In order to explain the SISF-related contrast that has been reported in post-mortem foils (section 3.1.2.5), the stress-induced I ~ II transition is in addition considered as fully reversible upon the release of the applied stress, to the extent where every superdislocation segment, irrespective of its character, is transformed as the stress is released. This seems quite improbable. In the hypothesis of Liu et al. [25, 167], domain B would take place at temperatures at which the transition from mode II to mode I is athermal. The interpretation of the positive TDFS is, as in other L12 alloys, in terms of the locking mechanism into the KW configuration [ 167]. So far, dislocation properties in domain A have been insufficiently studied by transmission electron microscopy. TEM information on the actual dissociation mode of superdislocations would be particularly crucial in Pt3AI in order to directly test the atomistic modelling of Tichy et al. [ 159].
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3.4. T h e d e f o r m a t i o n m i c r o s t r u c t u r e in d o m a i n B
In this section, we consider successively the characteristics of dislocation organization derived from the observation of slip lines (section 3.4.1) and of thin foils (section 3.4.2); then we examine the properties of selected features chosen for their possible relevance to the mechanisms of deformation. 3.4.1. Slip lines Whereas it is ascertained that primary slip is octahedral in nature, the geometry of slip is not completely trivial. The reasons for this are twofold: - additional systems may operate under certain orientations of the external load (see
section 2.2.10 on the evidence of cube slip in domain B). - slip may be very heterogeneous. Though extremely important and somehow evident from slip trace analyses, this latter aspect is rarely addressed in TEM studies. A striking illustration of this property has been recently presented by Korner [ 134] (section 3.4.2, see fig. 28). The evolution of the fine structure of slip traces has not been the object of a detailed study. It seems nevertheless established that slip lines are long, coarse and straight at temperatures lower than 0.6-0.8Tp and that they become fainter, shorter and eventually wavy when temperature is raised above these values; this latter property may however be partly accentuated by the thermal degradation of sample surfaces. There is no evidence so far of pencil glide similar to that found in b.c.c, crystals, where cross slip may occur so extensively that slip lines do not coincide with crystallographic planes. Octahedral slip dominates the whole temperature interval of domain B. More than one octahedral slip plane are often activated. Octahedral cross slip is rarely reported, though. TEM evidence of the latter slip system is in fact available in NiaAl-based alloys deformed below 77 K, but its operation is limited to short distances (Korner [116], Korner and Veyssibre [155]). Its occurrence has been suggested by some in-situ deformation tests, but it is then restricted to changes in dissociation modes (Caillard et al. [190]; see however section 3.4.3.2); the cross-slip distances are modest, e.g., of the order of the superpartial separation, hence consistent with the fact that they are not observed within the slip traces, at low and moderate temperatures. With regards to cube slip, it is important to distinguish between the primary cube slip plane and the cube cross-slip plane of the primary octahedral plane, which can be both activated to such an extent that they can be detected macroscopically. In samples loaded along [114] and at temperatures significantly below the peak, Thornton et al. [41] were the first to report extensive activity of cube slip within domain B, both on the primary and on the cube cross-slip planes. Lall et al. [28] have found that cross-slip on the cube plane increases as the test temperature approaches the peak one, while Bontemps-Neveu [23] has reported that, for load orientations located near [123] and [111], primary cube slip can be activated at 250 K below the peak temperature (section 2.2.10). On the other hand, Staton-Bevan and Rawlings [89] have claimed that cube slip does not operate at all within domain B. At the TEM scale, there is evidence of a cube slip (section 3.4.3.5). In-situ experiments have indeed revealed the operation of a primary cube slip at intermediate m
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temperatures (C16ment et al. [112, 113], Mol6nat and Caillard [114]), from which it was claimed that this slip system also exhibits a flow stress peak (see however section 2.2.10). Similarly, deformation on cube planes has been observed in post-mortem of Ni3(A1, Ti) samples, in the first consistent TEM investigation of deformation microstructures by Thornton et al. [41] and in a more recent study of Korner [180]. In the former, cube slip was identified after a deformation of 4% at 400~ (i.e., within domain B2), whereas Korner has given evidence of cube slip after deformation along [111] to 9% at room temperature. 3.4.2. D i s l o c a t i o n organization
The literature on observations of deformation microstructures is abundant in L 12 alloys. An unambiguous interpretation of TEM observations needs some precautions since the volume of material strained in a mechanical test is of the order of 100 mm 3, while the maximum volume of thin foils that can be prepared out of it7 is at most of 10 -5 mm 3. In addition, total dislocation lengths of the order of a few hundreds of ~tm are currently analyzed by TEM (orientation, Burgers vector, dissociation mode), as opposed to the several tens of kilometers of dislocation line contained in one deformed sample. On the other hand, regarding the characterization of collective effects within the slip plane, which will be shown in the following to be critical in L12 alloys, it is only when foils are sectioned very accurately parallel to a given slip plane that one can obtain a good representation of dislocation organization in this plane, otherwise, one favours dislocation directions near the direction of intersection between the slip plane and the plane of the thin foil. Despite the experimental difficulties, our perception of deformation microstructures in post-mortem L12 specimens may be considered as reasonably comprehensive, as indicated by the consistency between the conclusions of several research groups on the organization of dislocations and on its evolution with temperature. From very low temperatures to well above the peak, deformation is achieved by (110) superdislocations. Dissociation occurs essentially under mode I but there are quite a number of situations where reactions yield defects containing stacking faults (section 3.4.3.8). (100) dislocations appear rather frequently as junctions between the above (110) superdislocations (Veyssi~re and Douin [191]). It has been claimed that (100) is the main direction of deformation at elevated temperatures, that is, at and above the high-temperature peak of WHR (Sun and Hazzledine [154]). This range of deformation temperature has not been sufficiently investigated and there is nothing clearly established on this subject. Observations by different research groups of the temperature dependence of the deformation microstructure conform quite consistently to the trends outlined by Korner [134, 192]. Some differences may be noted from one work to the other; they arise from changes in experimental conditions such as sample composition, load orientation, total deformation, 8 heterogeneity of deformation, choice of thin foil orientation and, rather frequently, from the investigator's subjectivity. 7The total volumeof thin foils that have been ever inspected by TEM over the last 35 years of investigations on mechanical properties of L I2 alloys can be estimated to a few tenths of mm3, that is, only a fraction of one deformed sample. STEM studies of deformation microstructure are thus very difficult to compare when conducted in samples deformed under too different experimental conditions. The examples given in this review from Shi's work [283] were all taken from samples deformed to about 3%. In [134], the analysis of the deformation microstructure
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Important and/or original features deformation-wise will be analyzed in detail in the next sections. The most important properties can be listed as follows: (i) At low temperatures, ternary Ni3A1 single crystals deformed to several percent of permanent strain contain a significant density of mixed segments (Mulford and Pope [22], Saburi et al. [83], Korner [116], Karnthaler et al. [193], Jumojni et al. [73]), consisting mainly of dipoles and bundles of dipoles. A similar observation has been reported by Pak et al. [ 117] in Ni3Ge single crystals, with some indication of a preferred elongation along (110) directions. It is important to realize that differences from one work to another may just result from experimental conditions; in fact, at low temperatures the larger the permanent strain, the less prominent the elongated screw segments in the microstructure (see footnote 8). Screw superdislocations may be dissociated on the crossslip octahedral planes [116, 134], on the cube cross-slip plane forming the well-known Kear-Wilsdorf configuration (section 3.4.3.1) or else they may straddle the primary octahedral and the cube plane, forming incomplete KW locks (section 3.4.3.2). There is indication that the density of Kear-Wilsdorf locks is decreased when the orientation of the load is such that the shear stress in the cube cross-slip plane is decreased; due to strain heterogeneities, this is of course subjected to significant variations in the same sample. Very elongated KWs have been observed in Ni3(Si, Ti) after deformation to 1% at 4.2 K (Takasugi and Yoshida [29]). Interestingly enough in these observations, the microstructure is sufficiently distinct from others conducted at higher temperatures in the same alloy to indicate that it stems from the deformation at 4.2 K 9. (ii) Still at low temperatures but in samples deformed to modest permanent strains, dislocations elongated along (110) directions at 60 ~ from the screw orientations are quite frequent, yet less numerous than the straight screw segments (section 3.4.3.10); the 60~ disappear gradually as the temperature is raised up to, or a little above, room temperature. (iii) The microstructure of deformation at room temperature contains superdislocation dipoles whose distribution reminds that of macrokinks within superdislocations (fig. 26, 27). (iv) As deformation temperature is raised in the domain of positive TDFS, KWs tend to be increasingly bent in the cube plane in which they are spread (section 3.4.3.5). Nevertheless, as made clear by slip trace analysis, glide in the cube cross-slip plane contributes very little to the total strain (section 2.2.10). The importance given to the bending in the cube cross-slip plane varies from one author to the next: Korner [192] has been conducted on samples deformed to about 10%. The permanent strain is 4% in the work of Karnthaler et al. [193] except in the [123] orientation at room temperature for which strain was 12%. Total strain is more modest in the experiments of Dimiduk and Parthasarathy [80] and of Takasugi and Yoshida [29] (about 1% in both cases). In these conditions, it is extremely difficult to claim a one-to-one correspondence between the general morphology and the organization of superdislocations and the deformation temperature, especially when it is realized that, as indicated by the distribution of slip lines, significant strain heterogeneities may have taken place in a sample. N13($1, Ti) shows a pronounced positive TDFS starting at 4.2 K (section 2.2). Its deformation microstructure is somewhat different from the above trends in that (i) it consists of elongated screw superdislocations throughout domain B, that is, from 4.2 K to 600-800 K (the peak temperature depends upon load orientation) and that (ii) from the examples shown by Yoshida and Takasugi [29, 150], this particular property of screws is almost insensitive to temperature, at least relative to other L12 alloys. 9
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Fig. 26. Low magnification bright-field view of the organization of superdislocations in Ni3(A1, Hf), deformed at room temperature (compression along [123] to a permanent strain of 2%). The foil was sliced parallel to the primary (111) slip plane. The microstructure consists of kinked superdislocations elongated in the screw orientation (s) together with a number of long (d) and short (arrowheads) superdislocation dipoles. The screw segments are in the form of incomplete and complete KW locks. Here the mean distance between consecutive KW locks perpendicular to the screw orientation, is about 2 ~tm but it may be more in other deformed areas of the same sample. In the presence of a strong long-range obstacle, such as a subgrain boundary, mixed segments together with accompanying debris tend to accumulate and to dominate the microstructure, otherwise they would be eliminated during deformation. w
Fig. 27. Low magnification dark-field view of another typical aspect of the organization of superdislocations in Ni3(Al, Hf)(same deformation conditions as in fig. 26). The length of long superdislocation dipoles (d) is about the same as that of single superdislocations (s). On the latter, evidence of kinking including APB jumps (section 3.4.3.4)can be found; some 60' superdislocations are stabilized. Finally, note the presence of short superdislocation dipoles (cd) in large quantities under the form either of double or of single kinks.
W N W
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insists markedly on this property while Dimiduk [49] claims that cube bending, if any, is moderate (see also section 3.4.3.5). (v) Important enough in the behaviour of superdislocations is the fact that, in domain B, friction is larger on the cube plane than on the octahedral plane (sections 2.2.10 and 4.2.1.2). Dislocations of the primary cube slip are activated at temperatures already below the peak temperature. Their density increases markedly as temperature is increased above Tp, consistent with the facts that dislocation mobility in the cube plane increases with temperature and that it becomes comparable to that in the octahedral plane at temperatures close to the peak temperature Tp and above. (vi) A little above the peak, cube slip replaces octahedral slip except for load orientations located in the near vicinity of [001]. (vii) Diffusion-assisted phenomena have been reported below Tp, but they are in general not considered to be rate-controlling. For example, after creep above Tp, Schneibel and Horton [109] have not observed any evidence of subgrain formation, which they ascribe to the moderate role of non-conservative processes. This has been confirmed by Nathal et al. [194]. In brief, the presence of segmented superdislocations, elongated essentially along the screw direction, can be regarded as the dominant feature in the deformation microstructure in domain B. This segmentation attests to a somewhat sessile energetically favoured configuration in the screw orientation (fig. 26). It should be realized that observations cannot be taken too deterministically since not only does the frequency of alignment in the screw direction depend critically on test temperature and orientation but, in view of deformation heterogeneities, it depends also on the amount of strain that the sample has undergone locally. Rectilinearity in the screw direction is itself affected by the test temperature and load orientation (see fig. 39). As to properties of individual superdislocations, it is worth recalling also that depending on deformation temperature, superdislocations may be elongated in other crystallographic orientations. For instance, in the vicinity of the peak temperature and above (domain C), there is a composition-dependent tendency for superdislocations to be locked in the edge orientation (section 3.1.1.2), whereas the low-temperature substructure contains a fraction of superdislocations elongated in the 60~ of the octahedral plane (section 3.4.3.10). Information on dislocation densities is scarce, although it is certainly a critical issue. Korner [ 134] has shown that after a deformation at room temperature, at a relatively large permanent strain (about 10%), foils sliced parallel to the cube cross-slip plane exhibit a succession of bands, 5 to 7 ~tm thick, where deformation has taken place alternately on the cube plane and on the primary octahedral plane (fig. 28). If this microstructural finding +___ Fig. 28. Ni3(A1, Ti) deformed to 10% in compression nearly along [356] at room temperature. The foil was sliced parallel to the cube cross-slip plane. The deformation microstructure is heterogeneous, consisting in a succession of layers, 5 to 7 ~m thick. Two distinct mechanisms of deformation have taken place as indicated here by striking differences in dominating dislocation characters. In the top layer, the microstructure is dominated by mixed dislocations that clearly expand in the plane of the foil e.g. (010). Next to it, the microstructure consists of kinked screw segments elongated in the [101] direction, together with short debris. This is exactly how a section of the primary octahedral slip plane looks like when cut parallel to the (010) plane. It can be remarked in addition that it is only in the latter layer that evidence of APB tube contrast can be found under the form of faint striations oriented parallel to the [101] Burgers vector (courtesy of A. Korner).
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were applicable to lesser strains, it would have evident implications on the manner one should envision the work-hardening processes and perhaps the yielding in Ni3A1 alloys. It is then clear that much more work is needed in this particular direction (recent work on dislocation densities can be found in [69, 283]). Additional information on dislocation distribution in Ni3Al-based L12 alloys has been derived from creep experiments (Hemker et al. [54]). At high temperatures (about 1250 K), when deformation is dominated by cube slip, the dislocations are homogeneously distributed throughout the sample (Nathal et al. [194], Hemker [110]). At lower temperatures and higher stresses, Schneibel and Horton [ 109] together with Hemker [ 110] report that the distribution of dislocations is significantly heterogeneous, but Hemker et al. [79] indicate that the dislocations are in fact homogeneously distributed in the regions where they are found.
3.4.3. Specific configurations Deformation in domain B is accompanied by a wealth of specific features. It is not sure that these are all identified yet. 3.4.3.1. Kear-Wilsdorfconfigurations. The basic ingredients of the strengthening properties of L12 were set up in the early 60's. On the one hand, based on a crude theoretical estimate of APB energies in the octahedral and on the cube planes, Flinn [172] has pointed out that a dissociated superdislocation can reduce its energy by changing its habit plane, in particular from an octahedral to a cube plane. On the other hand, the properties of sessile screw segments are contained, to a very large extent, in the classical paper of Kear and Wilsdorf [63]. In view of the rudimentary TEM methods and instruments that were available at that time, the interpretation of their observations can be regarded as remarkably perspicacious "... since the [cube cross slip plane] happens to be normal to the foil, it is clear that cross slip of screw components of superlattice dislocations from (111) to (010) can account for the fact that only the short edge segments remaining in the slip plane are resolved as a pair while the long screws are not. This explanation also accounts for the straightness of the long screw segments since following cross slip into (010) the screw dislocation pair would become relatively immobile". Figure 29 is actually derived from Kear and Wilsdorf's representation of the cross-slip transformation of a screw APB-coupled superdislocation, dissociated in the primary octahedral plane in its non-screw part, into a Kear-Wilsdorf lock. 3.4.3.2. Incomplete Kear-Wilsdorf configurations. The successive steps of the transformation that yields a KW lock are schematized in fig. 30. Within the screw portion of an expanding loop, the simplest way to visualize the transformation is to consider that the leading screw superpartial forms locally a double kink (CJ-CJ) in the cube cross-slip plane (fig. 30(a)). Quantitative details on the forces involved in that process are given in section 4.2.2. Once nucleated, the leading superpartial can further expand by lateral motion of each CJ in opposite directions along the screw direction; simultaneously, the screw part may advance in the cube plane. As the APB in the cube plane widens, that located in the octahedral plane narrows (fig. 30(b)); this is because octahedral slip shows
Microscopy and plasticity of the L12 ~ phase
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almost no friction so that the trailing superpartial can equilibrate its position freely upon any change in position of the leading (section 4.2.2.1). The transformed screw segment lengthens as the terminating mixed portions (macrokinks (MKs), see section 3.4.3.3) keep moving in the octahedral plane (fig. 30(c)) in which they are dissociated. The widening of the APB in the cube plane may occur by sudden local jumps, by double kink nucleation and further zipping (fig. 30(d)). The transformation proceeds until the trailing partial reaches the common intersection of the two slip planes (fig. 30(e)), at which point the transformation into a KW configuration is complete. However, because lattice friction in the cube plane is significant in domain B, the transformation may not occur at the same rate for both superpartials. Differences are accentuated by the fact that friction on mixed segments is not negligible, which may prevent the double kinks from propagating homogeneously along the screw segment (fig. 30(d)-(e)). Consistent with the nature of cube slip (section 4.2.1.2), the lower the test temperature, the greater the probability for incompletely formed KWs to be left after deformation. The observation of a significant fraction of incomplete KWs is made possible because, at room temperature which is the temperature of storage and of TEM observations, the kinetics of glide on the cube plane is slow, thus the transformation rate is reduced (the absence of incomplete KWs after deformation at some intermediate temperatures may originate from the fact that samples had not been cooled sufficiently rapidly after deformation and that there was enough time for the screw segments to adopt the complete KW, which is their equilibrium configuration (section 4.2.2.2)). Experimentally, the non-planar nature of dissociation can be identified only by tilting the foil about the screw orientation, in keeping the weak-beam conditions comparable from one orientation to the next, until the images of the companion superpartials are seen edge-on (fig. 31). The fact that, due to unfavourable structure factors, APBs are in practice impossible to image in Ni3A1, prevents one from determining whether the APB is planar or consists in one or several steps (section 4.2.2.3). Hence, one cannot obtain precise experimental evidence on the trajectory of the leading superpartial on the cube and octahedral planes during the cross-slip transformation in Ni3Al-based alloys
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@
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Fig. 30. Transformation of a screw superdislocation by cross slip in the cube plane (see text). At each step, the superpartial positions in the previous step are superimposed as grey dotted lines. CJ and MK stand for the jog that closes the KW in the cube plane and for the macrokink, respectively. IKW and KW stand for incomplete and complete Kear-Wilsdorf, respectively. For the sake of clarity, the progression of the leading superpartial in the cube plane is represented as if it occurred by nucleation of CJ pairs.
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Fig. 31. Details of the dissociation of a superdislocation in Ni3(AI, Hf) deformed at room temperature along [123] to a permanent strain of 2% (courtesy of C. Bontemps-Neveu). The superdislocation consists of straight screw parts interrupted by kinks at A and C. (a) Foil section parallel to the octahedral cross-slip plane /3D = [111], such segments as those indicated by large white arrowheads are seen edge-on. These segments are the incompletely formed KW locks. (b) Foil section perpendicular to the cube cross-slip plane/3D = [101], the segments that were edge-on in the previous micrograph are now clearly resolved, while others (indicated by large white arrowheads) exhibit a single contrast, indicating that they are under complete KW configuration or close to it. Note the markedly cusped aspect of macrokink A. (c) Foil section parallel to the primary octahedral slip plane /3D = [111], the screw segments that showed different separations in the other foil orientations exhibit about the same width (a and b spot elementary kinks that result from the process of APB jump). m
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(the e x p e r i m e n t should h o w e v e r be feasible and w o r t h w i l e doing in alloys with m o r e f a v o u r a b l e structure factors). F u r t h e r m o r e , the habit plane of the c o m p a n i o n superpartials c a n n o t be d e t e r m i n e d with great accuracy f r o m the above tilting e x p e r i m e n t . In fact, the i m a g e width of a superpartial is a significant fraction of the A P B width ( l n m and a few nm, respectively), introducing s o m e uncertainty on the foil orientation at w h i c h the e d g e - o n condition is satisfied. In particular it cannot be ascertained that a screw s e g m e n t
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lies exclusively in the cube or in the octahedral plane, since a deviation of one of its superpartials over up to 4 to 5b in the other plane would be hardly detectable under weak-beam conditions. Incomplete KWs have been first identified in Ni3(A1, Hf) by Bontemps and Veyssi~re [195] and their existence confirmed in the same alloy by Mol6nat and Caillard [196], in Ni3(A1, Ti) by Korner [197] and in Ni3(Si, Ti) - d e f o r m e d at relatively high temperatures- by Yoshida and Takasugi [ 150]. The elongated KWs which are reported in Ni3(Si, Ti) after deformation at 4.2 K as being fully contained in the primary octahedral plane may in fact consist in incomplete KWs whose extension is limited to a few interatomic distances, that is, within experimental uncertainties. Indirect evidence of the dynamical formation of incomplete KWs is provided by the observation of APB jumps (section 3.4.3.4). With regards to the scheme of fig. 32, during a sequence of APB jumps the extent of APB lying on the cube plane should be minor compared to that on the octahedral plane 1~ These remarks enable us to re-visit the first weak-beam in-situ observation of dislocation motion in Ni3A1 by Caillard et al. [190]. Observing sudden changes in projected dissociation widths, these authors have concluded to the reversible change of the APB plane from (111) to (111), ignoring the possibility that the superdislocation might have assumed an incomplete KW configuration which, in the conditions of projection of this experiment, happens to be undetectable. In view of what we know now on incomplete KWs (section 3.2.3.1) and APB jumps (section 4.2.3.2), it might be that the asserted cross-slip step on (111) plane corresponds in fact to the narrowing of the APB in the primary octahedral plane, while the leading superpartial is under transit in the cube plane. Further in-situ observations have been produced since by Mol6nat and Caillard [196] in samples stressed differently, in which now cross-slip path through the (010) plane is invoked (see however Chou and Hirsch [199]). w
3.4.3.3. Kinks on K W configurations. In the pioneering work of Kear and Wilsdorf [63], the following excerpt: "... Certain of the long, straight dislocation segments in (fig. 6) can be seen to be connected to one another in a stepwise manner. This observation is again consistent with a model of cross slip occurring at different positions on an expanding dislocation loop .... " contains implicitly the idea that kinks, which, following Dimiduk [46], we shall call macrokinks (MKs), could be left in the primary slip plane between successive screw segments deviated on the cube plane. Macrokinks on superdislocations have been identified by Sun and Hazzledine [149] within the deformation microstructure of a 7/')" superalloy. It is important to distinguish between two categories of MKs, that is, "simple" and "switch-over" MKs (fig. 33) for they result from two distinct mechanisms of formation whereby the relative positions of the companion superpartials are conserved or reversed
1~ should be noted that this is, however, geometrically inconsistent with the results of post-mortem observations in which locked screw segments containing small kinks indicative of APB jumps (section 3.4.3.4) are sitting to their greatest extent on the cube plane (see for instance fig. 4(f) in [72]. This implies either that an APB jump happens to involve a distance longer than the few b's expected in the cube plane or, more likely, that the screw superdislocation has relaxed gradually towards the complete KW after it is locked in between two consecutive elementary kinks (EKs), for a definition see section 3.4.3.4).
Microscopy and plasticity of the L12 7' phase
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(OLO) Fig. 32. Sketch of APB jump. Once cross slip is nucleated (a), instead of proceeding in the cube plane until a complete KW lock is formed (fig. 30), the leading (L) superpartial may double cross slip via the cube plane back to a primary octahedral plane where it glides more easily (b). Then the trailing (T) superpartial reaches the point where it will have to slip in the cube plane under the driving force of the APB surface tension and shear stress, L becomes immobilized at an equilibrium position under stress (c). If again L cross-slips on the cube plane before T has erased the APB on the cube plane (d), then the conditions for a second APB jump are met (e) and (f). The height of the kink that sits in between the two equivalent incomplete KWs scales with the APB width Ao in the octahedral plane (APB jump). It can be noted that the locking of the screw superdislocation is ensured by the leading superpartial, while its unlocking is controlled by the trailing superpartial.
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T~ 2 MK
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Fig. 33. Comparison between a simple and a switch-over MK. The companion superpartials 1 and 2 are represented by different line thicknesses. On the left-hand side of the switch-over MK, superpartial 1 has crossslipped first in the cube plane, whereas this superpartial becomes trailing on the other side of the same MK. Superpartial 2 remains in the leading position on both side of the simple MK. The connection between two consecutive KW locks joined by a switch-over MK is sketched in fig. 62(c). CJ is a geometrically necessary short length of dislocation that connects the two consecutive branches of the same superpartial, when noncoplanar. CJ is neither a kink, for it does not slip in the primary octahedral slip plane, nor a jog (with respect to the screw part in the cube plane). Following Hirth and Lothe [288], one may evoke here a jink or a kog.
(Bontemps and Veyssi~re [ 195]). This point will be discussed in detail in section 4.2.3.2. Both categories have been identified within the deformation microstructure of L 12 alloys. A simple macrokink connects two KWs which are spread in opposite directions with respect to the octahedral plane of the kink [149] (fig. 33). The same superpartial is leading (or trailing) on both KWs. At each end of a switch-over MK, the KW segments have formed on the same side of the MK plane (Mills et al. [200], Tounsi et al. [201]) (fig. 33). The leading/trailing role of a superpartial is interchanged from one KW to the next. The first experimental identification of MKs [149] has been analyzed in terms of the simple kink configuration. MKs have been studied in detail by Bontemps and Veyssi~re [195] and it was concluded that the switch-over nature of MKs dominates the microstructure after deformation at room temperature (fig. 34). Couret et al. [72] have made the first systematic study of the population of MKs and have reported that about 25% of them are in fact switch-over in nature and that no variation of this proportion has been detected with temperature. This is however the only study available so far on this important matter which deserves ampler investigation. --+ Fig. 34. Stereographic TEM analyses of macrokinks (these observations can be better understood by comparison with fig. 33) (courtesy of C. Bontemps-Neveu). (a) and (b): B D -- [121] in both micrographs. The superpartial images intersect at one extremity of the kink just as in the left-hand side of the switch-over MK of fig. 33 (Ni3(A1, Hf) compressed along [123] at room temperature). (c) and (d): By comparison between images taken under g and - g , it can be checked that the superpartials never intersect in projection along the [ 111 ] beam direction as in the simple macrokink on the right-hand side of fig. 33 (there is a mistake in the labelling since only g, not b, is reversed between (c) and (d)). Note in addition, that this pair of images provides a further example of the reversal of the image asymmetry upon reversal of g (see fig. 21 and corresponding text). (Ni3(AI, Hf) compressed along [123] at 150~ (e) to (g): A switch-over MK imaged symmetrically with respect to the (010) orientation. (e) B D -- [121], the kink height together with its splitting width are the largest. (f) B D = [010], the dissociation width in the screw orientation is at a maximum, that of the MKs has decreased. (g) B D -- [121], the MKs are seen edge-on while the KW segment is still clearly dissociated. m
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15
Kink height (L o)
Fig. 35. Distribution of kink heights measured in units of APB widths in the octahedral plane, Ao. (a) Distribution of kinks at room temperature, comparison between Ni3(A1, Hf) (Shi [283]) and Ni3Ga (after Couret et al. [72]) for about the same total number of kinks. (b) Semi-log representation of (a); note the uncertainty on the determination of an average slope in both distributions. In Ni3(AI, Hf) the number of kinks that are smaller than one APB jump, deviates significantly from the average line (according to the analytical development of section 4.2.2.2, these kinks are produced after a jump in the cube plane /Jcump that is somewhat larger than the average jump at this temperature, a situation which results in a less value of lot). In addition to the deviant point for a kink height of about one APB jump, one can notice a deviation for a MK height that scales with about two APB jumps. (c) Comparison between the MK distributions in Ni3Ga at 20~ 200~ and 400~ (after Couret et al. [72, 203]). Note however that the increase in slope with increasing temperature is actually very modest and possibly not significant.
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After a KW is formed, as the mixed superdislocation keeps slipping in the octahedral plane, it may bypass the closing kink CJ and further expand (see fig. 65). Another KW segment may then be nucleated by repetition of the above process (section 4.2.3.2.1). Whichever the nature of the macrokink being formed, e.g., switch-over or simple, once a screw segment is immobilized, the height of the MK, hMK, which ensues, depends on the frequency of locking by cross-slip per unit length, fcs, of the still mobile neighbouring screw part
Yo hMK -- L f c s ' where Vo is the dislocation velocity in the octahedral plane and L the mobile segment. Expression (9) suggests the existence of a relationship height and length of KW segments. Couret et al. [72, 203] have found that MKs decreases exponentially with their height, consistently with the fact is thermally activated (fig. 35).
(9) length of the between kink the number of that cross-slip
3.4.3.4. APB jumps. A careful analysis of the kink distribution has enabled Couret et al. [203, 72] to point out an anomaly in the MK distribution for a kink height that scales with the APB width (fig. 35). This anomaly, which has important implications for the understanding of locking in L12 alloys, results from the mechanism of crossslip transformation schematized in fig. 32. Note that it is in fact Vitek [1] who should be credited for the first explicit sketch of an APB jump (see his fig. 9). The fact that superdislocations could undergo a jerky motion by steps of the order of their APB width is visible dynamically in the in-situ experiments of Caillard et al. [ 190] (see sections 3.4.3.2 and 3.5). This process of APB jump may repeat itself several times and it is known then as a repeated APB jump. Under weak-beam TEM this sequence gives rise to detectable steps on screw superdislocations (fig. 36) whose magnitude is of the order of the APB width in the octahedral plane. These particular macrokinks will be named hereafter elementary kinks (EKs). EKs and simple MKs proceed from the same basic dynamical double crossslip process (section 3.4.3.3). Similarly, the transformation of a glissile superdislocation into a KW lock, the formation of incomplete KWs, the processes of APB and repeated APB jumps are related. They will be discussed more quantitatively in section 4.2.2. Although the question is not widely documented yet, it appears that the distribution of EKs is polarized: in the reported events of repeated APB jumps, successive elementary kinks exhibit almost systematically the same sign (figs 36 and 37). It can be understood from fig. 38 that this implies that EKs are mobile. Consequently, one can never ascertain that a macrokink a few tens of nm height is not the result of the coalescence of several elementary kinks. Since the KWs bordering an EK do not belong to the same octahedral plane, each includes a jog (CJ) in the cube plane (see fig. 32) and, after coalescence of EKs to form a longer kink, this is what gives rise to the cusped aspects of the mixed segment in fig. 37. Hence, with regards to the understanding of the flow stress anomaly, analyses of the distribution of kinks including APB jumps and repeated APB jumps in post-mortem samples are very promising. This particular question will be further discussed in sections 4.2.3.2 and 4.3.1 in relation with superdislocation multiplication.
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Fig. 36. TEM evidence of APB jumps and repeated APB jumps in Hf-doped Ni3Al-based single crystals (courtesy of C. Bontemps-Neveu). (a) and (b): Repeated APB jumps in Ni3(A1, 1.5at.% Hf) deformed at 150~ along []23], same orientation and reflecting plane. In (b), note the presence of a succession of short dipoles so that the upper extremity of a given one is aligned in the screw direction with the lower extremity of its left-hand side neighbour (compare with figs 22 and 41). (c) Repeated APB jumps in Ni3(A1, 1.5at.%Hf) deformed at 150~ along [001], even in this (010) projection, the KW segments are rigorously straight, whereas they are already curved in (b) in a foil orientation that reduces by projection the amplitude of the bending (see fig. 39). (d) A sequence of repeated APB jumps in Ni3(AI, 0.25at.% Hf) deformed along [123] at 400~ the curvature of the KW segments is clearly visible. (e) Same as in figure (d) but projected such that the cube cross-slip plane is viewed on its edge, to show that the KW segments are actually curved in the cube plane.
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In samples deformed at intermediate and even at the highest temperatures in domain B, elongated KWs still dominate. However, instead of remaining aligned rigorously along the screw direction, they tend to be curved in the cube cross-slip plane, that is, in their dissociation plane (fig. 39). In fact, both superpartials have cross-slipped some distance off the octahedral plane from which these segments originate. A tendency of the screws to be deviated from the screw direction by as much as 20 ~ has been first reported by Douin et al. [139], but it is only in the work of Tounsi et al. [201] on polycrystalline Ni3Si, that the bending in the cube plane was shown to accompany primary octahedral slip. Strictly speaking, such curved segments are no longer KW locks though, with respect to octahedral slip, they may be considered as such. The bowing out increases with increasing deformation temperature. Its magnitude varies from one sample to another depending upon composition and load orientation; it may also vary quite dramatically within the same thin foil as a function of stress concentrations and depending whether KWs were formed in the early or later stages of deformation. For instance, whereas it is generally reported that deformation at room temperature results in well defined rectilinear KWs (figs 26, 27, 31 and 39), Korner [180, 204, 205] has clearly demonstrated that cube cross slip could be activated locally- though quite significantly - in samples oriented near [111]. The property of cube bending has also been reported by Mills et al. [200]. It is illustrated beautifully in the experiments of Hemker [ 110], where it is demonstrated that during inverse creep, that is, when octahedral slip is exhausted (section 2.2.9), deformation proceeds by expansion of superdislocations in the cube cross-slip plane. At variance with the above results, Dimiduk [49] has reported very scarce evidence, if any, of the bending of KWs. It seems nevertheless well established that the higher the temperature and the larger the shear stress in the cube plane, the more pronounced the bowing out of KWs in this plane. The effect of the shear stress is clearly manifested in the work of Karnthaler et al. [193]. Bontemps-Neveu [23] has stated that in Ni3(A1, Hf) single crystals, as far as the bending is concerned, the microstructure of samples strained along [123] at a given temperature looks very much the same as that resulting from the deformation along a direction located at 10 ~ from the cube orientation provided that, in the latter test, the temperature is 150-200~ higher. In fact, bent KWs should be regarded as some microyield stage on the cube cross-slip plane. When thermal activation becomes sufficient, these KW segments will eventually transform into dislocation sources for slip on the cube cross-slip plane as shown by the transition from primary to inverse creep (section 2.2.9, Hemker et al. [54, 79]). Hence, cube slip is concomitant with the operation of primary octahedral slip. It contributes very little to the strain since it occurs at a very slow rate compared to octahedral glide. Due to the nature of cube friction, there is no real threshold stress for cube slip (section 4.2.1.2). Quantifying the magnitude of bending in a more refined manner is uncertain since the major difficulty relies in relating curvatures of segments that have started to bend at different times during the experiment and have relaxed upon unloading. Of course, bent KWs are not in equilibrium, especially in thin foils thus providing another clear manifestation of the strong lattice friction on the cube plane by which bent KWs are maintained in position after the external load is released (Saada and
3.4.3.5. The bending in the cube plane.
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Veyssi~re [206]). In such occurrences of local cube activity, it can be verified that no dislocation character is clearly favoured even after deformation at room temperature (see for instance fig. 6 in Korner [180], see also section 3.1.1.2). In other words, friction in the cube plane is large irrespective of the dislocation character which implies that, contrary to what is often envisioned, lattice resistance does not necessarily originate from a tendency of the screw superpartials to be sub-dissociated into Shockley partials exclusively. 3.4.3.6. APB tubes. Since the early 60's, APB tubes have been associated with the large work-hardening rates measured in ordered alloys. It is thus quite clear that a good understanding both of the formation and of the properties of these defects is needed. Several mechanisms of formation of APB tubes have been proposed, such as
the jogged superdislocation process in the early work of Vidoz and Brown [207], a variety of situations where companion superpartials do not follow exactly the same cross-slip path (Kear [65]), - the annihilation of screw superdislocation dipoles (Chou et al. [208]), - the bypassing from pinned kinks (Hirsch [99]).
-
-
The formation of APB tubes will be addressed in some detail in section 4.3.3.2. Though APB tubes are not a new concept, they were identified unambiguously for the first time in B2 ordered FeA1 (Chou and Hirsch [209]). The experimental evidence of APB tubes is quite abundant in L12 alloys. In Cu3Au, thermal APBs sit on cube planes but deformation introduces new APBs, which are elongated along the primary Burgers vector direction (Kear [64]), whose contrast has been attributed to APB tubes (Hazzledine and Hirsch [210]). The discovery of APB tubes in Ni3A1 has been announced with some emphasis about a decade ago (Chou et al. [211]) and later reported in Ni3Al-based alloys by Mills et al. [200], Baluc [135] and Bonneville et al. [130]. It was accompanied by similar observations in Ni3Ga (Sun [62, 189]) and in Fe3Ge (Ngan et al. [188]). It is interesting to realize that whereas APB tubes were searched for and found with great difficulty in Ni3A1 [211], they became easy features to spot in the same alloy after their contrast properties were first described by Sun [189] and then by Ngan et al. [188, 212, 213]. Evidence of APB tubes can now be found in many TEM studies +-_ Fig. 37. The cusping of mixed segments in Ni3(A1, 0.25 at.% HI) deformed along [123] at room temperature. (a) to (d): While the screw segments contain a number of kinks, the long mixed segments are cusped; the distance between cusps scales with the kink height. This is interpreted as the result of kink sliding along the screw direction and of their coalescence to form longer mixed segments. Because EKs connect KW segments located on parallel but distinct octahedral planes, the long mixed segments must contain jogs in the cube plane (CJ in figs 32 or 33). These jogs are less mobile than the mixed segments in the cube plane giving rise to the cusping. It is clear that the overall mobility of such jogged mixed segments must be significantly less than that of a fully coplanar mixed segments of the same height; furthermore, the proximity of the pinning points make it impossible for a length of jogged segment to expand as a regular kink-source of the same length but jog-free. Finally, the presence of jogs provides an explanation for the existence of mixed segments bowed out in the octahedral slip plane, with a sometimes significant curvature, while lattice friction in this plane is certainly modest. Figures (a) to (d) stress the effect of EK accumulation but one can find similar examples of MK accumulation. A close inspection of these figures show that in fact both EKs and MKs may have coalesced to form a long mixed segments.
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Fig. 38. Development of microstructure when APB jumps are formed repeatedly. Consider the near-screw parts of a superdislocation that expands under stress. Only those APB jumps that are nucleated in the portion of the loop whose mean curvature is the closest to the screw orientation contribute two EKs with opposite signs to the distribution. An APB jump nucleated elsewhere does not add to the number of kinks because any EK exhibiting the "wrong" sign with respect the mean local curvature of the superdislocation will annihilate with its nearest neighbour (stars represent annihilations of kinks with opposite signs). This figure compares, in addition, the two ways of observing a succession of EKs along a superdislocation line by TEM: post-mortem and in-situ experiments correspond to space and time coordinates, respectively. In both cases, there is little chance of seeing nucleation events. of L12 compounds when dislocation imaging is performed under a diffraction vector non-parallel to the primary slip direction (fig. 40, see also fig. 45). Further examples of APB tubes have been published recently ([130, 212, 214]). One of the most striking properties of APB tube contrast is that it is observed with considerable magnitude under a fundamental reflection, provided g is not collinear with b. This suggests very strongly that some relaxation takes place [189] that gives rise to a dilatation strain field [212]. There is some controversy as to the origin of this contrast. Sun [189] claims that APB surface tension is responsible for distortions just as in a continuous elastic medium, while Ngan et al. [213], in order to account better for the magnitude of the contrast, add a relaxation component A R to the APB displacement vector R - ~1 <110) (a similar method had been employed by Lasalmonie et al. [ 187] in order to account for the residual contrast of APBs in rapidly solidified alloys). A theory of contrast from APB tubes in the absence of strain has been given by Chou and Hirsch [215] in the particular case of B2 alloys. This theory used in the case of APB tubes with complex cross-section gives very good match with experimental observations (Chou el al. [208]). The extent to which this theory applies to APB tubes in L12 alloys, where these defects are visible under fundamental reflections, is still however unclear.
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Fig. 39. The effect of bending of Kear-Wilsdorf segments in the cube plane in Ni3(AI, 0.25 at.% Hf) compressed along [123] (courtesy of C. Bontemps-Neveu). (a) Low-magnification microstructure of deformation at room temperature, image projection occurs along [111]. The KW segments are straight and exhibit a kinked shape. (b) Deformation at 150~ foil section and projection along the (010) cube cross-slip plane. At this rather moderate deformation temperature the bowing out of KW segments in the cube plane is already clear. The amplitude of bending is quite variable in the same foil, as attested by the differences in curvature exhibited by segments of about the same length (pointed to by letters). This stems from the time dependence of cube slip, itself dictated by lattice friction. Note also the existence of a large number of short dipoles attesting, as in (a), to the occurrence of annihilation processes. w
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Fig. 40. Production of APB tubes by a dragging mechanism originating at a single superdislocation in Ni3(AI, 0.25 at.% Hf) deformed along [123] at room temperature (courtesy of X. Shi). The presence of an APB tube appears to be in registry with that of a cusp on the dislocation line (small arrowheads). The APB tubes are always rigorously rectilinear. This observation rules out the hypothesis where-by APB tube result from the direct annihilation of screw segments by cross slip. Note that APB tubes may also originate from parts of the superdislocation located at the transition between the mixed and the screw segment (large arrowhead). The superdislocation may also be crossed by APB tubes which originate from the passage of other superdislocations. The shape of the mixed segment, as determined by the trailing of APB tubes, suggests that the tip of the hairpin in fig. 37(a) is not the leading segment, but the trailing part of the configuration. Several properties of APB tubes that may inform on their production should be noted m
- in Ni3(A1, Hf) deformed along [123], APB tubes are present after deformation up to a temperature of 450~ [283], indicating that the mechanism of APB tube formation operates at least over a large fraction of domain B. In fact, Shi pointed out that the disappearance of APB tubes from the microstructure at some intermediate temperature coincides with the rather sudden drop of work-hardening rate (fig. 5(a)). A similar remark can be m a d e from the work by Sun [62] on Ni3Ga. APB tubes may appear in larger densities than elongated screw superdislocations. - A P B tubes can remain rigorously rectilinear over microns, that is, over distances significantly longer than the mean length of K W locks. -
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APB tubes are attached to the near edge part of a single superdislocation (fig. 40), rather than to superdislocation dipoles as envisaged in some models. there is certainly a correlation between the presence of APB tubes and that of SISF dipoles (section 3.4.3.8, see fig. 44).
Experimental observations of APB tubes have been published recently [283-285]. The exact mechanism of APB tube formation is not elucidated yet. Finally, the finding of a significant density of APB tubes in an alloy such as FeaGe that exhibits a negative TDFS only [188] should be considered with some attention in the debate on the mechanical properties of L12 alloys (section 4.3.2).
3.4.3.7. Dipoles. The situation is confusing since the deformation microstructure of L12 alloys deformed in domain B contains a variety of dipolar defects. We distinguish between superdislocation- and superpartial-dipoles. SISF-containing dipoles constitute an important category of superpartial dipoles whose analysis is deferred to section 3.4.3.8. 3.4.3.7.1. Superdislocation dipoles.
Superdislocation dipoles are encountered rather frequently (figs 26 and 27). Since the microstructure contains a number of KWs, it is natural that a density of screw dipoles be built up upon expansion of fresh superdislocations during the straining. As the distance between the superdislocations approaching one another on parallel octahedral slip planes is decreased, the screw dipole may either transform into an APB tube or annihilate completely (section 4.3.3.2). Mixed and edge dipoles are also encountered in large densities at all temperatures; they have been indicated by Korner [192] as a prominent feature of the deformation microstructure at room temperature. These dipoles appear in the form of elongated prismatic superdislocation loops, with a length scaling with that of kinks. One remarkable property is that often the extremity of one mixed superdislocation dipole is rigorously aligned with another extremity of the neighbouring dipole along the screw direction (fig. 41). These defects are currently encountered in thin foils and they represent a direct manifestation of the mechanisms of annihilation that take place in the material (section 4.3.3.1), but no study of their actual fine structure or of their distribution has been conducted to date.
3.4.3.7.2. Superpartial dipoles.
Figure 42 shows that the two branches of the superdislocation dipole border a region of crystal containing a number of narrow linear defects. We show in the following that they are actually debris that appear in the microstructure when more that one slip system operates during deformation. Although they do not seem to play a role in the mechanisms that give rise to domain B (since domain B occurs even though only one octahedral slip system operates), they show some unusual and intriguing properties. These defects, which show a strain contrast, may connect the two branches of the superdislocation dipole. By appropriate + g / - g experiments this contrast can be shown to be dipolar in nature. This particular geometry suggests that the mechanism by which the narrow dipoles were produced at one superdislocation has been compensated by subsequent interaction with the next superdislocation with opposite Burgers vector. Under weak-beam conditions, it appears that the narrow dipoles are connected to only one of the two companion superpartials. In the first experimental report of such dipoles (Tounsi et
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Fig. 41. Two examples of a succession of closed mixed superdislocation dipoles in Ni3(AI, 0.25at.% Hf) deformed along [123] at room temperature. The upper extremity of any given dipole is aligned quite exactly with the lower extremity of its left-hand neighbour in the screw direction. (a) A curved dipole similar to that located in the middle of fig. 26 has self-annihilated in places. (b) In this example the matching between dipole ends in the screw direction is embodied by a dashed white line. Such configuration is interpreted as the result of the meeting of two glissile superdislocation and of the local self-annihilation of the resulting curved dipole by cross-slip. These examples of the annihilation of two superdislocations are representative of this category of debris at low temperature. At room temperature and above, alignments of dipoles contain usually a lesser density of mixed dipoles relative to that of annihilated screw segments (see figs 22(c) and 36(b)), with the length of mixed dipoles being of the order of the kink height within sessile KW configurations. For these reasons, the latter category of alignment, which results from the annihilation between a mobile dislocation and a kinked KW lock, is rather difficult to spot in TEM samples. Using appropriate imaging conditions (not shown here), it can be determined that there is in general no evidence of an APB tube that would connect two consecutive mixed dipoles as predicted by some models of APB tube formation [285]. m
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Fig. 42. Narrow dipolar segments in Ni3(A1, 0.25 at.% Hf) deformed along [001] at room temperature (courtesy of C. Bontemps-Neveu). (a) A large density of dislocation filaments, nearly all located in between a dipole of superdislocations itself elongated in the screw direction. All the filaments are parallel to the [ 110] direction, that is, to the trace of the (111) slip plane. The arrowheads point to configurations where the connection between a filament and a superdislocation can be seen most easily, there is indication that filaments are linked to only one of the two companion superpartials. (b) The dipolar nature of these filaments is evidenced by comparison between the 9 and - 9 reflections. From this observation, it is implausible that these filaments could originate from a dragging mechanism resulting from the motion of the superdislocations to which they are connected. Instead, it is likely that these filaments reflect an intersection process where the screw superdislocations play the role of trees: the effect starts upon intersection with the first superdislocation with Burgers vector b and it is entirely compensated by superdislocation - b . Similar to fig. 36, note in (a) that the upper MK which is located at the left-hand side of the micrograph is cusped, with an inter-cusp distance that corresponds approximately to the height of an EK. m
al. [201 ]), these have been shown not to be straight but to be bent out of the slip plane of the superdislocation to which they are attached, the bending is contained entirely either in a cube or in an octahedral plane. Furthermore, one can find many examples where these dipolar features show several preferential orientations, sometimes directed on either side of the superdislocation (fig. 43). Although they are too narrow to be identified for certain by application of the usual invisibility criterion, these dipoles should be formed of 1 1 (110) superpartials since a Burgers vector of the .~ (112) type would imply that they are always parallel to an octahedral plane, which is not what one observes experimentally. All these features rule out the idea that such narrow superpartial dipoles could be formed as a result of a jog dragging mechanism, upon motion of the observed superdislocation. After careful contrast experiments, Korner [216] has concluded that these dipoles do not have the same Burgers vector as the superpartial to which they are connected. Korner has extended to the L12 structure the explanation in terms of the intersection by a mobile dislocation proposed by Saada [217, 218] for f.c.c, metals and by Court et
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Fig. 43. Narrow dipolar segments in Ni3(A1, 0.25at.%Hf) deformed along [111] at 200~ (courtesy of C. Bontemps-Neveu). As in fig. 42, filaments can be seen to originate from a given superdislocation, essentially in its screw or near-screw parts (the near edge parts are not shown here). The filaments can be longer than shown in fig. 42, their length being in particular limited by intersection with the foil surfaces. (a) They show several preferential directions attesting to the simultaneous operation of two, if not three, distinct slip systems. (b) Filaments are elongated on both sides of a given superdislocation which rules idea a mechanism of formation by a dragging process that would result from the motion of the superdislocation.
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al. [219] for TiaA1 (D019). In this mechanism, a moving superdislocation forms a sessile junction upon cutting an immobile superdislocation. If the Burgers vector of the forest dislocation is not parallel to the line of the mobile dislocation, the two half-branches of the mobile superdislocation will not sit on the same atomic plane after they have interacted with the tree. They should thus leave a dipole in their wake. Accordingly, the presence of these narrow dipolar branches attests to the simultaneous operation of several slip systems and to the fact that the superdislocations to which they are attached were already immobile at the time they had been intersected. These features could thus be regarded as mixed imperfect APB dipoles in which superpartials have not completely annihilated. What is surprising is that, since under the model of Court et al. [219] the distances between superpartials should amount to at most one or two Burgers vectors, they do not annihilate by a fast climb process (Friedel [220]). These dipoles can even be observed after deformation at intermediate temperatures, which confirms their stability. Such stability could originate from the fact that the energy of self-diffusion is rather high in L 12 alloys. In fact, it has been observed by Tounsi [182] that dipoles do pinch out into loop rows, but at a very slow rate (several months at room temperature).
3.4.3.8. SISF-containing defects. In a variety of L12 alloys, there is ample evidence of elongated SISF-containing dipoles which, by contrast with the narrow dipoles examined in the preceding section, may be several tens of nm wide (SISF-containing dipoles can be seen in fig. 14(a) of Thornton et al. [41 ]). SISF dipoles are prominent in the vicinity of cracks, suggesting that their formation requires a high level of stress (Veyssi~re et al. [140], see also Jumojni et al. [221]). This latter property has been recently studied in some details in Ni3Al-based alloys by Dimiduk and Parthasarathy [80], and the close correlation between sample mishandling and the density of fault-containing defects is clearly assessed. This study shows also that SISFs are formed as indicated by formula (8), but this is restricted to localized slip bands. l There is strong and general indication that .~(112) superpartials have little mobility. This is for instance suggested by morphology studies that have pointed to the existence of favourable core configurations in the (011) dense directions (see for instance Veyssi~re et al. [140], Yan et al. [222], Dimiduk et al. [80]). The very strong friction to which 1 g(112) superpartials are submitted from the lattice is confirmed by the presence of SISF dipoles- which are intrinsically unstable- even after deformation at intermediate temperatures [ 140]. There are in fact two categories of SISF dipoles which differ by whether their habit plane coincides with the slip plane of the parent (110) superdislocation or not. The first category, which has been already discussed in section 3.1.2.3, is exemplified in fig. 44. These particular SISF dipoles seem to play an important role in domain B 1 as suggested by the low-magnification view of fig. 45. It has been determined that the deformation microstructure of Ni3(A1, Hf) deformed in single slip contains two categories of SISF dipoles in the primary octahedral plane (Shi [283]) that are bounded by superpartials 1 [~11] or tO g[121]. 1 with a Burgers vector either parallel to .~ So far there is no convincing model for their formation. We now focus on SISF dipoles that lie on an octahedral plane other than the primary slip plane [140]. The density of such SISF-containing defects depends on deformation
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Fig. 44. Examples of the correspondence, both in direction and width, between APB tubes and SISF dipoles in Ni3(A1, 0.25 at.% Hf) deformed at room temperature along [123]. (a) Rows of elongated loops, about 20 nm wide with a Burgers vector bs perpendicular to the direction of loop elongation (when these loops self annihilate during the TEM observation, they leave an APB tube in their wake). (b) Elongated SISF dipole with a 30 ~ character, note that whereas the SISF loop width is more than 50 nm over its longest part, it reduces to about the same width as that shown in figure (a) when the SISF transforms into the APB tube. While SISFs exhibit some kinks, APB tubes remains strictly straight. This observation of a width reduction at the transition is quite frequent (Shi [283, 284]). m
temperature; it is the lowest at high temperature, suggesting a thermally-activated annihilation of 51 (112) superpartials. The SISF density also depends upon alloy composition. For instance, SISF loops which lie off the primary slip plane are encountered in much larger densities in the deformation microstructure of Co3Ti (Oliver [147]) than in Ni3A1based alloys. SISF-containing defects have been reported to occur in places in a variety of alloys such as Ni3Ga (Takeuchi and Kuramoto [81]), Ni3Al-based alloys (Kear et al. [223], Oblak and Kear [224]) and Co3Ti (Liu et al. [25]). They have been interpreted as evidence of the dissociation under mode II (formula (8)). Several mechanisms have been put forward in order to account for the formation of SISF-containing dipoles (Giamei et al. [136], Suzuki et al. [181], Baker and Schulson [225]). These models are inconsistent with TEM analyses under weak-beam conditions that reveal that SISF loops are in many cases attached to one of the companion 1 (110) superpartials within a superdislocation dissociated under mode I (fig. 46). Accordingly, a formation mechanism based on stress concentration, such as arises upon superdislocation intersection, has been proposed (Veyssi~re et al. [140]). In view of this mechanism, the fact that in some alloys such as Co3Ti the density of SISFs appears larger than in others, such as Ni3Al-based
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Fig. 45. Low magnification observation of the distribution of SISF and APB tube debris in Ni3(AI, Hf) deformed at room temperature along [123], imaged under a reflection that sets superdislocations out of contrast. The samples contain a density of APB tubes that is often larger than that of elongated screw superdislocations. SISF dipoles are also present though in significantly smaller density than APB tubes. SISFs tend to transform into APB tubes (fig. 44). APB tubes are rigorously rectilinear over several microns. In other words their shape is not reminiscent of that of kinked superdislocations as they should be if they resulted from the direct cross-slip annihilation of a pair of screw superdislocations. (a) Though kinked the SISF dipoles are essentially parallel to the [101] direction of the APB tube. m
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Fig. 45 (continued). (b) The SISF dipoles are wavy and oriented at 60 ~ from the [101] direction. The nature of the curved fringed debris labelled 'x' is not elucidated.
alloys, does not necessarily imply that the dissociation mode has changed from mode I in the latter alloy to mode II in the former. Rather, differences in SISF densities can be explained by the fact that the conditions of SISF formation may vary from one alloy to another due to changes in SISF energies ll (TSlSF), or else by a change in mobility of the 1 .~(112) superpartials between both alloys. Confusion in TEM analyses may thus arise 1 liThe stress required to produce an SISF in the wake of a ~(112) superpartial is given by "rSlSF --
3,SiSF/b 89
For 25 m J m - 2 < 3'SISF < 50 mJm - 2 [140], then 85 MPa < "rSISF < 175 MPa. This stress
is much smaller than that needed for the creation of an APB, e.g., for 100 mJ m -2 < 7APB < 200 mJ m -2, then 400 MPa < "rApB < 800 MPa. In fact, "rSlSF compares with the level of resolved shear stresses currently applied to samples deformed in domain B, whereas "rApB remains significantly smaller than these stresses up to the peak.
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Fig. 46. SISF dipoles in Ni3(AI, 0.25 at.% Hf) deformed at room temperature along [123]. (a) The SISF dipole lies in the (111) octahedral plane as shown by the intersection, in projection of the superpartials (the detail of the connection between the SISF dipole and the superdislocation marked by a white arrowhead, is presented l [2-i1] Burgers vector. at a higher magnification in the insert); the SISF is surrounded by a superpartial with .~ (b) A SISF dipole contained in the (111) primary slip plane; the Burgers vector of the outer partial is 1 [211] (see also fig. 22(a) to (d)). Note that the markedly cusped aspect of the mixed superdislocation to which the SISF is connected, is somewhat similar to the observation of fig. 40 in the instance of the trailing of APB tubes. The correspondence between two mixed dipoles is also outlined by a white dashed line (Shi [283]). m
1 from the fact that, unlike a 89 superpartial, the passage of an individual .~(112) superpartial produces a striking fringe contrast in TEM. In a foil, even a few of such events creates an overwhelming fringe contrast. This may cause, and has indeed resulted in, overestimates of the relative contribution of 89(112) superpartials to the strain, a mistake also encountered in the literature on 7 / 7 ~ superalloys regarding the mechanisms of shearing of the L12 precipitates. The case of CoaTi deserves special attention since the observation of the SISF contrast is at the origin of core-controlled explanation of the negative TDFS (section 3.3). In the only unambiguous evidence of an SISF contrast given by Liu et al. [25] (their fig. 6), superdislocations are aligned in localized slip bands which contrast with the
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more homogeneous distribution in post-mortem samples. It must be noticed that the dissociation mode is changed from one surface to the other: superdislocations are widely SISF-coupled but this occurs systematically on one side of the slip band, whereas they are APB-coupled under a more narrow extension between the central part of the foil and the emergence with the other surface (see fig. 24(h)-(i)). In-situ straining experiments have shown (Oliver [147]) that when these dislocations are made mobile under application of a stress, their APB-coupled parts, which now extend throughout the foil, proceeds leaving behind SISF-containing half-loops, such as the few ones that are present in fig. 6 by Liu et al. [25]. The fact that superdislocations dissociated under mode II can be mobile in L 12 alloys has been shown to occur in Zr3A1 by Howe et al. [226] and by Douin [227]. The latter author has shown in addition that in Zr3A1 the choice of the dissociation mode is intricately dependent upon the character of the superdislocation, as first discussed quantitatively by Suzuki et al. [ 181 ]. Finally for this section, it may be worth mentioning that whereas the intrinsic nature of the superlattice stacking fault is generally taken for granted, there are enough crystallographically different situations of stacking fault-containing defect to let one believe that some of these stacking faults may be extrinsic. A non-conservative process which seems incompatible with a relaxation after unloading (section 3.1.2.6) has been reported in Ni3A1 polycrystals deformed well below the peak temperature (Veyssi~re et al. [171 ]). It has been confirmed in a Ni3(A1, Ti) single crystal deformed at 270~ (Korner [204]). In these cases, a slight bending of KW locks is seen to occur in the primary octahedral plane, which implies some diffusion-controlled motion of the APB (APB dragging, fig. 47). It should be noted that as slip of superpartials proceeds on parallel octahedral planes and as the APB is dragged behind them, mixed superdislocations apparently climb-dissociated can be formed without having involved any climb motion of the companion superpartials. APB dragging is by nature a high-temperature process which should be favoured under low strain rates and/or under creep conditions. It has been indeed reported in single phase Ni3(A1, Hf) crept at 760~ along the [001] orientation. Slip then occurs mainly on a dodecahedral { 110} plane which is actually the most stressed but which constitutes an unusual slip plane in this system 12 (Caron et al. [228]). Extensive APB dragging by octahedral slip has been recently reported in Ni3(Si, Ti) deformed above the peak temperature (Takasugi and Yoshida [39]). Note that in both tests the samples have not been cooled down to low temperatures very rapidly after the deformation, and so it cannot be fully ascertained that these observations account unambiguously for the processes that have taken place during deformation (section 3.1.2.6).
3.4.3.9. A P B dragging.
As emphasized earlier (section 3.4.2), the microstructure after deformation at and below room temperature of some L 12 alloys, possibly not limited to those showing a positive TDFS, contains a significant, though not dominant, density of
3.4.3.10. 60 ~ dislocations.
12It is however a non-close-packed plane just as the cube plane is, and by analogy with f.c.c, metals, one should not a priori exclude such slip systems to occur under favourable thermomechanical conditions.
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Fig. 47. APB dragging in polycrystalline Ni3(A1, 0.5 at.% Hf, 0.24at.% B) deformed at 650~ and quenched rapidly. (a) g = 200, B D = 010, the dissociation width of the elongated superdislocations with [101] Burgers vector is at a maximum. (b) g = 202, 13D --- 121, inclination of the foil intermediate between those of figures (a) and (c). Although still wavy, the superdislocations are the most rectilinear, especially the left-hand side one, when the (111) plane is edge-on. This indicates that each superpartial is roughly contained in a (I 11) plane. (c) a -- 202, B D = 101, the cube cross-slip plane is set on its edge. Though significantly curved, the superdislocation is projected edge-on along its screw and non-screw parts, attesting to the motion of the companion superpartials in parallel slip planes. If the kinks pointed by white arrowheads were split in the octahedral plane, this would show as exemplified in (d) where the KW parts are edge-on while the mixed parts exhibit a double contrast. m
D
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rectilinear superdislocations elongated along the 60 ~ dense atomic rows of the octahedral plane (fig. 48). There is indication that the dissociation width in this case is somewhat smaller than that usually observed for this dislocation character, at higher temperatures, when this particular character is no longer favoured. First reported by Tounsi et al. [201 ] in Ni3Si, the existence of 60 ~ superdislocations has been confirmed by several authors in Ni3Al-based alloys (Dimiduk [46], Bontemps-Neveu [23], Mol6nat and Caillard [229]) and in Co3Ti (Oliver [147]). 60 ~ dislocations have not been sufficiently studied. Calculations in the framework of anisotropic elasticity have failed to support an hypothetical explanation in terms of a further sub-dissociation of one superpartial or of a combination of both, which would be located in the inclined octahedral or cube planes, that is, off the reference octahedral plane. There remains the possibility that the observed preferred segmentation originates from an intrinsic core configuration such as some directional bonding, as sometimes invoked in order to explain properties of dislocations in Ti-A1 intermetallics. However, in view of the variety of L12 alloys where 60 ~ superdislocations have been observed, the hypothesis of a core structure controlled by directional bonds seems also quite unlikely to explain 60 ~ dislocations. Atomistic simulations using embedded atom interatomic potentials have not shown any tendency of the core of 60 ~ superpartials towards a possible extension off the octahedral plane (Pasianot et al. [230]). It should be noted that one of the first clear evidence of 60 ~ dislocations in metals has been provided by Karnthaler et al. [178] in a low stacking-fault energy disordered Cu-A1 alloy, an observation which casts a serious doubt on the relevance of the bond directionality thesis in this case. Veyssi~re and Douin [ 120] have proposed that 60 ~ dislocations could be responsible for the low-temperature negative TDFS. The difference between the mechanical properties of Ni3A1 and Co3Ti alloys at low temperature, say at the temperature of liquid nitrogen, would stem from the fact that stabilization along the 60 ~ orientation is stronger in the latter alloys.
3.5. Dynamical behaviour The first in-situ observations of moving dislocations in L 12 alloys have been reported by Suzuki et al. [231,232] and Nemoto et al. [233] in Ni3Ga and Ni3(A1, Ti), respectively. These papers can be read with benefit for they establish very clearly the situation. Both alloys show the same typical difference in dislocation behaviour depending upon whether octahedral or cube slip operates. At low temperatures cube slip hardly occurs. In Ni3(A1, Ti), there is some evidence of its operation at room temperature. It is viscous and stress-dependent, just as in b.c.c. metals at low temperatures. Suzuki et al. [231] have nevertheless pointed out that, at variance with b.c.c, crystals, this sluggish motion occurs in the absence of preferred dislocation orientations. These authors have in addition demonstrated that 1 (110) superpartials move independently in cube slip at 873 K, leaving an APB in their wake (under this orientation Tp ~ 700 K, Takeuchi and Kuramoto [81]). To the authors' knowledge, this is the only observation of this kind. It is possible that similar phenomena occur in Ni3Al-based alloys, but due to extremely unfavorable structure factors they are hard to identify by TEM.
Fig. 48. A 60' dislocation in Ni3(A1, 0.25 at.% Hf) deformed at room temperature along [T23] (courtesy of X. Shi). The dissociation of 60° segments is usually difficult to resolve. In this particular case, one can detect a succession of widenings (shown by yhite arrowheads), whose nature is not elucidated. The SISF dipoles (sf) are surrounded by a superpartial with [ 1121 Burgers vector.
4
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Fig. 49. A schematic possible representation of a pinning point. The jogs CJ terminate cross-slipped dislocation in the cube plane. For the sake of clarity, the scale is not respected since the lateral extension of the pinning point is not more than 3b, the cross-slipped distance is b while the dissociation width in the octahedral plane is 20b.
On the other hand octahedral slip occurs in avalanches at almost any temperature. Individual superdislocations have an extremely high velocity in octahedral slip, their motion is jerky [233]. Both research groups have reported that, once stopped, long screw superdislocations remain immobile until the end of the test. This conclusion is of course true within the resolution of such experiments, but it has been subsequently questioned based on in-situ observations carried out under weak-beam conditions (see below). The study of Nemoto et al. [233] shows evidence of the in-situ build-up of heavily kinked microstructures. Suzuki et al. [231,232] have in addition pointed out some behaviour which, in these tests, is inherent to the near vicinity of free surfaces. Of interest is the fact that superdislocations undergoing cube slip in the thickest parts of the foils tended to cross slip on an octahedral plane as they approached the foil edge and vice versa. A tendency towards change in dissociation plane in the close vicinity of foil free surfaces has been pointed out in post-mortem studies [ 171 ]. For the sake of completeness, the in-situ study of Jumojni et al. [221] where some quantitative data on the fraction of screw superdislocations as a function of stress and temperature is available, should be mentioned. This experimental work suggests that screw locks formed at room temperature may be destroyed upon slight increase of the applied stress and that these locks are stronger at elevated temperature. The above general trends have been supported subsequently in a series of experiments, all carried out by the same research group, but under weak-beam conditions. For this reason, considerable progress has been achieved in the analysis of the subtle properties of individual superdislocations. Furthermore, a significant effort has been devoted by ClEment et al. [112] and Couret et al. [198] to eliminate the possible sources of artefacts which, in view of the near vicinity of free surfaces, are unfortunately highly possible during such experiments (Veyssib~re [3, 4]). We shall now focus on octahedral slip (Caillard et al. [ 190], Mol6nat and Caillard [ 196, 229]). A major achievement of in-situ straining tests is the unambiguous observation of the dynamical locking of screw superpartials by cross slip [190], with no indication for it to be limited laterally to 2 or 3b as envisioned in the point pinning models (fig. 49). Instead of being curled between (invisible) pinning points as implied by these models (see also fig. 50 and section 4.1), mobile screw superpartials are shown to lock themselves
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(a)
~
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b
--- trace of (010) plane
segments pinned (b)
on
,,,
~
(olo) ''"q
/~I~
/'q~-\segments
on (111 )
segments on (010)
Fig. 50. In order for the dynamical configuration of (a), as postulated in the frame of dislocation motion controlled by point pinning, to relax into the scheme of (b), which conforms to experimental observations, the pinning points must be rigorously aligned along the screw orientation over the extent of the future relaxed KW segment, just before the applied stress is released (after fig. 5 in Pope [156]). If the pinning points were slightly misaligned under stress, even by a few Burgers vectors - and there is no reason why they should not be - this would be detected by TEM on the relaxed configurations down to a shift of about l nm.
over significant lengths, almost instantaneously (in fact within the time scale of video recording, under the magnification usually set up during the experiments the resolution on kink velocity is of about several hundreds of ~m per second), and to remain very much rectilinear during the locking and unlocking transformations. Another rewarding observation in these experiments is that of mobile screw superdislocations extended in the primary octahedral slip plane, a configuration which is hardly accessible in post-mortem samples. In-situ straining tests conform, sometimes closely, to post-mortem observations (Caillard et al. [131], Mol6nat and Caillard [229]). This is for instance the case of the observations of incomplete KWs (see, however, footnote 10) and of the mechanisms of repeated APB jumps (figs 4 and 5 in [196]) and of double cross-slip [229]. In the first of this series of observations, Caillard et al. [190] have demonstrated the jerky nature of superdislocation motion (it should be however noted that in this case, just as in any situation where the superdislocation is mostly parallel to the foil, the resolved shear stress was almost nil, unless of course heterogeneities of stress distribution in the foil). An accurate analysis of changes in dissociation widths during superdislocation motion have led these authors to claim that screws could either cross-slip from one octahedral plane to the other or from the octahedral plane to a cube plane. The latter motion has been confirmed later [196]. The former remains nevertheless unclear since as discussed in section 3.4.3.2, the corresponding narrowing could well be due to the instantaneous formation of an incomplete KW, whose APB on the cube plane is set edge-on in this case. Caillard et al. [190] did not consider a possible transition through the cube plane, which, as we now know, occurs extensively, and concluded in this preliminary study that (i) the locking/unlocking process occurs through a series of core transformations in which the APB remains in the some octahedral plane and that (ii) the jerky motion is a consequence of the superpartial core spreading in the cross-slip
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octahedral plane with its line still lying in the primary slip plane. By analogy with a similar observation in Be (Couret and Caillard [234, 235]), the origin of the positive TDFS was postulated to stem from an increased lattice resistance- just as for cube slip (C16ment et al. [ 113]) - as a result of CSF energy supposedly decreasing with increasing temperature. When observations are conducted under foil orientations which enable one to identify an extension in the cube plane, as did Mol6nat and Caillard [196], the situation is however significantly at variance from the previous one, since the contribution of double cross slip through the cube plane becomes easier to point out. Nevertheless, some ambiguity seems to remain as to exact plane of the cross-slip path, as pointed out by Chou and Hirsch [199]. The better identification of the contribution of cube cross slip has led Mol6nat and Caillard [196] to propose a new sessile configuration in which both superpartials have glided in the cube plane in opposite directions over more than b/2 (their fig. 13). This configuration was supported by an analytical treatment which forgets that, under stress, one of the superpartials must relax against the applied force. A more realistic point of view has been subsequently derived by Paidar et al. [236] (see also Mol6nat et al. [237]) in which the positions of both superpartials are consistent with the bias imposed by the applied stress. As a final remark regarding the phenomenon of repeated APB jumps, we find it important to point out that in many of the examples referred to above, dislocations are nearly parallel to the foil surfaces. Since they are screw in character, the corresponding Schmid factor is modest, which in turn suggests that this process would tend to operate still under rather modest applied stresses (see the discussion of section 4.2.3.2.2). The exception of fig. 6 in Mol6nat and Caillard [ 196] should be mentioned where the dislocation labelled D, which undergoes up to 10 successive repeated APB jumps, is clearly inclined to the foil. Its emerging left-hand part remains remarkably aligned with the rest of the dislocation up to jump number 8, with occasional indication of the bending of its leading superpartial, as if cross slip could be initiated at the free surface, by virtue of image forces. The fact that pile-ups of elementary kinks have been observed to lie on a same line in post-mortem samples, which substantiates many-step APB jumps, should be nevertheless acknowledged. In-situ tests have been carried out in the whole range of flow stress anomaly and above [229,237,238], from which it was stated that deformation above 573 K (~ domain B2) differs from that typical of lower temperatures (domain B 1): (i) Superdislocation motion is essentially jerky at low temperature, consisting mostly in APB jumps wherein the path in the cube plane is a significant fraction of that in the octahedral plane [196]. This analysis in terms of pencil glide is however inconsistent with the observation of slip traces that are known to be crystallographic in nature over the corresponding domain of temperature. Kink motion is rarely observed. (ii) At higher temperatures, slip occurs in avalanches. Superdislocations slip very long distances and undergo a double cross-slip mechanism via the cube plane. Mol6nat and Caillard distinguish this process from that controlling APB jumps, not withstanding the fact that both processes are essentially the same process (sections 4.2.2.3 and 4.2.3.2.1). Once locked in the screw orientation, superdislocations tend to bow out slowly in the cube plane, conforming to post-mortem experiments (section 3.4.3.5). Superdislocations
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are also reported to pile up against pre-existing KWs [237], a property clearly at variance with post-mortem observations in which no pile-ups have ever been reported. Although very localized slip seems to be a major problem in many of the in-situ observations, this latter difference with post-mortem observations could nevertheless be due to the extensive and uncontrollable relaxation that is known to take place in postmortem observations after sample unloading and cooling. A further difference between both techniques is that in-situ tests are inherently unable to reproduce the expansion of a superdislocation to form the characteristic kinked microstructure, neither can they give evidence of its further evolution, such as that now envisioned by several research groups (sections 4.2.3 and 4.3). This is since, in order for kinks and expanding superdislocations not to disappear at free surfaces within a fraction of ~tm, the primary slip plane would have to be parallel to the foil, thus in a situation where it is essentially unstressed. It should in addition be remarked that over a significant fraction of domain B, the average length of KW segments is larger than the average thickness of a thin foil. Mol6nat and Caillard [229] have nevertheless reproduced the temperature dependence of the flow stress in Ni3(A1, Hf) by local determination of operating stresses based on in-situ measurements of the curvature of mobile dislocations. However, the magnitude of the flow stress measured locally is substantially higher than in the bulk material, a discrepancy which, regarding the fact that dislocations are almost isolated in thin foils, is quite difficult to explain in terms of work hardening but may just attest to the fact that dislocations are more difficult to move in thin foils than in bulk samples for the conditions explored. In addition to the direct evidence of repeated APB jumps, one of the most important findings of Caillard and co-workers' in-situ experiments is the operation of jerky flow in connection with a positive TDFS, e.g., both in Ni3A1 and in beryllium. The implication of this observation is that dislocation motion cannot be treated as a steady process (section 5). For the sake of completeness, the in-situ observations of Baker et al. [239] and of Horton et al. [240], of slip bands containing a large density of dislocation pile-ups in Ni3A1 should be mentioned. These slip bands are very localized and contain evidence of - the gradual widening of APB-coupled superdislocations from one side of the pile-up to the other, which has possible implications on the processes of disordering after intensive coldwork, their eventual transformation into SISF-coupled superdislocations, a density of non-conservative debris, such as prismatic loops, suggesting that nonconservative processes do occur at room temperature after the repeated passage of superdislocations. -
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4. The i m p l i c a t i o n s o f m i c r o s t r u c t u r a l o b s e r v a t i o n s in d o m a i n B Based on microstructural observations, this section is devoted to offering a tentative frame for a discussion of the positive TDFS and related mechanical properties. It will be shown that the evolution of the dislocation microstructure of L12 alloys deformed
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in domain B differs significantly from what has been reported so far in other materials. Though our point of view is derived essentially from experimental observations, it is not possible to ignore in the discussion of their implications, the various underlying hypotheses that have led to the design of the several available models of flow stress anomaly. For this reason, this section will contain a number of unavoidable overlaps with the arguments developed in section 5 where the models in question are critically reviewed. The scenario for the microstructural evolution which we propose in the following is inspired by several works such as those of Mills et al. [200], Veyssibre [3], Hemker et al. [54], Mills and Chzran [241], Hirsch [99, 100, 242], Korner [134] and Saada and Veyssibre [70, 243].
4.1. Summary of facts After the publication of the first set of point pinning models (Takeuchi and Kuramoto [81], Paidar et al. [86]), dedicated TEM studies have gradually drawn the attention of the scientific community to ideas and concepts more specific to L12 alloys, such as incomplete KW locks, macrokinks, repeated APB jumps, APB tubes and octahedral versus cube slip. These properties have been reviewed in preceding sections (sections 2.2.10 and 3.3). Their identification and their analyses - be it under static or under dynamical conditions - have greatly contributed to making one think of the deformation of L12 crystals under a more microstructurally-oriented perspective (Mills et al. [200], Veyssi~re [3], Hirsch [98-100], Ezz and Hirsch [281], Shi et al. [286]). The vast majority of electron microscopists now agree upon the fact that there is no serious direct evidence for the locking and/or unlocking to occur under the form of pinning points, and this has been made particularly clear from in-situ straining tests (section 3.5). To our knowledge, the only case of a regular cusping of screw superdislocations is in Co3Ti deformed at 873 K (Liu et al. [168]), but this has been later shown to result in fact from radiation damage during TEM observations (Oliver and Veyssi6re [244]). Rather recently though, Pope [156] discussed the consistency between the PPV model and microstuctural observations and concluded positively. His view on in-situ TEM observations was at that time that "... jerky motion could be the result of the rapid motion of superkinks along the dislocation line, from one foil surface to the other [where] such kinks may nucleate more easily." On the other hand, Pope took as granted the absence of superdislocation bowing in the cube plane, still during in-situ straining tests (cube bending has however been reported in such tests, section 3.5). Regarding studies conducted on post-mortem samples, Pope argued that the observed features are but relaxed configurations which, "alive", i.e., under stress, should be bowing around pinning points (fig. 50). In other words, according to Pope, if the pinning point model cannot be supported by TEM, it is simply because this process, which is purely a dynamical one, is by nature not observable by means of this technique (section 4.2.3.1). Regarding macroscopic properties, although Pope stated that certain properties related to the flow stress anomaly, e.g., essentially the orientation dependence of the flow stress and the tension-compression asymmetry, "are adequately explained by the cross-slip
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model" (see sections 2.2.6 and 2.2.7), we think it reasonable to consider more firmly the implications of microstructural observations on "dead" samples. We aim to show that the conjunction of independent analyses of deformation byproducts both in post-mortem and in in-situ experiments yields a consistent though still incomplete picture of the deformation mechanisms which take place during domain B. First suggested by de Bussac et al. [245], the fact that the pinning process of screw superpartials must be fast relative to their unpinning is now generally agreed. The remark of de Bussac et al., which was made within the frame of point pinning, applies of course to longer locks. This is why it is important to study the properties under stress of incomplete KWs with varied extension in the cube plane and to address the problem of kink dynamics. This is what section 4.2 is aimed at. TEM observations indicate on the other hand that, in the L12 structure, both dislocation multiplication and organization are rather peculiar. The above considerations on isolated superdislocations will thus be complemented by some remarks on the organization of the deformation microstructure and on collective effects (section 4.3), with reference made to macroscopic tests.
4.2. The contribution of individual dislocations
In the domain of positive TDFS, the microstructure contains elongated complete and incomplete KWs. In the latter, the degree to which APB is extended in the cube plane is a function of test temperature and sample orientation (section 3.4.3). The KW configurations, which result from cross-slip process of the leading superpartial onto the cube plane, are sessile with respect to primary octahedral slip. Of considerable importance is the fact that they are connected by kinks, all extended in the primary octahedral plane. We shall show in section 4.2.2 that the cross-slip process that yields KW configurations is controlled by a set of parameters which are intrinsic to the crystal, including elastic constants and the APB energies in the octahedral and in the cube plane, "7o and "7c, respectively. In addition, the cross-slip process is obviously influenced by the applied stress. Regarding the mechanism of cross slip, however, differences may arise from the hypotheses that can be made on the core structure of superpartials. This is what we shall examine first.
4.2.1. Effects of superpartial subsplitting: cross slip and frictional stresses The nature of superpartial core spreading has major implications on two physical processes that are expected to effect the strengthening of L12 alloys by formation of KW locks: (i) the nucleation of cross slip and (ii) lattice friction in the cube plane. We shall now discuss these two aspects in order to show that it is not crucial to use the representation of the core in terms of a subdissociation into two Shockley partials. 1 Considering that a screw 7(110) superpartial may subdissociate 1 into Shockley partials in an octahedral plane, in the same way as a perfect 7(110) dislocation would do in a f.c.c, lattice, the activation energy for cross slip has been the object of several analytical estimates in which superpartial extension/constriction plays a crucial role (Lall et al. [28], Paidar et al. [86], Vitek and Sodani [246], Khantha et
4.2.1.1. Cross slip.
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Fig. 51 9Projection of the L12 lattice along [101], - the symmetry of the core of 1[10T] superpartials depends on that of neighbouring atomic rows in position c~, it can be dissociated either on (111) or on (1]-1), the simulations of Yamaguchi et al. [157] have even suggested that the spreading could occur simultaneously on both planes; - in positions/3 and 7, the core tends to spread in (1T1) and in (111), respectively. By definition, the distance between 13 and 3' is b/2.
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al. [88], appendices 1 to 3 in Hirsch [99], Schoeck [247]). In many studies, this further spreading, which is assimilated as a true splitting where the partials and the stacking fault are clearly defined, is thought to be at the origin of the large friction stress in the cube plane, of the cross-slip rate and of the TC asymmetry (see section 5). Copley and Kear [67] were probably the first authors to envisage the effect of spreading for L12 alloys. In view of the local crystal symmetries, the potential that the core of an immobile superpartial has to spread in some directions is dictated by the atomic row whereon this superpartial is sitting (fig. 51). Under the assumption of a finite core spreading, the motion of a superpartial would involve little resistance from the lattice as long as it can remain dissociated in the same plane, which is the case for octahedral slip (paths c~--y-ce--y or c~-/3-c~-/3). By contrast, lattice friction is large in the cube plane because dislocation motion then consists in a succession of steps, with a magnitude equal to b / 2 (b - 89 on which the core spreads alternately on ( 1 1 1 ) a n d (111) (path/3--y-/3-~,).
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This point of view on core symmetries 13 represents a quasi-static approach to the motion of superpartials in which the time required by the core to relax into the octahedral cross-slip plane is small with respect to that needed to slip over b in the cube plane. By contrast, if superpartials can reach such a velocity that they travel through the lattice sites with not enough time to relax off their slip plane with a magnitude significantly larger than thermal fluctuations, then the above considerations on core-spreading controlled motion become pointless. We give below the reasons why we believe that the quasi-static approach cannot be appropriately applied in order to mimic the motion of superdislocations in the L12 lattice and why the results which have been derived from this approach, are debatable, especially at temperatures where thermal activation facilitates the operation of cube slip, such as in domain B2. According to Schoeck [247], during the process of cross slip in the cube plane (path c~--7-fl-7-fl-7), the dissociated configuration on (111) at fl (e.g., at the next fl site when cross slip on the cube plane originates at 7), becomes energetically unfavourable when the stacking-fault energy tends to be large. In these conditions, the dislocation will move unhindered to the next "7 site and motion in the cube plane will proceed by steps of length b (7-7-3'-7). This question of the relative energies of configurations at sites /3 and 3' has also been studied by atomistic simulations under various approximations of interatomic potentials (for additional information the reader should consult Yamaguchi et al. [157], Vitek [1], Yoo et al. [248] and Pasianot et al. [230, 249]). In addition to the designers of the point-pinning models, a number of authors ascribe an instrumental role to the subsplitting of superpartials in the conditions of locking. This is mostly because this particular core spreading is regarded as containing some of the necessary ingredients in order to account for the orientation dependence of the flow stress (see for instance Hazzledine and Hirsch [210], de Bussac et al. [245], Hirsch [99, 242], Mol6nat and Caillard [196], Mills and Chzran [241], Korner [134]). Furthermore, superpartial subsplitting is sometimes invoked to explain the composition dependence of the flow stress: as the CSF energy is for instance increased by solute addition, the splitting is decreased and the cross slip becomes easier to activate, resulting in an increased tendency towards KW lock formation (Korner [ 134], Hemker and Mills [50]). Differences in superpartial widths appear to explain consistently why Ni3Fe shows a modest positive TDFS, if any, relative to other L 12 alloys. The subsplitting is in fact comparatively wider in this particular system (section 3.1.2.2) and KW formation thus becomes markedly inhibited (Korner [153]). It is first recalled that e l o n g a t e d - complete or i n c o m p l e t e - KWs can be observed after deformation at almost any temperature between 4.2 K in Ni3(Si, Ti) and 900 K (or even more in some Ni3Al-based alloys) with no reported differences in the widths of superpartial cores. In addition, within experimental uncertainties, the subsplitting of screw 89 (110) superpartials remains in fact unobserved: - it is hardly visible under HREM (section 3.1.1.1); 13
-Core-symmetry controlled dislocation motion is an underlying assumption in a number of models of dislocation motion in L12 alloys, such as in Paidar, Pope and Vitek's model [86] (section 5.4.2), in those explanations derived from in-situ experiments (section 5.5) as well as in Hirsch's approach (section 5.6.2). The latter is based on the computer simulations of Chou and Hirsch [199] which relies entirely on the quasistatic assumption of the motion of a superpartial.
P. Veyssidre and G. Saada
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Ch. 53
- based on weak-beam observations conducted on near-edge segments and provided an extrapolation to the screw orientation based on linear elasticity is valid, it is at most of 3b (section 3.1.2.2). Hence, if mechanical properties were actually determined by the width of superpartial cores, as envisaged in a large majority of studies, changes in core spreading within 3b or less would have to explainl4: the extreme variety of mechanical behaviour that is available in the L12 structure, i.e., a positive TDFS which is already pronounced at 4.2 K in Ni3(Si, Ti) (Takasugi and Yoshida [29]) and which may either culminate at 1000 K such as in Ni3(A1, Ti) or a little above room temperature as in nearly-stoichiometric polycrystalline Ni3Si (Suzuki et al. [20]); - the fact that in some compounds, the magnitude of the TC asymmetry remains very pronounced up to the peak (Heredia [37]).
-
In fact, the core of a g (110) superpartial in Ni3A1 is about as narrow as the core of a screw dislocation in aluminium or in similarly high stacking-fault energy f.c.c, crystals. We thus find it reasonable to c o n s i d e r - at least to explore the i d e a - that the activation energy for superpartial cross slip in the L12 structure is small enough for the process to be already athermal at temperatures significantly below the peak, which may include room temperature (Saada and Veyssi~re [70]). This particular idea that the details of the core structure of 1 (110) superpartials are of minor importance in the overall process of transformation into KW configurations will be further discussed in section 5.2, based on considerations on the validity of theoretical estimates of the activation enthalpy for cross slip in the L12 structure. In the following, we shall therefore assume that, in those alloys which exhibit a positive TDFS, the spreading is so modest that it is even difficult to ascertain that it does c o n t r i b u t e - at least to the extent which had been envisaged so f a r - to the flow stress anomaly. Since the transformation by cross slip of a screw segment into a KW configuration is certainly controlled by a thermally activated process, we now discuss the hypothesis that lattice friction in the cube plane is enough to control the dynamics of cross slip. 1
4.2.1.2. Lattice friction in the cube plane. In domain B, deformation on the octahedral plane cannot be studied separately from cube slip, where a superpartial experiences by far the largest slip resistance (see sections 2.2.7, 2.2.9, 2.2.10, 3.4.1 and 3.4.3.1 to 3.4.3.5). Experimentally, the importance of friction in the cube plane is indicated by the following observations:
14Nabarro [250] has pointed out that under adequate physical hypotheses on the rate-controlling mechanism such as the assumption that the flow stress is proportional to the fraction of locked segments - and provided the activation energy for cross slip is sufficiently small, then the strengthening by cross slip remains effective over the whole temperature range of domain B, that is, over several hundreds of K, as is the case. However, with regards to the published estimates of the apparent activation energy for the transition from the { 111) to the (I00} configuration of lower energy, Nabarro objects that "under normal conditions of observations, all segments will jump readily between configurations, with a majority in the low-energy { 100} configuration even at room temperature, contrary to the premise of the [point pinning] model."
-
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Microscopy and plasticity of the L12 "/~ phase
365
(i) The bending of KWs in the cube plane is more pronounced as the temperature is raised because friction in this plane is highly thermally activated (section 3.4.3.5), and it is for the same reason that the bending is maintained after deformation in postmortem samples observed at room temperature (Veyssi~re [3]). It is clear that backward relaxation of the bent segments is favoured by low cooling rates after deformation; (ii) When superdislocations are significantly bent in the cube plane, they adopt a shape that can be reproduced quite closely by calculations conducted under anisotropic linear elasticity (Douin et al. [139]) while, if friction were controlled by a subdissociation into Shockley partials in an octahedral plane, then one should observe some segmentation of the bent segments along the screw orientation. This is not the case even when cube slip is activated at room temperature (section 3.4.3.5); (iii) Incomplete KW configurations, which are mechanically unstable (section 4.2.2.2), are present after deformation at low and moderate temperatures (section 3.4.3.2); (iv) Slip lines attesting to the operation of cube slip under constant strain rate occur well below the peak but mostly in modest amounts relative to traces of octahedral slip (for a more detailed discussion, see section 2.2.10); (v) Below the peak temperature, whereas deformation at a constant strain rate is dominated by octahedral slip with occasional activity on the cube cross-slip plane, deformation under creep is mostly ensured by slip on this latter plane once octahedral slip is exhausted (sections 2.2.9 and 2.2.10). On the other hand, the smooth curvature of dislocations in the cube plane (sections 3.1.2.2 and 3.4.3.5) indicates that lattice friction acts on dislocations somewhat irrespective of their character at least for a range of alloy compositions which nevertheless exhibit a TDFS. To our knowledge, there is no atomistic simulation available for a non-screw superpartials undergoing cube slip, except in the edge orientation where the superpartial is spread simultaneously on the cube plane and on the two octahedral planes (Pasianot et al. [230]). The large frictional stress in the cube plane thus does not appear to result just from some core spreading of screw superpartials along a (110) atomic row, but from an "intrinsic difficulty to shear these atomistically rough planes" (Hemker et al. [54]). In view of the reduced extension of superpartial cores, slight changes in atomic positions are required to let dislocations slip on the cube plane or undergo cross-slip transformation. Hence, instead of the Friedel constriction/extension or of a similar mechanism, it seems reasonable to treat the problem of cube slip under the usual Peierls-Nabarro approximation. It could be equivalently described in terms of a difference in core thicknesses between close-packed and non-close-packed planes, that is, of atomic re-shuffling during superpartial motion, irrespective of dislocation character. Hence, it may not be strongly dependent on the local symmetries of dislocation cores. The sessile configurations which form in some alloy compositions (for instance of the Lomer-Cottrell types, section 3.1.1.2), just contribute to accentuate the lattice resistance to shear. In L12 alloys, the situation thus shows a marked similarity with slip of perfect 1 ~(110) dislocations in the {001}, {011} and {112} non-close-packed planes of f.c.c. crystals (Carrard and Martin [251,252], Bonneville et al. [253]). In spite of the light that in-situ experiments have shed on lattice friction on the cube plane (C16ment [112, 113], Mol6nat and Caillard [114]), our knowledge of this effect remains in fact very poor. Rather than a mechanical force, it seems nevertheless clear that
366
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friction should be treated as a viscous term. A friction stress is indeed better defined as the stress required to move a dislocation at a fixed velocity. In other words, the friction stress cannot be treated as a threshold stress below which, given the temperature, a dislocation would be frozen-in in the cube plane. This point of view, first introduced for L12 alloys by Copley and Kear [67] - in order to justify an hypothetical increasing difficulty for octahedral slip with increasing temperature- can be illustrated based on the following expression of the stress dependence of dislocation velocity V-
()m
Vo --w To
(10)
(Gilman [254]) where Vo and To are temperature-dependent phenomenogical parameters. Consistently, a dislocation should be able to cross slip over a few nanometers in the cube plane within the duration of the mechanical test, even under a modest applied stress. TEM observations imply however that the temperature dependence of the dislocation velocity in the cube plane is sufficiently high that motion is virtually suppressed at room temperature. In summary for section 4.2.1: (i) With regards to the known dimensions of superpartial cores in L12 alloys showing a positive TDFS, and to superdislocation properties in cube slip, there is indication for the cross slip of these superpartials not being controlled by a thermally activated process of constriction/extension. Neither is superpartial motion in the cube plane. (ii) Rather than a temperature-dependent threshold stress, lattice friction appears to be more appropriately accounted for as a viscous resistance by which every dislocation character is to some extent affected. This is of course entirely consistent with a classical (Peierls-Nabarro) treatment of the thermally-activated motion of dislocations. (iii) One should keep in mind that lattice friction on the cube plane is probably composition-dependent. Finally, it is worth emphasizing that no theoretical treatment of the thermally-activated nature of slip in the cube plane has been made available yet. As shown in the following, such a treatment constitutes a necessary step towards a closer description of the transition from the glissile to the Kear-Wilsdorf configuration. 4.2.2. The mechanics of cross slip of a screw superdislocation Experimentally, KW segments are significantly longer than the pinning points envisaged by Paidar et al. [86] (see section 4.2.3.1). The mechanics of a KW segment are thus tractable under the assumption of an infinitely long dislocation. This section is devoted to an analytical treatment of the properties, under stress, of a dissociated straight screw segment and to the theoretical analysis of the potential mobility of this segment from a mechanistic standpoint. 4.2.2.1. General considerations. As indicated by post-mortem and in-situ observations and in agreement with the above point of view on cube lattice friction, one should consider the possibility that during the process of cross slip a superpartial glides some distance in the cube plane as opposed to a step of b (via a saddle point at b/2) in the point
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Microscopy and plasticity of the LI 2 7 ~ phase
pinning models or else in the low-temperature version of Hirsch's model (Hirsch [99], section 5.6.2.2). Just before the superpartial cross slips in the cube plane, it has some kinetic energy which is spent to work against lattice friction. The superpartial, which is then gradually slowed down, glides a mean distance/jump before it is immobilized. This distance is of course temperature-dependent. Given the test temperature, /Jcump should be nevertheless varied because cube slip is thermally activated; in particular, it is not necessarily equal to b. It has long been thought that, in the absence of stress, the driving force for KW formation just resulted from the difference of APB energies between the octahedral and the cube plane (Takeuchi and Kuramoto [81], Paidar et al. [86]). It was Yoo [255] who first pointed out the role played by elastic anisotropy in the early step of KW formation by cross slip. Yoo's aim was in addition to make use of mechanical torques in order to model spontaneous changes in atomic stacking that would yield the transformation of the APB into a CSE Several mechanistic analyses of the dissociated screw superdislocation followed some years later (Mol6nat and Caillard [196], Schoeck [256], Hirsch [99, 242], Paidar et al. [236], Saada and Veyssi~re [243, 257], Chou and Hirsch [199]). By making use of the {/o,/c} cartesian coordinates (fig. 52) instead of polar {p, 0) ones, we have shown that one can derive an entirely analytical solution which eases
IC:/L ~+
(a)
........!...o............/..........
(b)
...ji i~i~" ~e ,:ii!f:
(c)
(d)
Fig. 52. The four non-equivalent dissociation configurations that can be adopted by a screw superdislocation. In (a) and (b), the superdislocation straddles the cube and the octahedral plane. (c) and (d) are the planar glissile and KW lock configurations, respectively. The conventions adopted for the positive directions of motion in the cube and in the octahedral plane are indicated by arrows. Expressions (11 a) and (11 b) relate to the motion of L in the positive direction in the octahedral and in the cube plane, respectively; they apply to every configuration of fig. 52. On the other hand, expression (1 lc) relates to the motion of T in the positive direction for the octahedral plane in (a) and (d), while expression (11 d) gives the force on T as it moves in the positive direction of the cube plane in the same figures.
P. VeyssiOre and G. Saada
368
Ch. 53
significantly the necessary direct comparisons between the several situations of interest (Saada and Veyssib~re [257]). This particular treatment under the {lo, lc} cartesian coordinates is summarized in the following. Consider the incomplete KW configuration of fig. 52(b) and assume that the applied stress is such that the superdislocation as a whole moves from left to right and from bottom to top. Let T and L be trailing and the leading superpartials, respectively. The forces FiJ, which act on dislocation J (J standing for L or T) in plane i (i _= o or c) are written
FL - r o b + E (al~162 +/c)-%=E[s-lAo
FcL = feb+ E (l~ + rv/~/c) - T c - E FT=rob_E(alo+lc) r
FTc = rcb- E(l~ + x/~lc)4)
-7o=E
+al~162
N s~o -( [s+l Ao
(lla)
+ lo + rV~/c] ' alo+Ic] r
+ Tc - E[ Ns + ( - l~ + r
,
(llb)
(llc)
.
(lld)
A necessary condition for a given superpartial J to move in the positive direction of plane i (fig. 52) is that FiJ is positive. In view of its complex nature (section 4.2.1.2), lattice friction, which is essentially a dynamical process, cannot be included analytically in the expressions of the forces. It must be discussed separately from the other terms. In the expressions of the forces, the first term is the external force, ri is the shear stress resolved in plane i. Given the projection plane, this term is the same for both dislocations and with regards to the above axis orientation (fig. 52) it is always positive. The second term is the elastic interaction between companion superpartials, it is of opposite signs for T and L. The third term in FiJ represents the back force exerted by the APB and its sign depends on the motion envisaged for the dislocations (fig. 52). The meanings of the different quantities in FiJ are as follows. E is the prelogarithmic factor of the superpartial self-energy. The expression of r is written r
-
al2o+ x/~l 2 + 2/o/c,
(12)
where A+2 OZ ~---
(~3)
Microscopy and plasticity of the L12 "Yt phase
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369
Table 4 Values of the elastic coefficients A, the Zener anisotropy parameter, and c~ as defined by eq. (13), in selected LI2 alloys (the values for Pt3AI and Ni3Si are calculated from interatomic potentials). These elastic coefficients are useful for the study of the properties of a dissociated superdislocation in the L I2 structure. Alloy A c~ Reference Ni3Fe 2.54 0.97 Korner et al. [152] Ni3AI 3.31 1.08 Bontemps-Neveu [23] Ni3(A1, Ti) 2.54 0.97 Yoshidaand Takasugi [ 183] Co3Ti 3.25 1.08 Oliver [ 147] Ni3Mn 2.9 1.02 Koiwa 1991: unpublished results Ni3Ga 3.17 1.06 Yasuda and Koiwa [289] Ni3(Si, Ti) 2.79 1.01 Koiwa 1991: unpublished results Ni3Ge 1.72 0.80 Koiwa 1991: unpublished results Pt3AI 1.30 0.68 Fu and Yoo [258] Ni3Si 1.91 0.52 Fu and Yoo [259] is an elastic parameter of the crystal 15 (A is the Zener anisotropy parameter). The APB energy ratio r is defined as 16 7c 3'0
m
r
)~o )~c
(14)
where the length )~i -- E/3'i is the equilibrium width under no applied stress of the planar configuration in plane i (i - o, c). The external stress is entirely defined by two parameters, e.g., the dimensionless shear stress resolved in the octahedral plane %b
s = ~,
(15)
3'0 and the usual load orientation factor N (eq. (2)). Insofar as numerical estimates are concerned, note that -0<~
- s i n c e A ~ 3.3 in d o c u m e n t e d Ni3Al-based alloys, then a ,,~ 0.93 and c~ ~ 1.08 (table 4), - ~ is, in the same alloys, a little less than c~ (Saada and Veyssi~re [243]; see tables 3 and 4). during straining experiments, both quantities s and N are usually less than unity. -
Given the set of parameters {~, 3'o, if, s, N } , the state of the system is entirely defined by the APB widths lo and lc. In a {lo, lc} diagram, the system is represented by a point located in the first quadrant and the condition Fi J - 0 is associated to an ellipse, n a m e d 2 J (fig. 53; Saada and Veyssi~re [243]). 15For the sake of simplicity, we shall also make use of the elastic constant c~ which is the reciprocal of a (~ = ~-:). 16Similarly, the APB energy ratio ( is the reciprocal of z introduced by Saada and Veyssi~re [257].
370
Ch. 53
P. Veyssidre and G. Saada
(a)
IC
~'c
Ec
~'0
Io (b)
~o
Ic
Eo
Io aZ'c ~'o (c)
Ic
~'c~ C ~ S Eo
~c
~'o
Fig. 53. The three possible domains for the parameter (. Z'o and X'c, represent the configurations {/o,/c} where the total forces F,L and F L are cancelled. These forces are positive for a representative point located inside of Z'o and Zc, respectively. (a) ~ < c~, the two ellipses do not intersect and there is no equilibrium configuration in which these forces cancel simultaneously. The stable configuration is planar in the cube plane (KW). (b) ~ > v/3, again a twofold configuration cannot be in equilibrium, and the system should evolve towards the planar configuration in the octahedral plane. (c) c~ < ~ < v/3, the twofold configuration is under equilibrium at S{/,~',/c }, but this equilibrium is unstable: when lc > l c (or/o < /,~), a twofold configuration transforms into a KW (outer ellipse) whereas when lc < l~' (or l,, > l,]), the eventual configuration is planar in the octahedral plane. It can be remarked that for/Jcump > Ao/x/r3, a screw superdislocation should remain split in the cube plane./lcUmp is defined in fig. 32(d).
Microscopy and plasticity of the L12 ~ff phase
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4.2.2.2. The unstressed case (Saada and VeyssiOre [257]). Consider the twofold configuration of fig. 52(a) in which L and T slip on the octahedral and on the cube plane, respectively. Of course, L and T play entirely symmetrical roles in the unstressed case, but this notation is nevertheless kept for consistency with the stressed situation. Equilibrium occurs when the relations
r
(alo +/c)Ao = 0,
(16)
r
(lo + lcV )r
(17)
= o
are simultaneously satisfied, that is, when the two corresponding ellipses intersect one another within the first quadrant of the {lo, lc} representation (fig. 53). Three distinct situations may occur according to the value of ff relative to c~. (i) ff < c~: there is no ellipse intersection (fig. 53(a)) and the only stable situation is the complete KW. A superdislocation initially extended in the octahedral plane, crossslips spontaneously in the cube plane under the action of the force FcL which is written as
F~ = 7o(O~ - ~).
(18)
(ii) ff > x/~: again the ellipses do not intersect (fig. 53(b)). Hence, the only stable situation is the planar octahedral configuration. A superdislocation extended in the cube plane cross-slips in the octahedral plane under the action of the force FoL given by Fo
-
"yo(r -
.
(19)
(iii) c~ < ff < x/~: the two ellipses intersect (fig. 53(c)) at the point S with coordinates {lo, 1c }, where
lo -
Ao ~-2 _ 2 o ~ + c~x/3'
(20a)
~--O~
Ic - Ao ff2 _ 2o~ + c~v~"
(20b)
The incomplete KW is an equilibrium solution but, as a saddle point of the energy, its equilibrium is unstable. A planar superdislocation extended in either plane is metastable. However, the activation energy to transform one planar configuration into the other is in general prohibitive, so that the planar configuration cannot cross-slip without the help of an applied stress. The dissociated screw superdislocation can thus be regarded as a bistable system. So far every L12 alloy that has been found to correspond to case (i) does exhibit a positive TDFS. Unfortunately, the APB energies have not been measured for L12 alloys which do not exhibit a positive TDFS. The transformation from a planar configuration located in the octahedral plane into a Kear-Wilsdorf lock is represented graphically in fig. 54.
P. VeyssiOre and G. Saada
372
o Ic A ; L e n A "
Ch. 53
TO___ )~o L ................................. "......... 0 Xo ::.::::::::::::::::::::::::::::::::::::::::::::::::::::::
Fc = 0
(b)
O
.:i: .i5:
"r
.i!'
Io
A '0
...............! ! ;..... ! ! : " ' :..... "?
3.0 o ~ c (a)
O .:~: #? 3-' .:v
B , O .....................:.~:
? ;57
:;i: .5?;"
C
:?"0 .:iF
i ,q:
61o
:?:"
d .iif
/!;i: ~"C .ii:
Xo o~ ~c
C O
(c) Fig. 54. Graphical analysis of a possible evolution of a dissociated screw superdislocation, the intermediate configurations represented end-on in (b). Each of the schemes (O, A, B, A t, B t, A ' , C t and C) relates to a representative point in the chart of (a). In (b) heavy lines represent the difference between the situation under consideration and the preceding one located immediately on top of it in figure (b). We consider a "usual" L12 crystal, i.e., such that ~ < c~, in which a screw configuration is initially dissociated in the O configuration represented by the point O{Ao, 0} on L'o. In this case Fc~ = 0 and FcL > 0, which promotes the cross slip of L in the cube plane. Suppose the position of T is fixed momentarily, then L cross-slips onto the cube plane over some distance until the representative point of the twofold configuration hits the outer ellipse, Zc, and this defines a point A{Ao, lc } which is located outside Z,,. At this stage, F~ = Fcx = 0 and FoT > 0. If, for the sake of simplicity, it is now assumed that L can be fixed in turn and T is freed, then T moves in the positive direction and the representative point shifts horizontally until it hits the inner ellipse, S,,, at B{lo, ld}. The screw segment has now adopted a twofold shape and the signs of the forces are the same as for the initial configuration O. Configuration C{0, Ac }, which is the only stable one in the crystal, can be reached after several A -+ B --~ A t -+ B t steps, whose number depends on the relative positions of the two ellipses. (c) A more realistic, yet oversimplified, path for the transformation of (a): when it is considered that lattice friction is large in the cube plane and negligible in the octahedral plane, the excursion 61c, of the representative point off the inner ellipse Zo is modest.
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Microscopy and plasticity ~( the LI 2 ,.yt phase
373
4.2.2.3. The effect of an applied stress (Saada and Veyssikre [243]). We focus on alloys where ~ < a, that is, such that the KW configuration is the most stable in the absence of an external stress. The effect of an applied stress can be treated along the same lines as in the above development. It should first be noted that when both superpartials are allowed to glide freely in the same plane, the equilibrium width /~i of the planar superdislocation is maintained during its motion. The fact that friction is by far the larger in the cube plane provides a simple priority rule for slip in which octahedral slip can be regarded as the fastest, while cube slip is delayed. Nevertheless, since friction is thermally activated, there is a finite probability for cross slip on the cube plane to occur at any stage, which we shall however consider only when the appropriate force F J is positive. In their simulations Chou and Hirsch [199] have chosen to adopt strict geometrical (crystallographic) constraints whereby the spreading symmetry of superpartial core may hinder slip in one or other of the octahedral planes; the choice of the forbidden plane is determined by the atomic environment of the (110) atomic row where the superpartial is sitting (fig. 51). We have preferred to relax these constraints, conforming to the above assumptions on superpartial cores (section 4.2.1.1) and on lattice friction (section 4.2.1.2). Furthermore, ignoring core symmetries agrees better with the idea of a dynamical transformation wherein mobile cores are not necessarily localized with precision in planes. We now consider the transformation schemes of fig. 55. In the course of the motion of a superdislocation in the primary octahedral plane, one of its superpartials, say L, cross slips over the distance/jump in the cube plane before it may be regarded as immobilized by viscosity (step 1). lJcump can be of varied magnitude, consistent with the analysis of lattice friction made in section 4.2.1.2. When T has reached its equilibrium position in the octahedral plane, the force on L becomes FoL - 2rob, which implies that the configuration in step 1 is in principle always unstable. In view of the above priority rule between slip planes, and although the formation of a complete KW by motion of L all the way through in the cube plane (step KW) cannot be totally excluded, L would most likely cross-slip onto the octahedral plane and then glide over some distance; T is then provisionally freed in the octahedral plane and both superpartials are now simultaneously mobile on two parallel octahedral planes (step 2). As soon as it has erased all the APB in its own octahedral slip plane, T becomes locked and is subjected directly to the driving force from the APB in the cube plane; meanwhile, L proceeds until FoL vanishes (step 3), the APB width on the octahedral plane, lot, is given by solving FoL{/JcUmp}-- 0 (expression (1 l a)). It should be noted that the distance L has slipped in the octahedral plane may then be larger than Ao (i.e., for small/jump, the slip distance is approximately [Ao/(1 - s)]). The screw segment has thence achieved a so-called APB jump by a double cross slip of L. In three dimensions, this process is accompanied by the nucleation of a pair of elementary kinks (EKs, section 3.4.3.4) at both ends of the cross-slipped segment (fig. 32). At step 3, what matters is the force F L that pushes L onto the cube plane:
- If F L > 0, L may cross-slip before T has erased the APB on the cube plane. The smaller s (i.e., the distance between T and L) and the larger N (i.e., the external shear
374
P. Veyssidre and G. Saada
Ch. 53
stress resolved in the cube plane), the larger FeE. As a result both superpartials glide in the cube plane (fig. 55, step 4) until either L is stopped by friction, and/or T has completely erased the APB on the cube plane, at which stage the system transforms into a configuration similar to that of fig. 55, step 1. Provided the motions of L and T are adequately synchronized, this sequence (fig. 55, step 1 to step 4) may then repeat itself several times giving rise to the process of repeated APB jumps and, in three dimensions, to multiple pairs of EKs with opposite signs (fig. 32). If FcL < 0, L is ascribed to the octahedral plane until the configuration becomes planar by motion of T in the cube plane. Then the superdislocation is freed in the octahedral plane where it automatically assumes its equilibrium width under no stress, )~o (step 3 to step 0). It is then liable to undergo a jump of unpredictable length. In three dimensions, this motion results in the formation of a simple macrokink (fig. 56, step (a), then step (e) and section 4.2.3.2). Note that, just like an EK, a simple MK requires that the leading superpartial has undergone a double cross slip. The sign of FcL is not only dictated by the magnitude of the stress in the cube plane, but also by that in the octahedral plane. This is because, given/jump, the interaction force between superpartial is a function of lo, which is itself affected by the resolved shear stress %. FcL is always positive, including small stresses, when N > (, which, in the unit triangle, corresponds to the [111] side of the (111) great circle. When, on the other hand, N < (, for Fct" to be positive, s has to be smaller than a critical stress s' given by n
s'= c~-(
oe-N"
(21)
This implies that, in this particular range of orientations, repeated APB jump, will occur for small stresses (see section 4.2.3.2.2). When now N < ( still and s' < s < s", where
x/3-r , v~-N
s" = ~
(22)
the sign of F~ depends on the extension of the incomplete KW in the cube plane,/jump, relative to a stress-dependent critical length, 1c {s, N}, (fig. 53(c)). FeE is negative when lieump < lc{s , N } and otherwise positive. The critical length lc{s , N } is obtained from eq. (20b) by substituting ff for the adequate effective APB energy ratio, which in this particular case, expresses as
~=
~-Ns 1-s
(~ - (, for s = 0). Finally, when s > s", F~ is always positive.
(23)
Microscopy and plasticity of the L12 ~/~ phase
w
(~
\
X
"
i ..................... o ......................
i
.~0
Icjump i ~"
~
1st APB jump
,j'"
375
(IKW)
.:i!ii:
(0)
2nd APB jump
Fig. 55. Transformations of a superdislocation initially glissile and mobile in the octahedral plane (step 0). At step (1), the leading dislocation is being immobilized by friction after it has cross-slipped in the cube plane over a distance, vc/Jump, which is controlled by thermally-activated lattice friction. The trailing dislocation proceeds to a distance Lom which is smaller than the equilibrium dissociation distance in the octahedral plane under no stress, )~o. Then the configuration may evolve towards an incomplete or a complete KW configuration (IKW and KW, respectively) or else L can be freed in the octahedral plane (step 2). Then both partials glide simultaneously on parallel though distinct octahedral planes until T meets the cube cross-slip plane where an APB had been created by the leading partial (step 3). At this stage the position of the leading superpartial is fixed at LoM > /~o, which is dictated by the applied stress. Again, there is a choice in the further evolution of the configuration. If the trailing superpartial erases the strip of APB on the cube plane before the leading itself cross-slips on a second cube plane, the superdislocation can become fully glissile in the primary octahedral plane (step 0). This part of the superdislocation will then slip over a distance dictated by the probability of cross slip of L on the cube plane, resulting in a MK. If now the leading superpartial has jumped in the cube /.jump plane over vc , which is not necessarily the same as in step (1), then the process of APB jump can repeat itself. At step (3), the choice between step (0) and step (4) is biased by the applied stress. This can be roughly understood as follows. If the stress is small, LoM is small, thus the force F L on the leading dislocation in the cube plane is positive. There is a driving force for the leading dislocation to pursue its motion in the cube plane. If, on the other hand, the stress is large, LoM becomes too large for the interaction between superpartial to help the leading dislocation to cross-slip in the cube plane.
376
Ch. 53
P. Veyssidre and G. Saada
(111
(olo) !
(c)
i (olo)
~~,~
i
(o~o)
zU
(e)
~
j
,j~~
(z zz
{010) Fig. 56. Comparison between the process of a repeated APB jump (a) to (d) and that of the formation of a simple MK of height hMK (a) and (e). In fact, fig. 56(a) is the next step of the last configuration of fig. 32, represented in fig. 32(f) after the short central KW segment is unzipped. Both the repeated APB jump and the simple macrokink are formed by double cross slip of L and T. The end-product depends on the velocity of T relative to L.
Microscopy and plasticity of the L12 "7~ phase
w
377
S t
S!
ffl t__
"0 N
~>~/3
I
"
-
E 0
z
S III
I ,
o
i
N*
i ,
I
r~
~
~3
Orientation factor N Fig. 57. Orientation dependence of the critical stresses s ~, s" and s H~. Repeated APB jumps may occur for stresses below s/, while they cannot occur above s H. When s ~ < s < s', the occurrence of APB jumps will depend on the value of lc. For N larger than N*, the microscopic saturation stress, s H~, is always smaller than s ~/, so that KW destruction occurs first. When N < N*, s ~ is smaller than s~; in this domain of stress and orientation, the process of APB jumps is the most favourable. Obviously, the same formalism can be used in order to calculate the stress, s Ht, required to destroy a complete KW
s'" =
v/3-~ v~+N
(24)
which may be considered as a microscopic saturation stress (sections 4.2.2.3 and 4.2.3.5) above which KW locks all become unstable, that is, the strongest obstacles against slip are all destroyed. The stress s' is larger than s'" when the load orientation is such that N passes a critical value N* of the order of 0.4. The orientation dependencies of the stresses s', stt and s H~ are represented in fig. 57, which maps the domains of operation of the different mechanisms of transformation of a screw segment. Finally, let us consider L le alloys where screw superdislocations are bistable, that is, alloys such that c~ < ~ < v ~ , if any. The above formalism predicts that a KW lock may be stabilized as well; however, the transformation is no longer spontaneous (Saada and Veyssi~re [243, 257]). In this case, metastable KWs can be nevertheless stabilized by adjusting the adequate effective APB energy ratio ( to a value exceeding v/3. This can be actually achieved by means of an appropriately oriented external load. Considering a superdislocation gliding in the octahedral plane in the bistable situation, the smaller the shear stress in the cube plane, the more difficult the creation of a KW lock. In these conditions, L12 alloys such that a c~ < ~ < v/3, if they exist, should exhibit almost no flow stress anomaly in a region of the unit triangle located in the neighbourhood of the [001] direction. On the other hand, it is even possible to adjust the applied load so as to obtain ( > v ~ , in which case an incomplete KW should always tend towards the octahedral configuration. Since superdislocations are then no longer liable to form locks, such alloys, if any, should not exhibit a flow stress anomaly.
378
P Veyssidreand G. Saada
Ch. 53
This simple reasoning thus predicts a category of L12 alloys (~ < ~ < v/3) whose TDFS might appear and disappear according to the orientation of the load axis. 4.2.3. The motion of an individual superdislocation in an octahedral plane This section is restricted to analysing the elementary mechanisms that give rise to a single kinked superdislocation (at this stage, no hypothesis is made on the nature of superdislocation sources). For this aim, we first consider the formation of KW segments (section 4.2.3.1). We then detail different mechanisms by which this screw superdislocation eventually becomes kinked (section 4.2.3.2), and we explore the implications of the hypothesis that macrokinks may be moved in the octahedral plane by application of an external load (section 4.2.3.3). The relationship between the nature of macrokinks and planarity of slip is discussed in section 4.2.3.4. Finally, the strength of KW locks as well as its influence on mechanical properties is analysed in section 4.2.3.5. The expansion of a superdislocation within a pre-existing microstructure, including source operation, is analysed in section 4.3. 4.2.3.1. The locking in the screw orientation and the formation of KW locks. As mentioned earlier, TEM observations consistently show that locked segments are by far l o n g e r - at least by one order of magnitude - than implied in the pinning point models. Pope [156] has offered a tentative rationale of this contradiction. His explanation is based on the transformation by relaxation of the microstructure into a succession of KWs and MKs (fig. 50). It relies essentially on a scheme that assumes that the pinning points are rigorously aligned along the screw orientation over the extent of the future relaxed KW segment 17. Although unlocking controls the motion of an isolated superdislocation, we have found it instrumental to understand the conditions of formation of incomplete KW segments, that is, the conditions of locking, since they dictate the further evolution of the superdislocation in its near-screw orientation. The following shows that the locking is not a completely trivial process. Let us consider first a circular dislocation loop of radius R and a chord AB of length X, parallel to its Burgers vector (fig. 58). The chord is located at a distance ( R - h) from the centre of the loop. Provided h is small enough, say of the order of the Burgers vector of a complete superdislocation (2b ~ 0.5 nm), it can be assimilated to a screw segment. It can be considered as undertaking a favourable situation for cross slipping in the cube plane when its length X is long enough to hinder the backward effects of line tension that are exerted by the lateral jogs (CJ). When h << R, the corresponding radius of the arc is
)C2 R -- - - . 8h
(25)
17At first sight, this might nevertheless look in accord with outputs of the subsequent computer simulation of Mills and Chzran [241], but it should be noted that the simulation did not assume a steady-state breakaway mechanism. The simulation takes into account, in addition, some correlating rule between consecutive pinning points.
w
Microscopy and plasticity of the LI 2 ~/i phase
379
,,..._
"-
A
~
B
Iq-h
O Fig. 58. Determination of the critical length beyond which cross slip may occur. Although the reasoning on cross slip from an octahedral to a cube plane in L12 alloys cannot be conducted exactly along the same lines as that in f.c.c, crystals, it is still reasonable to assume that at minimum, X should be of the order of three to four times the splitting width of the superdislocation, Ao, that is about 20 nm. Hence, a near-screw dislocation segment whose radius of curvature is larger than about 80 nm is liable to cross slip. In view of the stresses which are applied in domain B, this condition should be always fulfilled in practice. In L12 alloys such that ~ < c~, which so far is the most general situation, a screw superdislocation is not stable when dissociated in the octahedral plane. Under dynamical conditions, it has nevertheless a given lifetime in this glissile configuration. The driving force for its transformation into a KW configuration depends upon several factors: - an elastic torque, proportional to ( a - ~), which drives the leading superpartial onto the cube plane (eq. (18)); - lattice friction which tends to hinder the transformation in a viscous manner (section 4.2.1), which call in turn for the following comments: (i) The driving force for cross slip is not accurately known. Since in all cases of interest c~ is close to 1.1 and since the average value of ( amounts to 0.8 + 0.25, the uncertainty in the actual driving force, under no external stress, may be in fact as large as one order of magnitude (i.e., 0.05 < FcL/7o < 0.55). This difficulty, which stems from an inherent experimental inaccuracy in the determination of ~ (section 3.1.2.1), should be kept in mind when testing a theoretical model, in particular when the driving force enters an exponential. A theoretical treatment that includes this driving force can hardly claim accuracy. (ii) The flow stress anomaly of a L 12 alloy is usually analysed in terms of its propensity to produce KW locks. This, in turn, is sometimes discussed directly through the Zener anisotropy ratio A, with reference to the torque term (see for instance, Baluc et al. [ 143], Yoshida and Takasugi [150] and Takasugi and Yoshida [39]). It is in fact, incorrect to make use of this single criterion because what actually counts is the difference between and a. (iii) APB energies on the cube and on the octahedral planes are usually given with similar accuracy (section 3.1.2.1). With regard to lattice friction on the cube plane, this
380
P. VeyssiOre and G. Saada
Ch. 53
is a little surprising, especially at moderate and low temperatures where a sometimes significant fraction of incomplete KWs can be observed within post-mortem deformation microstructures. That KW locks are in fact at equilibrium is hard to ascertain. Why, despite the magnitude of lattice friction, a dissociation width in the cube plane can still be measured in practice on screw segments with some reliability may be explained in the following way: (i) the uncertainty in separation widths is of the order of the image width of a superpartial, approximately lnm, while this separation amounts to 4 to 6 nm, which implies a precision in the equilibrium separation amounting to up to 20%; (ii) because the friction stress is not a threshold stress (section 4.2.1.2), KW configurations are initially formed incompletely by motion of the leading partial in the cube plane over/Jcump. Their subsequent evolution towards complete KWs is gradual. This is achieved at a rate determined by the thermal history of samples between the moment at which KWs are formed during mechanical testing and the time of their observation. What the finding of a nominal separation suggests is that this period of time, of several tens of hours, is sufficient for the transformation to be completed (within the accuracy of TEM observations) even at room temperature. However, one should still envisage the possibility that at moderate temperatures, Ic is systematically a little less than the equilibrium width )~c, which would cause r to be systematically overestimated. If the locking rate in the screw orientation were small, a superdislocation would expand as a roughly elliptical loop, with a shape itself determined by anisotropic elasticity (Douin et al. [139]). A higher locking rate alone does not however explain the observed elongation of the microstructure parallel to the screw orientation. In order for long KWs to be formed, CJ segments must propagate laterally, or zip, in the screw direction fast enough in order not to be bypassed by the expanding dislocation (Mills et al. [200], Veyssi~re [3]). This critical part of this process is sketched in fig. 59. The competition between the bypassing and the zipping processes is controlled by the difference in velocities between the closing kink CJ and the bypassing MK [3]. Its treatment is not trivial. So far, the only attempt at predicting the distribution of KW lengths in the microstructure has been made by Mills and Chzran [241] by means of a computer simulation (section 5.2.4.1). Depending on line tension, a KW nucleus such as that schematized in fig. 49 could either lengthen or collapse by mutual annihilation of the pair of kinks in the cube plane, CJ. Conditions for the zipping/unzipping of cross-slipped segments have been discussed quantitatively by Hirsch [99, 242] whose own estimate is that a pinned segment should be unstable beyond a critical angle ~ of about 2 ~ Hirsch [100] then emphasized that in the most recent version of the point pinning model (Khantha et al. [101,260]), in view of the expected mean distances between pinning points, the bowing of screw segments must be far too pronounced for the pinning points to be stable. In this section we first recall very briefly the differences between the three possible categories of kinks in the octahedral plane (section 4.2.3.2.1). We then address directly the mechanism of repeated APB jumps (section 4.2.3.2.2), which we examine in three dimensions (section 4.2.3.2.3). Finally we discuss the implications of these considerations on kink distribution.
4.2.3.2. Kink properties.
Microscopy and plasticity of the L12 ~ phase
w
381
Ca
(a)
(b)
Fig. 59. As a kink (MK) moves under an applied stress, two competing mechanisms control the evolution of the structure of an individual superdislocation. (a) A length of KW segment is produced by a zipping mechanism. In this case, the closing segment CJ glides more or less at the same velocity as the kink MK. (b) The KW segment is bypassed as the mobile kink expands in the octahedral plane faster than the closing jog (CJ) can slip in the cube plane.
4.2.3.2.1. The mechanisms of kink formation, the planarity of slip.
We have seen in section 3.4.3, that kinks may be sorted out into two different families according to whether their height is of the order of the APB width in the octahedral plane (elementary kinks, EKs, corresponding to the mechanism of APB jump) or much longer (macrokinks, MKs, consisting in simple or in switch-over MKs). However, with regard to formation mechanisms, it appears more appropriate to group simple MKs and EKs and to distinguish them from switch-over MKs. Both an EK and a simple MK result from the double cross-slip motion of the leading superpartial (figs 32 and 55). By contrast, when the leading superpartial remains in the cube cross-slip plane, so that it is lost for octahedral slip (fig. 30), a switch-over MK can be produced by the bypassing mechanism that has been briefly outlined in the preceding section (fig. 59(b)). Couret et al. [72] have measured the frequency of kinks of different kinds and found that approximately 25% of the MKs are switch-over in nature and that this percentage is almost independent of temperature. These differences have implications on the conditions of operation of octahedral slip. For both EKs and simple MKs, a portion of the moving superdislocation line is transferred by a double cross-slip from a given octahedral plane to another parallel one (figs 30, 32 and 55). When double cross slip operates repeatedly on an expanding superdislocation, it gives rise to a loop that is stepped over a number of parallel octahedral planes (fig. 60(a)). By contrast, in the bypassing mechanism (switch-over MK), a superdislocation remains confined to its initial octahedral slip plane (fig. 60(b)).
382
P. VeyssiOreand G. Saada
Ch. 53
(a)
(b)
Fig. 60. The relationship between the mechanism of KW lock formation and the planarity of slip. (a) Simple MKs, octahedral slip is distributed amongst several parallel slip planes (see also fig. 61). (b) Switch-overMKs, all the mixed segments are contained in the same octahedral slip plane. 4.2.3.2.2. The repetition of APB jumps. For the sake of simplicity, the threedimensional organization of kinks will be examined in the next section. For APB jumps to occur one immediately after another, the motion of companion superpartials would have to be synchronized appropriately. Typically, the leading superpartial must have cross-slipped in the second cube plane before the trailing has succeeded in completely erasing the APB strip on the first cube plane (section 4.2.2.3, compare figs 56(d) and 56(e)). Paidar et al. [236], Hirsch [99], Chou and Hirsch [199] and Mol6nat et al. [237] have postulated that equality between the forces on L and T ensures that companion superpartials will slip at the same velocity on the cube plane and that, once the process is initiated at the level of step (3) in fig. 55, it repeats itself indefinitely. Furthermore, in their computer simulations of the transformation on a dissociated screw superdislocation, Chou and Hirsch [199] have postulated that cross slip would occur only when superpartials are assigned fixed positions in the lattice (for more details, see fig. 51 and the discussion in section 4.2.1.1). For small/jump, Chou and Hirsch [199] find that the condition F L - FcT implies that APB jump repetition occurs at "sufficiently large stresses", of the order of 160 MPa. They in fact determine a minimum stress for the operation of repeated APB jumps, a conclusion which conflicts with the fact that this mechanism is part of the microstructure of deformation at room temperature (section 3.4.3.4.1, Couret et al. [72, 203]) that is, in part of domain B 1, when the flow stress is the lowest. We recall that the necessary condition for the repetition of APB jumps (e.g., F L > 0 section 4.2.2.3) is fulfilled at any level of the dimensionless stress s, provided N > (. When now N < (, - APB jumps may occur repeatedly as long as s < s'. Taking N = 0 and the parameters used by Chou and Hirsch [199], this defines a maximum stress approximately equal to 100 MPa; - when s' < s < s", the system is bistable. When /Jcump > lc{S , N}, the system has a finite probability to undergo the transition from the incomplete to the complete
w
Microscopy and plasticity of the L12 "y~phase
383
KW (step (3) to step (4)) in fig. 55). On the other hand, when/jump < l c { s , N } , the incomplete KW cannot evolve towards a complete KW, which implies that it will undergo an APB jump sooner or later. We expect the cross-slipped distance/jump to increase as lattice friction on the cube plane weakens (sections 4.2.1.2 and 4.2.2), i.e., as temperature is raised. At low temperatures, /jump is a small number of interatomic distances, which favours the production of APB jumps. As the temperature is increased, /jump becomes eventually larger than 1c {s, N}, at which stage the twofold configuration should evolve towards the KW lock. Since cross slip is a thermally activated process, there is a finite probability for L to cross-slip before T has erased the APB on the cube plane, and there is still a finite probability, however smaller than for a single APB jump, for the sequence (1) to (4) in fig. 55 to occur several consecutive times. In other words, APB jumps together with repeated APB jumps should be regarded statistically rather than deterministically, as has been done elsewhere. The origin of the discrepancy between the present results and others stems from the fact that the synchronization of companion superpartials is not reduced to the equality between the force applied onto the leading and the trailing superpartials. In favour of a statistical point of view is the observation that the number of sequences of repeated APBs which involve a given number of EKs, decreases with this number (Couret et al. [72]). It may be remarked in addition, that though sequences of EKs are observed in the largest quantities after deformation at modest temperatures, yet a few examples of successive EKs in Ni3(A1, Hf) have been found after deformation at 650~ (Bontemps-Neveu [23]).
4.2.3.2.3. The polarization of elementary kinks. The EK distribution is almost systematically polarized (figs 36 and 37). By kink polarization, we mean a situation where kinks of both signs are not equiprobable on the same dislocation line. A microstructure where EKs of both signs are interspersed along the same screw superdislocation would attest to low EK mobility. What occurs in L12 alloys is that in the course of superdislocation expansion in the primary octahedral plane and because of the favourable elongation along the screw direction, the distribution of mixed parts - in particular of E K s - self-organizes so as to accommodate the mean local curvature imposed by local stresses (fig. 38). Hence, the polarization of the EK distribution is a local property of line tension dictated to a large extent by the applied stress; it originates from the ease of EK propagation along the screw superdislocation. Experimentally, EK mobility is further supported by room temperature in-situ experiments wherein very long portions of screw superdislocations are seen to undergo the APB jump process "as a whole", that is, within the time period of a video frame 18 (Caillard et al. [190]). 4.2.3.2.4. Remarks on kink distribution. As a screw superpartial slips in the closepacked octahedral plane, it has a finite probability to deviate (over llcump) into the cube plane. Whatever its formation mechanism, i.e., by double cross slip or bypassing, the 18EKs are in general not seen in situ since standard recording does not allow to resolve features moving faster than approximately 250 ~m s-l under the magnifications that are usually employed under weak-beam conditions.
384
P. Veyssidre and G. Saada
Ch. 53
height of a macrokink, hMK, depends directly on the free flight distance of a screw segment in the octahedral plane (fig. 56; steps (a) then (e)). Let us consider a straight superpartial segment of length L, let fcs be the cross-slip probability per unit time and per unit length (eq. (9)). The probability per unit time of cross slip of the dislocation segment is given by p = fcsL. Therefore, provided the time required to nucleate cross slip is much larger than the flight time, the number nMK of segments whose free flight distance is hMK can be written as an exponential decay nMK{T , N, s} = no exp {
p
-- ~oohMK
}
(26)
where Vo is the dislocation velocity in the octahedral plane. This form of the kink distribution is exemplified in fig. 35. As was first verified experimentally in Ni3Ga deformed at 20~ 200~ and 400~ by Couret et al. [72, 202], p increases with increasing temperature. However, the analysis of MK distribution may remain quite uncertain when it is affected by subsequent interactions. (i) As for EK polarization (section 4.2.3.2.3), there is a possibility that two neighbouring MKs, heights h~K and h~K, will coalesce and leave an MK whose resulting height, hMK -- h~K + h~K, is not related to the thermally-activated process of locking. If kink dragging takes place to some significant extent (section 4.2.3.3), this process should result in noticeable flattening of the MK distribution and the question as to why histograms fit an exponential decay so well remains open. We show in the following that the coalescence rate of kinks is probably quite significant, at variance from what the agreement between the experimental distribution and expression (26) indicate. (ii) Suppose that screw superdislocations undergo repeated APB jumps only. As the superdislocation continues to expand in the octahedral plane, more EKs are pushed to both extremities of the screw segment, where the deviation from screw orientation increases as the average distance between EKs gets gradually smaller. At some stage, consecutive EKs should eventually coalesce and yield macrokinks with a height equal to an integer number of elementary heights hEK. The macrokink of height mhEK that is formed after the coalescence of m EKs differs in structure from a simple MK of the same height by the fact that it contains ( m - 1) jogs in the cube plane (CJ), regularly distributed along its line. Such a macrokink (e.g., a multiple EK), is exemplified in figs 36 and 37 and schematized in fig. 61. In the micrographs, a succession of cusps whose length scales with that of the APB jumps can be distinguished along the mixed segment. The process of kink coalescence offers a general explanation of why some mixed segments often show considerable cusping on a fine scale. Cusped MKs must indeed be significantly less mobile than the coplanar simple MK because of the distribution of the jogs that would slip exclusively in the cube plane. The mobility of such MKs is in addition lessened by the process of trailing of APB tubes [285] (fig. 40). Hence, the coalescence of EKs and simple MKs may explain the presence in thin foils of a number of mixed segments whose length is much larger than the critical length lc -- c~#b/rc and also explain why the mixed segments may remain heavily bowed out after the applied load is removed. It should be noted that because coalesced switch-over MKs are planar while coalesced simple MKs or EKs are jogged, the latter should be much less mobile than the former (sections 4.3.1 and 4.5).
Microscopy and plasticity of the LI 2 ~! phase
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385
(a)
9IcJump on (01 O)
(b)
Fig. 61. Formation of a jogged MK upon coalescence of EKs. (a) The individual EKs are pushed by the applied stress along the screw direction. (b) The leading EKs have coalesced to form a jogged MK, five EKs long. Since the mobility of jogs in the cube plane is less than that of dislocation kinks in the octahedral plane, the MK adopts a cusped aspect with a periodicity that scales with the amplitude of an APB jump. (iii) By contrast with the distribution of EKs, that of MKs does not seem to be clearly polarized (figs 26 and 27), which may imply that MKs are less mobile than EKs. This could also originate from the preceding argument on the jogged (sessile) fine structure of some MKs. It can therefore be concluded that kinks are not straightforward features. Their properties are far from being understood satisfactorily. Some theoretical work remains to be done in order to understand the relationship between the distances between kinks and their height. Many more determinations of MK (which would discriminate between jogged and unjogged MKs) and EK distributions- heights and d i s t a n c e s - are needed before we have a clear understanding of the experimental situation. So far, the dynamic simulations of Mills and Chrzan [241 ], constitute the only attempt at predicting kink distribution, based on three-dimensional properties of superdislocations in the L 12 structure. This work will be analysed in more detail in section 5.7.1.
4.2.3.3. The viscous sliding of macrokinks (Saada and Veyssik.re [206]).
The applied force on a given MK is proportional to its length hMK. On the other hand, it is subjected to a drag force, itself being the resultant of two contributions:
386
Ch. 53
P. Veyssidre and G. Saada
~
~
o
Fo Fig. 62. The viscous sliding of MKs upon bending of KWs. At low temperatures, superdislocations are not significantly bent in the cube plane (a) so that MKs (examplified here as a simple MK) slide freely on the octahedral plane (b). As a result KWs are transported conservatively by a zipping/unzipping mechanism. At higher temperatures, thermal activation becomes enough to allow for some bending of the KWs in their dissociation plane (c). Then for MKs (examplified here as a switch-over MK) to undergo the same zipping/unzipping mechanism, the bent superdislocation has to be brought back at the intersection between the cube and the octahedral planes ((d) dashed line). Work has to be produced against the force Fc which tends to how out the KW segment in the cube plane. - the friction force in the octahedral plane which is proportional to hMK and which is small, - the friction force due to the bending in the cube plane of the KW segment (fig. 62), as explained below. This latter force is independent of hMK and is large. Since the longer a MK, the larger the force that is applied onto it, MKs of varied heigths should slip at different velocities under a given applied stress (the longer a MK, the faster). This should result in a finite coalescence rate especially in regions of the dislocation line where the curvature has to increase. When two MKs with the same sign coalesce, a longer kink is formed that may operate as a source when its length is larger than a critical height, provided the resulting kink is coplanar (section 4.3.1). It is thus important to discuss the conditions under which MKs may migrate along KW segments. We have seen in section 3.4.3.5 that as the test temperature is increased, sessile KWs tend to deviate from the screw orientation by bending themselves in the cube plane. Accordingly, Mills et al. [200] postulated that the bending prevents MKs from pursuing their displacement in the octahedral plane. We have nevertheless explored the possibility that MKs can still move by pulling the curved KW segment back to the screw orientation (fig. 62); this takes place against the Peach-Koehler force that is responsible for their curvature (Saada and Veyssi~re [206]). Hence the applied stress, which forces MKs to move in the primary slip plane, is also responsible for an additional resistance against M K motion by promoting cube bending. This is formally equivalent to some extrinsic friction exerted onto mixed dislocations during octahedral slip. Calculations indicate that (fig. 63): - MKs may slide and KW segments can thus be transported as long as the applied axial stress is less than a critical stress cra,
w
387
Microscopy and plasticity of the L12 "7t phase
GF II
IV
|
IJ a
"Cfc
r
O'a*
~fc + ~c
%
Fig. 63. Dependence of the effective friction stress O'F that is experienced by a mobile kink upon the applied axial stress cra, under the assumption that bent KW segments oppose kink slip (Saada and Veyssi~re [206]). qo,, and 99c are the Schmid factors on the octahedral and on the cube plane, respectively. When O'a is larger than a critical stress cra, the stress resolved in the octahedral plane, qo,,cra, is lower than crF. The diagram consists of four domains. In region I, KW configurations cannot bend in the cube plane. In region II, the applied stress is responsible for the viscous glide of kinks. In region III, which starts at a stress given by eq. (27), kinks are locked and octahedral slip is exhausted. In region IV, deformation proceeds by multiplication of Frank-Read sources in the cube plane. - above that stress and below another critical stress given by 2#b qOcaa = "rfc + ~ . LKW
(27)
KWs are sufficiently bent in order to exert a drag stress larger than the external resolved shear stress here Cpc is the Schmid factor in the cube cross-slip plane. MKs are then immobilized (LKw is the length of a K W segment between two consecutive M K s and "rfc represents the friction in the cube plane); -above the stress given by expression (27), bent KWs act as dislocation sources in the cube cross-slip plane. At this stage cube slip may compete with octahedral slip: when the test temperature is increased, the average M K height decreases whereas the bending is increased. As a result, M K motion is unfavoured so that new sources are more difficult to form by M K coalescence, whereas cube slip is facilitated as friction is reduced. The drag stress increases with decreasing kink height and increasing K W length. It is in addition partially reversible with temperature and it exhibits a non-Schmid dependence on the orientation of the external stress, which roughly conforms to experimental observations [206]. The viscous motion of macrokinks constitutes one of the numerous effects that may contribute to work harden L12 alloys, but it alone cannot explain the positive T D E It is
388
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P. Veyssidre and G. Saada
S,,,
/Sfc(T )
(a)
S
(b)
Fig. 64. Analysis of the peak stress. (a) The critical resolved shear stress on the octahedral plane becomes larger than the stress required to activate slip on the cube plane sfc(T) (considered as roughly controlled by lattice friction), before the microscopic saturation stress (s'" in eq. (28)) is reached. (b) Lattice friction on the cube plane is still larger than the microscopic saturation stress at the temperature it is intersected by the ascending CRSS on the octahedral plane.
not a model in itself. It is in particular unable to account for the low strain-rate sensitivity of L12 alloys, whose interpretation might be more appropriately related to the operation of MKs as sources (section 4.3.1). 4.2.3.4. The microscopic saturation stress. Other than being transported by moving MKs, KWs can be eliminated mechanically upon application of a sufficiently high stress (section 4.2.3.4, expression (24)). In this case, KWs are eliminated as soon as they are created which, in the absence of further strengthening mechanism, should correspond to the ultimate mechanical resistance of a given L12 alloy deformed in domain B. In fact, the stress at the peak "rp could originate from two different mechanisms (fig. 64). On the one hand, the applied shear stress is large enough to overcome lattice friction on the cube plane (fig. 64(a)) and the KW segments proceed preferentially on these planes. This agrees with properties derived from slip line analysis (section 3.4.1), except in the [001] orientation. In addition, it seems quite evident that this scenario applies to creep tests carried out below the peak temperature (section 2.2.9). As to the operation of cube slip, note however the significant difference between creep and constant strain-rate tests. When cube slip dominates in a test of the latter type, sources operate in the primary cube plane, whereas under creep deformation proceeds in the cube cross-slip plane, once primary octahedral slip is exhausted (sections 2.2.9 and 2.2.10). On the other hand, if friction on the cube plane cannot be overcome below the critical stress, s'", to destroy a KW (eq. (24)), the so-called microscopic saturation stress (section 4.2.2.3), the theoretical upper strength of these obstacles is reached. Instead of a peak (fig. 64(a)), this process should in principle give rise to a plateau (fig. 64(b)). Based on the analysis outlined in section 4.2.2.3, we have shown that both the peak flow stress 7-p and the peak temperature Tp should decrease with increasing N (Saada
Microscopy and plasticity of the L12 "y~phase
w
389
and Veyssi~re [261]). The predicted orientation dependence of the peak stress is given by the following expression
s'"{~,N}-s'"{~,O}~
,/3
(28)
N+V~' which agrees surprisingly well with some experimental data taken from a few binary or weakly alloyed L12 compounds. However, it disagrees with many more results obtained not only in alloys containing significant amounts of ternary additions but also in a binary Ni3A1 compound (see for example figs 8, 9 and 11). Possible causes of these discrepancies have been discussed by Saada and Veyssibre [70].
4.3. The collective behaviour of dislocations in domain B
We now aim to show that the organization of superdislocations and their interactions in the slip plane might well be as important as the behaviour of individual dislocations. Neither the strain rate that can be expected to result from APB jumps nor the strain resulting from the motion of screw superdislocations assumed to slip over some reasonable mean free path appears sufficient to achieve significant compressive ductility under usual deformation rates (i.e., larger than 10 -5 s -1) as is the case in L12 alloys. As expected, some superdislocation multiplication must occur. On the other hand, the tremendously high values of the WHR, in particular below 2% of permanent strain, together with the wide extension of the microplastic stage [286] further suggested by the strain dependencies of the WHR and of the activation volume (sections 2.2.3 and 2.2.8.2)- imply that multiplication of dislocations is difficult and/or that the annihilation rate is high. This is confirmed by the exhaustion of octahedral slip during primary creep (Hemker et al. [79], section 3.4.2), which indicates that as time proceeds the rate of superdislocation multiplication in octahedral slip is much smaller than the overall locking rate. From this point of view, the situation in L12 alloys closely resembles the stage of microplasticity in a regular alloy where strain stems from the exhaustion of mobile dislocations before multiplication has yet taken place.
4.3.1. Dislocation multiplication From the observed microstructure, it is reasonable to follow Mills et al. [200] and to consider that every MK is a potential source of new dislocations which contain in turn a fresh distribution of kinks, that is, of potentially operating new sources (fig. 65). A MK will operate as a source when its length becomes larger than a critical length given by
#b
h~K ~ - - . To
(29)
Following the results of Sun et al. [124] (section 3.1.1.2), Hirsch [99] postulated that the closing jogs in a KW segment (CJ in fig. 30) transform into a sessile Lomer-Cottrell
P. Veyssi~re and G. Saada
390 i
Ch. 53
j
h g
e
f
c
d
a
i -k.
J
h
k~l
e
a
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h
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6 e,
f
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5 c a
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a
b
b Fig. 65. Development - the figure reads from bottom to top - of a deformation microstructure from a kink M K (1) whose height has become large enough to operate as a source (2-3) is sitting at every junction between a kink and a K W portion such as at a and b. At some stage, which is governed by the probability of cross slip in the cube plane, the screw part starts to transform into a KW lock, which is marked with a star between c and d in (3). Then the kinks ac and db slip apart. The former is too small to operate as a source, whereas bd is large enough. The same process as in (2-3) repeats in different places, giving rise to kinked KWs.
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Microscopy and plasticity of the L12 "y~ phase
391
configuration (see however section 5.6.2.2.2 and fig. 74), thus anchoring the MK at both extremities (assuming that KWs are not bowed too much, Hirsch also proposed that the critical stress above which a MK is unstable should be smaller, by a factor of 6, than that for the operation of a Frank-Read source). Note that a MK could also operate as a source when the closing jogs (located at a, b, c, d in fig. 65) are significantly less mobile than the MK segment itself. Worthly of consideration is that when a kink operates as a source, the lateral part of the expanding dislocation is very often placed in dipolar situation with the adjacent outer immobile segments. It interacts, in particular, with its nearest own pre-existing branches that are located within or in the close vicinity of its slip plane. We shall see in the following section that this results in significant annihilation and/or immobilization of mobile superdislocation with immobile segments (section 4.3.2, see figs 36(b) and 66). As a consequence, a given kink cannot operate as a source more than once, by contrast with a regular source such as a Frank-Read or a pole source. It is remarkable that, in the microstrain stage (~ ..~ 10 -6 to 10-5), Ni3A1 deforms plastically at stresses which are one or two orders of magnitude less than the conventional flow stress "r0.2, while the WHR is dramatically high. This suggests that at a stress of say 10 -2 7-0.2, the amount of mobile dislocations which can be produced from the available one-cycle sources is not sufficient and that the microstructure exhausts its reservoir of mobile segments more rapidly than it generates kinks with appropriate lengths; still a large fraction of the strain is of elastic origin. In Ni3Al-based alloys 7-0.2 ranges between 10 -3 # and 10 -2 # in domain B. Thus multiplication should be restricted to those mobile segments whose length hMK is above 300 nm at room temperature and 30 nm at the peak. Within a freshly formed kinked screw line there must be some provision of adequate kinks for new sources, but it is not ensured that this provision is enough (that would explain the strong tendency for a primary octahedral slip to exhaust during primary creep). In order for the sample to deform plastically at a given strain rate, the stress must be raised to such a level that the rate of production of mobile segments is positive. We now suppose that MKs have some mobility in the primary octahedral plane as discussed in section 4.2.3.3 thus conferring the microstructure an additional degree of freedom. Then MKs larger than the critical length h~K are formed permanently by coalescence of pre-existing kinks of sub-critical size and multiplication is enhanced (we consider the coalescence of switch-over MKs which is the only one to yield fully glissile kinks, section 4.2.3.2.1). Therefore, increasing the external load at a given temperature enhances the mobility of kinks and favours the creation of new sources. As the temperature is increased, the bending introduces a larger effective friction against MK motion in the octahedral plane (section 4.2.3.3) and multiplication by kink coalescence is decreased. We conclude for this section that, even though the number of potential sources is finite, deformation is difficult because dislocation multiplication is largely inhibited in L12 alloys. Multiplication does not occur by bursts of superdislocations originating from the same source which itself operates repeatedly. Instead the operation of a source is restricted to the formation and expansion of one arc of superdislocation whose expansion in the slip plane is in addition hindered by strong interactions with the neighbouring superdislocations that belong to the same slip system (this is actually at variance from in-situ experiments where the avalanche production of dislocations has been reported
Ch. 53
P. V e y s s i k r e a n d G. S a a d a
392
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switch-over MKs ii
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Microscopy and plasticity of the L12 ,yt phase
393
to occur at high temperatures (section 3.5). This discrepancy is another indication that in-situ experiments do not always reproduce the two-dimensional expansion of a source in its slip plane to a sufficiently long free-flight distance). 4.3.2. Dislocation annihilation That the density of KW locks per unit volume does not appear to increase dramatically with strain attests to some significant annihilation rate during deformation. There are at least three potential mechanisms that are liable to confer some mobility to a KW segment and/or to help its annihilation: (i) By mechanical transformation into a glissile superdislocation, itself entirely contained in the octahedral plane. This requires a level of stress (the microscopic saturation stress, s m, eq. (24)) which is comparable to the flow stress peak; (ii) By means of the lateral slip of MKs (Veyssi~re [3], Saada and Veyssi~re [206]). KW zipping, which is schematized in fig. 62, offers the advantage to avoid the prerequisite destruction of the KW lock into a form that would be glissile in the octahedral plane. In this mechanism of MK sliding, a length of KW equal to the distance over which the kink has slipped is simply transported over a distance equal to the MK height; (iii) By interaction with a mobile superdislocation with opposite Burgers vector. These two superdislocations need not belong to the same octahedral plane and the interaction is governed by their approach distance, that is, the distance between the two parallel octahedral planes. Mechanisms (i) and (ii) have already been addressed in section 4.2.3.3 and section 4.2.3.4, respectively. We now concentrate on mechanism (iii). Consider a mobile superdislocation that originates from a MK, operating itself as a source. As schematized in fig. 66, the expanding loop is subjected to a number of attractive interactions with neighbouring segments, including the adjacent kinked immobile parts of the parent superdislocation. The subsequent transformations are governed by varied approach distances (for one double cross-slip event involving an excursion over /jump in the cube plane, the approach distance for self-annihilation is ljumP/V/-3). In fact, depending upon the approach distance and on the applied stress, an immobile kinked KW line may be
- bypassed by the expanding superdislocation, totally annihilated, -
+___ Fig. 66. Multiplication and annihilation. (a) For the sake of clarity, it is assumed that switch-over and simple MKs are not formed on the same side of the expanding loop. They all move from left to right under the effect at the applied stress. (b) Cusped and coplanar MKs are formed upon coalescence of simple and switchover MKs, respectively. Only the latter can expand as a source, provided of course it is longer than the critical height h~K. (c) As the first source expands, it produces new kinked screw segments and sources. (d) and (e): Depending upon the shear stress and upon the cross-section of interaction (determined by/jump), dislocations may just proceed or annihilate totally or partially with the surrounding microstructure, and give rise to APB tubes as proposed by Chou et al. [208], and to mixed dipoles. The geometry of sessile segments implies that consecutive mixed dipoles exhibit aligned extremities (represented by dashed lines in (e)). In the bottom part of figure (e), the mobile segment, which is too far from the sessile KW, is by-passed. Some similarities between this figure and fig. 9 by Jumojni et al. [73] should be noted.
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bound to the expanding superdislocation in the form of a stepped dipole, when the approach distance differs from 0. Screw parts are still liable to annihilate by cross slip. This annihilation may occur either completely or incompletely; in the latter case, the product takes the form of APB tubes according to a mechanism similar to that proposed by Chou et al. [208, 211] for B2 alloys, but which remains to be clarified for L lz alloys (see section 4.3.3.2). Hence, as the mobile segment expands, the prior distribution of obstacles that surrounded the kink source is replaced in part by a morphologically identical succession of screw and mixed dipolar branches (figs 65 and 66). Annihilation debris are shown in figs 36(b), 41 and 46. They have been recently studied at length by Shi et al. [284, 287]. Since a screw segment has a finite probability per unit length to undergo a double cross slip, both the mobile segment and the obstacles should be stepped over several parallel octahedral planes (section 4.2.3.2). Hence, the approach distance varies from place to place. Along the line of a given outer immobile superdislocation, mixed dipoles together with APB tubes and screw superdislocation dipoles - now of varied widths should again be formed in places. Elsewhere, the approach distance is too large for the mobile dislocation to be arrested. This is represented in fig. 66. As to the characteristic dimensions of mixed dipolar loops, it can be roughly inferred that the larger the applied stress, the narrower the dipoles. Only very narrow mixed superdislocation dipoles are stable against the levels of stress applied in domain B. In NiaAl-based alloys, the maximum distance between parallel octahedral slip planes, below which edge superdislocations dipoles are stable against the applied stress, is about 30 nm and 10 nm at 300 K and Tp, respectively. At room temperature, for instance, this implies that it is only when a superdislocation has achieved a total path in the cube plane of more than about 100b (as a result of several double cross-slip events, each involving a varied length/Jcurnp in the cube plane) that this part of the dislocation can proceed without being arrested by the segments of the same dislocation that belong to the octahedral plane where the motion of this superdislocation was initiated. The probability of immobilization by attractive interaction with pre-existing kinked KWs is increased as bending in the cube plane starts to proceed, that is, as the test temperature is increased in domain B 1. KWs that would be ineffective for the locking if they were rectilinear become part of the active obstacles as the approach distance is slightly varied by the effect of bending in the cube plane. However, when the bending gets pronounced, the obstacles are no longer nearly parallel to the octahedral slip plane but steeply inclined to this plane, so that interactions between the mobile dislocation and the surrounding obstacles become much weaker and gradually resemble those of a classical forest mechanism. 4.3.3. Further remarks on deformation debris
The nature of debris, especially their Burgers vectors, is representative of the interactions that are taking place during deformation. We shall restrict our discussion to debris resulting from the annihilation process within the primary slip system described in the previous section, that is, to mixed superdislocation dipoles and APB tubes. It is first worth noting that the presence of mixed dipoles suggests that cross slip has operated but deformation microstructures containing dipoles are not specific to L12
w
Microscopy and plasticity of the L12 .[t phase
395
alloys. In fact, mixed dipoles should be present in any material where cross slip and in particular double cross-slip may operate, since then a given dislocation is liable to share segments over several not too distant parallel slip planes for these segments to interact mutually. Regarding L12 alloys, every model of the flow stress anomaly involving a double cross slip - i.e., except point pinning models where dislocations are not transferred to a parallel neighbouring slip plane - predicts implicitly a distribution of dipolar loops as well as of APB tubes. Hence, the presence of either of these cannot be invoked in order to validate the details of a particular model. 4.3.3.1. Mixed superdislocation dipoles. Because of the particular microstructure of L12 alloys deformed in domain B, the distribution of superdislocation dipoles scales with that of pre-existing kinks (section 3.4.3.7.1). However, neither the structure of these dipoles which should be that of a prismatic superdislocation loop, nor their distribution, has ever been checked systematically by TEM (see however Shi et al. [287]). The mechanism of dipole pinching-off into loop rows may occur at relatively low temperatures since it requires pipe diffusion only. Point defects injected into the crystal as a result of dipole annihilation may help promote non-conservative processes among surrounding superdislocations, such as climb dissociation and APB dragging. As the test temperature is raised in domain B 1, mixed dipoles may still be formed, but they should disappear more readily from the microstructure, simply because diffusion becomes faster. Note that the extensive formation and annihilation of dipoles and the resulting production of point defects in large quantities explain the tendency of Ni3AI alloys towards disordering in localized shear bands [239]. 4.3.3.2. APB tubes. L12 alloys contain a distribution of narrow defects elongated along the screw orientation of the primary slip system, that show a typical faint contrast under specific diffraction vectors (section 3.4.3.6). So far, it is envisaged that these defects are result from the fact that the trailing superpartial does not follow the same path as the leading during the annihilation process of a mobile superdislocation with a locked screw segment (Chou et al. [208, 211], Sun [189], Hirsch [100]). In fact, the situation in L12 alloys is not as simple as that schematized for B2 alloys, since in order for the Chou et al.'s annihilation process to take place the two superdislocations should be lying on parallel habit planes according to the scheme of fig. 67. In the case of the annihilation of an incomplete or of a complete KW segment by interaction with a superdislocation gliding in an octahedral plane, one expects that in most cases annihilation occurs favourably by direct annihilation of the leading superpartials (fig. 67(c)), rather than through the double leading/trailing annihilation that is required to form an APB tube (fig. 67(d)). There is however a possible contribution of torque forces arising from anisotropic elasticity. The fact that APB tubes have also been observed in Ni3Fe (Ngan et al. [188]) raises additional questions as to the formation process of these defects. This particular alloy deforms essentially under cube slip with little evidence, if any, of cross slip and of the resulting locking of screw segments (Ngan et al. [31, 32]). According to Ngan et al. [188], the presence of APB tubes in Ni3Fe indicates that screw superdislocations may nevertheless annihilate by a local and limited cross slip (which, in this case, must
P. VeyssiOre and G. Saada
396
Ch. 53 APB
APB
(a)
\
X
APB
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Fig. 67. The annihilation of screw superdislocation dipoles and the formation of APB tubes (after a mechanism designed for B2 alloys by Chou et al. [208]). (a) Two dissociated screw superdislocations are in position to annihilate mutually such that the leading superpartial of one meets with the trailing of the other, by cross slip directly on the appropriate plane. (b) In a more realistic mechanism, annihilation is allowed to occur through multiple cross slip; the path is determined by computer simulations according to priority rules for slip in the B2 lattice (for details, see [208]). In the L12 lattice, the annihilation proceeds onto two octahedral planes and one cube plane, with markedly different dislocation mobilities. When annihilation occurs between a mobile superdislocation and a KW lock, APB tubes may not be formed systematically. (c) In this configuration of the slip plane of the incident dislocation, the trailing superpartial should follow the same path as its companion and annihilation should be total. (d) In this configuration, the attractive force between the trailing superpartials may be larger than the surface tension that tends to drive them in the wake of the leading superpartials. Recent work by Hirsch [279] and by Paidar and Veyssi~re (unpublished) point to the role of elastic anisotropy in determining the trajectory of superpartials during annihilation, showing that APB tubes tend to form more systematically than expected from the above schemes.
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Microscopy and plasticity of the L12 "y' phase
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occur on octahedral planes). However, in the absence of long preexisting locked screw segments, the origin of rectilinear APB tubes, a few ~tm long, is difficult to explain based on the above cross-slip annihilation process. Concerning APB tubes, it is worth pointing out that according to the mechanism of formation proposed by Chou et al. [208], the mean APB tube and KW lengths should scale and they should be interrupted or terminated by mixed dipolar configurations, themselves scaling with the MKs in size and distribution. According to our experience, this is rarely the case: it is very rare to spot a mixed dipole that would unambiguously connect two finite APB tubes shifted with respect to one another. The features named APB tubes usually appear as extremely rectilinear. They are at least one order of magnitude longer than the average length of KW segments located in their immediate neighbourhood (section 3.4.3.6). This suggests that APB tubes might not formed as a result of the annihilation of a mobile superdislocation onto a sessile kinked KW lock but as a particular dragging process (see fig. 36 and the recent work of Shi et al. [286]). Finally, the APB tube width has been observed to coincide, at least within the resolution of the weak-beam method, with that of SISF dipoles to which they seem to be parallel (fig. 44). This suggests that those elongated SISF dipoles that lie in the primary slip plane, which are seen in large quantities in the deformation microstructure at moderate temperatures (section 3.4.3.8), may be more important than initially anticipated in order to understand the organization of the deformation microstructure. Unfortunately, SISF dipoles, which are intrinsically unstable, are still poorly documented. To date, APB tubes have only been analysed in terms of their properties of contrast. Their actual shape is not known and their distribution is not documented. Their mechanism of formation is not elucidated (see however [287, 288]).
4.4. Work hardening We may now address in a preliminary form the question of the unusually large workhardening rates that are measured in domain B (section 2.2.3). In f.c.c, metals, for instance, dislocation interactions result in WHRs of at most 5 x 1 0 - 3 # under a wellorganized multiple slip, that is, the most stringent mechanical conditions for these materials. This is to be compared to WHRs of the order of # / 1 0 at the peak of WHR in Ni3Al-based alloys (section 2.2.3). In the latter alloys, deformation is ensured by the operation of one-step sources at kinks, a process by which a kinked KW line destroys itself permanently after moving some finite distance. As deformation proceeds and because of the pronounced planarity of slip, such an expanding superdislocation will eventually encounter an obstacle with which it interacts in a manner that is specific to L12 alloys. This obstacle consists in a distribution of provisionally immobile kinked superdislocations that are distributed in the slip plane or in a plane parallel to the slip plane and located in its immediate neighbourhood. Since the obstacle consists in a continuous superdislocation line, the mobile superdislocation has a large probability to be blocked, and even to be annihilated over large fractions of its length by interactions with segments with opposite Burgers vector (fig. 66).
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This dynamical behaviour can be contrasted to the situation met in "usual" alloys, where a collection of dislocations originating from the same source interact with trees interspersed some distance apart in the crystal; the mean resistance that arises from the microstructure is then inversely proportional to the inter-tree distance, which lies typically between 0.2 and 1 ~tm. Increasing deformation increases the forest density, that is, the stress that opposes dislocation motion. When the applied stress is maintained at a constant level, the density of mobile dislocations decreases. One may thus consider the work hardening rate as resulting from the increase in the internal stress due to the dislocation forest, and it is well known that the estimated work-hardening rate is in this particular case in good agreement with experiments. The situation in L12 alloys is quite different from the preceding: the mean free path of the superdislocation is rather small. In fact, it can be inferred from TEM observations that the blocking interaction should occur after a free flight distance of a few microns. This may be due either to the transformation of screw dislocations into KW, or, as has been pointed out above, by local interactions with neighbouring superdislocations. Then the exhaustion rate, therefore the work-hardening rate, is not the result of the increase of the long-range stress field of the deformation substructure. Note that, because of the occurrence of double cross-slip processes, a kinked screw superdislocation is in fact stepped over parallel octahedral slip planes with two consecutive KW segments being connected by a pair of jogs CJ whose length is of the order of /Jcump. When a mobile superdislocation interacts with a locked superdislocation consisting in a succession of KW segments, its expansion should be in principle hindered by two categories of interactions: (i) Dipolar interactions, with a strength of about #b/(27rl jump) per unit length of dislocation; (ii) Tree interactions with the jog pairs that are separated from the next by a distance equal to the KW length. Since both /jump and the density of jog pairs are small, it can be reasonably anticipated that the tree interaction is much less potent that the former. As the test temperature is increased, however, KWs tend to bend gradually in the cube plane (section 3.4.3.5). In a first stage, the bending is slight so that the tree interactions are not significantly modified but the cross section of blocking by dipolar interactions should increase. This would result in an increased WHR, conforming to the experimental observations (section 2.2.3). However, as the test temperature is increased further, that is, beyond a given mean radius of curvature, KW segments tend, on average, to escape in the cube plane, which results in a microstructure that resembles the usual discrete forest (sections 3.4.3.5 and 4.2.3.3), conforming to the "exhaustion" hardening model of Thornton et al. [41] (see section 5.3). Accordingly, the strength of interactions in octahedral slip should decrease, gradually adopting a magnitude corresponding to that of a forest mechanism. It is recalled that the peak of WHR does not coincide with the flow stress peak but is shifted towards lower temperatures, approximately at the level of the inflection point of r(T). The present qualitative description of dislocation interactions in domain B relies on two microstructurally-based properties: - the development of very potent dipolar obstacles in the immediate vicinity of each source;
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Microscopy and plasticity of the L12 "[~ phase
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the rapid decay of the density of mobile dislocations, itself due to the fact that sources can operate only once. In the absence of a quantitative analysis, it is however too early to conclude on the validity of the present tentative explanation of the origins of the work hardening in domain B. It has been also suggested [285] that the dragging of APB tubes by jogged mixed segments (fig. 40) may contribute quite significantly to the work-hardening rate. The drop of WHR at high temperature in domain B would coincide with the disappearance of APB tubes, known to occur at these temperatures (section 3.3.3.6). This description is in addition compatible with the property of flow stress reversibility (section 2.2.5), although this may not be obvious at first inspection since TEM observations indicate that the bent dislocations that have been formed during the prestraining stage do not disappear during the cooling stage (section 3.4.3.5). That, upon restraining, superdislocations do not experience the strengthening interactions with the bent superdislocations may in fact originate from the property that after modest permanent strain deformation has not proceeded evenly, so that superdislocations are not homogeneously distributed in the sample, even to the point that one encounters significant fractions of crystal containing no dislocations. Upon restraining, deformation would start at the edges of sheared zones and may expand more or less freely with respect to the high-temperature strengthening obstacles allowing only a limited effect of these, except maybe during some transient when the self-strengthening microstructure is not yet established in the free zones. When then prestrain amounts to a very large magnitude, samples are now filled with dislocations and the flow stress is no longer reversible.
4.5. Summary
and concluding remarks
As long as the radius of curvature of the nearly screw parts of an expanding superdislocation is less than a critical value of about 80 nm, these segments have a finite probability to cross slip onto the cube plane and to nucleate KW locks. Depending essentially on local stress conditions, the cross-slipped segment may be (i) Locked in an incomplete KW configuration that is given enough time to evolve gradually towards a complete one. The transition and the evolution of screw superdislocations towards the KW configuration depend critically on several factors, such as the relative value of the crystal parameters c~ and ff (section 4.2.2), a large lattice friction in the cube plane, - the ability to cross-slip in the cube plane, which is a consequence of the finite core size of superpartials.
-
Since the stress required to destroy a complete KW lock is of the order of the peak stress (section 4.2.3.4), the cross-slipped segment is permanently immobilized at any temperature within domain B. As the connecting mixed part slips in the octahedral plane, the locked screw segment increases its length by a zipping process, but it can be bypassed by this mixed part (fig. 59). This process of forming locked screw segments repeats itself as superdislocations expand, to give rise to a kinked KW configuration with varied kink heights. The macrokink height decreases with the probability of cross-slip
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into the cube plane at a given temperature. In this process where a double cross slip of screw segments is impeded, the macrokinks, named switch-over MKs, are coplanar, that is, the entire superdislocation expands in one single atomic octahedral plane (fig. 60(b)). Under the applied stress, these MKs may slide and coalesce to form larger MKs, which are themselves fully planar. The new MKs should move somewhat faster since the force applied to them is increased. (ii) Locked momentarily, that is, liable to undergo a double cross-slip process corresponding to an APB jump. This mechanism produces a pair of elementary kinks (EKs) with opposite signs bordering an incomplete KW. If the external stress is small enough, there is a finite probability for an APB jump to occur repeatedly. Otherwise, the superdislocation is freed in the octahedral plane. It then slips over some distance in this plane until it resumes cross slip. This produces simple MKs whose heights depends on the probability of cross slip in the cube plane at a given temperature. The higher the probability, the smaller the kink height. The cross-slipped distance is itself controlled by lattice friction: the larger the temperature, the longer/jump. The further evolution of KW segments is determined by the distance the leading superpartial has jumped in the cube plane, /Jcump, or equivalently by the distance between the two octahedral planes. Multiplication occurs at MKs longer than a critical length which is inversely proportional to the applied stress. Although each site of multiplication operates only once, superdislocation expansion may give rise to a number of kinks with a sufficient length (hMK > h~K) each of which can be activated as a source, which can operate only once, but multiplication still may not exceed exhaustion. This behaviour is similar to a microplastic deformation, where irreversible motion of dislocations occurs but where exhaustion is efficient enough in order to maintain the density of mobile dislocations at a low level. Another means for the microstructure to form new sources is by the coalescence of several mobile switch-over MKs with sub-critical heights. MK mobility is enhanced by the applied stress, the larger To, the larger the coalescence rate. This mechanism may help multiplication to overcome exhaustion under constant strain rate, but not under creep conditions. By contrast, the coalescence of two simple MKs does not contribute to producing new sources since the resulting MK is non-planar, exhibiting a pair of jogs at the points of impingement of the parent MKs. Note also that long mixed segments should be formed in order to provide the strain. It is likely that such segments exhaust rapidly. They are by nature included in kink-based models. As the temperature is raised, the bending of KW segments in the cube plane hinders MK mobility, which implies that the stress at which multiplication takes over exhaustion is increased. Beyond a certain temperature and/or a certain stress KWs are bent so far into the cube plane that kink coalescence does not operate any more. Under creep, this tends to favour slip on the cube cross-slip plane, while under constant strain-rate, this results in an increase of the applied stress so that kinks of smaller dimensions can be activated as sources. Meanwhile, the KW locks become less stable under stress, and multiplication may further occur upon destruction of incomplete KWs unless they have bent too far in the cube cross-slip plane, at which point octahedral slip is exhausted. As it stands, the present description of the possible microstructural evolution is not complete. It nevertheless helps understand the following puzzling experimental facts: (i) Since the multiplication rate is small and annihilation efficient, the microplastic stage is unusually extended [286].
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(ii) The small strain-rate sensitivity in domain B. In fact, the dislocations that carry most of the strain are neither the screw parts, which can be considered as locked up to the peak, nor the MKs, whose motion can be significantly viscous (section 4.2.3.3), but the longest mobile mixed segments formed in the course of source activation at MKs. These mixed segments are intrinsically mobile (fig. 26) and can respond to a strain-rate jumps in the same way as dislocations do in usual f.c.c, metals. (iii) The fact that slip is localized implies that the rate of locking/annihilation with the surrounding microstructure is large it can provide some hint for the large work-hardening rate in domain B, that could nevertheless decrease as a forest tends to take over in the crystal, by gradual exhaustion of the KWs in the cube plane. The possible contribution of APB tube dragging to the WHR should nevertheless not be underestimated [285].
5. Analysis of theoretical contributions on the positive temperature dependence of the flow stress This section is aimed at discussing several models of the flow stress anomaly in L12 alloys which have been selected for their representativeness of the variety of theoretical approaches in this domain. In doing this we have deliberately chosen to account for the arguments of the various authors. As a consequence, this section may seem redundant, contradictory, or even inconsistent, since it reflects the outcome of about 30 years of imaginative work by many scientists on various hypotheses. A patient reader will find however an expression of our views at the end of each section.
5.1. Introduction
Despite the wealth of theories that have been designed in the recent years, the flow stress anomaly and related properties are far from being fully explained from a theoretical standpoint. This frustrating though challenging situation results from the repeated attempt to adapt to situations encountered in L12 alloys methods and reasonings that had been successful so far in the case of more usual crystals. Reference to the much better known case of f.c.c, crystals - which in addition we shall use to discuss the mechanism of cross slip of a superpartial- may help clarify this remark. In f.c.c, crystals, the fundamental microscopic mechanisms are well identified and the relevant physical parameters satisfactorily documented - except for the actual core structure of dislocations in high stacking-fault energy systems. The microscopic properties of f.c.c, crystals can be thus regarded as conceptually understood. On the other hand, thermally activated glide is described at a macroscopic level by means of a simple equation that, under some sound physical assumptions, represents the process and captures qualitatively its phenomenological behaviour. However, due to our insufficient knowledge of the core structure of dislocations and of its behaviour under stress, the precise calculation of the activation parameters is still not achieved. Nevertheless, its is quite generally acknowledged that the macroscopic plastic behaviour of f.c.c, crystals
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is consistent with microscopic observations (dislocations interaction and distribution, insitu experiments, and slip-line distribution). In order to go one step further, that is, to model quantitatively the plastic flow of f.c.c, crystals, a complex dynamical problem has to be solved. With regards to the number of parameters, i.e., to the different types of interactions that are involved, a rigorous mathematical analysis of the phenomena seems beyond reach. Nevertheless, one may try to model the phenomenon with the help of numerical simulation. Despite some promising achievements (Devincre and Kubin [262]), a comprehensive modelling of the behaviour of f.c.c, crystals at a mesoscopic scale is still not available. For the following discussion on L12 alloys, it is worth noting in addition that in attempting to understand the mechanisms of plastic deformation of f.c.c, metals, the field has been obscured by a few inadequate methods such that using concepts beyond their limit of validity. This point is well illustrated by the crossslip models initially designed within the frame of linear elasticity and subsequently applied as a quantitative theory to dislocations exhibiting a splitting width of less than say 1 nm (Saada and Douin [137]; see also section 5.2.1); - fitting a single macroscopic observation to a microscopic analysis in view of establishing a correlation between theory and experiment. This argument can be exemplified by the several theoretical attempts at correlating the stress for the onset of stage III to the occurrence of cross-slip e v e n t - the so-called 7"ii I - - that have been made but only with limited success (Saada [263]); - failing to recognize that the proper function of experiments is to test theory, not to verify it (Popper [264]). -
Finally, it happens in many fields of research that uncertain analyses become legitimated by being inadequately quoted in subsequent papers. With regard to L 12 alloys, most of the debate on the causes of the flow stress anomaly has focused on properties of the KW configuration, that is, on the analysis of cross slip in the L12 structure. In a number of cases, the screw segments retain some mobility and the flow stress anomaly is assumed to be controlled by a pinning/unpinning process, itself governed by the local cross-slip locking of one of its superpartials. The presence of elongated though kinked KW segments and the organization of these, which characterize the deformation microstructure of L12 alloys in domain B so well, may be largely ignored or just regarded as dead debris (section 4.1, for a review see Pope [156]). In other analyses, some priority is given to accounting more closely to the deformation microstructure; the difficulty then consists in discriminating between the features that are relevant to the rate-controlling mechanism and the random debris. However, prior to getting to these analyses, we need to outline the way the orientation dependence of the flow stress has been modelled.
5.2. O n t h e e l e m e n t a r y
cross-slip process
There is ample experimental evidence that, at temperature below Tp, the flow stress in tension differs from that in compression. However, contrary to early beliefs, the
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orientation dependence of the flow stress is so much affected by composition that the properties that should be actually modelled are hard to define (section 2.2.6). To date, there has been a general consent that the TC asymmetry originates from the subsplitting of superpartials into two Shockley partials separated by a complex stacking fault. This subsection is aimed at questioning the theoretical foundations of this assumption that we have already criticised on experimental grounds in section 4.2.1.1.
5.2.1. The case of fc.c. crystals In f.c.c.-related structures, the elementary process of cross-slip between two octahedral planes is generally regarded as resulting from the creation of a constriction locally along a screw dislocation, then allowed to expand in the cross-slip plane (fig. 68(a)). Microscopic cross-slip mechanisms have been proposed by Schoeck and Seeger [265] and by Friedel [266]. The latter process, which appears to be the more realistic one, has been further developed in an analytical form by Escaig [267]. It is orientation-dependent, with its activation enthalpy differing according to whether the sample is deformed in tension or in compression. Under its Escaig formulation, the Friedel cross-slip model is a central ingredient that has been incorporated by Paidar, Pope and Vitek [86] to their modification of the original model of Takeuchi and Kuramoto [81], in order to predict the TC asymmetry of L12 alloys deformed in domain B. Obviously, deriving a correct estimate of the activation enthalpy is the critical step. As we shall see, this goal has not been achieved. In the Escaig formulation, notwithstanding the detailed process under study, the subtle and critical part of the calculation of the activation enthalpy requires a determination of the constriction energy Wc. This is achieved based on a general method outlined by (a)
Fig. 68. Cross-slip processes of dislocations with 51 (110) Burgers vector. (a) The Friedel mechanism in f.c.c. crystals. (b) In L12 alloys, the superpartial has to slip over b/2 in the cube plane before it reaches an atomic row where it is liable to spread in the cross-slip octahedral plane. The configuration is then highly asymmetric and this is even reinforced by the application of a stress.
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Stroh [268]. The analysis of the cross-slip process relies on a few approximations that are worth recalling (Saada [263], Duesbery et al. [269]): (i) In the calculation of Wc, the configuration under study does not have to be symmetric as assumed in Escaig's calculations. The corresponding uncertainty on the activation enthalpy associated with the asymmetry has been evaluated to about 40%. (ii) Wc cannot be calculated precisely for small separations, since it is critically affected by the choice of the cut-off radius. The uncertainty in this case may be as large as 100%. The smaller the dissociation width, the larger the error. (iii) The double constriction requires an energy Wd ~ Wc(1 -or/Ors), where Os has to be evaluated numerically [263]. (iv) How the theoretical cross-slip rate compares with the experimental plastic behaviour of crystals is not clearly established. Since constrictions may pre-exist on dislocations, the activation enthalpy may vary between Wd and Wd + Wc, that is, by a factor of about 2. Since the activation enthalpy enters in the calculations as the argument of an exponent, the theory can be regarded as quite flexible, to say the least. 5.2.2. L12 alloys Considering L12 alloys, it should first be noted that superpartials exhibit a core width of at most 2 to 3b, which implies a very small activation enthalpy even at low temperatures. In addition, predicting activation parameters for cross slip from linear elasticity, appears in this case very uncertain (Saada [263], Saada and Douin [137]). The elementary crossslip configuration envisaged in the PPV model differs indeed quite significantly from Friedel's mechanism in that the cross-slipped segment does not actually lie along the same atomic row as the rest of the screw segment but one atomic row farther, b/2, in the cube plane (fig. 68). In view of the above remark on the role of shape effects, one may anticipate that the intrinsic asymmetry of cross slip in L12 alloys should result in significant though unpredictable corrections to the activation enthalpy. Hence, utilizing Escaig's or an equivalent formalism in order to study the cross-slip transformation of a dissociated screw superdislocation in L 12 alloys that exhibit a positive TDFS, may at most provide a hint of what the actual cross-slip process could consist of. Such calculations are intrinsically quite flexible. We recall that the successive attempts at predicting the experimental orientation dependence of the flow stress through appropriate orientation factors, i.e., N, Q a n d / ( (eqs (2), (3) and (4), respectively) reveal themselves more or less inadequate to account for the wide variety of experimental responses (section 2.2.6). The fact that some correlation could be found between experiments and predictions based for instance on Q or K suggests that the hypothesis of stress-induced core deformations may be partly relevant in order to explain the orientation dependence of the flow stress but, in the present state, it appears impossible to tell which part of this hypothesis might be correct. The poor quality of the fits led Khantha et al. [101 ] to give up the idea of introducing a refinement such as the Friedel process in the description of cross slip, and to return to the treatment of Lall et al. [28]. Hence, designing a detailed theoretical model of this property turns out to be more complex than initially anticipated; it might well be pointless. Further theoretical effort is needed for this question and it might be worth exploring alternative mechanisms, not necessarily based on core effects. In this light, anomaly-related
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macroscopic mechanical properties other than the violation of the Schmid law are worth considering in priority; these properties include the dramatic magnitude of the workhardening rate (section 2.2.3), the absence of strain-rate sensitivity (section 2.2.8), the partial reversibility of the flow stress with temperature (section 2.2.5) .... Finally, further calculations of core-controlled properties in L 12 alloys that have been conducted under linear elasticity should be mentioned. These have been achieved by Hirsch in order to compare the double jog and the critical bow-out mechanisms for cross slip and for unpinning (appendices 1, 2 and 3 in Hirsch [99]), by Schoeck [247] on a set of infinitely long dislocations; in this case the theoretical analysis was conducted within the frame of anisotropic elasticity in order to account for torque effects. In view of the above objections, Hirsch's results are evidently uncertain. On the other hand, Schoeck's work provides some unexpected information on properties of pinned configurations. According to his analysis, it is only when the separation between Shockley partials is less than about 1.3b that the critical part of the transformation of a glissile configuration into a PPV lock (e.g.,/Jcump - b/2, with the subsplitting occurring in the cross-slip octahedral plane, that is, from site "7 to site /3 in fig. 51) would be energetically favoured. Schoeck acknowledges, however, that in the particular case of a modest dissociation width the concept of linear elasticity whereon his calculation is founded clearly breaks down. Interestingly, when the subsplitting exceeds atomic dimensions, that is, when elasticity calculations become reliable, the PPV lock is always energetically, unfavourable. Hence, cross slip on the cube plane is still possible but by a step height/lcump = 2b/2, so that the superpartial subsplitting would not spread into the cross-slip octahedral plane (direct jump from 7 to 3', in fig. 51).
5.3. Miscellaneous models
Flinn [172] was the first author to propose a microscopic explanation of the increasing strength of L12 alloys. Though the premise of Flinn's model was r i g h t - e.g., a favourable APB energy on some atomic plane such as the cube plane favours the subsequent locking of superdislocations out of the primary octahedral p l a n e - the development of the model was incorrect. This is because the explanation was applied very generally to superdislocation lines with any character, so that climb had to be implied in their dissociation process. For dislocation motion to proceed, Flinn proposed a diffusion-controlled mechanism involving APB dragging. This mechanism is inappropriate because of its pronounced strain-rate sensitivity, its independence on thermal history, and of the fact that the positive TDFS is already significant at temperatures too low for diffusion processes to be important (Davies and Stoloff [74]). It is nevertheless interesting to note that (i) Flinn's analysis inspired the concept of the Kear-Wilsdorf lock, which is actually contained in Flinn's paper as a particular case of the dissociation out of the octahedral slip plane (in fact, the lowest APB energy was determined by Flinn to be a conservative APB in the cube plane, that is, that of a KW configuration),
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(ii) climb dissociation and APB dragging have been identified in L 12 alloys deformed in domain B2 (section 3.1.2.6 and 3.4.3.6). Davies and Stoloff [74] and Johnston et al. [75] suggested that the lattice opposes an increasing resistance to dislocation motion as temperature is raised, but they did not address the actual origin of lattice/dislocation interaction. Subsequently, Copley and Kear [67] proposed that this latter effect could originate from the subsplitting of superpartials in the primary slip plane which, as a result of the temperature dependence of lattice constants, would decrease in magnitude as temperature is increased. In the frame of a Peierls-Nabarro model of lattice friction, the narrowing of dislocation cores implies an increasing slip difficulty. Copley and Kear's hypothesis has been ruled out quite elegantly by the experiment on the strain dependence of the flow stress anomaly performed by Thornton et al. [41] (section 2.2.4, fig. 6). At variance from the ideas and concepts that were prevailing at that time and in order to account for differences observed especially in strain-rate jump transients (section 2.2.8.1), the latter authors proposed the first two-stage mechanism for the positive TDFS. Below about 400~ that is, approximately the temperature of the inflection point of r ( T ) , the density of mobile dislocations decreases as a result of the increasing production of KW locks. Above 400~ where cube-slip activity becomes significant, dislocations of the primary slip plane exhibit an increasing frequency of intersections with trees that belong to the cube cross-slip plane. These low and high temperature mechanisms are known under the names of "exhaustion" and "debris" hardening mechanisms, respectively [41 ]. The "debris" hardening mechanism was later criticized by Staton-Bevan and Rawlings [30, 89], on the grounds that cross slip on the cube plane is not expected in samples deformed along [001] 19 and along []-23]. The "debris" hardening mechanism has nevertheless been supported since by Korner [204] as a result of careful weak-beam TEM observations. In fact, in order for the hardening intersection mechanism to take place, cube slip need not operate extensively, and it seems difficult to rule out the idea of a "debris" hardening based on the argument that cube slip is suppressed on a macroscopic scale. Rather, significant KW bending appears to be sufficient to provide a large density of trees, as observed in L 12 alloys deformed in domain B2. However, there remains the problem of accounting for the anomalous work-hardening rate in domain B 1 and of its peak at the temperature where the flow stress undergoes an inflection. Almost every subsequent approach to the flow stress anomaly is based on some a priori point of view on the motion of a dissociated screw superdislocation in view of its eventual elimination in the crystal either at a free surface or else by annihilation with another superdislocation. This is what we shall examine in the following sections. 19Some controversy will probably remain as to how accurately a given orientation such as [001] can be controlled in a real deformation experiment. Furthermore, it cannot be ensured that the stress tensor does remain the same throughout the sample. In fact, whatever the precision of sample orientation, local rotations due to crystal imperfections or after some permanent strain, due to end-effects (section 2.2.2) or else to local stress concentrations (section 3.4.2) should be enough in order to promote some bending in the cube plane. It is almost hopeless to suppress local cube slip by accurately orientating samples along [001], but it can be significantly inhibited.
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S.4. Local pinning models Early theories of the positive TDFS (Takeuchi and Kuramoto [81], Lall et al. [28], Paidar et al. [86]) rely on the hypothesis that plastic deformation is controlled by the locking rate of a screw dislocation, whose motion is arbitrarily forced to be steady (section 5.4.1). Based on otherwise reasonable assumptions, these pinning models were aimed at reproducing the variation of the flow stress with temperature, the violation of the Schmid law and facets of the tension/compression asymmetry. However, as we shall see in the following (sections 5.4.1 and 5.4.2), none of these early theories accounts for the actual strain-rate sensitivity of the flow stress (Hirsch [98, 99], Nabarro [250]) nor do they explain transient tests (Hemker et al. [54]). More recently, Vitek and Sodani [246] and Khantha et al. [88, 101,260] have re-visited the TK and PPV theories in an attempt to circumvent their most serious conflicts with macroscopic properties (sections 5.4.3 and 5.4.4). We now outline and discuss this family of models based on the point-pinning mechanism; several of the following arguments are inspired by the papers of Dimiduk [49], of Hirsch [98-100] and of Nabarro [250]. 5.4.1. Takeuchi and Kuramoto (1973) The Takeuchi and Kuramoto (TK) model [81 ], which is the first comprehensive statement of the local pinning models, can be summarized as follows. Consider a screw superdislocation containing a distribution of pinning points located a distance g apart according to fig. 69. Under the effect of the applied stress To, the dislocation bows out at a velocity V = "rob/B
(30)
until its centre is displaced by a distance d, at which point breakaway occurs. Since d scales with g, which itself scales with the breakaway stress To, the time d / V required to reach the breakaway position scales with (To)2. Assuming that the pinning process is steady allows one to write s ~ d/V = 1
(31)
where u is the frequency per unit length of dislocation of pinning point formation by cross slip. The TK model is founded on the assumption that breakaway is steady. In its original form, it does not address the question of the strain-rate sensitivity. Equation (31) simply states that the yield stress is the stress required to create a steady density of mobile dislocations, e.g., one pinning point is formed over the length t during the time required to travel over d. It implies in turn that the frequency u scales with ('to) 3. As a result, when it is assumed that the point pinning is thermally-activated, then
% -- A exp
HI } 3kT
(32)
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I iiii,,'"'"'"'"'"'"'""'"'"'"'"'"'"'"'""'"'"'"'"'"'"'"", iiiIII,'"'"'"'"'"'"""""'"'"'"'""'"'"'"'"'"' '/ '"'""'""'"'"'tlll till',,' ill!o,'""'"'"'"'"'"'"""'"'"'"" r v Y
-'V"-
,,
' |~
A g
-X
; i1,,,'-|
Fig. 69. The dynamical breakaway mechanism proposed by Takeuchi and Kuramoto [81]. The black and dotted grey lines represent the dislocation positions at two consecutive steps of the breakaway process.
where A is a constant that has been estimated by Takeuchi and Kuramoto (TK), and H1 is the activation energy for pinning 2~ which, in general is expressed as H1 -- H ~ - - T c V . From their experimental data, Takeuchi and Kuramoto obtained sound values for the enthalpy (H1~ - 0.3 eV) and for the activation volume (v = 6b3). Given the temperature and the shear stress in the cube cross-slip plane, HI determines a distribution of equidistant pinning points. It is the fact that the frequency of cross-slip events, that is, the density of pinning points per unit length of screw dislocation, increases with temperature that causes the the flow stress increase. Hemker et al. [54] argued that expression (32) implies the existence of a critical stress for octahedral slip, To. Below To dislocation motion is automatically impeded by KW formation while, above that stress, dislocations break away from the pinning points with no difficulty and they move subsequently at the free flight velocity. Hemker et al. [54] pointed out that the TK model is inherently inconsistent with the absence of a critical stress for octahedral slip in domain B as can be deduced from the shape of stress/strain curves and from the fact that primary creep on the primary octahedral plane can be activated at stresses below 7o.2. In view of the apparent activation energies listed by Liang and Pope [270], Nabarro [250] argued that, e v e n at r o o m t e m p e r a t u r e , all segments will readily jump between sessile and glissile configurations, with the majority in the low-energy sessile state. This prediction, deduced from the TK model, is clearly at odds with the premise of the model. Even more serious is Hirsch's remark [98] that taking B - 5 x 10 -5 Pas, To - 200 MPa and a strain-rate of 10 -4 s -1 yields a density of mobile dislocations of approximately Pm ~ 2 x 10 .2 cm -2. As pointed out by Hirsch [98], the stress dependence of the dislocation velocity postulated by Takeuchi and Kuramoto (eq. 30) implies that the strain rate depends linearly on stress, which conflicts severely with the fact that the flow stress is actually insensitive to strain-rate changes (section 2.2.8). Note that the failure of Flinn's explanation to account for the absence of strain-rate sensitivity of the flow stress has been regarded as sufficient to rule out this explanation (Pope and Ezz [271 ]) but, for some reason, this argument has not been used against the Takeuchi and Kuramoto theory, nor against the refinements of this theory achieved by Lall et al. [28] and by Paidar et al. [86]. 2~ should be aware that in most subsequent experimental studies of the mechanical behaviour of L12 alloys deformed in domain B, the activation energy for pinning has been estimated through eq. (32).
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Finally, the pinning point model requires implicitly that the cross-slipped segments exhibit some stability against stress; the stability in question is measured by the critical angle q~ beyond which breakaway occurs (fig. 69). This question has been discussed quantitatively by Hirsch [100]. In view of the predicted distances between pinning points and of the levels of stress that are applied in practice over domain B, the idea that the local pinning event could be strong enough to provide the desired strengthening appears to be unplausible.
5.4.2. Paidar, Pope and Vitek (1984) Paidar et al. [86] have made a gallant attempt at using what was available on crossslip calculations (section 5.2.1) in order to re-evaluate the activation enthalpy of the TK model. By introducing the asymmetrical cross-slip behaviour of a screw dislocation dissociated into Shockley partials, these authors were able to account for the TC asymmetry that was documented at that time. Paidar et al.'s method follows that introduced by Lall et al. [28] wherein the details of the pinning process are modified. As discussed at length earlier, the fact that examples of a significant disagreement between the PPV predictions and TC asymmetry experiments are at least as numerous as those of agreement (section 2.2.6) should be taken into consideration. The PPV model suffers from the same drawbacks as the original TK model in that it cannot account satisfactorily for the kinetic aspects (section 5.4.1). The PPV expression of the activation enthalpy has been subsequently corrected for anisotropic elasticity by Yoo [255]. In view of the arguments developed in section 5.2 on the validity of elastic calculations of the cross-slip process in the limit of small splittings, the claimed agreement between the orientation dependence of the flow stress predicted by the PPV model and experimental observations is certainly not as convincing as one would expect considering the widespread acceptance of the model. 5.4.3. Vitek and Sodani (1991) Several new theories including some of the most recent approaches the flow stress anomaly have followed the suggestion that the pinning process should be very fast compared to the unpinning process (de Bussac et al. [245]). The model proposed by Vitek and Sodani [246] is one of the earliest of this series. These authors have designed an analytical formulation, again relying on the pinning point principle, that is aimed at accounting for the absence of strain-rate sensitivity. In doing so they ignored, without justification, the in-situ observations of Caillard et al. [190] that indicate quite clearly that the motion of a screw superdislocation does not proceed through a steady mechanism of local pinning/unpinning. The Vitek and Sodani form of the TK-PPV model relies on the following assumptions (i) as postulated in TK-PPV, the distance g between two consecutive pinning points is a decreasing function of temperature of the form -goexp
~
,
(33)
where go is of the order of 2 to 3b. In fact, this expression is equivalent to assuming that the density of pinning points is under thermal equilibrium, which offers the advantage
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of implicitly accounting for the thermal reversibility of the flow stress (section 2.2.5). As emphasized by Hirsch [98], this assumption cannot however be justified by any thermodynamic consideration. (ii) the unpinning process is thermally activated, with an activation enthalpy Hu Hu
-
H ~ -
~-[b2e,
(34)
where ToT is the temperature-dependent part of the stress. The strain rate is given by the usual expression g = io exp
-
gu }
(35)
and "r[ = k T l n { ~ / ~ o } + H ~
bZgoexp { ~ }
(36)
The model reproduces the positive TDFS in compression (the orientation dependence of the flow stress is by nature similar to that of the PPV model) and the small strain-rate sensitivity, up to 800 K. As for the TK model, expression (36) implies the existence of a critical stress for octahedral slip. Using the expression of the activation enthalpy for pinning derived by Paidar et al. [86], but corrected for the torque effect, Vitek and Sodani [246] published a value of Hi~ = 4.9 x 10 -19 J, at zero stress. This yields a distance between consecutive obstacles that is far too large. This was a misprint which was corrected in a subsequent paper (Sodani and Vitek [272]) where, with an activation enthalpy for pinning ten times smaller than in the previous work, the distance between pinning points below 500-600 K is still too large to make sense (> 10 gm). 5.4.4. Khantha, Cserti and Vitek (1992, 1993)
Khantha et al. [88, 101, 260] have produced a model in which "the deformation is mainly controlled by the mobile screw dislocations moving on the (111) planes" and which relies on the existence of localized pinning points. This model differs from those of Paidar et al. [86] and of Sodani and Vitek [272] in many respects. In order to correct for discrepancies between the predictions of the PPV model and experimental measurements of the orientation dependence of the flow stress (section 2.2.6), Khantha et al. [88, 101] postulated that both the pinning and the unpinning mechanisms are thermally activated. Regarding the pinning process, the general trend is about the same as in the PPV model with the following modifications: (i) The derivation of the constriction energy and in particular, that of the saddle-point configuration, involves only the constriction in the primary slip plane. In other words, the role of the subsplitting in the octahedral cross-slip plane, which constitutes one of the most remarkable differences between the model of Paidar et al. [86] and that of Lall et al. [28], is no longer included.
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(ii) The effect of ternary additions is accounted for through an ad-hoc adjustable parameter/3c, whose value is zero for binary alloys. As to the unpinning mechanism, Khantha et al. [88, 101] consider that the breakaway process may take place simultaneously over more than one pinning point (fig. 70(a)). The technique of calculation is inspired from methods developed in the treatment of internal friction. The method is based on the idea that the release through a different path is necessitated whenever, along a given path, repinning by backward motion from the saddle point configuration becomes more likely than the forward transition. Since the breakaway path depends on the applied stress, different processes are liable to occur according to the applied stress, or, equivalently, according to the test temperature. As a result, the activation volume exhibits discontinuities at each transition between one breakaway path and the next (fig. 70(b)). The breakaway process is still postulated to occur steadily, which introduces a correlation between the processes of pinning and unpinning. In practice, this is inserted in the equations through the condition d = g, which, at high temperatures, is quite arbitrarily replaced by d = g/3 for a better fit. Not only did Khantha et al. obtain a good fit with the data on activation volumes but, by inserting these into the rate eq. (35), they were able to derive a variation of the flow stress with temperature in excellent accord with experimental data in Ni3A1, Ni3(A1, Ta) and Ni3Ga (although in the latter, the chosen value of the APB energy on the cube plane of 20 mJ m -2 is certainly much too small). So far, Khantha et al.'s model represents the most elaborate version within the pointpinning theories. It relies on the assumption of a steady-state breakaway process as the result of the "balance between pinning and unpinning rates" although it is "governed by two thermally-activated processes with different waiting times". It is stated that "the pinning process ... is dominant over the unpinning process," but then these authors force the rate of pinning to be equal to that of unpinning in order to avoid exhaustion. It is finally considered that the flight distance scales with the distance between pinning points, which seems rather arbitrary. Regarding comparison with experiments, it must be noted that the in-situ observations of Caillard et al. [190] together with those of Mol6nat and Caillard [196] are ignored. In Khantha et al.'s work, the comparison with experiments focuses on the activation volume (Khantha and Vitek [260]) which, in Ni3(A1, Ta) does show some anomalous behaviour between domains B1 and B2 (section 2.2.8.3). At variance from the PPV and the SV point-pinning models, the necessity of accounting for the orientation dependence of the flow stress is no longer given top priority in Khantha et al.'s thesis. As a matter of fact, the ultimate expression of the activation enthalpy is now much more similar to the Lall et al. form [28]. This enthalpy is flexible in order to account for the wide diversity of mechanical behaviour as alloy composition is varied (section 2.2.6). It should be noted that, as in the previous point pinning models, that of Khantha et al. implies the existence of a critical stress for an octahedral slip within a given sequence of major breakaway which, as emphasized by Hemker et al. [54], is in marked contradiction with experiments (especially with the activation of an octahedral slip in primary creep below to.2 within he range of temperature of domain B, section 2.2.9). Finally, the very recent work of Sp~itig [282] suggests that there is no discontinuity of the activation volume in Ni-rich Ni3A1 and Ni3(A1, Hf), so that introducing the multiple breakaway concept loses partly its relevance.
412
Ch. 53
P. Veyssidre and G. Saada
(a) !
S
......
-~- ...............
.
.
.
.
.
.
.
.
.
.
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.
.
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m=l
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200
!
600
('c I
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800 Temperature
(K)
Fig. 70. (a) In the analytical form of Khantha et al. model [88, 101], the dislocation breakaway corresponds to the release from m equidistant pinning points (average distance ~). It is illustrated here is the case m -- 2. There are 3 possible displacement states at a pin (pinned, saddle-point and unpinned) and 6 non-equivalent equilibrium configurations. Major breakaway occurs via one of the two sequences illustrated in (a) and (/3). The saddle configuration is denoted by S in both cases. When the inter-pin distance ~ is large (low temperatures), the simultaneous unpinning at two points is unlikely; thus, path (c~) is rate-controlling. When ~ decreases, the probability of re-pinning becomes larger than that of breakaway, at a certain stress. Hence, major breakaway becomes possible and it will dominates via path (fl). The average free-flight distance d (i.e., the distance after which, according to Khantha and Vitek [260], "a sufficient number of glissile-sessile core transformations are nucleated at random positions such that the dislocation returns to the pinned configuration shown" on top of the figure) is approximated to the mean separation between pinning points. (b) Temperature dependence of the activation volume in the major breakaway approximation. The m = 1 path is controlling between 300 K and approximately 600 K (To). A little above Tc(~ -- 140b), the m = 2 path takes over and the activation volume exhibits a discontinuity. The m = 2 remains dominant up to about Tc~ = 825 K (the distinction made between paths (c~) and (fl), fig. 65(a), does not play a role in this domain). At 825 K, the activation volume undergoes a second discontinuity (t? = 49b) and the m - 3 sequence starts to operate. Note that the average lengths between pinning points at the temperatures of discontinuity, i.e., 140b and 49b, would be of the adequate order of magnitude in order to be observed during in-situ experiments.
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Microscopy and plasticity of the L12 ~/~phase
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5.5. Locking/unlocking Various direct interpretations of dislocation motion during in-situ straining tests have been already discussed in section 3.5. Couret and Caillard [202] have proposed a model based on the group's in-situ observations of dislocation motion in a choice of alloys that all exhibit a positive TDFS. Couret and Caillard's analysis of the causes of the flow stress anomaly in Ni3A1 is a straight adaptation of their work on Be and Mg (Couret and Caillard [234, 235]) to the L12 structure. These authors noticed that in both systems the positive TDFS is manifested microscopically by the jerky motion of dislocations. By contrast, the negative TDFS is associated with much smoother dislocation displacements. The key observation in Be is that between 240 K (7- = 54 MPa) and 300 K (7- -- 64 MPa), the waiting time, or equivalently, the unlocking probability of screw segments, is constant. This observation serves as the foundation of the so-called locking/unlocking mechanism which relies on thermally-activated transitions between two core configurations. The formulation of the model is questionable in places, such as the use of probabilities for locking and for unlocking that are not defined on a per unit length basis. Couret and Caillard [202] propose that the cause for the apparently paradoxical behaviour of dislocations in Be is that the stacking-fault energy decreases with temperature, such that the increase in the activation for unlocking happens to compensate exactly for the increasing thermal fluctuations. We recall that since splitting in Be is small (..~ 2b), we are again facing calculations of thermally-activated core transformations conducted beyond the limit of applicability of linear elasticity. There is little justification for applying the Be model to the case of Ni3A1 on the sole basis that dislocation motion is similarly jerky in both systems. In fact, locking/unlocking may occur by different elementary processes and it should be realized that the specific mechanism of repeated APB jumps (Mol6nat and Caillard [196]) is different from what Couret and Caillard [234, 235] have proposed in order to describe the behaviour of Be. As a result of the claimed analogy between Be and Ni3A1, Couret and Caillard [202] were led to conclude that the shape of kinks in Ni3A1 is not important in the determination of the parameters that govern dislocation motion. These difficulties were later acknowledged by Mol6nat and Caillard [229] who stated that in fact "no conclusion can be drawn from a direct comparison [of ordered alloys] with pure metals ... since the sessile configurations have different origins". In the most recent models of the flow stress anomaly of L12 alloys based on in-situ experiments (Caillard et al. [131 ], Mol6nat et al. [237]), a distinction is made between the bypassing of obstacles and their intrinsic unlocking (fig. 71). At variance with the kink models (section 5.6), the intrinsic unlocking is claimed be the more likely mechanism 21. At present, Mol6nat et al.'s explanation for the cause of flow stress anomaly in L12 alloys [237] remains focused on the mobility of screw segments alone. It states that, in 21By nature in-situ experiments make it very difficult to observe the bypassing mechanism. Observing it would require that the slip plane is nearly parallel to the foil, and when this is achieved, the resolved shear stress on the slip plane is made negligible. Since deformation still has to proceed, alternative deformation mechanisms specific to the boundary conditions imposed by the proximity of free surfaces may take place in the thin foil (section 3.5).
P. VeyssiOre and G. Saada
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KW
Ch. 53
KW
1
MK
b
MK ,...__ ,,._.-
(a)
(b)
Fig. 71. Overcoming a KW lock (glissile and sessile segments are represented with a double and a thick line, respectively): (a) bypassing by an expanding MK; (b) unlocking by transformation of the sessile into a glissile configuration.
the double cross slip mechanism of formation/destruction of incomplete KW locks, the amount of cross slip on the cube plane upon locking, /jump, increases with increasing temperature. As a consequence, the stress that is required to allow for the motion of screw superdislocations, that is, for the destruction of incomplete KW locks with increasing extent in the cube plane, has to be increased. In conclusion for this section and as a transition for the next, we have found it worth citing the following excerpt taken from Kear and Wilsdorf [63], because of its strong analogies with Mol6nat et al.'s new model, and since it contains many of the issues that we have discussed in detail in section 4: "... possibly screw dislocations pairs that have undergone cross slip into a cube plane, became reactivated ... so that the slip will propagate from one plane to another by a double cross-slip mechanism. This model seems reasonable since the length of the screw segments are in some cases as large as 1 ~m, so that these segments could bow out between their pinning points at ordinary stresses; here the pinning points are taken to be the superjogs formed at the ends of the segments that have undergone cross slip ...". These superjogs are of course the macrokinks (MKs) that we have described in sections 3.4.3.3 and 4.2.3.2.
5.6. Kink models
5.6.1. Foundations Mention of kinks on KW locks can be found in the above excerpt of the paper of Kear and Wilsdorf [63] (see also fig. 29). In fact, MKs can be spotted in almost every early TEM study of deformation microstructures; however, neither had they been identified as such, nor had their importance been really emphasized until Sun and Hazzledine [149] pointed to their existence in the "7' precipitates of a superalloy (section 3.4.3.3). Mills et al. [200] then proposed that MKs could serve as a source of dislocations. On the other hand, Veyssi~re [3] suggested that transport of KWs by MK sliding becomes more
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Microscopy and plasticity of the L12 ~/' phase
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difficult as temperature is increased, and as the bending of KWs in the cube plane gets more pronounced (fig. 62). These arguments and their microstructural consequences were outlined and discussed in section 4.
5.6.2. Hirsch's approach The processes that Hirsch proposed [98-100, 242] in order to explain the flow stress anomaly of L12 alloys, and related macroscopic and microscopic properties, include a very sophisticated piece of complex geometry. As a whole, Hirsch's model is markedly microstructure-oriented. Detailed discussion of it requires a more lengthy development than can reasonably be outlined here. We advise a motivated reader to refer first to Hirsch's simplified presentation [100], before getting into the details of his earlier works [99, 242]. In the following account of Hirsch's 1991-1993 contribution, we shall distinguish between dislocation mechanisms (sections 5.6.2.1 and 5.6.2.2) and their theoretical treatments (section 5.6.2.3). 5.6.2.1. Generalities. The evolution of the microstructure is assumed to take place under the following assumptions" (i) Elongated incomplete KWs are formed dynamically by a zipping mechanism: "In contrast to all [point-pinning] models, the elementary cross-slipped segments extend rapidly laterally ... before the critical angle is reached [q5 in fig. 49], further cross slip along the same segment stabilises the long cross-slipped screws .... generating structures ... where long cross-slipped screws ... are linked by glissile superkinks ..." (figs 65 and 72). This mechanism of a competition between expansion and KW lengthening had already been described by Veyssi~re [3]. (ii) Immobile screws do not move again after being locked, they are bypassed instead: "... The yield stress is controlled by [MKs] bypassing the screw dislocation locks, and the increase of the yield stress with increasing temperature is due to the decrease of the lengths of [MKs] ...", which is exactly the same as Mills et al.'s original argument 22 (Mills et al. [200]). Note that this hypothesis is strictly at variance from that deduced by Mol6nat et al. [237] from their in-situ experiments (section 5.5). (iii) Slip proceeds by bursts, by unpinning of MKs, according to a sequence reproduced in fig. 72(a). Although there is no direct evidence for the bypass mechanism, the first two hypotheses are consistent with post-mortem microstructures. What in Hirsch's model differs from pre-existing points of view is the third hypothesis, which we now consider in some detail since it determines the development of the model. Starting from a distribution of kinked screw superdislocations, new slip is nucleated when the stress is sufficient to move the longest MKs. Hence, several conditions have been postulated: (1) There should be some provision for pinning MKs (at A, in fig. 72). Otherwise, nothing would prevent MKs from slipping and zipping out further KW segments (similarly to the schemes of figs 32, 38 and 59), so that bypassing would actually be impeded. 22Compare fig. 5 in [100], to fig. 5 in [200].
P. VeyssiOreand G. Saada
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Ch. 53
(a) G
D
.........
-4 . . . . . . . . . ---,,*P... g ,
v
.....
'~
Ls
\
~--'
(b) 6
" q ~
B
Fig. 72. (a) The basic assumption in Hirsch's model is that screw superdislocations move through a succession of locking/unlocking events. The locking occurs at A, where kink AB connects with a KW segment of length Ls. The unlocking occurs by means of the bypassing of A following the sequence (1 to 6); the dislocation is again immobilized at CD through cross-slip transformation into a KW segment. Note that in this process, B is implicitly assumed to be locked, otherwise the unlocking should rather proceed by a zipping process according to scheme (b) (this latter scenario would actually conform better to Hirsch's statement (i) in section 5.6.2.1). The sequence represented in (a) bears some similarities with the original hypothesis of Takeuchi and Kuramoto [81] in that it is considered that there is a steady breakaway process in which segments of length Ls move over a free-flight distance ~ (d in fig. 69), before they are locked again. However, the details of the locking/unlocking (pinning/unpinning) differ significantly. (2) MK immobilization (pinning) occurs through local dipolar configurations (at E in fig. 73). Only MKs formed by a double cross-slip process - e . g . , EKs and simple MKs (section 4 . 2 . 3 . 1 ) - are considered, switch-over MKs are ignored. (3) The pinning will differ according to whether KWs are incomplete (lJeump - b) or complete (then lJeump scales with ~c, the equilibrium APB width in the cube plane). (4) There is a thermally-activated process of bypassing the locked structure that is triggered at MKs (at A, in fig. 72). Since the nature of pinning differs with temperature, two different unlocking mechanisms would be rate-controlling at low and high temperatures, respectively (roughly domains B 1 and B2, in the definition of section 2.2). (5) Just as in the point-pinning models, the locking/unlocking process is steady. The question of dislocation multiplication is not addressed while annihilation is largely debated to explain for instance the formation of APB tubes. Hence, as a result of the steady evolution of the microstructure, there is a critical stress for dislocation motion in an octahedral slip, determined as a fraction of #b/hMK. This stress is essentially athermal, but the bypassing mechanism involves a thermally-activated process that introduces an additional small strain-rate dependent stress.
5.6.2.2. Local dislocation mechanisms. In the following, these pinning/unpinning mechanisms, which are unusually complicated in Hirsch's full presentation, will be discussed in a purposely simplified manner.
Microscopy and plasticity of the L12 .yt phase
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(a)
(b) 7-,
I j um P = 2b
(c) 7"
(d)
(e)
(f)
(g)
Fig. 73. Locking and bypassing at low stress/low temperature (Hirsch [99]). Some of the notations of the original paper have been reproduced for comparison. Note that the jogs E and Y (which becomes B) are supposedly so sessile that the dipole cannot slip on the octahedral plane. The critical steps of the bypassing process which corresponds to the cross slip of L towards L' is represented in the insert between (c) and (d).
P Veyssidreand G. Saada
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5.6.2.2.1. Pinning and bypassing at low temperature.
In the low-temperature mecha-
418
nism (fig. 73), /jump is taken as b23 (in fact, it is not considered that, as temperature is raised in domain B 1, the expected effect of temperature is to increase /jump gradually between b and a significant fraction of Ac). Locked screw segments are terminated by EKs (in the APB jump configuration of fig. 32, with the leading superpartial being located on the primary octahedral plane, two atomic planes next to the initial slip plane). Hirsch then considers the reaction between an EK and a regular MK located in the initial octahedral plane (figs 73(b) and (c)). Within the general frame of fig. 72, the stress to overcome the pinning is inversely proportional to the MK height hMK; it is to a large extent athermal (AB in fig. 72). Nevertheless, it is controlled by cross slip event, the critical step is between fig. 73(c) and 73(d), when the leading superpartial has to react with itself (L and L'), to start the unzipping process schematized in figs 73(d)-(g). According to Hirsch, the critical length for the cross-slip annihilation is b. More serious is the fact that if we follow Hirsch's hypothesis, and assume that this thermally-activated cross-slip process requires the motion of L over 2b in order to annihilate with L' (fig. 73, this distance is within the dimensions of the core of L and L'), the force of attraction between these branches is very large, corresponding to a stress Oatt = ~/271" of the order of the theoretical elastic limit. In other words, the critical step in question must be essentially athermal. More complicated situations were proposed by Hirsch [99] wherein, as the magnitude and the orientation of the applied stress vary, the leading superpartial experienced multiple double cross slip - conforming to the computer simulations of Chou and Hirsch [ 199], discussed in section 4 . 2 - so that the lateral MKs consist of a number of jogs extended successively in the octahedral and in the cube plane. Examples of such configurations are given in the following section (see figs 75 and 76). Insofar as debris is concerned, the low temperature bypassing mechanism predicts rows of vacancies and of interstitials, such as DBFE in fig. 73. In addition, Hirsch states that APB tubes joined by mixed dipoles are formed when an expanding superdislocation meets an immobile kinked one. However, as stated earlier, the observation of such debris in the low-temperature regime (section 3.4.3.6) cannot validate any particular hypothesis other than that of the occurrence of cross slip.
5.6.2.2.2. Pinning and bypassing at high temperature.
We first consider the simplest configuration (e.g., fig. 11 in [ 100]). The difference from the preceding case is that the mixed segment (CJ in fig. 74) that closes the now complete KW configuration (/Jcump --- Ac) is entirely contained in the cube plane and that neither can dipoles such as DBFE in fig. 73 be formed to further immobilize this segment, nor can annihilation processes, such as between L and L' in the same figure, operate. 23Hirsch postulates that the path of superpartials is dictated by core symmetry (see section 4.2.1.1) and that in order for the superpartial to cross-slip again in the primary slip plane to yield an APB jump, it must have travelled in the cube plane over a distance equal to an even number of elementary steps of b/2, at which the core spreads alternately on the primary and on the cross-slip octahedral planes (fig. 51). This assumption is also instrumental in the simulations of Chou and Hirsch [199] on which some of Hirsch's crucial postulates rely. This is in particular the origin of the complex kink geometries of figs 75 and 76.
Microscopy and plasticity of the L12 ,yt phase
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(a)
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419
L.
-
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H ! (d)
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Fig. 74. The h!gh temperature bypassing process in the Hirsch model [99] for a complete KW lock formed in one jump (/jump = Ac). In order for the bypassing process to operate (see fig. 59(b)), the closing jog CJ must be sessile (at point A, in fig. 72(a)). It is accordingly postulated that the further zipping of a length of KW segment is impeded when CJ has adopted a Lomer-Cottrell type structure as schematized in (b). Then, the LC serves as a pivot for the rotation of HI in the octahedral plane. As the leading superpartial H has turned by 180 ~ it experiences an attractive interaction with the trailing superpartial T in the primary slip plane (circled part in (c)). In this particular case of KW formation, there is in principle no obstacle to oppose the annihilation of these two segments. The configuration schematized in (d) should form athermally. More complicated situations, which differ from the present one by the fact that the KW lock is postulated to have formed after several double cross-slip events, have been introduced by Hirsch in order to allow for a thermally-activated bypassing process. These alternative mechanisms are schematized in figs 75 and 76. Finally, the configuration shown in (e) reproduces Hirsch's sketch of the motion of the superdislocation as in (d) but when the KW lock is bent in the cube plane. Based on this picture, Hirsch claims that the bending is unimportant in the bypassing process. There are in fact several reasons to believe that this bending causes some further drag stress. Firstly it is surprising that the transformation of the additional pair of edge segments, N, into LC locks (as postulated for CJ in (b)) is not considered in this case. In addition, and according to the directions of motion of CJ in (a), the force that is exerted onto segments N due to the external load, acts against the motion of the superdislocation HI in the octahedral plane. Finally, since cube slip is impeded by lattice friction in the cube plane, the longer the segments N, the larger the drag force.
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Because of the postulated pinning/unpinning sequence of fig. 72, Hirsch did not consider a still possible unzipping mechanism (his point (i) in section 5.6.2.1, fig. 72(b)). He introduced another bypassing mechanism wherein the "long" jogs with edge component (CJ), which had to be glissile during the zipping of the KW lock, "transform into pure edge relatively sessile configurations such as Lomer-Cottrell locks, when they come at rest" (fig. 74(b)). Based on the published literature on deformation microstructures in L12 alloys, on our own TEM experience, and on the fact that line tension opposes the formation of Lomer-Cottrell configurations at equilibrium, we consider the hypothesis on Lomer-Cottrell locks as untenable. In particular, since the presence of these edge locks is markedly composition-dependent (section 3.1.1.2), it cannot be safely utilized within a model of the flow stress anomaly that has to be applicable to every L12 alloy with a positive TDFS. In order to follow Hirsch's subsequent arguments, let us nevertheless assume that the closing segment CJ could be immobilized under a Lomer-Cottrell lock. In the configuration of fig. 74, a critical step of the bypassing process appears to be when the leading superpartial is disconnected from the point where it was attached to CJ (fig. 74(c) to 74(d)), yet the "bypassing mechanism involves the trailing superpartial" (Hirsch [242]). The difference from the low temperature process is that it is now required that superpartials be switched and that, at some stage, the superdislocation be constricted (the most detailed explanation of the latter statement is certainly in section 5 of Hirsch [242]). However, in the simplest case of KW bypassing represented in fig. 74 where the KW lock is formed in one jump of )~c (according to Chou and Hirsch [199]) this should be quite a general situation at least within a range of load orientations), we have not been able to identify the critical constriction step claimed by Hirsch (in the circled part of fig. 74(c), H and I are located in the same octahedral plane, they should annihilate driven by their attractive interaction). Again, there is apparently no thermally activated step, though the reason for the present objection is different from the one we have offered to the low-temperature bypassing mechanism. Furthermore, it is difficult to reconcile the assumption that the bypassing by superpartial switch-over should be a mechanism of high temperatures and/or of high stresses [242], with the observation that, according to Couret et al. [72], about 25% of the MKs are already switch-over in nature after deformation at room temperature and up to 400~ (section 3.4.3.3), i.e., all over domain B 1 and in particular when the flow stress is the least in domain B. According to Hirsch, when more complicated paths of KW formation occur as a consequence of the operation of a multiple double cross slip, much more complicated geometrical configurations have to be considered. This is because, based on the computer simulations of Chou and Hirsch [199], macrokinks should be stepped between several (111) planes. In order to illustrate the postulated origin of the critical activation step in the high-stress/high-temperature case, these situations of stepped MKs are considered here. We start from the second simplest high-stress/high-temperature situation depicted in fig. 75, where the formation of the KW lock is preceded by a succession of double cross-slip events. We have used the same notations as in the original papers. In our understanding, the jogs E and F stem from the first cross-slip process (of height, b) after which the transformation into a KW lock should occur irreversibly [199], i.e., lieump -- b as in fig. 73 (section 5.2.2). The double cross slip yields a pair of jogs labelled 7 in
Microscopy and plasticity of the L12 "7~phase
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(a)
421
cj (Olo)
zl) b
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~a3 "LC ~H
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t
Fig. 75. Hirsch's high temperature bypassing mechanism (see also figs 74 and 76). (a) The KW lock is assumed to have been formed completely; however, different from fig. 74, this occurs through several consecutive double cross-slip processes, resulting in a multi-jogged MK. (b) The jogs are assumed to be sessile in the LomerCottrell type configuration (J3, o~, fl and "7). When the gliding superdislocation passes behind this MK, it first experiences the pinning effect of the attractive interactions with the MK with which it is placed in a dipolar situation (c). This is just as in the low temperature mechanism (fig. 73(b)-(d)). Then, selected jogs must transform into a glissile configuration in such a manner that a length of dislocation becomes glissile again in the cube plane. It is unclear how, while/3 and "7 are not modified, it is c~ that transforms into c~* (in fact, this should involve some additional cross slip in the cube plane). The thermally activated part of the high temperature bypassing process corresponds to the transformation of a length of jog locked in the Lomer-Cottrell configuration into a glissile segment. The same mechanism is represented in fig. 76 in the case of a single jogged MK; the difficulty in defining the transformation of c~ into c~* then disappears. From comparison of figs 75 and 76, it seems difficult to define with precision the critical length lct in the high temperature activation volume (it is nevertheless given as 2.5b by Hirsch [99]).
422
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(a)
L
(b)
LC
Fig. 76. Redrawing of fig. 15 by Hirsch [99]. Most of the notations are unchanged (except that superpartials L and M become L and T). The jogs E, F (height 2b in the (010) plane) are apparently required as the first step of KW formation by multiple double cross-slip. The long jogs J1 and J2 are assumed to be immobilized by formation of Lomer-Cottrell type configurations. J l and J2 become glissile again at the end of the process of bypassing, by means of a thermally activated process with an activation volume proportional to the jog length. A debris that consists in a pair of elongated prismatic loops joined by an APB ring is left behind (marked E, F, J~, J~). If should be noted that at step (c), the forces applied onto the bypassing segment located beneath the stepped kink act in a direction opposite to those applied to segments C and D. fig. 75 (J1C and J2D in Hirsch's notations, see also fig. 76) in the cube plane. Like J3, jogs 7 , / 3 and c~ are sessile under the form of Lomer-Cottrell segments. It is worth emphasizing that whereas the whole presentation of the high-temperature mechanism is founded on this type of locking, all that is required is that slip on (010) is slow compared with slip on (111) (section 4.3.1). For the sake of simplicity, the switch-over of superpartials H and I is not represented here (see however fig. 76). The unlocking corresponds to the situation where (Hirsch [99]) "the sessile jogs J1C and J2D [7] are transformed into a series of glissile elementary jogs on (010), by a thermally-activated process. Dipoles J1E and J2F will be pinched off and HI can then advance, with the screw [T] being replaced by a screw of opposite sign [T']. The activation energy for the unlocking process [which then corresponds to the transformation of a LC lock into a mixed segment glissile in the cube plane] will be greater than that for unlocking the dipole illustrated in figure (73)" [99]. Hirsch "would
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Microscopy and plasticity of the L12 ~/' phase
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expect the critical length l'c to be large for this mechanism, of the order of the lengths J1C and JzD [')'] of the jogs lying in (010)". It should be remarked then that in view of the multiplicity of double cross-slip paths, the uncertainty on l'c is large. In fact, ltc can amount to between b and )~c and this is very important with regards to a search for a consistency between the predictions of the model and experimental determinations of activation volumes (6.5.6.2.3). When it is considered that the formation of a KW lock is completed after several double cross-slip steps (fig. 75(a)), complex coalesced MKs are formed (fig. 75(b)). At some stage, the edge segments located in cube planes transform into Lomer-Cottrell sessile configurations which provides the pivot on which the MK turns (A in fig. 72). This pivot is required for the above bypassing mechanism to operate (fig. 75(c-d)). The process shown in fig. 75 requires a clever, though arbitrary, choice of locking and unlocking events: while some edge segments are locked (e.g., J3, ol, /3 and -y, in fig. 75(c)), others must be mobile for the by-passing process to operate (c~* in fig. 75(d)) The same remark can be made for the sessile jogs J1 and J2 that must become glissile, again rather arbitrarily. Specific debris such as mixed dipoles and APB tubes are produced directly by a mechanism represented in a simplified manner in fig. 76 (EFJ1J2; see also fig. 15(c) in [99]). Regarding the bending of KW locks in the cube plane, with an increasing magnitude as temperature increases (section 3.4.3.5), Hirsch [99, 100] considers that since the bowed segment consists of a succession of screw parts and of elementary jogs - the latter being in principle mobile in the cube p l a n e - a bowed out screw may not be significantly more difficult to bypass than a straight screw. Hirsch nevertheless acknowledges the fact that work will have to be done against the applied stress on (010), as discussed by Saada and Veyssi~re [206]. However, the jogs in question have a total length which scales with the magnitude of the bending, which itself increases markedly with temperature, a point not considered by Hirsch. Hence, long jogs may have to move in the cube plane against friction. Finally, it is unclear why in Hirsch's explanations, these particular jogs, which are generally longer than the CJ jogs, are regarded as glissile while, for the hightemperature bypassing process to occur, the jogs CJ themselves are said be locked by transformation into a Lomer-Cottrell type configuration. 5.6.2.3. Theoretical analysis. Hirsch's calculations [99, 242] are very involved. Deriving them in detail would require the definition of a large number of variables. For the sake of simplicity, we shall restrict the present account to what we consider as the main hypotheses and equations. In a preliminary step, the model theoretically addresses the paths followed by companion superpartials during the transformation into an incomplete or a complete KW lock, occurring by multiple double cross-slip events. In particular, under particular assumptions, it is shown that repeated APB jumps should occur under large stresses (later supported in the numerical simulations of Chou and Hirsch [ 199] and by Paidar et al. [236]). This is inconsistent (i) with the fact that, experimentally, repeated APB jumps are observed most frequently in domain B 1, that is, at the lowest possible stresses within the domain of positive TDFS (see also section 4.2.3.2.2) and (ii) with calculations by Saada and Veyssi~re [243]. As mentioned above, the postulate is that the locking/unlocking mechanism is steady, which yields the following expression for the strain-rate
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P. Veyssidre and G. Saada
~=~~
- (Hu-Hl))kT
"
Ch. 53 (37)
The locking activation enthalpy H1 is taken from Paidar et al. [86], which then establishes the orientation dependence. Activation enthalpies for unlocking Hu are calculated in accord with the above low- and high-temperature mechanisms, both of which can be treated under the same formalism. The expression of the activation volumes is written v = c~b hMK l/c
(38)
where c~ is a geometrical factor that depends essentially on the shape of the expanding MK. The critical part of this expression is l'c, which represents some activation length that corresponds to the distance the superpartials have moved during the activation (section 5.6.2.2). (i) In the low-temperature process, l'c is taken as b, that is, for cross slip of superpartial L to occur over/jump between fig. 73(c) and (d) (see insert). As mentioned earlier, this critical length could be several times longer (in a similar situation in f.c.c, crystals, it is usually taken as 3 to 4b). Moreover, in view of the attractive forces that should prevail during cross-slip annihilation, it is unclear whether this step is actually thermally activated. (ii) In the high-temperature regime, the critical length would correspond to the constriction of a length of Lomer-Cottrell lock (fig. 74, steps (b) and (c)). In this case, l'c is taken as 2.5b (Hirsch [100]). Again, the adjustment for the critical distance is quite flexible since it scales with/jump, which is itself orientation and temperature dependent. In such conditions, it is quite easy to theoretically fit the experimental activation volumes which, in domain B 1, are about 3 times larger than in domain B2 (Baluc et al. [276], fig. 14(a)). In addition, it should be kept in mind that the experimental activation volumes are only apparent ones and, in L12 alloys, corrections for work hardening are a significant fraction of the direct measurement (section 2.2.8.2). Consequently the meaning of the agreement claimed by Hirsch may be questioned both on the theoretical and on the experimental sides. Finally, note that within Hirsch's hypotheses, the activation volume should be orientation-dependent since when N is varied from large values to zero, the high-temperature bypassing should evolve from that represented in fig. 74 to that in fig. 75 (in view of Chou and Hirsch [199] simulations, the smaller N, the more stepped the cross-slipped path that yields the formation of a KW lock). Under some simplifying assumptions, Hirsch derives an equation relating the stress to temperature that eventually conforms to the expected flow stress anomaly-related properties, such as the positive TDFS, the small strain-rate sensitivity and the tension/compression asymmetry. However, in addition to the above-mentioned problems posed within detailed postulated rate-controlling dislocation mechanisms, Hirsch's model appears somewhat unsatisfactory under the following respects: - the assumption of a steady locking/bypassing process is not established; it is somewhat in conflict with the in-situ observations of the jerky nature of dislocation motion,
Microscopy and plasticity of the L12 .yt phase
w
425
- in order to account for the orientation dependence of the flow stress, the PPV formulation of the activation enthalpy for cross slip is incorporated, which, as argued by Khantha et al. [88, 101], is questionable, - the average length of KW locks Ls, which is instrumental in the derivation of the expression of the flow stress dependence upon temperature (eq. (49) in [99]), is given an arbitrary stress dependence, - the role of switch-over MKs is not considered, whereas these constitute a significant fraction of the population of MKs. It is recalled that since switch-over MKs do not involve a double cross slip events, considering switch-over MKs instead of simple MKs could save a number of the many complications of Hirsch's model, - the dislocation velocity is expressed as V lOTob3 - - ,~,
Ct
kT
(39a)
or more simply as V
~
23-o T'
(39b)
where Ct is the sound transverse velocity (% and T are expressed in MPa and Kelvins, respectively). For reasonable values of ro and T, V is of the order of the sound velocity which is by far too large.
5.7. Modelling One reason why the various attempts to work out a theory of the mechanical behaviour of L 12 alloys have failed arises from the difficulty of modelling. This difficulty is manifested at two levels: (i) While there is no reasonable doubt that the transformation of a glissile screw dislocation into a KW occurs by cross slip, the description of the dynamical behaviour of a screw superdislocation under stress requires the knowledge of the properties of its kinks and more precisely of their distribution (distance and size) and of their nature (simple, switch-over, coalesced), which themselves determine kink mobility (section 4.2). (ii) Once the dynamics of a single screw dislocation are known, the dynamics of a large number of dislocations remains to be treated adequately (section 4.3). (iii) Strain proceeds mostly by motion of non-screw dislocations. Up to now there have been few attempts to solve these difficult problems. Three such attempts are summarized below.
5.7.1. Mills and Chrzan (1992) Since it appears that the general idea of steady state and of an average distribution of obstacles fail to explain the observed features, and since a completely dynamic calculation is extraordinarily complicated, Mills and Chrzan [241] have worked out a dynamic simulation relying on the following assumptions:
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(i) The velocity V of a dislocation segment of curvature n is given V = rl(Tob- F n ) ,
(40)
where F is the line tension, t~ the curvature, and 77is a constant (Frost and Ashby [273]). (ii) The probability of pinning per unit length depends upon the orientation of the dislocation segment and upon temperature. Note that in this simulation, the pinning points have the length of a Burgers vector as in the TK and subsequent models. (iii) A correlation between neighbouring locking events is allowed in order to immobilize long dislocation segments. (iv) Overcoming of pinning points depends on the angle between the two segments adjacent to the pinning point. The results of the simulation are quite stimulating. Starting from a circular dislocation loop, one obtains loops elongated along the screw direction and these loops contain kinks. At low stresses the motion of the screw parts rapidly exhausts while, at large stresses, multiplication may occur. Intermediate stresses correspond to intermediate situations. It is rewarding to obtain pictures that correspond qualitatively to the observed situation, simply based on a pinning/unpinning mechanism and a few adequate rules, with no hypothesis made on the steadiness of the process. The details are even more interesting when analysis of the size distribution of kinks is made. There are indeed two categories of kinks: - s h o r t ones (h < h~K ~ #b/To), which are in fact immobile and which obey an exponential distribution law as confirmed by electron microscopy (Couret et al. [203]), - large - m o b i l e - kinks, which follow an exponential law different, however, from the previous one. Kink distribution depends on the applied stress. The simulation strongly suggests that during deformation the effect of pinning is to reduce the number of moving dislocations, not to slow them. It suggests that collective processes can be very complex. There remain however, several difficulties with this work. Firstly, the simulations are conducted under constant stress, i.e., samples are in fact deformed under creep conditions and we have seen in section 2.2.9 that this yields quite specific mechanical properties. Secondly, the time constants are far too small, of the order of 10 -7 seconds (see fig. 3 in [241]). The elongated screw segments are in fact approximated by a succession of local pinning points, and whether this allows for the complete transformation into a KW, which is observed experimentally both in post-mortem and in-situ experiments, remains to be validated. At present, the simulations do not allow the jogs CJ to compete with the expanding MK (fig. 59). Finally, there is no provision for cross slip, hence for double cross slip, so that neither the formation of simple MKs and of EKs (with the process of repeated APB jumps) nor the processes of annihilation on neighbouring slip planes (discussed extensively in section 4), can be accounted for. In view of the importance of these features in the deformation microstructure, the simulations of Mills and Chrzan in their present state should be regarded as a first step toward a representation of the dynamic processes.
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5.7.2. Greenberg and Ivanov (1992) These authors [274] have addressed the issue of the collective behaviour of dislocations through a set of differential equations, with the simplest being dng = --ngVgs -F nsVsg + M, dt
(41)
dns -- ngVgs -- nsVsg, dt
(42)
qo"r = a # b (ng + ns) 1/2 q- "rf,
(43)
dE
d--t = b qong V,
(44)
where ng and ns are the glissile and sessile dislocation densities respectively, Ugs and Usg are the transition probability from the glissile state to the sessile state and conversely, M is a multiplication term, qo is the Schmid factor, c~ is a coefficient comprised between 0.2 and 1, "re is a friction stress and V is the dislocation velocity assumed to be stressindependent which is questionable. More complex situations, which would involve more than two dislocation species, could be treated with the same method. The kinetics is very simple, the only non-linear term is in eq. (43). Curiously enough, both sessile and glissile dislocations appear to contribute to the internal stress that resists dislocation motion (eq. (43)). By means of a clever choice of parameters, a numerical solution of these equations provides a fair representation of the temperature dependence of the flow stress for Ni3A1, without any input, however, from the actual microstructural mechanisms that control the flow stress anomaly. In fact, there is nothing in the above hypotheses that can be clearly associated to some microstructural properties which would be specific to L12 alloys. In addition, this model as it stands has no provision to account for the flow stress reversibility (section 2.2.5).
5. 7. 3. Cuitino and Ortiz (1993) This work [275] constitutes a nice attempt at modelling the mechanical behaviour of L12 alloys in domain B, by associating physical assumptions on dislocation mechanisms to a comprehensive and solvable set of equations. Although some of these physical assumptions are questionable, Cuitino and Ortiz' method deserves particular attention since it might well be validated under physical assumptions that would better conform to the actual microstructural organization. The following overview is restricted to outlining some general features of the model, it does not pretend to cover it completely. The density of mobile dislocations on a given slip system 'a' is assumed to take the following form
pa--Psat[1--(1--/9~ Psat
~a }] Esat
(45)
P. VeyssiOre and G. Saada
428
Ch. 53
where Ca is the shear strain provided by slip system 'a' and Csat a critical shear strain, named saturation shear strain by Cuitino and Ortiz. This expression simply expresses the fact that due to multiplication and annihilation, the dislocation density in the slip system increases with strain, from po to Psat. The number na of obstacles per unit surface encountered by a dislocation moving in this slip plane is the sum of two terms: (i) a forest term which scales with the dislocation density in the intersecting planes, (ii) the pinning points which nucleate directly on the dislocation line itself, as postulated in the point-pinning models. The latter is calculated in the following way. Let L be the mean free path between cross-slip pinning events and n cs the number of obstacles per unit surface created by the cross-slip process. The rate of obstacle creation per unit time and unit surface is dn cs dt
=
/gaVa
(46)
L '
which integrates as CS
na
Ea
(47)
-- 2bL"
Assuming that L may be expressed as L - Lo exp
{H~} ~
(48)
,
where/-/1 is the activation enthalpy for pinning, one gets
na = ~-'~(Ca,~ p~) nt- k - Ea ~,
(49)
w h e r e Ca,/3 and k are constants. Finally the strain rate is related to the stress by
-- ~o
7"a %
-- 1
]
(50)
for 7"a > %, and 0 otherwise. Here % characterizes the hardness of the crystal. In fact, these expressions should also contain a work-hardening term, which is included in the work of Cuitino and Ortiz, but which, since it is unnecessary for our purpose, we have ignored for the sake of clarity. The assumption contained in expression (50) is a very strong one, in that it implies a threshold stress below which no deformation could occur. This model has been solved numerically and compared to experimental data. The authors claim a good fit. However, the comparison is not thorough as it stands and the physical foundation of the model is definitely not acceptable. In particular, it does not distinguish clearly between mobile and immobile dislocations (eqs (47) and (49) imply
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429
that no pinning is created on dislocations at rest). Besides, pinning points are assumed to form on dislocations irrespective of their character. Furthermore, there is no provision for unpinning. Finally, the model makes use of the PPV formulation for the activation enthalpy for pinning, which is considered as rate-controlling. Nevertheless, as stated earlier, the method looks quite promising, and should be developed and used under more reasonable hypotheses.
6. G e n e r a l
conclusion
In spite of the considerable amount of experimental and theoretical effort devoted over 35 years to the positive TDFS of L12 alloys, it is quite frustrating to realize that the mechanisms that are involved this anomalous behaviour remain so unclear. The present state of understanding of the mechanical properties of L12 alloys stems largely from a tradition of employing former paradigms in order to fit new situations.
6.1. Analogies and differences with previously analysed systems The scientific community has very early convinced itself that the flow stress anomaly of L12 alloys is a manifestation of effects that are localized at the level of superdislocation cores. The validity of this idea has been supported as early as in the mid-sixties when superdislocations elongated under the form of Kear-Wilsdorf locks were observed directly by TEM. As in other cases of atypical mechanical behaviour, such as those encountered in b.c.c, and h.c.p, crystals, and by analogy with these, an explanation relying on cross-slip properties of a single superdislocation has been imagined for L12 alloys. The design of the so-called pinning point models indeed proceeds from achievements of several theories of the flow stress, mainly of b.c.c, crystals, and to some extent of f.c.c, and h.c.p, metals and alloys. By analogy with b.c.c, metals, it has been postulated that the core of screw superdislocations in L 12 alloys tends to expand out of the slip plane. Hence, when mechanisms of superpartial cross slip, specific to the L12 structure, are incorporated in a given model, a violation of the Schmid law ensues quite naturally. A few parameters have then to be appropriately adjusted. Besides, it is now quite clear that in L12 alloys the locking process is thermally activated and much faster than the unlocking. The activation enthalpy for unlocking by cross slip would be very large indeed except for the levels of stress attained at the flow stress peak. It is thus reasonable to postulate that the self-destruction of KW segments occurs through a by-passing process originating from the expansion of adjacent kinks. Striking enough is the similarity between the positive TDFS of L12 alloys and that of beryllium deformed under prismatic slip. At the dislocation core level, the analogy between these is quite close. As slip proceeds on prismatic planes in Be, screw dislocations tend to cross-slip onto the basal plane while, in L 12 alloys, superdislocations undergoing octahedral slip are subjected to a finite driving force for cross slip in the cube plane.
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However, conducting too close a comparison with L12 alloys would be quite uncertain since the splitting widths of the stable configurations in these two cases differ by about one order of magnitude. As a consequence, the rate of unpinning should be far larger in Be than it is for instance in Ni3A1. Finally, it is worth noticing that within the range of negative TDFS both of b.c.c. metals and of Be, as well as during the operation of cube slip L12 in alloys, in-situ straining experiments have revealed that dislocation motion is smooth. This is consistent with the operation of a steady thermally-activated process. By contrast, in the domain of positive TDFS of both Be and L12 alloys, in-situ observations indicate that dislocation motion is jerky, which is at odds with every model devised so far, with the exception of the computer simulations of Mills and Chzran [241 ].
6.2. The evolution of the microstructure
In L12 alloys, the thermally-activated locking of superdislocations involves cube slip. The locking starts to operate at temperatures below 300 K - though this does not apply to every L12 alloy - to form incomplete and complete Kear-Wilsdorf barriers. KW segments are kinked as a consequence of the finite probability of locking exhibited in places by superdislocations in the course of their expansion. As the test temperature is raised, KW segments are increasingly bent in the cube plane, the mean magnitude of the bowing out depending on load orientation. Since cube slip is sluggish, the magnitude of the bowing of the KW segments in the cube cross-slip plane is time dependent, hence not necessarily homogeneous all through a given sample. Considering that the deformation microstructures in post-mortem samples are not prohibitively affected by relaxation, one is led to envisage that, once a KW is formed, the subsequent motion of screw superdislocations does not proceed via their thermallyactivated self-unpinning. In fact, the building-up of the deformation microstructure can be explained through transformations originating at kinks. The model of bypassing published by Hirsch in the early nineties is however probably not dealing with the adequate rate-controlling elementary mechanisms. The TEM evidence of kink coalescence suggests that the motion and the organization of kinks play an instrumental role in the evolution of the microstructure. This conclusion is quite at variance from the underlying hypotheses in the models ~ la b.c.c, of pinning and unpinning of screw superdislocations. It is also in conflict with implications of in-situ experiments which, by their nature, cannot account for the three-dimensional organization of the microstructure. As a superdislocation expands and forms kinked KWs or as a result of the coalescence of pre-existing kinks, some kinks reach eventually a critical length h~K ~ a#b/'ro, beyond which they may in turn operate as sources and subsequently bypass KWs (in L12 alloys, because of the close neighbourhood of locked superdislocation segments, a kink-source is not liable to operate more than once). The fact that kinks are liable to coalesce in order to provide new sources is consistent with the fact that a distinct critical stress for octahedral slip is ill-defined. Following Hemker et al. [54], but with some significant differences in the mechanisms involved, we propose that a hint f o r - if not the clue to - a microstructure-based
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Microscopy and plasticity of the L12 .yl phase
431
description of the positive TDFS of L12 alloys relies on the dynamic evolution of the kink distribution. The situation may then seem rather gloomy since, to date, there is no direct method either to observe or to model this type of phenomenon, and since possible causes of mistakes are numerous both theoretically and experimentally (post-mortem TEM cannot allow for an unambiguous study of a dynamical problem and in-situ TEM is restricted to crystallographically limited situations involving too small a sample, where in addition end-effects at free surfaces are not under control).
6.3. Consequences on mechanical properties The scenario that we have briefly outlined in section 4.5 and in the preceding part of this section is, at present, essentially qualitative. It nevertheless enables one to reconcile several seemingly conflicting experimental results.
6.3.1. Creep The following ideas on creep properties are, to a large extent, contained in the works of Hazzledine and Schneibel [106] and of Hemker et al. [54]. Prior to deformation, the mobile dislocation density pom in the octahedral plane is significantly larger than that in the cube plane, pm. Besides, superdislocation mobility is larger in the octahedral plane. Plastic deformation then proceeds mostly on octahedral systems and the creep rate in the octahedral plane is larger than that in the cube plane. What controls pm is the rate of creation of KWs and the building-up of a substructure consisting of kinked screw superdislocations within slip bands, pm diminishes as the number of mobile kinks that can coalesce in order to provide new sources gradually exhausts. Meanwhile, the provision of sources for slip on the cube cross-slip plane, which originates at KW segments, is increased. The exhaustion of octahedral slip followed by a progressive increase of slip in the cube cross-slip plane is consistent with the fact that within domain B, superdislocation mobility is by far the least in the cube plane. Since the locking rate is an increasing function of test temperature, the mechanical behaviour during the exhaustion regime is atypical; the higher the temperature, the smaller the creep rate. On the other hand, the creep rate on the cube plane is normal in that it accelerates as a result of the gradual increase of pm with deformation, which is thermally activated. Primary creep and the exhaustion stage correspond to a situation where the density of mobile dislocations in the cube cross-slip plane stems from the locking of dislocations undergoing octahedral slip; that is, a situation where the slip processes act in series (sequential processes). Therefore, in the early stages of creep, the creep rate is controlled by octahedral slip. On the other hand, when the density of mobile dislocations is large enough in the cube cross-slip plane, multiplication may occur independently in this plane. At that stage cube slip and octahedral slip act in parallel and the creep rate is controlled only by cube slip. Although, and maybe because, creep tests have considerably clarified our understanding of dislocation behaviour in domain B, we feel that creep properties are not yet sufficiently covered.
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6.3.2. Stress relaxation
Stress relaxation tests have revealed several difficulties that cast some doubt on the validity of their current interpretations in L12 alloys. Firstly, beyond the conventional strain of 0.2%, the activation volume undergoes a steady decrease with increasing prestrain though much less rapid than in the microdeformation stage. Secondly, the activation volume corrected for work hardening is far too small relative to the apparent volume. Finally, the existence of a discontinuity of the activation volume at some temperature appears to exist in some alloys only (it has been found in Ni3(A1, Ta) but neither in Ni3(AI, Hf) nor in Ni-rich binary NiaA1, while, apart from that, the three alloys exhibit similar mechanical and microstructural properties [282]). These difficulties suggest very strongly that, in the domain of positive TDFS, L12 alloys do not meet the requirement that the density of mobile dislocations remains constant, as is usually postulated when analyzing transients in "normal" materials. In other words, it is likely that the behaviour of L 12 alloys undergoing stress relaxation attests to the rapid exhaustion of mobile dislocations. In order to improve our understanding of the conditions under which relaxations takes place and possibly of the transients in general, it seems now worthwhile, though probably painful, to follow and to analyze the kinetics of relaxation over the widest possible period of time and to include exhaustion in the analysis of relaxation. Efforts in this direction have been initiated just recently [282]. In conclusion, the fact that data such as activation volumes derived from stress relaxation do not coincide with those originating from strain-rate jumps is not at all surprising considering that the periods of time during which these transients take place may differ quite significantly.
-
6.3.3. Deformation at a constant strain rate
As already emphasized, applying to L12 alloys paradigms that have been shown appropriate to the analysis of the plasticity of "normal" alloys, leads to complicated and somewhat inconsistent results. This is particularly true regarding the analyses of tests conducted under constant strain rate. There is little doubt that superdislocations move at very small stresses, and that the corresponding microelastic limit Tin is almost independent of temperature. On the other hand, the stress Ts required to destroy a KW should also be temperature-independent; it is of the order of magnitude of the flow stress at the peak. In the domain of positive TDFS, the conventional macroelastic flow stress 7o.2 lies between 7"m and, of course, 7-s. Besides, 7-o.2 is at least partly reversible, indicating that the yield stress is not governed by immobile obstacles. In fact, 7-o.2reflects a dynamical process that, in otherwise normally thermally-activated situations, stems from the thermally-activated glide of slow dislocations (such as for the low-temperature mechanical behaviour of b.c.c, metals). This reasoning cannot apply to L12 alloys, since it implies a relatively small activation energy and a significant strainrate sensitivity. Since the strain-rate sensitivity is small in L12 alloys showing a positive TDFS, the dislocations which carry the plastic strain, i.e., the actually mobile dislocations, behave as dislocations in alloys showing similarly little strain-rate sensitivity. Since the screw parts have a finite probability of locking, the modest strain-rate sensitivity should originate from the mixed parts.
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Microscopy and plasticity of the L12 7t phase
433
On the other hand, and contrary to otherwise similar systems, superdislocations generate locks whose destruction requires a level of stress comparable to that of the flow stress peak. The creation of a distribution of such screw sessile segments during superdislocation expansion is at the origin of the exhaustion of the population of mobile dislocations. This population may decay very rapidly either by self-locking or else by annihilation with neighbouring immobilized segments. Under such conditions, the permanent strain that is obtained from the initial density of mobile dislocations should be modest, which is at variance with the observed ductility of single crystals. Thence, the system must have the potential to create fresh dislocations. One likely possibility is that these originate from the macrokinks, which are themselves a byproduct of the above process of locking during superdislocation expansion. Analysing the conditions under which multiplication occurs not only requires a good model of the effect of a stress on the development of the kink, but it also implies that the kink distribution is well characterized. Then, the various factors that contribute to establish the distribution of kinks should be critically envisaged. Amongst these, kink sliding, because of the resulting coalescence, introduces a serious complication in the analysis of the kink distribution after deformation. In addition, the bowing of a kink under stress is accompanied by annihilating interactions with the neighbourhood, be it with adjacent segments or with independent superdislocations (this is why kink-sources cannot operate twice). Furthermore, as a result of annihilation, a fraction of the kink population which was sitting on the neighbouring lines is destroyed. There is no means to predict whether these annihilated kinks will be less or more numerous than those newly created upon kink development as a source. In fact, there is no sensible reason to believe that the kink distribution should remain steady as deformation carries on. It is probably vain to describe work hardening in L12 alloys as a result of the motion of a single screw dislocation or even as the effect of the long-range static (internal) stress exerted by the microstructure. Regarding the evidence for exhaustion under creep and under stress relaxation, i.e., when the stress is kept constant or decreased, respectively, it seems logical that, in order to ensure a constant strain rate, the applied stress is raised. We believe that the strain hardening of L12 alloys results from the necessity to compensate for the massive exhaustion of mobile dislocation by adequate increase of the applied stress. Exhaustion results from (i) an active annihilation with strong obstacles, such as already immobilized kinked KW segments located in the slip plane or in its close vicinity, and from (ii) the trailing of APB tubes by jogged mixed segments. The increase of the WHR with increasing temperature may then stem from an increasing difficulty of dislocations to move within the KW locks, which get gradually more bowed out in the cube plane, as well as from an increasing density of jogs. The drop of WHR occurs when the bowing out of KWs is so large that the hardening becomes gradually closer to that of a forest mechanism and/or with the sudden disappearance of APB tubes. What seems to conform best to the microstructural properties of L12 alloys is a theory that would take into account dynamical phenomena such as the self-exhaustion properties as well as the contribution of the nearby microstructure to the overall exhaustion rate. A complete analysis of the positive TDFS and of related mechanical properties requires
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therefore a thorough analysis of the processes that control the overall mobility, coalescence and expansion of the mixed segments within a distribution of KW configuration. This, in turn, necessitates careful description of the dynamics of multiplication as well as of those of locking within a many-body system, and eventually to write and to solve a set of differential equations, a task which no one has succeeded in doing so far. One of the difficulties arises in addition from the need for some reliable knowledge of the kink population. This direction of investigation is surely not easy. At the present state of investigation of the atypical properties of L12 alloys, it is not clear whether the only hope relies on computer simulations or on a sophisticated theoretical analysis. Although the description of dislocation dynamics in L12 alloys that we have outlined in section 4 appears to be the closest to experimental facts (encountered in post-mortem samples), we cannot exclude that its theoretical developments run into problems as serious as those experienced by the previous models that we have abundantly criticized throughout the present review. A few complementary remarks should be added: Since the conventional flow stress ~-0.2is large, stable plastic deformation in tension requires a large work hardening rate (dot/de > tr, Consid~re criterion) and this represents an intrinsic limitation to ductility. - We have no systematic knowledge of the effects of alloying and of the deviation from the stoichiometric Ni/A1 ratio. These effects have been shown to be indeed significant and difficult to correlate quantitatively with macroscopic quantities relevant to the flow stress anomaly such as the flow stress, the peak stress and with the orientation dependence of these. It could be that the motion of EKs or CJs would be affected by their interaction with point-like obstacles (point defects or impurity atoms), but the extent of this effect is unpredictable. - A s it stands, the kink-based interpretation presently proposed for L12 alloys might be generalized to situations of a positive TDFS in other ordered structures, when the locking of screw segments is similarly quite irreversible. It appears quite clear that, contrary to a belief that dominated research on L12 alloys over the eighties, the questions of the orientation dependence of the flow stress and of the tension-compression asymmetry are not of primary importance in the understanding of the positive TDFS. - It is probably time that several hypotheses on which reliance has been placed, such as a steady dislocation behaviour, a mechanism based on point-pinning, a quasi reversible locking-unlocking process, a critical stress for octahedral glide, be carefully reconsidered.
-
-
Acknowledgements
We are glad to acknowledge the contributions of Drs J. Oliver, A. Korner and M. Mills for having provided us with illustrations, Dr X. Shi for having allowed us to use part of her PhD results, K. Hemker, D. Dimiduk, J. Douin and F.R.N. Nabarro for careful and critical review of the manuscript and B. Willot for invaluable technical help.
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CHAPTER 54
Enhancement of Dislocation Mobility in Semiconducting Crystals by Electronic Excitation K. MAEDA Department of Applied Physics Faculty of Engineering The University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113 Japan and
S. TAKEUCHI Institute for Solid State Physics The University of Tokyo Roppongi, Minato-ku, Tokyo 106 Japan
Dislocations in Solids 9 1996 Elsevier Science B.V. All rights reserved
Edited by F. R. N. Nabarro and M. S. Duesbery
Contents 1. 2.
Introduction 445 Dislocation motion in semiconducting crystals 447 2.1. Dislocation types in semiconducting crystals 447 2.2. Characteristics of dislocation motion in semiconducting crystals 449 2.3. Two schemes of the Peierls mechanism 451 2.4. Discrimination of the two schemes 456 3. Effects of electronic excitation on dislocation glide 461 3.1. Velocity measurements 462 3.2. Macroscopic plasticity 473 3.3. Miscellaneous facts 477 4. Mechanism of the REDG effect 481 4.1. Phonon-kick (recombination enhanced) mechanism 482 4.2. Interpretation of experimental results by the phonon-kick mechanism 487 4.3. Other mechanisms 492 5. Summary 500 References 500
1. Introduction Ordinary factors that govern the plasticity of crystalline matter are temperature, the content of foreign atoms and the density of defects. The roles of these metallurgical factors in dislocation glide are well understood not only in metals but also in non-metallic crystals. However, it is commonly known that in non-metals an additional factor, the electronic state of the crystals, significantly affects the plasticity of the crystal: doping of electronically active impurities alters the mobility of dislocations in semiconductors (doping effect [1]), application of an electrical field induces change in the crystal strength (electromechanical effect [2, 3]), and electronic excitation such as that caused, for instance, by light illumination brings about remarkable effects on the dislocation motion in the crystals (photomechanical effect [4-15]). These non-metallurgical effects are considered to be related to the fact that dislocations in non-metals are usually electrically active and their electronic states are quite sensitive to external electrical disturbances. However, the microscopic mechanisms of these effects have rarely been elucidated. Soon after Kuczinski [4] reported as early as 1957 the photomechanical effect (softening under illumination) on Ge for the first time, strong objections were raised to the presence of the effect [16-18]. After many careful tests were repeated, the effect was denounced to be erroneous [17] or due to nothing other than heating of the crystals by strong illumination [18]. Thus the photomechanical effect appeared to have been overlooked for many years, although some experiments [13] were left unexplained. Meantime, similar optoplastic effects caused by light illumination were discovered also in alkali halide crystals [ 19], in various IIb-VIb compound semiconductors [20], in I-VII crystals of CuC1 [21] and CuBr [22], and even in organic crystals as anthracene [23]. The effects, called the photoplastic effect (PPE), were, however, qualitatively different from the photomechanical effect in elemental semiconductors in the point that in most cases illumination induces hardening of the crystal rather than softening. The PPE is characterized by the reversible hardening on illumination, the specific wavelengths inducing the effect, the significant hardening by rather weak illumination and the saturation of the effect with increasing illumination intensity. Since dislocation motion in these crystals of relatively high ionicity is dominated by point obstacles, the entity that is primarily affected by electronic excitation due to illumination is considered to be the point defects (or impurities). Hence, the central problem is thought to be in the state of the point defects rather than dislocations themselves. We will not devote much space to the PPE in this article but will only briefly touch on the effect in relation to the main topic of this article, the radiation enhanced dislocation glide (REDG). The readers who are interesting in the PPE are referred to review articles [24-26]. The REDG effect treated in this article is an enhancement of the dislocation velocity caused by an electronic excitation of the crystal with electron beam irradiation or with light illumination. The phenomenon is considered to be due essentially to a modification
446
K. Maeda
and S. Takeuchi
Ch. 54
of the Peierls mechanism and hence intrinsic in nature, in contrast to the photoplastic effect in alkali-halide crystals which is undoubtedly the modification of defect hardening and therefore extrinsic in nature. The REDG effect is a softening effect and is at variance with the generally observed hardening effects or the positive PPE in II-VI crystals. Independently from the academic activities mentioned above, some researchers in industries who were developing optoelectronic devices such as light-emitting diodes and semiconductor lasers had noticed the presence of a phenomenon similar to the photomechanical effect [27-33]. One of the technological difficulties that was most serious in the early stage of the development of these devices was rapid degradation of the devices that occurred immediately after they began to operate. It was soon clarified that the degradation is caused by anomalously rapid multiplication of dislocation networks that is induced concurrently with the forward biasing which is necessary for device operation [34]. In most cases the anomalous dislocation multiplication was achieved by dislocation climb that was enhanced by forward biasing; however, in some cases evidence was found for dislocation multiplication in the glide mode [30, 35, 36]. Later, when the entire features were revealed by systematic experiments, it turned out that some of the photomechanical effects are the same phenomenon as the REDG effect. To elucidate the mechanism of the REDG effect, we need full understanding of the fundamental mechanism of dislocation glide. Although it is beyond doubt that the Peierls potential is the main obstacle to dislocation motion in highly covalent crystals, there is controversy on the elementary processes: which process, kink nucleation or kink migration, controls the dislocation mobility? What are the magnitudes of the formation energy of a kink and of the activation energy for kink migration? To these basic questions no unanimous answer has been obtained up to date. In this article, however, a discussion is given about the mechanism of the REDG effect based on our own position on the above problems. The present article is organized in the following way. General features of the dislocation motion in semiconducting crystals are reviewed in section 2, where empirical dislocation mobility equations, doping effects and polarity effects are described. Then, the theory of the Peierls mechanism for a dislocation with high Peierls potentials both of the first and the second kinds is reviewed. According to the theory, there exist two schemes of motion of the dislocation depending on the magnitude of the migration energy of the kinks [37]: one scheme is that the kink velocity is so high that the dislocation migrates without kink collision, and the other scheme is that the dislocation migrates with the generated kinks colliding to annihilate with those of the opposite sign moving in the opposite direction. It is impossible to discriminate the two cases only by the mobility measurements, but the discrimination is essential in understanding the mechanism of REDG effects. Hence, in the final part of section 2, the present authors' view is presented on the elementary process of dislocation motion in the semiconducting crystals under consideration, based mainly on the results of the recent novel experiments for dislocation motion in strained thin films conducted by the group of one of the present authors (Maeda) [38]. Section 3 is the main part of the present article, in which experimental results are presented on REDG effects in III-V compounds of GaAs [39-42], InP [43] and GaP [44], those in elemental semiconductors of Si [45, 46] and Ge [46], those in the II-VI compound ZnS [47] and those in the IV-IV compound SiC [48, 49]. It is shown
w
Enhancement of dislocation mobility in semiconducting crystals
447
that the effect not only depends on the compound species but varies greatly with the type of the crystal, i.e. n-type or p-type, and also with the type of the dislocation. However, it has been clarified that the temperature and excitation-intensity dependencies of the effect can be described by a common formula for every case, strongly indicating that the mechanism of the effects is in common for all the crystals. The negative PPE is also described in this section, and is interpreted in relation to the REDG effect. Section 4 is devoted to the discussion of the mechanisms of the excitation enhancement of the dislocation mobility. At first, the so-called phonon-kick mechanism, the most plausible mechanism for the REDG effect, is briefly reviewed and then it is shown how the experimental results are interpreted by the model. Subsequently, other possible mechanisms for the REDG effect are also presented and critically discussed. Section 5 summarizes the previous sections.
2. Dislocation motion in semiconducting crystals 2.1. Dislocation types in semiconducting crystals Dislocations in covalent crystals differ from those in metallic crystals in various respects [50, 51 ]: (1) Along the core of the dislocation having an edge component, dangling bonds appear at every atomic distance; they may be reconstructed to form pairs of bonds. (2) Localized electronic states may accompany the atoms at the core of the dislocations; they can produce deep energy levels in the bandgap. (3) The Peierls potential of the first kind and also that of the second kind are high due to the covalency in the interatomic bonding. (4) In compound crystals, a variety of dislocation types exists depending on the atomic structure of the core. These peculiarities in covalent crystals yield characteristic dislocation behavior which is substantially different from that in metallic crystals. Some examples are doping and crystal polarity effects on the dislocation mobility, as described later. The materials treated in this article are crystals tetrahedrally coordinated due to sp 3 hybridization, i.e. elemental semiconductors of Si and Ge with the diamond structure, III-V and II-VI compounds with the zincblende or the wurtzite structure and SiC with the 6H hexagonal structure. Although the degree of the covalent bonding, or the ionicity, varies from crystal to crystal, dislocation behavior in these crystals has many common features: (1) The mobility of the dislocation, except at high temperature, is controlled by the Peierls mechanism. (2) Dislocations are generally dissociated into Shockley partial dislocations [52, 53]. (3) Two types of dislocations (c~- and t-dislocations) in compound crystals have different mobilities [54, 55]. These results suggest that the mechanism of the dislocation glide in these crystals is largely the same. Before going into detail, we describe here the nomenclature concerned with the dislocations in tetrahedrally coordinated crystals. The glide plane of dislocations is { 111 } in cubic crystals with the diamond or the zincblende structure and (0001) in hexagonal crystals with the wurtzite or 6H structure. The Burgers vector of the dislocations 1 1 corresponds to the shortest lattice vector; i.e. ~(110) in cubic crystals and g(1210) in hexagonal crystals. The close-packed glide planes do not stack equally-spaced but consist
448
K. Maeda and S. Takeuchi
Ch. 54
of alternate narrowly-spaced and widely-spaced planes; the ratio of these spaces is 1:3. It is established [56-58] that dislocations in any tetrahedrally coordinated crystals are dissociated into partial dislocations on the glide plane in their stable state, in the same manner as the dislocations in face-centered-cubic metals and in hexagonal-close-packed metals. The dissociation reactions are
1
~ ( 1 1 0 ) - 4 g1(121)-- -4- (211)
(2.1)
and
1 ~{1210)--+ ~1(1]00) +
1 (0]-10},
(2.2)
creating an intrinsic stacking fault between the partials. The plane of the stacking fault is in the narrowly spaced close-packed planes, and hence the glide plane of the dissociated dislocation must be in the narrowly-spaced (glide) planes, not in the widely-spaced (shuffle) planes [59]. It is also known that at low temperatures the glide dislocations tend to lie along close-packed directions due to deep troughs of Peierls potentials along these directions, i.e. in cubic crystals along (110) directions and in hexagonal crystals along (1210) directions. Thus, the total Burgers vector of a glide dislocation is either parallel or at 60 ~ to the dislocation line. The former dislocation is a screw dislocation and the latter dislocation is called a 60~ The Burgers vectors of both partial components of the screw dislocation make 30 ~ to the dislocation line and hence they are called 30~ dislocations or simply 30 ~ The Burgers vector of one partial dislocation of the 60~ makes 30 ~ and that of the other 90 ~ to the dislocation line, and hence they are called the 30 ~ and 90 ~-partials, respectively. Since the zincblende and wurtzite structures do not have a centre of symmetry, there exists a polarity for {111} and {0001} surfaces. As a result, two edge dislocations having the same Burgers vector but with dissimilar signs are not equivalent. In III-V and II-VI compounds, in the direction perpendicular to { 111 } and {0001 } planes are oriented pairs of unlike atoms, group III and V atoms or group II and VI atoms. The {111} or {0001} surface in the direction of the group V or VI atom is defined as the plus surface and that in the direction of the group III or II atoms as the minus surface. An edge dislocation whose extra-half plane resides on the side of the minus surface is conventionally called an a-type dislocation and that having the extra-half plane on the side of the plus surface is a /3-type dislocation. For any non-screw dislocations having an edge component, there exist a-types and/3-types in zincblende and wurtzite crystals. Thus, we have 60~ and 60~ 30~ and 30 ~ -partial, 90~ -partial and 90~ -partial. For dissociated dislocations, the extra half-plane ends at the narrowly-spaced plane; therefore, an c~-type dislocation ends at group V or group VI atoms at the dislocation core, and that of/3-type at group III or group II atoms. Hence, dangling bonds associated with a dislocation having an edge component are on the metalloid element in a-type dislocations and on the metallic element in/3-type dislocations. In fig. 1 we show an expanding hexagonal dislocation loop in the diamond structure (a) and that in the zincblende or the wurtzite structure
w
Enhancement of dislocation mobility in semiconducting crystals
,,
90" 30"
30"
30"
30" i
30"
30~
90"a"
30"13
449
''-~\\~
30"a
30"
90" 30"
30"~ ' - ' % \ \ \ ~ (a)
\ 90"1~
(b)
Fig. 1. Expanding hexagonal loops of dissociated dislocations in diamond structures (a) and in zincblende and wurtzite structures (b). (b), consisting of screw segments and 60~ segments each dissociated into partial dislocations. In every dissociated dislocation segment in (b), the combination of the leading partial and the trailing partial differ from that in any others, and hence we have six different types of gliding dislocations in compound crystals. In most of the experimental situations, screw dislocations, c~-dislocations and /3-dislocations are distinguished but no distinction is made for the sequence of the partials, e.g. between the dislocation with 30~ leading partial and 90~ trailing partial and vice versa. When we specify the sequence, we use the convention, e.g., 30c~/90c~ for the above case. It should be noted that in real crystals the dislocation mobilities are different for different dislocation types, and hence the shape of the dislocation loops is generally elongated in one direction.
2.2. Characteristics of dislocation motion in semiconducting crystals The dislocation motion in Si, Ge and III-V compound crystals is smooth and continuous, indicating a large lattice friction. The velocity has been measured as functions of stress and temperature by the double etching technique, or the X-ray Lang method [60]. The velocity is represented experimentally, in most cases, by the following equation which is the product of a stress term and a temperature term [50]. V-
Vo
7-
exp
( ) - ~Q
9
(2.3)
Here, 7- is the applied shear stress, Q is the activation energy, V0 and 7-o are constants having dimensions of the velocity and the stress, respectively. The stress exponent rn ranges between one and two. The activation energy varies from crystal to crystal and
K. Maeda and S. Takeuchi
450
2.5
Si
.
GaAs
>
-
E
o
1.5
..'"
InP I Ga Ge .
!,.. o t-
uJ
Ch. 54
"'"
o-'"0I~
InAs (~ " " ~ ' 0 -
inSb
..'"
0
._>
_
O
<
CdTe ..-'"
O ~
HgSe.T-"
O
(9 0.5
CuCI .-'" so S ' ~ o"
0
I
I
I
I
I
I
I
I
I
I
1
2
3
4
5
6
7
8
9
Kbp2h
10
(eV)
Fig. 2. Correlation between experimental glide activation energies Q in tetrahedrally coordinated crystals and the material parameters Kb2h, where K, bp, h denote the energy factor, the magnitude of the Burgers vector of the partial dislocation, and the period of the Peierls potential, respectively. The circles indicate values from velocity data and the bars from plastic deformation experiments. The dotted line indicates the relation Q = 0.25Kb~h (,,., 0.3Gb~).
even for the same crystal depends on the conduction type and also on the dislocation type. In fig. 2 are plotted experimental activation energies obtained for various materials as a function of where K is the energy factor, bp is the strength of the Burgers vector of the partial dislocation and h is the spacing between Peierls valleys [61, 62]. In the figure, in addition to the activation energies of the dislocation velocity (circles), the activation energies obtained from deformation experiments (bars) are also included. A correlation is seen to exist between Q and the implication of which will be discussed later. It should be noticed, however, that the strength difference generally seen among different groups of crystals, i.e. group IV crystals, III-V compounds and II-VI compounds (the strength decreases in this order) is due to the differences in the glide activation energy which is primarily determined by the lattice constant which gives the values of bp and h, and the magnitude of the shear modulus G which reflects the bonding strength of the crystals. As seen for the data for Si in fig. 2, the activation energies obtained are largely scattered. This is partly due to the difference of the stress levels at which the activation energy is obtained; the activation energy is a decreasing function of the stress [50, 63]. Also, the stress exponent m is not constant but a varying function of the stress. In particular, it is known in some cases that as the stress is decreased the m value increases to very large values, indicating that there is a threshold stress, often called the starting stress [64], for the dislocation motion. Thus, eq. (2.3) is only an approximate expression of the dislocation velocity for a narrow stress range.
Kb~h,
Kb~h,
w
Enhancement of dislocation mobility in semiconducting crystals
451
When the stress exponent m ~ 1, the concept of dislocation mobility g defined by the ratio of (dislocation velocity)/(dislocation driving force) is useful. The mobility of dissociated dislocations is determined by the mobility of the constituent partials according to 1/~t = 1/~q + 1/~t2, where g~ (i = 1,2) represents the mobility of the partial i. Wessel and Alexander [65] found that the 30~ in Si has a mobility order of magnitude smaller than that of the 90~ meaning that the 30~ controls the total dislocation motion. Consequently, the screw dislocation composed of two 30~ has a mobility similar to that of the 60~ composed of a 30~ and a 90~ which is actually observed in experiments [50]. As early as in the 1960s, it was discovered that the dislocation velocities in Ge and Si are affected, either enhanced or depressed, by impurity doping that changes the carrier type and its density [1, 66-71]; the dislocation velocities in these crystals are essentially a function of the position of the Fermi level. Such effects are called the doping effect. In III-V compounds, the velocity of the c~-type dislocation is always larger than that of the/3-dislocation by orders of magnitude [55]. The velocity of the screw dislocation is almost the same as that of the/3-dislocation. Hence, it is considered that the mobility of the 30~ which is the common constituent partial of the/3-dislocation and the screw dislocation, controls the velocities of the two dislocation types, and that the mobilities of 30~ - and 90~ are much larger than that of 30~ -partial. Such effects may be termed the polarity effect. The doping effect and the polarity effect observed in semiconducting crystals suggest that the velocity of the dislocation in these crystals is sensitively affected by the electronic state of the dislocation core. This is the most interesting feature of the dislocation motion in semiconducting crystals.
2.3. Two schemes of the Peierls mechanism
2.3.1. Dislocation velocity in the Peierls mechanism As mentioned in section 2.1, the dislocation glide in semiconductors at low temperatures is undoubtedly controlled by the Peierls mechanism. In the Peierls mechanism, three processes, the formation of a kink pair, the subsequent migration of the kinks along the dislocation line, and annihilation of two kinks or arrival of the kinks at the end of the dislocation line, constitute the unit cycle of dislocation motion. Quite generally, the steady velocity of a dislocation controlled by the Peierls mechanism is given by V = 2JAb,
(2.4)
where J is the kink-pair nucleation rate per unit length of dislocation and h is the kink height. The parameter A represents the average length of dislocation available for the nucleation of a kink-pair during the lifetime of the kinks and is therefore termed the kink mean free path (m.f.p.). If the kink migration is so slow that the kink ceases its propagation on collision with another kink of an unlike sign which has travelled from the opposite direction, the kink m.f.p, is determined by J and the kink velocity Vk as /~ = V/Vk/J.
(2.5)
452
Ch. 54
K. M a e d a a n d S. Takeuchi
2/1,
3/
=
/
vk
\--'r
/
(a)
\ vk
/
\
\
\
(b)
L
Fig. 3. Kink-pair nucleation and subsequent kink migration in the kink-collision regime (a) and in the kink-collisionless regime (b). In this case (kink-collision regime; fig. 3(a)) V = 2h vV/-~kJ,
(2.6)
which is independent of the dislocation length L. In contrast, if the kink migration is so fast that the kink sweeps out the whole dislocation segment without kink collision, A becomes equal to L/2. Therefore, in this regime (kink-collisionless regime; fig. 3(b)), the dislocation velocity is given by
V-
JLh,
(2.7)
which increases in proportion to the dislocation length L.
2.3.2. Point-obstacle model (theory of Celli et al.) A theoretical model considering the presence of point obstacles for kink migration in the course of kink-pair formation was first proposed by Celli et al. [72] and modified by Rybin and Orlov [73]. We shall refer to this model as the point-obstacle model. Figure 4 illustrates the potential energy of a kink pair as a function of the pair separation depicted with a full line for a dislocation under stress -i- and a broken line for that without stress. Using the stress-dependent formation energy of a kink pair, Fkp(T), the kink-pair nucleation rate is written in the form [72] J ~ U[ f ('r) eXP
-
Fkp (T))kT '
(2.8)
where f('r)-
( 1 + 7 __ - 1 + i f : c )e[ x p -
(T1 - - + ~'~)1 .
(2.9)
Here Zc is the kink separation at the saddle point and "q is a parameter defined by
~-I -- bh~'
(2.10)
Enhancement of dislocation mobility in semiconducting crystals
w
453
13_
r
...............
o
-
z'=0
L _
(1) cIll ~
r
~>0
o 13_
Xc
"~
Kink-PairSeparation
Fig. 4. The potential energy of a kink-pair as a function of the pair separation in the point obstacle
model [72, 73].
with Ed, the activation energy required for a kink to surmount the point obstacle, and denoting the average separation of the point obstacles on the dislocation. The kink migration velocity in this model is given by Vk--Uk~exp
--~
,
(2.11)
where Uk is the vibrational frequency of kink. Hence, one obtains from eq. (2.6) the dislocation velocity for the kink-collision case as
V "~ 2 h i ~ U~ukf (T) e x p ( - F k p ( r ) / 2 + E d ) k T
for
L >> ~,
(2.12)
and for the kink-collisionless case as V ~ L haus f (~) exp (
Fkp(r)]~ kT /
for
L << ~
"
(2 13) "
An important feature in these expressions is that the stress-dependence of dislocation velocity is in any case quite supralinear.
2.3.3. Kink diffusion model (Hirth-Lothe theory) The point-obstacle model above is based on the string approximation of dislocation line. On this approximation, the width of a kink is related to the height of the Peierls potential: with increasing potential height, the kink width decreases, thereby minimizing the kink formation energy. A simple calculation [74] shows that the kink width w is given by
w-
(E0 2Wp'
(2.14)
454
Ch. 54
K. Maeda and S. Takeuchi
f,
rl
&
a <-->. r~ .,,'\ t ' \
f.~
'z" = 0 f"~ /'~
t'\
Em
,,(
/'.'f'tl
0
\D
,',
..
..
I
o tLLI
2F k
.,,.,..
i,,0 o 13_
!
i
I
I
I
I
i
,
I
I
I
I
Kink-Pair Separation, x Fig. 5. The potential energy of a kink pair as a function of the pair separation in the kink diffusion model [37].
where E0 is the dislocation line energy and Wp is the height of the Peierls potential with a sinusoidal form which is related to the Peierls-Nabarro stress 7-p by
hbTp
Wp ~ ~ .
(2.15)
As E0 ~ 0.5Gb 2 and Tp in covalent semiconductors is considered to be around 0.1G [61 ], the kink width must be smaller than ~ 2a. Thus the kink is rather abrupt in tetrahedrally coordinated semiconductors in contrast to metals in which rp is generally much smaller and as a result the kink is quite smooth. Owing to the lattice periodicity, abrupt kinks would feel a large potential variation at every atomic distance during their migration along the dislocation line. Such an intrinsic potential, called the Peierls potential of the second kind (2nd PP), will form obstacles to kink migration. When the second PP is large, the kink migration proceeds in a diffusional manner with activation energy Era. Along with kink-pair formation and subsequent expansion of the pair, the potential energy of total system will vary as shown in fig. 5. The difference from the point-obstacle model in fig. 4 is that the potential barrier against kink motion is regularly spaced with a distance as short as an atomic periodicity a. For this model, referred to as the kink diffusion model, Hirth and Lothe [37] derived an expression for the rate of kink-pair nucleation
J~
a2kT Dkexp
-- ~
-- -~-T u~
-
kT
,
(2.16)
where we have used the diffusion constant of a kink Dk-
/,I0a2 exp
- ~
.
(2.17)
Enhancement of dislocation mobility in semiconducting crystals
w
455
V
~ ~~
"
Lb
L
Fig. 6. The dependence of dislocation velocity V on the length of dislocation line L expected in the Peierls mechanism. In the kink diffusion model, the length Lb separating the two glide regimes is given by eq. (2.21).
The kink drift velocity under stress r is related to the kink diffusion constant by the Einstein relation rbh rbha 2 (Em) Vk ~ D k - - - ~ ,-~ uO k T exp - ~
(218)
.
Equations (2.6), (2.16) and (2.18) are combined to yield the dislocation velocity for the kink-collision case as 2"ra b h 2
V ~ .-----~uoexp
(Fk+Em) -
for
kT
L > Lb.
(2.19)
For the kink-collisionless case, from eqs (2.7) and (2.16), one obtains
kT
uoexp
-
kT
for
L > Lb.
(2.20)
The critical dislocation length Lb that bounds the two regimes is given by
Lb--2aexp
~F k ) ,
(2.21)
which is equivalent to the average separation of thermal kinks. Thus also in the kink diffusion model, we have two schemes for the dependence of dislocation velocity on the length of dislocations as illustrated in fig. 6: in the dislocations shorter than Lb, the velocity increases linearly with the segment length L, and in the dislocations longer than Lb, the velocity is saturated to a value independent of L. A substantial difference from the point-obstacle model is that the dislocation velocity depends linearly on stress in both regimes.
456
K. Maeda and S. Takeuchi
Ch. 54
2.4. Discrimination of the two schemes
The elementary process of dislocation glide in semiconductors is a subject of a longlasting controversy which has not been settled satisfactorily as yet. The first experimental fact that should be compared with theories is the dependence of dislocation velocity on the dislocation segment length. The velocity of dislocations with macroscopic lengths is apparently independent of the dislocation length [63], which appears to rule out the kinkcollisionless case in ordinary experimental conditions. The length-independent velocity, however, does not necessarily mean that dislocations proceed with kink collision as discussed above. We shall return to this problem shortly. The second fact of substantial importance is the stress dependence of dislocation velocity, which was most intensively studied in Si [63, 67, 75, 76]. In situ X-ray topographic measurements by Imai and Sumino [75] showed clearly that the dislocation velocity in bulk Si single crystals decreases with decreasing stress quite non-linearly at low stresses and below a critical stress around 10 MPa dislocation motion is completely suppressed. At such low stresses the activation energy also exhibits an anomalous increase with decreasing stress [50, 63]. Since such non-linear stress dependence is often observed in presumably impure samples, the most probable cause of this starting stress is thought to be the presence of impurity atoms that block dislocation motion [64]. In more ionic crystals such as II-VI compounds for which the stress levels used in experiments are relatively low, the effect of point obstacles should be more pronounced. Thus as a general statement, dislocation mobility at low stresses is quite likely to be affected by point defects and hence the point-obstacle model described above may not be too far from reality. However, at high stress levels, this non-linearity disappears and instead the dislocation velocity exhibits a fairly linear dependence on the stress [75] 1. As pointed out above, the kink diffusion model and the point-obstacle model differ in the stress dependence of dislocation velocity, linear in the former and non-linear in the latter. Therefore at moderate stresses as used in ordinary experiments for covalent crystals, the kink diffusion model seems more relevant than the point-obstacle model. If we assume that the microscopic process of dislocation glide in semiconductors is described by the kink diffusion model and the dislocations proceed with kink collision and annihilation, the glide velocity should be described by eq. (2.19). This is the most conventional picture that has been accepted by majority of researchers in this field [77-81]. Nevertheless, as remarked by Louchet and George [82] and Heggie and Jones [83], there is a serious discrepancy in the magnitude of the pre-exponential factor between the theory and experiments: the theoretical value calculated from eq. (2.19) is always several orders of magnitude smaller than the experimental values as listed in [50]. An attempt was made to explain this discrepancy by considering an entropy factor [84]: precisely, the activation energy Fk + E m must be replaced by the activation free energy 1Velocity measurements in a wider stress range using the double etching method [76] showed that the stress exponent ranges from 1 to 2 depending on the dislocation type and the temperature. The disagreement with the results of X-ray measurements by Imai and Sumino [75] probably arose from the use of high stresses by Alexander et al. that must reduce the kink-pair formation energy and hence enhance the apparent stress exponent.
Enhancement of dislocation mobility in semiconducting crystals
w
457
Fk + Em - (Sk + Sm)T, where Sk and Sm denote the entropies of kink formation and kink migration, respectively; therefore, the pre-exponential factor in eq. (2.19) must be multiplied by a factor of exp{(S'k + Sm)/k}. Using a Keating-type potential, Marklund obtained S'k ~ 0.5k and Sm as large as 5k for 90 ~ partials. No calculation was made, however, for 30 ~ partials which are considered to control the overall motion of perfect dislocations. The magnitude of Sk + Sm that is required to account for the experimental pre-factors is as large as ~ 8k for the velocity data of Si obtained by Imai and Sumino [75] that reproduced the linear stress dependence compatible with eq. (2.19). Furthermore, another fact that is incompatible with the kink-collision picture was revealed by recent measurements, the experimental principle of which was first devised by Nikitenko et al. [85], of dislocation velocity in Si [86, 87] and GaAs [88, 89]. A remarkable fact was that dislocations would not move when the crystal was subjected to intermittent loading with pulse durations shorter than time required for a dislocation to advance by one period of the Peierls relief. This intermittent loading effect is difficult to explain by the kink collision picture. In the kink collision regime, the mean density of kinks Ck = 1/(2A) -- 1/Lb under a stationary load. This may be combined with eqs (2.18), (2.19) and (2.21) to give the expression for the dislocation velocity in terms of Ck
V = 2Ckhvk.
(2.22)
As long as the kink density is determined by kink collisions, this expression should still hold even under a time-varying load as well. Since the drift motion of kinks is caused by applied stress, the kink velocity Vk will quickly attain a steady value on loading and will diminish on unloading. In contrast, the kink density Ck will not respond to loading conditions immediately but takes some time to relax to the steady levels. The intermittent loading effect arises from the fact that the relaxation time for kink generation differs from that for kink annihilation. Since the kink migration energy controls the kink generation and annihilation, the magnitude of Em can be deduced from the time scale in which the intermittent loading effect is observed. A quantitative analysis [89], however, shows that a large value of Ern ~ 1.9 eV or a small value of Fk ~ 0.1 eV is required for Si to interpret the actual effect experimentally observed [85]. The value of Fk ~ 0.1 eV is unphysically too small. An atomistic calculation using the Keating potential for Si [90] showed that the formation energy of a double kink is at least 1 eV on the 30~ For b.c.c, metals and ionic crystals, the smooth kink model is considered to be a good approximation. This is supported by the fact that the glide activation energy evaluated by macroplastic deformation tests is in good agreement with the kink-pair formation energy calculated based on the observed Peierls-Nabarro stress [61]; that is, in these crystals, Q ,~ Fkp = 2Fk and Em ~ 0. In the smooth kink model with the assumption of a constant line tension, the energy of an isolated kink is given by [74]
[;'k '~ - - ( 2 Wp Eo ) 1/2 "~ - 7r 7r
2
Tp EO
(2.23)
458
K. Maeda and S. Takeuchi
Ch. 54
In semiconductors, however, this relation would not hold because the kinks are abrupt. Nevertheless, the kink formation energy should not be smaller than the value simply calculated from eq. (2.23) with appropriate parameters. Substituting E0 ~ 0.5Gb 2 and -rp ~ 0.1G for semiconductors, one obtains Fk ,~ 0.2Gb 3. If one considers that dislocation motion is controlled by the least mobile partial, the Burgers vector b in the above discussion should be replaced by that of the partial dislocation bp. Hence for Si (G 68 GPa, bp ~ 0.22 nm), we have Fk ,~ 0.2Gb 3 ~ 1.0 eV. An atomistic calculation of Fk by Marklund [189] for a kink on 30 ~ partial dislocations in Si showed that Fk ~ 0.5 eV O. 1Gb~. Therefore, the kink formation energy in Si should be reasonably around Fk ~ (0.1 ~ 0.2)Gb~.
(2.24)
Incidentally, it may be worth noting that the glide activation energy Q exhibits an experimental correlation with a material parameter Kb2h (~ 1.2Gb3). The overall fitting is achieved by Q - 0.25Kb2h (~ 0.3Gb~) as shown in fig. 2, which together with eq. (2.24) suggests that also in semiconductors Q ,~ 2Fk and Em << Fk as in the case with b.c.c, metals and ionic crystals. Dislocations shorter than Lb (eq. (2.21)) ought to glide with velocities proportional to the segment length (fig. 6). The existence of such a length-dependent regime is actually confirmed by TEM observations of very short dislocations [91, 92]. Figures 7(a)-(c) show some experimental results obtained for various systems by in-situ TEM observations of dislocation motion. In most cases, the velocity vs length relations are almost linear, which strongly suggests that the glide of dislocations in these lengths is in the kink-collisionless regime (eq. (2.20)). In 60 ~ dislocations in Ge, however, the relation exhibits an obvious tendency to saturation as expected from fig. 6. The border length demarcating the two regions appears to be ~ 1 gm in this case but in other cases it is much larger. If the border length is given by Lb in eq. (2.21), the kink formation energy Fk can be assessed by the experimental value of Lb. On the other hand, the glide activation energy must increase by Fk on the transition from the kink-collisionless regime (eq. (2.20)) to the kink-collision regime (eq. (2.19)). Thus to be consistent within the framework of the kink-collision model, the values of Fk evaluated from Lb and from the change of activation energy with the transition must coincide with each other. Systematic measurements of glide activation energy in both the length-dependent regime and the saturation regime were carried out by using specimens in a form of thin crystalline films epitaxially grown on a substrate [38, 93]. Hetero-epitaxial films having a lattice mismatch with the substrate are strained biaxially. If the temperature is high enough for dislocations to glide, the lattice mismatch can be accommodated by the introduction of dislocation loops in the epi-film in the mode as illustrated by fig. 8. Here, we focus on the motion of segments threading the epi-layer. Since the length of the threading segment is geometrically determined by the film thickness, we could experimentally obtain the length vs dislocation velocity relation by measuring the lateral propagation velocity of the threading segments as a function of the film thickness. It is convenient that the border length bounding the two glide regimes can be covered by the thickness range of common films (sub micron to microns). Figure 9 shows the results thus obtained for dislocations in completely miscible Sio.9Ge0.1 alloy films grown on
Enhancement of dislocation mobility in semiconducting crystals
w
(a)
(b) Ge
t/)
3
Screw
InSb screw
~
3
!
o~
E
E
-.9, o
v
459
2
0 0.0
,
~ - z ~ -~ ... ! ~ , , ~ ~
O
/
v
I
I
0.5
1.0
0 0.0
1.5
I
I
0.5 1.0 L (10- 6 m )
L (10-6m)
1.5
(c) GaAs 1.0
screw
t/}
,sE O v
0.5
0.0
I
0
1
I
2 L (10-6m)
3
Fig. 7. The dependence of dislocation velocity V on the length of dislocation line L experimentally observed on the TEM scale (a) in Ge at T = 703 K, 7- -- 40 Mpa [92], (b) in InSb at T = 523 K, "1- = 44 MPa [91, 190], and (c) in GaAs at T = 623 K, 7- -- 50 MPa [188]. a Si (100) substrate [38]. Owing to the well-defined specimen conditions, the resolved shear stress driving the dislocation (~- = 300 MPa) can be unambiguously evaluated from the lattice misfit (0.4%). The horizontal scale at the bottom is the length of threading dislocations calculated from the film thickness indicated on the top. The open circles present the dislocation velocities measured at 773 K and the closed circles are glide activation energy. Evidently the dislocation velocity depends on the dislocation length in the range below ,-~ 1 t,tm, close to ,,~ 0.4 lxm found in previous T E M experiments for pure Si [94]. The dislocation velocity exhibits saturation at large lengths, which apparently manifests the anticipated transition of glide mode. However, it should be noted
460
Ch. 54
K. Maeda and S. Takeuchi
I
(100)/
Slip Pla <,,,>
_
/
,
. . . . . . .
,
i
epi-film
77- 7
Substrate
! I
/
I I I
Fig. 8. Introduction of a dislocation loop into a hetero-epitaxial thin film lattice mismatched with the substrate. The length of dislocation segment T threading the epi-film is determined geometrically by the film thickness and the stress driving the dislocation is unambiguously given by the lattice misfit.
5.0
Film Thickness h (lxm) 0.5 1.0 1.5
0
I
i
2.0
t
4.0-
25
_ ~ _
3.0
---~. 2.O
~ 3.0
L_ c-
-
~.5
,,5
"~2.0
? / 1.0-
/
~'
/! I 0
1.0 "~
epi -
0.5
"
0
sub. I 0.5
I I I 1.0 1.5 2.0 Dislocation Length L (lam)
I 2.5
Fig. 9. The dependence of dislocation velocity and that of glide activation energy on the dislocation length measured for glides of threading dislocations in Sio.9Geo.1 films of various thicknesses [38, 93]. Note that the activation energy does not show significant variation while the length-dependence of velocity changes. that the activation energy does n o t show any significant c h a n g e on this transition. If the border length of ~ 1 g m represented Lb in eq. (2.21), Fk w o u l d be ~ 0.5 eV, large e n o u g h to be detected as the change of activation energy on the transition. This
w
Enhancement of dislocation mobility in semiconductingcrystals
461
contradiction clearly indicates that the velocity saturation is not brought about by the transition of the glide mechanism from the kink-collisionless regime to the kink-collision regime. All of the above facts lead us to the conclusion that the glide of long dislocations in bulk crystals is not in the kink-collision regime. A possible alternative interpretation for the velocity saturation is to consider the presence of obstacles on a dislocation line which block kink propagation but can be passed when another kink of opposite sign arrives at the obstacle from the other direction [38]. If dislocations are partitioned by such obstacles with a mean separation L*, this length L* plays a role of the dislocation length which determines the kink mean free path )~. In this case, the dislocation velocity is given by
kT
--~ u0exp
-
kT
for
L <
(2.25)
and
kT
uoexp
-
kT
for
L >
.
(2.26)
The expression (2.26) relevant to long dislocations in bulk crystals differs from the conventional expression (2.19) both in the activation energy and in the pre-exponential factor. The experimental activation energy, formerly interpreted as Fk + Em, should now be regarded as 2Fk + Em; the pre-exponential factor in the former theory must be multiplied by a factor of ~ L*/b, which is as large as 5 • 103 for L* ~ 1 ~tm. This factor removes the serious discrepancy in the pre-factor between experiments and theory. The point obstacles must be of intrinsic nature because it appears that the effective length L* does not depend much on the crystal purity. The insensitivity of L* to temperature suggests that the obstacles are such defects that are incorporated into dislocation lines for some reasons. Although geometrical jogs, for example, are candidates, we should reserve conclusions on the origin of the obstacles at present.
3. Effects o f electronic excitation on dislocation glide The effect of electronic excitation on dislocation glide has been studied at various levels, from the glide velocity of individual dislocations on microscopic scales, to the macroscopic plasticity of the crystal which reflects the dislocation mobility. The quantitative studies at the velocity level are limited to those common materials such as Si and GaAs for which bulk single crystals are available in dimensions large enough to allow velocity measurements under well defined conditions. For other materials, the experimental evidences are rather qualitative or indirect. Nevertheless, it seems worthy to overview all the experimental facts relating to the REDG effect.
462
K. Maeda and S. Takeuchi
Ch. 54
3.1. Velocity m e a s u r e m e n t s 3.1.1. Long-range motion
The principal advantage of using bulk samples is the reliability of the stress that can be assessed unambiguously for crystals with a controlled shape loaded in a simple mode (normally compression or bending). A drawback, however, may be the low resolution of the microscopies ordinarily used for observations of defects in bulk crystals. Nevertheless, the low resolution is rather convenient for viewing dislocations in low magnifications and tracking their motion over long distances. The REDG effect at this level has been investigated most intensively for III-V compounds. Among them, the most systematic study was performed on GaAs [39-42]. Si is another material for which convincing verification of the REDG effect was made [45, 46]. This section describes the results of experiments for such popular materials and for other covalent crystals, InP [43], GaP [44], InSb [95] and Ge [46]. The most conventional technique of dislocation observation in bulk crystals is optical microscopy which is used to observe dislocations usually revealed by chemical etching. The direct observation of dislocations is also possible by scanning electron microscopy (SEM) if the sample yields cathodoluminescence (CL) with a sufficient intensity [96]. The imaging of dislocations in the SEM-CL mode is based on the fact that radiative recombination of electron-hole pairs generated by the electron probe beam is locally reduced in the vicinity of dislocations and as a consequence the dislocations are observed as dark contrasts in the luminescent background. Figure 10 shows SEM-CL images of such dark contrasts of dislocations in an n-GaAs that emerge at the crystal surface at the sites imaged as dark spots [39]. In this series of images the crystal was being stressed in the SEM chamber so that the dislocations were driven to glide on the slip planes. Incidentally, it is worth remarking that the dislocations are visible all along their movement which allows us to make in situ continuous observation of dislocation motion even at a TV scanning speed [41]. The excitation source can be a laser light of appropriate wavelength [95] but in most cases the electron probe beam of an SEM accelerated at 30-35 keV was utilized to irradiate samples with various intensities. The use of SEM enables us not only to control very precisely the beam intensity (excitation intensity) and the beam position (excited dislocations) but also, in favorable cases, to directly image dislocations in the SEM-CL mode, as mentioned above. The non-destructiveness of the method thus makes it possible to measure dislocation velocities in a single specimen at different temperatures, stresses, and electronic excitation intensities systematically. Figure 11 demonstrates the effect of the electron beam irradiation on the displacement with time of a single c~-dislocation in an n-GaAs crystal investigated by the SEM-CL method [41]. In the duration indicated by "dark" the electron beam was switched off except for occasions for taking snapshots to reveal dislocation positions at a sufficiently low beam current. Facts evident in fig. 11 are: (1) the dislocation motion is continuous and smooth irrespective of electron irradiation, (2) the electron irradiation increases the dislocation velocity (the slope of the plot) simultaneously with the start of irradiation, and (3) the effect of irradiation is reversible with the enhanced velocity resuming its previous value in darkness immediately after the irradiation is stopped. This reversibility
w
Enhancement of dislocation mobility in semiconducting crystals
463
Fig. 10. Sequential SEM-CL images of individual dislocations in GaAs in situ deformed at T = 435 K, "r -- 40 MPa [39]. The dark spots are dislocations emerging at the surface. The dark contrast is obtained even at a TV scanning speed [41] indicating that the inherent dislocation core acts as non-radiative recombination center.
464
Ch. 54
K. Maeda and S. Takeuchi
0.3 n-GaAs screw dislocation r
E b t. ~O
0.2 -irrad. 1-
dark
O O . ~o'J
O t--
E
0.1
O ...,. .~
/o 0
I
i
I
I
I
1000
2000
3000
4000
5000
6000
Loading Time (s) Fig. 11. The effect of electron beam irradiation on the glide of a single dislocation observed by the SEM-CL method [41], demonstrating that the REDG effect is reversible. indicates that the effect is not due to an irreversible radiation damage which may be caused in non-metals by irradiation of electrons of subthreshold energy [97]. The increase of dislocation velocity is enhanced by increasing irradiation intensity. Figure 12 shows the dependence of dislocation velocity increment on beam intensity for s-dislocations in two III-V compounds [44, 98]. Each data set was obtained for a s i n g l e dislocation with other parameters such as temperature and stress being fixed. The velocity increases with the current density I as AV cx I "Y, where the exponent ? is around unity with a small variation depending on the dislocation systems (when the current density is much larger, the velocity increase may exhibit saturation as in the case of ZnS [99] described in the next section). The temperature dependence of enhanced velocity is quite characteristic. In figs 13(a)(f) we summarize representative results in various III-V compounds [40, 43, 44, 95] and elemental semiconductors [46]. The REDG effect was absent in Ge and InSb under the conditions employed. The absence of REDG effect in 90/30 dislocations in Si reported in the experiments using laser illumination [45] was not reproduced under electron irradiation [46]. The most notable fact is that the glide velocity under irradiation still commonly exhibits a temperature dependence of the Arrhenius type. The velocity enhancement is brought about by a substantial reduction of the activation energy. However, since the pre-exponential factor is simultaneously reduced by the irradiation, the effect can be observed only at temperatures below a critical temperature Tc.
w
Enhancement of dislocation mobility in semiconducting crystals
465
10 .4 10-5
-
10-6
-
GaP %"
E ;~
o
10-7 -
GaAs 10-8 I
10-9 10-3
10-2
I
I
10-1
1
10
I (A/m 2) Fig. 12. The dependence of dislocation velocity V on the electron beam intensity I used to induce the REDG effect in GaAs [40] and GaP [44].
Increase of irradiation intensity does not affect the slope of the velocity vs temperature relation in the enhanced state; in other words, irradiation reduces the glide activation energy by an amount independent of the irradiation intensity. Figures 14 (a) and (b) demonstrate such behaviors observed in two systems [40, 98]. The parallel shift of the enhanced velocity upward in the figure accordingly results in an increase of Tc. Different investigators [40, 95] gave contrary reports on the stress dependence of the REDG effect. The experiment using the SEM-CL technique [40] yielded the result as shown in fig. 15; i.e. the stress dependence is characterized by a stress exponent m ~ 1.5 irrespective of irradiation. Meanwhile the experiment using the etch-pit method with use of a laser light as excitation source [95] showed that the stress dependence under illumination was weaker than that in darkness and as a consequence the REDG effect disappears at large stresses. The stress exponent m ,.~ 3.5 in darkness, however, is anomalously large in comparison with typical values reported for GaAs [100-102]. For simplicity, therefore, we adopt the first result in the following discussion. The relation of the REDG effect to the doping effect was investigated in GaAs [42]. Figure 16 shows the results for fl-dislocations which exhibit a very pronounced doping effect. It should be noted that the irradiation always enhances the velocity irrespective of the conduction type or the position of the Fermi level. All of these facts are summarized by an experimental formula for dislocation velocity as a function of the temperature T, the stress T, and the irradiation intensity I: V-
Vt -r"to
exp
-
+ Ve
"/-
I
~' exp
- E t - AE kT
(3 1) "
with 7 ~ 1 and m ,.~ 1 ~ 1.5. The first term in the right-hand side indicates the velocity attained in the absence of irradiation and the second is the additive term representing the REDG effect. As schematically illustrated in fig. 17, at low temperatures the first term
466
Ch. 54
K. Maeda and S. Takeuchi
(a) 10.4 cz
(b)
n-GaAs
lO-S
n-lnP 13-dislocation
10-6 13
03
E
Qo o
~ ~
10.8
03
E
""~ .....iro:ad"
bo "'~.....
10-s
~176 ,.!rrad. ~
""" "'"bo
,dark , 10-1q dark,- "i"" 1.0 1.5 2.0 2.5 3.0 1/T (10-3K-1) 10 -3
(c)
" " O . . . . . ..
10.~0 -
i
I
3.5
n-GaP or-dislocation
irrad. ""-,9
1.4 (d)
"%,
i
1
1.8 2.0 I/T (10-3K-1)
1.6
2.2
2.4
lo-'
InSb
"b'--...
10.5
"'O ....
irrad. "--.o..........
10 -6
-
in E
E 10.7
dark
dark
"%~
10"s
10 -9 I
1.5 (e)
I
1.8 2.0 I/T (10-3K-1)
10 .8
2.2
2.2
I
I
I
I
2.4
2.6
2.8
3.0
3.2
1/T (10-3K-1)
Si 60"-dislocation
(f)
10" Ge
10-6 ~ N ~
6ir0"a-dd.iSl~176
"',
"7 It}
E
E;::=10"1c
10-12 .2
lightillumi. ~ i"'~ik...,. . "dark \ ele.irrad. dark 1.4 1.6 1.8 I/T (10-3K-1) I
I
I
10.8 10-10 I
1.2
1.4
I
I
1.6 1.8 1/T (10-3K 1)
,..
2.0
Fig. 13. The temperature dependencies of dislocation velocity in the dark (open marks) and under electronic excitation (filled marks) reported for various crystals" (a) GaAs [40], (b) InP [43], (c) GaP [44], (d) InSb [127], (e) light-illuminated p-Si [45] and electron-irradiated i-Si [46], and (f) Ge [46].
w
467
Enhancement of dislocation mobility in semiconducting crystals
(a)
10-~ "~
S"
n-GaAs a-dislocation
.
E
~ ; ~• ....""~z~.. ~"~-~,."........ I = 0 . 3 9 Am-~
~9 10-7o
> E 0 -.~ 0 0 09
I = 0.044 Am-2
o 10<
~
dark
I
1.5
I
2
I
I
2.5 3 1/T (10-3 K-l)
3.5
4
10-5
(b)
~ k
p-GaAs 13-dislocation
"'b "..
co
"".%
vE O O
>m 10-7
"",,. o'&.
E O
I= 2.9 Am-2
tl;I o o
r'~ o~
~'da~
"""
"'a ! = 0.45 Am-2
10-9-
1.5
I
I
2
2.5
3
1/T (10-3 K -1)
Fig. 14. Dislocation velocity vs. temperature relations for different intensities of electron beam irradiation on c~-dislocations in n-GaAs [40] and fl-dislocations in p-GaAs [98]. dominates and at high temperatures the second term takes over the first. The critical temperature Tc is given by equating the first and the second terms and expressed by
AE
Vt
}
"Y
(3.2)
The magnitude of AE is independent of intensity I and stress "r, and does not exceed Et. The magnitudes of Et and A E are both dependent on the material and the dislocation type. The parameters experimentally measured are listed in Table 1. We note that A E is
468
K. Maeda and S. Takeuchi
Ch. 54
10-5
5"
n-GaAs (z-dislocation
~
-
10-7 " ) / I
irrad.
~
E
v
J 10 -9
,of" 10.1
dark
i
i
10
100 -r (MPa)
Fig. 15. The velocity V vs stress "1- relations in the dark (filled marks) and under electron beam irradiation (open marks) for c~-dislocations in n-GaAs [40].
10-5
"7o~ E v
,-,~....
GaAs I~-dislocation
intrinsic o
10-7 bo.. "0,.
o uO
""-...
n-type
> tO
-q irracf.
dark
10-9
Q
O
El
"" b"
"'b'b'o-.., dark
10-13
" - ,.., " " " ' O ,- ,.,
10-11
I
1.5
""o....
dark
irrad.
irrad. I
2 lIT (10-3 K1)
I
2.5
3
Fig. 16. The REDG effect in different doping conditions illustrated for fl-dislocations in GaAs [42]. Note that the velocities in the dark differ considerably, demonstrating the doping effect on the dislocation mobility. in all cases smaller than the bandgap energy Eg. We should also note that fl-dislocations and screw dislocations in n-GaAs behave quite similarly, which strongly suggests that the mobility of these dislocations, both in darkness and under excitation, is controlled by the 30~ -partials [103], the c o m m o n component of the two dislocation types.
w
Enhancement of dislocation mobility in semiconducting crystals
469
In V
"......iiiiiii iiiiiiiiiiiiiiiiiiiiiiiii ............
~
dark
I
I =I]
I
rc2 Tel
1/T
Fig. 17. Schematic illustration of the REDG effect described in the form of eq. (3.1). Radiation causes reduction of the activation energy by an amount independent of the irradiation intensity. The pre-exponential factor increases almost linearly with the irradiation intensity, but is smaller than that in the dark, which results in the existence of a critical temperature Tc only below which the REDG effect is observed.
Table 1 Parameters describing the REDG effect in semiconductors. Dislocation type s means screw dislocation. Crystal Dislocationtype Et (eV) E t - AE (eV) AE (eV) Eg (eV) Ref. ZnS s, cz,/3 1.2 0.3 0.9 3.6 [99] n-GaP c~ 1.48 0.37 1.11 2.26 [44] n-GaAs cz 1.00 0.29 0.71 1.4 [39] n-GaAs /3 1.7 0.6 1.1 1.4 [40] n-GaAs s 1.7 0.6 1.1 1.4 [41] i-GaAs /3 1.7 0.4 1.3 1.4 [42] p-GaAs /3 1.3 0.7 0.6 1.4 [42] n-InP /3 1.6 0.7 0.9 1.3 [43] p-Si 60~ 1.82 1.14 0.68 1.11 [45] i-Si 60~ 2.07 1.25 0.82 1.11 [46] i-Si 60~ 2.07 1.28 0.79 1.11 [ 104]
3.1.2. Short-range motion The R E D G effect can be observed in thin films under the T E M electron beam. Even when the absolute magnitude of the stress is not known precisely, the effect of the T E M b e a m irradiation can be assessed by comparing the velocity of a dislocation in darkness with the velocity of the same dislocation under irradiation. Figure 18 shows the temperature variation of the velocity ratio between the two conditions obtained in n-GaAs [105]. The broken line shows the slope expected from the data obtained in the bulk crystals [40], in good a g r e e m e n t with the thin-film data. This fact indicates that the effect is operating also for short-range motion on the T E M scale and hence in the lattice friction controlled by the Peierls mechanism. Another advantage of using T E M is that we could investigate the R E D G effect on a short length scale, which would give an important insight into the microscopic m e c h a n i s m of the R E D G effect.
470
Ch. 54
K. Maeda and S. Takeuchi
T E M in-situ
n-GaAs ~-disl0cati0n
"o
10 2
O
O O LL c(1)
q3
o
101 o
E
o
0
C) C
0
tr UJ
9
i 9
~
,'"' o
6
10 0
o /
t"
10-1 1
I 1.5
i 2
1/T (10 -3
, 2.5
3
K-1)
Fig. 18. The temperature variation of the ratio of V~, the velocity under irradiation by a 350 kV electron beam, and V~, that in the dark, measured in situ by transmission electron microscopy [ 105]. The broken line indicates the slope expected from the data obtained in bulk crystals [40], showing that the REDG effect takes place in the short-range motion controlled by the Peierls mechanism.
TEM experiments using thin specimens have been conducted also in ZnS [47, 99, 106, 107] and ZnSe [ 191 ] of zincblende structure and 6H-SiC hexagonal crystals [48, 49] with attention to the effect of the electron beam on the motion of the dislocations. The most systematic study was performed by a French group for ZnS deformed in situ on a tensile holder [47, 99, 106, 107]. Figure 19 shows the velocity of a screw dislocation in ZnS measured as a function of the segment length for three different beam intensities [106]. Quite similar plots have also been obtained for a and /3 dislocations, and even for individual partial dislocations [107], though the polarity (c~ or/3) was not identified. The intensity dependence is shown in fig. 20 for various types of dislocations in ZnS [99]. In contrast to the results of III-V compounds shown in fig. 12 and the results obtained in SiC [49], the velocity exhibits a clear tendency to saturation. One should note that the current densities in the TEM experiments are much higher than those used in the SEM experiments. Figure 19 clearly indicates that the velocity enhancement by beam irradiation takes place in the length-dependent regime. This is the case for any type of dislocation in ZnS [ 106], but not the case in SiC [49] which also exhibits a linear increase of dislocation velocity with TEM electron beam intensity. Figure 21 shows video-recorded images of a trapezoidal loop of a partial dislocation in a 6H-SiC crystal that expands on a basal plane by irradiation with a 100 keV electron beam [49]. In this specific case, the segment length of the 90 ~ partial became shorter as it advances. Figure 22 plots the velocity of the 90 ~ dislocation as a function of the segment length, showing that the enhanced glide takes place in the saturation regime.
Enhancement of dislocation mobility in semiconducting crystals
w
0.8
471
5600 Am -2
ZnS screw dislocation T = 390 K 0.6
H
03
E to | O
/
0.4
v
o
0.2
280 Am -2
I
0
,
J
|
0.4
0.2
L (10 .6 m) Fig. 19. The velocity of a screw dislocation in ZnS at different electron beam intensities measured by Faress et al. [106] as a function of dislocation length L.
ZnS 60" left A ,,7 03
60" right
4
screw
0
"
0
I
I
1
2
I
I
3 4 I (103Nm 2)
I
5
Fig. 20. The irradiation intensity dependence of dislocation velocity per unit length V/L measured on the TEM scale for various types of dislocations in ZnS (after Levade et al. [99]). Note that the current density ! of TEM electron beam is much higher than those used in fig. 12.
472
K. Maeda and S. Takeuchi
Ch. 54
Fig. 21. Video-recorded images of a hexagonal loop of a partial dislocation expanding in the basal plane of a 6H-SiC crystal under TEM electron beam excitation [49].
w
473
Enhancement of dislocation mobility in semiconducfing crystals
,p-
20
II I,
E 03 i O 1"v ou r3 mO Q~
l
I, t ,
> O .m
tl:l O NO
,,r-
10
~
s
6HSiC
0
0
I
I
I
I
I
20
40
60
80
100
120
Dislocation Length (10-9 m)
Fig. 22. The velocity of the 90~
marked Dl in fig. 21 measured as a function of the segment length [49].
The activation energy of enhanced glide in ZnS was assessed from the temperature dependence with appropriate corrections taking into account the intensity dependence of the velocity and the effect of stress deduced from the curvature of the dislocation line [99]. The experimental values are listed also in table 1 together with the value of activation energy in darkness evaluated by extrapolating the data to zero intensity.
3.2. Macroscopic plasticity In comparison with the large number of materials exhibiting the positive PPE in II-VI and ionic crystals, observations of the negative PPE at the level of plastic behavior are limited to a few materials. Apart from the old data mainly obtained by indentation tests [4, 13, 15], the negative PPE has been observed in the form of the temporary reduction in the flow stress during light illumination in CdS [108, 109], CdTe [110, 111], CdSe [112] and GaAs [95]. Figure 23 shows the results for GaAs [95]. An analysis of the stressstrain curve in the negative PPE in CdTe [ 111] showed that the increase of dislocation velocity under illumination was several hundreds percent. The excitation spectra of the negative PPE have been obtained for CdS [109], CdTe [111] and GaAs [95] (fig. 24). The peak in the excitation spectra corresponds to the intrinsic absorption in the respective crystal, which indicates that the negative PPE is brought about by the electronic excitation that generates electron-hole pairs, as
474
Ch. 54
K. Maeda and S. Takeuchi
175
150
125 I ~" i o o ~ 5o
.
.
.
.
K 25
0 e (%)
--
Fig. 23. The negative photoplastic effect observed in GaAs excited by interband light illumination (Mdivanyan and Shikhsaidov [95]). 1.5 CdS
CdTe
GaAs
0.5 t~
7 0
-0.5 400
600
800
1000
1200
/~ ( n m )
Fig. 24. The excitation spectra of the negative photoplastic effect in CdS [109], CdTe [111] and GaAs [95]. The arrows indicate the band edge positions of the respective crystals. in the positive PPE [25, 26]. The decrease of the effect at wavelengths shorter than the band edge is due to the extremely strong absorption that blocks bulk excitation necessary for the effect. The spectra in CdS and CdTe have long tails extending to low energies with some structure, which resembles that obtained in the positive PPE. It is known that the positive PPE can be reduced by simultaneous illumination of infrared (IR) light in addition to the fundamental excitation [108, 109, 113, 114]. An interesting fact is
Enhancement of dislocation mobility in semiconducting crystals
w
475
I~Ron 1
I•off
13..
o
v
b
Interband Illumination on -2
Interband Illumination off Time
Fig. 25. The effect of additional infrared (IR) illumination on the positive photoplastic effect caused by interband illumination in CdS in the prismatic slip orientation (Shikhsaidov and Osip'yan [109]).
that, in some cases, the IR illumination even reverses the sign of the PPE as shown in fig. 25. Disappearance of the negative component on cessation of gap illumination indicates that the interband excitation is necessary for the negative PPE to occur under the IR illumination. The IR light alone induces only a minor PPE [109]. The negative PPE is induced by more intense light than that inducing the positive PPE. The light intensity inducing the negative PPE in CdTe (typically 0.1 W/cm 2 [111 ]) is not very different from that inducing the REDG effect in Si (,-., 1 W/cm 2 [45]), but if one takes into account the difference in the excitation depth (3 ~ 4 mm ~ the crystal size in CdTe [111] and 50 ~tm in Si [45]), the excitation densities per unit volume are much different. Although it is not easy to compare directly the electronic excitation level due to light illumination with that due to electron irradiation, a rough estimate of electron-hole generation rate density based on appropriate parameters involved in the excitation process [40] indicates that the light intensity used in the study of the REDG effect in Si [45] is about one order of magnitude weaker than the electron beam intensity used in studies of the REDG effects in GaAs [40]. Nevertheless, in the study of GaAs by Mdivanyan and Shikhsaidov [95], they presumably used the same light source (a monochromatic focused light from a 200 W xenon lamp with unknown intensity) to successfully observe the negative PPE and the REDG effect of almost the same magnitude as that induced by electron irradiation [40]. Generally, however, the excitation intensity in the bulk samples in the studies of PPE is much smaller, probably by a factor of 103~5, than in those inducing the REDG effect studied for velocity measurements. One may wonder whether the negative PPE due to the REDG effect is observable at such low excitation levels. Generally the macroscopic deformation is controlled by the least mobile dislocation c~mponent. For fl-dislocations, least mobile in n-GaAs, the critical temperature Tc at the light intensity used by Mdivanyan and Shikhsaidov [95] can be estimated from the velocity data available [40]: knowing the experimental results that the REDG effects for c~-dislocations are almost the same in the two experiments
476
K. Maeda and S. Takeuchi
Ch. 54
using electron beam [40] and light [95], one can estimate Tc for/3-dislocations at the same light intensity to be around 600 K, which is well above 423 K at which the negative PPE was observed. At a temperature T << Tc, the velocity at the flow stress rl under illumination with intensity I is given by the second term of eq. (3.1): V1-Ve
I
"r rl
exp
--
"i0
(3.3) kT
'
whereas the dislocation velocity at the flow stress rd in darkness should be given by the first term of eq. (3.1): Vd -- Vt ( ~ )
m exp ( - ~ ) .
(3.4)
Based on the assumption that the density of mobile dislocations in constant-rate deformation tests is determined so as to minimize the flow stress [ 115], Sumino [ 116] showed that when the dislocation velocity is expressed in such a form as eq. (2.3), the plastic strain rate depends on stress as g (X. 7"m+2
exp
Q) .
(3.5)
- ~
Equating eq. (3.3) and eq. (3.4) with m in both being replaced by m + 2, one obtains the ratio of the flow stress in the form
T1
exp{
(m + 2)k
(1
1
T
Using AE = 1.1 eV, m ~ 1 for/3-dislocations [40], and Tc = 600 K and T = 423 K, we expect a softening effect as large as ra/'rl ~ 20. This value is larger than the observed value of rd/'q ~ 1.3, but it is likely that the excitation density was lower than that presumed above especially in the interior of the specimen. In any case, this estimate encourages us to conclude that the negative PPE is observable at the light intensities used in the experiments. Therefore, in the following argument, we assume that the negative PPE and the REDG effects are of the same origin. In II-VI crystals, the negative PPE or the REDG effect can be detected more easily in the absence of simultaneous positive PPE, the presence of which would cancel the negative PPE. Such an example is found in the prismatic slip of wurtzite CdS crystals [109], as seen in fig. 26. For the basal slip exhibiting the positive PPE, the negative PPE will compete with the positive component. Actually, with increasing illumination intensity, the positive PPE was found to be overtaken by the negative PPE [ 109, 111 ], as mentioned above. Such competition is observed also in the temperature dependence in CdS [ 109] as shown in fig. 26: with decreasing temperature, the positive PPE in the basal slip increases as usual but it turns to decrease below 100 K. This is consistent with the view that the positive PPE is caused by an excitation-induced change in the interaction
w
Enhancement of dislocation mobility in semiconducting crystals
477
14 12 10
6 n b "<1
4
2
200
-4
~/
250
prismatic slip
T ('C) Fig. 26. The temperature dependence of the photoplastic effect in CdS in two different slip systems, suggesting competition of positive and negative components [109].
between dislocations and some point obstacles [25], while the REDG effect is caused by the modification of the Peierls mechanism, which controls the low-temperature deformation. Actually, dynamical observations of illumination-enhanced glide of prismatic screw dislocations in CdS at helium temperature [117] evidently showed that the dislocation motion is smooth and continuous proving that the enhanced glide occurs in the intrinsic regime. In { 111 } slip in CdTe [ 111 ] and prismatic slip in CdS [ 109], the negative PPE depends on the plastic strain; however, the dependence is opposite in CdTe and in CdS: in CdTe the negative PPE increases with increasing plastic strain, whereas in CdS it decreases. The former case is understood if one takes into account the experimental fact that the positive PPE decreases with strain [118]. The latter case may be explained by that the Peierls mechanism responsible for the REDG effect becomes unimportant as the density of point obstacles increases with deformation. The negative PPE in GaAs [95] does not show any significant delay effect which is often observed in the positive PPE [26, 119], whereas in CdTe [111] and CdS [109] the negative PPE manifests itself a little bit later than the positive PPE when the light intensity is intermediate.
3.3. Miscellaneous facts Although the studies conducted for device materials are rather qualitative, the presence of the REDG effect was observed in several systems. Table 2 lists these examples
Table 2 The materials and dislocation types for which the REDG effect has been studied under experimental conditions indicated. g and T denote the estimated injection rate of minority carriers per unit volume and the resolved shear stress driving the dislocations, respectively. Material
T / G(lop3)
Dislocation type screw, a, p
Excitation source EB
Typical intensity 200 kV, 500 A/m2
Sic
basal 90' -partial
EB
150 kV, 1 x lo4 A/m2
ZnSe
60'
EB
CdS
prismatic screw
Ar-Laser
GaP LED
ff?
Current
5 x lo5 A/m2
1030
0.7
n-GaP
a
EB
30 kV, 2 A/m2
2 x 1028
0.5
GaAIAsPI GaAS
ff?
=+-Laser
200 W/m2
1024
0.4
InGaAsPl InGaP
ff?
Current
3 mW output
1033?
GaAlAsl GaAs
ff?
a+-Laser
lo8 W/m2
GaAlAsI GaAs
a?
a+-Laser
GaAIAsl GaAs
ff?
GaAlAsl GaAs
ff?
ZnS
g (m-3~-1)
1031
0.1
1032
?
T (K) 290
450
N
REDG effect Yes
E, (eV) 3.6
300
Yes
2.8
-
8000 A/m2
1030-32
-1
300
Yes
2.6
<2 x
lo6 WIm2
1029
-1
<77
Yes
2.4
300+
Yes
2.3
Yes
2.3
300+
Yes
1.8
<0.1
300+
weak
1.6
1031
350
weak
1.5
lo8 W/mz
lo3'
0.7
370
Yes
1.5
fi+ -Laser
3 x lo4 WIm2
1028
1.6
300+
Yes
1.5
Current
2 x lo7 A/m2
338
Yes
1.5
190
6x
0.1
450
N
1
N
650
Ref.
Table 2 (Continued) Material
Dislocation type
Excitation source
Typical intensity
GaAlAs/
a?
EB
30 kV, 3 A h 2
n-GaAs
0
EB
n-GaAs
P
EB
30 kV, 3 A h 2
n-GaAs
screw
i-GaAs
g ( ~ ns-I) -~
1028
T/G
0.7
REDG effect
300?
Yes
1.5
700
yes
1.4
[40]
GaAs
30 kV, 0.5 A/m2 2 x
E, (eV)
T (K)
-
Ref.
".
9
0.6
300
1028
0.6
500
700
yes
1.4
~401
g
EB
30 kV, 20 A/m2 7 x lo2*
0.6
350- 700
yes
1.4
r411
g
cr
EB
30 kV, 3 A/m2
1028
0.6
300
450
yes
1.4
[42]
i-GaAs
P
EB
30 kV, 3 A h 2
1028
0.6
300
500
yes
1.4
1421
p-GaAs
P
EB
30 kV, 3 N m Z
1028
0.6
350-600
yes
1.4
[421
n-InP
P
EB
30 kV, 3 A/m2
lo2*
0.6
450-600
yes
1.3
r431
p-Si
60'
Nd-YAG Laser
lo4 W/m2
1oZ7
10
600
yes
1.1
[451
p-Si
screw
Nd-YAG Laser
lo4 W/m2
I 027
4
640-700
yes
1.1
[451
i-Si
60°
EB
30 kV, 40 A/m2
1
640-730
yes
1.1
[461
2
InGaAsP/InP
60' ?
Current
108 A/m2
300+
negligible
0.95
[I261
$
i-Ge
60'
EB
30 kV, 50 A/m2
loz9
0.9
500
700
no
0.67
[46]
InSb
?
light
?
?
?
350
500
no
0.18
[127]
-
-
-
%
b
700
'2
IE
Fs.
$f'
$
2
480
K. Maeda and S. Takeuchi
Ch. 54
together with the systems that exhibit no significant REDG effect. Light-emitting double heterostructures of InGaAsP/InP with light emission in the 1.3 ~tm wavelength range are famous for having long lifetime, reflecting the absence of degradation due to dislocation multiplication [ 120]. The absence of the REDG effect in other hetero-systems may be due to the lack of stress driving the dislocations owing to the lattice matching structures [98]: the lattice matched GaA1As/GaAs [121] with light emission in the 0.8 ~tm range, for example, does not exhibit the REDG effect, though GaAs is the most typical material showing the REDG effect. Actually, the REDG effect in GaA1As/GaAs presumably stressed to some extent was found to be induced by strong irradiation by the TEM electron beam [122]. It is important to note that the REDG effect in the photonic devices is induced by forward biasing the p-n junctions to inject minority carriers into the active layers from which the light is emitted on carrier recombination. This fact strongly suggests that the essential role both of the light illumination and of the electron irradiation in the REDG effect is played by the minority carrier injection. The materials in table 2 have been arranged in the decreasing order of the bandgap energy. We notice the clear trend that the REDG effect is observed more commonly in materials with relatively large bandgaps and not in those with narrow gaps. TEM experiments give us an insight to the glide mechanism. The dislocations in plastically deformed GaAs [128] can move under irradiation by the TEM beam in such a manner that a segment of screw dislocation, that had been elongated due to highstress deformation, bows out first, then kink-pair nucleation continues preferentially at the bowed-out segment, and finally the giant kinks of a-character which are formed propagate along the screw segment. A similar process is observed also in partial dislocations in the basal slip of 6H-SiC [49]. The propagation of giant kinks along dislocations may be simply due to the higher mobility of the dislocation component constituting the super-kink segments. Such preferential enhancement in the dislocations of a specific component in some cases leads to transformation of dislocation lines to serrated shapes. This type of shape change has been observed in Si [129], CdTe [130] and SiC (fig. 21). In Si [ 129], 90 ~ partials are severely serrated while 30 ~ partials remain straight. Similarly in ZnS [107], the enhanced mobility of 90 ~ partials is higher than that of 30 ~ partials. In all cases the 90 ~ partials have the highest mobility under irradiation. The intermittent loading effect is considered to be, as discussed in section 2.4, brought about by the difference between the time required for forward kink-pair formation and its expansion under stress and the time required for backward kink-pair contraction during unloading. Consequently, as one increases the unloading duration tp, dislocation motion becomes suppressed since the kink-pairs gain enough time to shrink back during unloading. The suppression of dislocation motion becomes most distinct when tp exceeds a critical value tp that is determined by the kink mobility during unloading since the kinkpair contraction is achieved only by kink migration. Therefore, if excitation enhances the kink migration, we should find a decrease of tp when we excite the crystal only in the unloading durations. However, experiments on a-dislocations in n-GaAs [98] did not show such an effect, indicating that at least in this system it is not the kink migration that is enhanced. Conclusions for ZnS [47] and SiC [49] are, however, different. We shall discuss this problem again in section 4.2.
w
Enhancement of dislocation mobility in semiconductingcrystals
481
4. M e c h a n i s m o f the R E D G effect The trivial but most common mechanism to enhance defect motion or reaction is heating of the lattice by irradiation of light or energetic particles. The main cause of the photomechanical effects [4, 13] and the enhanced crystallization of amorphous semiconductors under an intense laser light (laser annealing [131]) were attributed to the lattice heating. This mechanism, however, does not explain the REDG effect. The temperature rise of less than a few degrees directly measured by use of a thermocouple explains only a small fraction of the velocity increase [40, 45]. The temperature rise evaluated from the temperature dependence of the velocity differs definitely from one type of dislocations to another. Also the linear dependence of the velocity on the irradiation intensity is inconsistent with the exponential dependence expected from the temperature rise that would increase linearly with the irradiation intensity [40]. Thus, in every respect, the lattice heating effect is ruled out from the mechanisms of the REDG effect. Irradiation might induce formation or annihilation of point defects that would alter the mechanical or electrical environment of dislocations. However, this is quite unlikely from the complete reversibility of the effect and the quick response to the irradiation. Moreover, there is no case in which evidence for defect formation or annihilation was detected in association with the REDG effect. The defect motion and reactions stimulated by electronic excitations are observed in many non-metallic solids [ 132-135]. Phenomenologically, the electron-stimulated defect motion (ESDM) is interpreted in terms of the adiabatic potential and the configuration coordinate. The adiabatic potential is defined by the total energy of the system consisting of the lattice strain energy and the electronic energy established when the lattice motion is so slow that the electronic states can completely follow the quasi-static configuration of the lattice [136] (adiabatic approximation). The configuration coordinate is a parameter expressing the lattice configuration around the defect; it may be one of the vibrational normal modes or may be an appropriate combination of them. The ESDM occurs, as illustrated by configuration coordinate diagrams in fig. 27, when the adiabatic potential in the excited state becomes unstable with respect to some coordinate or it finds a path with a lower activation barrier along a coordinate leading to defect motion. The ESDM mechanisms are thus described commonly by an instability of the electron-lattice system against a specific lattice distortion which results in formation of a defect. The mechanisms of ESDM differ from one another only in the nature of the electronic excitations and the dynamics of the microscopic processes. The electronic processes following the primary excitation also differ substantially depending on the nature of excitation; they are classified into inner shell excitation, valence excitation within the centre, and ionization of the centre which causes a change in the defect charge. The atomic process can be coupled with the relaxation process of the excited electronic system in different ways. The first subsection section 4.1 is devoted to the description of the phonon-kick mechanism, which is, in our opinion, the most plausible mechanism of the REDG effect. Then in the next subsection section 4.2 interpretations of the experimental results are given in the framework of this model. Since the phonon-kick mechanism is rather untypical among various mechanisms often quoted in literature to interpret ESDM phenomena in
482
Ch. 54
K. Maeda and S. Takeuchi
TotalEnergy potential)
TotalEnergy 3otential)
(adiabatic
(adiabatic
state
f
excited state
groundstate i
nd~ate
o
Q
,1,=. .,..-
o
(a)
12
(b)
Fig. 27. Configuration coordinate diagrams illustrating two cases in which electron-stimulated defect motion takes place. In (a) the stable configuration in the ground state becomes destabilized in the excited state; in (b) the excited state has a path for defect motion with an activation energy lower than that in the ground state.
solids, in the final subsection section 4.3 we discuss other possible mechanisms that should be considered. For more details of the ESDM mechanisms, the reader is referred to review articles for each topic [ 137-141 ].
4.1. Phonon-kick (recombination enhanced) mechanism Generally electronic excitation brings the total electron-lattice system including the defect to a high energy state which is destined to relax to the ground state or to an intermediate excited state. The energy released on the relaxation may be dissipated in some cases in the form of heat or light radiation. However, there is another mechanism in which the relaxation energy is directly utilized to induce defect motion. The most classical model in this category is the MGR (Menzel-Gomer-Redhead) mechanism [ 142, 143] that was first proposed to account for electron-stimulated desorption from metal surfaces. In this mechanism, an adsorbed atom once excited to the anti-bonding state commences to gain kinetic energy descending a repulsive potential (adiabatic potential) of the excited state; if the kinetic energy, which has been gained until de-excitation (transition to the ground state) occurs, exceeds the energy necessary for the atom to leave the surface ascending the attractive potential of the ground state, desorption of the atom is achieved successfully. This mechanism, however, is directly applicable only to systems for which the configuration coordinate diagram describing the reaction can be drawn with a single coordinate (the atom-surface separation for the case of desorption). In solids, the kinetic energy, which may be accepted initially by some coordinate, must be transferred to a specific coordinate for reaction (reaction coordinate) which is generally different from the former. This type of mechanism, however, is most plausible as the mechanism of the REDG effect. We describe the basic idea in a condensed form. The first quantitative theory by Weeks et al. [ 144] used a concept of defect molecules: they regarded a defect in a crystal as a molecule made up of 5' coupled oscillators
Enhancementofdislocationmobilityinsemiconductingcrystals
w
483
neq (E)
n(E) ~
~ _
~R neq(E-Ep)
i
,/
< Ep >
E*
Fig. 28. The number distribution functions n(E) of defect molecules possessing vibration energy between E and E + dE. n(E) in the excited state energized with deposition of energy E,pto the molecular vibration at a rate R is given by a function proportional to Rneq(E-Ep) (broken curve) plus neq(E) (solid curve) where neq(E)represents n(E) in thermal equilibrium. exchanging energy with the surrounding lattice maintained at a temperature T. When a large amount of energy Ep has been deposited by some means into the local vibrations of the defect molecule, the excess energy will be eventually dissipated to the environment, but if the molecule is sufficiently isolated from the surrounding the deposited energy has enough time to be distributed within the molecule to establish a local equilibrium. Let n ( E ) d E be a number of defect molecules possessing vibrational energy between E and E + dE. We define n(E) = neq(E) for thermal equilibrium. As long as the energy deposition rate R is sufficiently low, n(E) in the energized state is shown to be given by neq(E) plus a function proportional to R times n e q ( E - Ep), an energy distribution simply shifted from the equilibrium spectrum to a higher energy range by an amount of Ep as depicted in fig. 28. If the migration of the defect occurs when the energy of the defect exceeds some critical value E*, the migration rate, being proportional to the hatched areas in fig. 28, is increased by the energy deposition. Using the classical theory of unimolecular reaction, Weeks et al. [ 144] derived an expression for the total migration rate Ktotal which is given by the sum of the two areas: if Ep < E*, Ktotal-Ktexp
-~
+r/Rexp
(
E*kT-EP) ,
(4.1)
- Epl >>(S- 1)kT,
(4.2)
-
where
K, (E*- Ep) ~7- ~
E*
for
IE*
and
Kt (2S-2)!(E*) ]-S m
K] ( S -
1)!
k-T
for
[ E * - E p [ <<
(S-1)kT.
(4.3)
484
Ch. 54
K. Maeda and S. Takeuchi
~ ' ~
10 -3
GaAs CurrentEnhanced eF
10 -4 m
lo-5 .l-.,
1.09 eV
n" .-= 10 -6
1
E t - E e = 1.06 0V
cO
.Thermal
<
\
10-7 Et t =1.4eV
i ~~
=
t
I
3.0
3.4
3.8
,6/~/VB/,
-.-///,,-//~//I
Injected Hole
10-8 1.8
2.2
2.6
1/T (10"3 K"1) Fig. 29. Minority carrier injection enhances annealing of point defects in n-GaAs with a reduction of activation energy (1.06 eV) of magnitude very close to the electronic energy (1.09 eV) released on hole capture by the non-radiative recombination center associated with the defect [145].
The factor Kt/Kl represents the ratio of the rate of internal equilibration within the defect molecule to the rate of equilibration with the lattice. If Ep > E*, it is shown that the enhanced term is given simply by setting E* = Ep in eq. (4.1), so that it is athermal. Equations (4.2) and (4.3) tell that the efficiency factor 7"/ increases with decreasing rate of energy dissipation Kl, as expected, and decreasing number of 5'. The last fact reflects that as the number of degrees of freedom of vibration ,5' becomes smaller, the more significant becomes the statistical probability for a particular mode (the reaction coordinate) to acquire sufficient energy by fluctuation to accomplish defect migration over the barrier E*. This means that the phonon-kick mechanism is effective in systems consisting of a small number of atoms. For the process of energy deposition, Weeks et al. [144] considered non-radiative recombination which occurs when the defect captures an excess carrier to a deep defect level in the bandgap. This was based on the experimental fact that many defect systems in semiconductors under minority carrier injection exhibit ESDM effects whose behavior can be described by experimental formulae very similar to eq. (4.1) with the reduction of activation energy being very close to the electronic depth of the defect level from one of the band edges [138]. Figure 29 presents an example taken from the results of a pioneering work by Kimerling and Lang [145], who showed that minority carrier injection enhances annealing of electron traps in n-GaAs with a reduction of activation energy of magnitude close to the depth of that centre measured from the valence band. The actual mechanism of non-radiative carrier capture that possesses this feature is multiphonon emission [146]. A typical situation is illustrated by a configuration coordinate diagram in fig. 30: the curve indicated by U(gr) is the adiabatic potential of the
w
Enhancement of dislocation mobility in semiconducting crystals
485
Total Energy
(adiabatic potential)
U(f.e.+f.h.)
.+t.h.) r
-
b..._
v
Fig. 30. Configuration coordinate diagram illustrating non-radiative carrier capture by the multiphonon emission mechanism, gr denotes the ground state, f.e. the free electron, f.h. the free hole, and t.h. the trapped hole.
ground state which has a stable point at the origin. Instead of carrier injection, we consider a pair excitation of a free electron and a free hole into their respective bands. The adiabatic potential U(f.e. + f.h.) corresponding to this excited state is separated from that in the ground state by the bandgap energy Eg. At this stage the lattice configuration along a representative coordinate Q is still stable at Q - 0. Now let us suppose that the crystal is of n-type and a centre initially occupied by an electron captures a hole. If there is a strong electron-lattice coupling, the adiabatic potential U(f.e. + t.h.) corresponding to the state involving a free electron in the conduction band and a trapped hole in the centre may have a minimum at a configuration different from Q = 0; the new stable configuration may be found along a different coordinate. Intersection of the potential U(f.e. + t.h.) with U(f.e. + f.h.) gives rise to a significant probability for electronic transition between the states f.e. + f.h. and f.e. + t.h., which is not mediated by radiative transition. Efficient transitions take place near the crossing point that can be accessed by thermal activation from the potential minimum. Therefore, nonradiative transition in this mechanism becomes significant if the electron-lattice coupling is so strong that the thermal barrier Ub for carrier capture in fig. 30 is sufficiently small. After a hole is captured by the centre, the system relaxes to the stable point along the potential U(f.e. + t.h.) releasing energy amounting to Ep (phonon-kick energy). The released energy is transformed to lattice vibrations excited in this coordinate (accepting mode), which may be viewed as emission of a large number of phonons in this mode. The energy deposition involved in the above mentioned model is considered to be achieved in this way with
486
K.
Maeda and S. Takeuchi
Ch. 54
E ~
r
iii r
._o .>__ o
<
! ".~
"O (3
('~-Igl'~-Ep)2[(1-g 2)
"t3
rr"
0
I
0
g2E*
E*
E*/g2
Phonon Kick Energy, Ep Fig. 31. Reduction of activation energy in the phonon-kick mechanism as a function of the phonon-kick energy Ep (after Sumi [148]).
a rate proportional to exp(-Ub/kT). The net reduction of activation energy AE to be observed experimentally is equal to Ep - Ub, which corresponds to the electronic depth (thermal depth) of the trap from the top of the valence band in the energy diagram. The concept of defect molecule is, however, somewhat unrealistic since real defects are embedded in the lattice and they should not be as isolated as it is assumed in the theory. A more realistic theory, though still classical and phenomenological, was proposed by Sumi [ 147], who considered that the real vibrations around a defect are described by combinations of normal lattice modes coupled with each other. The expressions for migration rate he obtained are, however, very similar to those derived by Weeks et al. [ 144]: the enhanced kinetics is of thermal activation type with an apparent activation energy reduced from that in thermal equilibrium. Figure 31 [ 148] shows the reduced activation energy as a function of the phonon-kick energy Ep. Here, 9 is a phenomenological parameter which expresses the strength of coupling between the reaction mode and the accepting mode. The value of 9 can vary from 0 (no coupling) to 1 (the reaction mode is identical to the accepting mode). A difference from the theory by Weeks et al. is that the energy reduction is exactly Ep only in the region Ep < 92E *. Enhanced migration of a defect requires repetitive carrier captures at the same centre. For this to happen, the first capture of a carrier must be followed by the capture of a carrier of opposite sign, thereby completing recombination of the excited electron-hole pair. Thus, the rate of nonradiative capture responsible for the enhancement is equal to the rate of repetitive recombination. At low excitation levels, the recombination rate increases linearly with the intensity I since the capture of minority carriers generated in proportion to I limits the rate, but at high intensities it saturates at a value determined by the density of majority carrier because the majority carriers capture now becomes the controlling process. Thus, the pre-exponential factor in the enhanced rate for saturation must depend proportionally on the majority carrier density, which is experimentally verified for point defects [ 145].
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Enhancement of dislocation mobility in semiconducting crystals
487
A prerequisite of the phonon-kick mechanism is that the defect acts as an efficient non-radiative recombination centre. This condition is fulfilled by most dislocations in semiconductors, GaAs [149-151], GaP [152, 153], ZnSe [154], CdS [155], CdTe [156], and ZnO [157]. It is pointed out in many cases [96] that impurities are involved in the non-radiative behavior of dislocations. Vardanyan et al. [158] proposed a model for the PPE which considers that carrier recombination at deep centres, which act as dislocation pinning obstacles as well, assists depinning of the dislocation by the phonon-kick mechanism. However, the fact that moving dislocations are observed as dark contrast (fig. 10) indicates that the inherent dislocation core itself acts as a nonradiative recombination centre. This means that the phonon-kick mechanism can operate in the intrinsic process of the Peierls mechanism. Knowledge of the electronic energy levels associated with fresh dislocations would offer a direct proof for the relevance of the phonon-kick mechanism to the REDG effect; unfortunately, however, we are still far from drawing a definite conclusion regarding this point. Nevertheless, the phonon-kick mechanism operating in the Peierls mechanism is a very plausible model capable of explaining many experimental facts, as shown in the next subsection.
4.2. Interpretation of experimental results by the phonon-kick mechanism The phonon-kick mechanism is likely to operate in defect processes in which only a small number of atoms are involved. Although dislocation glide requires a collective motion of a large number of atoms, the fundamental processes, the formation of a smallest double kink (SDK) and the kink migration, proceed by rearrangement of a small number of atoms. Now the first question is, what is the centre which provides a nonradiative path for carrier recombination? Kinks themselves possibly introduce electronic energy levels in the bandgap [78, 159]. Though this is not at all obvious, they may act as nonradiative recombination centres: if it is the case, the carrier recombination at the kink can enhance the kink migration. On the other hand, for SDK formation to be enhanced, there must be a site on a straight dislocation where nonradiative recombination can take place. Since the atomic configuration of such a straight dislocation site will be, of course, different from that of the kink, the energy levels associated with them will be also different. This also requires us to differentiate the magnitude of the phonon-kick energy Ep and the barrier height for carrier capture Ub for kink migration from those for SDK formation. As illustrated in fig. 32, let AEk and AEs denote the net reductions of activation energy in possible enhanced kink migration and possible enhanced SDK formation, respectively. The index i - 0 indicates the state before a SDK is formed, and other states i = 1 , 2 , . . . are those at which kinks can make diffusive jumps to the neighboring states over the intervening second PP barriers. The state i - p indicates the saddle point configuration. It is important to note that the energy reduction is different between SDK formation (0 --+ 1) and its reverse process (1 -+ 0) that is required for backward kink migration. The next question is which process is responsible for the REDG effect. To answer this question, we need to reanalyze the dislocation glide process, decomposing it into elemental components. Since eqs (2.6) and (2.7) are quite general, the problem is to
488
Ch. 54
K. Maeda and S. Takeuchi
em
!~
aAa,, ~/
__
-
i , , 1 2 ...
0
g-,-
T
"
,:
v V A A
~, , . . . . p Kink-Pair State, x/a
,
,
_
Fig. 32. The potential barrier profile in the course of kink-pair nucleation that proceeds by formation of a smallest double kink (state 0 --+ state 1) and kink migration over the saddle-point state p. The barriers in the dark are represented by a broken curve that is effectively reduced by the phonon-kick mechanism to the solid curve. Note that the reduced barrier between the states 0 and 1 differs depending on the direction of the state change [ 160].
derive expressions for J and Vk from a more fundamental basis [104, 160]. We rewrite the expression for Vk (eq. (2.18)) for convenience in the form "cbha 2 (Ek) Vk--Uk k T exp - ~
(4.4)
.
Here we use notations Ek instead of Em and Uk instead of u0 in order to stress that Ek may be reduced from Em by excitation and Uk may depend on the recombination rate. The kink-pair nucleation rate J is derived from a set of rate equations which describe the forward and backward jumps over the respective barriers shown in fig. 32. With E + and E/~_1 denoting respectively the barrier height for the forward jump from the state i to the state (i + 1) and for the backward jump from the state (i + 1) to the state i, J is written in the form
[
"rbh J-u~-~exp
-~
1
Z(
E+-E~+')+Ek
i=0
}]
(4.5)
,
where again we use Us instead of u0 to express its possible dependence on the recombination rate. Substituting eqs (4.4) and (4.5) into eqs (2.6) and (2.7), we obtain the generalized formulae for dislocation velocity: __2Tbh2a kT
x exp
[
-~
~~(E i=0
+-E~I
)+Ek
}]
for
L >~ Lb
(4.6)
Enhancementof dislocationmobilityin semiconductingcrystals
w
489
and
7-bh2 [ - ~-~ 1 {~P~o~ (E + - EL1 ) + Ek }1 V - Us--~---Lexp
for
L << Lb.
(4.7)
In darkness, since y~'~(E+ - E L , ) - Fkp and Ek -- Em, and if Us ,-~ Vk ,-~ vO (the Debye frequency), we arrive at the same formulae as those derived by Hirth and Lothe [37]:
V =
2rbh2a [ Fkp/2 + Em] kT voexp kT
for L >> Lb
(4.8)
L<
(4.9)
and v=
Tbh2Luoexp [ k---U
Fkp+Em]
-
kT
for
For the REDG effect, we have to analyze separately the respective cases differing in the process(es) that is (are) enhanced. Case (i). Both the SDK formation and the kink migration are enhanced, which means recombination occurs on both straight dislocation sites and kink sites. In this case, the frequency factors Us and 89 are related to the recombination rate R with an efficiency factor r/. Since R and r/ should depend on the process, we must set Us = r/sRs and Uk -- r/kRk with indexes expressing the specific process. For the energy terms, y~'~(E+ E/~_I) = F k p - AEs + AEk and Ek = E m - AEk. Therefore,
V=
2rbh2a ~x/-~-s~s~v/R~Rk kT x exp
Fkp/2 + Em -- (AEs + AEk)/2]
for
]
kT
L :>:>Lb,
(4.10)
and
"rbh2
V - ~ T s R s - ~ L exp
[
-
Fkp
+ Em - AEk]
kT
for
L << Lb.
(4.11)
J
Case (ii). Recombination occurs only at straight dislocation sites and hence only SDK formation is enhanced. In this case, Us -- r/sRs but 89 - uo, and ~ ( E + - E~+ 1) F k p - AEs and Ek = Em, yielding
V-
2rbh2a [ Fkp/2 + E m ~x/~-s~s~X/~s kT exp kT
-
AEs/2"
for
L >> Lb
(4.12)
and
rbh 2
V - ~TsRs~
L exp
[
-
Fkp
+ Em- AE~] kT
for
L << Lb.
(4.13)
K. Maeda and S. Takeuchi
490
Ch. 54
Table 3 Features of the REDG effect expected from the phonon-kick mechanism. L - dislocation length, SDK - smallest double kink formation, KM - kink migration, ! - excitation intensity ([104, 160]). Glide regime
Length dependence
Enhanced process SDK + KM
kink-collision
no dependence
SDK KM
kink-collisionless
Pre-exponential Reductionin factor activationenergy o( I I(AEs + AEk) I ~AEs
cx: x/7 (x
1 5AEk
V/I
SDK + KM
~ I
AEs
SDK
~ I
AEs
KM
no effect
no effect
o c L f o r L < L * no dependence for L > L*
Case (iii). Recombination occurs only at kink sites and hence only kink migration is enhanced. In this case, Uk = r/kRk but Us -- UD, and ~ ( E + - E ~ I ) - Fkp + AEk and Ek = Em - AEk, so that we obtain y
2.rbh2a
-
kT
exp
Fkp/2 + Em - AEk/2] kT
]
for
L >~ Lb
(4.14)
and
"rbh2Lexp[
V = UD k T
-
Fkp-[-Em] kT
for
L<
(4.15)
Since the recombination rates Rj (j = s, k) are, irrespective of the kind of centre, proportional to the excitation intensity I at sufficiently low excitation levels, the intensity dependence expected in each case is as listed in table 3 together with the expected reduction of glide activation energy. The absence of the effect for the case (iii) in the kink-collisionless regime may be surprising, but it comes from the fact that, unless SDK formation is enhanced, enhancement of kink drift beyond the saddle point is completely canceled by the reduction of J due to the enhanced backward jump from the state 1 to 0, or annihilation of the smallest double kink. The experimental linear dependence of velocity on excitation intensity (fig. 12) is obtained for case (i), both in the kink-collision regime and in the kink-collisionless regime, and for case (ii) in the kink-collisionless regime. The REDG effect in ZnS exhibits linear intensity dependencies at low intensities with a tendency to saturation at high intensities (fig. 20 [99]). Since the velocity saturation can be explained by the expected saturation of the recombination rate at high generation rates of electron-hole pairs, we could consider that the intensity dependence relevant to the above argument is essentially linear. In addition, the REDG effect in ZnS takes place in the lengthdependent or kink-collisionless regime (fig. 19, [106]); hence, two possibilities remain: enhancement takes place only in the SDK formation process, or it occurs both in the
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Enhancement of dislocation mobility in semiconducting crystals
491
SDK formation and the kink migration [47]. In contrast to ZnS, the REDG effect in 90~ in SiC is observed in the length-independent regime (fig. 22), meaning that case (i) (SDK formation and kink migration both enhanced) applies in the kinkcollision regime [49]. Although this appears in contradiction with the conclusion drawn in section 2.3 that the dislocation motion is always in the kink-collisionless regime, it is possible that extremely enhanced SDK formation brings the dislocation motion to the kink-collision regime. Nevertheless, as mentioned in section 3.3, for a-dislocations in n-GaAs electronic excitation does not affect the intermittent loading effect, indicating that the SDK formation is the only process that is enhanced by excitation. Table 3 shows that this corresponds to the case (ii) (only SDK formation enhanced) in the kink-collisionless regime [98]. Thus, the process enhanced may differ from crystal to crystal and probably from one type of dislocation to another. The interpretation of the net reduction of glide activation energy AE differs depending on which process(es) is (are) enhanced. In every case, however, since AEs < Eg and AEk < Eg in the phonon-kick mechanism, the magnitude of AE does not exceed the bandgap energy Eg. As pointed out before, the REDG effect is more likely to be observed in wide-gap semiconductors, which indicates a trend that the critical temperature Tc, below which the REDG effect is observable, increases with Eg. Since Tc increases with the energy reduction AE (eq. (3.2)), the above correlation means that AE increases with Eg. This is quite naturally understood if one considers that the level depth determining the phonon-kick energy Ep and hence the upper limit of AE (AEs and/or AEk) will be likely to increase with increasing Eg. Dissociated dislocations may be strongly reconstructed so that the dangling bonds along the dislocation cores are all eliminated with the electronic bandgap free from deep states [78, 161, 162]. Solitons or anti-phase defects bounding reconstruction domains give rise to dangling bond levels located deep in the bandgap. Heggie and Jones [83, 163] calculated energy levels of solitons on 30~ in Si to obtain a donor level at Ev + 0.6 eV (0.6 eV above the valence band top) and an acceptor level at Ev + 0.2 eV. The inverted order of donor/acceptor levels is a consequence of a strong electron-lattice coupling in this system that results in an effectively negative Coulomb interaction (the Anderson negative-U) between two electrons on the same soliton site. Thus, the solitons fulfill the conditions, deep levels and a strong electron-lattice coupling, favorable for them to act as non-radiative recombination centres operative of the phonon-kick mechanism. Hence it is likely that the solitons act similarly to straight dislocation sites for the enhanced SDK formation in the REDG effect. In this case, since the kink-pair formation occurs preferentially at the soliton sites, the kink-pair nucleation rate J so far considered must be modified by multiplying a factor of a/d, where d is the mean separation of solitons. As d is given by aexp(Fs/kT) with Fs, the soliton formation free energy, the kink-pair formation energy (free energy) Fkp must be interpreted as including Fs. Therefore, if the kink-pair nucleation in darkness can occur everywhere on a dislocation line while that under excitation occurs only at soliton sites, the energy reduction to be observed is smaller than the values listed in table 3 by Fs/2 in the kink-collision regime and by Fs in the kink-collisionless regime. Solitons may not be important in GaAs, for theoretical calculations [164, 165] showed a moderate reconstruction in/3-90~ but the absence of strong reconstruction in a-90~ hence, though this is not
492
K. Maeda and S. Takeuchi
Ch. 54
conclusive, most types of partials introduce gap states necessary for the phonon-kick mechanism. For kinks, on the other hand, Hirsch [78] assumed an unreconstructed kink, whereas Jones considered a reconstructed kink, both trying to account for the doping effect on dislocation mobility. While the unreconstructed kink having a dangling bond should give rise to electronic states in the bandgap, the reconstructed kink, according to calculations by Heggie and Jones [83, 166], has no gap states. It should be remembered that for the REDG effect to occur the SDK formation must in any case be enhanced. This suggests that kinks formed at solitons, conceivable sites of enhanced SDK formation in Si, necessarily have dangling bonds which may act as non-radiative recombination centres in the kink migration process. These speculative arguments, however, lack experimental evidences, especially the actual depth of electronic levels associated with the solitons and kinks have not been identified on a definite experimental basis. By electron beam irradiation or light illumination with a photon energy greater than the bandgap the electron-hole pairs are generated only in a thin layer below the irradiated surface and the region below the surface within the carrier diffusion length should be affected by the electronic excitation. However, it appears that the excitation effect on the dislocation motion extends much deeper than this diffusion length into the interior [45]. This may indicate that the effect of enhancement of SDK formation in the excited layer near the surface is propagated into the crystal interior in a depth equal to the kink mean free path.
4.3. Other mechanisms 4.3.1. Charge state mechanisms
Since defects in semiconductors in many cases form deep levels in the bandgap, the charge state can be varied depending on the position of the Fermi level. The formation
energy of a defect in the negatively charged state, for example, is different from that in the neutral state by an amount of ( E a - EF), where Ea and EF are the acceptor level associated with the defect and the Fermi energy, respectively, because an electron must be transferred from the electron bath kept at the energy EF to the defect level located at the energy Ea. The defect migration energy is given by the difference of the formation energy of the defect at the stable site and at the saddle-point; so that the defect migration energy can also be dependent on the Fermi level. The effect of doping on defect mobility is observed not only in the glide of dislocations but also in many other defects [ 167]. Although the charge state mechanism was originally used for defect motion in thermal equilibrium, it also provides a mechanism for ESDM when the barrier for migration is reduced in a modified charge state that can be induced by electronic excitation. However, to the authors' knowledge, there are no examples in which the enhanced defect process is still of thermally activated type. Athermal enhancement is observed in migration of interstitial boron atoms in Si induced by injection of minority carriers [168]. In usual ESDM phenomena, the enhanced rate is proportional to the excitation intensity because a single enhanced event occurs on every excitation. However, the intensity dependence in this case was found to be
Enhancement of dislocation mobility in semiconducting crystals
w
0
p-Si
493
1
k..
(D e."
tu*6 tT.
ID..
.o_ .o_ t._
LI_ v
T
B
T
B
Atom Position (configuration coordinate) Fig. 33. Formation energy of a Si interstitial along a possible migration path according to an ab initio calculation by Car et al. [170] for various charge states, suggesting athermal migration in the saddle-point mechanism under electronic excitation.
almost quadratic. This fact implies the presence of two charge states that differ by two electronic charges the transition between which by capturing two carriers induces enhancement of defect migration. The athermal behavior is understood by considering a special case of fig. 27(a) in which the stable configuration and the saddle-point configuration are exchanged on every change in the charge state of the defect (saddle-point mechanism [ 169]). Another example of this type is the migration of self-interstitial atoms in Si for which an ab initio calculation was made for formation energy of a self-interstitial at various lattice sites and for various values of Fermi energy [170]. Figure 33 shows the calculated adiabatic potentials for migration of an interstitial Si atom along possible diffusion paths. The potentials dependent on the Fermi energy not only account for the doping effect but also illustrate what happens when the charge state is changed on electronic excitation. In p-type, for example, the Si in the ground state is stable at the T site with a charge +2e, but when the charge state is changed to + 1e or neutral, the T site is no longer stable and spontaneously relaxes to the new stable site (B site) that was formerly the saddle-point of the interstitial. When the interstitial Si further captures holes at the B site to resume the initial charge state, the B site again becomes destabilized and the Si falls down to either the initial or the next T site. Thus on every alternation of charge state induced by electronic excitation, the Si finds itself at the saddle-point and diffuses athermally. This saddle-point mechanism accounts well for the anomalously high mobility of Si interstitials under electron irradiation at cryogenic temperatures as low as 4.2 K [ 171]. For the negative PPE, a model which can be classified into the charge state mechanism, though different in nature from the above, has been proposed by the G6ttingen group [111 ]. They considered that, with increase of dislocation charge on excitation, the formation energy of a kink-pair should decrease since the repulsive interaction between line charges on the dislocation favors bowing-out of the segment or nucleation of a kink-pair [172]. This model was originally proposed to interpret the doping effect on the dislocation mobility [77]. However, this model has a difficulty that the charge on a dislocation line should be largely shielded by the free carriers abundant in conductive materials and hence the effect should be marginal in such crystals.
494
K. Maeda and S. Takeuchi
Ch. 54
Another model for the REDG effect is that proposed by Mdivanyan and Shikhsaidov [95]. They considered that the dislocation glide in semiconductors is described by the point-obstacle model (Celli's model [72]) that assumes the presence of point obstacles that impede the kink propagation along the dislocation line. Assuming the kink-obstacle interaction to be of electrostatic nature, they attributed the cause of the REDG effect to a change in the charge state of the obstacle under irradiation. This model predicted the disappearance of the REDG effect at high stresses, which was, however, not reproduced experimentally. Minority carrier injection should shift the charge state of a centre in the same direction as that one would obtain when the crystal is doped with compensating impurities (acceptors in n-type, donors in p-type). In the charge state mechanism, therefore, we expect that if the minority carrier injection enhances the defect motion in one type of conduction, it suppresses the defect motion in another conduction type. Such a behavior is not observed in the REDG effect: since/3-dislocations in GaAs, for example, have much larger mobility in n-type crystals than in p-type crystals in darkness (fig. 16), the charge state mechanism predicts that minority carrier injection should result in glide enhancement in p-type but suppression in n-type crystals; however, experimentally the irradiation (minority carrier injection) in both cases induces e n h a n c e m e n t (fig. 16). Thus among the charge state mechanisms, the saddle-point mechanism is the only one in which enhancement is always obtained irrespective of the conduction type; nevertheless, in this case the enhanced motion will occur always athermally. This point is also in disagreement with the REDG effect in which the enhanced motion is, in most cases, still thermally activated. Thus, all charge state mechanisms are incompatible with the observed REDG effect. 4.3.2. Excited state mechanisms
A defect with an electron excited to an upper orbital may feel a migration barrier lower than that in the ground state. Reorientational motion of FA-centres (pairs of an anion vacancy and an isovalent cation impurity) in alkali halides serves as an example of this category [173]. The defect motion in this system is essentially a diffusional jump of the anion vacancy. In the stable configuration an electron is trapped by a single well potential formed by the anion vacancy, but in the saddle-point configuration it now feels a double well potential separated by an anion situated in between the two half vacancies. The orbital in the p-type excited state has a node at the middle potential hump while the orbital in the s-type ground state has none; therefore, the electronic energy difference between the s-state and the p-state becomes smaller at the saddle-point. Since the parity of the wave function is preserved in the process, the electronic contribution to the height of the migration barrier in the p-state is smaller than in the s-state. This results in a reduction of the migration barrier in the excited state. In some cases the barrier height becomes even negative in the excited state, which means that the stable site and the saddle-point are exchanged under excitation. In the last case, defect migration proceeds athermally in the saddle-point mechanism. It is generally expected that, when the excited orbitals are more extended in space, the electronic excitation will reduce the sensitivity of the electronic energy to the change in the local atomic configurations occurring in the course of defect migration and hence
w
Enhancement of dislocation mobility in semiconducting crystals
495
excited state
13r} k..
E
c5
O
ground state
thermal
=
LU
I---
minority carrier injected
v
Fig. 34. Diagram illustrating the excited state mechanism proposed by Sheinkman [174] for semiconductors. The defect is assumed to migrate with an negligible activation energy in an excited state above the conduction band edge.
reduces the migration barrier. Sheinkman [174] proposed a model of this category for defects in semiconductors. He considered a defect level in the lower bandgap at a depth Ed from the conduction band and a specific excited state of anti-bonding character in the conduction band separated by Eexc from the gap level (fig. 34). In n-type crystals, the gap level is occupied in the ground state. He assumed that the defect migration energy Et in the ground state (the gap level being filled) is high, but once the electron is thermally promoted to the excited state the defect can migrate with a very low activation energy much smaller than Eexc and Et. Hence the apparent activation energy in the thermal equilibrium is given by Eexc. When minority carriers (holes in fig. 34) are injected, they recombine with the electrons in the ground state. Since the gap level is now emptied, defect migration is possible if a thermal electron at the bottom of the conduction band is thermally promoted to the excited state. The apparent activation energy in this case is the sum of Ei - Eexc - Ed and the Fermi energy EF ,-~ 0 measured from the bottom of the conduction band. Therefore the enhanced migration rate is given by Voc/~exp(/~exc-/~d+EF) -
kT
o(Rnexp
( E e x c - E d ) ( 4 1 6 ) kT ' "
where R and n are the recombination rate and the majority carrier density, respectively. Although this formula reproduces features of the defect process enhanced by minority carrier injection, there has been no theoretical calculation which supports the existence of such an excited state. For the REDG effect, no detailed explicit model has been proposed. The electrons and holes in solids interact with the lattice in various ways. In nonmetals, with the wave function getting smaller in space the electrons (holes) tend to interact more strongly with a localized lattice distortion. Thus, the electron-lattice coupling may lead to shrinkage of free carriers to a localized state even in perfect crystals. This actually occurs for holes in alkali halides [175]: a hole is trapped at two adjacent halogen ions spontaneously distorting the lattice around the site, thereby forming
496
Ch. 54
K. Maeda and S. Takeuchi
AB
anti-bonding state
L.r
0
"6
8E
,
dangling bond state DB
bonding state
<:22>
,,.._ ,,.--
bond separation Fig. 35. Electronic energy levels associated with covalent bonds.
a singly ionized halogen molecule (Vk-centre). Such a self-trapped hole further attracts an electron forming a self-trapped exciton (STE) [175]. It is well established in alkali halides that illumination of light in the photon energy ranging from ultra-violet to gamma rays induces creation of Frenkel pairs (consisting of an F-centre, an anion vacancy trapping an electron, and an H-centre, a halogen interstitially trapping a hole) [141]. For this remarkable phenomenon many mechanisms have been proposed [176, 178, 179], among which the model based on STE [178, 179] has been most widely accepted. Later theoretical works [175, 180] have revealed that the defect formation is mediated by a spontaneous breaking of symmetry of the STE. In covalent semiconductors, self-trapping does not occur either for holes or for excitons, but they may be trapped at defect sites [175]. At present we have no definite example in which the excitonic mechanism is certainly operating in the defect process in semiconductors. Nevertheless, this mechanism is sometimes inferred in such systems as defect creation in amorphous SiO2 [181, 182] which is characterized by extreme structural disorder. The trapped exciton in ionic crystals is viewed as a state in which an electron is transferred from an anion to a cation, which causes electrostatic instability of the defect. In covalent crystals, the trapped exciton is regarded as a different form of excitation in which an electron in the bonding (B) orbital is promoted to the anti-bonding (AB) state. On breaking of the bond, the two states merge to a dangling bond (DB) state in which both the B and AB states originate, as illustrated in fig. 35. Since the B and AB states are located below and above the DB state, respectively, the electron occupying the B (AB) state contributes a positive (negative) energy to the bond-breaking process. In such circumstances, when the electron initially occupying the B state is excited to the empty AB state, the electronic contribution to the breaking energy is reduced by an amount 3E. The mechanism for the REDG effect proposed by Belyavskii et al. [183] is based on a similar idea with some modifications. They extended the same model to n-type and p-type crystals to explain also the doping effect on the dislocation mobility. The defect processes they considered were switching of bonds in the course of kink nucleation and kink migration. Though the original argument of Belyavskii et al. is concerned with the kink-pair formation process, we consider here the kink migration process for simplicity.
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Enhancement of dislocation mobility in semiconducting crystals
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Eg eke
Ekc
o r" Eka LM 0 C
0
Ek d
0
w
~Ekv Ekv
0
Kink Configuration
3
3
3
Fig. 36. The electronic energy diagram specific for kink migration proposed by Belyavskii et al. [183] to account for the REDG effect. The levels Ekc and Ekv are the states associated with the atoms 1 and 3 at the kink site bonding with each other initially, and the states Eka and Ekd arise from a dangling bond associated with the atom 2. On bond switching in the kink migration, the levels are mixed at the saddle point so that they change their positions as depicted. The electronic occupation affects the contribution to the kink migration energy.
The situation is illustrated in fig. 36: the energy levels Ekc and Ekv are the electronic states associated with the atoms 1 and 3 at a kink site bonding with each other initially. Due to a bond distortion at the kink, the states Eke and Ekv have been shifted slightly inwards to the bandgap from the bottom of the conduction band and from the top of the valence band, respectively. Owing to the bonding and anti-bonding nature of the two states, they are further shifted deeper into the bandgap on stretching the bond. Meanwhile, a dangling bond associated with the atom 2, that is situated next to the atom 1, now gets close to the atom 3, and as a result the DB states Eka and Ekd, initially located at the middle of the gap split into two B and AB levels which are shifted toward the respective band edges as the atom 3 approaches the atom 2. The gap U between /~ka and Ekd arises from the Coulomb repulsion of electrons at the same DB site. At the saddle point, the levels Ekc and Eka, and the levels Ekv and Ekd, both intersect, but in the adiabatic case, the two states repel each other due to the Coulomb interaction of electrons on the different atoms 1 and 2. Under such circumstances, filling the levels /~ka and Ekv increases the electronic contribution to the barrier height for bond-switching, whereas filling the levels Ekd and Ekc lowers the barrier. Therefore, if one takes into account the variations of level occupation, the exponential factor giving the kink mobility with an activation energy Em should be replaced by
498
Ch. 54
K. Maeda and S. Takeuchi
exp
_Era ~--~)
--+ exp ( - E~--~) m [1 + Celexp ( 6 k @ ) +
Chexp (~E~v)
+Cexexp(SEkc+6Ekv+e)] kT
(4.17)
Here, we have assumed that the Fermi level is at the mid-gap and hence Em represents the migration energy of a neutral kink. Cel, C'h and Cex are, respectively, the relative concentrations of the kinks capturing an electron in the state Eke, a hole in the state Ekv, and an electron in Eke excited from Ekv leaving a hole in Ekv. The energy values ~Ekc and 5Ekv are the quantities defined in fig. 36. The energy ~ represents a reduction of the barrier due to the exciton binding energy which is not explicit in fig. 36. In equilibrium, since C e l - e x p ( EF - Ekc ) kT '
Ch
=exp(Ekv-EF) kT
'
Cex = CelCh,
(4.18)
and the terms with 6~el, Ch and Cex in eq. (4.17) are all negligible compared to unity, only the factor exp(-Em/kT) remains as expected. When the crystal is electronically excited, however, Cel, Ch and Cex will increase as the kink captures excess carriers which are generated. In the case of intrinsic crystals the excitonic term with Cex dominates and then the kink velocity becomes Vk o( Cex exp (
-
Em - 6Eex) kT
(4.19)
where 5Eex - 3Ekc + ~Ekv + e. This mechanism proposed by Belyavskii et al. may be termed the bond-weakening mechanism. Another elementary stage of dislocation glide is the kink-pair formation. They assumed that nucleation of a SDK requires two such bond-switching events and one bond-breaking event, with their electronic energy diagrams identical to fig. 36. Based on the assumption that the dislocation velocity is given in the form of eq. (2.19), they obtained an expression for the ratio of the velocity under excitation Vi and that in darkness Vt Vi =
exp
(4.20)
Thus, according to their conclusion, the velocity increases with the excitation intensity I to a power of 3/2 because Cex in eq. (4.20) should be proportional to I, which is in contradiction to the experimental observation of the linear dependence shown in fig. 12. However, we should remember that a full analysis based on eqs (4.6) and (4.7) is necessary for deriving of correct expressions for enhanced dislocation velocity controlled by multiple processes in series. In the phonon-kick mechanism, we had to differentiate between the barrier height reductions in SDK formation process and in its annihilation
Enhancement of dislocation mobility in semiconducting crystals
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499
Table 4 Features of the REDG effect expected from the bond weakening mechanism. SDK - smallest double kink formation, KM - kink migration, I - excitation intensity. Glide regime
kink-collision
kink-collisionless
Enhanced process SDK + KM SDK KM
Pre-exponential factor cx I o( %/7 cx: V~
Kink migration energy E m - t~Eex Em Em -- ~Eex
Reduction in activation energy t~Eex 0 ~Eex
SDK + KM SDK KM
oc I cx I no effect
Em - 6Eex Em -
diEex 0 -
process. This peculiarity is absent in the bond-weakening mechanism. In the bondweakening mechanism, therefore, we have always y~'~(E+ - E~+ 1) -- Fkp irrespective of the cases discussed. Other parameters are: in the case (i), Us = r/sI, Uk = r/kI and Ek -- E m - t~Ek; in the case (ii), Us = r/s1, Uk = UD and Ek = Em; in the case (iii), Us = Up, Uk -- r/kI and Ek = Em-3Ek. Here r/i (i = s, k) now stands for a proportionality factor relating the enhanced frequency vi (i --- s, k) for bond-switching to the excitation intensity I. The features expected from this model are as summarized in table 4. Although this mechanism may be classified, in a sense, within the charge state effect discussed in the preceding section, the basic idea is an excitation-induced bondweakening, in which electronic charge plays no central role. For its applicability also to the doping effect, the bond-weakening model is attractive, and can be a possible alternative mechanism for the REDG effect. Generally, however, the change of the activation energy by doping is much less than those in the REDG effect, which tempts us to consider that the two effects have different origins.
4.3.3. Phonon softening mechanism Van Vechten et al. [ 184] proposed that a high density of electronmhole pairs (plasma) generated by interband excitation induces lattice softening which causes non-thermal atomic rearrangement. They used this model (plasma annealing model) to interpret a possible electronic effect in laser annealing [131] and suggested the same mechanism operating in optically induced dislocation glide found in GaA1As/GaAs double heterostructures [ 185]. A similar mechanism was proposed by a Russian group [ 127] to account for the doping and illumination effect on dislocation mobility in GaAs and InSb [95]. The model is based on the fact that an increase of free electrons leads to a shift of the frequency of bound plasmon-LO modes to high frequencies, which they considered to be accompanied by a change of the binding energy of the crystal atoms. The mechanism is, however, only qualitative and has not been applied to any other systems showing ESDM. Therefore, we have little basis to further evaluate this model at present.
4.3.4. Coulomb explosion mechanism This mechanism was first proposed by Knotek and Feibelman [186] to explain the desorption of atoms from surfaces of ionic crystals that is enhanced by excitation of inner-shell
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Ch. 54
electrons by means of ultra-violet light or soft X-rays. The inner-shell excitation of an ion (e.g., Ti4+ in TiO2) induces an Auger process in which the electronic energy nonradiatively released on filling the vacant inner level by a valence electron associated with the neighboring ion (02- in TiO2) is utilized to excite the other electrons in the same ion (0 2- in TiO2) resulting in its ionization to multiple positive charges. Since the ionized atom (O i+ in TiO2) becomes positively charged, instability arises under strong electrostatic (Coulombic) forces exerted by the surrounding positive ions (Ti4+ in TiO2) and pushed out from the lattice position in an explosive manner. Among various ESDM mechanisms, this Coulomb explosion mechanism is the most established on firm experimental bases. Due to its electrostatic nature, this mechanism is likely to operate most efficiently in ionic crystals, but it is claimed to be responsible for enhanced crystallization of amorphous silicon induced by very intense synchrotron radiating X-rays [187]. However, this mechanism is not relevant to the REDG effect which is definitely induced by the excitation of valence electrons.
5. Summary It has been established that the dislocation velocity in tetrahedrally coordinated semiconducting crystals with rather wide bandgaps is commonly enhanced by electronic excitation of the crystals with electron beam irradiation or with light illumination; the effect is termed the REDG (radiation enhanced dislocation glide) effect. The REDG effect is described by a universal equation as a function of temperature and excitation intensity; i.e. the velocity under excitation is represented by the sum of the usual thermal activation term and an enhancement term which is also expressed by an Arrhenius-type equation with a reduced activation energy and with a pre-exponential factor proportional to the excitation intensity. This fact indicates that a common mechanism, essentially the modification of the Peierls mechanism, is involved in the effect. The REDG effect can often lead to softening of the flow stress in the macroscopic plasticity. The REDG effect is well interpreted by the so-called phonon-kick mechanism of defect migration, where phonons emitted on nonradiative electron-hole recombination process at dislocation sites contribute to the activation process for the dislocation migration. Some other possible mechanisms have been also presented and discussed.
Acknowledgment The authours would like to thank Profs E Louchet, G. Vanderschaeve, D. Caillard, R. Jones and V.I. Belyavsky for valuable comments.
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CHAPTER 55
The Role of Dislocations in Melting Blh.LAJOOS Department of Physics University of Ottawa Ottawa, ON K I N 6N5 Canada
Dislocations in Solids 9 1996 Ebevier Science B.V. All rights reserved
Edited by F. R. N. Nabarro and M. S. Duesberv
Contents Abbreviations 507 1. Introduction 507 2. Bulk melting 508 2.1. Melting rules 509 2.2. The dislocation theory of melting 511 2.3. The experimental evidence 515 2.4. The role of dislocations and grain boundaries in melting 518 2.5. The role of dislocations and grain boundaries in amorphization 2.6. Discussion 520 3. Two dimensional melting 520 3.1. Two dimensional physics 521 3.2. The role of topological defects 522 3.3. Various realizations of 2D systems 540 3.4. Computer simulations on systems of particles 576 4. Conclusion 584 Note added in proof 584 References 585
519
Abbreviations 2D 3D BOO C Cd DTM DW DWL DWF HIC HNY IC /( KT LJ MC MD PO WCA potential
two-dimensional three-dimensional bond orientational order commensurate dislocation density dislocation theory of melting domain wall domain-wall lattice domain-wall fluid hexagonal (or honeycomb) incommensurate Halperin-Nelson and Young incommensurate defined in eq. 23 Kosterlitz and Thouless Lennard-Jones Monte Carlo molecular dynamics positional order defined in eq. 40
1. Introduction Melting is a thermodynamic process occurring when the free energy of the liquid is lower than that of the solid. This statement does not say anything about the kinetics. Many attempts over the years have been made to develop theories of melting. Several schools of thoughts emerged with sometimes diverging views. The substantial development in experimental and computational techniques which has occurred over the last two decades has made possible a more integrated picture of melting. For instance some consensus is emerging on bulk melting; it is a surface driven phenomenon. And a whole new field has opened up, that of two-dimensional (2D) melting, for which the kinetics are very different. 2D melting has been in the limelight since an elegant dislocation theory of melting (DTM) predicted a continuous melting transition. This theory made quite a sensation, since it was always thought that melting is a first order process, in view of the latent heat. The simplest topological defects creating disorder are the dislocations. It is natural then to want to relate disordering processes such as melting with dislocations, and this has been done since the early days of condensed matter physics. Sir Nevill Mott and
508
B. Jods
Ch. 55
Ronald Gurney in the late 1930s pictured a liquid as a polycrystal with an average grain diameter spanning just a few atoms [ 1]. Because grain boundaries are formally equivalent to arrays of dislocations, the Mott and Gurney model can be viewed as the ancestor of the three-dimensional (3D) DTM which sees melting as a catastrophic proliferation of dislocations. Even though this theory is not considered now to accurately portray the melting kinetics in bulk or 3D solids, the 2D version has been a lot more successful. One reason is purely dimensional: in 2D solids a dislocation is a point defect, and not a line defect, making it a thermal defect, and the elementary tolological disordering defect. The 2D theory was developed by Halperin, Nelson and Young [2-4] in 1978 from an idea by Kosterlitz and Thouless [5, 6], and Berezinskii [7, 8]. The litterature on 2D melting now far exceeds that on bulk melting. In view of the focus of this review on the role of dislocations in the melting process, a major part of the review will be devoted to 2D melting. However, maintaining the natural historical evolution, we will begin with bulk melting. This also gives us the opportunity to define a few terms and present the various approaches that have been used to study melting.
2. B u l k m e l t i n g A simple criterion for crystallinity is that the Fourier component of the density pfr is zero, except for at least one non-zero reciprocal lattice vector k:
lim p~ -
0 ~ 0
if k not a reciprocal lattice vector, -~ for at least one non-zero reciprocal-lattice vector k.
(1)
When a crystal melts, no reciprocal lattice vector can be identified for which p~ shows a significant value. The crystal loses its long range positional order (PO). Landau argued that: "Any transition from crystal to liquid is associated with a change in symmetry. Elements of symmetry are either present or absent. Intermediate states are not present and therefore a continuous transition in the sense of the gas-liquid transition at temperatures above the critical point is not possible" [9, 10]. It is an experimental fact that a critical end point does not exist for the solid liquid transition in bulk solids strengthening the general perception of a clear first order transition. In 2D we shall see that an intermediate state with persisting bond orientational order (BOO) is possible. The loss of long range PO by itself does not, however, guarantee a liquid phase. A general characteristic of a liquid is its absence of resistance to shear forces (the shear modulus c44 - 0). If the loss of resistance to shear forces is used as a melting criterion, known as Born's melting criterion, then melting can be described as an instability and in principle a critical end point would be possible. These comments lay down the two perspectives that had been used to study melting: as driven by the thermodynamics only or by a mechanical instability (for some recent reviews on melting see [12-16]). As a thermodynamic instability, there is a priori no prerequisite for a mechanical instability in the solid at the point at which the free energies of the liquid and the solid are equal. The transition is expected to be first order and the
w
The role of dislocations in melting
509
kinetics heterogeneous. If a mechanical instability drives the melting, then homogeneous kinetics should be observed. Such a principle is the most likely to be operative if the transition shows strong precursor effects which tend to smooth out the entropy jump [ 13], or is even continuous. As a final note, in a thermodynamic approach the two phase nature of the problem, a comparison of the free energies of the liquid and the solid, is emphasized. This is not the case in a mechanical instability perspective. From a thermodynamic perspective the melting point is defined as the point at which the chemical potentials of the liquid and the solid are equal,/~s = #l. From this equality the famous Clausius-Clapeyron relation can be derived (see for instance ref. [10]) giving the liquid-solidus line; dP 1 d--T = L TA----V'
(2)
where L is the latent heat given by TASm. The entropy of melting ASrn is positive for all substances except the highly quantum solid helium at low temperatures. It is natural to associate ASm with the configurational entropy gain resulting from the disordering occurring during melting. A large amount of data has been accumulated over the years on this quantity (for a review see [14]). Analysis of the data has been made easier by use of the "principle of corresponding states" which says, that it is possible to define reduced units of length, energy, volume, temperature, etc. for each substance, and that measured in terms of those quantities, the properties of all liquids and gases with the same interaction law will be identical. This principle works well on rare gases and alkali metals for instance, and gives credence to the expectation that similar solids will melt in the same way. Melting rules probe this expectation further.
2.1. Melting rules The two main melting rules are discussed in this section, Born's and Lindemann's (for a more extensive discussion see [12-14]). The first correlates the variation of elastic constants, in particular the shear modulus and the compressibility, with the melting point, and the second, the amplitude of the atomic vibrations. They have been promoted as instability theories, but we will argue that they are more profitably viewed as laws of corresponding states. 2.1.1. Born's melting rule As mentioned above, the absence of shear strength defines a characteristic of a liquid. Born used this observation to establish his approach to melting, that a theory of melting should consist in an investigation of the stability of the solid under shearing [ 17]. Similar observations were made earlier by Durand [18]. Tallon [19] made extensive compilations of the variation of the shear modulus with temperature. Plotted as a function of the dilatation he found that the extrapolation to zero shear modulus gave the volume of the liquid (see fig. 1). The shear modulus decreases with increasing volume. But no obvious evidence of instability is found below the melting temperature Tm [20].
B. Jods
510 )
i
Ch. 55 )
I
0
13.. 0
1 0 {,h ,.... :} "10
oO E
0
e"-
Or)
4
-k
1
_3
7.
I
0.1
....
I ....
0.2
or (true) Fig. 1. The shear modulus associated with melting as a function of true dilatation tr = In(V/V0). For all crystals V0 = V(T = 0) except silver bromide where V0 = V(T = 300 K). The succession of solid points are for solid, while the single open points on the dilatation axis are for zero shear modulus at the dilatation at the melt at the freezing point. (1) benzene, c66" (2) krypton, c44" (3) argon, c44; (4) copper, 89 - c12)" 1 1 (5) silver bromide, ~(Cll--cl2); (6) lead, ~(cll-cl2)" (7) zinc, c44; (8) aluminium, ~1 (c11-c12); (9) cadmium, 1 1 c44; (10) silver, i(Cll - c12)" (11) gold, ~(Cll - Cl2) [19]. The shear instability argument is closely related to the divergence of the compressibility which had been earlier proposed as a melting criterion by Herzfeld and GoeppertM a y e r [21]. Boyer [13] has investigated in detail the possibility of relating melting with a thermoelastic instability which would be manifested by precursor effects to melting in the compressibility. He argues from parameter-free equation of state calculations for alkali halides and published thermal expansion results for a variety of materials that such precursor effects do exist and that consequently melting may not be in many materials as strongly first order as usually believed. Boyer's study raises the interesting prospect that precursor effects may be important in some classes of materials with the ensuing questions about how it affects the kinetics of melting.
w
The role of dislocations in melting
511
2.1.2. Lindemann's melting rule The paper by F.A. Lindemann which was published in 1910 [22] is the first to relate properties of atoms in the lattice to the melting transition. His idea, which still holds considerable appeal today, is that there is a maximum vibrational amplitude Um that can be sustained by a crystal before it melts. The greatest success of this melting rule is to show that for each class of solids melting occurs when the root mean square amplitude of thermal vibration reaches a critical fraction f of the distance of separation of the nearest neighbours r,
f-
X/-~/r.
(3)
Reported values of f are 0.07 for f.c.c, metals [23, 24], 0.11 for b.c.c, metals [23] and 0.11 for rare gases [25] and alkali halides [24]. They were obtained either from lattice dynamics, using phonon spectrums to calculate (u 2) [23, 25], or from the decrease in the structure factor peaks [24]. Shapiro [23] summed eq. 4 for f.c.c, and b.c.c, metals using an elastic force model:
1 Ek,j (uZ>- 3row ~k,j %(k) 2
1
3raN ~
1
~j(k)
w~(k)(exp[n~~ kB
} _ 1)
(4)
Gupta [25] summed the same equation for rare gas solids from phonon spectra calculated by a rigid atom method. Martin and O'Connor [24] obtained results for two f.c.c, metals A1 and Cu and a number of alkali halides. Their values for AI and Cu are comparable with those obtained from the phonon spectrum. There are two ways to look upon the Lindemann rule, either as a catastrophe theory, beyond a certain amplitude of vibration of the atoms the solid breaks down, or as a law of corresponding state. As the evidence keeps growing about the possibility of superheating crystals (see section 2.3), it is becoming clearer that if thermal instability exists it is at a higher temperature than the melting point. Hence the latter perspective should be considered. From the point of view of a surface induced melting, for which there is a growing consensus, the Lindemann rule may give the amplitude of atomic vibration for the disappearance of the nucleation barrier to the melt at the surface.
2.2. The dislocation theory of melting The dislocation theory of melting (DTM) has been the most popular of the theories providing a microscopic perspective on melting kinetics. As mentioned in the introduction, it has its origin in the observation by Mott and Gurney in the late 1930s that a liquid can be viewed as a polycrystal with a grain size of only a few atoms [1]. Following Mott and Gurney, Bragg speculated that the dislocation core where the disturbance is the greatest is liquid-like [26]. His estimate of the core energy obtained from the latent heat of melting was found later to be quite accurate for an f.c.c, metal simulated with pairwise interatomic interactions [27]. A few years after Bragg's work Shockley accounted for the fluidity of a liquid by assuming it to be equivalent to a crystal with a high density of
512
B. Jo6s
Ch. 55
dislocations [28]. The theory close to its present form began to take shape in the 1960s with the work done independently by Mizushima [29] and Ookawa [30]. They proposed that after an initial rise due to contributions from core and elastic interactions, the free energy of a crystal as a function of dislocation density Cd will begin to decrease due to entropic effects, configurational and vibrational. The melting temperature is determined as the temperature at which the free energy of a crystal saturated with dislocations equals that of a perfect crystal. It was followed over the next two decades with various other approaches to the same idea. The DTM is a catastrophe theory. At the melting temperature the free energies of the perfect solid and of the solid saturated with dislocations are equal, so an instability occurs. This is how the theory has been promoted. But it could equally have been presented as a two phase theory where the liquid near melting is approximated by a solid saturated with dislocations and a first order phase transition takes place when the two free energies are equal. As we shall see, it is the second point of view which survives to this day. I will present a simplified version of Mizushima's theory, which gives the essence of the model, then summarize the other contributions [31-35].
2.2.1. The essence of the DTM: Mizushima's theory In Mizushima's theory [29], the crystal is divided into cubic domains of equal volume. Each domain is given a line element of dislocation assumed parallel to one of the [100] directions. Its size is therefore determined by the dislocation line density Cd = L / V , where L is the total length of the dislocation lines and V the volume of the crystal. It follows that Cd 1 is the area of the face of a domain and c-~1/2, the edge length of a domain or the average separation between dislocation lines. The increase in energy per unit volume of the crystal, AF0, due to the presence of dislocations comes from the core energy and the interaction energy between dislocation lines:
' In 7rl/2r~
AF0(cd) -- cdb3/z a + ~
{
= cdb3# a -- ~
{
' In (Trcdr~) 1 ,
(5)
where # is the appropriate elastic constant for the dislocation, b the Burgers vector, 7rr2 is the core area, and a is some constant of the order of one measuring the core energy. The interaction energy between dislocations has been chosen to be zero when the cores begin to overlap. As the temperature rises this energy is lowered by the increase in entropy. There are a variety of contributions to the entropy. Mizushima considered two; the configurational entropy of the dislocation lines Sconf and the entropy Svib due to the lowering of the lattice vibrational frequencies resulting from the anharmonic dilatation of the lattice: Scone --- kB In P, where P is the number of configurations. Since in this model the dislocation lines pass by each domain once, P is essentially of the form Z n, /2 where n is the total number of domains, n -- c~ , per unit volume, the inverse of the volume of a domain. Taking into account branches and nodes Z is found to be about 3/2 20. Consequently Sconf -- Cd kB In Z. The contribution of a mode of frequency w to the free energy is equal to kBTln{2sinh(hw/(2kBT))} (see, for instance, [36]), which in the high temperature limit reduces to kBTln(hw/(kBT)). There are as many modes
w
The role of dislocations in melting
513
LL T=T m
\ \ Cd
Fig. 2. Free energy F versus dislocation density em at three temperatures. The vertical broken line gives the saturation density of dislocations.
as atoms in the crystal. If the new vibrational frequency is w' for atoms in the cores, the change in free energy per atom in the core becomes - k B T l n ( ~ / ~ ) . ~o~ is assumed to be independent of the density of dislocations. The change in vibrational frequency contributes then a term linear in the total density of dislocation lines Cd. The total entropy contribution to the free energy is therefore: AF1 (Cd, T) -- - R T { o
-3 3/2 Cd
In Z + pb2cd ln(~o/~')},
(6)
where p is the number of atoms belonging to the core per atomic plane. Adding eqs (5) and (6) gives the total free energy of the dislocation network. Figure 2 shows the variation of AF - AF0 + AF1 with Cd for three temperatures up to the saturation maximum c~nax. The melting temperature curve is the middle one obtained when AF(0) - AF(c~nax). c~nax is not easy to determine accurately. It depends on the crystal structure and the core radius. Roughly c~nax - ( T r r 2 ) - 1 . Kuhlmann-Wilsdorf [31] made the additional suggestion that dislocations will appear in dipole pairs or loops. Siol [37] in a similar way had proposed earlier that, in the f.c.c. case, the dislocations should be of the shear type on { 111 } planes, and that they should appear as closed loops. Dipoles and loops have no long range strain fields, reducing the energy of the defect.
2.2.2. Other versions of the theory Over the years a variety of ways of estimating the different contributions to the free energy of a dislocated crystals have been published. If one collects them, the following general form can be written down: AF
-
a l C d -- a 2 c 2 -- a3Cd
In Cd
.-. 3 / 2 -- tt4c: d -- T ( a s c d
3/2 + a6c d )
(7)
514
B. Jo6s
Ch. 55
where a l, a2, a3, a4, a5 and a6 are functions of material constants and correspond respectively to the core and lattice energy (el), to the anharmonic contribution to the energy (a2), to the elastic interaction energy between dislocations (a3 and a4) and to the entropy terms (as and a6). These coefficients are dependent on the particular model used to represent the array of dislocations. Some terms differ actually in origin. The term in a4 is present only in Edwards and Warner [35] where it represents a term of interaction energy (screening) between dislocations. These authors also have a negative a2 term reflecting their assumption that the lattice compresses upon the introduction of dislocations except for the core region. Experimentally [38] exactly the opposite is observed, the lattice expands. The entropic term a6 corresponds to configurational entropy for Mizushima [29] but to the entropy of vibration of the dislocations for Ookawa [30]; it does not appear in other models. The volume of melting AVm is considered by only a few authors. Kuhlmann-Wilsdorf explains a volume increase upon melting by the "riding up (of slip planes) against each other" in the dislocation cores [31]. Ninomiya [33] explicitly calculates AVm as an anharmonic dilatation of the lattice. The elastic energy increases as the square of the strain e but the shear modulus decreases as: 1
reducing the contribution of the dislocations to the total energy. Minimizing the strain energy consisting of the energy required to expand the lattice and the contribution from the dislocations gives the dilatation ea induced by the presence of the dislocations. The main entropy contributions to the free energy come from the lattice vibrations which soften under dilatation and the vibrations of the segments of dislocation. The configurational entropy of the distribution of dislocations is negligible. The entropy terms add a temperature dependence to the dilatation. To determine Tm a value for the saturation density of dislocations ca is required. In units of percentage of atoms 0.33 seems to be a fairly universal number [48-50]. Ninomiya's model is complete enough to allow for the calculation of not only Tm and AVm, but also of the melting entropy AS'm [49, 50]. As a concluding note, field theoretical models have been developed for dislocation mediated melting assuming elastic properties for the dislocations [39-41]. These are mean field theories where the emphasis is more on the critical phenomena than on quantitative predictions. The work was also applied to 2D solids (see section 3.2.4). Mean field theories are however not known to give accurate predictions on critical behaviour. 2.2.3. Critical analysis of the theory The theory has been popular as one of the few theories giving a microscopic description of the melting process with a visually appealing scenario [54]. It also seems to have had some success. Kotz6 and Kuhlmann-Wilsdorf [42] argued that the solid liquid interfacial energy should be half a general dislocation grain boundary. Using Turnbull's experimental
w
The role of dislocations in melting
515
value [43], very good agreement was obtained with a simple theoretical formula for half of the energy density of an average grain boundary as determined by Read [44]. Bragg estimated the core energy from the latent heat of melting [26]. Years later with the advent of computers an actual calculation verified its accuracy [27]. Evidence of the loops were found in molecular dynamic simulations of a Lennard-Jones model crystal heated to melting [45]. But by the early 1980s doubts about its applicability had begun to arise [46]. The DTM is an instability theory in spirit and therefore advocates homogeneous melting kinetics. The reality, however, is that melting is an heterogeneous process initiated from surfaces and extrinsic defects, as discussed in the following sections. Couchman and Jesser [47] had already tried back in 1977 to reconcile the DTM with a surface initiated process arguing that the two perspectives are compatible. At the melting temperature the energy barrier to surface induced melting disappears. Dislocations will generate at the surface as a precursor and be in effect the liquid front sweeping through the solid. How good this picture is depends on how accurate it is to represent the liquid as a solid saturated with dislocations [48]. Interestingly the above "successes" actually rely on the accuracy of this view point and not on the DTM representing the correct kinetics of melting. The recent study by Poirier [49] and Poirier and Price [50] use this fact. They tested the predictions of Ninomiya's model [33] for the melting temperature, volume of melting and latent heat on a large number of metals, finding a good to excellent agreement with experiment. Kristensen et al. [51] derived a rate equation for the formation and annihilation of dislocation segments from which quantitatively both the melting and instability temperatures can be calculated. The criteria used to calculate Tm is that at Tm the crystal-liquid interface is in thermodynamic equilibrium or the time derivative of the number of dislocation segments being created should be zero. Agreement with experiment is of the same order of accuracy as that found by Kotz6 and Kuhlmann-Wilsdorf [42], and Poirier and Price [50]. If dislocations cannot drive the melting transition, maybe they could play a role in the disordering of metastable superheated solids. This is the idea put forward in a recent theory by Lund [155] inspired by the 2D DTM (see section 3.2.2.7).
2.3. The experimental evidence The main objection to instability theories leading to homogeneous melting kinetics is the experimental evidence that melting is initiated from the surface and that crystals can be superheated. Looking at how crystals can be superheated also provides some key information on how crystals melt. Bilgram in a 1987 review [14] discusses extensively the superheating of crystals. Since then interesting new developments in particular in the field of computer simulations have occurred (see [16, 113], for instance). 2.3.1. Surface tension
In view of the importance of surfaces in the melting process a crucial quantity allowing classification of melting behaviours is o- the work spent in forming a unit of the surface
516
B. Jo6s
Ch. 55
of discontinuity between two phases. ~r is usually known as the surface tension, even if this term would have been more appropriate to indicate the work required to stretch the surface [54]. The latter quantity is called surface stress. The role of the surface in melting depends on the relative values of crSL, CrLVand crsv where S, L and V stand for solid, liquid and vapour respectively. Expressed in terms of the concept of wetting, if crsv = crSL + CrLV we have perfect wetting. The liquid spreads on the solid surface without expense of surface free energy. If osv < crSL+ aLV then partial wetting occurs. The liquid will tend to form drops on the surface. This occurs in a few substances where the structure of the solid is very different from the short range order of the liquid. Gallium, mercury, cadmium, zinc, p-toluidine, salol, and the semiconductors of diamond cubic and zinc-blende structure belong to this class. Superheating should be easier in crystals which do not wet their surfaces. There is a nucleation barrier to melting. These materials melt laterally, i.e., they show facets or in other words smooth surfaces. In most cases crsv > aSL + aLV which means that the free energy of the solid-vapour interface is lowered if a layer of liquid is intercalated. These systems undergo what is known as surface melting which has been an intense subject of research in the last decade and a half. The low value of oSL combined with the fact that the solid-liquid interface has been shown both theoretically and experimentally to be diffuse (for a review see for instance [54, 55]) would explain why it is intrinsically difficult to superheat crystals.
2.3.2. Surface melting As it was mentioned earlier, in the majority of solids crsv > oSL _qt_O'LV. When this inequality holds the surface melts at a lower temperature than the bulk. This is known as surface melting which has been discussed in a number of reviews [55-60]. The most common example of a substance exhibiting surface melting is ice. Faraday, in attempting to explain the mechanical properties of ice, was the first to suggest the presence of a liquid like layer on the surface of the solid. Although there are similarities between the two effects, surface melting is not roughening. A rough surface is not a liquid surface. The origin of surface melting is that minimizing the surface tension. ~rsv > CrSL+ CrLV leads to surface melting. The theory so far does not give a clear picture of the conditions under which this occurs. Ice seems a special case. Experimentally, surface melting was mainly studied for simple metals (mostly lead) and rare gases (argon and neon). Lead has been the most studied [61, 62, 57]. It revealed the important result that the effect was dependent on the surface structure: the (110)-plane does show surface melting while the more densely packed (111) surface does not. The intermediate density (100) shows incomplete melting. This result seems fairly general. Lennard-Jones solids usually do show surface melting (for a review see ref. [55]). Systems where a significant rearrangement is necessary to go from the solid to the liquid such as diamond or zinc-blende semiconductor compounds show no surface melting.
2.3.3. Superheating crystals The difficulty in superheating crystals has often been used to support theories of melting advocating homogeneous processes. That superheating is difficult does not however mean
w
The role of dislocations in melting
517
that it is impossible. A variety of experimental approaches has been attempted. With free surfaces, superheating has been achieved by various methods in particular using thermal conduction and internal heating. The most successful approaches involved inhibiting nucleation of the melt at the surface. In the case of superheating by thermal conduction, the ideal situation is a highly viscous liquid with a structure differing from that of the solid. The rate of molecular rearrangement may then be slower than the transfer of heat allowing superheating of the inside of the crystal while the liquid solid boundary progresses inward. On melts of high viscosity the attained superheatings were: 50~ for phosphorous pentoxide [64], 185 ~ for albite [65], 450 ~ for quartz [65] and 40 ~ for crystobalite, a form of quartz [66]. In all cases melting initiated from the surface or from defects, and the superheating could last for tens of minutes if not an hour. As203 is a case underlying the need for a differing structure between solid and liquid to observe superheating. As203 has two phases, at low temperature, arsenolith, and above 0~ claudetit. The latter melts at 309 ~ while the former is metastable above 0 ~ and melts at 278 ~ but can be superheated to above 309 ~ Claudetit cannot be superheated, because its structure is very close to that of its melt, in spite of the fact that it is a viscous liquid. The case of gallium further emphasizes the need of differing structures between liquid and solid. In spite of the fact that liquid gallium has low viscosity, gallium in the partial wetting category can be superheated [67-70]. Other similar examples exist [14]. The conditions for superheating by thermal conduction being present in only a small class of substances, internal heating has been attempted, but with moderate success. An example is thin wires of Ga superheated by current pulses [71]. The Tyndall flowers are melt figures obtained from the superheating of ice by sunlight. In the second case the resulting figures depend upon the inclusions acting as nucleation sites. It has been argued [72] that air bubbles are the most likely for the regular dendritic types of growth known as Tyndall flowers. A clear indication of the importance of the surface in the melting process is the decrease in the melting temperature with size, a well-known fact. The origin of the decrease is that the surface tension trEy < ~rsv. The drop in the melting temperature of gold is shown in fig. 3 taken from [73, 74]. The most dramatic illustration of the role of surfaces is, however, given by experiments where the surface effects are inhibited. This is the case for micron-size particle inclusions and coated clusters. Several experiments involved rare gas bubbles in A1 [75-77] and some other metals. Melting is determined by the disappearance of the diffraction peaks related to the rare gas solid clusters. Superheating of several hundred degrees has been observed. The rare gases are highly pressurized in the bubbles [78]. A very beautiful experiment on single crystal spheres of Ag coated with gold has been performed by Daeges, Gleiter and Perepezko [79]. Gold's melting temperature is about 100 K higher than Ag. Superheating of up to 25 K was observed for times of the order of a minute. Melting eventually occurs from the voids formed by the Kirkendall effect, which expresses the fact that the diffusion of two species past each other can go at different rates. Passivating the surface with an epitaxial layer of a high melting point and low miscibility with the crystal seems the most promising direction to achieve high superheating. In another experiment Si wafers have been passivated by SiO2 and
518
Ch. 55
B. Jo6s I
T(OK)
I
I
9.
. ~ y o ' e e*
1ooo
I
m.p. bulk-----~
....
,-L--N---~..'~.~--'--'r'~'~
;
9
--
I,r
500
l
l
I
I
50
100
150
200
D(A)
Fig. 3. Experimental and theoretical values of the melting point temperature of gold particles as a function of the diameter of the clusters [73, 74].
then heated by laser light. Superheating of approximately 3 K has been achieved [80]. Broughton inspired by the Daeges, Gleiter and Perepezko experiment set up molecular dynamics simulations of coated clusters of particles interacting with truncated LennardJones (LJ) potentials, varying the pressure and the range of the interaction [113]. He observed significant superheating comparable to experimental findings and, in all situations studied, the melting involved the propagation of a first order interface and not a homogeneous process.
2.4. The role of dislocations and grain boundaries in melting As we saw above, melting is usually initiated from the surface, even for solids which do not wet their surface. Extrinsic defects such as grain boundaries, holes, inclusions, dislocation cores qualify as nucleation sites for the melt, a point which has been argued convincingly by Cahn and Johnson [15]. There has been speculation for a while that grain boundaries (GB) exhibit something similar to surface melting ([54], p. 68). The subject has been given a lot of attention in the metallurgical literature [81, 82]. This goes back to the perception that grain boundaries can be viewed as a liquid layer sandwiched between two solids and as such that the grain boundary free energy crCB --~ 2CrSL [83, 84, 42]. Experiments show crGB > 2CrSL [54, 14]; the extra energy comes from the spreading of the interface over several atomic layers. It has been argued than these high-energy regions are likely to melt sooner than the bulk on heating, and to solidify after the bulk on cooling. The attempts to observe the premelting at grain-boundaries have not been successful [85-88] however. Measurements on copper using the technique of rotating sphere-on-a-plate indicate an absence of premelting up to 0.95Tm to 0.99Tm [85, 87], while transmission electron spectroscopy results on aluminium bicrystals show no boundary melting up to 0.999Tm [86]. The absence
w
The role of dislocations in melting
519
of pre-melting does not exclude some amount of disordering below Tm while retaining essentially crystalline character [86, 88]. Molecular dynamics simulations done on copper simulated using an embedded-atom-method potential [89] and silicon simulated by the Stillinger-Weber empirical potential [90], also find no evidence of pre-melting at the grain boundary. Since these are rather different systems it gives credibility to the feeling that it is a rather general effect. An earlier study on a Lennard-Jones system did find a gradual disordering along the grain as the temperature was raised towards the melting point with the self-diffusion coefficient going continuously to the liquid value [91]. This may be a reflection of the larger self-diffusion in this central force system with no bond angle stiffness. The matter does not seem closed. A recent Monte Carlo study on a lattice gas model using the grand canonical ensemble finds pre-melting [92]. Even if no pre-melting is found, the grain boundary is an equally likely nucleation site for melting as the surface, which is what is implied by these previous studies and explicitly observed in some computer simulations [90, 89, 16]. Besides grain boundaries, voids and possibly dislocation cores could act as nucleation sites for the melt. Void melting has been studied by Lutsko et al. [89] by computer simulation. A number of experimental studies have looked at the effect of dislocations on the melting kinetics with rather different results. In GaAs, melting is simultaneous in regions free of dislocations and in those with many dislocations, and the dislocations remain stable during melting [93, 94]. In contrast in Si dislocations have a high mobility near the melting temperature. They consequently seem irrelevant during the melting process, annealing out rapidly as the temperature is raised [95, 96]. In both solids the melting was observed to occur from the surface. The technique used was X-ray transmission topography. Results similar to Si were observed for A1 using the same technique [97]. In the experiment on Si, nucleation sites for the melt were found inside the crystal in some samples. It was determined that this occurred at microdefects [95]. In the above studies where the presence of dislocations could be directly seen, there is no evidence of their involvement in the melting kinetics. Earlier works motivated by the DTM sought for precursor effects to melting resulting from a proliferation of dislocations. A few papers report claims of such findings. The inferences are, however, indirect. Associating sound attenuation to the increase in dislocation density, a proliferation of dislocations with an inverse ( - 1 ) power law divergence was inferred in ice [98], helium-4 crystals [98], and some intermetallics In2-Bi [99]. A similar measurement saw a small anomaly in the sound velocity of Bi-Sn [ 169]. Salol, or phenyl salicylate (chemical formula C13H1003, with 12 formula units per unit cell), has also been the subject of such studies. Sharp anomalies in the hypersonic velocity and attenuation were observed by Abdul Awal and Cummins [101] near melting (Tm ~ 41 ~ using the Brillouin scattering technique. Woodruff [54] had earlier reported seeing diamond like shaped features when looking at the melting of salol under a microscope. These structures were reminiscent of dislocation etch pits.
2.5. The role of dislocations and grain boundaries in amorphization There are strong similarities between melting and amorphization. Both are order-disorder transitions. The atomic disorder created in the amorphization process can lead to volume
520
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change and lattice softening [102]. This observation is the basis for viewing solid state amorphization as a disorder induced melting process, a point of view very amenable to a Lindemann rule [103]: by molecular dynamics study and experiment it is found that at the crystalline-amorphous phase transformation the total mean square displacement (U 2) , sum of a dynamic (thermal) (U2ib) and a static part (Usta), 2 reaches a critical value equal to that for the melting of the corresponding perfect crystal [ 104]. But contrary to melting the process can be either heterogeneous or homogeneous, depending upon the way the disordering is achieved [105]. Heterogeneous amorphization is observed in irradiation experiments. The nucleation originates in regions of higher atomic displacement efficiency, i.e., along dislocation cores [106, 107], grain boundaries and below a free surface [107] (for a review see [15]). Dislocations are quite favorable because they attract point defects such as vacancies. It seems that homogeneous amorphization can also be observed akin to the melting that would be achieved if a solid could be superheated to its point of mechanical instability. The evidence comes from electron irradiation at very low temperature [ 107] and from hydrogen-induced amorphization data [ 108].
2.6. Discussion
At this point, what seems clear about bulk melting is that it is a heterogeneous process. Surfaces are the most likely nucleation sites but grain boundaries, voids, and defect clusters can also initiate the melting. In some systems there are precursor effects whose origin is not yet clear. There is no strong evidence of dislocation involvement. The DTM remains as a model for the liquid close to melting.
3. Two dimensional melting Two-dimensional physics is a relatively young field of research, which grew out of considerable progress in surface science in the 1970s. The same time period saw some huge advances in the theory of phase transitions. One has just to think of the renormalization group technique for which K. Wilson received the Nobel prize in 1982. 2D systems became a particular target of theoretical research because of their ease of simulation. As the comments below show, profound qualitative changes occur with the reduction in dimension from three to two. For one, thermal fluctuations, which play a dominant role in the character of a phase transition, become more important as the spatial dimension is reduced [ 11]. One consequence of the increase in thermal fluctuations, which has been known for a long time [110, 112], is the absence of long range order. What brought the field of 2D melting to the forefront of science were the seminal papers of Kosterlitz and Thouless [6], published in 1973, which made use of the new renormalization group technique to look at disordering transitions in the 2D XY model, melting in 2D solids, and the superfluid transition. They showed that melting would be a continuous transition driven by the elementary point topological defects, the dislocations. Further excitement was provided by the work a few years later (1978-1979), of Halperin and Nelson [2, 3]
w
The role of dislocations in melting
521
and Young [4] specifically on the melting transition, who showed that melting will be a two-step process, involving two continuous transitions. Within a decade of the publication of these papers, the study of the melting of 2D systems produced a wealth of results and a very rich physics, which has been reviewed in several major reviews [115-122]. Unraveling the nature of the transition has become a much more daunting task than could have been imagined. For one, an experimental ideal 2D system is nearly impossible to realize, with imperfections, finite size effects, boundary effects, the potential field used to localize the particles in 2D dimensions not being homogeneous, and the existence of the third degree of freedom. The theory made assumptions about the properties of the dislocations and the solid, in particular that elasticity theory is applicable. Computer simulations, which in principle could produce the ideal system, are limited still by time and the size of the system. In spite of all this, some consensus can be drawn now, nearly two decades after 2D melting became the subject of intense research. We will show in this section that there is no universal behaviour but discernible correlations between the nature of the interparticle interactions, the dislocation properties and the melting transition. In the first section we briefly review what distinguishes 2D solids from bulk systems. We then discuss the role of topological defects in the disordering process through the description of the various theories put forward, and their predictions, in particular the Kosterlitz-Thouless (KT) and Halperin-Nelson and Young (HNY) theories which as a whole is known as the KTHNY theory of 2D melting. There follows a report of the search for the predictions of the KTHNY theory, namely that the transition is continuous and proceeds in two steps, first with a loss of positional order at a temperature Tm to an intermediate hexatic phase, and then at a higher temperature Ti, with the loss of orientational order. The report of this search forms the bulk of the review, and covers a fascinating array of 2D systems. The last part deals with computer simulations, the search for the hexatic phase in what could be called virtual reality, a medium gaining in familiarity.
3.1. Two dimensional physics As one goes from 3D down to 2D important changes occur. As mentioned above, increased thermal fluctuations reduce PO. There is no long range order (LRO) in the sense that ((r-~)2), the mean-square fluctuation of the distance r between two atoms separated by an average distance f, increases logarithmically with the size of the system. This is in contrast to the finite value obtained for a three dimensional solid and is mainly due to the fewer degrees of freedom available in 2D to equilibrate fluctuations. The absence of LRO was predicted many years ago independently by Peierls [110] and Landau [9, 10]. It was demonstrated rigorously later by Mermin and Wagner [111, 112]. But bond orientational order (BOO) can be present (for a recent review see ref. [113]). Peierls and Landau and others were aware of it but it was again Mermin who formally demonstrated it for a harmonic crystal [112]. From a purely topological perspective BOO is favored in 2D. Contrary to 3D systems, in 2D the average number of neighbours in the liquid and the solid are the same [114]. The loss of BOO requires the creation of topological defects, hence their importance in 2D melting theories.
522
B. Jods
Ch. 55
With the increased fluctuations due to the lowering of the dimensionality, the critical temperature Tc (the highest temperature at which a condensed phase is observed) and the triple point Tt (where the liquid, solid and gas phases can co-exist) are driven to lower temperatures. Typically, for simple substances on known substrates the ratios are [124],
Tc(2D)/Tc(3D)~0.4,
(9)
Tt(2D)/Tt(3D) ..~ 0.6.
(10)
Notable examples are the Kr and Xe monolayers on graphite [125, 126]. Willens et al. [127] show the gradual drop of the melting temperature Tm of a Pb film as its thickness is decreased. They also observe a broadening of the transition, indicating that the first order jumps at melting are weakened as the film approaches 2D character. This work raises the issue of what is the surface in a 2D system. That obviously depends very much on the system, and as to whether heat can be transferred directly to the monolayer uniformly, and not just at its edges. There are a variety of other signatures of a 2D system: T 2 dependence of the heat capacity instead of T 3, a blunt liquid vapour boundary (i.e., a wide range of densities over which there is little change in the vaporization temperature, or a small curvature of the boundary around Tc) [128, 129], etc.
3.2. The role of topological defects The 2D dislocation theory of melting (DTM) originates from the 3D theory. From the DTM comes the idea that the presence in the solid of free dislocations will remove its shear strength. At low temperatures, the dislocations appear in bound dipole pairs. The topological order and the shear strength are preserved in the crystal. When the dislocations unbind, the topological order is destroyed, and the shear strength is lost because dislocations are much easier to move. These ideas were first applied by Kosterlitz and Thouless [5, 6] and Berezinskii [7] to the superfluid transition; the unbinding of the vortices destroying the condensate wavefunction. The 2D character, point defects instead of line defects as in 3D, and the increased thermal fluctuations affecting the critical behaviour, led to the novel results for melting that we will now describe. We however start with the 2D-XY model, which will naturally lead us to the theory of melting. In the 2D-XY model, spins are placed on a 2D lattice and constrained to orient themselves within that plane but without any restriction in direction. There is one order parameter, the change in spin orientation, corresponding to the change in inter-particle distance in the solid. In the Halperin, Nelson and Young (HNY) theory the additional degree of freedom of BOO is considered.
3.2.1. The Kosterlitz-Thouless (KT) theory In the 2D-XY model, within the continuum limit of the spin-wave approximation where only small changes in directions are permitted, a power law decay of correlation function at all temperatures is observed, i.e., every point is a critical point (see for instance [123]). Normally a system has an ordered phase with infinite correlation lengths and a disordered phase with exponential decay in the correlation function. The spin-wave approximation
w
The
role
of dislocations
1
~,
f
\
\ k
/
~
-
4
/
i
i
|
\
523
in melting
,
,~
ik,, -
P
9 -r-
,
),
- / i
I !
i
\
- ~
I,
k
<,'
~.
\
\
x
~
/
/
/
/
/
i
/
I
J
F i g . 4. V o r t e x - a n t i v o r t e x
pair
~
in the
J
2D XY
system.
predicts one phase for the whole temperature spectrum, a line of critical points. Kosterlitz and Thouless proposed that if one goes beyond this approximation and includes topological defects, the line of critical points would end at a finite temperature Tc, above which the decay in the correlation function is exponential [5]. The transition is characterized by a change in the correlation function rather than by the appearance or disappearance of long range order. With or without defects the correlation decays at infinity. The topological defects are vortices (see fig. 4). Vortices are like charges. They come quantized in units of integer multiples of 27r in the winding number, the integral of the change in the spin orientation along a closed contour. The charges can be positive or negative. The energy of a single vortex can be calculated within the continuum approximation. We start from the Hamiltonian H - _j Z
;, . ;
- _s Z
cos(o, _ oj)
~_ J }~(o~ - oj) ~ ~ 7J / L~o 12 a~,
(11)
where J > 0 and the sum (ij> is over the nearest neighbours and the 0i are angles with respect to some reference direction; the angle changes are supposed to be small between
524
B. Jo6s
Ch. 55
the nearest neighbours, which allows for the continuum approximation in the last step. From the last expression, the energy of the vortex is equal to
E(vortex) = Ec +
f R27rr dr JIVOI 0
R = Ec + 7rJ In m ,
(12)
a0
where R is the size of the system and a0 is a distance of atomic dimensions characterizing the core. Ec is the core energy associated with the region r < a0. Notice the remarkable similarity with dislocations. As with single dislocations the energy of a vortex diverges in an infinite system. This is not the case for a vortex-antivortex pair. They can annihilate and their energy is given by an expression similar to that for a dislocation dipole; r
E(pair) - 2Ec + 27rJ In m ,
(13)
a0
where r is the separation. Although the creation of a vortex is energy expensive, there is an entropy associated with the fact that one can place the vortex in many different sites. This entropy is 2ks so that the free energy is
ln(R/ao)
F-E-TS R = (TrJ - 2 k s T ) I n - - ,
(14)
ao
implying that for T > TKT where 7rJ TKT = 2kS'
(15)
free vortices may indeed exist even in the thermodynamic limit. Based on the above arguments, KT proposed the following scenario. At low temperatures, only vortex pairs exist. These are thermally generated and ride over a spin-wave background. They do not cause serious perturbations at large distances and the correlation function is essentially that given by the spin-wave approximation. Above TKT the pairs unbind since the system can support isolated vortices and the correlation undergoes a major change in behaviour. The expression for TKT derived above ignores the screening from the pairs present at any given temperature. The notion of screening can be readily appreciated if one notices that the gas of vortices is equivalent to a Coulomb gas in two dimensions. In two dimensions the potential energy resulting from the Coulomb interaction is logarithmic. The thermally excited pairs act as a dielectric, screening the interaction between pairs of opposite "charges" or dipoles. The vortex unbinding occurs at a temperature when the "dielectric constant" characterizing the system diverges (or, equivalently the coupling
w
The role of dislocations in melting
,
t
I
T
525
I
..
r
TKT Fig. 5. Vortex contribution to the heat capacity in the 2D XY model. TKT is the KT transition temperature.
constant goes to zero). Focusing on a particular dipole of size r, the interaction between the two vortices will be screened by smaller pairs lying within the range of the field. These pairs will themselves be screened by others lying within their respective fields and so on. One thus has a scaling situation and renormalization group techniques are ideally suited to this problem. The screening renormalizes the vortex interaction modifying the critical temperature, in effect by renormalizing the effective strength of the interaction [5, 130]. The theory predicts a continuous transition from the low temperature phase (with quasi-long range order) to the high temperature disordered phase. The transition will be characterized by only essential singularities in the energy and the specific heat. There will be no peak at TKT [130]. This is because the density of vortices will be low at the transition and little entropy is associated with them. This is an important aspect of the theory especially when applied to melting. A broad maximum is, however, expected in the specific heat above TKT [ 131, 132] as the number of dissociated vortex pairs increases with T (see fig. 5). Computer simulations have confirmed these properties of the 2D-XY model [133] (and references therein). It is interesting to note that at the transition there is only one vortex seen in systems of up to 200 x 200 spins [133]. The theory has also been successfully applied to the 2D superfluid transition [134], superconductor vortices [135], and the Gaussian model related to the XY model [136]. These success have given it credence in its application to melting
3.2.2. Application to melting: measurable quantities There are great similarities between the XY model and the 2D solid. First analogous to the spin wave approximation, we have for a lattice the harmonic (phonon) approximation. And secondly the equivalent to the vortex is the dislocation, as the properties of the vortex discussed above show.
3.2.2.1. Positional order and structure factor of the solid phase. In 3D the correlation function C6(R) tends to a constant at large R ensuring LRO. Here the correlation
526
B. Jo6s
Ch. 55
function is defined as
CC(Ft) - (p~(R)p~(O)),
(16)
PC (/~) -- e id'(h+a(h)).
(17)
In two dimensions as we noted earlier if(R) diverges leading to an algebraic decay of the correlation function at large R [137, 3],
Cd([t) ~ R -,7~
(18)
There is no translational long range order (LRO). This is also reflected in the structure factor peaks which instead of delta function singularities will have power law singularities,
s(0") .-- I~- d1-2+"~(T).
(19)
For an isotropic solid r/d depends quadratically on the length of the Bragg vector (~ if the system is at the solid to hexatic transition. If A and # denote the Lam6 elastic constants we have 3#+A - kBT4~rp'2PL + A) 1012'
r/d(s~
(20)
leading to values in between 1/4 and 1/3 for the first Bragg point.
3.2.2.2. Predictions on the elastic constants. The defects in the XY model can be mapped to a 2D Coulomb gas. In the solid the vortices become dislocations characterized no longer by a winding number, a scalar, but by a Burgers vector b, a vector quantity. From scalar charges we go to vector charges. The interaction energy for a vector dipole such as the dislocation with Burgers vectors bl - -b2 has the form; E12 - 2Ecb2 + jb2 [ ln ~R12 -cos
20 ] ,
(21)
a0
where J - Y/47r, Y being the two-dimensional Young's modulus, a0 is the lattice parameter, b the modulus of the Burgers vectors, and 0 the angle between b'l and/~12, the vector joining the two dislocations. The Young's modulus Y can be expressed in terms of the two elastic constants ~ and p: y = 4#(# + ,~) 2#+A
9
(22)
The role of dislocations in melting
w
527
The temperature Tm - TKT at which the unbinding of dislocation pairs or the formation of free dislocations occurs can be shown to be still given by eq. 15 which in terms of the elastic constants becomes 4a 2 #(# + A) K -
kBTm 2# + A
=
Ya 2 kBTm
= 167r.
(23)
This condition, called the KT stability criterion has been obtained by Kosterlitz and Thouless [5] but also, it seems independently, by Feynman (see [138]). It has been used to test the validity of the DTM.
3.2.2.3. Bond orientational order (BOO) as noted by Halperin and Nelson [2, 3] and Young [4]. The fact that the elementary topological defects have now vector nature signifies that the 2D solid has an additional order parameter not present in the XY model. This order parameter characterizes bond orientational order (BOO) which for triangular lattices can be defined as ~(r')
- - e 6i0(~') ,
(24)
where 0(~') is related to the displacement field ff(~') by 1 (i~uu(~') i~ux (~')) 0(~) = ~ ~x ~Y "
(25)
The corresponding correlation function, the bond orientational correlation function,
c6<)- (r162
(26)
approaches a constant as r --+ c~ in a 2D solid. This implies that there is long range BOO in 2D solids. This makes the nature of the melting transition more complex than the disordering transition in the XY model. An additional defect has to be considered, the disclination which breaks the BOO. Disclinations are "rotational singularities", formed by taking a perfect triangular lattice and either removing or adding a triangular wedge (see fig. 6). In other words at the center of the core of the disclination an atom will have one neighbour less, or more, than required. As the orientational order is integrated around this atom, in the first case this will lead to a positive winding number, while in the second a negative one with the usual conventions used to define the Burgers vector. The interesting point is that a dislocation can be viewed as a pair of disclinations of opposite winding number (see fig. 7). Halperin, Nelson [2, 3] and Young [4] developed a two stage theory of melting. In the first stage the thermally created dislocation dipoles unbind destroying the LRO but preserving to a certain extent BOO in a continuous transition similar to the one observed in the XY model. Co(r ) decays exponentially while C6(r)
,',-, r -r/6(T)
(27)
528
Ch. 55
B. Jo6s
(a)
(b)
Fig. 6. Positive and negative disclinations in a triangular lattice.
o o
oQ
Oo
o(, c
7 O
oi__- 5-
Fig. 7. A dislocation can be viewed as a pair of five- and seven-fold disclination.
decays algebraically. As the temperature is further raised the dislocation cores eventually unbind at Ti leading to an exponential decay of C6(r) or a total loss of the BOO. The intermediate phase is referred to as the hexatic phase. There are definite predictions about
w
The role o f dislocations in melting
529
r/6. It is given by 18kBT r / 6 ( T ) - 7rKA(T)'
(28)
where KA is the bond-orientational stiffness parameter. KA(T) diverges near Tm on the solid side, and, analogous to the universal jump in the dislocation pair interaction strength K, has a universal jump at Ti from 72kBTi/rr to 0 above the transition. Consequently 7]6 at the hexatic-liquid transition takes the value: r/6(Ti) = 1/4
(29)
(same universality class as the XY model [130]). KA enters in the interaction free energy FH of the hexatic phase which is a function of bond deformation as;
d2r[VO(r)] 2.
ks = ~KA(T)
(30)
3.2.2.4. Local bond orientational order.
Because of the difficulty of determining the long range behaviour of the BOO correlation function, some authors have defined a local bond orientational order parameter [ 139, 140] ni
~6 -- ~i -- L E ni
.
ei60i'J"
(31)
.
Here Oi,j is the orientation of the nearest-neighbour bond between particle i and its jth neighbour relative to an arbitrary direction.
3.2.2.5. Effect of a substrate field. A modulating substrate field if weak and incommensurate with the preferred equilibrium spacing of the 2D solid will not totally destroy the transition. There may however be only one transition at Tm. The stability criterion for a lattice subjected to a substrate field has been shown by Nelson and Halperin [3] to be given by: K-
4a 2 [ # ( # + A ) #7 ] 2# + A + p + ' y - 16rr,
kBTm
(32)
where "7 measures the torque exerted by the substrate on bond orientation. These substrate fields can also produce a new class of solids, domain wall lattices, which will be discussed later (see section 3.3.4.2). The stability condition of eq. 32 has been used by Coppersmith et al. [282] to look at the phase diagram of such lattices and by Shrimpton and Jo6s for a more complete theory for the specific case of the Kr on graphite lattice [274] (see section 3.3.4.2). Another effect of a substrate field is to preserve some of the BOO even in the liquid phase. Instead of isotropic rings the diffraction pattern will now consist of diffuse spots
530
B. Jo6s
Ch. 55
whose radial and angular half-widths have a temperature-independent ratio near the transition [143] (for an experimental observation see [144]). The spots may not be aligned along the substrate symmetry direction but be rotated by a small angle. This effect known as "orientational epitaxy" has been predicted by Novaco and McTague [145, 146]. Its physical origin is that the stress produced by the lattice mismatch between the preferred lattice spacings of the substrate field and of the 2D solid is most easily equilibrated by a shear of the 2D lattice leading to the rotational effect. (Shears are usually energetically less costly than compressions.) An "orientational epitaxy" has been observed very clearly in the rare gas monolayers adsorbed on graphite ([147] and references therein), one of the most studied of the 2D systems (see section 3.3.4). 3.2.2.6. Structure factor of the hexatic phase. The behaviour of the BOO is a crucial question in 2D melting because it has been one of the key criteria used to argue in favour of, or against, the existence of a hexatic phase. Evidence of its existence from experimental observation (section 3.3) and from computer simulations (section 3.4) will be discussed in later sections. We will here focus on the expected structure factor profiles for a hexatic phase. Intuitively they will appear as a set of six ellipses as in fig. 4 of Brock et al. [148] (see fig. 8). A solid would have had six points, an isotropic liquid a ring. The ellipses are intermediate structures, longitudinal broadening for low positional correlation, and smaller transverse broadening for quasi long range BOO. More specifically, several approaches have been proposed for the theoretical description of the diffraction pattern from a hexatic phase. Scaling arguments [149-151] predict that the coefficients C6n in the Fourier expansion of the azimuthal intensity profile I(qS) -~ n C6n COS 6n~b behave as C6n - C~ 2 in the 2D hexatic, so that the azimuthal intensity profile of a diffraction peak is Gaussian. This was confirmed on thin free-standing smectic films using precise electron diffraction measurements [157]. Aeppli and Bruinsma [143] developed a phenomenological theory taking into account the coupling of the liquid density and the BOO parameter, and deduced the complete 2D peak shape. When the hexatic isotropic transition is second order, the radial profile in its vicinity is predicted to be square-root Lorentzian. Otherwise, the profile is closer to Lorentzian. This behaviour fits the X-ray diffraction radial intensity scans of smectic films [152, 153]. Recently Peterson and Kaganer [158] calculated the structure factor for the gas of interacting elastic dislocation dipoles such as is assumed present in the hexatic phase of the KTHNY theory. They find Gaussian line shapes in the radial and azimuthal directions. In the case of randomly distributed dislocations the ratio between the transverse and longitudinal half-widths depends only on the Poisson ratio and varies between 0.52 and 0.58 as u goes from 0 to 89 When the correlations produced by the elastic interactions are introduced, the radial widths are drastically reduced while the azimuthal widths remain essentially the same as for uncorrelated dislocations. The precise value depends on the dislocation density. This explains the 1:5 ratio observed in the thin smectic films [152, 156, 157]. 3.2.2.7. Extension of the KTHNY theory to bulk systems. In bulk systems the equivalent of the dislocation dipole is the dislocation loop. The dislocation dipole can be viewed as the condensation of the vacancies in a row, the dislocation loop as condensation in a disk. Using this analogy, Lund [154, 155] showed that thermally activated dislocation
The role of dislocations in melting
w
k Q=
(a)
(b)
4~
531
AQ ~ 2~ "~
/
h
h
Fig. 8. Structure factor patterns for (a) an isotropic liquid and (b) an orientationally ordered hexatic. Q is the wavevector in the maximum of the structure factor, with a the corresponding lattice constant in a triangular lattice, and AQ measures twice the inverse of the distance ~ over which the molecules are positionally ordered [ 149].
loops renormalize the elastic properties of a solid, and there is a transition temperature Ts above which the shear modulus # vanishes. Near Ts the shear modulus has a power law behaviour as a function of the reduced temperature t, #(t) ~ U, where u ~ 0.5. Since Ts is above the melting temperature, this disordering will be the melting of the superheated solid. 3.2.3. Dislocations It is implicitly assumed in the KTHNY theory of melting and in some of the alternate theories discussed in the following section that the dislocations have the properties predicted by elasticity theory; in short, an interaction energy varying logarithmically with distance and in effect zero Peierls stress, i.e., the dislocations are mobile. In addition in computer simulations and in experiments relying on Voronoy plots to identify defect locations, the cores are supposed to be small. These properties are not neces-
532
Ch. 55
B. Jods
x
./J"
+ vacancy
+
=4.26A
-
"--
~, g
0 -
+
~ i +
.+__x......-x,.~ 0=4.36A :
O't
.-'"'"
t
1
t
t
I
5
t t ttl
t
10
t
t
t
50
dipole seporolion (o) Fig. 9. Dislocation dipole energies for floating Xe monolayers for lattice parameters: a - 4.26, 4.36 and 4.45 ,~ [163].
Fig. 10. Dissociated dislocation core in Xe adsorbed on graphite [163].
sarily satisfied in actual systems and are dependent upon the nature of the interparticle interactions. It was first observed by Hoover, Hoover and Moss [159] that the dislocation cores in the Lennard-Jones monolayers were extended. Jo6s, Duesbery and collaborators studied extensively the properties of dislocations in this system [160, 162-165]. They found not only an extensive core approximately 30 lattice spacings in diameter but that the small dipoles, those with overlapping cores, behaved more like composite defects than elastic dipoles with energies, for the vacancy dipoles, significantly lower than elastic predictions (see fig. 9). These defects were more likely to be nucleated than the large dipoles. The small dipoles were pinned [164] although the large ones were mobile. The LJ potential can be considered to be an intermediate range hard core potential. The consequences of these properties on testing melting theories have been reviewed by Strandburg [119]. The main points are that finite size effects will be important and that what may appear as defect clustering may simply be disordering of a dislocation core. Duesbery and Jo6s [161-163] have also looked at the effect of a substrate modulating field simulating the situations occurring for Xe and Kr on graphite in the incommensurate phase, i.e., when the adsorbate is not in registry with the substrate. The density modulations in the adsorbate taking the form of a hexagonal domain wall lattice (see section 3.3.4.2) can have significant effect on the dislocation properties. Short dipoles form pinned complex defects as one would find in a lattice gas. In dipoles with sep-
w
The role of dislocations in melting (a)
8 -
Xe
x interstitial + vacancy
",,I
~o
6
-o
2
533
0
f
. . . . . . . . . .
1
5
I
I
I
5O
10
dipole separation (a)
(b)
3
Kr x interstitial +
o
vacancy
2
Q)
I
+
v OI
,
1
,
,
. . . . .
5
I
10
I
I
I
I
50
dipole separation (a) Fig. 11. Dislocation dipole energies for physisorbed Xe (a) and Kr (b) on graphite (a = 4.418 ,~ for Xe and a - 4.102 ,~ for Kr) [163].
aration equal to or larger than a domain-wall segment, the cores dissociate into two partials separated by a stacking fault, typically the length of the domain wall segment (see fig. 10). The vacancy dislocation dipole energies are quite different in the Xe and Kr monolayers. Kr expands upon adsorption on graphite while Xe compresses. The energy variations for Xe are quite similar to those of the ideal floating monolayer (see fig. 11 (a)). In Kr after an initial rise in the energy, there is a collapse showing the change from a dislocation in the Kr lattice to that of a dislocation in the hexagonal network of domain wall segments (see figs l l(b) and 12). This shows in one way that Xe on graphite is expected to behave very much like a floating monolayer while Kr will show behaviour reflecting the properties of the network of domain walls (see section 3.3.4.2). The properties of the dislocations have also been studied for the monolayer with the nearest neighbour piecewise linear force potential (PLFP) defined by the expres-
534
Ch. 55
B. J o d s
"
"
"~".".'"
~
"
"
"
~
"
*
2
"-%%"
~
"
~
~
~
"
.~:..:...-~.~!.~.~.~.~.~.~.~.~.?§247
".":':'."
~
~
"
~
~
~
~
~ 2"2"2
~
9
9 "
*
:..:.:•)•.C...............................•..............................•r.•.:::..: ..........•..%.r........................ 9
9
9 . . . . . . . . o
2
.
.
,,
.
.
9
9 ,,
•................................•.q:•`::....... .
9
9 .
.
9 ,,
.
.
,,
.
9
9 .
.
. .
9 .r
9
.............:.~...~.......~.........A...~...........A....~................~v........ . . . . . . . .. - ~ -
~?:. ~..:.9 . ~
:-:~~; ~, . . " . ~ . . : X..:. v , /".:..:.v.::::.:~~% : z .. . . . . "
:-::-,'-.'-'-:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . - . . . .
o . .
o
.
.
.
.
.
o
.
o
o
.
o
.
~.~.:.~ .
.
.
.
.
.
9 .
.
o
.
.
o
.
o
.
. .
~..J.~.~
....~:........~....~...............:...~..-..~-~..~..............~...............;~x.
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::: .~.~.~.~i..`.~:~:~.~.~.~.~.~.~4!~i.~.~.~.~.~.~.~.~.~.~.~?~..i~.~.~.~.~.~.~.~.~.~. Fig. 12. A dislocation dipole in the Kr on graphite monolayer, when the defect becomes a dipole in the domain-wall lattice (DWL). The "differential displacement" vector drawn between each nearest neighbour pair of atoms is that by which the relative displacement of the atom pair differs from the commensurate value [163].
sion: ~ a ( r - do): - /~W2~ VpLr~ (r) --
1
r <~ d o + w ,
- ~ ~ ( r - do - 2 w ) 2,
do + w < r ~< do + 2w,
O,
r>do+2w.
(33)
w is taken to be 0.15d0 as in reference [166]. For this value of w the dislocation core has only one broken bond. Ladd and Hoover [166] reported properties close to those predicted by elasticity theory but later studies by Jo6s, Ren and Duesbery [ 165] revealed that departure from elastic behaviour could be considered significant. Maybe because of the short range of the interaction combined with the absence of a hard core, the dislocations were pinned in the lattice. In addition away from the 0 = 0 direction studied in [166] the core region, although not as extensive as in the LJ monolayer, was still about 9 lattice spacings in diameter. In the long range Coulomb system, Fisher, Halperin and Morf [167] found properties close to those predicted by elasticity theory. If one could hazard to make general statements based on the three potentials studied, the following picture emerges on the effect of the range of the interactions on the properties of dislocations for central forces. Finite range hard core attractive potentials have a large dislocation core region significantly larger than the range of the interaction. Soft core potentials have a smaller core, the smaller the smoother the potential. For short range potentials the dislocations are pinned and for long range potentials they are mobile. These are obviously tentative conclusions and a more systematic study is required, but as we shall see later, these inferences seem consistent with findings reported in computer simulation studies.
w
The role of dislocations in melting
535
Beside assuming elastic properties for the dislocations, the density of dislocations is implicitly taken to be low, i.e., the gas of dislocations is dilute. This requires a fairly large core energy. With respect to Saito's simulations on a dislocation vector system, the three systems discussed above can be considered to have a large core energy (see section 3.2.5.1). 3.2.4. Alternative theories
Before we examine the available information on 2D melting, we will briefly review other theories or scenarios for melting which have been proposed. Most of them, as the dislocation dipole unbinding scenario, are based on elasticity theory or mean field theory. The one which has found the most support with experimental observations including computer simulations is Chui's grain boundary theory of melting. Fisher, Halperin and Morf [ 167] had found that in the electron lattice the free energy to generate a single small grain boundary goes to zero at the same temperature as the dislocations unbind. Within a low density approximation, Chui considered the effects on the grain boundary free energy of transverse and longitudinal fluctuations, crossings, couplings to dislocations and density fluctuations [168]. He concluded that the KTHNY melting mechanism will be pre-emptied by a first order transition involving the creation of the grain boundaries at a temperature T1 below the dislocation unbinding temperature Tc. That T1 < Tc is expected on the ground that Tc is a mechanical instability temperature. The transition will go from weakly first order for large core energies to strongly first order for small core energies. The main support for Chui's theory has come from the fact that it predicts a large defect density at the melting transition which has been observed in pictures of colloidal suspensions (see section 3.3.3) and in some computer simulations (see section 3.4.5). His prediction that the BOO is not preserved above T1 because the grain boundaries will not remain bound in pairs runs however counter to observation. This may be a limitation of the approximation made in developing the theory rather than a fundamental difficulty with the mechanism. As we have seen earlier, dislocations can be viewed as bound pairs of disclinations. Using this observation Kleinert developed a mean field theory where the disclinations are the elemental defects [169]. This permits the possibility of a simultaneous unbinding of both dislocation and disclination. But it ignores the reasonable assumption made in the KTHNY theory that disclinations are tightly bound in pairs, particularly in the solid phase. Apart from a first order phase transition, there are no specific predictions to compare with experiment. Mean field theory predictions are however often incorrect in predicting the nature of phase transitions because they ignore fluctuations. Ramakrishnan also developed a mean field theory predicting a first order melting transition [171] by calculating the free energy of the density fluctuations. This is an extension of an earlier 3D version developed with Yussouff [170]. In contrast to the previous theory there is a definite prediction to compare with computer simulations, the change in entropy upon melting. And agreement with such calculations has been reported (see section 3.4.4). In addition to these theories one could mention mechanisms suggested by the work of Jo6s and Duesbery [162, 163] on the dislocation dipoles in the rare gas systems. The very low energy of small vacancy clusters suggests that in such hard core attractive
536
B. Jo6s
Ch. 55
systems, the favored defects mediating a phase transition should be the vacancies. The vacancy energy rapidly drops to zero as the lattice is expanded driving the density driven low temperature solid to solid-gas co-existence. The small dipoles, acting as composite defects which can form large "disordered" core regions, can act as nuclei for the melt. Another defect mediated melting theory is the one developed by Glaser and Clark [140, 172, 173] which diminishes the importance of topological defects in favour of geometrical defects. Inspired by their molecular dynamics simulations on systems of particles interacting with a truncated LJ potential, the WCA potential (see section 3.2.5), they came to the conclusion that a variety of defects other than disclinations and dislocations can carry the thermal energy. The merit of their theory is to focus attention on the importance of these other defects. They have however not integrated cohesively in their theory the non-topological defects with the topological defects (see section 3.2.5).
3.2.5. Simulations on defect models These are not microscopic first principles simulations of the melting as will be described in a later section 3.4. The simulations discussed in this section assume at the onset which defects can be thermally activated in a system and what their properties are, usually those predicted by elasticity theory, then simulate usually by Monte Carlo simulations the macroscopic properties of a finite lattice whose state is characterized by the density and arrangement of the defects. 3.2.5.1. Dislocation vector systems The simplest examples are the ones done by Saito [174, 175] on dislocation vector systems. The dislocations are placed on a triangular mesh. Periodic boundary conditions are imposed which modify the logarithmic interactions. The key parameter determining the nucleation rate of dislocation dipoles and their growth is the ratio Ec/J. A large core energy Ec will limit the creation of new dipoles and favour the growth of existing ones. On the other hand a small ratio will favour the proliferation of small dipoles. Using Monte Carlo simulations Saito looked at the kinetics of melting for two values of the ratio, 0.57 and 0.82. For the larger value, he observed at the transition the dislocation unbinding of the pairs and the elastic constant parameter K going through 167r, the latter determined from the dislocation correlation function. With the smaller value of the ratio E c / J Saito observed a strong discontinuity and hysteresis in the energy and in the specific heat. The proliferation of dislocations was what one would expect for a grain boundary driven transition. The indications were that this was a first order transition although no size dependence study was made to make sure. The numbers 0.57 and 0.82 should not be taken at their face value because Saito used a grid with a lattice parameter of lattice constant 2a0. This coarsening of the grid boosts the critical value required for a continuous melting behaviour because it essentially doubles the minimum energy required for the growth of a pair from J ln(1 + 1/R12) ~ J/R12 to J ln(1 + 2/R12) ~ 2J/R12. The creation energy of a new pair is itself increased, but not as substantially, from 2Ec to 2Ec + J In 2. Taking these facts into consideration the 0.57 and 0.82 values would be closer to 0.46 and 0.58 for a lattice parameter of a0 when comparison are to be made with defect studies and computer simulations, placing Lennard-Jones [165], piecewise linear force [165] and electron [167] systems in the category of large core energies. To appreciate the significance of these results one should remember that in the scenario reproduced in the Saito
w
The role of dislocations in melting
(a)
I ,
Smooth oriented
O
(b)
I
Rough oriented
T1
Fluid
i
Hexatic
Ti
537
Rough [ unoriented I T2
[
Solid
=O
]
Tm
Fig. 13. The duality relation between (a) the interface phases of the Laplacian roughening model and (b) the phases predicted by the defect-unbinding theory of two-dimensional melting [120].
simulations disordering is achieved by the creation of dislocation dipoles with elastic properties. The dissociation of the dislocations themselves into a pair of disclinations is not allowed although that requires a higher temperature. Saito's work suggests the question: What are the other possible disordering mechanisms? The importance of the inelastic properties of dislocations is just one such aspect. Large core regions can act as nuclei for local melts. We will return to this subject later.
3.2.5.2. The Laplacian roughening model.
To consider the hexatic to isotropic transition, disclinations have to be included. This is possible through the Laplacian roughening model. Nelson [176] has shown that a duality transformation of its partition function leads to a model which in the long wavelength limit has a Hamiltonian equivalent to the disclination Hamiltonian. In a dual transformation there is an inverse relationship between the temperature in the two dual models (see fig. 13). This allows a low temperature approximation in one model to give the high temperature limit in the other. In Laplacian case, to replace a Hamiltonian with long range interactions with one with short range interactions, makes the model more amenable to simulations, in particular Monte Carlo. The Laplacian roughening model describes a solid-vapour interface in terms of the well-known solid-on-solid representation in which the interface is represented by columns of height h(g') arranged here in a triangular lattice. The interaction
H - -J~--~.
6h(~') - E h(~) + h(~'+ ~J)l
(34)
J A
(where ~j are the nearest-neighbour vectors) is invariant under uniform translations of the interface by an integer and appropriate uniform tilts of the interface. The system thus has the possibility of three phases; smooth oriented, rough oriented and rough unoriented (see fig. 13). The intermediate phase is dual to the hexatic phase. Strandburg et al. [178] found in their Monte Carlo studies height-height correlation functions in support of the existence of these phases and transitions in accord with the KTHNY theory. In a later study Strandburg [177] modified the model to be able to change the core energy. She observed using finite size scaling that at lower core energy the two continuous transitions
538
B. Jo6s
Ch. 55
combined into a direct first order transition. This is in accord with the findings of Saito that changing the core energy can have significant effect on the melting transition. Another group, however, disagrees with these findings [179-183], arguing that the transition is first order. A two-step melting is, however, observed when a certain degree of rotational stiffness is added [183]. This is to explain the continuous melting transition observed in monolayers made of large molecules such as C2H4 and hexane on graphite (see section 3.3.4.3).
3.2.5.3. Polygon tiling model. This model starts from a polygon construction which Glaser and Clark [ 140, 172, 173] used to investigate in detail the defect structure through the melting transition of a monolayer with a truncated Lennard-Jones potential interaction known as the Weeks-Chandler-Anderson (WCA) potential (see eq. 42) [184]. On configurations obtained from molecular dynamics, they applied the Voronoi and the Delaunay constructions. The Voronoi construction [ 141 ], which is perhaps the most frequently used method for characterizing local topological order in 2D particle systems, assigns to each particle a neighborhood in a manner similar to the construction of the Wigner Seitz cell. In the absence of topological defects a deformed hexagon is associated with each particle since each particle has six nearest neighbours. The number of sides of the polygon associated with a given particle determines the local topology. In the Delaunay triangulation all the nearest neighbour pairs are connected by "bonds". The focus is therefore more on the bond lengths and the bond angles than in the Voronoi construction [141 ] to which the Delaunay construction is dual. Glaser and Clark observe a condensation of defects at the transition. The defect structures show a great deal of variation. The general impression of polycrystalline disorder is more apparent in the Delaunay construction than in the Voronoi construction. These observations have led Glaser and Clark to set up a defect mediated melting theory [173] where the focus is on geometrical rather than topological defects. The theories discussed earlier in this section were based on the latter. A perfect triangular lattice can be viewed in the Delaunay triangulation as a tiling of triangles. Any departure from this indicates the presence of defects. In an attempt to identify defected areas from the configurations obtained by molecular dynamic simulations they chose to dilute the nearest bond network by removing bonds that are significantly longer than the most probable nearest neighbour separation. In practice this is done by removing bonds opposite bond angles exceeding 75 ~ (equivalent to removing bonds more than ~ 20% longer than the most probable nearest neighbour distance). The resulting patterns involve in addition to triangles, squares and polygons of more than four sides. These latter reflect regions of low density. A vacancy would be a polygon with six or more sides. They observed that most of the surface increase upon melting is due to these geometrical voids. Upon melting they also find a high level of correlation among the defects with chains, or ladder-like structures of squares, being common (see fig. 14). However, most aggregates were more complex. To characterize defect correlations they classified the structures in the WCA system according to their charge, proportional to the degree of deformation from perfect tiling at a vertex, with the unit of charge being a 60 ~ angle deformation. Glaser and Clark then used this classification scheme of defect vertices as a basis for a statistical mechanics model of melting [173]. A creation energy for nontriangular
w
The role of dislocations in melting
539
Fig. 15. Liquid phase configuration resulting from Monte Carlo simulation of a specific version of the polygon tiling model of Glaser and Clark [173].
540
B. Jo6s
Ch. 55
tiles, interactions between the defect tiles and energy for the creation of the tiling faults were used as input into Monte Carlo simulations. As can be seen in fig. 15, long range elastic interactions such as occur between topological defects, are not well accounted for, leading to unphysical configurations. Some additional feature is required in the model. It has at the moment the merit of focusing the attention on the importance of defects, other than the topological defects, in the structure of a liquid.
3.3. Various realizations of 2D systems The predictions contained in the KTHNY theory of melting created a great deal of interest in the nature of the melting transition in 2D systems. We just saw the theoretical investigations it has initiated. Now we will look at actual physical systems where a tremendous amount of ingenuity has been devised to create situations as close as possible to ideal 2D behaviour. In this section we will describe all aspects of these systems whether theoretical or experimental except for computer simulations. In these latter types of study the issues and the results are less system specific and raise questions of their own. For these reasons and for the sheer mass of the work done they will be treated separately in section 3.4. There has been a recent exhaustive review by Strandburg [119] and we will refer to it for a more detailed discussion for earlier work. We will focus on the more recent studies and on the role of dislocations. A 2D system is an ideal concept. In reality there are additional features or degrees of freedom which complicate or enrich the disordering transition. The experimental systems studied fall mainly in the following classes: (i) layered liquid crystal phases, (ii) electrons on the surface of liquid helium, (iii) lattices of microscopic objects large enough to be studied by optical imaging techniques such as monodisperse colloidal spheres in suspension, and (iv) atomic or molecular systems (mainly adsorbed such as rare gas monolayers on graphite) probed by a variety of methods including X-ray synchrotron, neutron scattering and heat capacity measurements. The interest in the first system is mainly in providing an example where two dimensional order provides great persistence in BOO as the lattice disorders. The electron system shows properties in accord with the KTHNY theory but no direct structural information on the nature of the liquid phase near melting or the thermodynamics of the transition. Those are not lacking in our two main classes of systems although they are of very different nature. In the lattices of microscopic objects we have direct pictures, but no thermodynamic information. The observed configurations can be interpreted as evidence of the existence of a hexatic phase. In the adsorbed monolayers, both thermodynamic and structure factor information is available. A great variety of situations has been created mainly as a result of misfit variations with the substrate, and the melting behaviour is very rich, some favoring KTHNY but mostly not.
3.3.1. Smectic liquid crystals 3.3.1.1. The structure of liquid crystals. Liquid-crystal materials are composed of anisotropic molecules with a complicated chemical make-up. The KTHNY has had a significant impact on the study of a subset of the thermotropic liquid crystals (thermotropic
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The role of dislocations in melting
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Fig. 17. Schematic illustration of the molecular order in various smectic liquid crystal phases [195]. meaning that the phase changes are triggered by temperature). These are usually constituted of rod-like molecules. One example which will be discussed below is racemic 4-(2-methylbutyl)phenyl 41-(octyloxy)-(1,1')-biphenyl-4-carboxylate which goes by the shortened appellation 8OSI (see fig. 16, Brock et al. [ 148]). We will refer to them by their shorthand names. Typical sizes of the molecules are 25 • 5 x 5 ~3. Liquid crystals form many phases which can be grouped into three categories: (i) nematics which have only a long range orientational ordering along a direction called the director r/, no layering, (ii) cholesterics, layered and within each layer a dominant orientational order which varies from layer to layer, and (iii) smectics, layered perpendicularly to the director (normal smectics) or with the perpendicular to the layer plane making a fixed angle to the director (tilted smectics). It is this last category (iii) which is mainly of interest to us. These are the systems where well defined phases with BOO have been observed. Figure 17 shows the various smectic phases which have been observed and the nature of their ordering. By 1978 at least eleven of the phases had been observed but there was no consensus on what differentiated these phases and specifically what different sorts of order these phases exhibited microscopically.
542
B. Jo6s
Ch. 55
3.3.1.2. Phases in the smectics. Smectic liquid crystals are distinguished by having an intermediate degree of positional order in addition to molecular orientational order and, in some cases, BOO. The term "smectic" derives from the Greek cr#r/7#c~ meaning "soap", since the smectic phases tend to have mechanical properties similar to those of layered phases of soap. Smectics have historically been identified by the textures they exhibit under a polarizing microscope and by miscibility studies with known phases [ 185]. The simplest smectic phase is the smectic A (SmA) phase. This phase has traditionally been described as a system that is a solid in the direction along the director and a fluid normal to the director, or equivalently, as stacked two-dimensional fluids. It is more properly described as a one-dimensional density wave in a three dimensional fluid with a density wave along the nematic director. In fact in real SmA materials, X-ray scattering shows that the higher spatial harmonics of the density wave are surprisingly weak. Thus the smectic planes should be interpreted, not as lattice planes, but as planes of a certain phase of a sinusoidal density wave. The smectic C (SmC) phase is similar except that the density wave vector makes a non-zero angle with the director; this angle is called the tilt and it functions as the order parameter for a SmA-SmC phase transition. The remaining smectic phases all possess more order than the SmA and SmC phases. Soon after the development of the melting theory by Halperin and Nelson [2, 3], Birgeneau and Litster [186] applied it to the 3D liquid crystal phases. They suggested that some of the more exotic phases of smectic liquid crystals might actually be 3D realizations of a BO ordered phase. In this scenario, each smectic layer is an independent, 2D BO ordered system: the HexB is the BO ordered version of the SmA and the SmI phase the BO ordered version of the SmC phase. A consequence of the fluctuation dissipation theorem is that algebraic decay of a correlation function implies an infinite susceptibility to an ordering field. Thus, an infinitesimal coupling between neighbouring hexatic layers should drive the quasi long-range hexatic BO ordering into a truly long-range BOO but maintain the short range PO in a stacked hexatic system. Instead of an algebraic decay in C6(~') this correlation function goes to a constant ~: 0. This conversion from algebraic decay to true long range order is an essential difference between 2D and 3D hexatics. There are other BO ordered hexatic tilted phases beside SmI, SmF and SmL which as fig. 17 shows differ by the direction in which the projection of the director points with respect to the nearest neighbours. Initially there was one SmB phase until it was discovered [188] that there are actually two, HexB mentioned above and a truly 3D crystal phase CryB. Their equivalents are found in the tilted phases CryJ, CryG and CryM. Further crystalline phases are arrived at by including herringbone order. For a recent detailed discussion see the book by Pershan [ 189]. As one can well imagine, a multitude of possible phase transitions can be observed and the defect structures in these systems would form a study in itself (see for instance [ 190-192]). The studies related to melting or disordering transitions have focused on the crystalline to the liquid phases in the smectics with in particular the goal of determining whether the KTHNY theory was applicable to these systems. The transitions corresponding to the 2D sequence solid-hexatic-liquid would be CryB-HexB-SmA for normal smectics or CryJ-SmI-SmC for an example of tilted smectics. The latter sequence has the added complication of the tilt. Nelson and Halperin [193] generalized their 2D melting theory to systems consisting of rigid rod-like molecules. The additional
w
The role of dislocations in melting
543
degree of freedom, the tilt, when coupled to the other degrees of freedom gives rise to four possible fluid phases: isotropic liquid, hexatic, locked tilted hexatic and unlocked tilted hexatic. One important result from this theory is that the molecular tilt degree of freedom is coupled to the BO order and induces finite BO order in the tilted phases. Thus there is no true thermodynamic tilted liquid phase unless the coupling constant between the tilt and BO degrees of freedom is smaller than a critical value. In short tilt favours BO order. Subsequently Bruinsma and Nelson [194] solved a mean field model with a coupling term between the BOO and the molecular tilt angle for bulk smectics. Their conclusion is that molecular tilt induces BOO through the coupling term. This suggests further that the SmC is not a true thermodynamic phase and that there is no real transition between SmC and the tilted hexatic phases SmI or SmE The SmC phase must exhibit some residual BOO. This is equivalent to the paramagnetic-ferromagnetic transition under a small applied magnetic field. The distinctive observed texture of the SmC phase as observed under optical microscopy indicates however that the coupling is so small that the SmC phase is treated as a separate phase (for discussion see [195]). 3.3.1.3. Structural studies of BOO. The structural studies of BOO have been done mainly by X-ray synchrotron source (see review by Brock et al. [148]). The first observations were made in the untilted phases. The very first attempt to observe the hexatic phase in a SmB phase failed [187] but revealed in the process a new crystalline phase now called CryB. The hexatic phase was later seen in freely suspended films by Pindak et al. (1981) [188] in a phase now called HexB. These experiments showed that the SmB is actually two phases, one is a bulk crystal phase CryB and the other the layered hexatic phase HexB. The samples in this 1981 experiment were multi-domain and no quantitative information about the BOO was obtained. A lot more success was achieved with the tilted phases which is not surprising in view of what has been said above; the tilt favours and stabilizes the BOO in large domains. The first studies of these phases were made by Dierker, Pindak and Meyer (1986) [197] in a very thin film of the liquid crystal compound commonly labelled 8SI (a compound very similar to 80SI). They found that a common defect in the SmC phase is a +27r point disclination (or winding number + 1). In it the director follows a circumferential path around the defect core. Upon cooling into the SmI phase, the large strain caused by the coupling of the tilt and the BO is released into five radial arms with 60 ~ disclinations at the ends (see fig. 18). These separate regions of relatively uniform director orientation. The length of the arms appear to diverge at the long range hexatic to crystalline SmJ (or CryJ) phase transition in a manner predicted by KTHNY: this reflects the similar behaviour in the Frank constant KA which characterizes bond angle rigidity, see eqs (28)-(30). That this happens indicates that tilt coupling is weak. In the cooling process the arms develop kinks alternately to + and - 6 0 ~ directions. An example of this behaviour is shown in fig. 19. Eventually in regions of continuously varying orientation, the dislocations coalesce at sharp grain boundaries separating uniform polygonized areas. Only large angle grain boundaries are observed in fully polygonized 2D crystals. The development of the fully polygonized crystal takes several days to transpire, giving an indication of the mobility of the dislocations. At the CryJ-CryG transition originally straight grain boundaries develop a 60 ~ zigzag modulation which grows in amplitude with decreasing temperature.
544
B. Jo6s
Ch. 55
Fig. 18. (a) The five armed star defect and (b) calculated pattern. The arrows represent the local orientation of the sixfold symmetric hexatic axes [197, 148].
Fig. 19. The observed 60~ kinks in the radial disclination arms [197]. A more exhaustive study of the hexatic phase itself was made possible when it was discovered that with the application of a small magnetic field, the tilt direction of the SmC phase could be oriented at the S m A - S m C phase transition and that as the liquid crystal cools from the SmC to the SmI phase a single domain phase could be achieved. The first measurement of BOO was made by Brock et al. (1986) [152] on such a tilted hexatic phase. There are two limitations in using a tilted phase. The first, mentioned above, is that the linear coupling with the tilt will destroy the continuous transition, and the second is that the two-fold symmetry of the tilt field would remove the six-
w
The role of dislocations in melting
545
fold symmetry of the hexatic axes. Fortunately these by themselves give measures of the importance of the coupling with the tilt. The measurements by Brock et al. (1986) were made on thick films of 8OSI. Their structure factor studies of the BOO parameter reveals in this system the absence of a phase transition. The SmC phase shows remnants of BOO which continuously grows into the SmI phase, a bulk hexatic phase. The coupling between tilt and BOO has in this case destroyed the sharp phase boundary. Repeating the experiments for thin films made of 22 layers [ 153] puts one in the two-dimensional limit for this material. There is a definite transition from a fluid phase SmC to a twodimensional hexatic phase SmI and finally to a crystalline solid phase SmJ. The BOO parameters seem to have the behaviour expected from a bulk to 2D evolution [150] (see section 3.3.2.6). In the 2D limit the increase in the amplitude of the fluctuations leads to algebraic decay in the correlation functions.
3.3.1.4. Thermodynamic studies of BO0 transitions. In contrast to-the structural studies the focus in the thermodynamic studies has been on the normal (untilted) phases. In bulk normal smectics the SmA-HexB transition has been found to be continuous in such compounds as 650BC and 370BC. The exponent ~ for the divergence of the heat capacity is however very different from a 3D XY model. Values of c~ -- 0.6 [198, 199, 195] are found instead of the expected -0.007 [200]. A conflicting measurement [201] argues for a discontinuous transition but the large exponent (c~/> 0.58) is a common feature. The origin of this anomalously large exponent is not clear. Is an additional ordering such as herringbone responsible as has been suggested [203]? Other compounds, such as PHOAB [202], have the first order SmA-HexB transition. Mixtures of PHOAB with 3(10)OBC, a compound exhibiting a continuous transition, show the same anomalies in c~ for concentrations of PHOAB (< 20% wt). For higher concentrations (> 40% wt) the hexatic phase is stable over a large temperature range [204] and the transition is first order. This argues against the presence of herringbone fluctuations near the transition which should be continuous. In bulk tilted smectics asymmetric heat capacity peaks have been found [205] in the SmC-SmI transition and the situation is, not surprisingly, even more complicated. In a beautiful set of experiments Geer et al. (1992) [206] and Stoebe et al. (1992) [207] made detailed heat capacity measurements on a series of free standing films ranging from thicknesses of 300 layers down to 2 layers to see the transition to 2D behaviour. The compound used 75OBC exhibits the following sequence of transitions: I (81~ SmA ( 6 5 ~ HexB (59~ - C r y E . The HexB-CryE transition involves a change in symmetry and is first order (see fig. 20). The SmA-HexB transitions appear continuous, but the heat capacity profiles as one goes to the very thin films does converge to the expected form for a KT transition. It has a sharp peak with c~ = 0.28, as for a three state Potts model [324], the symmetry of the 2D herringbone order, the model for spins on a lattice with three possible orientations, and also, as we shall see, for the most widely studied phase of adsorbed rare gases on graphite (see section 3.3.4) when one adsorption site out of three is occupied. The above exponent 0.28 was observed for He 4 on graphite. Equally intriguing is that the heat capacity profile is identical to the variation of dp/dT, where p is the in-plane molecular density (see fig. 21). The two observations suggest some additional ordering at the transition than the type involved in a KT transition.
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Fig. 20. Heat capacity Cp/A per unit area against temperature of a ten-layer free-standing 750BC film. The relevant phase transitions are indicated in this figure [205, 193]. This is only a very small sample of the work which has been done on phase transitions in smectics (for a recent review and details we refer to Huang and Stoebe [ 195] or Huang [196]). The focus was on transitions from the SmA or SmC phases to their corresponding hexatic phases. In summary, the ideas in the KTHNY theory and their generalizations to anisotropic molecules has greatly helped unravel the basic classifications of smectic phases and hint at the role of dislocations and disclinations in the ordering process but a lot more is going on here than those theories can explain. Heat capacity measurements reveal that the liquid to hexatic transitions in the smectics are not exactly XY. These differences must be caused by additional degrees of freedom, namely interactions between the layers and coupling between BOO and tilt, aspects which have not been explored theoretically sufficiently with respect to their consequences on the heat capacity. The richness in the microscopic structure of the defects (a reference to it has already been made in the previous section) is also expected to play a role. The study of these defects is a fascinating subject in itself (see for instance [190-192]). Hints at the complex role played by defects can be found in such works as by Benattar et al. (1979) [209] and Sirotta et al. [210] in which it is shown that the diffusive peak in the SmF phase and the Bragg peaks in the CryG phase have been observed to be coincident in high resolution X-ray diffraction studies. This indicates that the CryGSmF transition does not change the short range structure and that the SmF phase is a faulted version of the CryG phase. Furthermore, chemical changes can also play a role in such transitions [211].
The role of dislocations in melting
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More theoretical understanding is required. What complicates the picture is the extra degrees of freedom act differently in each class of system enriching the observed behaviour. But overall the transition from SmA to HexB exhibits the expected continuous behaviour of a liquid to hexatic phase transition in the limit of thin films.
3.3.2. Electrons on the surface of liquid helium This 2D system is realized by setting up an electric field perpendicularly to the surface of liquid helium with a capacitor immersed in the liquid. The perpendicular motion of the electrons is frozen into an isolated quantum level while the parallel motion is classical in the sense that T >> Tfermi because of the very low density of electrons. The interaction is pure Coulomb e2/r up to several hundred interparticle spacings. A liquid-solid transition is induced as the temperature is lowered. This was first observed by Grimes and Adams (1979) [212]. Because the electron Boltzmann factor is given by e x p ( - V / ( k B T ) ) , where V is a sum of pair potentials, the phase diagram can be expressed in terms of one combined surface density Ns and temperature T parameter s - ~x/-~--~se2/(kBT). For/-' < 1 the kinetic energy dominates and the system behaves like an electron gas. At intermediate densities 1 < F < 100 the electron motions become highly correlated or liquid-like. For large values of /-' a transition to a solid phase is expected as the Coulomb interactions dominate. No direct structural information is available showing a dislocation unbinding mechanism. The predictions on BOO have not been verified because they are not attainable. The location of the melting point has been used as the key test of the KT theory. Driving the electron lattice with an a.c. voltage, mobility [213] and power absorption [214] measurements were used to identify a melting point in accord with the KT stability criterion (eq. 23), and estimates from
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3.3.3. Monolayers of micron and sub-micron size objects 3.3.3.1. The systems. In contrast to the previously discussed system, the ones in this section can be viewed by video imaging with visible light so that one can obtain direct visual evidence of the lattice defects and their dynamics, and monitor BOO in real space with no limits on time scales. The main systems in question are suspensions of monodisperse (all of the same size) colloidal spherical particles of submicron size immersed in a solvent and usually confined between glass plates (see fig. 22). Such particles have been used for a while as model systems in condensed matter physics [218]. The ones most used in 2D experiments are polystyrene latex spheres which range from ,-~ 0.1-10 ~tm in diameter (for a review of the experiments see Murray [219]). The spheres are formed by emulsion polymerization techniques [220] in which polar species, most often carboxylate or sulphate groups, are appended to the polymer ends on the surface of each sphere. When immersed into a polar solvent such as water, these polar groups dissociate, leaving one electron surface charge and one oppositely charged screening ion in solution. The surface charge density can be very h i g h - as large as one electronic charge per 100 /~2 on the sphere surface. For high surface charge density spheres the total surface charge for a 0.1 #m diameter sphere can be typically 103 to 10a electronic charges. For a diameter range between 0.1 and 1 gm, the colloidal spheres in a solvent suspension exhibit Brownian motion due to collisions with the suspension medium. The collision time between particles is ~ 30 ms for a 0.3 ~tm diameter sphere [221]. The space between spheres is roughly 1 lxm. The Brownian motion permits thermodynamic equilibrium and the definition of a temperature. Both space and times are very amenable to observation. These are not the only advantages of these systems. It is possible to
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The role of dislocations in melting
549
create a perfectly smooth substrate and to confine the spheres to a plane, all this using electrostatic forces as we shall discuss below. The interaction between spheres is basically a screened Coulomb potential due to the excess charge on each sphere and the opposite charges in the solvent but there are other contributions, van der Waals, hard sphere and many body hydrodynamic [222]. In spite of the expected complexity of the interactions, molecular dynamics simulations using Yukawa (or Debye-Htickel) potentials with an effective charge much smaller than the actual titratable charge [223] reproduce the essential features of the 3D phase diagram [224]. In addition, a different kind of experiment can be made [225] when the solvent is replaced by a a ferrofluid made of much smaller particles (100-1000 ~,). The latex spheres then create a lattice of magnetic dipole "holes" [226]. Some other systems are discussed in section 3.3.3.4 which have similar advantages as the colloidal particles. 3.3.3.2. Experiments on colloidal spheres between flat plates in wedge geometry. Two groups have mapped out the complex phase diagram of these colloidal spheres between two flat smooth and repulsive glass plates [227-230]. The trick used is to make a small angle wedge between the glass plates in order to produce a gradual controlled density change along the wedge and to achieve simultaneous equilibration of the two-dimensional fluid, two-dimensional crystal, and a three dimensional crystalline reservoir directly in contact with each other (see fig. 22). One is then able to sample equilibrated phases of colloid at various average densities. As the system is in direct contact with both thermal and particle reservoir, constant chemical potential and temperature are maintained along the wedge in equilibrium. In 1987, Murray and Van Winkle performed a series of experiments on wedges with angles more than an order of magnitude smaller than the ones previously used and which allowed them a detailed study of the 2D melting transition [231-233,219]. Murray and Van Winkle used polystyrene sulphate spheres of d = 0.305 ~tm diameter with a a titratable surface charge of ~ 2 x 104 electrons. The average separation between spheres near the 2D melting transition was 2.5d. The wedge angle producing the density gradient was ~ 5 x 10 -4 rad. The gradient in density was 1% per image (of size ~ 30 sphere separations). In the other direction the wedge angle was 5 times smaller. The major disadvantage of a wedge geometry is the imposed density gradient in the direction of the wedge, which could smear out a density jump associated with the first order transition. The density gradient also favours the creation of dislocations. A 1% gradient produces on average a density of about 3 x 10 -4 dislocations per image, with a Burgers vector perpendicular to the gradient. In addition the two dimensional crystal nearby serves as an orientation boundary condition for a possible hexatic phase in direct contact. The main advantage is equilibration in a statistical sense of all phases simultaneously. In the experiments a charge-coupled device camera was used to image a region of size 25 x 38 ~tm2 with between 1000-2000 particles. Digital imaging was used to locate the particles. From the accumulated data, accurate determinations of 2D sphere density, instantaneous and time-resolved pair correlation functions, structure factors and orientational correlation functions were done. Figure 23 (this and following figures taken from [231]) gives the density variation vs. position along the wedge identifying the locations of the change in PO and BOO. Figure 24 (p. 158 of [219]) shows the change
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with density in the structure factor S(k), the pair distribution function g(r) and the BO correlation function g6(r). With increasing density n one sees first, for n > nl a large change in BOO with a slight change in PO, then, as n > n2, a large increase in PO while g6(r) becomes fiat. At that point the first peak in S(k) reaches 5, the limit at 2D freezing predicted by Ramakrishnan [171], giving a clear indication of the freezing transition. The variation in the correlation lengths is given in fig. 25. The observed behaviour of the translational and orientational correlation functions is remarkably similar to the predictions of the KTHNY theory with a definite two-stage melting or freezing scenario. The limiting ~76 obtained from power law fits to g6(r) was consistent with the predicted 1/4 value. But the imposed density gradient and accompanying lack of density resolution made the determination of the exponents imprecise. It should be noted that the intermediate hexatic phase occurs along the wedge over a distance ~ 4~, the correlation length making it hard to confidently characterize this phase. Later, time resolved correlation functions were calculated [232, 219]. They do show quite nicely how BOO decays much more slowly than PO in the intermediate hexatic phase. To complete the picture a Voronoi polyhedron analysis and the dual representation of Delaunay triangulations (see section 3.2.5.3) were used to study defect statistics and configurations in the three phases. Figures 26 show the instantaneous positions of particle centers for various average densities very close to melting.
The role of dislocations in melting
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In the Delaunay triangulation of bonds shaded areas mark regions with coordination numbers different from six. These are regions of disclinations. A dislocation is a nearest neighbour pair of 5 and 7 coordinated disclinations. These can be seen in fig. 26(a), the crystal phase close to melting. At n2, the melting point, the concentration of five- and seven-fold disclinations (an upper bound for dislocations) is ~ 5% and the concentration of vacancies is ~ 1%. Dislocation motion has been observed at that point. In the intermediate hexatic phase the defect structure and motion is very similar to but more complex and clustered than in the crystal (figs 26 (b) and (c)). As the density is decreased into the intermediate region below n2 there is a rapid increase in fiveand seven-fold disclinations followed by a smooth rise to a concentration of ~ 8% at nl. The defect clusters have become more numerous and have a wide distribution of separation. They tend to form elongated strings which however do not close up into grain boundary loops. Most dislocations appear free. In the high density fluid defect strings completely encircle the ordered regions. When this happens BOO is completely lost. Very close to freezing the 2D fluid is over 80% six-fold coordinated. This has also been observed in simulations (see section 3.4 and [234, 235]). This is basically a
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consequence of dimensionality. As mentioned earlier, there is a prevalence of BOO in 2D systems where the average number of neighbours in the liquid and the solid are the same, contrary to 3D systems [114]. There is no evidence of a melting process driven by grain boundary melting. There are loosely bound disclination pairs and unbound disclinations which are not included in Saito's model [174]. It is hard to see a two-phase coexistence either. In this Coulombic repulsive system, at the melting point the defect clusters are about l a - 2 a in size, they then become branched and by the density at which both BOO and PO is lost the clusters have percolated across the images. More recently Murray, Sprenger and Wenk [221 ] performed another set of experiments with spheres of the same size but five times the amount of charge. With stronger repulsive forces melting occurred at a third of the earlier densities. The purpose was to compare 3D and 2D melting behaviours, in particular to show the absence of two-phase separation or of grain-boundary like behaviour in the 2D system. Phase separation on the contrary is quite clearly seen in the 3D systems (see fig 27, also figs in [221]). The greater extent of BOO in 2D is also quite evident in this study (see fig. 28).
w
The role of dislocations in melting
553
Fig. 26. Delaunay triangulations (with the nearest neighborhoods of disclinations in the lattice darkened) for four densities ranging from crystalline to liquid phases: (a) crystal at density n -- 1.03nl, (b) hexatic at density n = nl + 0.6(n2 - nj).
554
B. Jo6s
Ch. 55
Fig. 26 (continued). Delaunay triangulations: (c) hexatic at density n = nl + 0.33(n2 - h i ) , and (d) liquid at density n = 0.98nl. Dislocations are tight 5-7 disclination pairs. Dislocation pairs are quadruplets of 5-7.7-5 disclinations and vacancies are usually eightfold disclinations with two fivefold neighbours [219].
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The role of dislocations in melting
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Fig. 27. Defect maps comparing 3D and 2D melting behaviour in experiment of Murray, Sprenger and Wenk [221]. Sphere centers are represented as vertices. Defects or non-sixfold coordinated neighborhoods are shaded. Left: three dimensional run. Density from top to bottom: crystal n -- 0.0599 -- nc; and intermediate region, n = 0.98nc, 0.965ne and 0.953nc. Right: two dimensional run. Density from top to bottom: crystal, n = 0.0606 - nc, and intermediate region, n = 0.974nc and 0.93nc.
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Fig. 28. Correlation functions for the Murray, Sprenger and Wenk experiment [221] shown in fig. 27. Top panels: g(r) and bottom panels: 9n (r). The arrows on each curve mark the translational correlation length for that density. Left: three dimensional run. The densities for the curves are as follows: (a) fluid at n -- 0.0560, (b) fluid at n - 0.0561, (c) intermediate region at n - 0.0571, (d) intermediate region at n -- 0.0578, (e) intermediate region at n = 0.0587, and (f) crystal at n - 0.0599. Right: two-dimensional run. The densities for the curves are as follows: (a) fluid at n = 0.519, (b) fluid at n -- 0.535, (c) intermediate region at n - 0.0563, (d) intermediate region at n = 0.0569, (e) intermediate region at n = 0.0590, and (f) crystal at n = 0.0606.
3.3.3.3. Expansions with uniform density.
Two other p r o m i s i n g g e o m e t r i e s h a v e b e e n used in studies with colloidal m o n o d i s p e r s e spheres: m o n o l a y e r s on the surface of water by A r m s t r o n g , M o k l e r and O ' S u l l i v a n [236] and freely e x p a n d i n g m o n o l a y e r s b e t w e e n flat parallel plates by Tang and the above authors [237]. In the first of these studies [236] m o n o l a y e r s of p o l y s t y r e n e m i c r o s p h e r e s are trapped by surface tension on the surface of water in a L a n g m u i r trough. T h e t r o u g h was m o u n t e d on an isolation box and had barriers used to sweep the surface clean and c o m p r e s s the m o n o l a y e r . T h e entire apparatus is installed on a m i c r o s c o p e . T h e spheres at the surface exhibit strong optical contrast so they could easily be i m a g e d due to the very different ratios of the index of refraction, 1.6 b e t w e e n those of p o l y s t y r e n e and air, c o m p a r e d to 1.2 b e t w e e n those of p o l y s t y r e n e and water. Being able to c o m p r e s s the m o n o l a y e r gives
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The role of dislocations in melting
557
it an advantage over the wedge geometry, but this is offset by the difficulty of keeping impurities out of the trough. Contamination of the surface produces aggregate particles and impurity induced free dislocations. Two microsphere sizes were considered, 1.01 ~m and 2.88 l.tm. The smaller sized spheres showed a proliferation of defects even in the solid phase and have a first order melting transition in which the impurity induced defects coalesce into a close network of grain boundaries completely disordering the system. The larger sphere system had a more typical two-stage melting sequence with the application of the pressure. The authors also comment on the tendency of defects to cluster and of multi-dislocation interactions and pairing. The correlation lengths ~ 10a-20a and the multiparticle defects raise here also the role of contaminants. The second experiment [237] with expansion of the monolayer of 1.01 ~tm particles between flat parallel plates circumvents the problem of contaminants, which was rather difficult to avoid in an exposed surface. By varying the separation of the plates a density driven melting could be produced in a system which still had uniform density. A twostage melting consistent with a KTHNY melting is observed. But the defect kinetics was nothing like a dislocation unbinding. In the intermediate phase there was spontaneous creation of clusters consisting of dislocations and dislocation pairs, or small dislocation loops that are characteristic of grain boundaries. Vacancies were also generated by the annihilation of two off-one-row dislocations with opposite Burgers vector. The relationship between dislocations and vacancies is discussed in [165]. These local vacancies tend to act as nuclei for the melt. 3.3.3.4. Magnetic or electric holes, bubbles, and flux lines. Helgesen and Skjeltorp [225] have made experiments on the lattice of magnetic dipole "holes" [226] obtained by suspending the spheres in a ferrofluid made of much smaller particles (1001000 A). In their experiment an in-plane rotating magnetic field HII binds the spheres in rotating pairs whereas a perpendicular field Ha_ produces a repulsive force between the pairs which form a triangular lattice. With only the rotating HII the pair-pair interaction averaged over time will give a net attractive potential between the pairs. On the other hand with only Ha_ the force between the pairs will be repulsive. The competition between the two fields creates a minimum in the pair-pair interaction. The equilibrium lattice parameter is determined by the ratio Ha_/HII; a ratio of ~ 0.3 would correspond to a soft core system and a ratio of ~ 1.0 to a hard core system. The solid-liquid transition is driven by the frequency of the rotating field. Increasing that frequency is equivalent to heating the pairs. A microscope connected to a video camera is used to observe the spheres. For a given value of the frequency the position of each sphere can be recorded as a function of time and analysis of positional and orientational correlations performed and diffraction patterns constructed as for the suspensions in polar solutions. Two-stage melting is observed but grain boundary melting is proned. There is also an interesting run showing melting nucleated at a grain boundary (see fig. 29). In a similar experiment by Kusner et al. [238] a 3.75 M.Hz oscillating electric field is applied to a monolayer of 1.6 ~m spheres confined between glass plates. The electric field polarizes the water but not the polystyrene spheres which act as "electric holes". Two-stage melting is proned on the basis of the variation of the correlation functions. No evidence of grain boundaries is found. The dislocations in the hexatic phase close to
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the solid phase show a number of isolated dislocations. Even well into the intermediate phase there are no elongated strings as have been observed by Murray et al. In an earlier experiment by the same group [239], a small density (1%) of larger spheres was added. The larger spheres act as pinning sites for the dislocations. The last experiment described in this section on macroscopic objects is about magnetic bubbles in a garnet, which can also be viewed by optical microscopy. In this experiment Seshadri and Westervelt [240] used a bismuth substituted iron garnet film with material composition: Fe3.91Y1.20Bil.09Gdo.95Gao.76Tmo.09012. The film had a strong growth induced magnetic anisotropy with easy axis perpendicular to the film plane. The anisotropy supports domains with magnetization parallel or antiparallel to the applied field HB. The domain walls are narrow (N 0.1 ~tm) compared to domain sizes (~ 10 ~tm). There is an interval of HB for which small domains called bubbles are energetically favorable. The bubbles move without substantial deformation in the lattice. Bubble size and density is controlled by varying the field. The techniques have been extensively studied since these bubbles are of high technological importance [241,242]. Arrays of ,.., 12000 bubbles with densities ~ 4000 mm -2 were produced. The bubbles form a lattice with a dipolar repulsive force 1/r 3. They can be viewed by optical microscopy using the Faraday rotation of transmitted polarized light. Questions about equilibration time make it hard to argue about defect configuration. A Lindemann criterion has been found to describe well the melting point: melting occurs when the root mean square of the difference between the displacements of adjacent bubbles equals ,.., 10% of the average spacing between bubbles. A related system which will only be mentioned here is the superconducting flux lattice. This is a lattice of tiny current vortices, induced in the mixed state of a sample in the presence of an applied magnetic field. They are sometimes called Abrikosov vortex lattices after the name of their inventor. There have been early suggestions that this lattice can undergo a solid-to-liquid transition, i.e., melt. A continuous transition has been been observed in the melting of the vortex glass in crystal samples with many defects [243]. It has been explained as the transition between a vortex liquid and a vortex glass [244, 245]. The transition has been shown both theoretically [245] and experimentally [243, 246, 247] to become sharper as the amount of disorder decreases. In clean samples the transition is first order [248, 249] according to theoretical suggestions [250]. In such samples the vortices can form a 2D vortex glass with hexatic order [251,252], and its melting is thought to involve dislocations [253]. Overall these experiments give similar results showing a two-stage melting process. The beauty of these experiments is that the kinetics of the melting can be followed in real time. Disagreements lie in the details of the kinetics of melting or the arrangements of the defects in the hexatic like phase near melting. In the latest measurements by Kusner et al. [238] defect configurations and BOO correlations follow KTHNY behaviour. Murray and co-workers see patterns not in disagreement with the KTHNY theory albeit noticeable clustering of the defects which is not a feature of the 3.3.3.5. Discussion.
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The role of dislocations in melting
561
theory [219]. The other groups [236, 237, 225] argue that the kinetics of the defects do not fit the dislocation unbinding mechanism and lean towards a grain boundary mechanism. The difference in perspective is mostly a matter of the degree of clustering. But from a thermodynamic point of view, as seen from Chui's theory [168], this can make the difference between a first and a second order transition. Grain boundary melting is always first order although weakly so for large core energies. In the Murray and van Winkle [231] experiment the core energy lies between values for which the strong first order and weakly first order transition is predicted. The core energy argument is however not a strong one. Other properties of dislocations such as mobility are equally important [165]. It has been argued [219] that when the systems are not well equilibrated grain boundary melting is favored. This may explain the different observed behaviours. It should be noted that these experiments do not provide the possibility of thermodynamic measurements. Interestingly whatever arguments are put forward to explain the different observations the question remains that all these systems, except for the small spheres on the contaminated surface, exhibit from a structural standpoint two-stage melting, with persistence of BOO in the intermediate hexatic phase, whether well equilibrated or not. 3.3.4. Physisorbed and intercalated monolayers This class of systems has been the most studied in relation to two-dimensional melting. They have also been the most controversial. The KTHNY theory was applied to it very soon after the Halperin and Nelson papers came out in 1978-1979 [2]. By then experimental techniques to study surface structures had made considerable progress and the groundwork was laid for tackling the 2D melting problem. In May 1980, a major conference on "Ordering in Two Dimensions"[115] reported early results. For over a decade following these promising beginnings, the subject was intensely studied by a variety of techniques; synchrotron X-ray diffraction, neutron scattering, heat capacity measurements, to mention the most important. The systems of choice were the rare gas monolayers adsorbed on graphite (Gr). But other systems were also considered, in particular some simple molecules on the same substrate, some intercalates also in graphite, and rare gases on metallic substrates such as Ag. We will focus on monolayers adsorbed on graphite, rare gases and some molecular systems. Additional references to other systems will be made at the end of the section. 3.3.4.1. Rare gases and graphite. The rare gas monolayers on graphite were attractive because they formed nearly ideal model systems (for a recent review see [254]). Rare gases are chemically inert and interact weakly through a van der Waals type of interaction which is accurately known [255-257]. The popularity of graphite is due to the marked mechanical stability of its basal plane and its chemical inertness. Little surface preparation is required compared to metallic surfaces. This is important in view of the large areas required for thermodynamic and scattering measurements. A common form of graphite which has been used in experiments is the exfoliated crystal. The brutal process of exfoliation leaves large basal plane areas intact. The estimated coherence length ranges from ~ 200-400 A for the Papyex and Grafoil to several thousand for the single crystal exfoliated UCAR ZYX graphite [258] (see fig. 30). More recently experiments on single crystal surfaces have been made with coherence lengths of at least
562
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L
P(O)
/ Fig. 30. Schematic diagram of a Grafoil sheet. L indicates the typical coherence length of the adsorbed layers while P(O) describes the distribution arising from the non-parallelism of the substrate surface [117]. 10000 ,~ [259]. This increase in coherence length comes at the expense of the specific area of adsorption. It is as high as 30 m2/g for Grafoil down to 2.8 m2/g for the UCAR ZYX samples, and still smaller in the single crystal. The area of the single crystal in the Specht et al. experiments [259] was about 2 • 3 mm 2 orders of magnitude smaller than a UCAR ZYX sample. The exfoliated graphites greatly stimulated the study of physisorbed systems because their high specific area provided a large number of surface atoms ( ~ 1021 g - l , for the case of Grafoil). An added advantage for thermodynamic studies is that overlayer excitations are generally much lower in frequency than those of the substrate. Also, graphite is nearly transparent to X-rays and neutrons. On the other hand the adsorption potential for rare gases is sufficiently strong to produce a well defined monolayer, of the order of 100 meV for Ar, Kr and Xe (to convert to Kelvins multiply by 11.6). An extensive bibliography of physical adsorption potentials can be found in [257]. And the "corrugation", the amplitude of the periodic part of the potential, is smooth enough in some cases to produce a legitimate 2D solid but strong enough in others to have dramatic consequences - commensurate (C) phases and highly incommensurate (IC) domain wall lattices (DWL). This corrugation creates a wealth of different behaviour which enriches the physics but can complicate comparisons with the KTHNY theory of melting. In discussing melting the effects of the motion of the adsorbed atoms perpendicular to the substrate and the exchange of these atoms with the gas phase have also to be considered. 3.3.4.2. Commensurate and incommensurate phases. The easiest way to understand the types of phases produced by the modulation in the substrate potential is to consider the simple one dimensional model known as the Frenkel-Kontorowa [260] or Frank and van der Merwe [261] model, in which a chain of atoms interacting with springs, and
w
The role of dislocations in melting
563
Fig. 31. A chain of particles in a periodic potential.
having equilibrium separation a, is subjected to a sinusoidal substrate potential of period b (see fig. 31). Its classical Hamiltonian is: /z
a) 2 i
W(
1-cos
27rxi) b
(35)
i
where xi is the position of the ith atom. This model is known in the field of dislocation theory as the simplest model for dislocation motion. If a r b, there is a competition between the order preferred by the 1D lattice and that preferred by the substrate. The equilibrium configurations are given by the set of difference equations: r
- 2r + r
7rW = ~ sin 2"ar tto"
(36)
where r is the displacements of atom from some equilibrium periodic configuration in registry with the substrate (a commensurate state). The location by xi = b ( i n + r If the change in separation from atom to atom is small, this set of equations can be approximated by a differential equation which is none other than the familiar sineGordon equation with soliton-like solutions: d2r di 2
71-W
= ~
/Lb2
sin 27rr
(37)
The solutions to the difference equations or the sine-Gordon equation yield two types of equilibrium configurations for the chain of atoms, the already mentioned commensurate (C) and the incommensurate (IC). In a C phase the atoms lie at minima in the substrate field, the substrate wins out. In an IC phase, soliton-like domain walls (DW) separate C regions (see fig. 32). If the density of DW is low, each wall and each C region is well resolved, we have what is known as a domain wall lattice (DWL). If the domain walls overlap, a weakly modulated "smooth" structure is obtained which structurally retains the essential features of the original lattice. This is usually what is obtained when the corrugation is very weak relative to the strength of interparticle interaction, or the two lattice constants are very different. This simple 1D model predicts an exponentially decaying repulsive interaction between walls [263, 262]. The approximation leading to the sine-Gordon equation is equivalent to assuming a continuous mass density in the
564
B. Jo6s
Ch. 55
(a)
no
(b)
q,
/-- / J
_./
Fig. 32. The displacement ~b of an adsorbate atom from a commensurate site as a function of atom number n in the 1D Frenkel-Kontorowa model; (a) an isolated soliton-like domain-wall, (b) a periodic domain-wall structure.
chain; the discreteness is lost. In the continuum model the DW are free to slide in the lattice. If the lattice discreteness is restored, a pinning potential exists, the Peierls potential, which decays exponentially with the width of the walls [262]. Basically if the walls are a few lattice constant wide, the substrate pinning force is negligible. In generalizing to two dimensions, the essential physics remains the same. The interplay between the preferred substrate spacing and that preferred by the monolayer can lead to soliton like DW and the existence of C and IC phases. The 2D model was developed by Shiba [264] for the graphite substrate which has hexagonal symmetry. In the IC phase two types of wall ordering can be observed on a graphite surface, a striped phase, with walls parallel to each other, and a honeycomb phase with the walls forming hexagons intersecting at three-pronged vertices (see fig. 33). DWL form a fascinating subject. Extensive reviews of their properties exist, in particular for striped phases (see reviews [263,268-270, 121]). The honeycomb phase is now also fairly well understood, studied in some detail by Shrimpton et al. [271-274]. From energetic considerations, striped phases are expected to be favored at low DW densities over honeycomb phases because of the compression or expansion at the vertices which usually makes wall crossing energies positive [275, 276]. At higher DW densities, the larger average distance between DW reduces the repulsion energy between the walls in a honeycomb array making that phase more favorable. In reality, however, as Villain [263] has shown, entropy makes the honeycomb phase favored in isotropic systems even at low DW densities. Consider the diagram of such a phase in fig. 34(a). The total DW length and the total number of intersections does not change if we shrink one of the honeycombs in the network. This configuration can fluctuate into the one shown in fig. 34(b) by repeated breathings of the different honeycombs in the network.
w
565
The role of dislocations in melting
(a)
(b)
(c)
(d)
Fig. 33. Schematic pictures showing honeycomb array of domain walls (a) (expanded) and (c) (compressed), and striped arrays (b) (expanded) and (d) (compressed) on a graphite substrate when the walls are unrelaxed or very narrow. Expansions and compressions lead naturally to the walls shown, which have been given the name super-light and super-heavy respectively. The presence of DW means an excess or a shortage of atoms in the monolayer with respect to the relevant C phase. In the systems that we will be mostly discussing, the C phase of interest is the phase known as V~ x v~. This means that the nearest distance between atoms is v/3 times the distance between nearest adsorption sites and it corresponds to a filling factor of one third (see fig. 35). Figure 35 also shows the different possible wall geometries in IC phases close to a v/3 x v/3 C phase. Light and super light DW correspond to decreased densities, and heavy and superheavy DW to increased densities. Light and heavy DW are actually stacking faults in the triangular lattice and for this reason are not favored for spherical adsorbates. This has been demonstrated by computer simulation [275, 277] and experiment [278] for Kr/Gr. In this system DW
566
Ch. 55
B. Jo6s
(o1
(b)
Fig. 34. Honeycomb domain wall networks: (a) before breathing; (b) after breathing.
heovy
light
super heavy
super light
Fig. 35. Possible types of domain walls (DW) for an adsorbed monolayer on a graphite substrate. This a schematic representation based on infinitely narrow DW. The thick solid line corresponds to a super heavy domain DW, the double thin lines to a heavy DW. The thick dashed line is a superlight DW, while the double thin dashed lines a light wall [254]. have been shown by the same authors to be fairly wide, 5 to 10 Kr atomic rows, making them very mobile. Heavy DW are only observed in a hexagonal IC phase of the highly quantum D2 [321]. A great deal of attention has been given to the C-IC transition. Theories based on Fermi statistics have been particularly useful in understanding this transition in striped phases [270]. The configurational meanderings of the striped DW are analogous to the motions of a 1D lattice of fermions. DW energies replace fermion mass; line tensions, fermion hopping probabilities. Pauli exclusion principle represent the hard core repulsion of the wall. Within this framework Pokrovsky and Talapov showed that the C to striped IC transition should be continuous [280]. Arguments based on the entropy associated with the configurational meanderings of the DW reproduced their results [281-283]. The transition is continuous because collisions between the DW reduce the total entropy of the system.
The role of dislocations in melting
w
567
In contrast the C-IC transition is predicted to be first-order in the honeycomb DWL. The entropy associated with the breathing of the DW network is sufficient to reduce the total free energy at the onset of the transition [263, 282]. A heat capacity study of CO on graphite seems to confirm this prediction [285]. The DW networks in both the striped and honeycomb phases can be treated as renormalizable objects with elastic properties [281-283,273,274]. A C phase is not an elastic medium because of the exponential decay of any particle displacement. The DW in an IC phase restore its elasticity to the lattice. The low energy excitations of the lattice are now those of the DW network. In the striped phase it is dominated by the meandering and the repulsion of the DW. In the honeycomb phase, a more complex set of DW excitations can be observed, in addition to the lowest breathing mode. These are similar to the vibrational modes of a network of strings tied at three pronged vertices as has been shown by Shrimpton and co-workers [271-273]. The elastic properties are mediated by the repulsive interactions between the vertices carried by the connecting DW. This vertex repulsion produces in Kr on graphite a variation of the misfit e with chemical potential p, = A l l z - pc(T)l b,
(38)
where b ,-., 0.33 and A = 8 • 10-4K -b [272]. The misfit e is defined as a ~ ac]
= ~ ,
(39)
ac
where a is the average lattice constant of the monolayer and ac the lattice constant of the closest C phase. Equation (38) is in agreement with the experimentally observed relationship found at much higher temperatures [284]. In a striped phase a 1/2 power law has been predicted by Pokrovsky and Talapov [280] and observed in Br intercalated in graphite [286, 287]. As any lattice, DWL are prone to defects and they can melt. The elementary topological defects are the dislocations, diagrams of which are shown in fig. 36 for striped and honeycomb DWL. A prime example of DWL melting is the intermediate DW fluid phase (DWF) between a C and an hexagonal IC phase (HIC) which has been predicted by Coppersmith et al. [282]. As the average DW separation increases the DWL weakens, reflected by an increased meanderings of the walls until a mechanical instability occurs which can be represented by the KTHNY theory. This instability is not temperature driven, so the re-entrant liquid phase produced could extend all the way to zero temperature (fig. 37). The DWF phase has been observed in several systems, including Kr on graphite, which are discussed in the next section. In the limit of strongly overlapping DW, the smooth IC monolayer is found. It is in this limit that another substrate effect exists which has been predicted by Novaco and McTague [145], known as "orientational epitaxy". Novaco and McTague showed that the orientation of the adsorbed monolayer need not match that of the substrate, i.e., that of the C phase. Since shears are usually energetically more favorable than compressions, more adsorbed atoms can be placed close to adsorption sites if the monolayer rotates with
568
Ch. 55
B. Jo6s
b
13
(Q)
(b)
Fig. 36. Domain wall dislocation dipoles; (a) in a striped phase, and (b) in a honeycomb domain wall lattice. When the dislocations appear unpaired, a domain wall fluid is obtained. Compare figure (b) with fig. 12.
%
LIQ UID
,,,
COMM~I~NCOMM SOLID V
SOLID
| Fig. 37. Possible phase diagram for an adsorbate with C and IC phases, as a function of temperature T and chemical potential ~. respect to the substrate. The degree of rotation depends on the misfit of the monolayer e. The onset of the rotation is a function of the strength of the corrugation [264], the density modulations within the monolayer [272] and temperature since the energy variation with respect to rotation is small. In the opposite limit lies the C phase. It is not an elastic medium and its melting is more accurately viewed as a disordering process in a lattice gas (see for instance [288, 289]). When considering the role played by dislocations in the melting of the adsorbate lattice, the range of lattices going from smoothly modulated to DW-like will be considered.
w
The role of dislocations in melting
569
3.3.4.3. Melting of monolayers adsorbed on graphite.
In most experiments the graphite sample is placed into a cell in which gas is pumped. The pressure in the cell controls the chemical potential of the gas, and hence the adsorption on the surface and the lateral pressure on the 2D solid. Together with temperature it is the key thermodynamic variable. Using the equation of state for the gas and the known number of adsorption sites on the graphite sample (obtained using the C phase of krypton), a coverage can be calculated and used instead of the pressure as the other thermodynamic variable. In terms of coverage, C and IC phases are easier to characterize. One way to classify the many adsorbed monolayer films on graphite is to group them according to the general features of their coverage vs temperature phase diagrams, because these features are a very good measure of the effect of substrate corrugation. If we follow this principle, developed in detail by Moses Chan [129] for the adsorbed monolayers on graphite, four categories are found: (i) Monolayers of Ne, Xe, CD4 and CH4 which are incommensurate at melting and have phase diagrams resembling those of their bulk counterparts (see fig. 38), (ii) monolayers of Ar and C2D4 similar to (i) but with a very narrow or non-existent solid-liquid coexistence region, (iii) the monolayers Kr, N2 and CO which have a very stable x/~ x x/~ C phase persisting up to temperatures higher than the "expected" liquid-vapour critical temperatures, and a C-IC phase proceeding via intermediate DW fluid regions (see fig. 39), and (iv) light molecules such as 3He, 4He, H2 and D2 similar to (iii) but with a less stable C phase and a C-IC transition which can involve a striped DW phase (see fig. 43). We will not attempt an exhaustive review of each of these categories but focus on some well characterized examples. Our interest is in the melting kinetics of the elastic lattices, i.e., of the IC phases. /5, it
;I
,?
sltl I S
~I i s s l sS /
YF+ S. . . .
""
i I
I
LI +, V,
S+V
i
i
F
I
] L.. S i
i
T Fig. 38. Schematic coverage n versus temperature T phase diagram of a physisorbed monolayer exhibiting first order melting such as Xe on graphite. S, L, V and F stand for solid, liquid, vapour and fluid phases respectively. Dashed lines stand for first order phase boundaries.
570
B. Jo6s
Ch. 55
IC
f / !
C+F
,
! !
I
] /
I
/
T Fig. 39. Phase diagram of Kr, N2 and CO on graphite. The tricritical and critical melting points are shown as solid triangle and solid circle respectively. Solid and dashed lines stand for continuous and first order phase boundaries. The nature of the transition along the dotted line is less certain [129].
(i) The most studied example in this category is Xe. Xe favors a lattice spacing of 4.40 A, noticeably larger than 4.26 A, the lattice spacing of the ~ x v/3 C phase on graphite. Upon adsorption, the Xe monolayer has a tendency to compress. This compression preserves essentially its harmonicity [290] and the properties of its topological defects [163]. Its phase diagram (see fig. 38)closely resembles that of a bulk solid [129]. Its superlight DW are wide (~ 100 A) and consequently atomic separations on the walls are close to those preferred in the ideal monolayer [291]. At submonolayer coverages Xe patches exhibit typical first order melting behaviour [292, 293]. In the compressed monolayer region, the melting temperature increases rapidly from the triple point at 100 K. High resolution X-ray scattering experiments claim to find a crossover from the first order to continuous melting behaviour with increasing pressure near 125 K [294]. The continuous melting regime has been touted as an example of KTHNY melting and has been the subject of a number of careful X-ray scattering experiments [ 144, 293-297], some on single crystal substrates [144, 294]. These experiments find correlation lengths o up to 2000 A in the solid phase. The variation of the BOO and PO has been argued to be as predicted by the KTHNY theory and not to be due to the substrate field [296], in particular since similar results are observed on a much smoother substrate Ag(111) [298]. The transition is however fairly sharp [293]. A similar crossover from first order to continuous has been observed in vapour pressure isotherm measurements, albeit at a higher temperature, 155 K in one experiment [299, 300] and 147 K in another [301]. A combined heat capacity and vapour pressure isotherm experiment on much larger graphite samples finds that the melting transition is always first order [302]. This conclusion is based on the persistent sharp heat capacity peak that appears at melting. So thermodynamically the melting is first order but there is persistence of BOO. Abraham has argued
w
The role of dislocations in melting
571
on the basis of molecular dynamics (MD) studies that the third degree of freedom gives the appearance of a continuous transition; there is a continuous exchange of atoms between the first and second layer producing near melting a small temperature interval of solid-liquid coexistence [303, 304]. In computer simulations (see section 3.4.2) it is now believed that the ideal Lennard-Jones (LJ) monolayer has a first order transition, although also some form of intermediate phase may exist (a Xe monolayer is for all practical purposes a LJ monolayer) [305]. (ii) Ar is a much smaller atom than Xe. Its preferred interatomic spacing on graphite is 3.8 ,~, significantly smaller than the 4.26 A of the nearest C phase. Ar is always IC and its phase diagram also resembles that of an ideal monolayer. The difference with Xe is that it does not seem to have a solid-liquid coexistence region. A series of intriguing results on this system show a unique melting behaviour. A heat capacity study finds a small sharp peak beside a broad anomaly, at the melting temperature predicted by X-ray scattering studies [307]. The small peak is consistent with an "abrupt" density change of about 0.2% over a temperature range of 0.3 K. But scattering studies of melting reveal no evidence of density discontinuity at melting, just a continuous change in correlation length from 900 A to 200 A [308] in one study and 1500 ,~ to 400 ,~ in the other [309]. In addition, although the Novaco-McTague orientational epitaxy present in the solid phase decreases as the solid melts, some of it persists into the liquid phase [309]. This behaviour is consistent with the KTHNY theory where the creation of dislocation dipoles destroys the PO but maintains the BOO of the solid. Zhang and Larese [310] have recently managed to conciliate to a degree the two types of studies. From vapour pressure isotherm studies they found evidence of two-stage melting. They observe a narrow peak in the temperature derivative of the lattice constant. It is not clear whether this peak makes the transition weakly first order or can be explained within the context of a KTHNY scenario. The intermediate phase between the solid and isotropic liquid exhibits short range PO and solid-like properties such as the persistence of the orientational epitaxy, phonons and a compressibility anomaly. What seems quite clear is that this unique behaviour is the result of the substrate corrugation, and may reflect the existence of two competing types of ordering, that of an atomic lattice and of the DWL. The adsorbed Ar monolayer is, after all, an IC lattice with DW, even if they are fairly dense. In monolayers expanding upon adsorption the DW maintain separations close to those favored by the adsorbate while within domains the lattice is expanded. DW tend for this reason to be more important than in monolayers compressing upon adsorption. The intermediate phase could be a defected DWL. Shrimpton et al. [311] illustrated this point using MD simulations. Argon's melting behaviour bridges the floating monolayers with the well resolved DWL such as Kr which is discussed in the following section. The melting of C2D4 or C2H4 shows all evidence of being continuous [312, 313]. The computer evidence, however, shows that it is not of the KTHNY type but is associated with the gradual tilting of the long molecular axis towards a direction perpendicular to the surface [314]. The reduced local density produces a gradual disordering of the monolayer. Similar explanations have been given for the continuous melting of hexane as opposed to butane, a similar molecule, which has a first order melting transition (chemical formula CH3(CH2)n_2CH3, n = 4 for butane and n = 6 for hexane). Hexane, the longer molecule, goes from a trans- to a cis- or gauche-configuration [315, 316].
572
B. Jo6s
Ch. 55
(iii) The Kr monolayer is probably the most interesting of the adsorbed rare gases because it illustrates a lot of the physics discussed in section 3.3.4.2 in particular in relation to the C-IC transition and DWL. It has a very stable C phase, so stable actually that its melting temperature is comparable to the critical temperature Tc of the liquid gas coexistence region. Its phase diagram is described as having an incipient triple point [317, 259] (see fig. 39). This C phase has been used to determine the number of adsorption sites on graphite samples, using the fact that the amount of adsorbed gas is easily determined from the gas pressure in the cell and the number of moles introduced into the cell [258]. In the C phase the Kr atoms are separated well beyond their equilibrium lattice parameter of 4.02 ,~, to 4.26 ,~ as mentioned earlier. So, although this phase is very stable, by increasing the gas pressure more atoms can be easily adsorbed. These additional atoms form mobile DW. Close to the C phase their interactions are too weak to stabilize them into a regular lattice and they form a DWF as explained above. As the density of DW increases they solidify into a honeycomb array of walls. Kr shows clearly the predicted fluid phase between the C and IC phases [259]. Upon further compression the DW merge and the monolayer rotates with respect to the substrate [ 147] exhibiting the orientational epitaxy discussed in section 3.2.2.5. X-ray scattering finds the melting of the DWL to be continuous. The DWL seems a prime candidate for a dislocation unbinding melting mechanism [162, 163,274]. The IC solid-DWF boundary was mapped out by Shrimpton and Jo6s applying the KTHNY stability criterion of eq. 32 to the string model discussed in the previous section. At lower coverages or lower misfits, the IC solid melts into the DWF which, with increased temperature, evolves into a conventional fluid. At higher coverages the IC to fluid transition is still continuous. These X-ray scattering results are confirmed by heat capacity measurements on a very similar system, CO on graphite [285]. The entire submonolayer and monolayer phase diagram of CO on graphite is surprisingly similar to that of Kr on graphite. A statistical model called the "Potts model", a generalization of the Ising model (for a review on Potts models see [318]), has been successfully used by Caflish, Berker and Kardar [320] to map out the overall phase diagram of the Kr on graphite system. A given C phase can be in any one of three sublattices, a, b or c (see fig. 40). If the adsorption sites are split into triplets of sites, an order parameter can be defined which specifies which type of site is occupied by an adsorbed atom. The relevant Potts model is therefore the three-state Potts model. In the submonolayer regime, a "dilute Potts model" is used with an order parameter determining whether a triplet is occupied [319]. In the IC phase, hexagonal domains are used as units, and they can be in any of the three sublattices, a, b or c. The interaction energy between two domains will depend upon the type of wall separating them (see fig. 40). It is actually most effective to consider triplets of domains because the vertex configurations determine the energy of the lattice and they depend on the three adjacent domains. To correctly account for this topology "chirality" has to be introduced in the Potts model, hence the name "helical Potts model". "Chirality" is the dependence of the interaction energy between the three nearest neighbour domains on the order of appearance of the sublattices: U(a, b, c) = U (b, c, a) ~: U (a, c, b). Renormalization group techniques are used to solve for the phase diagram. This last model is very interesting from the perspective of this review because
w
573
The role of dislocations in melting
Dislocation ~, subl.c occupied
( 0
Super-
2ovy
Heavy__..l crossing
crosstng
subl.a
t
Heavy wall
subl. b
t
subl.a occupied
Super-heavywall
Fig. 40. Structures in a compressed adsorbed layer on graphite in the IC phase near the x/~ x x/~ C phase. It shows the possible wall crossings in the honeycomb phase and the neighboring occupied sublattices (subl). [320]. it takes into account dislocations in a natural way (see fig. 40). The full phase diagram of Caflish, Berker and Kardar [320] for Kr on graphite is shown in fig. 41. Figure 42 shows one of their fully dislocated configurations which occurs in the DWF phase. The alternance of superheavy and heavy wall segments reflects the dissociation of the dislocations in the DWL into two partials separated by a stacking fault, the heavy wall, as discussed in section 3.2.3 and shown in fig. 10. This picture complements the "string model" of Shrimpton et al. discussed in the previous section. (iv) The light molecules 3He, 4He, H2 and D2 adsorbed on graphite have been the subject of a great deal of interest partly because of their highly quantum character. Their phase diagrams (see fig. 40) are exceedingly similar to each other (for the most recent studies and references see [321,322], and for a review [129]). They are also dominated by the x/~ x x/~ C phase (see fig. 40). The melting of the C phase into the fluid (F) phase is a vacancy driven continuous transition of the 3-state Potts model type with its characteristic 1/3 specific heat exponent [323, 325-327]. With increasing coverage, the C monolayer becomes incommensurate (IC); at low temperature, first forming a striped phase evolving into a hexagonal IC (HIC) phase [321, 322]. At higher temperature, the C-IC transition proceeds with a hexagonal DWF which solidifies with increasing density into a HIC phase. The transitions from the DWF to the HIC and striped phases are continuous [129] This is very similar to Kr on graphite, with the difference that Kr does not show a striped IC phase as we have just seen earlier. For Kr high entropic effects produced by wide mobile DW favoured the HIC phase. Theoretical studies support the observation of the striped phase in the monolayers of light molecules [328-330]. The domain walls are fairly narrow although the pinning stress appears to be still fairly small [330].
574
Ch. 55
B. Jo6s
60 / '
Temperature(K) I00 ' ' ~~
b 0]- Commensurate
'
'
..aox~ d
%
-
o-5~- , ~
Fluid - ~
I
-
I/,
,
,,
, ,........
1
-
l/ I~1
0
,,
I
50 Temperature (K)
"
i
I00
v
"", "
Incommensurate / / Commensurate 9J solid// solid 2 /
-
0
400
(1)
i
200
if) o3
~' (3..
"
0
Fig. 41. Reentrant phase diagram (curves) in the pressure and temperature variables calculated by Caflish, Berker and Kardar for Kr on graphite within the helical Potts-lattice gas model. Also are shown experimentally determined phase transition points (for details see [320]).
The above overview is far from being exhaustive. A number of systems have not been mentioned such as CF4 on graphite [331 ] which exhibit interesting phase diagrams with similar physics. 3.3.4.4. 2D melting on other substrates. Graphite has not been the only substrate used to study 2D melting. We noted already earlier that Xe has been adsorbed on Ag(111) to check the effect of the substrate corrugation on the melting transition. Other substrates involve a range of transition metals with rare gas adsorbates, such as Kr [335] and Xe [336] on Pt(111), and alkali metal adsorbates, in particular K and Cs on a wide range of substrates, Ru(001), Pt(111), Ni(111), Rh(100), etc. Adsorption in these systems is induced by the high degree of electroposivity of the alkali-metal atoms (for reviews, see [332-334]). The alkali atom has an induced dipole as it draws close to the surface which is of opposite direction to the surface dipole which produces the work function of the surface. A variety of phase diagrams has been observed. The ones of interest to us are those exhibiting significant incommensurate phases such as K [337] and Cs/Cu(111) [338], K on Ni(111) [339] and on Ni(100) [340] which are all on substrates of hexagonal symmetry. The melting of the incommensurate phases of these systems has been studied in the references quoted. The literature has been reviewed by R. Diehl, the leader of one of two groups reporting work on this subject. LEED measurements show evidence of a gradual loss in ordering evidenced by the increased widths of the diffraction peaks in the first three of these systems. So far it appears to be the only evidence available for
w
The role of dislocations in melting
575
Fig. 42. One of the many fully-dislocated configurations which can occur in the zero-temperature dense fluid. Heavy and superheavy-wall segments are drawn with thin and thick lines respectively. Walls meander with such segments alternating between dislocations. They are maximally packed but never cross [320].
a continuous melting transition. There is not enough detail to argue whether these are more like DWL or floating monolayers. In the last system, desorption occurred before disordering could be observed. 3.3.4.5. Intercalates. We mentioned earlier Br intercalated into Gr as an example of a striped DWL [286, 287] and that in situ high resolution X-ray scattering experiments have found the 1/2 power law for the variation of the misfit with temperature. The melting transition from the X-ray scattering perspective is found to be continuous, consistent with the picture of a striped DWL disordering in a KTHNY way. But Mochrie et al. [341 ] find that the decay of positional correlations as measured by a sine wave order parameter has a power law exponent evolving continuously from 0.04 to ~ 2.0 contrary to KTHNY theory. For the hexagonal DWL of K and Rb layers intercalated in graphite, the melting transition is found to be smooth with a power law decay with temperature of the elastic constant C44 [342] as observed by neutron scattering. With nitric acid-intercalated graphite the dislocation density was followed by relating it to the variation of the correlation length and the diffusion constant [343]. 3.3.5. Discussion In summary, the best candidates for dislocation mediated melting are the IC systems, where a DWL melts continuously, in a manner consistent with a scenario involving a mechanical instability, similar to the one forming the basis of the KTHNY theory. This should not come as too much of a surprise. DWL, in contrast to the actual adsorbed monolayers, form an ideal 2D elastic medium. The adsorbed monolayers have to contend with the effects of the third degree of freedom (effectively turning the whole system into a surface), and substrate effects (modulation in the adsorption potential, most notably).
576
B. Jo6s
Ch. 55
3.4. Computer simulations on systems of particles 2D melting offers a major challenge to computer simulation studies. At first sight it seems like a simple enough problem, to set up a lattice, heat it up and follow either particle trajectories in molecular dynamics (MD) or thermodynamic quantities in Monte Carlo (MC) simulations, but quickly the limitations of the size of the system, the boundary conditions and the length of time of the runs raised questions about the significance of the results. There have been nearly as many simulations done on 2D melting as experiments. A number of extensive reviews have been published on different aspects of these calculations (see [118-120, 173]) and we will refer to those for more details, in particular for those not directly relevant to the focus of this review. 3.4.1. The methods
Two methods of computer simulation are used: the Monte Carlo (MC) and the molecular dynamics (MD) methods. In both methods results for quantities of interest are obtained as averages over configurations generated by the repeated application, of basically simple algorithms, which govern the motion of the particles. The physics is classical. Quantum effects, except for those incorporated into the interparticle interactions or the interactions with a substrate are ignored. In the MD method a classical hamiltonian is set up and the particle motions are determined as a discretized integration of the equations of motion. The time increments are called time steps and they are a key parameter in ensuring a reliable solution. Various algorithms have been set up for different statistical ensembles with the conserved quantities incorporated into the Hamiltonian or added as additional constraints. Considerable developments in the field have occurred over the last two decades. The melting studies have used mainly two types of algorithm: (i) a canonical ensemble with the system coupled to a heat bath (some care is required in the choice of constraint to ensure that a canonical distribution is maintained [344, 345, 118]), and (ii) a constant pressure ensemble usually implemented by coupling the system to a "piston" [346]. The mass of the piston is arbitrary but the algorithm works well if it is chosen so as to have area fluctuations on the same time scale as the time it takes for sound to propagate through the sample [346]. In the MC method [347], as its name implies, particle motion is random. It is, however, governed by a probability distribution which in the simplest case of the canonical ensemble is the Boltzmann distribution. In this case particle moves are determined by a random number between 0 and 1. If this number is lower than exp(-/3AE) the move is rejected. In principle the MC method offers considerably more flexibility in implementation than the MD method. It is not restricted, as the MD method is, to obeying the time evolution imposed by the equations of motions. Phase space can be sampled much faster by a clever choice of moves. For example a constant pressure ensemble can be implemented simply by using volume changes, as attempted moves with the appropriate statistical weight. Thermodynamic quantities are therefore more easily calculated. The loss of dynamical information is however not trivial.
w
The role of dislocations in melting
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The two methods are complementary. It is hard to discriminate against one or the other. When discussing the effectiveness of each method in the study of melting, there are some crucial considerations discussed below. 1. Choice of the ensemble. A constant density ensemble gives the opportunity to investigate whether or not coexistence of the liquid and solid phases can be achieved, a signature of the first order transition, or whether a hexatic phase with long range BOO is possible. Particle trajectories have been invoked to demonstrate coexistence [348-350]. True separation is however not observed but an alternance of solidlike and liquidlike patches. In the 1/r 12 system at the apparent coexistence temperature the solid part was found by Broughton et al. [368] to be stable but to melt and reform during the simulation indicating a low interracial energy between the solid and liquid phases. In very long runs on the same system Naidoo, Schnitker and Weeks [353] find similar results but they temper their conclusions with arguments of limitations of time, sensitivity to boundary conditions, and finite size effects. In both calculations the energy versus density diagrams show a "van der Waals" type loop characteristic of a first order transition (see fig. 44). In principle the long distance behaviour of the BOO should be able to distinguish hexatic and two-phase coexistence. However in a finite system both may exhibit persistence of BOO, and long range behaviour due to the periodic boundary conditions may be elusive. In a constant pressure ensemble there is no coexistence region. If no hexatic phase is found, then none exists. But the width of the transition could be quite narrow in the path chosen to cross the transition. The order of the transition is not easily determined either if the transition region is narrow. Implementing the constant pressure may induce an artificial effect from pressure waves as argued by Toxvaerd [351 ]. But Toxvaerd and Abraham and Koch [352] disagree on this issue. 2. Choice of boundary conditions. Periodic boundary conditions are usually chosen. They eliminate edge effects but they favour superheating by stabilizing the solid and preventing the nucleation of defects such as vacancies [349, 118]. In a recent study on the 1/r 12 system, Naidoo et al. [353] have used different boundary conditions and found significant differences in the variation of the energy with density when changes in boundary conditions were made (see fig. 44). 3. Finite-size effects. It is hard to believe that these would not be important especially in systems where the dislocation cores are large such as the Lennard-Jones system [162, 163]. In spite of this the subject has been explored in only a few studies. Toxvaerd had observed significant size dependence on defect density and elastic constants [354, 355]. Udink and van der Elsken have also come to a similar conclusion when looking at PO and BOO in the Lennard-Jones system [356]. And so have Zollweg, Chester and Leung [235], and Zollweg and Chester [360] on the hard-disk and the 1/r 12 systems. All these studies indicate that the first order behaviour is weakened as the system size is increased. But, is that any proof of a continuous transition? Larger systems will have larger kinetic barriers to transform into the liquid phase, leading to the next question, the effects of finite time. Ultimately what is required is finite size scaling, such as was done by Lee and Strandburg [357] on the hard disk system. 4. Effects offinite time. The greatest frustration to any one involved in computer simulations is the constant reminder that MD runs correspond typically to picoseconds and at
578
Ch. 55
B. Jo6s
HIC
i~lO-.---~,
9 o 9 o,,,,,
.....................
,..,
C+F /
0
[="
t t i s
qFig. 43. Typical phase diagram for a light molecule adsorbed on graphite, namely 4He, 3He, H2 and D2, SIC and HIC stand for unaxial incommensurate with stripes of domain walls and a hexagonally symmetric incomensurate phase respectively. C, F and DWF stand for the ~ • v ~ commensurate (C) phase, fluid and domain wall fluid phases. Solid lines stand for continuous transition, dashed lines for first order, and dotted line, uncertain. Solid triangle and circle locate positions of 3-state-Potts critical and 3-state-Potts tricritical points [129].
best nanoseconds, rather short times compared to what is available to experimentalists. Novaco and Shea [370] observed in the 1/r 5 system relaxation times longer than their longest runs (100000 time steps). Naidoo et al. [353] found evidence of kinetic bottlenecks which prevented the break-up of the 1/r 12 system set up in an unstable hexatic phase, even after five million time steps. This suggests that long times on a small system may be better than a large size without good enough statistics, which takes us back to the need for finite size scaling. Having made all these precautionary statements let us now look at what has been found for some key systems. 3.4.2. H a r d core p o t e n t i a l s
In hard core potential systems a repulsive core keeps the particles apart. We will present results on the hard disk system, the Lennard-Jones monolayer, the 1/r 12 potential and the Weeks-Chandler-Andersen potential (WCA) [ 184] systems (the last potential is just the repulsive part of the LJ potential). Weeks [358] has shown that for central potentials the steeper the repulsive part the larger the density discontinuity in a first order transition. For this reason we will discuss the systems in increasing order of the smoothness of the interparticle interactions. (i) The melting of the hard disk system has been the first studied, even before the predictions of the KTHNY theory were published. In 1962 MD simulations were performed by Alder and Wainwright [359]. From the variation of the density versus pressure
The role of dislocations in melting
w
579
2.9
2.7 E! 0
F4
,:,
2.5
U
0
2.3
2.1
1.9
9
0
oO
9 9
13
m
9
I t
0.950
t
t I I 0.970
t
, 1 t 0.990
,
• I , 1.010
,
,
I , 1.0"50
t
, 1.050
Fig. 44. E n e r g y - d e n s i t y d i a g r a m , for l i q u i d - s o l i d traverse of periodically replicated s y s t e m P (filled circle), of s y s t e m F1 (open circle) w i t h s u r r o u n d i n g solid-like walls, and s y s t e m F 4 (squares) w i t h s u r r o u n d i n g l i q u i d - l i k e walls [353].
curve they deduced a first order transition. There have been a few other studies since, although the attention moved more towards the other hard core systems. Recently it has been given renewed attention to elucidate the controversy concerning the order of the transition in 2D systems [235, 360, 357]. Being the system with the steepest potential interaction, evidence of continuous melting in this system would be a strong indication of similar behaviour in the other hard core systems. The simplicity of the potential also allows for the study of large systems or long times, or both. Zollweg and Chester [360] showed a significant decrease in the first order character when going from system size 1024 to 16 384 and even up to 65 536 in their MC simulations. They observe in the pressure density curve a shorter tie-line than had been previously observed by Alder and Wainwright in their 870 disk system [359]. The supposition is that for even larger systems it will vanish. The intermediate phase has not been identified. They contest on the grounds of finite size effects the contention by Lee and Strandburg [357] that the transition is first order. However, the jump in density between the two phases may decrease with system size but that does not mean it will go to zero in the thermodynamic limit. Lee and Strandburg use an elegant approach developed by Lee and Kosterlitz [361]. It involves a finite size scaling argument based on a MC calculation of the bulk free energy barrier between the liquid and solid phases. Lee and Strandburg arrive at the opposite conclusion to the previous authors. The energy barrier increases with increasing size of the sample. The method only requires small systems, much smaller than the correlation length at the transition and therefore long runs with good statistics can be achieved. The method has been tested with success on the 3-state Potts model [362]. It is however not clear that, even the largest system studied, is sufficiently large for the melting kinetics. (ii) The Lennard-Jones monolayer received a great deal of attention not only because it is the most physical of the simple potentials but because it is readily identifiable with the rare gas monolayer. And as we know, this class of monolayers has been the subject of
580
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extensive experimental investigations. A study by Frenkel and Mctague [363] following soon after the KTHNY theory, reported observation of the hexatic phase. But quickly the evidence in favour of a first order transition began accumulating [364, 365, 349, 118]. At this stage the situation is less clear but quite interesting and revealing about two dimensional melting. The first evidence of unusual behaviour came with the observation of a low interface free energy in the coexistence region [364]. This was followed by the observation of noticeable size dependence which indicated weakening of the first order transition with increasing sample size [354, 355]. The coexistence region proved the most revealing. It does seem to exist. Particle trajectory pictures of Abraham [348-350] show patches of solidlike and fluidlike regions. His constant pressure simulations show hysteresis [349, 118]. Loops were observed in the variation of the pressure with density [366, 364, 354] (see fig. (45)). Such loops are characteristic of a first order transition. In spite of this Udink and Frenkel find the solid-liquid coexistence region to be a well defined homogeneous state with a BOO transition [305]. The hexatic-like phase has the predicted properties of a KTHNY state but it seems rich in defects. The KTHNY theory has definite predictions about the elastic constants specifically at melting. K defined in eq. 23 should be equal to 167r. Using K, Abraham [349] had early proposed that the KTHNY melting point was a point of mechanical instability and hence the melting point of a superheated solid. He based his conclusion on free-energy calculations that placed the transition at a lower temperature than the point where K 167r. The more recent calculations of Udink and Frenkel [305] find melting temperatures for which K ~ 167r. Matters are complicated by evidence that the shear modulus depends sensitively on the size of the system, and hence so does K [355]. The LJ system seems to be in an intermediate situation with some possibility of hexatic like character but thermodynamically with the first order melting transition. A van der Waals equation of state argument when applied to this system would still prefer the first order transition [367]. (iii) The work of Broughton et al. [368] on the 1/r 12 system shows coexistence but also finds a small interface energy between the solid and liquid phases. Recent studies further confirm the very weak nature of the first order phase transition and the existence of large correlations in the coexistence region" Zollweg et al. [235, 360] with their study of size dependence and Naidoo et al. [353] with their extensive studies on the effects of boundary and simulation time. Naidoo et al. [353] find in the intermediate region very similar defect patterns as have been observed in the photographs of the colloidal spheres discussed previously (see section 3.3.3.2). There is a greater clustering of the defects than expected in an hexatic phase, a point to which we will return later. (iv) The WCA potential is a repulsive potential obtained by truncating the LJ potential at its minimum and shifting it upwards by an amount c equal to the depth of the LJ potential. As a result, both the potential and its derivative are continuous at the cut-off. The WCA potential is given by
VWCA(r) =
{ 4e[(~r/r) 12 - ( o r / r ) 6] + e 0
for r < rc, for r ~> rc.
(40)
Glaser and Clark have carried out extensive calculations on the WCA system. [140, 172, 173]. Their thermodynamic studies on 896-, 3584-, and 8064-particle systems show the typical first order melting transition. The most interesting aspect of their
w
581
The role of dislocations in melting ,
i
I
I
'
I
'
I
i
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'
/
i i
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i
i /-
i
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I
/
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/ /"
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I
./1 I,LI
I
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!
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I
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i
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Fig. 45. Pressure (unit e/tr 2) as a function of density (unit cr-2 for the Lennard-Jones potential system at a temperature T* : 1.0 [354]. The full line and full circles for a 3600 particle system, and the dotted line and empty circles for the smaller 256 particle system. Also shown is the extension of the fluid pressure and the line in the small system (dotted line) together with the virial expansion pressure and its tie line (dotted and dashed line). work is their study of the defect structures in the liquid. It starts from the Voronoi polygon construction which Glaser and Clark used to investigate in detail the defect structure through the melting transition. They also used the dual construction, the so-called Delaunay triangulation where all nearest neighbour pairs are connected by "bonds", which therefore focuses more on the bond lengths and the bond angles. They observe a condensation of defects at the transition. The same is observed in the dual construction. The defect structures show a great deal of variation at the transition. The general impression of polycrystalline disorder is in particular apparent in the Delaunay triangulation. This observation has lead Glaser and Clark to set up the defect mediated melting theory, where the focus is on geometrical rather than topological defects, which was discussed earlier in section 3.2.5.
582
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Ch. 55
3.4.3. Intermediate strength interactions There have been a few studies on intermediate strength interactions such as the potentials 1/r 6 [369], 1/r 5 [370, 371] and 1/r 3 [372-374]. These studies also point to a weak first order transition. But concerns about long time fluctuations and pre-transitional effects left Allen et al. unwilling to draw a conclusion on their study of the 1/r 6 system [369]. Novaco and Shea [370] arrived at similar conclusions. 3.4.4. Soft core interactions Using a scaling property inherent to 1/r n potentials, which he applied to the ClausiusClapeyron equation, Weeks [358] showed that when n < d (where d is the dimensionality of the system) the system upon melting cannot have a density discontinuity. It is expected that this would apply to any similar long range potential with a soft core, such as the Coulomb interaction. It, however, does not exclude the possibility of an entropy driven first order transition but it would soften its character. We have already considered the experimental realization of such a system in the 2D electron lattice on the surface of liquid helium (see section 3.3.2) and reported good evidence of a continuous transition of the KTHNY type. There is however no agreement among the various computer simulations. First order transitions have been reported which agree in the location of the transition with experiment [375] or in the entropy jump [376, 373] with the density wave theory of Ramakrishnan [171]. Locating the transition is not itself proof of predicting the correct nature of the transition especially in view of the small difference in free energies between the liquid and solid phases. The theory of Ramakrishnan is a mean field treatment so its reliability as to the order of the transition is not clear. Cheng et al. [377] calculated by MD simulations pressure-area isotherms and observed hysteresis leading them to conclude also to the first order transition. The discontinuities in the isotherms are almost imperceptible in accord with Weeks' predictions. These first order predictions remain challenged by the studies of Morf [ 167,378,379] and more recently Naidoo and Schnitker [380]. In the Fisher et al. paper [167], the dislocations were found to obey elastic behaviour down to very close separations. On the other hand, because of the long range of the potential interaction, the dislocations are expected to be mobile. These two properties favour dislocation mediated melting. The most revealing results are however those of Naidoo and Schnitker [380] who made extensive simulations on the Yukawa system in an attempt to explain the experiments on the colloidal suspensions. Their simulations similar to the ones mentioned above on the 1/r 12 system [353] were run for very long times, often millions of time steps. Their study, however, relies on the absence of a volume change on melting, a conclusion based on the similarity of the Yukawa potential with the 1/r potential, and on a calculated equation of state. The absence of a volume change permits a search for the hexatic phase by varying the density. By identifying the onset of BOO and PO with the densities at which the orientational correlation length exceeds 30a0 (where a0 is the lattice parameter) and the positional correlation length 15a0, in a similar way as Murray and Van Winkle [231] did for the colloidal suspensions, they find an intermediate phase ~ 1% wide in density which may be a hexatic phase. The decay in the correlation functions Cd(~' ) and C6(~) is algebraic as predicted by the KTHNY theory although not with the same exponents. The discrepancy is in particular large for ~Td, more than a factor of 2 (0.7 instead of
w
The role of dislocations in melting
583
1 / 4 - 1/3). The analysis of defect configurations shows in general a very complex structure usually with clustering of dislocations as has been observed in some other simulations [353, 381] and experiments on the colloidal suspensions (see section 3.3.3.2). Interestingly the "clumping" of defects is more similar to that of Tang et al. [237] than that seen by Murray et al. [219]. Murray [219] argued that the clumping was the result of a lack of equilibration and that it takes hours for dislocation climb to be completed. This time is not reasonably possible in computer simulations. It should also be mentioned that the sphere sizes are not the same in the Murray et al. and Tang et al. experiments, 0.305 ~tm in the former and 1.01 ~tm in the latter. 3.4.5. The nearest neighbour piecewise linear force interaction This is also a soft core potential. But in addition it is only nearest neighbour. It is defined in eq. 33. Combs [382] considered the melting of the monolayer with this interaction. He finds in his MD studies first order grain boundary type of melting similar to what had been proposed in the theory of Chui [168]. Runs were fairly short, ~ 4007- after equilibration. 3.4.6. Synthesis Computer simulations, although having some limitations in particular of time, boundary conditions, and system size, point at a 2D melting behaviour which is different from bulk melting, and does not appear universal. The nature of the interparticle interactions is important. Weeks [358] seems to have been the first to draw attention to this fact, when he argued that the density discontinuities decreased with decreasing n for 1/r n interactions. The sensitivity of the dislocation properties to the range of the interactions [165] also bears evidence to this fact. Computer simulations show a gradual evolution of behaviour as one goes from the hard disk system to the soft core potentials. As the core softens recent evidence points more and more towards the existence of an intermediate phase. It is not seen in the hard disk system, but BOO is found in the solid liquid coexistence region of the Lennard-Jones monolayer [305]. In the 1/r 12 system, and in the screened Coulomb system [380], defect patterns similar to those seen in the colloidal suspensions are observed. The calculated patterns show some evidence of not being totally thermalized. Kinetic bottlenecks require simulation times still beyond present capabilities. The observed intermediate phase is not as expected from the hexatic phase of the KTHNY theory. There is more defect clumping and an overall higher density of them. That the intermediate phase is not identical to the hexatic phase proposed by the KTHNY theory should come as no surprise since this theory assumes elastic dislocation dipoles with point-like cores. Clumping of the defects may at least partially be due to extended cores, which have been demonstrated to exist in the Lennard-Jones monolayer [ 162, 165]. The theory also excludes the possibility of aggregation of dislocations as it is derived within the dilute limit. This aggregation could lead in some cases to the first order grain boundary transition predicted by Chui [ 168]. The evidence for such melting behaviour is the strongest for the nearest neighbour piecewise linear force interaction system. But some key features of the hexatic phase predicted by the KTHNY theory are seen in the smoother potential systems: the persistence of BOO and algebraic decay of correlation functions.
584
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Computer simulations overall still tend to favour a first order melting with loop like energy-density and pressure density variations, but the homogeneous hexatic-like intermediate phases found in the smoother systems including the LJ system indicates behaviour quite different from that observed in bulk systems. This intermediate phase is only seen in constant density simulations. The clue in the melting dynamics lies in this coexistence region which needs further probing with attention to the boundary conditions.
4. Conclusion It seems that the melting of bulk systems is dominated by the surface, and defect cores. In the latter category belong dislocations, but especially grain boundaries. There is no strong evidence of dislocations being involved in the kinetics of melting. Imaging techniques are progressing rapidly and new insight may be available in the near future if a renewed interest in this subject occurs. The DTM appears useful mainly as an approximate theory of the liquid state near melting. In 2D systems, the situation is quite different. Increased thermal fluctuations will tend to wash away sharp phase boundaries. And topological constraints brought about by dimensionality favour the persistence of BOO into the liquid phase. As discussed by Collins [114] the average number of neighbours in both the liquid and solid phases is six in a triangular lattice [114]. These two facts favour a continuous phase transition. The persistence of BOO may however in itself not be a sufficient proof of a dislocation unbinding process. It is also compatible with a grain boundary process [168]. There is sufficient evidence to suggest that the loss of translational order can be in many cases attributed to the appearance of topological defects. Whether they form clusters, grain boundaries or remain dilute as assumed by the KTHNY theory, will affect the width of the intermediate phase, and will have a great influence on the thermodynamic signature of the transition. There is experimental evidence of continuous melting, in micron and submicron size colloidal suspensions, with screened Coulombic interactions, and similar systems (magnetic or electric holes, magnetic bubbles ...), electrons confined on the surface of liquid helium, incommensurate adsorbed monolayers with DWL character. Liquid crystals may be added to this list. Computer simulations, on the other hand, do not yet offer a clear picture. Do they reveal the present limitations of computer simulations? The melting behaviour seems dependent on the potential interaction. The nature of the melting transition in 2D is still not very well understood. Profound differences from bulk behaviour are found. In addition to dependence on the interaction potential, the melting kinetics may also be more sensitive than bulk systems to boundary conditions, imperfections, and overall the way the 2D system is created.
Note a d d e d in p r o o f There are two recent computer simulation studies further confirming the special character of 2D melting: ref. [383] on the 1/r 12 system (increased evidence of a hexatic phase),
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The role of dislocations in melting
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and ref. [384] on the hard disk system (one stage continuous transition). The occurrence of a hexatic phase is predicted in refs [385] and [386] for systems with solid-solid transitions. On the experimental front, ref. [387] reasserts the persistence of BOO through the melting transition of Xe on graphite, while refs [388] and [389] present two new systems with hexatic phases, some Langmuir-Blodgett films, and "plasma crystals" respectively. ("Plasma crystals" are layered materials made of colloidal particles introduced into a charge neutral plasma.)
Acknowledgments This work has been supported by the Natural Sciences and Engineering Research Council of Canada. Helpful discussions are acknowledged with M.S. Duesbery and B. Grossmann.
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Author Index
Abbaschian, G.J., 586 Abraham, EE, 587, 591-593 Abraham, EE, s e e Barker, J.A., 593 Abraham, F.E, s e e Broughton, J.Q., 592 Abraham, F.E, s e e Koch, S.W., 593 Ackland, G.J., s e e Vitek, V., 183 Adams, G., s e e Grimes, C.C., 590 Aeppli, G., 588 Aeppli, G., s e e Bruinsma, R., 589 Ageeva, V.A., s e e Shishokin, V.P., 25 Agnolet, G., 588 Agullo-Lopez, F., 503 Aharony, A., 588 Aharony, A., s e e Brock, J.D., 588 Ahlers, M., 131 Ahlquist, C.N., s e e Bush, F., 503 Ahlquist, C.N., s e e Carlsson, L., 503 Ainslie, G., 586 Akita, K., s e e Ueda, O., 503 Alder, B.J., 593 Aleinikova, I.N., s e e Deryagin, B.V., 501 Alexander, H., 502, 503 Alexander, H., s e e Gottschalk, H., 502 Alexander, H., s e e Kuesters, K., 501 Alexander, H., s e e Wessel, K., 502 Alexander, W.O., 24 Allen, C.W., s e e Kuczinski, G.C., 501 Allen, M.P., 593 Allen, N.P., s e e Pfeil, L.B., 24 Almin, A., s e e Westgren, A., 25 Alonso, J.J., s e e Fernandez, J.F., 594 Alper, T., s e e Saunders, G.A., 587 Als-Nielsen, J., s e e McTague, J.P., 592 Als-Nielsen, J., s e e Nielsen, M., 592 Als-Nielsen, J., s e e Pluis, B., 586 Andersen, H.C., 593 Andersen, H.C., s e e Bagchi, K., 594 Andersen, H.C., s e e Weeks, J.D., 589 Andrei, E.Y., s e e Glattli, D.C., 590 Androussi, Y., s e e Caillard, D., 504 Anisimov, V.I., s e e Greenberg, B.A., 133 Anstis, G.R., s e e Chou, C.T., 68, 439 Antolovich, S., s e e Webb, G., 132 Antolovich, S., s e e de Bussac, A., 132, 251,440 Anton, D.L., 25, 133, 437 Anton, D.L., s e e Giamei, A.F., 25 Antonova, O.V., s e e Greenberg, B.A., 133 Aoki, K., 184, 436 Ardell, A.J., 25 Ardley, G.W., 26
Argon, A.S., s e e Pollock, T.M., 25, 26 Arko, A.C., 436 Armstrong, A.J., 590 Armstrong, A.J., s e e Tang, Y., 590 Arrell, D.J., s e e Vall6s, J.L., 26 Aruga, T., 593 Asaro, R.J., 185 Ashby, M.F., s e e Frost, H.J., 252, 441 Auerbach, D.J., s e e Abraham, EF., 591 Aust, K.T., 587 Awal, M.A., 587 Axe, J.D., s e e Sirota, E.B., 590 Aziz, R.A., 591 Bacon, D.J., 131 Bagchi, K., 594 Bain, E.C., 25 Bak, E, 591 Baker, I., 132, 439, 440 Baker, I., s e e Horton, J.A., 440 Baker, I., s e e Munroe, P.R., 134 Bakker, A.E, 590 Ballufii, R.W., 587 Balluffi, R.W., s e e Chart, S.W., 587 Baluc, N., 67, 132, 183, 184, 436--439, 441 Baluc, N., s e e Bonneville, J., 132, 183, 437 Baluc, N., s e e Mills, M.J., 67, 132, 251,439 Baluc, N.L., 251 Barker, J.A., 593 Barnby, J.T., 131 Barrett, C.S., 26 Basinski, S.J., 437 Basinski, Z.S., s e e Basinski, S.J., 437 Baskes, M.I., s e e Yoo, M.H., 184, 440 Bassani, J.L., 183 Bassani, J.L., s e e Qin, Q., 182, 183 Bassani, J.L., s e e Wu, T.-Y., 185 Batallan, E, s e e Simon, Ch., 593 Beardmore, P., 131 Beauchamp, E, 134, 438 Beauchamp, P., s e e Dirras, G., 134, 435 Beauchamp, E, s e e Douin, J., 67, 184, 437, 438 Beauchamp, E, s e e Lasalmonie, A., 439 Beauchamp, E, s e e Tounsi, B., 132, 439 Beauchamp, E, s e e Veyssi~re, E, 68, 438 Bedanov, V.M., 593 Behrensmeier, R., s e e Kung, J., 503 Belyavskii, V.I., 504 Benattar, J.J., 590 Benattar, J.J., s e e Moussa, F., 590 595
596
Author
index
Borel, J.-E, 586 Benton, W.J., s e e Kusner, R.E., 590 Borel, J.-P., s e e Buffat, Ph., 586 Berezinskii, V.L., 585 Born, M., 585 Bergersen, B., s e e Gooding, R.J., 591 Bouard, A.D., s e e Fortini, A., 501 Bergersen, B., s e e Jo6s, B., 592 Bourgoin, J.C., 504 Bergersen, B., s e e Shrimpton, N.D., 591,592 Boyer, L.L., 585 Berker, A.N., 588, 592 Bradshaw, A.M., s e e Bonzel, H.E, 593 Berker, A.N., s e e Caflisch, R.G., 592 Bragg, W.L., 585 Besold, G., 587 Brechet, Y., s e e Louchet, E, 502 Betteridge, W., 24 Bretz, M., 592 Beuers, J., 131 Brezin, E., 591 Beuers, J., s e e Jonsson, S., 131 Brinkman, W.E, 589 Bhadra, R., s e e Okamoto, P.R., 587 Brinkman, W.E, s e e Coppersmith, S.N., 591 Biggers, R., s e e Mahmood, R., 589 Britun, V.E, s e e Pilyankevich, A.N., 501 Bilgram, J.H., 585 Brock, J.D., 587, 588 Binder, K., 593 Brock, J.D., s e e Aharony, A., 588 Birgeneau, R.J., 588, 589, 591 Broughton, J., 586 Birgeneau, R.J., s e e Aharony, A., 588 Broughton, J.Q., 592 Birgeneau, R.J., s e e Brock, J.D., 588 Broughton, J.Q., s e e Weeks, J.D., 593 Birgeneau, R.J., s e e D'Amico, K.L., 588 Brown, G.S., s e e Heiney, P.A., 592 Birgeneau, R.J., s e e Dimon, P., 592 Brown, G.S., s e e Stephens, EW., 591 Birgeneau, R.J., s e e Erbil, A., 592 Brown, L.M., s e e Vidoz, A.E., 68, 439 Birgeneau, R.J., s e e Heiney, P.A., 592 Brown, N., 133, 182, 185 Birgeneau, R.J., s e e Mochrie, S.G.J., 592, 593 Brown, N., s e e Marcinkowski, M.J., 25, 66 Birgeneau, R.J., s e e Nagler, S.E., 588 Brown, ED., s e e Loginov, Y.Y., 502 Birgeneau, R.J., s e e Nuttall, W.J., 594 Brown, S.A., s e e Kumar, K.S., 184 Birgeneau, R.J., s e e Specht, E.D., 591,592 Bruch, L.W., s e e Gottlieb, J.M., 592 Birgeneau, R.J., s e e Stephens, P.W., 591 Bruin, C., s e e Bakker, A.E, 590 Bishop, D.J., 588 Bruinsma, R., 589 Bishop, D.J., s e e Gammel, P.L., 590 Bruinsma, R., s e e Aeppli, G., 588 Bishop, D.J., s e e Grier, D.G., 591 Brtimmer, O., 504 Bishop, D.J., s e e Murray, C.A., 591 Buffat, Ph., 586 Bishop, D.J., s e e Safar, H., 591 Burstall, H.E, 24 Bishop, J., s e e Betteridge, W., 24 Burta-Gapanovich, L.N., s e e Deryagin, B.V., 501 Bishop, J.F.W., 185 Bush, E, 503 Bjurstrom, M.R., s e e Jin, A.J., 592 Butler, D.M., 588, 592 Blackburn, M.J., s e e Shechtman, D., 133 Bladon, P., 594 Blair, D.G., 184 Caflisch, R.G., 592 Blanchard, C., s e e Levade, C., 504 Cahn, J.W., 587 Boas, W., s e e Schmid, E., 182 Cahn, J.W., s e e Hilliard, J.E., 587 Bohr, J., s e e D'Amico, K.L., 592 Cahn, J.W., s e e Kikuchi, R., 438 Bohr, J., s e e McTague, J.P., 592 Cahn, R.W., 585 Bohr, J., s e e Nielsen, M., 592 Caillard, D., 131-134, 437, 439, 504 Bolle, C.A., s e e Grier, D.G., 591 Caillard, D., s e e Bonneville, J., 440 Bollman, W., 25 Caillard, D., s e e Cl6ment, N., 67, 131, 132, 437 Bondarenko, I.E., 502 Caillard, D., s e e Couret, A., 131, 133, 439, 440 Bonneville, J., 67, 132, 182-184, 252, 437, 440 Caillard, D., s e e Farenc, S., 131, 134 Bonneville, J., s e e Baluc, N., 132, 183, 436, 439, 441 Caillard, D., s e e Fnaiech, M., 502 Bonneville, J., s e e Spfitig, P., 132, 183, 436, 437 Caillard, D., s e e Levade, C., 503 Caillard, D., s e e Louchet, F., 504 Bonneville, J., s e e Stoiber, J., 252 Caillard, D., s e e Mol6nat, G., 67, 131-133, 251,252, Bontemps, C., 67, 131, 132, 251,436, 439 437, 439, 440 Bontemps-Neveu, C., 435 Caillard, D., s e e Paidar, V., 132, 440 Bonzel, H.P., 593 Caillard, D., s e e Vanderschaeve, G., 501 Booker, G.R., s e e Titchmarsh, J.M., 504
Author
Car, P., 504 Carlsson, L., 503 Cames, C.P., s e e Kimerling, L.C., 504 Caron, P., 440 Caron, P., s e e Khan, T., 436 Carrard, M., 440 Carrard, M., s e e Bonneville, J., 440 Carter, C.B., s e e Chiang, S.-W., 438 Carter, C.B., s e e Kuesters, K., 503 Castaldini, A., s e e Cavallini, A., 501 Catlow, C.R.A., s e e Agullo-Lopez, F., 503 Cavallini, A., 501 Celler, G.K., 587 Celli, V., 502 Chaikin, P.M., s e e Sirota, E.B., 590 Chakravarty, S., s e e Gann, R.C., 593 Champier, G., s e e George, A., 502 Chan, M.H.W., 588 Chan, M.H.W., s e e Feng, Y.P., 591 Chan, M.H.W., s e e Jin, A.J., 592 Chan, M.H.W., s e e Kim, H.K., 588, 592 Chan, M.H.W., s e e Migone, A.D., 592 Chan, M.H.W., s e e Shrimpton, N.D., 591 Chan, M.H.W., s e e Zhang, Q.M., 592 Chan, S.W., 587 Chandavarkar, S., 593 Chandler, D., s e e Weeks, J.D., 589 Chatelain, A., s e e Borel, J.-P., 586 Chaudhuri, A.R., s e e Patel, J.R., 500 Chen, C.Q., s e e Zhang, Y.G., 133 Chen, S., s e e Hu, G., 438 Chen, X., s e e Hu, G., 438 Chenal, B., s e e Lasalmonie, A., 439 Cheng, A., s e e Shrimpton, N.D., 592 Cheng, H., 593 Cheng, M., 588, 589 Chester, G.V., s e e Gann, R.C., 593 Chester, G.V., s e e Strandburg, K.J., 588, 589 Chester, G.V., s e e Zollweg, J.A., 590, 593 Chevenard, P., 24 Chiang, S.-W., 438 Chikawa, J., 587 Chikawa, J., s e e Sato, F., 504, 587 Chin, A.K., 501 Chin, S., s e e Lall, C., 67, 131, 182, 435 Chinone, N., s e e Kishino, S., 501 Chinone, N., s e e Nakashima, H., 501 Choi, S.K., 502, 503 Chou, C.T., 68, 132, 439 Chou, T., 594 Choyke, W.J., s e e Dean, P.J., 503 Christian, J.W., 185 Chrzan, D.C., 132, 184, 251 Chrzan, D.C., s e e Mills, M.J., 132, 184, 251,440
index
597
Chui, S.T., 589 Chui, S.T., s e e Ma, H., 591 Ciccotti, G., 587 Cladis, P.E., s e e Brinkman, W.F., 589 Clark, N.A., s e e Glaser, M.A., 589 Clarke, R., s e e Nagler, S.E., 588 Clarke, R., s e e Rosenbaum, T.F., 592 Cl6ment, N., 67, 131, 132, 437 Cl6ment, N., s e e Caillard, D., 131-133, 437, 439, 504 Cl6ment, N., s e e Couret, A., 131,439 Cockayne, D.J.H., 67 Cockayne, D.J.H., s e e Gom6z, A., 502 Cockayne, D.J.H., s e e Korner, A., 67, 437 Cockayne, D.J.H., s e e Ray, I.L.F., 502 Cole, M.W., s e e Shrimpton, N.D., 591 Cole, M.W., s e e Vidali, G., 591 Colella, N.J., 592 Colella, N.J., s e e Gangwar, R., 592 Collett, J., s e e Sirota, E.B., 590 Collins, R., 587 Combs, J.A., 594 Comsa, G., s e e Kern, K., 593 Conte, R., s e e Groh, P., 437 Conway, C.G., s e e Pfeil, L.B., 24 Cooman, B.C.D., s e e Kuesters, K., 503 Copley, S.M., 182, 251,436 Copley, S.M., s e e Tien, J.K., 26 Coppersmith, S.N., 591 Corbett, J.W., s e e Bourgoin, J.C., 504 Corey, C.L., s e e Vogel Jr., EL., 25 Cormia, R.L., 586 Cormia, R.L., s e e Mackenzie, J.D., 586 Costa, P., s e e Naka, S., 134 Cotterill, R.M.J., 585 Cottrell, A.H., 67, 251 Couchman, P.R., 586 Couderc, J.J., s e e Levade, C., 503 Couderc, J.J., s e e Vanderschaeve, G., 501 Coujou, A., s e e Caillard, D., 132, 439 Coujou, A., s e e Cl6ment, N., 132 Coujou, A., s e e Couret, A., 131, 133, 439 Coujou, A., s e e Lours, P., 132 Coulet, A.L., s e e Grange, G., 587 Coulomb, P., s e e Lours, P., 132 Couret, A., 67, 131-133, 251,436, 439, 440 Couret, A., s e e Caillard, D., 131-134, 437, 439, 504 Couret, A., s e e Cl6ment, N., 67, 131,437 Couret, A., s e e Farenc, S., 131, 133, 134 Couret, A., s e e Fnaiech, M., 502 Couret, A., s e e Levade, C., 503 Couret, A., s e e Mol6nat, G., 132, 133, 252, 440 Couret, A., s e e Sun, Y.Q., 67, 131,437 Court, S.A., 133, 439 Crabtree, G.W., s e e Kwok, W.K., 591
598
Author
Crary, S.B., 592 Crawford, R.C., 25, 134 Crestou, J., s e e Couret, A., 131,439 Crimp, M.A., 66, 67, 437 Crimp, M.A., s e e Sun, Y.Q., 67, 68, 131,437 Cserti, J., s e e Khantha, M., 132, 183, 251,436, 437 Cserti, J., s e e Vitek, V., 182, 183 Cui, J., 591,592 Cuitifio, A.M., 185, 441 Cummins, H.Z., s e e Awal, M.A., 587 Curwick, H., 435 Czernuszka, J.T., 504 D'Amico, K.L., 588, 592 D'Amico, K.L., s e e Specht, E.D., 591,592 Dadras, M.M., s e e Morris, D.G., 134 Daeges, J., 587 Dahl, K., s e e Tammann, G., 25 Dahm, A.J., s e e Guo, C.J., 590 Dahm, A.J., s e e Kusner, R.E., 590 Dahm, A.J., s e e Mehrotra, R., 590 Damgaard Kristensen, W., s e e Cotterill, R.M.J., 586 Darinskii, B.M., s e e Belyavskii, V.I., 504 Dash, J.G., 586, 588 Dash, J.G., s e e Ecke, R.E., 592 Davey, S.D., s e e Pindak, R., 589 David, R., s e e Kern, K., 593 Davidson, S.M., 504 Davies, R.G., 26, 132, 182, 251,436 Davies, R.G., s e e Beardmore, P., 131 Davies, R.G., s e e Stoloff, N.S., 134 Davies, R.G., s e e Thornton, P.H., 26, 67, 132, 182, 251,435 Daw, M.S., s e e Yoo, M.H., 184, 440 de Bussac, A., 132, 251,440 de Bussac, A., s e e Webb, G., 132 de Leeuw, S.W., s e e Kalia, R.K., 593 De'Bell, K., s e e Piercy, P., 592 Dean, P.J., 503 DeAngelis, H.M., s e e Kimerling, L.C., 504 den Nijs, M., 591 Denier van der Gon, A.W., s e e Frenken, J.W.M., 586 Denier van der Gon, A.W., s e e van der Veen, J.F., 586 Deryagin, B.V., 501 Deville, G., s e e Gallet, E, 590 Deville, G., s e e Glattli, D.C., 590 Devincre, B., 440 Diaz, J.O., s e e Nathal, M.V., 25, 439 Diehl, R.D., 593 Diehl, R.D., s e e Chandavarkar, S., 593 Diehl, R.D., s e e Fisher, D., 593 Dierker, S.B., 589 Dimiduk, D.M., 67, 131, 132, 182, 184, 251,436 Dimiduk, D.M., s e e Simmons, J.P., 133
index
Dimiduk, D.M., s e e Sriram, S., 133 Dimiduk, D.M., s e e Stucke, M.A., 133 Dimiduk, D.M., s e e Yoo, M.H., 26 Dimon, P., 592 Ding, E.J., 251 DiPietro, M.S., s e e Kumar, K.S., 184 Dirras, G., 134, 435 Dirras, G., s e e Beauchamp, P., 134 Donnelly, S.E., 587 Donnelly, S.E., s e e Rossouw, C.J., 586 Doucet, J., s e e Benattar, J.J., 590 Douin, J., 67, 184, 437, 438, 439 Douin, J., s e e Beauchamp, P., 438 Douin, J., s e e Saada, G., 438 Douin, J., s e e Veyssi~re, P., 26, 67, 68, 437-439 Dowling, W.E., 436 Downey, J., s e e Kwok, W.K., 591 Doyoma, M., s e e Cotterill, R.M.J., 585 Dresselhaus, M.S., s e e Erbil, A., 592 Drose, K., s e e Loginov, Y.Y., 502 Duburcq, V., s e e Glattli, D.C., 590 Ducastelle, F., 437 Duesbery, M.S., 26, 133, 134, 183, 184, 435, 440, 589 Duesbery, M.S., s e e Grossmann, B., 589 Duesbery, M.S., s e e Jo6s, B., 589 Dulieu, D., 24 Dumrongrattana, S., s e e Pitchford, T., 589 Dupouy, J.M., s e e R6gnier, P., 131 Durand, M.A., 585 Dutta, P., s e e Cheng, H., 593 Dzhafarov, T.D., 503 Echigoya, J., s e e Nemoto, M., 132, 440 Ecke, R.E., 592 Edwards, S.F., 586 Einstein, T.L., 592 Ekwall, R.A., s e e Brown, N., 185 Eldrup, M., 586 Elgin, R.L., 588 Ellis, D.E., s e e Cheng, H., 593 Eng, S., s e e Sanchez, J.M., 133, 438 Erb, U., 587 Erbil, A., 592 Erofeev, V.N., 502 Erofeev, V.N., s e e Bondarenko, I.E., 502 Erofeeva, S.A., 502 Ertl, G., s e e Bonzel, H.P., 593 Escaig, B., 67, 131, 184, 437, 440 Escaig, B., s e e Bonneville, J., 67, 184 Escaravage, C., s e e George, A., 502 Eschenfelder, A.H., 590 Esquivel, A.L., 504 Estrin, Y., s e e Kubin, L.P., 131 Etienne, B., s e e Glattli, D.C., 590
Author
Evans, J.H., s e e Eldrup, M., 586 Evans-Lutterodt, K.W., s e e Brock, J.D., 588 Ezz, S.S., 67, 68, 131, 182-184, 251,435-437, 441 Ezz, S.S., s e e Pope, D.P., 25, 66, 131, 182, 440 Fahey, D.A., s e e Crary, S.B., 592 Fain Jr., S.C., s e e Cui, J., 591,592 Fain Jr., S.C., s e e Taub, H., 588 Fan, W.C., 593 Farber, B.Y., 502 Farber, B.Y., s e e Nikitenko, V.I., 502 Farenc, S., 131, 133, 134 Farenc, S., s e e Caillard, D., 131 Farenc, S., s e e Couret, A., 131, 133, 439 Farenc, S., s e e Faress, A., 503 Faress, A., 503 Faress, A., s e e Levade, C., 503 Faress, A., s e e Vanderschaeve, G., 501 Farkas, D., 134, 183 Farkas, D., s e e Pasianot, R., 184, 440 Fat-Halla, N.K., s e e Takasugi, T., 131,436 Faust, W.L., s e e Williams, R.T., 504 Feibelman, J., s e e Knotek, M.L., 504 Feng, Y.P., 591 Fem,Sndez, J.E, 588, 594 Ferraz, A., 586 Ferreira, M.E, s e e Femfmdez, J.F., 588 Ferreira, O., s e e Tejwani, M.J., 590, 592 Fink, J., s e e vom Felds, A., 587 Finnis, M.W., 183 Finotello, D., 588 Fiory, A.T., 588 Fiory, A.T., s e e Hebard, A.F., 588 Fisher, D.S., 251,589, 590, 593 Fisher, D.S., s e e Coppersmith, S.N., 591 Fisher, D.S., s e e Fisher, M.E., 591 Fisher, D.S., s e e Narayan, O., 251 Fisher, M.E., 591 Fisher, M.E., s e e Huse, D.A., 591 Fisher, M.P.A., 590 Fisher, M.P.A., s e e Fisher, D.S., 590 Fisher, M.P.A., s e e Koch, R.H., 590 Fisher, R.M., s e e Marcinkowski, M.J., 25, 66 Flannery, B.P., s e e Press, W.H., 252 Fleischer, R.L., 26 Fleshier, S., s e e Kwok, W.K., 591 Flinn, P.A., 26, 67, 182, 251,438 Floyd, R.W., s e e Taylor, A., 24 Fnaiech, M., 502 Foglietti, V., s e e Koch, R.H., 590 Forbes, K.R., s e e Hernker, K., 132, 251 Foreman, A.J.E., 25 Fortini, A., 501 Fourdeux, A., s e e Kubin, L.P., 134
index
Foxall, R.A., s e e Duesbery, M.S., 184 Frahm, R., s e e Greiser, N., 592 Frangois, A., 435 Frank, F.C., 591 Frank, V.L.P., s e e Lauter, H.J., 592 Fraser, H.L., s e e Court, S.A., 133, 439 Fraser, H.L., s e e Vasudevan, V.K., 438 Freeland, P.E., s e e Patel, J.R., 502 Frenkel, D., 593 Frenkel, D., s e e Allen, M.P., 593 Frenkel, D., s e e Bladon, P., 594 Frenkel, D., s e e Udink, C., 592 Frenkel, J., 591 Frenken, J.W.M., 586 Frenken, J.W.M., s e e Pinxteren, H.M., 586 Frenken, J.W.M., s e e Pluis, B., 586 Friedel, J., 184, 439, 440 Filsch, H.L., 502 Filtzlen, G.A., 24 Frost, H.J., 252, 441 Fu, C.L., 435, 440 Fu, C.L., s e e Yoo, M.H., 26 Fujii, Y., s e e Sirota, E.B., 590 Fujita, H., s e e Moil, H., 587 Fujita, K., s e e Yamashita, Y., 501 Fujita, M., s e e Moil, H., 587 Fujiwara, T., 501,503 Fukatsu, S., s e e Yamashita, Y., 501 Gadijak, G.V., s e e Bedanov, V.M., 593, 594 Gal'vides, N.M., s e e Deryagin, B.V., 501 Gallagher, P.C.J., 131 Gallagher, W.J., s e e Koch, R.H., 590 Gallet, E, 590 Galligan, J.M., s e e Kung, J., 503 Gammel, P.L., 590 Gammel, P.L., s e e Grier, D.G., 591 Gammel, P.L., s e e Murray, C.A., 591 Gammel, P.L., s e e Safar, H., 591 Gangwar, R., 592 Gann, R.C., 593 Gao, Y., 438 Gaspailni, EM., s e e Finotello, D., 588 Gastaldi, J., s e e Grange, G., 587 Gay, J.M., s e e Pluis, B., 586 Geer, R., 590 Geer, R., s e e Huang, C.C., 589 George, A., 502 George, A., s e e Louchet, F., 502 George, E.P., 438 George, E.P., s e e Yoo, M.H., 26 Giamei, A.E, 25, 68, 438 Giamei, A.E, s e e Kear, B.H., 439 Gibala, R.G., s e e Dowling, W.E., 436
599
600
Author
Gibbs, G., s e e D'Amico, K.L., 592 Gierlotka, S., s e e Pluis, B., 586 Gignac, W., s e e Allen, M.P., 593 Gilman, J.J., 440 Gilman, J.J., s e e Johnston, W.G., 25 Gilman, J.J., s e e Westbrook, J.H., 500 Gilmer, G.H., s e e Broughton, J.Q., 593 Ginsberg, D.M., s e e Safar, H., 591 Glaberson, W.I., s e e Fiory, A.T., 588 Glaser, M.A., 589 Glattli, D.C., 590 Gleiter, H., s e e Daeges, J., 587 Gleiter, H., s e e Erb, U., 587 Godfrin, H., s e e Lauter, H.J., 592 Goeppert-Mayer, M., s e e Herzfeld, K.E, 585 Goldheim, D.L., s e e Westwood, A.R.C., 501 Gomer, R., s e e Menzel, D., 503 Gom6z, A., 502 Gondi, P., s e e Cavallini, A., 501 Goodby, J.W., s e e Geer, R., 590 Goodby, J.W., s e e Gray, G.W., 589 Goodby, J.W., s e e Huang, C.C., 589 Goodby, J.W., s e e Pindak, R., 589 Goodby, J.W., s e e Pitchford, T., 589 Goodby, J.W., s e e Stoebe, T., 590 Gooding, R.J., 591 Goods, S.H., 251 Goods, S.H., s e e Mills, M.J., 251 Goodstein, D.L., s e e Elgin, R.L., 588 Gordon, M.B., 591 Gorid'ko, N.Y., 501 Gornostyrev, Yu.N., s e e Greenberg, B.A., 132, 133 Goto, K., s e e Sato, E, 504 Gottlieb, J.M., 592 Gottschalk, H., 502, 503 Gottschalk, H., s e e Alexander, H., 503 Granato, A.V., 184 Granato, A.V., s e e Holder, J., 586 Granato, A.V., s e e Teutonico, L.J., 184 Grange, G., 587 Gray, G.W., 589 Greenberg, B.A., 132, 133, 441 Greene, R.L., s e e Greiser, N., 592 Greiser, N., 592 Grest, G., s e e Robbins, M.O., 590 Grier, D.G., 591 Griffiths, R.B., s e e Butler, D.M., 592 Grimditch, M., s e e Okamoto, P.R., 587 Grimes, C.C., 590 Griscom, D.L., 504 Griscom, D.L., s e e Tsai, T.E., 504 Groh, P., 437 Grossmann, B., 589 Guan, D.L., s e e Veyssi6re, P., 438
index
Guedon, J.Y., s e e Kubin, L.P., 134 Guenin, B.M., s e e Mehrotra, R., 590 Guillon, D., s e e Huang, C.C., 589 Guillop6, M., s e e Ciccotti, G., 587 Guiu, E, 252 G0nther, S., s e e Morris, D.G., 133, 435 Guo, C.J., 590 Gupta, A., s e e Koch, R.H., 590 Gupta, N.P., 585 Gurney, R.W., s e e Mott, N.F., 585 Gutmanas, E.Y., 503 Gutzow, I., 586 Haasen, P., 131,502, 504 Haasen, P., s e e Gutmanas, E.Y., 503 Haasen, P., s e e Jendrich, U., 502 Haberman, R., 184 Hagel, W.C., s e e Sims, C.T., 24, 26 Hahn, Y.D., s e e Whang, S.H., 133 Haken, H., 251 Hall, R.N., 501 Halperin, B.I., 585 Halperin, B.I., s e e Coppersmith, S.N., 591 Halperin, B.I., s e e Fisher, D.S., 589 Halperin, B.I., s e e Nelson, D.R., 585, 589 Halpin-Healy, T., 592 Hamana, T., s e e Saburi, T., 131,436 Hamao, N., s e e Mishima, Y., 436 Hammonds, E.M., s e e Birgeneau, R.J., 588, 591 Hamona, T., s e e Saburi, T., 184 Hanada, S., 134 Hanada, S., s e e Takasugi, T., 134 Hanneman, R.E., 501 Hansen, F.Y., 592 Hanson, D., s e e Alexander, W.O., 24 Hardcastle, S.E., s e e Zabel, H., 593 Harding, W., s e e Titchmarsh, J.M., 504 Harker, A.H., s e e Itoh, N., 504 Hart, E.W., 587 Hartig, C., s e e Schr6er, W., 134 Hartman, R.L., s e e Petroff, P., 501 Haussermann, E, 502 Havner, K.S., 184 Havner, K.S., s e e Hill, R., 185 Hayes, W., 503 Hazzledine, P.M., 67, 68, 437, 439 Hazzledine, P.M., s e e Chou, C.T., 68, 439 Hazzledine, P.M., s e e Couret, A., 67, 251,439 Hazzledine, P.M., s e e Crimp, M.A., 66, 437 Hazzledine, P.M., s e e Karnthaler, H.P., 439 Hazzledine, P.M., s e e Schneibel, J.H., 182, 437 Hazzledine, P.M., s e e Stucke, M.A., 133 Hazzledine, P.M., s e e Sun, Y.Q., 67, 68, 131, 133, 251,437, 438
Author
Hebard, A.E, 588 Hebard, A.E, s e e Fiory, A.T., 588 Hedges, J.N., 25 Heggie, M.I., 502, 504 Heggie, M.I., s e e Jones, R., 504 Heideman, A., s e e Larese, J.Z., 592 Heidenreich, R.D., 25 Heiney, P.A., 592 Heiney, P.A., s e e Birgeneau, R.J., 588, 591 Heiney, P.A., s e e Stephens, P.W., 591 Held, G.A., s e e Greiser, N., 592 Helgesen, G., 590 Hemker, K.J., 67, 132, 133, 184, 251,436, 437 Hemker, K.J., s e e Baluc, N., 67, 132, 184, 436, 438 Henderson, D., s e e Barker, J.A., 593 Henry, C.H., 503, 504 Heredia, EE., 131, 182, 183 Heredia, H.E, 435, 436 Hersh, H., 504 Herzfeld, K.F., 585 Hess, W., 590 Heuser, F.W., s e e Teutonico, L.J., 184 Hignett, H.W.G., 24 Hiki, Y., 587 Hilhorst, H.J., s e e Bakker, A.F., 590 Hill, R., 185 Hill, R., s e e Bishop, J.EW., 185 Hilliard, J.E., 587 Hilliard, J.E., s e e Cahn, J.W., 587 Hirakawa, S., s e e Takasugi, T., 133, 437 Hirsch, P.B., 25, 67, 68, 132, 184, 251,437, 438, 440, 441,502 Hirsch, P.B., s e e Chou, C.T., 68, 132, 439 Hirsch, P.B., s e e Couret, A., 67, 132, 251,436 Hirsch, P.B., s e e Ezz, S.S., 68, 183, 436, 437, 441 Hirsch, P.B., s e e Gom6z, A., 502 Hirsch, P.B., s e e Hazzledine, P.M., 439 Hirsch, P.B., s e e Silcox, J., 67 Hirth, J.P., 25, 67, 441,501,502 Hitzenberger, C., s e e Korner, A., 438 Ho, J.T., s e e Cheng, M., 588, 589 Hobart, R., 591 Hochman, R.E, s e e Kuczinski, G.C., 501 Holder, J., 586 Holt, D.B., s e e Yacobi, B.G., 502 Homann, A., s e e Melzer, A., 594 Hondros, E., s e e Chou, C.T., 68, 439 Hoover, N.E., s e e Hoover, W.G., 589 Hoover, W.G., 589 Hoover, W.G., s e e Ladd, A.J.C., 589 Horn, P.M., s e e Birgeneau, R.J., 591 Horn, P.M., s e e Brock, J.D., 588 Horn, P.M., s e e D'Amico, K.L., 588 Horn, P.M., s e e Dimon, P., 592
index
601
Horn, P.M., s e e Greiser, N., 592 Horn, P.M., s e e Heiney, P.A., 592 Horn, P.M., s e e Mochrie, S.G.J., 592, 593 Horn, P.M., s e e Nagler, S.E., 588 Horn, P.M., s e e Rosenbaum, T.E, 592 Horn, P.M., s e e Specht, E.D., 591,592 Horn, P.M., s e e Stephens, P.W., 591 Hornbecker, M.E, s e e Kear, B.H., 67, 132, 183 Home, R.W., s e e Hirsch, P.B., 25 Horton, J.A., 440 Horton, J.A., s e e Baker, I., 132, 440 Horton, J.A., s e e George, E.P., 438 Horton, J.A., s e e Schneibel, J.H., 437 Horton, J.A., s e e Veyssi~re, P., 438 Horton, J.A., s e e Yoo, M.H., 132, 184 Howe, L.M., 439 Howie, A., s e e Hirsch, P.B., 438 Hsieh, T.E., s e e Baluffi, R.W., 587 Hsiung, L.M., 68 Hu, G., 438 Huang, C.C., 589 Huang, C.C., s e e Geer, R., 590 Huang, C.C., s e e Pitchford, T., 589 Huang, C.C., s e e Stoebe, T., 590 Huang, C.C., s e e Viner, J.M., 590 Hug, G., 131, 133 Hug, G., s e e Lasalmonie, A., 439 Hui, S.W., s e e Cheng, M., 588, 589 Huse, D.A., 591 Huse, D.A., s e e Fisher, D.S., 590 Huse, D.A., s e e Safar, H., 591 Hutchinson, J.W., 185 Hutchinson, T.S., s e e Blair, D.G., 184 Ichihara, M., s e e Maeda, K., 502, 503 Ichihara, M., s e e Suzuki, K., 25, 68, 132, 439, 440 Ichihara, M., s e e Suzuki, T., 25, 66 Ichihara, M., s e e Takeuchi, S., 251 Ignatiev, A., s e e Fan, W.C., 593 Ihm, G., s e e Vidali, G., 591 Imai, H., s e e Fujiwara, T., 501,503 Imai, M., 502 Imura, T., s e e Nohara, A., 133 Indenbaum, V.N., s e e Greenberg, B.A., 133 Inui, H., 438 Iqbal, M.Z., s e e Davidson, S.M., 504 Ishida, K., s e e Kamejima, T., 501 Isozumi, S., s e e Ueda, O., 503 Ito, R., s e e Kishino, S., 501 Ito, R., s e e Nakashima, H., 501 Itoh, H., s e e Murakami, K., 503 Itoh, N., 503, 504 Iunin, Y.L., s e e Farber, B.Y., 502 Iunin, Y.L., s e e Nikitenko, V.I., 502
602
Author
Ivanov, M.A., s e e Greenberg, B.A., 132, 441 lwamoto, M., 501 Iwanaga, H., s e e Takeuchi, S., 501,502 Iyer, K.R., s e e Kuczinski, G.C., 501 Izumi, M., s e e Nohara, A., 133 Izumi, O., s e e Aoki, K., 184, 436 Izumi, O., s e e Hanada, S., 134 Izumi, O., s e e Kawabata, T., 133 Izumi, O., s e e Liu, Y., 133, 435, 438 Izumi, O., s e e Takasugi, T., 131, 133, 134, 436, 437 Jackson, K.A., s e e Celler, G.K., 587 Jancovici, B., 588 Janke, W., 589 Jendrich, U., 502 Jenkins, M.L., s e e Cockayne, D.J.H., 67 Jensen, E.J., s e e Cotterill, R.M.J., 586 Jesser, W.A., s e e Couchman, P.R., 586 Jian, N., 439 Jiang, N., 68 Jin, A.J., 592 Johnson, D., s e e Mahmood, R., 589 Johnson, W.D., 503 Johnson, W.L., s e e Cahn, R.W., 585 Johnston, T.L., 436 Johnston, T.L., s e e Beardmore, P., 131 Johnston, T.L., s e e Thornton, P.H., 26, 67, 132, 182, 251,435 Johnston, W.D., 501 Johnston, W.D., s e e Chin, A.K., 501 Johnston, W.G., 25 Jones, I.P., s e e Ngan, A.H.W., 68, 131,435, 439 Jones, I.P., s e e Yan, W., 439 Jones, R., 502, 504 Jones, R., s e e Heggie, M., 502, 504 Jonsson, S., 131 Jonsson, S., s e e Beuers, J., 131 Jo6s, B., 589, 592 Jo6s, B., s e e Duesbery, M.S., 589 Jo6s, B., s e e Gooding, R.J., 591 Jo6s, B., s e e Grossmann, B., 589 Jo6s, B., s e e Shrimpton, N.D., 591,592 Jorgensen, P.J., s e e Hanneman, R.E., 501 Jourdan, C., s e e Grange, G., 587 Joyce, J.C., s e e Morris, D.G., 133, 435 Jumojni, K., 436, 439 Kabler, M., s e e Celli, V., 502 Kaganer, V.M., s e e Peterson, I.R., 589 Kalia, R.K., 593 Kalia, R.K., s e e Cheng, H., 593 Kalia, R.K., s e e Vashishta, P., 593 Kamejima, T., 501
index
Kamioka, H., 587 K a n ~ f i , T . , s e e Kawabata, T., 133 Kancheev, O.D., s e e Mirkin, I.L., 25 Kapitulnik, A., s e e Grier, D.G., 591 Kapitulnik, A., s e e Murray, C.A., 591 Kardar, M., s e e Aharony, A., 588 Kardar, M., s e e Caflisch, R.G., 592 Kardar, M., s e e Halpin-Healy, T., 592 Kardar, M., s e e Paczuski, M., 588 Karkina, L.E., s e e Greenberg, B.A., 133 Karnthaler, H.P., 67, 439, 441 Karnthaler, H.P., s e e Baluc, N., 67, 184, 437, 438 Kamthaler, H.P., s e e Hazzledine, P.M., 437 Karnthaler, H.P., s e e Korner, A., 438 Karnthaler, H.P., s e e Mills, M.J., 67, 132, 251 Karnthaler, P., s e e Mills, M.J., 439 Karsten, K., 25 Kasami, A., s e e Iwamoto, M., 501 Kashyap, B.P., 131 Kato, M., s e e Jumojni, K., 436, 439 Kawabata, T., 133 Kawabata, T., s e e Takasugi, T., 134 Kear, B.H., 25, 67, 132, 133, 183, 436, 437, 439 Kear, B.H., s e e Copley, S.M., 182, 251,436 Kear, B.H., s e e Giamei, A.E, 68, 438 Kear, B.H., s e e Oblak, A.E, 439 Kelly, P.J., s e e Car, P., 504 Keown, S., 24 Keramidas, V.G., s e e Chin, A.K., 501 Kerins, J., s e e Kusner, R.E., 590 Kern, K., 593 Kestel, B.J., s e e Meng, W.J., 587 Khan, T., 436 Khan, T., s e e Caron, P., 440 Khantha, M., 132, 183, 251,436, 437, 440 Khantha, M., s e e Vitek, V., 182 Kikuchi, M., 501 Kikuchi, R., 438 Kim, H.-Y., s e e Shrimpton, N.D., 592 Kim, H.-Y., s e e Vidali, G., 591 Kim, H.K., 588, 592 Kim, H.K., s e e Zhang, Q.M., 592 Kimerling, L.C., 504 Kimerling, L.C., s e e Weeks, J.D., 503 Kimura, K., s e e Maeda, N., 501 Kishino, S., 501 Kishino, S., s e e Nakashima, H., 501 Kisielowski-Kemmerich, C., s e e Alexander, H., 502 Kisliuk, A., s e e Voronel, A., 587 Klassen, N.V., 503 Klein, M.L., s e e Jo6s, B., 592 Klein, M.L., s e e Moiler, M.A., 592 Klein, R., s e e Hess, W., 590 Kleinert, H., 586, 589
Author
Kleinert, H., s e e Janke, W., 589 Kluge, M., s e e Wolf, D., 587 Knotek, M.L., 504 Koch, R.H., 590 Koch, S.W., 593 Koch, S.W., s e e Abraham, EE, 591,593 Koehler, J.S., 25, 66, 184 Kohlstedt, D.L., s e e Chiang, S.-W., 438 Koiwa, M., s e e Yasuda, H., 441 Kojima, K., 501 Komiya, S., s e e Ueda, O., 503 Kontorowa, T., s e e Frenkel, J., 591 Koren, G., s e e Koch, R.H., 590 Kornblit, A., s e e Willens, R.H., 588 Korner, A., 67, 68, 131, 132, 251,437-439 Korner, A., s e e Schoeck, G., 439 Kortan, A.R., s e e Erbil, A., 592 Kortan, A.R., s e e Mochrie, S.G.J., 592, 593 K6ster, W., 26 Kosterlitz, J.M., 585, 588 Kosterlitz, J.M., s e e Lee, J., 593 Kotz6, I.A., 586 Kravchenko, V.Y., s e e Vardanyan, R.A., 504 Kremer, K., s e e Robbins, M.O., 590 Kristensen, J.K., s e e Kristensen, W.D., 586 Kristensen, W.D., 586 Krug, J., 252 Kubin, L.P., 131, 134 Kubin, L.P., s e e Devincre, B., 440 Kubin, L.P., s e e Naka, S., 134 Kubin, L.P., s e e Potez, L., 133, 437 Kubo, A., s e e Maeda, K., 501 Kuczinski, G.C., 501 Kuesters, K., 501,503 Kuhlmann-Wilsdorf, D., 586 Kuhlmann-Wilsdorf, D., s e e Kotz6, I.A., 586 Kulkarni, S.B., 502 Kumar, K.S., 184, 435 Kung, J., 503 Kuramoto, E., 133, 182, 184, 436 Kuramoto, E., s e e Suzuki, K., 25 Kuramoto, E., s e e Takeuchi, S., 67, 132, 182, 251, 436 Kuramoto, M.E., s e e Suzuki, K., 440 Kurnakov, N.S., 26 Kusner, R.E., 590 Kuz'menko, P.P., s e e Gorid'ko, N.Y., 501 Kwok, W.K., 591 Kyser, D.E, s e e Wittry, D.B., 504 Ladd, A.J.C., 589 Laird, C., s e e Wu, T.-Y., 185 Lall, C., 67, 131, 182, 435 Lam, N.Q., 587
index
Lambert, M., s e e Benattar, J.J., 590 Lan~on, E, s e e Gordon, M.B., 591 Landau, L.D., 585 Lang, D.V., s e e Henry, C.H., 504 Lang, D.V., s e e Kimerling, L.C., 504 Lapasset, G., s e e Potez, L., 133, 437 Larese, J.Z., 592 Larese, J.Z., s e e Zhang, Q.M., 592 Lartigue, C., s e e Simon, Ch., 593 Lasalmonie, A., 439 Lasalmonie, A., s e e Naka, S., 134 Lauter, H.J., 592 Lauter, H.J., s e e Taub, H., 588 Lee, J., 593 Lee, P.A., s e e Coppersmith, S.N., 591 Lefebvre, A., s e e Caillard, D., 504 Legrand, B., 131 Legrand, J.E, s e e Simon, Ch., 593 LeGuillon, J.C., 589 Leibfried, G., 184, 251 Left, R., 133, 438 Left, R., s e e Morris, D.G., 133, 435 Leung, C.H., s e e Williams, R.T., 504 Leung, P.W., s e e Zollweg, J.A., 590 Levade, C., 503, 504 Levade, C., s e e Faress, A., 503 Levade, C., s e e Louchet, F., 504 Levade, C., s e e Vanderschaeve, G., 501 Levelut, A.M., s e e Benattar, J.J., 590 Lewis, M., s e e Mahmood, R., 589 Li, J.C.M., 251 Li, Z.C., 133 Li, Z.C., s e e Whang, S.H., 133 Li, Z.R., s e e Migone, A.D., 592 Li, Z.X., 133 Li, Z.X., s e e Whang, S.H., 133 Liang, J.C., s e e Brock, J.D., 588 Liang, S.-J., 440 Lifshitz, E.M., s e e Landau, L.D., 585 Lin, D., s e e Wen, M., 132 Lin, W.N., s e e Esquivel, A.L., 504 Lindemann, EA., 585 Linker, G., s e e vom Felds, A., 587 Lipsitt, H.A., s e e Shechtman, D., 133 Lischner, D.J., s e e Celler, G.K., 587 Litster, J.D., s e e Aharony, A., 588 Litster, J.D., s e e Birgeneau, R.J., 589 Litster, J.D., s e e Brock, J.D., 588 Litzinger, J.A., 592 Litzinger, J.A., s e e Butler, D.M., 588, 592 Liu, C.T., 25 Liu, C.T., s e e Yoo, M.H., 132, 184, 436 Liu, C.Y., s e e Veyssi6re, P., 438 Liu, J.L., s e e Chan, S.W., 587
603
604 Liu, Y., 133, 435, 438 Liu, Y.H., s e e Arko, A.C., 436 L6fvander, J.P.A., s e e Court, S.A., 439 Loginov, Y.Y., 502 Loiseau, A., s e e Ducastelle, F., 437 Loiseau, A., s e e Hug, G., 133 Lomak, N.V., s e e Negrii, V.D., 503 Lomer, W.M., 67 Loretto, M.H., s e e Court, S.A., 439 Lothe, J., s e e Hirth, J.P., 67, 441,501,502 Louat, N.P., s e e Duesbery, M.S., 440 Louchet, F., 502, 504 Lours, P., 132 Lours, P., s e e Caillard, D., 132, 439 Lours, P., s e e Cl6ment, N., 132 L6wen, H., 586 Lowrie, R., 435 Lozovik, Y.E., s e e Bedanov, V.M., 594 Lu Da 24 Lu, Y.N., s e e Ding, E.J., 251 Lticke, K., s e e Granato, A.V., 184 Lticke, K., s e e Teutonico, L.J., 184 Lund, F., 586, 588 Lutsko, J.F., 587 Lutsko, J.F., s e e Phillpot, S.R., 587 Lutsko, J.F., s e e Wolf, D., 587 LUty, F., 504 Luzzi, D.E., s e e Inui, H., 438 Lyakhovets, V.D., s e e Martynyuk, M.M., 586 Lye, R.G., s e e Westwood, A.R.C., 501 Lyuksyutov, I., 588 Ma, H., 591 MacDonald, J.E., s e e Pluis, B., 586 Mackenzie, J.D., 586 Mackenzie, J.D., s e e Ainslie, G., 586 Mackenzie, J.D., s e e Cormia, R.L., 586 Madsen, J.U., s e e Cotterill, R.M.J., 586 Madsen, L.L., s e e Viswanathan, R., 594 Maeda, K., 501,502, 503 Maeda, K., s e e Nakagawa, K., 504 Maeda, K., s e e Nakano, K., 501 Maeda, K., s e e Takeuchi, S., 501,502 Maeda, K., s e e Yamashita, Y., 501 Maeda, N., 501,503, 504 Maeda, N., s e e Maeda, K., 502 Maeno, N., 586 Magerl, A., s e e Zabel, H., 593 Mahajan, S., s e e K. Chin~ A., 501 Mahmood, R., 589 Maitland, G.C., 591 M a k , A . , s e e Specht, E.D., 591 Makin, M.J., s e e Foreman, A.J.E., 25 Malozemoff, A.P., 590
Author
index
Mann, J.A., s e e Kusner, R.E., 590 Maradudin, A.A., 586 Marcinkowski, M.J., 25, 66, 436 Maree, P.M., s e e Frenken, J.W.M., 586 Marklund, S., 502, 504 Marklund, S., s e e Jones, R., 504 Marsh, J.H., 24 Martens, A., 25 Martin, C.J., 585 Martin, J.L., s e e Baluc, N., 132, 183, 436, 439, 441 Martin, J.L., s e e Bonneville, J., 132, 182, 183, 252, 437, 440 Martin, J.L., s e e Carrard, M., 440 Martin, J.L., s e e Sp~itig, P., 132, 183, 436, 437 Martin, J.L., s e e Stoiber, J., 252 Martynyuk, M.M., 586 Mast, D.B., s e e Guo, C.J., 590 Masuda, K., s e e Murakami, K., 503 Matsui, J., s e e Kamejima, T., 501 Matsui, M., 503 Matsui, M., s e e Sato, F., 587 Matsumoto, A., 133 Maurer, R., s e e Balluffi, R.W., 587 McClain, B.R., s e e Brock, J.D., 588 McEvily, A.J., s e e Johnston, T.L., 436 McLean, M., s e e Chou, C.T., 68, 439 McQueency, D.C., s e e Agnolet, G., 588 McTague, J.P., 588, 592 McTague, J.P., s e e Allen, M.P., 593 McTague, J.P., s e e Frenkel, D., 593 McTague, J.P., s e e Nielsen, M., 592 McTague, J.P., s e e Novaco, A.D., 588 Mdivanyan, B.E., 502 Meakin, J.D., 437 Mecking, H., s e e Schr6er, W., 134 Mehrotra, R., 590 Mehrotra, R., s e e Guo, C.J., 590 Melzer, A., 594 Meng, W.J., 587 Menzel, D., 503 Mera, Y., s e e Yamashita, Y., 501 Mermin, N.D., 587 Merzhanov, K.I., s e e Deryagin, B.V., 501 Meyer, R.B., s e e Dierker, S.B., 589 Migone, A.D., 592 Mihara, M., s e e Choi, S.K., 502, 503 Mikhieva, V.T., s e e Shishokin, V.P., 25 Mikolla, S.E., s e e Gao, Y., 438 Miller, B.I., s e e Johnston, W.D., 501,503 Miller, M.M., s e e Kwok, W.K., 591 Mills, M.J., 67, 132, 184, 251,439, 440 Mills, M.J., s e e Baluc, N., 67, 184, 437, 438, 441 Mills, M.J., s e e Chrzan, D.C., 132, 184, 251
Author
Mills, M.J., s e e Hemker, K.J., 67, 132, 133, 184, 251, 436 Mills, M.J., s e e Yoo, M.H., 26 Miner, R.V., s e e Nathal, M.V., 25, 439 Miracle, D.B., s e e Farkas, D., 134 Mirkin, I.L., 25 Mishima, Y., 26, 435, 436 Mishima, Y., s e e Miura, S., 182, 436 Mishima, Y., s e e Suzuki, T., 131, 182, 435 Mishima, Y., s e e Tounsi, B., 132, 439 Mitchell, J.W., s e e Hedges, J.N., 25 Mitzi, D.B., s e e Grier, D.G., 591 Mitzi, D.B., s e e Murray, C.A., 591 Miura, S., 182, 436 Miura, S., s e e Suzuki, T., 25, 66, 131, 182, 435 Mizushima, S., 585 Mochrie, S.G.J., 592, 593 Mochrie, S.G.J., s e e Nagler, S.E., 588 Mockler, R.C., s e e Armstrong, A.J., 590 Mockler, R.C., s e e Tang, Y., 590 Mol6nat, G., 67, 131-134, 251,252, 437, 439, 440 Mol6nat, G., s e e Caillard, D., 131, 134 Mol6nat, G., s e e C16ment, N., 131,437 Mol6nat, G., s e e Couret, A., 131,439 Mol6nat, G., s e e Paidar, V., 132, 440 Moiler, M.A., 592 Moncton, D.E., 589 Moncton, D.E., s e e D'Amico, K.L., 588, 592 Moncton, D.E., s e e Dimon, P., 592 Moncton, D.E., s e e Heiney, P.A., 592 Moncton, D.E., s e e Nagler, S.E., 588 Moncton, D.E., s e e Nielsen, M., 592 Moncton, D.E., s e e Pindak, R., 589 Moncton, D.E., s e e Specht, E.D., 591,592 Moncton, D.E., s e e Stephens, P.W., 591 Monemar, B., 501,504 Montoya, K., s e e Kumar, K.S., 184 Morf, R.H., 590, 593 Morf, R.H., s e e Fisher, D.S., 589 Morrill, G.E., s e e Thomas, H.M., 594 Moil, H., 587 Morris, D.G., 133, 134, 435, 438 Morris, D.G., s e e Left, R., 133, 438 Morris, D.G., s e e Veyssi6re, P., 67, 438 Morris, M., 133 Morris, M.A., s e e Morris, D.G., 134 Moss, S.A., 24 Moss, W.C., s e e Hoover, W.G., 589 Mott, N.F., 585 Mouritsen, O.G., s e e Besold, G., 587 Moussa, F., 590 Muchy, D.C., s e e Louchet, F., 502 Mtilbacher, E.Th., s e e Kamthaler, H.P., 67, 439, 441 Mulford, R.A., 68, 183, 435
index
Muller, E.W., 25 Mtiller-Heinzerling, Th., s e e vom Felds, A., 587 Muller-Krumbhaar, H., s e e Saito, Y., 589 Munroe, P.R., 134 Murakami, K., 503 Murakami, K., s e e Umakoshi, Y., 133 Murata, Y., s e e Aruga, T., 593 Murayama, Y., s e e Maeda, K., 501 Murray, C.A., 590, 591 Murray, C.A., s e e Grier, D.G., 591 Murray, C.A., s e e Van Winkle, D.H., 590 Nabarro, F.R.N., 133, 440, 586 Nadeau, J.S., 501 Nadgoryni, E., 502 Nagler, S.E., 588 Nagler, S.E., s e e D'Amico, K.L., 588 Nagler, S.E., s e e Rosenbaum, T.F., 592 Nagler, S.E., s e e Specht, E.D., 591,592 Naidoo, K.J., 593, 594 Naka, S., 134 Naka, S., s e e Khan, T., 436 Nakagawa, K., 504 Nakagawa, K., s e e Takeuchi, S., 501 Nakahara, S., s e e Willens, R.H., 588 Nakamura, N., 134 Nakano, K., 501 Nakashima, H., 501 Nakashima, H., s e e Kishino, S., 501 Namba, Y., s e e Umakoshi, Y., 133 Narayan, O., 251 Nathal, M.V., 25, 439 Naumovets, A.G., s e e Lyuksyutov, I., 588 Nazmy, M., s e e Anton, D.L., 133 Negrii, V.D., 503 Neite, G., s e e Nembach, E., 25 Nelson, D.R., 585, 589 Nelson, D.R., s e e Berker, A.N., 588 Nelson, D.R., s e e Brezin, E., 591 Nelson, D.R., s e e Bruinsma, R., 589 Nelson, D.R., s e e Chou, T., 594 Nelson, D.R., s e e Halperin, B.I., 585 Nembach, E., 25 Nembach, E., s e e Mol6nat, G., 134 Nemoto, M., 132, 440 Nenno, S., s e e Pak, H.R., 68, 184, 437 Nenno, S., s e e Saburi, T., 131, 184, 436 Neubert, M.E., s e e Mahmood, R., 589 Neumann, D.A., s e e Zabel, H., 593 Newton, J.C., s e e Hansen, F.Y., 592 Ngan, A.H.W., 68, 131,435, 439 Nicholls, J.R., 437 Nicholson, R.B., s e e Hirsch, P.B., 438 Nielsen, M., 592
605
606
Author
Nielsen, M., s e e McTague, J.P., 592 Nikitenko, V.I., 502 Nikitenko, V.I., s e e Bondarenko, I.E., 502 Nikitenko, V.I., s e e Erofeev, V.N., 502 Nikitenko, V.I., s e e Farber, B.Y., 502 Ninomiya, T., 586 Ninomiya, T., s e e Celli, V., 502 Ninomiya, T., s e e Choi, S.K., 502, 503 Nishitani, S.R., s e e Yamaguchi, M., 182 Nix, W.D., s e e Hemker, K.J., 132, 251,436, 437 Noguchi, O., 26, 436 Noguchi, O., s e e Wee, D.M., 131, 182, 435 Noh, D.Y., s e e Brock, J.D., 588 Noh, D.Y., s e e Nuttall, W.J., 594 Nohara, A., 133, 134, 435 Northrop, D.C., s e e Davidson, S.M., 504 Nos6, S., 593 Notkin, A.B., s e e Greenberg, B.A., 133 Nounesis, G., s e e Huang, C.C., 589 Nounesis, G., s e e Pitchford, T., 589 Novaco, A.D., 588, 593 Novaco, A.D., s e e McTague, J.P., 588 Novikov, N.N., s e e Gorid'ko, N.Y., 501 Nuttall, W.J., 594 Oberg, S., s e e Jones, R., 504 Oblak, A.E, 439 Oblak, J.M., s e e Giamei, A.F., 68, 438 Oblak, J.M., s e e Kear, B.H., 437, 439 Ochiai, S., s e e Mishima, Y., 26, 435, 436 Ochiai, S., s e e Miura, S., 182 Ochiai, S., s e e Suzuki, T., 435 O'Connor, D.A., s e e Martin, C.J., 585 Ohta, H., s e e Liu, Y., 438 Okamoto, H., 25 Okamoto, P.R., 587 Okamoto, P.R., s e e Lam, N.Q., 587 Okamoto, P.R., s e e Meng, W.J., 587 Okamoto, P.R., s e e Wolf, D., 587 Oliver, J., 438, 440 Ono, S., s e e Liu, Y., 435, 438 Ono, S., s e e Takasugi, T., 133, 134, 437 Ookawa, A., 585 Orlov, A.N., s e e Rybin, V.V., 502 Ortiz, M., s e e Cuitino, A.M., 185, 441 Oshiyama, A., s e e C a r , P., 5 0 4 Osip'yan, Y.A., 501,503 Osip'yan, Y.A., s e e Erofeeva, S.A., 502 Osip'yan, Y.A., s e e Klassen, N.V., 503 Osip'yan, Y.A., s e e Negrii, V.D., 503 Osip'yan, Y.A., s e e Vardanyan, R.A., 504 Oslund, S., s e e Berker, A.N., 592 O'Sullivan, W.J., s e e Armstrong, A.J., 590 O'Sullivan, W.J., s e e Tang, Y., 590
index
Osvenskii, V.B., s e e Erofeev, V.N., 502 Ou-Yang, H.D., s e e Sirota, E.B., 590 Ourmazd, A., s e e Hirsch, P.B., 502 Oya, N., s e e Noguchi, O., 436 Oya, Y., s e e Miura, S., 182 Oya, Y., s e e Noguchi, O., 26 Oya, Y., s e e Suzuki, T., 133, 435 Oya, Y., s e e Wee, D.M., 131, 182, 435 Pace, N.G., s e e Saunders, G.A., 587 Pacheva, E., s e e Gutzow, I., 586 Paczuski, M., 588 Paidar, V., 67, 132, 183, 251,436, 438, 440 Paidar, V., s e e Ezz, S.S., 67, 131, 182, 251,435 Paidar, V., s e e Mol6nat, G., 133, 440 Paidar, V., s e e Yamaguchi, M., 66, 183, 438 Pak, H.R., 68, 184, 437 Pak, H.R., s e e Saburi, T., 131, 184, 436 Pansu, B., 590 Pansu, B., s e e Pieranski, Pa., 590 Pantelides, S.T., s e e C a r , P., 5 0 4 Paris, E., s e e Glattli, D.C., 590 Parthasarathy, T.A., s e e Dimiduk, D.M., 436 Pashley, D.W., s e e Hirsch, P.B., 438 Pasianot, R., 184, 440 Pasianot, R., s e e Farkas, D., 134 Passell, L., s e e Larese, J.Z., 592 Patel, J.R., 500, 502 Patel, J.R., s e e Frisch, H.L., 502 Pearson, D.D., s e e Anton, D.L., 437 Pearson, J., s e e Okamoto, P.R., 587 Peierls, R., 587 P61issier, J., s e e Louchet, E, 502, 504 Pelz, J.P., s e e Birgeneau, R.J., 591 Perepezko, J.H., s e e Daeges, J., 587 Pershan, P.S., 589 Pershan, P.S., s e e Sirota, E.B., 590 Peters, C., s e e Specht, E.D., 591 Peterson, I.R., 589 Petrenko, V.E, s e e Osip'yan, Y.A., 501,503 Petroff, P., 501 Petzer, G., s e e Gottschalk, H., 502 Petzow, G., s e e Beuers, J., 131 Peyrade, J.P., s e e Louchet, E, 504 Pfann, W.G., s e e Vogel Jr., EL., 25 Pfeil, L.B., 24 Pfliiger, J., s e e vom Felds, A., 587 Pfn•r, H., s e e Piercy, P., 592 Phillpot, S.R., 585, 587 Phillpot, S.R., s e e Lutsko, J.F., 587 Pickens, J.R., s e e Kumar, K.S., 184 Piel, A., s e e Melzer, A., 594 Pieranski, P., s e e Pansu, B., 590 Pieranski, Pa., 590
Author
Pieranski, Pa., s e e Pansu, B., 590 Pieranski, Pi., s e e Pansu, B., 590 Piercy, P., 592 Piggins, N., s e e Pluis, B., 586 Pilyankevich, A.N., 501 Pindak, R., 589 Pindak, R., s e e Cheng, M., 588, 589 Pindak, R., s e e Dierker, S.B., 589 Pindak, R., s e e Huang, C.C., 589 Pindak, R., s e e Moncton, D.E., 589 Pinxteren, H.M., 586 Pirouz, P., s e e Hirsch, P.B., 502 Pitchford, T., 589 Pleiner, H., 589 Pluis, B., 586 Pluis, B., s e e Frenken, J.W.M., 586 Pluis, B., s e e van der Veen, J.F., 586 Poirier, J.P., 586 Pokrovsky, V., s e e Lyuksyutov, I., 588 Pokrovsky, V.L., 591 Pollock, T.C., 131 Pollock, T.M., 25, 26 Polvani, R.S., s e e Strutt, P.R., 26 Ponomarev, M.V., s e e Greenberg, B.A., 133 Pontikis, V., s e e Ciccotti, G., 587 Pooley, D., 504 Pope, D.P., 25, 66, 131, 133, 182, 438, 440 Pope, D.P., s e e Ezz, S.S., 67, 131,182, 184, 251,435, 436 Pope, D.P., s e e George, E.P., 438 Pope, D.P., s e e Heredia, EE., 131, 182, 183 Pope, D.P., s e e Heredia, H.E, 435, 436 Pope, D.P., s e e lnui, H., 438 Pope, D.P., s e e Kuramoto, E., 133, 182, 184, 436 Pope, D.P., s e e LaU, C., 67, 131, 182, 435 Pope, D.P., s e e Liang, S.-J., 440 Pope, D.P., s e e Liu, C.T., 25 Pope, D.P., s e e Mulford, R.A., 68, 183, 435 Pope, D.P., s e e Paidar, V., 67, 132, 183, 251,436 Pope, D.P., s e e Tichy, C., 438 Pope, D.P., s e e Tichy, G., 67, 183, 185 Pope, D.P., s e e Umakoshi, Y., 67, 131, 182, 436 Pope, D.P., s e e Wee, D.M., 131, 182, 435 Pope, D.P., s e e Wu, Z.I., 435 Pope, D.P., s e e Wu, Z.L., 182 Pope, D.P., s e e Yamaguchi, M., 66, 67, 183, 438 Popper, K.R., 440 Porter, W.D., s e e George, E.P., 438 Porter, W.P., s e e Inui, H., 438 Potemski, R.M., s e e Monemar, B., 504 Potez, L., 133, 437 Pr~estgaard, E., s e e Kristensen, W.D., 586 Pratt, N., s e e Czernuszka, J.T., 504 Pratt, P.L., s e e Guiu, E, 252
index
607
Preparata, EP., 588 Press, W.H., 252 Price, G.D., s e e Poirier, J.E, 586 Puls, M.P., 26 Puschl, W., s e e Couret, A., 131 Putnam, EA., s e e Berker, A.N., 592 Qin, Q., 182, 183 Rabier, J., s e e George, A., 502 Rabier, J., s e e Veyssi~re, P., 438 Rabinovich, S., s e e Voronel, A., 587 Rainville, M.H., s e e Howe, L.M., 439 Ralph, B., s e e Taunt, R.J., 184 Ramakrishnan, T.V., 589 Ramaswami, B., s e e Sastry, S.M.L., 133, 438 Rand, W.H., s e e Giamei, A.F., 68, 438 Rao, S., s e e Dimiduk, D.M., 436 Rao, S.I., s e e Simmons, J.P., 133 Ravitz, S.F., s e e Abbaschian, G.J., 586 Rawlings, R.D., s e e Nicholls, J.R., 437 Rawlings, R.D., s e e Staton-Bevan, A.E., 67, 183, 251, 435, 436 Ray, I.L.F., 502 Ray, I.L.F., s e e Crawford, R.C., 25, 134 Ray, I.LF., s e e Cockayne, D.J.H., 67 Read, W.T., 25, 586 Redhead, P.A., 503 R6gnier, P., 131 Rehn, L.E., s e e Meng, W.J., 587 Rehn, L.E., s e e Okamoto, P.R., 587 Reichl, L.E., 184 Ren, Q., s e e Jo6s, B., 589 Rentenberger, C., s e e Karnthaler, H.P., 67, 439, 441 Reppy, J.D., s e e Agnolet, G., 588 Reppy, J.D., s e e Bishop, D.J., 588 Reusch, E., 25 Reynaud, F., s e e Fnaiech, M., 502 Rice, J.P., s e e Safar, H., 591 Rice, J.R., 185 Rice, J.R., s e e Asaro, R.J., 185 Rice, J.R., s e e Hill, R., 185 Richardson, G.Y., s e e Duesbery, M.S., 26, 183 Richter, D., s e e Larese, J.Z., 592 Ricolleau, C., s e e Ducastelle, F., 437 Riedel, E.K., s e e Solla, S.A., 588 Rieu, J., s e e Kubin, L.P., 134 Rigby, M., s e e Maitland, G.C., 591 Riste, T., 587 Rivaud, G., s e e Levade, C., 504 Robbins, M.O., 590 Robinson, McD., s e e Celler, G.K., 587 Roccasecca, D.D., s e e Chin, A.K., 501
608
Author
Rogers, D.H., s e e Blair, D.G., 184 Rosenbaum, T.E, 592 Rosenbaum, T.F., s e e Nagler, S.E., 588 Rosenman, I., s e e Simon, Ch., 593 Rtisner, H., s e e Mol6nat, G., 134 Rossouw, C.J., 586 Rossouw, C.J., s e e Donnelly, S.E., 587 Roy, J.A., s e e Schulson, E.M., 435 Ruan, Y.-Z., s e e Guo, C.J., 590 Rudge, W.E., s e e Abraham, EE, 591 Russell, W.B., 590 Rybin, V.V., 502 Saada, G., 67, 131, 132, 134, 251,436--440 Saada, G., s e e Bontemps, C., 131,436 Saada, G., s e e Shi, X., 441 Saburi, T., 131, 184, 436 Saburi, T., s e e Pak, H., 68 Saburi, T., s e e Pak, H.-R., 437 Saburi, T., s e e Pak, H.R., 184 Sadananda, K., s e e Duesbery, M.S., 440 Safar, H., 591 Sahoo, D., s e e Venkataraman, G., 587 Saito, M., s e e Kikuchi, M., 501 Saito, Y., 589 Saka, H., 435 Saka, H., s e e Matsumoto, A., 133 Saka, H., s e e Nohara, A., 133 Saka, H., s e e Zhu, Y.M., 133, 134, 435 Sakamoto, K., s e e Maeda, K., 501 Sakka, Y., s e e Nakamura, N., 134 Sanchez, J.M., 133, 438 Sass, S.L., s e e Yoo, M.H., 26 Sastry, S.M.L., 133, 438 Sato, A., s e e Jumojni, K., 436, 439 Sato, E, 504, 587 Sato, E, s e e Chikawa, J., 587 Sato, M., s e e Maeda, K., 501 Sato, T., s e e Hanada, S., 134 Saunders, G.A., 587 Sauthoff, G., 26 Savchenko, I.B., s e e Osip'yan, Y.A., 501,503 Saville, D.A., s e e Russell, W.B., 590 Savino, E.J., s e e Farkas, D., 134, 183 Savino, E.J., s e e Pasianot, R., 184, 440 Sch~iublin, R., s e e Baluc, N., 67, 132, 184, 436, 438 Schaumburg, H., 501 Schaumburg, H., s e e Haussermann, E, 502 Scheerer, B., s e e vom Felds, A., 587 Schildberg, H.P., s e e Lauter, H.J., 592 Schmid, E., 182 Schmidt, O., s e e Volmer, M., 586 Schneemeyer, L.E, s e e Gammel, P.L., 590 Schneibel, J.H., 182, 437
index
Schneibel, J.H., s e e George, E.E, 438 Schneibel, J.H., s e e Hazzledine, P.M., 437 Schnitker, J., s e e Naidoo, K.J., 593, 594 Schoeck, G., 439, 440 Schoeck, G., s e e Couret, A., 131 Schoeck, G., s e e Korner, A., 439 Schowalter, W.R., s e e Russell, W.B., 590 Schreiber, J., s e e BrOmmer, O., 504 Schr0er, W., 134 Schrtiter, W., s e e George, A., 502 Schr/Ster, W., s e e Schaumburg, H., 501 Schulson, E.M., 435 Schulson, E.M., s e e Baker, I., 132, 439 Schulson, E.M., s e e Howe, L.M., 439 Schulze, D., 25 Schwartz, D.K., s e e Viswanathan, R., 594 Seeger, A., s e e Schoeck, G., 440 Seitz, E, s e e Koehler, J.S., 25, 66, 184 Sen, S., s e e Esquivel, A.L., 504 Seshadri, R., 590 Shamos, M.I., s e e Preparata, EP., 588 Shang-Keng Ma 251 Shapiro, J.N., 585 Shea, EA., s e e Novaco, A.D., 593 Shechtman, D., 133 Sheinkman, M.K., 504 Shi, X., 441 Shiba, H., 591 Shikhsaidov, M.S., 503 Shikhsaidov, M.S., s e e Klassen, N.V., 503 Shikhsaidov, M.S., s e e Mdivanyan, B.E., 502 Shikhsaidov, M.S., s e e Osip'yan, Y.A., 503 Shinohara, T., s e e Jumojni, K., 439 Shirai, S., s e e Chikawa, J., 587 Shirai, Y., s e e Yamaguchi, M., 182 Shiraki, Y., s e e Yamashita, Y., 501 Shishokin, V.E, 25 Shockley, W., 585 Shrimpton, N.D., 591,592 Shu, Q.S., s e e Ecke, R.E., 592 Silcox, J., 67 Simmons, G., 585 Simmons, J.P., 133 Simon, Ch., 593 Sims, C.T., 24, 26 Sinclair, J.E., s e e Finnis, M.W., 183 Sinha, S.K., 587 Sinha, S.K., s e e Sirota, E.B., 590 Siol, M., 586 Sirota, E.B., 590 Sitch, P., s e e Jones, R., 504 Skjeltorp, A.T., 590 Skjeltorp, A.T., s e e Helgesen, G., 590 Slonczweski, J.C., s e e Malozemoff, A.P., 590
Author
Small, M.B., s e e Monemar, B., 504 Smallman, R.E., s e e Ngan, A.H.W., 68, 131,435, 439 Smallman, R.E., s e e Yan, W., 439 Smimov, L.V., s e e Greenberg, B.A., 133 Smith, E.B., s e e Maitland, G.C., 591 Snow, D.B., s e e Anton, D.L., 133, 437 Sodani, Y., 132, 184, 440 Sodani, Y., s e e Vitek, V., 68, 182, 184, 251,440 Solla, S.A., 588 Solla, S.A., s e e Strandburg, K.J., 589 Somerscales, E.F.C., 24 Song, A.K., 503 Song, K.S., s e e Williams, R.T., 504 Sorensen, L.B., s e e Sirota, E.B., 590 Sp~itig, P., 132, 183, 436, 437, 441 Sp~itig, P., s e e Baluc, N., 132, 436 Sp~itig, P., s e e Bonneville, J., 183, 437 Specht, E.D., 591,592 Specht, E.D., s e e D'Amico, K.L., 588 Sperner, E, s e e K~ster, W., 26 Spohn, H., s e e Krug, J., 252 Sprenger, W.O., s e e Murray, C.A., 590 Spring, M.S., s e e Karnthaler, H.P., 439 Sriram, S., 133 Stan, M.A., s e e Guo, C.J., 590 Stankiewicz, J., s e e Femfmdez, J.E, 588, 594 Stanley, H.E., 251,588 Staton-Bevan, A.E., 67, 68, 183, 251,435, 436 Staubli, M., s e e Anton, D.L., 133 Steele, W.A., s e e Shrimpton, N.D., 591,592 Steinberg, V., s e e Voronel, A., 587 Stephens, P.W., 591 Stephens, EW., s e e Birgeneau, R.J., 588, 591 Stephens, EW., s e e Heiney, P.A., 592 Stephens, P.W., s e e Nielsen, M., 592 Stephenson, G.B., s e e Brock, J.D., 588 Sternbergh, D.D., s e e Hemker, K., 132, 251 Stewart, G.A., s e e Butler, D.M., 588, 592 Stewart, G.A., s e e Litzinger, J.A., 592 Stoebe, T., 590 Stoebe, T., s e e Geer, R., 590 Stoebe, T., s e e Huang, C.C., 589 Stoiber, J., 252 Stoiber, J., s e e Baluc, N., 183, 439 Stokes, R.J., s e e Cottrell, A.H., 251 Stoloff, N.S., 25, 26, 68, 134 Stoloff, N.S., s e e Davies, R.G., 26, 132, 182, 251,436 Stoloff, N.S., s e e Hsiung, L.M., 68 Stoloff, N.S., s e e Sims, C.T., 26 Stoneham, A.M., 503 Stoneham, A.M., s e e Hayes, W., 503 Stoneham, A.M., s e e Itoh, N., 504 Strandburg, K.J., 588, 589 Strandburg, K.J., s e e Lee, J., 593
index
609
Stroh, A.N., 184, 440 Strudel, J.L., 131 Strutt, ER., 26 Strzelecki, L., s e e Pansu, B., 590 Strzlecki, L., s e e Pieranski, Pa., 590 Stucke, M.A., 133 Sullivan, T.S., s e e Ecke, R.E., 592 Sumi, H., 504 Sumino, K., 503 Sumino, K., s e e Imai, M., 502 Sumino, K., s e e Yonenaga, I., 503 Sun, Y., 251 Sun, Y., s e e Couret, A., 251,436, 439 Sun, Y., s e e Mol6nat, G., 132, 252 Sun, Y.Q., 25, 67, 68, 131, 133, 134, 251,436-439 Sun, Y.Q., s e e Couret, A., 67, 132, 251 Sun, Y.Q., s e e Ezz, S.S., 68, 183, 436 Sun, Y.Q., s e e Hazzledine, P.M., 67, 68 Sun, Y.Q., s e e Hirsch, P.B., 68 Sun, Y.Q., s e e Jian, N., 439 Sun, Y.Q., s e e Jiang, N., 68 Sun, Y.Q., s e e Korner, A., 67 Sun, Y.Q., s e e Komer, A.K., 437 Surrendranath, V., s e e Mahmood, R., 589 Suter, R.M., s e e Colella, N.J., 592 Suter, R.M., s e e Gangwar, R., 592 Suter, R.M., s e e Greiser, N., 592 Suto, H., s e e Nemoto, M., 132, 440 Sutton, M., s e e Dimon, P., 592 Sutton, M., s e e Nagler, S.E., 588 Sutton, M., s e e Specht, E.D., 591 Suzuki, H., 586 Suzuki, K., 25, 68, 132, 439, 440 Suzuki, K., s e e Maeda, K., 502, 503 Suzuki, K., s e e Takeuchi, S., 251,502 Suzuki, K., s e e Yamashita, Y., 501 Suzuki, M., s e e Zabel, H., 593 Suzuki, T., 25, 66, 131, 133, 182, 435 Suzuki, T., s e e Mishima, Y., 435, 436 Suzuki, T., s e e Miura, S., 182, 436 Suzuki, T., s e e Noguchi, O., 26, 436 Suzuki, T., s e e Takeuchi, S., 502 Suzuki, T., s e e Tounsi, B., 132, 439 Suzuki, T., s e e Wee, D.M., 131, 182, 435 Sverbilova, T., s e e Voronel, A., 587 Sviridov, V.V., s e e Belyavskii, V.I., 504 Swalski, A.T., s e e Alexander, H., 502 Swope, W., s e e Bagchi, K., 594 Taggart, K., s e e Kashyap, B.P., 131 Tajbakhsh, A.R., s e e Brock, J.D., 588 Takagi, N., s e e Fujiwara, T., 501,503 Takahashi, T., s e e Liu, Y., 133, 438 Takasugi, T., 131, 133, 134, 435-437
610
Author
Takasugi, T., s e e Liu, Y., 133, 435, 438 Takasugi, T., s e e Yoshida, M., 134, 435, 438, 439 Takeuchi, S., 67, 132, 182, 251,436, 501-503 Takeuchi, S., s e e Maeda, K., 501-503 Takeuchi, S., s e e Maeda, N., 501,503, 504 Takeuchi, S., s e e Nakagawa, K., 504 Takeuchi, S., s e e Nakano, K., 501 Takeuchi, S., s e e Suzuki, K., 25, 68, 132, 439, 440 Takeuchi, S., s e e Suzuki, T., 131 Takita, K., s e e Murakami, K., 503 Takusagawa, M., s e e Fujiwara, T., 501 Takusagawara, M., s e e Fujiwara, T., 503 Talapov, A.L., s e e Pokrovsky, V.L., 591 Tallon, J.L., 585 Taluts, G.G., s e e Greenberg, B.A., 133 Tammann, G., 25 Tang, Y., 590 Tangri, K., s e e Kashyap, B.P., 131 Taub, H., 588 Taub, H., s e e Hansen, EY., 592 Taunt, R.J., 184 Taylor, A., 24 Taylor, F.W., 24 Taylor, G., s e e Hirsch, P.B., 441 Taylor, G.I., 184 Tejwani, M.J., 590, 592 Tessier, C., 592 Testardi, L.R., s e e Patel, J.R., 502 Testardi, L.R., s e e Willens, R.H., 588 Tetelman, A.S., s e e Johnston, T.L., 436 Teukolsky, S.A., s e e Press, W.H., 252 Teutonico, L.J., 184 Thiaville, A., s e e Brezin, E., 591 Thibault-Desseaux, J., s e e Louchet, E, 502 Thomas, G., s e e Vogel Jr., EL., 25 Thomas, H.M., 594 Thompson, A.W., s e e Dimiduk, D.M., 67, 184 Thompson, L.J., s e e Meng, W.J., 587 Thomson, N., s e e Loginov, Y.Y., 502 Thomson, R., s e e Celli, V., 502 Thornton, P.H., 26, 67, 132, 182, 251,435 Thouless, D.J., s e e Kosterlitz, J.M., 585 Tichy, G., 67, 183, 185, 438 Tichy, G., s e e Heredia, EE., 131, 182 Tien, J.K., 26 Tien, J.K., s e e Sanchez, J.M., 133, 438 Titchmarsh, J.M., 504 Toennies, J.P., s e e Frenken, J.W.M., 586 Tomizuka, A., s e e Takeuchi, S., 501 Toporov, Y.P., s e e Deryagin, B.V., 501 Torzo, G., s e e Taub, H., 588 Tosatti, E., 586 Tounsi, B., 132, 439 Toussaint, D., s e e Janke, W., 589
index
Townsend, ED., s e e Agullo-Lopez, E, 503 Toxvaerd, S., 593 Toyozawa, Y., 504 Travitzky, N., s e e Gutmanas, E.Y., 503 Trimble, L.E., s e e Celler, G.K., 587 Troxell, J.R., 504 Truman, J., 24 Tsai, T.E., 504 Tsien, F., 593 Tsurisaki, K., s e e Takasugi, T., 134 Tully, J.C., s e e Weeks, J.D., 503 Turnbull, D., 586 Turnbull, D., s e e Ainslie, G., 586 Turnbull, D., s e e Cormia, R.L., 586 Ubbelohde, A.R., 585 Udink, C., 592, 593 Ueda, O., 503 Ueda, O., s e e Maeda, K., 501 Ueta, S., s e e Jumojni, K., 436 Uhlmann, D.R., 586 Umakoshi, Y., 67, 131, 133, 182, 436 Umakoshi, Y., s e e Yamaguchi, M., 131, 134, 183 Umebu, I., s e e Ueda, O., 503 Umerski, A., s e e Jones, R., 504 Usami, N., s e e Yamashita, Y., 501 Vald6s, A., s e e Gallet, E, 590 Valleau, J.E, s e e Tsien, E, 593 VaU6s, J.L., 26 van der Elsken, J., s e e Udink, C., 593 van der Merwe, J.H., s e e Frank, EC., 591 van der Veen, J.E, 586 van der Veen, J.E, s e e Frenken, J.W.M., 586 van der Veen, J.E, s e e Pluis, B., 586 Van Vechten, J.A., 504 Van Winkle, D.H., 590 Van Winkle, D.H., s e e Murray, C.A., 590 Vanderschaeve, G., 501 Vanderschaeve, G., s e e Caillard, D., 504 Vanderschaeve, G., s e e Faress, A., 503 Vanderschaeve, G., s e e Levade, C., 503, 504 Vanderschaeve, G., s e e Louchet, E, 504 Vardanyan, R.A., 504 Vashishta, P., 593 Vashishta, P., s e e Kalia, R.K., 593, 594 Vasudevan, V.K., 438 Vasudevan, V.K., s e e Court, S.A., 133 Vasudevan, V.K., s e e Sriram, S., 133 Vechten, J.A.V., s e e Monemar, B., 504 Veilin, V.M., 501 Vekilov, Y.K., s e e Veilin, V.M., 501 Venkataraman, G., 587
Author
Ver Snyder, EL., 24 Vetterling, W.T., s e e Press, W.H., 252 Veyssi~re, P., 26, 67, 68, 131, 132, 183, 251, 435, 437-439 Veyssi~re, P., s e e Beauchamp, P., 134, 438 Veyssi~re, P., s e e Bontemps, C., 67, 131, 132, 251, 436, 439 Veyssi~re, P., s e e Caron, P., 440 Veyssi/~re, P., s e e Dirras, G., 134, 435 Veyssii~re, P., s e e Douin, J., 67, 184, 437, 438 Veyssi~re, P., s e e Francois, A., 435 Veyssi~re, P., s e e Hug, G., 133 Veyssii~re, P., s e e Korner, A., 438 Veyssii~re, P., s e e Oliver, J., 440 Veyssi~re, P., s e e Saada, G., 67, 131, 132, 134, 251, 436, 439, 440 Veyssi~re, P., s e e Shi, X., 441 Veyssi~re, P., s e e Tounsi, B., 132, 439 Vidali, G., 591 Vidoz, A.E., 68, 439 Vignia, B., 133 Viguier, B., s e e Hemker, K.J., 133 Vilches, O.E., s e e Ecke, R.E., 592 Vilches, O.E., s e e Tejwani, M.J., 590, 592 Villain, J., 591 Viner, J.M., 590 Viner, J.M., s e e Huang, C.C., 589 Viner, J.M., s e e Pitchford, T., 589 Vinokur, V.M., s e e Kwok, W.K., 591 Viswanathan, R., 594 Vitek, V., 68, 182-184, 251,435, 440 Vitek, V., s e e Ezz, S.S., 131, 184, 436 Vitek, V., s e e Gom6z, A., 502 Vitek, V., s e e Heredia, EE., 131, 182 Vitek, V., s e e Inui, H., 438 Vitek, V., s e e Khantha, M., 132, 183, 251,436, 437, 440 Vitek, V., s e e Paidar, V., 67, 132, 183, 251,436 Vitek, V., s e e Pope, D.P., 182 Vitek, V., s e e Sodani, Y., 132, 184, 440 Vitek, V., s e e Tichy, G., 67, 183, 185, 438 Vitek, V., s e e Umakoshi, Y., 67, 131, 182, 436 Vitek, V., s e e Wee, D.M., 131, 182, 435 Vitek, V., s e e Wu, Z.L., 182 Vitek, V., s e e Yamaguchi, M., 66, 67, 183, 438 Vogel Jr., EL., 25 Volmer, M., 586 vom Felds, A., 587 Voronel, A., 587 Voronoi, G.E, 588 Wagner, H., s e e Mermin, N.D., 587 Wainwright, T.E., s e e Alder, B.J., 593 Wakao, K., s e e Ueda, O., 503
index
611
Wakeham, W.A., s e e Maitland, G.C., 591 Wang, H., s e e Simmons, G., 585 Warner, M., s e e Edwards, S.F., 586 Watanabe, S., s e e Takasugi, T., 131, 133, 436, 437 Watanabe, W., s e e Hanada, S., 134 Watkins, G.D., 504 Watkins, G.D., s e e Troxell, J.R., 504 Webb, G., 132 Webb, G., s e e de Bussac, A., 132, 251,440 Wee, D.M., 131, 182, 435 Wee, D.M., s e e Suzuki, T., 133, 435 Weeks, J.D., 503, 589, 593 Weeks, J.D., s e e Broughton, J.Q., 593 Weeks, J.D., s e e Naidoo, K.J., 593 Wells, B.O., s e e Nuttall, W.J., 594 Welp, U., s e e Kwok, W.K., 591 Wen, M., 132 Wenk, R.A., s e e Murray, C.A., 590 Wessel, K., 502 Westbrook, J.H., 24-26, 182, 435, 500 Westervelt, R.M., s e e Seshadri, R., 590 Westgren, A., 25 Westwood, A.R.C., 501 Whang, S.H., 133 Whang, S.H., s e e Li, Z.C., 133 Whang, S.H., s e e Li, Z.X., 133 Wheeler, R., s e e Vasudevan, V.K., 438 Whelan, M.J., s e e Hirsch, P.B., 25, 438 White, M., s e e Taylor, F.W., 24 Whittenberger, J.D., s e e Kumar, K.S., 184 Whitworth, R.W., s e e Osip'yan, Y.A., 501 Widested, J.P., s e e Larese, J.Z., 592 Wight, D.R., s e e Titchmarsh, J.M., 504 Willens, R.H., 588 Williams, A.A., s e e Pluis, B., 586 Williams, C., s e e Moussa, E, 590 Williams, EI.B., s e e Gallet, F., 590 Williams, F.I.B., s e e Glattli, D.C., 590 Williams, J.C., s e e Dimiduk, D.M., 67, 184 Williams, P.M., 504 Williams, R.T., 504 Williams, R.T., s e e Song, A.K., 503 Williams, W.S., s e e Kulkami, S.B., 502 Wilsdorf, H.G., s e e Pollock, T.C., 131 Wilsdorf, H.G.E, s e e Kear, B.H., 25, 67, 133, 183, 436 Winter, E., s e e Hazzledine, P.M., 437 Wittry, D.B., 504 Wittry, D.B., s e e Esquivel, A.L., 504 Wolf, D., 587 Wolf, D., s e e Lutsko, J.F., 587 Wolf, D., s e e Phillpot, S.R., 585 W611, C., s e e Frenken, J.W.M., 586 Woodruff, D.P., 586
612
Author
Woolhouse, G.R., s e e Monemar, B., 501,504 Wu, F.Y., 592 Wu, T.-Y., 185 Wu, X., s e e Hu, G., 438 Wu, Y.P., s e e Sanchez, J.M., 133, 438 Wu, Z.I., 435 Wu, Z.L., 182 Xu, Q.,
see
Zhang, Y.G., 133
Yacobi, B.G., 502 Yakovenko, L.I., s e e Greenberg, B.A., 132 Yamaguchi, A., s e e Ueda, O., 503 Yamaguchi, M., 66, 67, 131, 134, 182, 183, 438 Yamaguchi, M., s e e Inui, H., 438 Yamaguchi, M., s e e Paidar, V., 183 Yamaguchi, M., s e e Umakoshi, Y., 133 Yamakoshi, S., s e e Ueda, O., 503 Yamashita, Y., 501 Yamashita, Y., s e e Maeda, K., 502 Yan, W., 439 Yasuda, H., 441 Ye, Y.-Y., s e e Fu, C.L., 435 Yip, S., s e e Lutsko, J.F., 587 ~ p , S . , s e e Phillpot, S.R., 585, 587 Yip, D., s e e Phillpot, S.R., 587 Yip, S., s e e Wolf, D., 587 Yodagawa, Y.M., s e e Mishima, Y., 26 Yodogawa, M., s e e Mishima, Y., 435, 436
index
Yoffe, A.D., s e e Williams, P.M., 504 Yokoyama, T., s e e Matsui, M., 503 Yonenaga, I., 503 Yoo, M.H., 26, 67, 132, 133, 184, 251,436, 440 Yoo, M.H., s e e Baker, I., 440 Yoo, M.H., s e e Fu, C.L., 435, 440 Yoo, M.H., s e e Hazzledine, P.M., 67 Yoo, M.H., s e e Horton, J.A., 440 Yoo, M.H., s e e Veyssi6re, P., 438 Yoshida, M., 134, 435, 438, 439 Yoshida, M., s e e Takasugi, T., 134, 435 Yoshinaga, H., s e e Suzuki, T., 131 Young, A.P., 585 Yussouff, M., s e e Ramakrishnan, T.V., 589 Zabel, H., 593 Zaretskii, A.V., s e e Osip'yan, Y.A., 501 Zasadzinski, J.A., s e e Viswanathan, R., 594 Zeppenfeld, P., s e e Kern, K., 593 Zhang, Q.M., 592 Zhang, Q.M., s e e Kim, H.K., 588, 592 Zhang, Y.G., 133 Zhemchuzhnii, S.F., s e e Kumakov, N.S., 26 Zhu, J., s e e Gao, Y., 438 Zhu, Y.M., 133, 134, 435 Zhu, Y.M., s e e Saka, H., 435 Zinn-Justin, J., s e e LeGuillon, J.C., 589 Zollweg, J.A., 590, 593 Zollweg, J.A., s e e Strandburg, K.J., 588
Subject Index
abrupt kink 454, 458 accepting mode 485, 486 activation energy 449, 461 for kink migration 446 for pinning 364, 408 activation enthalpy 154, 411 activation volume - apparent 277, 282, 424 -discontinuity 283, 411,412, 432 - effective 281-283 - strain dependence 284 408, 412, 424 adiabatic approximation 481 potential 481 Ag2MgZn 119 A13Ti 106, 290, 305, 317 L12-stabilized 258, 306 alignment of dipolar loops 305, 336, 343, 344, 393, 394 alkali halides 445, 495 alloy: - B2 alloy 114 - L12 alloy 80, 84, 103, 107 amorphization 517 amorphous silicon 500 anomalous slip mode 45 anomalous temperature dependence of flow stress 15-17 anthracene 445 anti-bonding (AB) state 481,496 anti-phase defect 491 anti-phase boundary (APB) 12, 29, 30, 59, 127, 138, 144 dissociated 62 -dragging 342, 352, 353, 399, 405 - energy 37, 316 composition dependence 314, 315 -energy ratio 299, 369-371,374 energy temperature dependence 315 - j u m p 323, 329-331,334-336, 373, 385, 389 relaxation 315-317, 340 - t u b e 49, 51, 55, 65, 318, 339, 342, 348, 349, 393-395, 397, 418, 433 - tube contrast 325, 340 wetting 291, 315 s e e anti-phase boundary applied shear stress 449 Ar 571 Arrhenius type of temperature dependence 464
athermal relaxation 484 athermal behavior 493, 494 atomistic alloying 23 studies 147 atoms at the dislocation core 448 Auger process 500
-
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-
-
-
t
h
e
o
r
e
t
i
c
a
B2 alloy 114, 119, 122, 255 bandgap energy 468, 480 basal slip 476 bending in the cube plane 337 Be 358, 413 beryllium 75 bifurcation 176 bond orientational order (BOO) 508, 521 structure factor 530 -correlation function 527, 529, 582 bond switching 496 bond-weakening mechanism 498 bonding (B) state 496 Br intercalated into graphite 575 breakaway 164, 378, 408, 411,412 Brown mechanism 117 Brown's model 104 bulk crystal 461 Burgers vector 447
l
-
-
-
-
capture of minority carriers 486 carrier capture 484, 486 cartier diffusion length 492 cathodoluminescence (CL) 462 C2D4 on graphite 571 C2H4 on graphite 571 CdS 473, 476--478, 487 CdSe 473 CdTe 473, 475, 477, 480, 487, 450 celular automata 220, 245 centre: - F-centre 496 - FA-centre 494 H-centre 496 Vk-centre 496 centre of symmetry 448 charge state 492 mechanisms 492 CL: s e e cathodoluminescence climb 446 closing jog (CJ) 326, 328, 335, 339, 356, 380, 381, 389, 398, 418, 419
-
-
e
n
e
r
g
y
-
-
-
-
-
A
P
B
-
:
613
614
Subject index
clusters 518 CO on graphite: phase diagram 570 CoHf 122, 255 Co3Pt 291 collision of kinks 451 colloidal suspensions 546, 548, 553-555 between flat plates wedge geometry 549 with uniform density 556 commensurate (C) phase 559, 562, 563 complex stacking fault (CSF) 30, 144 composition effects 260, 266, 364, 433 compound - I-VII 445 - II-VI 446--448, 456, 473 - IIb-VIb 445 - III-V 446--448, 451 - IV-IV 446 compounds 137 computer simulations 15, 421,576 - boundary conditions 577 continuum-based 193, 243 disrete model 220, 245 of finite time 577 finite-size effects 577 Lennard-Jones monolayer 579 configuration coordinate 481 diagram 481,484 constitutive model 170 constrictions 154 continuum theory of plastic deformation 170 core -configuration 260, 273, 274, 291,292, 294, 309, 314, 404, 429 - of configuration 362 structure of dislocation 138, 147 correlation functions 541 suspensions 549, 551 Yukawa system 583 CoTi 122 Co3Ti 103, 256-259, 289, 309, 314, 317, 318, 350, 354, 369 Co3(Ti, Ni) 259, 316 Cottrell-Stokes experiment 207, 268 Coulomb explosion mechanism 499 Coulomb system - colloidal particles 546 -dislocation properties 534 on liquid helium surface 547 Yukawa potential simulation 582 covalency 447 covalent crystals 446 CoZn 122 CoZr 255
-
- e x p a n s i o n s
-
-
- e f f e c t s
-
-
o
n
-
-
- c o l l o i d a l
-
- e l e c t r o n s
-
creep exhaustion 389, 433 - inverted (inverse) 286, 296, 337 - primary 389 rate temperature dependence 287 285, 388, 426, 431 critical dislocation length 455 resolved shear stress 140 stress 214 - temperature of RADG effect 464, 467, 491 cross slip 88, 107, 126, 138, 366, 402 -distance in cube plane 366, 375, 382, 383, 393, 394, 398, 400, 414, 416, 419 -double 78, 89, 93, 100, 128, 331,358, 373, 376, 381,394, 398, 400, 414, 418, 420, 421 - mechanism 189, 314, 361,364, 366, 379, 402 - transformation 327 Cs on alkali metals 572 CSF: see complex stacking fault 30 Cu-A1 alloy 354 Cu3Au 103, 263, 301,314, 316, 339 cube - cross slip 286, 306, 319, 325, 341,358 glide 80, 99 - plane 257 cube slip - primary 258, 276, 286, 289, 297, 319 strain rate dependence 288 CuBr 445, 450 CuCI 445, 450 cumulated strain 263 fl-CuZn 114, 212, 255 -
-
- t e s t
-
-
-
-
-
2D electron lattice on liquid helium surface 582 D2 on graphite 573 2D-XY model 522 dipole dislocation 325 dangling bond (DB) 447, 471,492, 496 -state 491,497 dark contrast 462, 487 debris 394 deep level 447, 492 defect - charge 481 mediated melting theory 536 molecule 482 deformation microstructure 317, 320, 359 degradation of electronic devises 446, 480 Delaunay triangulation 538, 550, 581 - colloidal suspensions 553-555 delay effect in PPE 477 density fluctuations 535 density wave theory of melting (Ramakrishnan) 535, 550, 582 -
-
Subject index
dependence of REDG on bandgap energy 480 length 455, 458, 460, 470, 490 - dislocation type 467 intensity 464, 490, 492, 498 - material 467 stress 465 temperature 464 desorption 499 diamond structure 447 diffusion 255, 286, 325, 352 dilute alloys 73 dipolar loops alignment 302, 335, 343, 347, 393 dipole dislocation 100, 322, 323, 522 directional coarsening 22 disclination 527, 535 dislocation 29 - (100) 320 - 6 0 ~ 302, 310, 323, 325, 352, 355, 448 30 ~ partial 448 90 ~ partial 448 - annihilation 281,339, 389, 393, 416, 428, 435 -bistable 371,377, 382 -bypassing 339, 416, 418-421 charge 493 collective behaviour 193, 212, 391 contrast 296, 299 - core 144 280, 284, 325, 408, 431 - dipole 340, 343 dissociation 146 -dynamical behaviour 354 - exhaustion 202, 203 - forest 66, 281,347, 398, 428 - j e r k y motion 356-358, 360 - kinked 322 - length 452 - locking 296, 297, 312, 357, 378, 416, 417, 423 - loops 449 mobility 446, 451 multiplication 279, 361,389, 393, 400, 426, 428, 433 -organization 320, 322, 361 pile-up 359 sub- 410 - t y p e 447 - a - t y p e 447, 448, 451 - ~ - t y p e 447, 448, 451 -unlocking 357, 416, 423 - vector systems 536 dislocation theory history of 9-15 of melting (DTM) 522 - of melting 2D, see KTHNY theory 520 -
-
d
i
s
l
o
c
a
-
e
x
c
i
t
a
t
t
i
-
-
-
-
-
i
o
o
n
n
of melting 3D 511 dislocation velocity 152, 202, 335, 356, 366, 425427 - fluctuations 202 intensity dependence 464 - length dependence 455, 458, 460 linear stress dependence 456 non-linear stress dependence 456 temperature dependence 464 disordering 317, 359, 395 dissociation 447, 491 composition dependence 297 - c l i m b 295, 309, 315, 352, 395, 405 dipole 321 - mode I 291,301,305, 311,312, 317, 320 - mode II 291,301,305, 309 reaction 448 - sub- 271,294, 295, 297, 300, 354, 361 D03 structure 119, 120 dodecahedral slip 352 domain A 257 domain B 257, 259, 285, 315 domain C 257, 264, 286, 325 domain wall (DW) 563-566 domain-wall lattice (DWL) 529, 534, 562 Ar on graphite 571 Br intercalated into graphite 567 dislocations 532, 568 - honeycomb 567 - honeycomb phase 563, 564 - Kr on graphite 572 - striped phase 563, 564, 573 doping effect 445-447, 451,465, 492, 493, 496 double cross slip 78, 89, 93, 100, 128 double etching technique 449 double kink formation energy 457 drag stress 387 ductilization 21 DWL: see domain wall lattice dynamical simulations 192 continuum-based 193, 243 -discrete model 220, 245, 250 -
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
d
e
n
s
i
t
y
615
-
-
edge dipoles 51, 55, 100 efficiency factor 484 Einstein relation 455 elastic - coefficients c~ 368, 369 constants 580 - torque 379, 395 elasticity - anisotropic 299, 367, 380, 409 298, 300, 404, 413 electric holes monolayers 557 -
-
l
i
n
e
a
r
616
Subject index
l~tromechanical effect 445 electron beam REDG excitation 443, 469 electron-hole generation rate density 475 electron-hole pair 473, 499 recombination 486 electron lattice 2D electron lattice on liquid helium surface 582 coupling 485, 491 system 482 electron-stimulated defect motion (ESDM) 481, 492 electron-stimulated desorption 482 electrons on liquid helium surface 547 electrostatic instability 496 elementary process 446 embrittlement 21 energized state 483 energy distribution 483 energy factor 450 entropy - DWL honeycomb 564 - DWL striped 566 of dislocations 512-514 - of kink formation 457 of kink migration 457 - vortices 524 ESDM: see electron-stimulated defect motion Eshelby twist 294 excitation - of REDG 490 - of PPE 473 excited state 481 mechanisms 494 excitonic mechanism 496 exhaustion 266, 283, 286, 337, 398, 400, 431,433 experimental uncertainties 261,267, 270, 271,273, 274, 279, 290, 298, 300, 320, 329, 357, 359, 380, 406 extrinsic photoplastic effect 446
-
-
-
-
-
-
-
face-centered-cubic metal 448 faulted dipole 300 f.c.c, crystal 364, 365, 379, 397, 401 Fe3A1 120 FeCo 119 Fe3Ga 257 Fe3Ge 256--258, 314, 317, 318 Fermi level 451,465, 492 flow stress 139, 473, 476 - anomaly 259, 269, 401 -conventional strain 267, 283 of stoichiometric deviation 17 in compression 269 in tension 269 -
-
-
e
f
f
e
c
t
s
- orientation dependence 258, 260, 270, 271,274, 277, 286, 317, 377, 425 - reversibility 264, 267, 286, 399 strain rate dependence 258, 266, 280, 401,407, 432 - temperature dependence 255, 257, 258, 359 dependence anomalous 15-17 flow stress peak 265, 276 orientation dependence 389 - temperature 276, 325 forest dislocation 66 formation energy of a double kink 457 of a kink 446 of a kink pair 452 of a soliton 491 forward biasing 446, 480 Frank-van der Merwe model 562 Frenkel-Kontorowa model 562 Frenkel pairs 496 fresh dislocations 487 -
-
t
e
m
p
e
r
a
t
u
r
e
-
-
-
-
-
GaAIAs/GaAs 478, 499 GaAIAsP/GaAs 478 GaAs 446, 457, 459, 462-469, 473, 479, 480, 484, 487, 491,494, 499 GaP 446, 450, 462, 465, 466, 469, 478, 487 Ge 300, 445-447, 450, 451, 459, 462, 464, 466, 479 Giamei lock 58, 64 glide 446, 448 activation energy 450 - plane 447 glissile 138 grain boundaries 519 in 3D melting 518, 519 grain boundary melting 552, 557 - theory of 535, 561,583 grain boundary pre-melting 519 -
-
6H hexagonal structure 447 H2 on graphite 573 6H-SiC 470, 480 Halperin, Nelson and Young theory, see KTHNY theory hard core potentials 578 hard disk system 578 hardening by illumination 445 3He on graphite 573 4He on graphite 573 heat capacity 525 Ar on graphite 571 - CO on graphite 572 -
Subject index - 2D-XY model 525 smectic liquid crystals 544-547 Xe on graphite 570 hetero-epitaxial films 458 hexagonal DWL 573 hexagonal-close-packed metal 448 hexatic phase 521,530 colloidal suspensions 551 - electric holes 557 LJ monolayer 580 Yukawa system 583 hexatic transition 526 HgSe 450 Hirth-Lothe theory 453, 489 honeycomb phase 567 HREM 32, 291,296, 302 hybridization: sp 3 hybridization 447 -
o
f
617
ionization 481 IR 474
-
-
-
-
InAs 450 incommensurate (IC) phase 559, 562, 563 indentation test 473 infrared (IR) light 474 InGaAsP/InGaP 478 lnGaAsP/InP 479 inner shell 499 - excitation 481,500 InP 446, 450, 462, 466, 469, 479 InSb 450, 459, 462, 464, 466, 479, 499 in-situ deformation experiments 73, 89, 116, 122, 312, 330, 335, 340, 354, 409, 411, 413, 415, 424, 430 instability - mechanical 509 - mechanical DWL 567 of the electron-lattice system 481 - shear 509, 510 - theory DTM 512, 515 - thermodynamic 508 thermoelastic 510 intensity dependence of REDG 464, 490, 492, 498 mterband excitation of PPE 475 intercalated monolayers 556 intercalates in graphite 575 intermediate region 580 mtermetallic compounds 7 mtermittent loading effect 457, 480 internal stress 277, 281,433 interstitial boron 492 intrinsic REDG 446 - absorption 473 stacking fault 448 ionic crystal 456, 473, 499 ionicity 445, 446 -
-
-
jerky flow 73, 83, 89, 113, 116, 129, 289 jog dragging 345 K on alkali metals 574 Kear-Wilsdorf (KW) configuration 47, 88, 326 Kear-Wilsdorf (KW) lock 12, 37, 40, 64, 91, 95, 105, 138, 190 - as sources for cube slip 339, 387 bending 386 -bending in cube plane 321, 336, 337, 341, 358, 365, 387, 400, 423, 430 -by-passing 380, 381, 393, 413, 417, 422, 429, 430 -complete 309, 311,322, 326, 328, 329, 331,361, 363, 373, 395, 399, 419, 421 - destruction 377, 414 - formation 328, 375 - incomplete 88, 92, 101,306, 309, 311,321,322, 326, 328, 329, 331, 357, 361, 363, 365, 368, 371,395, 399, 415 - kinked 390, 397, 399, 433 - transport 386 unlocking 413 - unzipping 376, 386 zipping 380, 381,386, 399, 415 Keating-type potential 457 kink - as source 389, 390, 397, 433 -coalescence 339, 384-387, 391, 400, 423, 425, 430, 433 collision 446 case 453, 455 regime 452, 461,490 -collisionless case 453, 455 -collisionless regime 452, 461,490 - configuration 330 -cusped/jogged/stepped 329, 339, 342, 345, 385, 393, 421,433 diffusion constant 454 model 453, 454, 487 -distribution 334, 335, 381,383, 385, 426, 431 - double 326, 373 dragging 397 -elementary (EK) 335, 339, 358, 381,383, 418 formation 381 formation energy 446 height 335, 384, 391,451 - macro- 327, 330, 381 - mean free path 451,461,492 - migration 416, 446, 451,487, 489, 490 -
-
-
-
- c o l l i s i o n
- c o l l i s i o n
-
- d i f f u s i o n
-
-
-
-
618
Subject index
migration energy 446 mobility 386 nucleation 446 pair formation 154, 451 - pair formation energy 452, 457 pair nucleation rate 451,452, 488 - polarization 335, 383 - simple 332, 376, 381,382, 384, 386, 393, 425 site 487, 488 -sliding 339, 385-387, 393, 414, 415, 433 - switch-over 332, 381, 382, 384, 387, 393, 400, 420, 425 -velocity 451 width 451 kink-based models 414 Kosterlitz and Thouless (KT) theory 520, 522 correlation function 522, 523 stability criterion: electrons on liquid helium surface 547 stability criterion: ideal 2D system 527, 580 stability criterion with a substrate 529, 572 - vortices 523 Kr 574 Kr on graphite - dislocations 532, 534 - experiment 572 - phase diagram 570, 573 KT: see Kosterlitz and Thouless KTHNY theory 520, 521 - b u l k systems 530 -correlation function, BOO 527 function, PO 526 - for ideal 2D systems 527 hexatic phase 528 of melting 2D 520 with molecular tilt 542 with substrate field 529 -
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Lennard-Jones -monolayers: dislocation properties 532 system clusters 518 system with grain boundaries 519 level occupation 497 lifetime of kinks 451 light illumination 445 light intensity for PPE and REDG 475 light-emitting diodes 446 light-emitting double heterostructures 480 Lindemann rule 520, 560 line energies 59 linear stress dependence of dislocation velocity 456 liquid crystal 540 liquid helium: electrons on the surface 547 local pinning 95, 407 local vibration 483 localization of slip 176 localized electronic states 447 locking process 93, 294, 312 locking-unlocking mechanism 79, 82, 413 Lomer 59, 294, 296 Lomer-Cottrell lock 31, 41, 59, 295, 365, 389, 419423 long dislocation 461 long range - order 138 motion 462 low temperature - mechanical properties 257-259, 269, 272, 288, 290 - microstructural properties 260, 302, 305, 306, 310, 319, 321,330, 363 -
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Llo structure 108 L12 alloy 80, 84, 103, 107 L 12 ordered 7 t phase 29 L12 structure 7, 71, 108, 137ff., 255ff. Laplacian roughening model 537 laser annealing 481,499 lattice friction 449 - o n cube plane 325, 337, 362, 365, 372, 379, 383, 399 - Peieds-Nabarro model 258, 365, 406 lattice heating 481 lattice matching 480 leading partial 449 length dependence of dislocation motion 455, 458, 460, 470, 490
macrokink, see also superkink 75, 80, 88, 98, 100 macroscopic properties 139, 156 magnetic bubbles, monolayers 560 magnetic dipole holes, monolayers 557 magnetic holes 557 many-body potential 147 MC method 576 MD method 576 mean field theories 514, 535 mechanical instability 509 melting rules 509, 560 Born's melting rule 509 Lindemann's melting rule 511 Mg 415 MGR mechanism 482 microdeformation 266, 281,389, 400, 432 microstructural stability 14, 23 minority carrier injection 480, 484, 492, 494, 495 mobility of dissociated dislocation 451 model: 2D-XY model 522
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Subject index molecular dynamics (MD) method 576 Monte Carlo (MC) method 576 Monte Carlo (MC) simulations 576 multi-modal particle size distribution 21 multiphonon emission 484 narrowly-spaced plane 448 nearest neighbour PLFP dislocation properties 533 PLFP computer simulations 583 negative PPE 473, 476, 493 negative TDFS 317, 343 negative-U (Anderson) 491 new compound bases 23 Ni3AI 33, 41, 51, 84, 256, 259, 274, 276-278, 291, 296, 297, 302, 314, 316, 348, 352, 354, 369, 411 Ni3A1 ('7~) 6-8, 137ff. Ni3A1Ni-rich binary 270, 274, 289, 294, 315, 316, 411,432 Ni3(A1, B) 260, 274, 275, 277, 316 Ni3(AI, Cr) 278 Ni3(A1, Hf) 259, 262, 263, 265, 268, 274, 275, 284, 288, 289, 302, 305, 306, 316, 322, 323, 329, 330, 332, 334, 336, 337, 339, 341, 342, 344-349, 351, 352, 355, 359, 383, 411,432 Ni3(AI, Hf, B) 264, 278, 280, 284, 285, 287, 289, 296, 353 Ni3(AI, Mo) 289 Ni3(A1, Nb) 259, 272, 273, 277, 289 Ni3(AI, Sn) 316 Ni3(A1, Ta) 260, 264, 272, 273-277, 281,283, 286, 288, 289, 296, 316, 411,432 Ni3(A1, Ti) 263, 264, 271,276, 278, 289, 296, 300, 316, 320, 325, 330, 352, 354, 369 Ni3(A1, V) 316 Ni3(AI, W) 266, 271,288, 289 Ni3(A1, Zr) 260, 271,274, 275 nickel-based alloys 84, 87 "7t phase 80 Ni3Fe 263, 290, 301, 316, 317, 369, 395 Ni3Ga 41, 84, 88, 264, 269, 273, 275, 277, 290, 291, 296, 301, 302, 305, 316, 334, 348, 354, 369, 411 Ni3Ge 257, 289, 296, 321,369 Ni3Ge-Fe3Ge 256-258 Ni3Mn 369 Ni3Si 256, 312, 316, 337, 354, 369 Ni3(Si, Ti) 259, 272, 290, 301,316, 321,330, 352, 363, 369 non-glide components 140 non-linear stress dependence 456 non-planar 148 -
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619
non-radiative recombination 484, 487, 491,492 non-Schmid effects 139, 170, 269 normal slip mode 33 n-type 447 octahedral - cross slip 272, 306, 319, 403 glide 84 - slip primary 258, 286, 319, 325, 357 order-disorder transition 15, 114, 121 ordered alloys 137 orientation factor - K 272, 404 - N 271,276, 373, 374, 404 - N t 276 - Q 271,275, 404 orientational epitaxy 530, 567 Ar on graphite 571 -
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partial Shockley 271,291,301 peak strengthening 21 peak temperature 30, 115, 119, 122, 139, 162, 255, 276 Peierls mechanism 76, 79, 82, 446, 487 schemes 451 Peierls potential 446 of the first kind 446, 447 - o f the second kind 446, 447, 454 Peierls stress 122 Peierls-Nabarro stress 454, 457 Peierls-type frictional forces 112, 129 phase: - 3' phase 7 - 3,t phase 80, 288, 330 -),t phase L 12 ordered 29 L12 phase 137 phonon kick energy 485, 487 -mechanism 447, 481,482 phonon softening mechanism 499 photoplastic effect (PPE) 445, 473, 476, 493 photomechanical effect 445 physisorbed monolayers 561 pinning 138 -correlations 195 pinning/depinning transition 192 pinning obstacle 487 planar defects 314 plasma 499 annealing model 499 PLFP (piecewise linear force potential) - dislocation properties 533 computer simulations 583 -
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620
Subject index
PO correlation function (Yukawa) 582 point obstacles 444, 452, 456 point-obstacle model 452, 456, 494 point pinning 356, 357, 380, 407, 426, 428 polarity 448 -effect 446, 447, 451 polygon tiling model 538 polysterene latex suspensions 548 Portevin-Le Chatelier (PLC) effect 74, 106, 129 positional order (PO) 508, 521,525 correlation function 526 positive TDFS 473 deformation microstructure 317, 319 inflection point in 274, 278, 281,289, 406 models 401 - strain dependence 266 post-mortem TEM observations 75, 82, 91, 116, 121,340, 430 Potts model 572 - helical 572 PP: see Peierls potential PPE: see photoplastic effect pre-exponential factor 461 precursor effects 510 to melting 519 prestraining 264, 267, 286 primary creep transient 143, 216, 229 principle of corresponding states 509, 511 prismatic slip 476 - in Be 429 Pt(111) 574 Pt3A1 31, 81,256-259, 318, 369 Pt3Cr 256 Pt3Ga 256 Pt3In 256 Pt4Sb 256 Pt3Sn 256 Pt3Ti 256 Pt3X 255 p-type 447 excited state 494 -
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radiation enhanced dislocation glide (REDG) 445 radiative recombination 462 rafting 22 rare gas monolayers - adsorbed on graphite 561,569 - dislocation properties 532 reaction coordinate 482 mode 486 recombination enhanced mechanism 482 reconstructed kink 492 reconstruction of dangling bonds 447, 491 -
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REDG: see radiation enhanced dislocation glide reduction of activation energy in REDG 484, 486, 487, 490, 491 relaxation process 480 repeated APB jump 335, 336, 340, 357, 358, 374, 376, 377, 382, 400, 414, 423 restricted slip 175 reversibility of PPE and REDG 445, 462, 481 roof type barriers 113 saddle-point mechanism 493, 494 saturation of PPE and REDG 445, 470, 486, 490 regime of dislocation motion 458 - stress in L12 278, 379, 390 scale invariance 214 areal-velocity scaling function 218 scanning electron microscopy (SEM) 462 schemes of Peierls mechanism 451 Schmid factor 269, 358 Schmid law violation 85, 269, 273, 429 screw dislocation 82, 112, 121, 139, 148, 448 see smallest double kink self-interstitial atoms 493 self-trapped exciton (STE) 496 self-trapped hole 496 SEM: see scaning electron microscopy SEM-CL 462, 463 semiconductor lasers 446 serrated flow, see also jerky flow 278, 289 sessile 138 shear band 176 shear instability 509, 510 shear modulus 450 Shockley partial 31, 146 - dislocation 447 shuffle planes 448 Si 446, 447, 450, 451, 456--458, 462, 464, 466, 469, 475, 479, 480, 491--493 SiC 446, 447, 478, 480, 491 - 6H-SiC 470, 480 Sio.9Geo. l 460 Si0.9Geo. 1 alloy 458 simulations - atomistic 305, 317, 354, 363 - deformation 378 sine-Gordon equation 563 SiO2 496 SISF 31, 43, 55, 59, 144 SISF dipole 302, 305, 348, 349, 351 SISF-dissociated 62 slip lines 72, 122, 317, 319 - localization 347, 351 - multiple 263, 288, 313, 347 -
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S
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D
K
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Subject index
-planarity 378, 381,382 single 263 smallest double kink (SDK) 487 formation 489, 491 smectic liquid crystal 540 - BOO structure factor 543 - defects 546 - phases 540, 541,542 studies of BOO 543 - thermodynamic studies of BOO 545 smooth kink 454 - model 457 softening by REDG 446 soliton 491 formation energy 491 site 491 solution hardening 19, 260 sp 3 hydralization 447 specific heat, s e e heat capacity splitting 138 stacking fault 144, 533, 573 - CSF 291,295, 298, 316 - energy 298 - SISF 291,343, 347 stainless steel 73 starting stress 450, 456 STE: s e e self-trapped exciton steady velocity 451 straight dislocation 489 487, 489, 491 strain - hardening 87, 110, 191,207 localization 139 strain rate 139 - constant 215, 432 - j u m p s 278, 279, 280 -sensitivity 86, 109, 121, 140, 163, 206, 258, 269, 277, 282, 408, 432 strength of coupling 486 stress -exponent of REDG effect 449, 450, 456, 465 increment in Cottrell-Stokes tests 268 - increment in strain-rate jump experiments 279 - relaxation 234, 267, 279, 281,434 stress-strain curve 473 string approximation 453 striped phase 564, 567 structure: L 10 structure 108 LI2 structure 7 structure factor 531 - 2D solid 525 - hexatic phase 530 s-type ground state 494 -
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621
superalloys 137, 143 - history of 3-9 superconducting flux lattice 560 superdislocation 12, 146, 290 - APB-coupled 313, 318, 326, 352 - dipole 343 - SISF-coupled 352, 359, 317 superheating 515-517 solids 515 superkink, s e e a l s o macrokink 48, 55, 190 distribution of lengths 198 superlattice extrinsic stacking fault (SESF) 45 superlattice intrinsic stacking fault (SISF) 29 superpartial 146 1 - 5(110) 291,300, 354 -
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1 - 5(110) dipole 343, 345, 346 1 - .~(112) 296, 302, 313, 317, 351 surface: "),-surface 144, 145 surface damage 261 melting 515, 516 switched partials 49, 53 system - l / r 12 577, 580, 582 - l / r 5 578 -
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T o, peak temperature 30, 139 temperature dependence 15, 93, 113, 123, 124, 138, 168, 237, 250, 255, 258, 464 tension--compression (TC)asymmetry 140, 158, 173 - composition dependence 273 - general 270, 364, 409 - in cube slip 287, 317 - maximum 273, 275 - neutrality or inversion line 273-275, 277 - temperature dependence 275 ternary compounds 20, 162 tetrahedral coordination 447 theory of Celli et al. 452 thermal - activation 237, 245, 250 depth of trap 486 kink 455 - reversibility 139, 207 thermodynamic instability 508 thermoelastic instability 510 thermomechanical treatment 23 threshold stress for octahedral slip 267, 285, 289, 408, 410, 416, 430 TiA1 108 TiAI (Llo) 255 -
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622
Subject index
Ti3AI (D019) 347 TiO2 500 trailing partial 449 transformation ofthe core 138 transients 277, 284 transmission electron microscopy (TEM) 29 twinning 109 type: - n-type 447 - p-type 447 unimolecular reaction 483 unlocking processes 93 unreconstructed kink 492
work hardening rate 65, 263, 268, 279, 283, 389, 397, 401 - composition dependence 263, 264, 276 - drop 342 orientation dependence 262, 264, 280 - peak 265 reversibility in temperature 264 work softening 264 wurtzite 447 -
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Xe 574 Xe on Ag(111) 570, 574 Xe on graphite - dislocations 532, 532 - experiment 569, 570 phase diagram 569 X-ray Lang method 449 yield 139 - anomaly 137, 189, 205 - criteria 170 - drop 279 function 172 - strength anomaly 189, 205 - stress anomaly 30, 45, 71, 93 -
vacancies 536 valence excitation 481 velocity - fluctuations 202 - measurement 462 Voronoi construction 538, 550 WCA - liquid 539 potential 536, 580 system 538 weak-beam image shift 298, 299 method 32 multiple images 300 resolution 298 - TEM 48 Weeks-Chandler-Anderson (WCA) potential 538 wide gap semiconductors 480, 491 widely-spaced plane 448 work hardening 143, 280, 397 -
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Zener anisotropy parameter A 369, 379 zincblende 447 ZnO 487 ZnS 446, 469, 470, 478, 480, 490 ZnSe 470, 478, 487 Zr3A1 256, 263, 352
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/3-CuZn 114, 212, 255 "7 phase 7, 71 "7 surface 144, 145 "7~, see Ni3A1, L 12 6H structure 447, 470, 480
