Dislocations in Solids, Volume 13

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Author: Frank R.N. Nabarro | John P. Hirth

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Dislocations in Solids Volume 13

This page intentionally left blank

Dislocations in Solids Volume 13

Edited by

F. R. N. NABARRO† School of Physics University of the Witwatersrand Johannesburg, South Africa and

J. P. HIRTH Hereford, AZ, USA

Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sidney • Tokyo North-Holland is an imprint of Elsevier

North-Holland is an imprint of Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK

First edition 2007 Copyright © 2007 Elsevier 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 Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: permissions@ elsevier.com. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice 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. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-444-51888-0 ISBN-10: 0-444-51888-6 ISBN-13: 978-0-444-85269-4 (set) ISBN-10: 0-444-85269-7 (set) ISSN: 1572-4859 For information on all North-Holland publications visit our website at books.elsevier.com Printed and bound in Italy 07 08 09 10 11

10 9 8 7 6 5 4 3 2 1

Preface As this volume neared completion, Professor Nabarro passed away on July 20, 2006. He worked on this volume until the very end with his usual insightful comments. The volume is dedicated to him in recognition of the foresight and drive that have led to the many volumes in the series. The focus of the present volume is on dislocations in a variety of materials, dislocation motion and observations of dislocations. Deshpande et al. present a model of dislocation plasticity in friction and wear. This represents the first in-depth dislocation dynamics modeling in the tribology area. Dislocations in piezoelectric crystals are discussed by Nowicki and Alshits: dislocation theory is just now starting to be applied in detail in this important class of materials. Ananthakrishna provides an analysis of the collective behavior of dislocations. Both statistical approaches and nonlinear dynamic approaches are covered. The role of dislocations and disconnections in martensitic transformations is presented by Pond et al. The focus is on atomistic mechanisms at a growing interface. Niewczas gives a detailed description of the conversion of dislocations as they are traversed by a twin boundary, and discusses a number of associated hardening mechanisms. The motion of dislocations in quasicrystals is discussed by Edagawa and Takeuchi. Spence presents TEM studies of dislocation cores, including remarkable observations of kinks and their motion. Minor et al. discuss in situ TEM observations of dislocation behavior during nanoindentation. The analysis of dislocation gradients by X-ray microdiffraction is described by Ice and Barabash. Shilo and Zolotoyabko discuss X-ray imaging of photon interactions with dislocations. J.P. Hirth

v


Contents Volume 13

Preface

v

Contents

vii

List of contents of Volumes 1–12

ix

71. V.S. Deshpande, A. Needleman and E. Van der Giessen Discrete Dislocation Plasticity Modeling of Contact and Friction 72. J.P. Nowacki and V.I. Alshits Dislocation Fields in Piezoelectrics

1

47

73. G. Ananthakrishna Statistical and Dynamical Approaches to Collective Behavior of Dislocations 81 74. R.C. Pond, X. Ma, Y.W. Chai and J.P. Hirth Topological Modelling of Martensitic Transformations

225

75. M. Niewczas Dislocations and Twinning in Face Centred Cubic Crystals 76. K. Edagawa and S. Takeuchi Elasticity, Dislocations and their Motion in Quasicrystals 77. J.C.H. Spence Experimental Studies of Dislocation Core Defects

263 365

419

78. A.M. Minor, E.A. Stach and J.W. Morris, Jr. In situ Nanoindentation in a Transmission Electron Microscope 79. G.E. Ice and R.I. Barabash White Beam Microdiffraction and Dislocations Gradients 80. D. Shilo and E. Zolotoyabko X-Ray Imaging of Phonon Interaction with Dislocations

Author Index Subject index

641 659

vii

499 603

453


CONTENTS OF VOLUMES 1–12 VOLUME 1. The Elastic Theory 1979, 1st repr. 1980; ISBN 0-7204-0756-7

1. 2. 3. 4.

J. Friedel, Dislocations – an 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. Schöck, 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.

R.W. Balluffi and A.V. Granato, Dislocations, vacancies and interstitials 1 F.C. Larché, Nucleation and precipitation on dislocations 135 P. Haasen, Solution hardening in f.c.c. metals 155 H. Suzuki, Solid solution hardening in body-centred cubic alloys 191 V. Gerold, Precipitation hardening 219 16. S.J. Basinski and Z.S. Basinski, Plastic deformation and work hardening 261 17. E. Smith, Dislocations and cracks 363 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. Schröter, Electrical properties of dislocations in semiconductors F.R.N. Nabarro and A.T. Quintanilha, Dislocations in superconductors 193 M. Kléman, The general theory of disclinations 243 ix

127

x 23. 24.

CONTENTS OF VOLUMES 1–12 Y. Bouligand, Defects and textures in liquid crystals 299 M. Kléman, 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.P. 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. Neuhäuser, 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.

G. Bertotti, A. Ferro, F. Fiorillo and P. Mazzetti, Electrical noise associated with dislocations and plastic flow in metals 1 34. V.I. Alshits and V.L. Indenbom, Mechanisms of dislocation drag 43 35. H. Alexander, Dislocations in covalent crystals 113 36. B.O. Hall, Formation and evolution of dislocation structures during irradiation 235 37 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 interfaces 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.

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 46. N. Narita and J.-I. Takamura, Deformation twinning in f.c.c. and b.c.c. metals 135 47. A.E. Romanov and V.I. Vladimirov, Disclinations in crystalline solids 191

CONTENTS OF VOLUMES 1–12

xi

VOLUME 10. Dislocations in Solids 1996; ISBN 0-444-82370-0 48. J.H. Westbrook, Superalloys (Ni-base) and dislocations 1 49. Y.Q. Sun and P.M. Hazzledine, Geometry of dislocation glide in L12 γ ′ -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 L12 structure 52. D.C. Chrzan and M.J. Mills, Dynamics of dislocation motion in L12 compounds 187 53. P. Veyssière and G. Saada, Microscopy and plasticity of the L12 γ ′ phase 253 54. K. Maeda and S. Takeuchi, Enhancement of dislocation mobility in semiconducting crystals 443 55. B. Joós, The role of dislocations in melting 505

135

VOLUME 11. Dislocations in Solids 2002; ISBN 0-444-50966-6 56. 57. 58. 59. 60. 61. 62. 63.

M. Zaiser and A. Seeger, Long-range internal stress, dislocation patterning and work-hardening in crystal plasticity 1 L.P. Kubin, C. Fressengeas and G. Ananthakrishna, Collective behaviour of dislocations in plasticity 101 L.M. Brown, Linear work-hardening and secondary slip in crystals 193 D. Kuhlmann-Wilsdorf, The LES theory of solid plasticity 211 H. Mughrabi and T. Ungár, Long-range internal stresses in deformed single-phase materials: The composite model and its consequences 343 G. Saada and P. Veyssière, Work hardening of face centred cubic crystals. Dislocations intersection and cross-slip 413 B. Viguier, J.L. Martin and J. Bonneville, Work hardening in some ordered intermetallic compounds 459 T.M. Pollock and R.D. Field, Dislocations and high-temperature plastic deformation of superalloy single crystals 547

VOLUME 12. Dislocations in Solids 2004; ISBN 0-444-51483-X 64. 65. 66. 67. 68. 69. 70.

W. Cai, V.V. Bulatov, J. Chang, J. Li and S. Yip, Dislocation core effects on mobility 1 G. Xu, Dislocation nucleation from crack tips and brittle to ductile transitions in cleavage fracture 81 M. Kleman, O.D. Lavrentovich and Yu.A. Nastishin, Dislocations and disclinations in mesomorphic phases 147 C. Coupeau, J.-C. Girard and J. Rabier, Scanning probe microscopy and dislocations 273 T.E. Mitchell and A.H. Heuer, Dislocations and mechanical properties of ceramics 339 R.W. Armstrong and W.L. Elban, Dislocations in energetic crystals 403 M.M. Chaudhri, Dislocations and indentations 447


CHAPTER 71

Discrete Dislocation Plasticity Modeling of Contact and Friction V.S. DESHPANDE Cambridge University, Department of Engineering, Trumpington Street, Cambridge CB2 1PZ, UK

A. NEEDLEMAN Brown University, Division of Engineering, Providence, RI 02912, USA and

E. VAN DER GIESSEN University of Groningen, Department of Applied Physics, Nyenborgh 4, 9747 AG Groningen, The Netherlands

© 2007 Elsevier B.V. All rights reserved

Dislocations in Solids Edited by F. R. N. Nabarro and J. P. Hirth

Contents 1. Introduction 3 2. Background 5 3. Modeling frameworks 11 3.1. Phenomenological plasticity 13 3.2. Discrete dislocation plasticity 15 4. Discrete dislocation plasticity modeling of single asperity contact and friction 4.1. Single asperity indentation 21 4.2. Single asperity sliding 29 4.3. Single asperity coefficient of friction 41 5. Concluding remarks 42 Acknowledgements 44 References 44

21

1. Introduction Contact and frictional sliding of a surface between two solids governs a range of technologically important mechanical behaviors and failure mechanisms. Frictional dissipation significantly contributes to the energy needed for material processing and to that used by rotating machinery. Wear and surface fracture limit the lifetime of machines and components. Because of the increased surface to volume ratio as size scales decrease, surface effects play a key role in governing the performance and reliability of micro-scale devices. Also, contact and friction at small scales is involved in recently developed technologies such as nanoscale imprinting and cold welding. At the tectonic scale, abrupt frictional sliding along a fault is what constitutes an earthquake. Thus, contact and frictional sliding between two solids is a basic mechanics problem with a wide range of applications. The basic understanding of friction can be traced back to Leonardo da Vinci in the 16th century but with the statement of the classical frictional law in its current form being due to Amontons and Coulomb in the 17th and 18th centuries. The classical Amontons– Coulomb description of friction states that the shear force along an interface is proportional to the normal force, with the proportionality constant being the coefficient of friction. Two coefficients of friction are identified: (i) a static coefficient of friction that governs the onset of sliding and (ii) a dynamic coefficient of friction that governs the response during sliding. The Amontons–Coulomb description has been widely used both to determine service behavior and to organize experimental measurements, i.e. to characterize friction between two solids the coefficient of friction is measured. In their classic work, Bowden and Tabor [1]1 rationalized the Amontons–Coulomb description of friction in terms of the role of surface roughness. Contact only takes place at asperities so that the true area of contact is a small fraction of the apparent area of contact and the friction force depends on the true area of contact, not the apparent area of contact. The normal force dependence then arises from its effect on the number of asperities in contact and the evolution of the stress carried by the asperities. Despite its successes, there are fundamental problems with the Amontons–Coulomb description of friction: (i) experiments have shown that friction is both time and slip distance dependent (e.g., Dieterich [2], Ruina [3], Prakash and Clifton [4]); and (ii) analyses have shown that with this description sliding along an interface between dissimilar elastic solids is, in a wide range of circumstances, an ill-posed problem (Renardy [5], Adams [6], Martins et al. [7], Ranjith and Rice [8], Rice et al. [9]). Accordingly, there is strong motivation for seeking a more adequate description of frictional sliding. Motivated by the experimentally observed rate dependence, phenomenological rate and state frictional constitutive relations have been developed, largely in the geophysics literature, that capture key features seen experimentally, Dieterich [2], Rice and Ruina [10], 1 No attempt is made to give a complete bibliography for the topics covered. Rather, the references cited are intended to provide an entry into the various literatures for the interested reader.

4

V.S. Deshpande et al.

Ch. 71

Prakash and Clifton [4], Prakash [11]. The key features incorporated into the frictional constitutive relation are: (i) that the coefficient of friction instantaneously changes in response to a step increase in sliding speed and then subsequently changes to a steady-state value that depends on sliding speed; (ii) the approach to this steady state value occurs over a characteristic sliding distance; (iii) the interface restrengthens in stationary contact; and (iv) the shear force does not change instantaneously when there is a step change in normal force. The influence of both sliding velocity and normal force for the sliding of steel on steel has been mapped out by Lim et al. [12] in terms of a friction-regime map. At sufficiently low sliding velocities, the coefficient of friction is reported to depend mainly on the surface roughness and the plastic properties of the surface. At higher velocities, the coefficient of friction is found to become strongly dependent on the sliding velocity. In this regime the effects of oxidation and local heating can play important roles. To establish a framework for considering the physical basis of frictional sliding, we follow Bowden and Tabor [1] in writing the frictional force, Ff , as Ff =

τf Af

(1)

where τf is the contact shear stress, Af is the contact area of an asperity and the summation is over the true contact area. Surfaces are generally rough, having asperities over a wide range of size scales from the atomistic to some surface specific upper limit. What needs to be determined is the evolution of the asperity population, the contact area of each asperity and the contact shear stress. The most straightforward issue to address, apparently at least, is the evolution of the shear force for a single asperity. Contributions to the shear force come from adhesion between the contacting solids and from asperity deformation. The relative importance of these depends on the particular solids in contact and on the asperity size. Deformation of the asperity may be elastic or inelastic and, if sufficiently stressed, asperities can fracture. Also, the deformation and stress field of the asperity under consideration can be affected by the proximity and size of neighboring asperities. An even more difficult issue is the characterization of the contact area and its evolution. Due to the multi-scale nature of rough surfaces there are some subtle issues that remain to be resolved. The classical characterization of rough surfaces is in terms of a collection of individual asperities (Greenwood and Williamson [13]), but over a wide range of size scales, surfaces often have a self-affine fractal character in the sense the surface of each asperity is rough and has asperities which have asperities, etc., see, for example, Bouchaud [14]. An analytical advantage to representing a contact surface as one having a self-affine fractal character is that such a surface is size scale-independent so that if the material constitutive relation does not involve a size scale, attractive scaling relations emerge, see Gao and Bower [15], Pei et al. [16]. For example, with no constitutive length scale, from dimensional considerations there cannot be a direct dependence on the amplitude of the surface roughness. However, real surfaces have both small size and large size cutoffs to the fractal representation, and even between these cut-offs the fractal representation may only be approximate. In any case, when cut-offs are introduced, the dependence of the predictions on the choice of cut-off needs to be determined. For a range of models, the

§2

Discrete dislocation plasticity modeling of contact and friction

5

contact area is found to be proportional to the normal force, see, e.g., Archard [17], Gao and Bower [15], Pei et al. [16], thus providing support for the Amontons–Coulomb normal force dependence presuming that τf in eq. (1) is independent of the normal force. Here, we confine attention to consideration of the contact and sliding of unlubricated surfaces of crystalline solids that deform plastically by dislocation glide so that the analyses pertain mainly to ductile metals. Furthermore, we only consider circumstances where environmental effects do not play a dominant role. Even with these restrictions, the range of relevant issues is much wider than will be covered here. We focus on the range of size scales from, say 0.1 µm to 100 µm, which is much larger than an atomistic size scale but sufficiently small that the discreteness of dislocations comes into play. There have been relatively few discrete dislocation analyses of contact and friction and what can currently be predicted is quite limited. Nevertheless, the tools are being developed for significant progress to occur in the not-too-distant future.

2. Background Motivations for developing a predictive micro-mechanical theory of contact and friction include providing a physical basis for the development of phenomenological frictional constitutive relations and providing the basis for the design of surfaces with desirable frictional properties and improved wear resistance. In either case, a framework is needed that defines what properties appropriately characterize frictional response. Within the classical Amontons–Coulomb framework the frictional response of a surface can be characterized via Ff = μFn

(2)

where Ff is the frictional force resisting sliding, Fn is the normal force and μ is the coefficient of friction which has two possible values; μs that governs the onset of sliding and μd that governs continued sliding. Thus, in this context, a micro-mechanical theory that could determine the values of μs and μd is needed. From a theoretical perspective, a fundamental problem with the Amontons–Coulomb description of friction stems from the predicted instability of steady sliding of one linear elastic half space over a different elastic half space. When the magnitude of the shear traction, |Tt |, is perturbed in a mode of the form |Tt | = Q(t) exp ikx

(3)

where Q(t) is some arbitrary function of time, x is a coordinate in the interface and k is a wave number, the variations in the slip velocity, Vslip , obtained via a linear stability analysis are found that have the form Vslip ∝ exp(a|k|t)

(4)

with t being time, Renardy [5], Adams [6], Martins et al. [7], Ranjith and Rice [8]. The values of friction coefficient for which a is positive cover a wide range of practical interest.

6


Ch. 71

With a > 0, an arbitrarily small perturbation will lead to an arbitrarily large variation and, in this sense, sliding of elastic solids with Amontons–Coulomb friction leads to ill-posed problems. The effect, if any, of inelastic behavior of the sliding solids on stability remains to be investigated. Thus, one motivation for frictional relations that go beyond the Amontons–Coulomb description, eq. (2), is to regularize the governing equations so that initial/boundary value problems become well-posed. In addition, experimental observations have shown significant deviations from eq. (2). Observations that μd depends on sliding velocity date back at least to the late 1950s (Bowden and Freitag [18], Bowden and Persson [19]), but the origin of the current rate and state framework for characterizing frictional sliding stems from the work of Dieterich [2] and Ruina [3]. Within the rate and state framework, frictional sliding is characterized by constitutive relations of the form Ff = μ(θi , Vslip ) κj ψj (5) j

where Vslip is the slip velocity, the θi are internal variables as are the ψj and the κj are a set of constitutive parameters. Evolution equations need to be specified for the θi and the ψj with the restrictions that a steady-state is approached where μ(θi , Vslip ) → μ(Vslip )

and

j

κj ψj → Fn .

(6)

As an example of a rate and state friction law, the Dieterich–Ruina (Dieterich [2], Ruina [3]) relation, which has a single state variable, can be written as μ = μ0 + A ln

Vslip V0 θ1 + B ln V0 L0

(7)

with the evolution equation θ˙1 = 1 −

Vslip θ1 L0

(8)

where (˙) = ∂( )/∂t, and μ0 , A, B, V0 and L0 are constitutive parameters. The evolution equation, eq. (8), has the property that when θ1 = L0 /Vslip , θ˙1 = 0. Then, from eq. (7) μ = μ0 + (A − B) ln

Vslip . V0

(9)

The relation (9) gives the dependence of μ on sliding velocity in the steady-state. When Vslip = V0 , then μ = μ0 . Rate weakening, i.e. a decreasing coefficient of friction with increasing sliding velocity, occurs for B > A. The Prakash–Clifton normal force dependence has the form (Prakash and Clifton [4], Prakash [11]) Ff = μ(ψ1 + ψ2 ).

(10)

§2


7

Hence, eq. (10) is of the form of eq. (5) with two state variables and with κ1 = κ2 = 1. The evolution equations for ψ1 and ψ2 are taken to be of the form ψ˙ 1 = −

1 [ψ1 − C1 Fn ]Vslip , L1

ψ˙ 2 = −

1 ψ2 − (1 − C1 )Fn Vslip . L2

(11) (12)

In the steady state ψ˙ 1 = ψ˙ 2 = 0, ψ1 + ψ2 = Fn C1 Vslip so that eq. (6) is satisfied and the normal force dependence of the Amontons–Coulomb friction law emerges but with a sliding velocity dependent coefficient of friction. Although the logarithmic dependence in eq. (7) fits a wide range of experimental data, it cannot hold generally since it predicts an infinite coefficient of friction when either Vslip or θ1 vanish. A physically based relation that avoids the singularity but has a dependence very close to that in eq. (7) is given in Lapusta et al. [20] while a phenomenological power law form was proposed by Povirk and Needleman [23]. The extent to which the number of state variables and the form of the evolution equations depends on the materials involved and the sliding conditions is not known. The experimental basis for eqs (7) to (9) comes mainly from measurements on rock, Dieterich [2], but the main features of the rate and state description are seen to hold for sliding along a variety of interfaces, Dieterich and Kilgore [21], Baumberger and Berthoud [22]. The experiments leading to eqs (10) to (12) were carried out for the interface between a tungsten-carbide tool material and a structural steel or titanium alloy, Prakash and Clifton [4], Prakash [11]. The form of the frictional constitutive relation can play a role in predictions of stick-slip instabilities, Povirk and Needleman [23], and in predictions of the sliding mode; i.e. whether progressive sliding takes place continuously behind some sliding front or in propagation of a sliding pulse, e.g., Coker et al. [24]. Within the rate and state framework, the task of a micro-mechanical model of friction is to predict the sliding velocity dependence of the coefficient of friction, the appropriate internal variables and their evolution. For example, the frictional response is dependent both on time and sliding distance. Micro-mechanical analyses can, in principle, identify those aspects that are rate dependent and those that depend on sliding distance. This is clearly a much larger task than predicting a coefficient of friction. Because of its multi-scale nature, analyses of friction have been carried out at scales from the atomic, e.g., Landman et al. [25], Müser et al. [26], Robbins et al. [27], to the continuum, e.g., Chang et al. [28], Tworzydlo et al. [29], Molinari et al. [30], Kogut and Etison [31]. There are some issues that probably are only properly addressed by analyses at the atomic scale, for example, the effects of chemistry on the evolution of the friction force, but here attention is restricted to deformation effects. In particular, we consider circumstances where at least one of the solids in contact undergoes inelastic deformation due to dislocation glide and, if the other solid does not deform by dislocation glide, we presume it can be characterized as linear elastic or rigid. The main relevance of these analyses is to ductile metals. With the frictional force written in the form of eq. (1), what needs to be predicted is the number of contacts, the contact area of each, the associated frictional force and the

8


Ch. 71

evolution of each of these quantities as sliding progresses. Characterization of the evolution of the contact area is difficult due to the multi-scale nature of rough surfaces. One approach, stemming from the classical work of Greenwood and Williamson [13] is to represent the topography of a rough surface as a collection of discrete asperities having some specified size distribution. However, rough surfaces typically have a self-affine fractal character over a wide range of size scales, e.g., Bouchaud [14], and for a fractal surface the procedure of Greenwood and Williamson [13] breaks down. Continuum mechanics analyses of the evolution of contact area for surfaces with fractal scaling under normal load have been carried out for elastic solids (Ciavarella et al. [32], Hyun et al. [33]) and more recently for elastic-plastic solids (Gao and Bower [15], Pei et al. [16]). For an elastic solid with a self-affine fractal surface and for nonadhesive contact, the true contact area consists of an infinite number of contact spots, each of zero size so that the total contact area is zero, but the pressure on each contact is infinite, Ciavarella et al. [32]. Scale independent plasticity does not essentially change this picture, Gao and Bower [15]. Pei et al. [16] found that introducing a small length scale cut-off did give rise to physically meaningful predictions in the limit of increasing surface area. Indeed, regardless of the roughness description, for roughness amplitudes small compared to any other characteristic geometric dimension, with a scale independent constitutive description, the quantities characterizing the roughness that affect the response must be dimensionless, e.g., a slope, a height-to-wavelength ratio, etc. Real surfaces are not completely self-affine having a small length cut-off, for example at the atomic scale, and also a large length cut-off. Thus, there are difficulties both with describing realistic surfaces as a collection of discrete smooth-surfaced asperities and as a self-affine fractal and the appropriate characterization of realistic surface roughness is an open issue. Persson et al. [34] discuss the characterization of surface roughness and implications for contact and friction, with a focus on rubber friction. As discussed by Pei et al. [16], continuum analyses based on a broad range of assumptions about rough surfaces give a contact area that increases roughly linearly with normal load. These analyses of contact area pertain mainly to conditions at the initiation of sliding, i.e. conditions relevant for the static coefficient of friction. What remains to be modeled is the evolution of contact area under continued sliding where large deformation and fracture of asperities takes place. Analyses of the dependence of these processes on surface morphology are needed to provide a physical basis for the transition from static friction to dynamic friction as well as to determine the time dependence and sliding distance dependence of that transition. Analyses based on a conventional continuum plasticity constitutive relation assume that the plastic response of the material is size independent. A wide variety of experiments have established that plastic deformation in ductile metals is size dependent in a size scale range from about 0.1 µm to 100 µm, Fleck et al. [35], Ma and Clarke [36], Stölken and Evans [37], Swadener et al. [38], Florendo and Nix [39], Greer et al. [40] and Dimiduk et al. [41]. Mechanisms giving rise to size dependence include: strain gradients leading to geometrically necessary dislocations, constraints on dislocation glide leading to boundary layers, the limited number of dislocation sources and a limited number of obstacles to glide leading to ‘dislocation starvation’. All these mechanisms can potentially operate during the deformation of micrometer scale asperities. This size dependence may have important

§2


9

effects on how load and deformation are distributed among asperities at various size scales and on the scaling relations that emerge. One framework for incorporating a size scale dependence into plasticity analyses is through the use of a plastic constitutive relation that incorporates a length scale. A considerable number of such non-local plastic constitutive relations have been proposed, e.g., Gao et al. [42], Acharya and Bassani [43], Fleck and Hutchinson [44], Gurtin [45], Gudmundson [46]. Different formulations lead to different governing equations and boundary conditions; some formulations preserve the structure of conventional continuum plasticity boundary value problems, e.g., Acharya and Bassani [43], while some introduce additional balance equations and boundary conditions, e.g., Gurtin [45]. All of these constitutive relations have focused on representing in a phenomenological constitutive relation size effects stemming from geometrically necessary dislocations (Nye [47], Ashby [48]) arising from strain gradients. Generally, those without additional boundary conditions cannot capture boundary layer effects while those with additional boundary conditions can. None of these can represent the size effects arising due to limited dislocation sources or those due to ‘dislocation starvation’. Nevertheless, because strain gradients do play a significant role in contact, analyses of the size dependence of indentation using phenomenological size dependent plastic constitutive relations have yielded insight and scaling relations, e.g., Nix and Gao [49], Begley and Hutchinson [50], Wei and Hutchinson [51], Qu et al. [52]. An alternative approach is discrete dislocation plasticity. In this approach, dislocations are modeled as line singularities in an elastic solid. The key idea is to write displacements, strains and stresses as superpositions of fields due to the discrete dislocations, which are singular inside the body and known either analytically or from a separate calculation, and image fields that enforce the boundary conditions (Van der Giessen and Needleman [53]). The long range interactions between dislocations are obtained directly while constitutive rules are needed for relating dislocation motion to the local stress state, for dislocation annihilation and, depending on the level of modeling, possibly for dislocation nucleation and interaction with obstacles. The formulation in Van der Giessen and Needleman [53] was for infinitesimal deformations, but the framework has been extended to finite deformations by Deshpande et al. [54]. Analyses that account for the full three dimensional character of dislocations have been carried out, but due to the large computational demands what can be calculated is still quite limited. Some examples of what can be done in three dimensions are illustrated in Devincre and Kubin [55], Schwarz [56], Zbib et al. [57], Weygand et al. [58], Han et al. [59] and Cai and Bulatov [60]. With the exception of the work of Fivel et al. [61] discrete dislocation plasticity calculations related to contact and friction have been limited to two dimensional plane strain analyses. In Fivel et al. [61] three-dimensional indentation of a single crystal was simulated and, while good qualitative agreement between the experimentally observed and predicted dislocation structures was seen, the range of indentation depths considered was too small for size effects to emerge. The remaining discussion will be confined to two dimensional model problems, even though there are aspects of dislocation plasticity that cannot be modeled within the twodimensional framework. For example, realistic hardening behavior, such as the scaling of the flow strength with the square root of the dislocation density in stage II hardening is

10


Ch. 71

not obtained. A means of obtaining more realistic hardening behavior, still retaining the computational advantages of only solving two dimensional boundary value problems was introduced by Benzerga et al. [62] through special dislocation constitutive rules. While initial results are promising (Benzerga et al. [62,63]), the extent to which such a framework extends the range of realistic behaviors that can be modeled by two dimensional analyses remains to be determined. Even in a restricted two dimensional framework, accounting for the discreteness of dislocations allows one to explore the implications of aspects of the plastic deformation of crystalline solids for friction that one cannot consider using any available phenomenological framework, for example the finite spacing of dislocation sources. In addition, large deformations by dislocation glide leading to the sliding off of parts of asperities by highly localized slip with no fracture involved can, in principle, be modeled. Furthermore, because the elastic energy stored in the dislocation structure can be calculated, a discrete dislocation plasticity analysis permits the plastic dissipation at an asperity (and therefore the heat generated) to be directly calculated. Although the evolution of contact area is of key importance for the micro-mechanics of friction, discrete dislocation plasticity modeling of this has not yet been undertaken. What discrete dislocation plasticity modeling has been carried out has been confined to the prediction of τf in eq. (1) for a fixed asperity. Furthermore, attention in discrete dislocation plasticity analyses have only focused on the initiation of sliding, i.e. on static friction. The importance of discrete dislocation effects for contact and friction was recognized by Kuhlmann-Wilsdorf [64] who noted that the plastic response of an asperity would be size dependent when the asperity size was of the order of the dislocation source spacing. In a pioneering study, Polonsky and Keer [65,66] used discrete dislocation plasticity to model elastic-plastic deformation of micro-contacts on scales too small to apply conventional continuum plasticity theory. Their analysis indeed revealed that when the asperity size becomes comparable to the dislocation source spacing, the asperities can sustain considerably higher loads than those predicted by continuum plasticity. Hurtado and Kim [67,68] analyzed size scale effects using a discrete dislocation model and found three regimes in the variation of τf with contact size. For relatively large contacts, τf is equal to the Peierls stress while for very small contacts τf is equal to the theoretical shear strength of the solid. The transition between these two regimes is governed by the stress to nucleate a dislocation loop at the edge of the contact and they obtained a scaling of τf ∝ a −1/2 , where a is the contact size in the transition regime. Hurtado and Kim [67,68] modeled mica contacts and assumed that no dislocation activity occurs in the bulk, with slip solely due to dislocations along the interface. Arguing that the size of the interface zone over which plasticity occurs scales with the contact size, Bhushan and Nosonovsky [69] obtained a scaling for τf similar to that in Hurtado and Kim [67,68]. The analyses of sliding of Deshpande et al. [70] which will be discussed in detail in Section 4 presumes that dislocations are generated by Frank–Read sources in the bulk, not surface dislocations as in the model of Hurtado and Kim [67,68]. Nevertheless, a similar scaling with asperity size is obtained. Although very few dislocation analyses of even the initiation of sliding have been carried out, analyses of indentation are relevant. Two dimensional discrete dislocation plasticity

§3


11

simulations of the indentation of crystalline solids by Pippan et al. [71,72] have shown size effects that mainly result from source-limited plasticity. Widjaja et al. [73] carried out two-dimensional simulations of cylindrical indentation of a single crystal and, over the parameter range considered in [73], the length scale associated with the cylindrical indenter was more dominant than the material length scale. In the following, we will discuss a discrete dislocation plasticity framework for analyzing both the static and dynamic frictional response of ductile crystalline solids. However, the few results currently available pertain to the static frictional response of single crystals.

3. Modeling frameworks In a continuum mechanics framework, a body is a collection of material points and its mechanical response depends on the time history of the positions of each material point. Material points can be identified by their position in some suitably chosen reference configuration. Adopting Cartesian tensor notation, let xi (i = 1, 2, 3) denote the position of a material point in the reference configuration relative to a fixed Cartesian frame. Then, in the current configuration, this material point is at x¯i . The Cartesian components of the displacement field are ui (xj , t) = x¯i (xj , t) − xi

(13)

where t is time. In conventional continuum mechanics it is assumed that the displacement ui is single valued and continuously differentiable at each material point so that for any closed curve C in the body

C

∂ui dxj = 0 ∂xj

(14)

because the two end points are the same physical point. When the material response is elastic, the displacement of the material points is continuous. However, when plasticity occurs by dislocation glide in an otherwise elastic solid, eq. (14) no longer holds throughout the body, since the displacement is discontinuous across the glide plane. Thus, eq. (14) is satisfied piecewise in the body but not everywhere. One important implication of the lack of satisfying compatibility is that interior points can be become exterior points; this cannot occur when eq. (14) holds everywhere. In the context of asperity deformation, this means that in the discrete dislocation framework the possibility of part of an asperity sliding off can occur whereas in a conventional continuum formulation this is not possible. Also, note that the occurrence of slip steps on the surface obviously affects the predicted surface roughness. In particular, since there is a displacement jump, the slope of the surface profile can contain delta functions. In a contin-

12


Ch. 71

uum plasticity description where the average response of dislocations over some volume is modeled, the displacement field is taken to be continuous and, in this context, eq. (14) holds. With attention confined to quasi-static deformations (i.e. inertia plays a negligible role) and with no body forces, equilibrium inside the body is expressed by ∂σij =0 ∂ x¯j

(15)

for the components of the symmetric Cauchy stress tensor σij . On its boundary, equilibrium specifies the traction vector as T¯i = σij n¯ j , with n¯ j the unit normal. The boundary conditions are imposed on the current surface of the body the determination of which is an outcome of the solution. Since the balance laws of continuum mechanics hold in the current configuration, the displacement discontinuities that can occur when the discreteness of dislocations is accounted for can have a profound effect. For example, as a slip step develops on a free surface, boundary conditions need to be applied to material points that previously were interior material points. In circumstances where the change in geometry between the unloaded state and the loaded state is small, the divergence in eq. (15) can be carried out in the unloaded (reference) configuration, i.e. ∂σij =0 ∂xj

(16)

and boundary conditions are applied on the known unloaded configuration of the body. Hence, when geometry changes are neglected in a discrete dislocation analysis, the difficulties associated with the current geometry being non-smooth can be avoided since the field equations are taken to hold in the initial, reference configuration. The balance laws need to be supplemented with constitutive equations which describe how the stress evolves with deformation. In a conventional continuum framework, the stress depends on the history of x¯i only through the deformation gradient Fij given by Fij =

∂ x¯i ∂ui = δij + , ∂xj ∂xj

(17)

with δij the Kroneker delta (δij = 1 if i = j , δij = 0 otherwise). One consequence of this, is that there is no length scale in the governing equations. Hence, the predicted material response is scale independent. There are a wide range of experimental observations showing that plastic deformation at the micrometer scale is size dependent. To account for this, a variety of non-local or gradient plasticity theories have been proposed that give rise to a size dependent response. In many of these theories additional balance laws and associated boundary conditions are introduced.

§3.1


13

3.1. Phenomenological plasticity Conventional plasticity is a continuum formulation where dislocations are averaged out. p Their average effect is incorporated phenomenologically into the plastic part, Dij , of the rate of deformation tensor Dij , which, for finite strains, is defined as Dij =

1 ˙ −1 Fik Fkj + F˙j k Fki−1 . 2

(18)

The remaining elastic part Dije is related to the Jaumann rate of stress by Hooke’s law when elastic strains remain small, as can appropriately be assumed for most metal deformation processes. The Jaumann stress rate is a measure of the rate of change of the stress tensor that properly accounts for changes due to finite rotations. The plastic strain rate is given by a flow rule, the most widely used being that of an isotropically hardening Mises solid for which p

Dij =

p 3 ε˙ e 1 σij − σkk δij , 2 σe 3

(19)

p

where ε˙ e is the effective plastic strain rate and the effective stress σe is defined by σe2 =

3 1 1 σij − σkk δij σij − σkk δij . 2 3 3

(20)

p

The effective plastic strain rate, ε˙ e , is taken to be a function of σe that represents the uniaxial stress–strain response of the material. This function may be rate dependent or rate independent. For the purposes here, one of the most salient features of this constitutive relation is that it is scale independent. At size scales where the orientation of individual crystals matter, a plastic flow rule that accounts for the discrete orientation of slip systems has the form p

Dij = (α)

1 (α) (α) (α) , ¯ (α) ¯ j + s¯j(α) m γ˙ s¯i m i 2 α

(21) (α)

where s¯i specifies the current slip direction and m ¯ i the current slip plane normal for slip system α. The quantity γ˙ (α) is a specified function which gives the slip system stress-strain response. Although the discreteness of slip systems and their reorientation due to deformation is accounted for, the response is still scale independent. Overviews of continuum slip theory and its physical background are given by Asaro [74], Bassani [75] and Cuitiño and Ortiz [76]. A variety of plasticity theories have been introduced that incorporate a material length scale that is assumed to affect the plastic properties. There is, however, no consensus at present on the way the length scale enters. Most of the currently available size-dependent plasticity formulations are based on the notion that hardening is enhanced by gradients of plastic strain, or gradients of plastic slip in the context of crystal plasticity. Examples where

14


Ch. 71

strain gradients are intrinsic to the deformation problem include bending as well as indentation, so that non-locality is pertinent to contact and friction. The relevant slip gradient is related to the notion of lattice incompatibility, as introduced in the 1950s by Nye [47], which is equivalent to Ashby’s [48] concept of geometrically necessary dislocations. In contrast to discrete dislocation plasticity, a basic assumption of these size-dependent phenomenological plasticity theories is that the displacement field is compatible. As a consequence, interior points always remain interior points. While the total displacement field satisfies compatibility, neither the elastic nor the plastic parts of the deformation field are compatible, i.e. satisfy eq. (14). In crystal plasticity, for small deformations, incompatibility is measured by αij = ej kl

∂γ (β) β

∂xk

(β)

(β)

si ml ,

(22)

where ej kl is the permutation symbol.2 The way in which slip gradients (or strain gradients in an isotropic plasticity theory) enter the constitutive relations varies among the available theories. In one class of theories, the effect of slip or strain gradients is to enhance the hardening. An example is the crystal plasticity theory of Acharya and Bassani [43] in which the instantaneous hardening rate of a slip system not only depends on the current slips γ (α) but also is a function of ℓαij , where ℓ is a material length scale that commonly is taken to be constant. The structure of boundary value problems for this class of theories is the same as for a conventional, size-independent plasticity theory and no additional boundary conditions are introduced. In another class of theories, additional field quantities are introduced. These are governed by additional field equations that supplement those for the continuum displacement field, e.g., Fleck and Hutchinson [44], Gurtin [45], Gudmundson [46]. In the crystal theory of Gurtin [45], for example, an additional field variable has a direct effect on plasticity by playing the role of a back stress originating from the development of geometrically necessary dislocations. Boundary value problems for this class of theories have a different mathematical structure from a conventional, size-independent plasticity theory and additional boundary conditions are introduced. These boundary conditions can lead to the evolution of plastic strain gradients in problems for which local theories predict uniform straining but where limiters of dislocation motion, such as particles and interfaces, hinder plastic slip. In such cases, size dependence of the response evolves as deformation proceeds. Nix and Gao [49] proposed a theory of size effects in indentation based directly on the density of geometrically necessary dislocations as a function of indentation depth. This density is added to a density of statistically stored dislocations that one has in uniaxial tension, and the total density is assumed to scale with the square of the flow stress according to the Taylor law. The resulting scaling law will be compared to the predictions of discrete dislocation simulations in Section 4.1.1. The use of dislocation densities is also central to a new class of plasticity formulations based on a statistical-mechanics treatment of ensembles of dislocations pioneered by Groma [77]. These theories involve a set of coupled evolution equations for the total 2e

123 = e312 = e231 = 1, e132 = e213 = e321 = −1, all other ej kl = 0.

§3.2


15

dislocation density and the density of geometrically necessary dislocations, which enter as new field variables in the crystal plasticity theory of Yefimov et al. [78]. With parameters fit to one particular boundary value problem, this model has been successful in predicting the response under other loading conditions, including bending, with good agreement with discrete dislocation simulations, see Yefimov et al. [79]. An important contribution to the enhancement of hardening in this model arises from a back stress connected to the gradient of plastic slip – a feature that the model shares with the theory of Gurtin [45]. However, a key feature of the theory of Yefimov et al. [78] is that it incorporates an explicit source for dislocations. This means that it is, at least potentially, capable to deal with source-limited plasticity in small volumes, Van der Giessen and Needleman [80]. This capability is of importance for modeling contact and friction because discrete dislocation plasticity calculations indicate that the effects of both geometrically necessary dislocations and the finite spacing of dislocation sources come into play for micrometer size asperities. The development of phenomenological theories that incorporate relevant size dependent mechanisms is of importance because carrying out analyses with such phenomenological theories is generally much less computationally intensive than with discrete dislocation plasticity. This is apt to be particularly the case for three dimensional calculations.

3.2. Discrete dislocation plasticity The theory of dislocations is a well-established subject (see Nabarro [81], Hirth and Lothe [82]) that has had a major impact on the development of materials science, for example, through the development of a quantitative characterization of the interaction between dislocations and elements of a material’s micro-structure such as precipitates and through the prediction of the stress-strain response of a representative volume element. Moreover, in discrete dislocation plasticity, general initial/boundary value problems are formulated and solved, as in conventional continuum plasticity, but plastic flow is directly represented by the collective motion of discrete dislocations instead of in terms of a phenomenological constitutive relation. In discrete dislocation plasticity, plastic deformation is described by the nucleation and glide of dislocations represented as discrete line singularities in an elastic medium. Once dislocations nucleate, field quantities are computed by means of superposition, Van der Giessen and Needleman [53]. The key idea is illustrated in Fig. 1. The displacement and stress fields are written as the superposition of two fields, ui = u˜ i + uˆ i ,

σij = σ˜ ij + σˆ ij

in V ,

(23)

respectively. The (˜) fields are the superposition of the fields of the individual dislocations, in their current configuration. The singular displacement and stress fields of the individual dislocations are known either analytically or from a separate calculation. Any convenient representation can be used as long as it contains the correct singular terms. The sum of the individual dislocation fields does not in general satisfy the prescribed boundary conditions. The (ˆ) fields are smooth image fields (provided of course that the only singularities in the boundary value problem are those associated with the dislocations) that correct for the

16


Ch. 71

Fig. 1. Decomposition of the boundary value problem into the sum of the singular (˜) fields associated with the individual dislocations and the image (ˆ) fields that enforce the prescribed boundary conditions.

actual boundary conditions. Since the (ˆ) fields are smooth, the boundary value problem for them can be solved by any convenient numerical method such as the finite element method or the boundary element method. This superposition is quite general. It can be used to solve fully three-dimensional boundary value problems as in Weygand et al. [83], but the methodology has been mostly applied to two-dimensional plane strain problems because of the large computational demands of three-dimensional discrete dislocation plasticity. The basic superposition idea can also be used when geometry changes are accounted for, although then an iteration procedure is required [54]. In any case, the orientation of slip planes and possible slip directions and the elastic constants need to be specified. In addition, various constitutive parameters are required, but these can vary depending on what level of modeling is used; two-dimensional or threedimensional dislocation dynamics, small deformation theory or finite deformation theory as well as, to a certain extent, the detail with which the elastic interaction between dislocations is computed. In any case, at least a constitutive relation is needed to specify the change in position of the dislocation in response to the imposed loading. As in conventional plasticity theory, the deformation history needs to be calculated in an incremental manner. At each time step, a boundary value problem of the type sketched in Fig. 1 needs to be solved. In the following, we describe aspects of the boundary problem formulation and solution for small deformation theory (where geometry change effects are neglected) and for finite deformation theory (where geometry change effects are taken into account). 3.2.1. Small deformation theory At time t, the displacement and stress fields are known throughout the body as is the position of each dislocation. An increment of loading is prescribed and the aim is to compute the displacement and stress fields at time t + dt as well as the positions of all dislocations

§3.2


17

at that same time. Since the stress field of each dislocation is an equilibrium field, i.e., satisfies eq. (16), substitution of eq. (23) into eq. (16) gives ∂ σˆ ij = 0, ∂xj

(24)

with the boundary conditions uˆ i = u0i − u˜ i

on Su ,

(25)

Tˆi = Ti0 − T˜i

on ST .

(26)

and

Here, u0i and Ti0 are the prescribed values of displacement and traction on Su and ST , respectively, while u˜ i and T˜i are the corresponding surface values arising from the discrete dislocation displacement and stress fields (see Fig. 1). The stresses and strains are related by σij = Lij kl εkl ,

(27)

∂uj 1 ∂ui , εij = + 2 ∂xj ∂xi

(28)

with

provided the elastic moduli are uniform throughout the body. If the elastic moduli vary, as in a composite material, a polarization stress term needs to be included in eq. (27) (see Cleveringa et al. [84]). When the elastic properties are homogeneous, the constitutive relation for the (ˆ) field takes the form σˆ ij = Lij kl εˆ kl ,

εˆ kl =

1 ∂ uˆ k ∂ uˆ l . + 2 ∂xl ∂xk

(29)

In order to solve the boundary value problem, the current positions of all dislocations need to be known. To determine the change in dislocation structure a constitutive relation is needed that gives the velocity of a point on the dislocation line. This velocity is generally taken to be a function of the Peach–Koehler force resolved on the glide plane of the dislocation. The Peach–Koehler force is the configurational force corresponding to the change in potential energy with the change in dislocation position. Contributions to the Peach–Koehler force arise from: (i) the stress fields of all other dislocations in the body; (ii) the interaction of different points along the dislocation line (the self-interaction); and (iii) the image field contribution. Because the dislocation singularity at a point on the dislocation line does not change as the point moves, the Peach–Koehler force is finite on the dislocation line. However, it is important to note that the energy change due to a change in

18


Ch. 71

the length of dislocation line is not accounted for by the Peach–Koehler force and, within the framework of discrete dislocation plasticity, enters through a constitutive relation. The calculations to be discussed in Section 4 are two-dimensional plane strain calculations. Consistent with plane strain, all dislocations are edge dislocations and an explicit expression for the Peach–Koehler force on edge dislocation I is f

(I )

(I )

(I ) = ni

(J ) (I ) σ˜ ij bj , σˆ ij +

(30)

J =I

(I )

with ni the slip plane normal and bj the Burgers vector of dislocation I . In the calculations to be discussed, constitutive relations are specified for: (i) the nucleation of new dislocations; (ii) the annihilation of dislocations of opposite sign and (iii) the possible pinning of dislocations at obstacles. Specific forms of the constitutive relations used in the calculations will be presented subsequently. One important difference between discrete dislocation plasticity and phenomenological continuum plasticity, whether conventional or size dependent, is that the strain εij entering the equations of discrete dislocation plasticity is the elastic strain. Plastic deformation is associated with displacement jumps across slip planes. An ‘average’ plastic strain can be defined by taking an average over some specified volume, but the value obtained for the plastic strain then depends sensitively on the particular volume chosen. 3.2.2. Finite deformation theory Although at the time of this writing, no finite deformation calculations of friction and contact have been published, the large deformations of asperities potentially have a significant effect on what is predicted. Therefore, we briefly summarize the framework for finite-strain dislocation plasticity presented by Deshpande et al. [54]. This framework assumes that (i) lattice strains remain small away from the dislocation cores and (ii) the elastic properties are unaffected by slip. Thus, dislocations are still represented by their linear elastic fields. The geometry changes that are accounted for are those due to dislocation motion. Specifically, the finite-strain framework accounts for: (i) finite deformation-induced lattice rotations and (ii) the effect of shape changes due to slip on the momentum balance. As in the small-strain formulation, the total displacement rate and stress fields are given by a superposition of the analytically known (˜) fields of dislocations in an infinite medium and the (ˆ) fields that enforce the boundary conditions. In contrast to the small-strain analysis, the image problem for the (ˆ) fields is nonlinear and iteration is required for the solution. Also, the geometry changes affect the dislocation dynamics and the formulation of constitutive rules. Deshpande et al. [54] have opted to solve the nonlinear problem by using an updated Lagrangian formulation, that is with the reference configuration being taken as the current configuration. To account for the effect of lattice rotations, the anisotropic elastic constitutive relation needs to be specified on base vectors that rotate with the lattice. Equilibrium is expressed by eq. (15) so that an iterative solution procedure is required since neither the stress, nor the positions of material points nor the positions of dislocations are known. In particular, the iterations need to incorporate (i) glide of dislocation I which is obtained

§3.2


19

from a constitutive rule; (ii) rotation of the lattice at the current position of dislocation I ; (iii) the change in the position of dislocation I due to the glide of other dislocations; and (iv) the change in the position of dislocation I due to the image field. One consequence of accounting for finite deformations is that dislocations are no longer confined to a fixed slip plane due to slip on intersecting slip systems. In addition, because of finite rotations, the orientation of a nucleated dislocation loop varies with the local deformation state. Furthermore, if the material is elastically anisotropic then due to nonuniform lattice rotations, the elastic moduli become functions of position so that the constitutive relation for the (ˆ) stress field involves a polarization stress term. As in the small deformation formulation, all strains are elastic and plastic flow arises from slip which involves displacement jumps. The existence of these jumps needs to be accounted for during the iteration procedure to determine the updated state. A conventional finite element formulation cannot account for these displacement jumps. In Deshpande et al. [54], an approximate formulation was introduced that involved smearing out the jumps over a finite element. This has significant limitations for analyzing contact, friction and wear. The predicted surface profile inherently depends on the size of the region over which the averaging is done, interior points are precluded from becoming surface points (the introduction of new surface is important when environmental interactions are accounted for) and the sliding-off of part of an asperity is ruled out. Hence, there is a strong motivation for developing alternative numerical methods that can fully account for these finite deformation discrete dislocation plasticity effects. 3.2.3. Energy dissipation A portion of the mechanical energy expended in plastically deforming a solid is elastic and recoverable, while the remainder is the plastic work. Generally most, but not all, of the plastic working is converted into heating. Within a purely mechanical theory, conservation of energy implies Wappl = + Wplas ,

(31)

where Wappl is the work of the external loading, is the elastic energy stored and Wplas is the plastic dissipation. The elastic energy stored in the material is given by =

A

φe dA,

1 φe = σij εij , 2

(32)

where εij = εˆ ij + ε˜ ij . Because of the 1/r singularity in the dislocation stress and strain fields, is infinite when one or more dislocations are present. In order to obtain a finite value of a core cut-off needs to be introduced and some finite core energy associated with the excluded region around the dislocation line. In discrete dislocation plasticity, the dislocations contribute in two distinct ways to the energy balance: (i) dislocation glide is what gives rise to plastic dissipation and (ii) the elastic fields associated with the individual dislocation stress fields (and the core region) contribute to the stored elastic energy. Thus, when dislocations are present the energy density φe depends on the dislocation positions. Of particular relevance for friction is that the

20


Ch. 71

energy dissipation associated with dislocation glide serves as a heat source. In discrete dislocation plasticity the proportions of the applied work going into stored elastic energy and into plastic dissipation can be directly calculated (see Benzerga et al. [63]). With the local plastic dissipation rate determined, the temperature distribution can be calculated from ∂ ∂ ∂ ρcp κij + w˙ plas , (33) = ∂t ∂xi ∂xj where ρ is the density, cp is the heat capacity, is the temperature, κij is the heat conductivity matrix and w˙ plas is the local rate of plastic dissipation per unit volume. Under adiabatic conditions, i.e. when the time scale is too short for the temperature gradient term in eq. (33) to come into play, the temperature rise can be computed directly from the plastic dissipation obtained from the discrete dislocation plasticity analysis. The temperature rise during frictional sliding can be quite large, leading to local melting as observed by Tabor and Persson [19]. Even if melting does not occur, the large temperature rise can affect mechanical properties and permit mechanisms, for example dislocation climb, to come into play that would be ruled out at lower temperatures. Also, high localized heating can induce the formation of adiabatic shear bands in an asperity, see Molinari et al. [85]. The implications of asperity heating for frictional sliding are discussed by Rice [86], mostly in the context of rock mechanics, but the basic mechanics of the effect of localized heating on friction pertains to metals as well. 3.2.4. Statistical aspects At small scales, statistical effects play an import role in the behavior predicted by discrete dislocation simulations: (i) because of the limited number of dislocations present, the overall behavior can depend on the specific location of dislocation sources and/or obstacles to dislocation glide; and (ii) because dislocation dynamics is inherently chaotic in the sense that the evolution of the dislocation structures is sensitive to unavoidable (and uncontrollable) small perturbations, Deshpande et al. [87]. A consequence of the limited number of dislocation sources and/or obstacles is that the results of computations can depend on the precise location of the sources and obstacles. Hence, for a given source and/or obstacle density (which is generally all that can be presumed known), several computations with different realizations are needed to assess the sensitivity of a quantity of interest to variations in source and/or obstacle location. Such sensitivity is to be expected in circumstances, such as deformation of an asperity, where stress variations are large over short distances. While the extreme sensitivity of discrete dislocation predictions to small perturbations has a negligible effect on the material behavior averaged over relatively large volumes (e.g., uniaxial tension of macroscopic specimens), it may have a significant effect on behavior that depends on local fields. For example, Deshpande et al. [87] showed that discrete dislocation predictions for crack growth differed by about a factor of two when the location of dislocation sources was varied by 10−3 b. This is a fundamentally different type of statistical variation than that associated with the discreteness of dislocation sources and obstacles, since it is not controllable. At present, which responses are sensitive to each of these types of statistical variation is not known and, if a response is sensitive, it is not known how to quantify that sensitivity.

§4


21

The sensitivity of the frictional response of micrometer scale asperities to either of these sources of statistical variation remains to be quantified.

4. Discrete dislocation plasticity modeling of single asperity contact and friction The aim is to use the framework outlined in Section 3.2 to simulate frictional sliding. That has not happened yet. What has taken place are discrete dislocation plasticity calculations of indentation for various shaped indenters and some calculations of the initiation of sliding for an isolated contact. Furthermore, these calculations are two-dimensional, plane strain calculations for a single crystal and, in addition, finite deformation effects are neglected. However, some of these normal contact and sliding results can be combined to explore the predicted size dependence of the coefficient of static friction for a single asperity. As in any continuum mechanics formulation, what needs to be specified are the constitutive properties and the boundary conditions. Then, these together with the balance laws of continuum mechanics give a system of equations that determine the evolution of the dislocation structure and the overall response. The properties that need to be specified in the studies discussed here are the Burgers vector b; the elastic constants (since elastic isotropy is assumed: Young’s modulus E and Poisson’s ratio ν); the drag coefficient B that linearly relates the dislocation glide velocity to the Peach–Koehler force; the annihilation distance Le ; the number, location and strength of dislocation sources; and the number, location and strength of any dislocation obstacles. Dislocation sources and possibly obstacles are distributed randomly with a specified density on slip planes at angles φ (α) . The dislocation sources are two-dimensional analogs of a Frank–Read source which emit a dipole when the Peach–Koehler force on the source reaches τnuc b over a time tnuc . A dislocation obstacle releases a pinned dislocation when the Peach–Koehler force on that obstacle exceeds τobs b. Representative values used in the calculations are: E = 70 GPa; ν = 0.33; b = 0.25 nm; Le = 6b; and B = 10−4 Pa s. Dislocation sources are randomly assigned a nucleation strength τnuc from a Gaussian distribution with average τ¯nuc . The nucleation time tnuc , is taken to be 10 ns, which is associated with the time for a loop (a dipole in two dimensions) to expand to a stable separation. What typically varies between calculations are the density (and therefore the average spacing) of the sources and obstacles, the source and obstacle strengths and the number and orientation of slip systems. In the calculations discussed here, there are three slip systems and representative values of τ¯nuc and τobs are, respectively, 50 MPa and 150 MPa.

4.1. Single asperity indentation We summarize some results of Balint et al. [88] on the indentation of a film of thickness h by a rigid wedge that are relevant for modeling frictional contact. The faces of the wedge are inclined at an angle ω with respect to the indented surface and the boundary value

22


Ch. 71

Fig. 2. (a) Sketch of the indentation boundary value problem analyzed. (b) The definitions of the actual contact length a, the nominal contact length aN and the indentation depth δ.

problem analyzed is sketched in Fig. 2. Perfect sticking between the film and the wedge was assumed so that the rate boundary conditions along the contact were u˙ 1 = 0,

˙ u˙ 2 = −δ,

(34)

where δ is the depth of the indentation. The actual contact length is a 3 and the nominal contact length is aN ≡ 2δ/ tan ω. The single crystal films were specified to have three slip systems at φ (α) = ±54.7◦ and 0◦ with respect to the x1 -axis and source and obstacle densities ρsrc = 48 µm−2 and 98 µm−2 , respectively. Calculations were carried out for films of thickness h = 2 µm, 10 µm and 50 µm and for wedge angles of ω = 5◦ and 10◦ wedges. Convergence studies were carried out to ensure that the contact was accurately modeled and that the gradients in the (ˆ) fields due to the indentation were accurately captured. A numerical issue in discrete dislocation plasticity is that small time steps are required to capture the dislocation dynamics. Thus, in order to limit the computing time needed to reach significant plastic deformations an artificially high loading rate is used in most discrete dislocation plasticity calculations. The situation in this regard is much better for discrete dislocation plasticity than for molecular dynamics but is still a limitation. For example, in the indentation calculations of Balint et al. [88] an indentation rate of δ˙ = 0.4 ms−1 was used with a time step of t = 0.5 ns. The indentation force Fn versus applied indentation depth δ response of the h = 2 µm and 10 µm films is shown in Figs 3a and 3b for the ω = 5◦ and 10◦ indenters respectively. 3 What is termed the actual contact length here is referred to as the end-to-end contact length in Widjaja et al. [110].

§4.1


23

Fig. 3. Indentation force versus displacement response for the (a) ω = 5◦ and (b) ω = 10◦ wedges. Results are shown for h = 2 µm and 10 µm films. From Balint et al. [88].

In these calculations, ρnuc = 48 µm−2 and ρobs = 98 µm−2 . The indentation force was computed as Fn = −

L/2

−L/2

T2 (x1 , h) dx1 ,

(35)

where T2 = σ2j nj , with nj being the normal to the indented surface. The responses of corresponding elastic films (i.e. films without any dislocation activity) are included in Fig. 3 for comparison purposes. The indentation force increases approximately parabolically with displacement in all cases. For a given indentation depth, the indentation force increases with decreasing film thickness and wedge angle ω. The variation of the actual contact length a with the applied displacement δ is plotted in Figs 4a and 4b for the ω = 5◦ and 10◦ wedges, respectively. Results are included in Fig. 4b for all three film thicknesses. The variation of the nominal contact length, aN , with δ is also included in Fig. 4. Jumps in the actual contact length a occur as a consequence of material sink-in: as part of the deformed surface outside the contact length becomes nearly parallel with the surface of the indenter, continued indentation results in the contact length increasing in bursts. Comparison of the results for two calculations with the same dislocation source and obstacle densities in Balint et al. [88] showed that the jumps are stochastic in nature but trends remain the same. The actual contact length a is less than the nominal contact length for the h = 10 µm and 50 µm films (especially so for the ω = 10◦ indenter, Fig. 4) due to material sink-in. For the h = 2 µm films, sink-in occurs in the early stages of deformation but switches to pile-up when the plastic zone under the indenter reaches the rigid substrate (see Fig. 5). When pile-up occurs, the actual contact length a is greater than the nominal contact length aN = 2δ/ tan ω. The contact pressure p = Fn /a, the ratio of the indentation force to the actual contact length, is plotted in Fig. 6. Since for a sufficiently thick elastic film and for sufficiently shallow indentation depths, p is independent of indentation depth, the results of an elastic

24


Ch. 71

Fig. 4. Actual contact length a versus indentation depth δ for indentation with the (a) ω = 5◦ and (b) ω = 10◦ wedges in films with different thickness h. Results for two distributions with the same densities of dislocation sources and obstacles are included in (b) for the h = 10 µm film. The dashed lines show the variation of nominal contact area, aN , with δ. From Balint et al. [88].

Fig. 5. The deformed mesh (displacements magnified by a factor of 2) for the ω = 10◦ wedge indentation of the (a) h = 2 µm (pile-up) and (b) h = 50 µm films (sink-in) at a = 4 µm. From Balint et al. [88].

Fig. 6. Contact pressure p versus actual contact length a relation for indentation with the (a) ω = 5◦ and (b) ω = 10◦ wedges. Results are shown for three film thicknesses h. The continuum predictions of the nominal contact pressures for a single crystal are also shown (dashed lines). From Balint et al. [88].

§4.1


25

analysis serve as an indication of when mesh effects come into play. In the calculations in Balint et al. [88], size independence for an elastic calculation was found for a > 0.05 µm which is four to five times the element length. Thus, the numerical results for smaller indentation depths are affected by the discretization and are excluded from consideration. The contact pressure versus contact length relations in Fig. 6 for the h = 10 µm and 50 µm films are almost identical with the difference mainly due to different stochastic jumps in contact length so that for h 10 µm the response is essentially independent of film depth. The value of p decreases with increasing contact size, before reaching a contact-size independent value for the h = 10 µm and 50 µm films. Also shown in Fig. 6 are the contact-size independent nominal pressures, pN = Fn /aN , obtained using conventional crystal plasticity (see Bouvier and Needleman [89]). The results for the h = 2 µm film are qualitatively different from those for the two thicker films in that the contact pressure initially decreases with increasing contact length, but when the plastic zone reaches the substrate x2 = 0, the pressure increases with increasing contact length. To gain insight in the origin of the difference between the continuum and discrete dislocation predictions, Fig. 7 compares the plastic zones. This is done in terms of distributions of slip Ŵ defined as Ŵ=

3 (α) γ ,

(36a)

α=1

where the slip γ (α) on a slip system α is given by (α)

(α)

γ (α) = si εij mj . (α)

(36b)

(α)

Here si and mj are unit vectors tangential and normal to the slip plane. In discrete dislocation plasticity Ŵ is not the actual slip since, on a given slip plane, the contributions

Fig. 7. Distribution of the total slip Ŵ in the h = 50 µm film indented by the ω = 5◦ wedge indenter to an indentation depth δ = 0.4 µm. (a) Discrete dislocation prediction and (b) continuum crystal plasticity prediction. The distributions are shown in the deformed configuration (no magnification of the deformation). From Balint et al. [88].

26


Ch. 71

Fig. 8. The dislocation structure in a central 20 µm × 20 µm region around the indenter tip in the h = 50 µm film indented to (a) δ = 0.2 µm and (b) δ = 0.4 µm by the ω = 5◦ wedge. From Balint et al. [88].

from dislocations on all slip planes are contained in γ (α) . Nevertheless, it does give a picture of the deformation mode. Plasticity is restricted to only under the indenter in Fig. 7b. The discrete dislocation plasticity results in Fig. 7a give a larger plastic zone size than corresponding continuum crystal plasticity results. Two stages in the evolution of the dislocation structure are shown in Fig. 8 for the ω = 5◦ wedge indentation of the h = 50 µm film. There are geometrically necessary dislocations (GNDs) arising from the plastic strain gradients, but there are also significant numbers of statistically stored dislocations so that the structure associated with the GNDs cannot readily be distinguished. Previous two dimensional discrete dislocation plasticity simulations of the indentation of crystalline solids, Polonsky and Keer [66], Kreuzer and Pippan [71,72] have shown an indentation size effect for sinusoidal and wedge shaped indenters, with source-limited plasticity playing a significant role. The effect of a much reduced dislocation source and obstacle density is illustrated in Fig. 9a. Curves of contact pressure p versus actual contact length a are plotted for the ω = 5◦ wedge and the h = 10 µm and 50 µm films with a much lower density of dislocation sources and obstacles (ρsrc = 9 µm−2 , ρobs = 18 µm−2 ). For comparison purposes, the corresponding results from Fig. 6a are included. With the lower density of sources, the contact pressure is higher and a size independent contact pressure is reached at a ≈ 8 µm as compared with a ≈ 4 µm in Fig. 6. However, the contact pressures converge to the same limit at large contact sizes. A comparison between the contact length a versus indentation depth δ relation for the h = 10 µm and 50 µm films for the two sets of source and obstacle densities is shown in Fig. 9b. The reduced plasticity in the crystal with the lower source densities results in increased material sink-in although again the curves converge for large indentation depths. Discrete dislocation plasticity predictions of the indentation of films by a rigid cylinder were reported by Widjaja et al. [73] and included in Fig. 10 is the discrete dislocation plas-

§4.1


27

Fig. 9. A comparison between the ω = 5◦ indentation of the ρsrc = 48 µm−2 and ρobs = 98 µm−2 films with the ρsrc = 9 µm−2 , ρobs = 18 µm−2 films. (a) Actual contact pressure versus actual contact length and (b) actual contact length as a function of indentation depth. Results are shown for the h = 10 µm and 50 µm films. From Balint et al. [88].

Fig. 10. Discrete dislocation predictions of the contact pressure p versus indentation depth δ relation for three selected film thicknesses h indented by the ω = 5◦ wedge and also for the indentation of a h = 10 µm film by a cylindrical indenter of radius R = 5 µm. The results are shown for the ρsrc = 48 µm−2 and ρobs = 98 µm−2 materials. Wedge indentation data from Balint et al. [88].

ticity prediction (Widjaja [90]) for a h = 10 µm film with ρsrc = 48 µm−2 indented by a cylinder of radius R = 5 µm. Since the finite element mesh for the cylindrical indentation

28


Ch. 71

calculation was significantly finer than that for the wedge simulations the data for depths smaller than ∼0.05 µm are meaningful and therefore are not eliminated. Clearly, the indentation response with the cylindrical indenter is qualitatively different from that with the wedge indenters. In particular, the contact pressure initially increases with indentation depth before decreasing and leveling off at p ≈ 900 MPa. Thus, not only is the indentation size effect qualitatively different but the size-independent contact pressure is also significantly higher. These differences arise because of the interaction of the length scale associated with the indenter (i.e. the radius of the indenter) with the material length scales. By contrast, no length scale is associated with the wedge indenter. Similar observations of the effects of the indenter shape on the indentation response of single crystals were reported by Kreuzer and Pippan [71] for wedge and cylindrical indenters and by Polonsky and Keer [66] for sinusoidal indenters. Experimentally, a dependence of the indentation size effect on indenter shape was seen by Lou et al. [91]. Thus, it is expected that the effect of asperity size will depend on asperity shape. 4.1.1. Comparison with the Nix–Gao model Nix and Gao [49] developed a model to capture the variation of the contact pressure with indentation depth based on the role of geometrically necessary dislocations. They first calculated the density of geometrically necessary dislocations from geometric arguments and then employed the Taylor relation for the scaling of strength with dislocation density to derive the contact pressure versus indentation depth relation as pN = p0

1+

δ∗ . δ

(37)

Here δ ∗ is a characteristic length that depends on the wedge angle and p0 is the contact pressure in the infinite indentation depth limit. We note that in the Nix–Gao model neither p0 nor δ ∗ are intrinsic material properties; the characteristic length δ ∗ depends on the density of statistically stored dislocations and p0 depends on the wedge angle. Balint et al. [88] found that eq. (37) does not provide the best fit to the discrete dislocation predictions of the p–δ relation. The discrete dislocation predictions of the nominal contact pressure, pN = Fn /aN versus δ were found to be well represented by a similar relation having the form pN δ∗ n . = 1+ p0 δ

(38)

The best fits to the discrete dislocation predictions using eq. (38) (with p0 , δ ∗ and n as free parameters given by the least-squares fit) are shown in Fig. 11 for the h = 50 µm films. The best-fit value of n for the ω = 5◦ wedge is ≈0.7, with δ ∗ = 0.10 µm for the higher source density material and 0.37 µm for the lower source density material. For the ω = 10◦ wedge, the best-fit value of n is 0.64. Consistent with the Nix and Gao [49] model, the discrete dislocation calculations predict that δ ∗ increases with increasing wedge angle ω. The linear dependence of the square of the contact pressure, (p/p0 )2 , on the inverse indentation depth, 1/δ, as predicted by Nix and Gao [49] is in good agreement with a wide

§4.2


29

Fig. 11. Comparison between discrete dislocation predictions for the nominal contact pressure, pN , for the h = 50 µm films and the best fit power laws of the form of eq. (38). Data from Balint et al. [88].

body of experimental evidence for indentation depths δ > 1 µm. Typically, the Nix–Gao relation (37) gives a less good fit to indentation data over a wider range of indentation depths; see for example Poole et al. [92], Gerberich et al. [93], Swadener et al. [38]. This is consistent with the conical indentation simulations of Begley and Hutchinson [50], based on the Fleck and Hutchinson [94] strain gradient plasticity theory, which do not predict the Nix–Gao [49] type scaling over the full range of indentation depths. Begley and Hutchinson [50] suggested that alternative descriptions of the effective strain (which governs the interaction between the statistical and geometrically necessary dislocations) in the Fleck and Hutchinson [94] strain gradient plasticity theory could be employed to obtain the Nix– Gao [49] type scaling. Recently, Abu Al-Rub and Voyiadjis [95] have shown this. There are several possible reasons why the best fit scaling in the discrete dislocation simulations differs from the scaling given by Nix and Gao [49]: (i) the indentation depths in the calculations are restricted to δ 0.4 µm, while the Nix–Gao scaling is most appropriate for indentation depths δ > 1 µm (see [95]); (ii) the model of Nix and Gao [49] essentially assumes no interaction between the statistically stored and geometrically necessary dislocations while this interaction occurs in the discrete dislocation plasticity calculations; and (iii) the square-root scaling in eq. (37) arises from three-dimensional dislocation interaction effects not accounted for in the two-dimensional simulations of Balint et al. [88].

4.2. Single asperity sliding The sliding of a single asperity has been investigated by Deshpande et al. [70] by means of the boundary value problem sketched in Fig. 12a. The analysis focuses on the initiation of sliding between a flat-bottomed indenter and a planar single crystal substrate. Contact

30


Ch. 71

Fig. 12. (a) Sketch of the boundary value problem analyzed in Deshpande et al. [70]. Softening cohesive relation (b) and non-softening cohesive relation (c).

between the indenter and the substrate was modeled using a cohesive shear traction versus sliding displacement relation, Fig. 12b,c. Friction being associated with adhesion is an old idea, which Bowden and Tabor [1] used to develop their concept of a ‘plastic junction’. Various researchers have argued that the adhesion arises from the inter-atomic attractive forces between the two contacting surfaces. However, at present there is no fundamental basis for choosing the form of cohesive relation to use at the size scale at which discrete dislocation plasticity is relevant. Therefore, Deshpande et al. [70] used two cohesive relations between shear traction and slip, as sketched in Figs 12b and 12c. One is a softening cohesive relation with a finite cohesive energy, sketched in Fig. 12b which is similar to that suggested by Xu and Needleman [96] and used in fracture analyses, while the other, Fig. 12c, is a non-softening relation in which the shear traction reaches a maximum and then remains constant.

§4.2


31

In molecular dynamics studies of friction, some appropriately chosen interaction potential is taken to connect atoms across the contact surface and provide ‘surface adhesion’. This interaction involves irreversible ‘snap-throughs’ into local energy minima. As noted by Johnson [97], a model of interlocking elastic asperities, such as that proposed by Caroli and Nozieres [98], would give analogous behavior, albeit at larger length scales. The nonsoftening cohesive relation in Fig. 12c, which can be regarded as the Maxwell construction to an oscillating shear traction versus displacement relation, can represent either of these scenarios. The indenter and crystal are subjected to a relative displacement U (t) in the x1 -direction to give relative sliding of the contacting surfaces. The focus in the calculations is on the initiation of sliding, with the definition of initiation depending on the cohesive relation used; for the softening cohesive relation, Fig. 12b, initiation of sliding is identified with vanishing shear traction, whereas with the non-softening cohesive relation, Fig. 12c, initiation of sliding is identified with the attainment of a specified value of U . In either case, the value of U at initiation is denoted by Uf , and the friction stress τf is the corresponding value of τ , the average shear traction over the contact length a, i.e.

1 T1 dx1 . (39) τ =− a Scoh Computed curves of average shear stress over the contact length, τ , versus applied displacement, U , are shown in Fig. 13 for a crystal with three slip systems at φ (α) = ±60◦ and 0◦ with respect to the x1 axis and a density of the point sources and obstacles of ρsrc = 72 µm−2 and ρobs = 124 µm−2 , respectively. The cohesive properties in this calculation were taken to be τmax = 300 MPa and δt = 0.5 nm. In Fig. 13a, for the non-softening cohesive relation, τ first increases approximately linearly with U for contact sizes a 4 µm, and then reaches a plateau, while for contact lengths in the range 0.04 µm a 2 µm, τ continues to increase with increasing U . For the two smallest contacts, a = 0.1 µm and a = 0.04 µm, no dislocation activity occurs before the cohesive strength is attained. The horizontal line at τ = 300 MPa is the response of the a 0.6 µm contacts after the contact shear stress attains the cohesive strength τmax = 300 MPa. For a sufficiently small contact the initiation of sliding is identified with τ attaining the cohesive strength τmax , while for a sufficiently large contact τ reaches a nearly constant value and this plateau value is identified with τf . In the transition regime, for the non-softening cohesive relation, the identification of the values of Uf and τf are somewhat arbitrary. The contact stress τ versus sliding displacement U response for the case of a softening cohesive relation is shown in Fig. 13b for contact sizes in the range 0.04 µm a 20 µm. As in Fig. 13a, the contact shear stress τ reaches the cohesive strength τmax for small contact sizes and attains a size independent limit for large contact sizes (a 4 µm). For contact sizes in the transition regime, the value of τ in Fig. 13b drops to zero rather abruptly at the points marked by filled circles (the drop is not shown in Fig. 13b). Separation with no dislocation activity occurs for sufficiently small contacts, while considerable dislocation activity precedes attaining the cohesive strength in the transition regime. Comparison of the softening and non-softening cohesive relation curves of friction stress τf versus contact size a in Fig. 14 shows that, while the small contact size and large contact

32


Ch. 71

Fig. 13. Contact shear stress versus displacement response for (a) the non-softening cohesive law and (b) the softening cohesive relation, for selected values of the contact size a. The filled circles in (b) indicate the value of τ just before it abruptly drops to zero. From Deshpande et al. [70].

size limiting values of the τf –a curve are unaffected by the form of the cohesive relation, the values of τf obtained for the case of the softening cohesive relation are slightly lower than those obtained for the non-softening relation for intermediate contact sizes. However, the qualitative features of the τf –a curve are the same for both cohesive relations, with the friction stress approximately proportional to a −1/2 for the intermediate size contacts. The effect of the contact size a on τf using two values of Uf , 0.05 µm and 0.1 µm, is shown in Fig. 14. The dependence of τf on the choice of Uf is weak, at least for the two values in Fig. 14. The form of the dependence of friction stress on contact size in Fig. 14, and in the analysis of Hurtado and Kim [67,68], is consistent with the low values of friction stress seen experimentally in the surface force apparatus (Homola et al. [99]), where the contact area is relatively large, and the much higher friction stress values seen for contact with an atomic force microscope tip (Carpick et al. [100]), where the contact area is relatively small. Only very recently have measurements been made in the transition regime, in a case

§4.2


33

Fig. 14. Friction stress as a function of the contact size. The friction stresses for two choices of the displacement Uf are shown for the non-softening cohesive relation while τf is defined as either the plateau or the value of τ just before the abrupt drop to zero for the softening cohesive relation. From Deshpande et al. [70].

where surface effects likely dominate so that the model of Hurtado and Kim [67,68] is applicable, and the experimental measurements are in good agreement with a square root dependence on contact size (Li and Kim [101]). The number of dislocations per unit contact length N/a provides an indication of the amount of induced plasticity, and is plotted as a function of contact size in Fig. 15 at a fixed value of displacement U . The value of N/a is maximum at a ≈ 0.6 µm and then decreases with no dislocations nucleated for the a = 0.1 µm and a = 0.04 µm contacts.

Fig. 15. Number of dislocations per unit contact length, N/a, at U = 0.05 µm as a function of the contact size a for sliding with the non-softening cohesive relation. From Deshpande et al. [70].

34


Ch. 71

Fig. 16. Deformed mesh at U = 0.05 µm for the cases with (a) a = 10 µm and (b) a = 1 µm for sliding with the non-softening cohesive relation. The displacements are magnified by a factor of 10 in both plots. The arrows indicate the direction of sliding of the indenter relative to the substrate. From Deshpande et al. [70].

Contacts with a 0.1 µm are dislocation source-limited elastic contacts for which τf is controlled by the cohesive strength. The greater dislocation density for smaller contacts results in more localized deformation as seen from the deformed meshes (at U = 0.05 µm) shown in Figs 16a and 16b, for the a = 10 µm and 1 µm contacts, respectively. These plots also show that material is being transported from the rear of the contact (the positive x1 -direction) to the front of the contact (the negative x1 -direction) which results in material piling-up in front of the contact region (the indenter is sliding in the negative x1 -direction relative to the substrate). This shows that plastic deformation can lead to the creation of asperities even when the surface is initially flat. The variation of the friction stress τf (using Uf = 0.05 µm) with contact size a is shown in Fig. 17a for three sets of material parameters where the ratio ρobs /ρsrc is maintained at 124/72 ∼ 1.7. The friction stress τf of the ρsrc = 20 µm−2 material is slightly higher than that of the ρsrc = 72 µm−2 and ρobs = 124 µm−2 material for intermediate size contacts, but for both materials τf ∝ a −1/2 in that regime. The results for the high dislocation source material (ρsrc = 155 µm−2 and ρobs = 267 µm−2 ) differ in that: (i) the friction stress is not proportional to a −1/2 for intermediate size contacts and (ii) the contact size, a, at which τf reaches the cohesive strength is smaller. In Fig. 17b the dependence of the dislocation density on a is similar for ρsrc = 20 µm−2 and ρsrc = 72 µm−2 , with the dislocation density first increasing with decreasing contact size and then reducing to zero for contact sizes a < 0.4 µm. However, with ρsrc = 20 µm−2 , the dislocation density is about a factor of three smaller than that with ρsrc = 72 µm−2 for all contact sizes considered. On the other hand, the dislocation density with ρsrc = 155 µm−2 is greater than that with ρsrc = 72 µm−2 for large contacts but lower for intermediate and small contact sizes. The difference in deformation mode between the crystals with ρsrc = 72 µm−2 and ρsrc = 155 µm−2 for the a = 2 µm contact is illustrated by the distribution of total slip Ŵ in Fig. 18. Large amounts of slip occur to about 0.5–1 µm below the contact surface for the lower source density material. On the other hand, slip is concentrated at the surface for the high source density material with the φ (α) = 0◦ slip system most active. Provided that there are sufficiently many dislocation sources, the discrete dislocation results are consis-

§4.2


35

Fig. 17. (a) Effect of source (and obstacle) density on the friction stress and (b) the corresponding number of dislocations per unit contact length. In both cases the results are plotted at U = 0.05 µm. The non-softening cohesive law is used and the results for ρsrc = 72 µm−2 are the same as those shown in Fig. 14. From Deshpande et al. [70].

tent with the behavior predicted from non-hardening isotropic continuum plasticity where, asymptotically, plasticity is concentrated in a vanishingly narrow band beneath the contact. Conventional models for friction typically assume that the intrinsic friction stress τf is independent of the normal contact pressure with the dependence of the macroscopic frictional force on the contact pressure a result of the change in actual contact area, Bowden and Tabor [1]. The variation of the friction stress with contact area is plotted in Fig. 19a for the non-softening cohesive relation. In these calculations a normal pressure given by T2 = −p

on − a/2 x1 a/2, x2 = 0

(40)

is first increased to the specified level and then the sliding displacement prescribed while keeping the pressure constant. For comparison purposes, the zero contact pressure case from Fig. 14 is also shown. With p = 100 MPa, only a small amount of dislocation activity

36


Ch. 71

Fig. 18. Distribution of the total slip Ŵ at U = 0.05 µm for the a = 2 µm contact. (a) The material with ρsrc = 72 µm−2 and (b) material with ρsrc = 155 µm−2 with the non-softening cohesive relation. The contact surface is indicated in each case. From Deshpande et al. [70].

takes place prior to the application of the sliding displacement. Fig. 19a shows that τf is largely unaffected by a pressure p = 100 MPa. When the contact pressure p is increased to 200 MPa, the value of the friction stress is considerably reduced for larger contacts, where τf is governed by plastic flow, but is unaffected for smaller contacts, for which there is negligible dislocation activity. The influence of varying the cohesive strength τmax on the friction stress τf is shown in Fig. 19b. The friction stress is negligibly affected by the cohesive strength for the larger contacts, since for large contacts the τf –a curve is governed by the plastic properties of the substrate. Although the size of the transition region increases with increasing cohesive strength, the value of τf in the transition region is relatively unaffected by the cohesive strength, which indicates that the size effects in this region are mainly due to substrate plasticity. Although Hurtado and Kim [67,68] considered circumstances where dislocation nucleation only occurred at the edge of the contact, while in Deshpande et al. [70] all dislocations originate from Frank–Read sources inside the material, the results of the two analyses share several common features. Hurtado and Kim [67,68] predicted a size dependent friction stress τf separated into three regimes: (i) for small contacts, τf is size independent and equal to the theoretical shear strength; (ii) for sufficiently large contacts, multiple

§4.2


37

Fig. 19. (a) Effect of normal pressure on the friction stress (τmax = 300 MPa, Uf = 0.05 µm) and (b) the effect of the cohesive strength on the friction stress (p = 0, Uf = 0.05 µm). These calculations were carried out with ρsrc = 72 µm−2 and with the non-softening cohesive relation. From Deshpande et al. [70].

dislocation-cooperative slip takes place, which is a mobility controlled process and τf is equal to an effective Peierls stress; (iii) in the transition between these two regimes, the friction stress is governed by nucleation of a dislocation at the edge of the contact, which gives rise to an inverse square root size dependence. The analyses in Deshpande et al. [70] exhibit a similar dependence on contact size, including an inverse square root size dependence in the transition regime, but the governing parameters differ. For example, the Peierls stress is zero and the large contact size regime is governed by dislocation nucleation occurring through internal sources so that the source spacing (or density) is a key parameter and the square root size dependence in the transition regime is lost for a sufficiently large source density. The transition from substrate plasticity governed friction to interface cohesion gov-

38


Ch. 71

erned friction is a consequence of source limited plasticity in Deshpande et al. [70]. The cohesive strength mainly determines the upper-shelf in the friction stress versus contact size relation; otherwise the value of τf is controlled by substrate plasticity. 4.2.1. Energy dissipation in frictional sliding An issue of interest is the extent to which the plastic dissipation depends on contact size. This has particular importance regarding frictional heating which could result in the formation of adiabatic shear bands and/or the welding of asperities during frictional sliding. Here we present selected results from Deshpande et al. [102] to illustrate the dependence of energy dissipation and storage on contact size. In the presence of a cohesive surface, part of the external work, in addition to being stored elastically or dissipated as heat, is dissipated in the cohesive surface. Thus the energy balance eq. (31) becomes Wappl = + Wplas + Wcohes ,

(41)

where is the elastic energy stored in the material, Wplas the plastic dissipation and Wcohes the energy expended in the cohesive surface. The evolution of the partitioning of energy during sliding is shown in Fig. 20 for three contact sizes. The evolution of the total plastic and strain energy is plotted in Fig. 20a while Fig. 20b shows the evolution of the cohesive energy. For the 0.1 µm contact, where τf = τmax , the energy is partitioned into elastic energy and cohesive energy, with the elastic energy dominating the early stages of sliding and the cohesive energy the latter stages of sliding. For the 10 µm contact, where τf is about equal to the yield strength, the cohesive energy is a negligible fraction of the total work W and the partitioning is mainly between the elastic energy and the plastic dissipation. Some of the elastic energy is associated with the ‘overall’ stress state and some is associated with the dislocation structure present in the material. To gain insight into the fraction of the elastic energy associated with the dislocation structures, unloading calculations were carried out for the 10 µm and 1 µm contacts until the average shear stress across the contact τ ≈ 0. The evolution of , Wplas and Wcohes during unloading are included in Fig. 20. While about 20% of the total external work done is stored as elastic energy associated with the dislocation structure for the 10 µm contact, this stored energy rises to about 40% in case of the 1 µm contact (and is zero for the 0.1 µm contact since there is no dislocation activity). In addition to the above overall energy considerations, the spatial distribution of dissipation is relevant in relation to local heating. An issue in the computation of this in discrete dislocation plasticity is that since the plastic part of the deformation is associated with the evolution of displacement jumps across the slip planes, the displacement gradient field needed to compute strains involves delta functions. In Deshpande et al. [102] an approximation was used to calculate the spatial distribution of plastic dissipation. Deshpande et al. d , in each finite element that was computed by [102] introduced a smooth strain rate field, ε˙ ij differentiating the total displacement rate field u˙ i in that element using the finite element

§4.2


39

Fig. 20. Normalized (a) plastic dissipation Wplas and strain energy and (b) cohesive energy Wcohes as a function of the displacement U for three contact sizes a. The evolution of the energies upon unloading from U = 0.5 µm for the a = 10.0 µm and 0.1 µm contacts is also included. Wplas , and Wcohes have been normalized by the total work W . These calculations correspond to the data presented in Fig. 13a. From Deshpande et al. [102].

shape functions. Then, at each point within an element, the plastic dissipation, wplas , is the d minus the energy stored, i.e. stress working through ε˙ ij wplas =

0

t

1 d ∗ dt − σij εij σij ε˙ ij . 2

(42)

As illustrated in Fig. 21, the spatial distribution of the plastic dissipation is strongly nonuniform and depends on the size of the contact region. For the 1 µm contact, the extent of the region near the surface with high dissipation is several times the extent of the contact

40


Ch. 71

Fig. 21. Distribution of the plastic dissipation wplas per unit volume at U = 0.05 µm for the cases with (a) a = 10 µm and (b) a = 1 µm. These distributions correspond to the calculations of Fig. 13a. The contact surface is indicated in each case. From Deshpande et al. [102].

area and the high dissipation region extends more than 0.5 µm into the material. On the other hand, for the 10 µm contact, plastic dissipation mainly occurs directly below the contact surface. These differences are consistent with the slip distributions in Fig. 18 and imply that the temperature distribution is contact size dependent. No plasticity induced temperature rise is predicted to occur for the 0.1 µm contact since there is no dislocation activity. Calculations of the temperature distribution would be of much interest since it is difficult to experimentally determine temperature distributions in sliding contact, especially below the sliding surface. The capability of computing the temperature distribution has been developed (at least in the two-dimensional context) but no such calculations have been carried out. The circumstance discussed here pertains to the initiation of sliding. What would be of greater interest is the temperature distribution, and its asperity size dependence, under continued sliding.

§4.3


41

4.3. Single asperity coefficient of friction The single asperity normal contact and sliding simulations presented in Sections 4.1 and 4.2 can be combined to obtain an estimate of the adhesional friction coefficient for single wedge-like asperity contacts as follows. For a given indentation force Fn , the actual contact length a is given by the indentation simulations while the frictional sliding force τf a is obtained from the sliding simulations. This procedure presumes that the indentation and sliding processes are decoupled, and in fact the normal indentation and sliding simulations are not performed sequentially with the dislocation structure and normal pressure distribution from the indentation used as the starting point for the sliding simulations. Rather, the friction stress τf is determined as in Section 4.2 where a sliding simulation for a flat contact of size a is carried out for an initially dislocation-free crystal. Additionally, the effect of the indentation pressure on the friction stress τf is neglected. Thus, the estimate of the coefficient of friction is highly approximate, but hopefully suggests possible trends. Bhushan and Nosonovsky [69] have presented a similar analysis based on scaling relations derived from nonlocal plasticity considerations. In their model, the material length scales that give rise to the size dependence of the contact pressure and friction stress are inputs. In the discrete dislocation simulations presented here, the size dependence is a natural outcome. The discrete dislocation predictions of the actual indentation pressure p versus actual contact area a for the three materials considered in Section 4.2 are presented in Fig. 22 for the ω = 5◦ wedge. These computations were carried out in the manner described in Section 4.1 on the h = 50 µm films. The friction coefficient μ ≡ τf /p

(43)

Fig. 22. (a) The contact pressure p versus actual contact length a relation for the ω = 5◦ wedge indentation of the three materials analyzed in Section 4.2. (b) The associated coefficient of friction, μ ≡ τf /p, for the initiation of sliding at a single asperity contact with the adhesive strength τmax = 300 MPa. Results from Deshpande et al. [103].

42


Ch. 71

for the single asperity contact is then obtained from a combination of the results in Figs 17a and 22a (where the adhesive strength is τmax = 300 MPa). The predicted values of μ are shown in Fig. 22b as a function of the contact size a. The coefficient of friction μ decreases with decreasing contact size for contact sizes a 1 µm and is essentially independent of contact size (or, equivalently, indentation force) for larger contact sizes. For the three materials considered here, the large-contact size value of the single asperity coefficient of friction μ is between 0.25 and 0.35. The reduction in the coefficient of friction at small contact sizes is a result of the friction stress τf being limited by the adhesive strength τmax for small contacts, while the contact pressure continually increases with decreasing contact size due to the indentation size effect seen in Section 4.1. Experimental data does suggest that the coefficient of friction decreases with decreasing contact sizes; see for example the nano and macro single asperity coefficient of friction measurements in Ruan and Bhushan [104]. The indentation and sliding material length scales ld and ls , respectively, in the model of Bhushan and Nosonovsky [69] can be adjusted to give results similar to those obtained here. In particular, the choice ld > ls , predicts that the friction coefficient decreases with decreasing scale similar to the discrete dislocation plasticity predictions presented here. However, the physical interpretation of the length scales in the model of Bhushan and Nosonovsky [69] is quite different from those in the discrete dislocation plasticity analyses. In particular, ls in Bhushan and Nosonovsky [69] is associated with the mean length over which a dislocation climbs. Climb of dislocations is not accounted for in our analyses. Rather, the size dependence in the sliding calculations of Section 4.2 is a result of the effects of geometrically necessary dislocations and source limited plasticity.

5. Concluding remarks For the circumstances considered in Section 4, for a sufficiently large asperity, say 10 µm or larger, plastic deformation is reasonably well represented by continuum plasticity and discrete dislocation effects do not play a major role. For a sufficiently small asperity, say 1 µm and smaller, adhesion dominates so that dislocation plasticity does not play a significant role in determining the frictional response. In the intermediate regime, discrete dislocation effects are significant and lead to a size dependent response for both the normal pressure and the shear response. However, because the scaling with size of the normal and tangential responses is similar, the single asperity coefficient of friction in Fig. 22 is not strongly size dependent in the intermediate regime. On the other hand, the size scales at which a transition between regimes occurs is determined by discrete dislocation properties, e.g. densities and strengths of sources and obstacles. The circumstances considered in Section 4 are limited to the onset of sliding and static friction. Advances in both modeling and computation are needed to develop the capability to carry out discrete dislocation plasticity analyses of friction in a broader range of circumstances and discrete dislocation effects may play additional roles when continued sliding is considered. In addition, plastic deformation mechanisms other than dislocation motion, e.g., twinning and/or phase transformations, may come into play and the coupling of these with dislocation plasticity may need to be considered.

§5


43

Impediments to further progress stem from a lack of understanding of physical mechanisms as well as from a lack of a suitable computational methodology. Among the key modeling issues are: (i) the development of criteria for dislocation nucleation and (ii) the modeling of grain boundaries, which can serve as sources and sinks for dislocations. In the discrete dislocation plasticity simulations discussed in Section 4, dislocations nucleate at Frank–Read sources in the bulk. On the other hand, in the model of Hurtado and Kim [67, 68], dislocations nucleate on the surface. Nucleation from surfaces, say from surface steps, is apt to play an important role for smaller size asperities (if only because of the increasing surface to volume ratio with decreasing size). There is a need to incorporate a surface-step dislocation nucleation criterion into discrete dislocation plasticity formulations. The recent analyses of Xu and Zhang [105] and Yu et al. [106] provide a step in that direction. Wang et al. [107] have observed a tensile residual stress layer with a thickness of a few tenths of micrometers after contact on a polycrystalline aluminum surface. The existence of this tensile layer is predicted to be a consequence of dislocation nucleation from surface steps and the existence of such a tensile surface layer has clear implications for wear mechanisms. The extent to which such a layer is affected by dislocations nucleated in the bulk remains to be determined. The results presented in this article have been restricted to single crystals. In a wide variety of circumstances of interest, sliding distances are greater than the grain size. In polycrystalline materials, the grain boundaries play an essential role by limiting slip distances and can also act as dislocation sources and sinks. Furthermore, high local temperatures in frictional sliding can lead to grain boundary deformation modes occurring that are not possible at lower temperatures. An obvious computational challenge is to extend the two-dimensional analyses to threedimensions. But even in the two dimensional context, the current computational capabilities are not up to the task. Frictional sliding often leads to the formation of rubble along the sliding surfaces. This rubble can form by asperity fracture or by extensive plastic slip. Computational methodologies that can allow for both kinds of separation are being developed (e.g., Remmers et al. [108], Rabczuk and Belytschko [109]) but are not yet available. The successful development of such a methodology is key for micro-mechanically based simulations of wear. Despite the limitations, current modeling and computational capabilities for simulating frictional sliding using discrete dislocation plasticity have not been fully exploited. For example, analyses to date have neglected finite deformation effects. Even if separation does not occur, asperities undergo large deformations which affects their stress carrying capacity and, accordingly, the frictional response. Implications of discrete dislocation effects for multi-asperity contact and friction, in particular for the evolution of contact area during frictional sliding, have yet to be explored. A discrete dislocation plasticity calculation of the temperature distribution in a deforming asperity remains to be carried out. The stress–strain response and the evolution of the dislocation structure are intimately coupled in discrete dislocation plasticity. The calculation of the effective coefficient of friction in Section 4.3 is based on separate calculations of indentation and sliding. The dislocation structures that form when normal pressure and sliding occur together will differ from those that are present in the uncoupled analyses. In particular, the effects of this coupling on the source limited plasticity regime in which the friction stress is set by the adhesive

44


strength are unclear. This may result in qualitative differences in the size dependence of the friction coefficient μ seen in Fig. 22. The limited number of analyses that have been carried out show that there are strong size effects at the scale where discrete dislocation plasticity is relevant. These size scales may play a prominent role in setting length scales that determine surface roughness and, therefore, the size distribution of asperities. Discrete dislocation plasticity analyses of contact and sliding have the potential to provide a basic understanding of friction and wear and to play a significant role in the development of physically based friction constitutive descriptions. At present, that potential is unfulfilled. Indeed, one may say that the surface has barely been scratched!

Acknowledgements AN is pleased to acknowledge support from the MRSEC Program of the National Science Foundation under award DMR-0520651 and the support and hospitality provided by the Kavli Institute for Theoretical Physics (National Science Foundation Grant No. PHY990794) during his participation in the program on Fracture, Friction and Earthquakes. We are pleased to acknowledge Professors A.F. Bower and K.-S. Kim of Brown University, Professor M.O. Robbins of Johns Hopkins University and Professor N. Lapusta of Caltech for helpful comments and criticisms.

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CHAPTER 72

Dislocation Fields in Piezoelectrics J.P. NOWACKI The Polish–Japanese Institute of Information Technology, ul. Koszykowa 86, 02-008 Warsaw, Poland and

V.I. ALSHITS Institute of Crystallography RAS, Leninskii prospect 59, 119333 Moscow, Russia e-mail: [email protected]

© 2007 Published by Elsevier B.V.


Contents 1. Introduction 49 2. A dislocation and its electrostatic analogue as a combined line defect in a piezoelectric medium 2.1. Three definitions of a dislocation 51 2.2. Electrostatic analogue of a dislocation 53 2.3. Combined electro-elastic line defect – a 4D dislocation 55 3. Fundamental properties of 4D dislocations of an arbitrary shape 56 3.1. Generalized Green’s tensor for the problem considered 56 3.2. “Displacement” and “distortion” fields of a generalized line defect 58 3.3. Energy of a 4D dislocation and its interaction with external fields 62 4. Electro-elastic fields of straight 4D dislocations 65 4.1. The integral representations 65 4.2. The matrix representations 69 4.3. The Stroh-like algebraic representations 71 5. Some concluding remarks 76 Acknowledgements 77 References 77

51

1. Introduction In recent years, due to technological applications of piezoelectric materials in multiple devices (resonators, filters, sensors, delay lines, etc.) the problem of electro-elastic fields produced in such materials by different sources has received considerable attention from researchers (see, for instance, [1–14]). The reason for such popularity is rather evident: piezoelectric devices of a new generation designed for functioning at very low signal levels call for materials and constructions of increasingly high quality and sensitivity. Such devices might be very dependent on the level of electric field noise excited by various sources (e.g., by dislocations) in piezoelectric media due to electro-mechanical coupling. From this point of view, any progress in theoretical description of electro-elastic fields in piezoelectric structures, apart from its ordinary scientific aspect, might have a more wide address related to possible applications. And it is not accidental that the majority of publications mentioned above is related to piezoelectric fields of line defects, in particular, of dislocations, in different single-, bi- or multi-crystalline structures. Indeed, a dislocation is one of the most dangerous sources of piezoelectric fields, which dramatically increase their strength in its neighborhood. It was Merten [15–18] who made the first theoretical approach to the problem of the piezoelectric field generated by a straight dislocation in an infinite medium. Using an approximate theory (neglecting piezoelectric stresses and assuming elastic isotropy), Merten found the first analytical results for electro-elastic fields excited by a dislocation. It has been found that the distortion of elastic strain field caused by the inverse piezoelectric effect represents a correction of only a few percent to the primary elastic strain. However, in Merten’s approximation the electric potential, though sensitive to direction, was independent of the distance from the dislocation; that looked to be unrealistic. The next series of theoretical studies of the same problem was accomplished by Kosevich, Pastur and Feldman [19], Saada [20] and Faivre and Saada [21]. In order to avoid Merten’s approximations, they extended the exact plane-strain solution given by Eshelby, Read and Shockley [22] for a straight dislocation in anisotropic purely elastic medium to include the complete piezoelectric coupling into the constitutive equations. It was shown that the electric potential excited by a dislocation depends not only on direction but also (logarithmically) on distance. Based on this theory the detailed calculations of the magnitudes and configurations of piezoelectric fields around dislocations in crystals of the wuertzite structure have been made by Smirnova [23]. For example, it has been shown that a screw dislocation lying in the basal plane and dislocations oriented along the c axis are not piezoactive, i.e. do not excite any electric field. On the other hand, the electric field of an edge dislocation lying in the basal plane with a Burgers vector also belonging to the same plane has a considerable radial component. The further study of piezoelectric fields surrounding a static dislocation, accomplished by Huston and Walker [24], was devoted to

50

J.P. Nowacki and V.I. Alshits

Ch. 72

scrutinizing the previous results by means of computer calculations. However in this paper we shall concentrate our attention basically on an analytical approach to the problem. The specific character of theories related to piezoelectricity consists in the principal impossibility of simplifications related to such a powerful approximation as assumption of isotropy of a medium. Indeed, the piezoelectric effect is a purely anisotropic phenomenon, which cannot occur in media with a center of symmetry. Meanwhile a theoretical description of elastic fields caused by line defects (e.g., by dislocations) in anisotropic (even purely elastic) media looks to be an extremely complicated, almost hopeless problem. Direct attempts to get explicit solutions lead to results only for very symmetric situations and even then final expressions turn out to be so cumbersome that their analytical analysis becomes practically impossible. The real progress came to this theory only after the recognition of advantages provided by an implicit approach, when solutions are expressed in terms of some parameters, which often cannot be found explicitly and need additional computing. This was the basic idea, which allowed solving many problems of anisotropic elasticity. The theory of dislocations in media of unrestricted anisotropy is almost completely based on an implicit description. The principally important step here was made by Stroh [25,26], who developed his famous sextic formalism reducing the description of generalized plane strain in an arbitrary anisotropic medium to an eigenvalue problem, so that the 2D field of a straight dislocation was expressed in a very compact form in terms of eigenvalues and eigenvectors of the 6 × 6 Stroh matrix (see also [27]). Of course, these eigenvectors and eigenvalues are not known explicitly, but they are the same for any observation point and their computing is not a problem [28]. On the basis of the Stroh theory the so-called integral formalism was developed by Barnett and Lothe [29] (see also [30]), which in many cases has advantages over the eigenvector theory. The other non-trivial implicit idea was related to using the field of a straight dislocation itself as a “brick” for building the more complex 3D dislocation fields. In particular, Indenbom and S. Orlov [31] have reduced the field of an arbitrary dislocation loop to a superposition of the fields of straight dislocations. And Lothe [32,33] has proved that the self-action of a semi-infinite straight dislocation inclined to a planar free surface of an anisotropic half-space is directly determined by the orientational derivative of the line energy of a straight dislocation in the unbounded medium with the same elastic constants. These and many other beautiful theorems and a complete list of references can be found in the book edited and to large extent written by Indenbom and Lothe [34]. A separate story is associated with a description of the fields of straight dislocations in bounded anisotropic media. The Stroh approach has proved to be efficient also for 2D problems of this kind. In terms of the sextic formalism there were found the fields of straight dislocations in a half-space (Barnett and Lothe [35]), in the structure of a layer on a substrate (Barnett [36]), and within an infinite strip (Wu and Chiu [37]). Alshits et al. [38, 39] extended the Stroh formalism for a description of anisotropic elasticity in multilayered media and found dislocation fields in inhomogeneous strips, single, on substrates or inside sandwiches. The results obtained in [37,38] became an extension of the previous theories for isotropic homogeneous strips [40–43]. As was shown by Jergan [44] (see also [45]), the Stroh theory can be modified for a description of 2D electro-elastic fields in piezoelectrics. In this case the formalism becomes

§2

Dislocation fields in piezoelectrics

51

eight-dimensional, so that the 6 × 6 Stroh matrix is replaced by a similar 8 × 8 matrix. Later Alshits, Darinskii and Lothe [46] introduced a ten-dimensional modification of the Stroh method for media with coexisting piezoelectric, piezomagnetic and magneto-electric couplings. In both cases the extended Stroh formalism has proved to be very fruitful in the theory of surface acoustic waves [46–49] in such media. Recently Nowacki et al. [11–14], by combining the methods developed in [38,39,45], have solved a series of problems related to plane fields in bounded piezoelectric media with line sources in the bulk or at the surface. There have been found Green’s functions describing 2D electro-elastic fields in a piezoelectric strip [11,12] and in a piezoelectric layer-substrate structure [13,14] of unrestricted anisotropy. Actually, these results represent an extension of early papers [19–21], where the fields of straight dislocation were found for an unbounded piezoelectric body. In this paper we are going to return to electro-elastic dislocation fields in unbounded piezoelectric media, however basically for dislocations of arbitrary shape exciting general 3D coupled fields. We shall develop an appropriate formalism which will allow us to extend for piezoelectrics practically all classical results of the modern theory of dislocations in purely elastic media of unrestricted anisotropy. In next Section 2 we shall introduce a combined electro-elastic line defect in a piezoelectric medium which formally may be considered as a 4D dislocation representing a combination of an ordinary dislocation with its electrostatic analogue. Then in Section 3 we shall develop a four-dimensional formalism of the theory and on this basis derive a series of representations for electro-elastic fields of extended 4D dislocations of an arbitrary shape and consider the problem of their interactions with external fields. Section 4 will be devoted to an extension for the piezoelectric case of the integral, matrix and algebraic approaches to the theory of straight dislocations in purely elastic media. Finally, in the last Section 5 we shall sum up some qualitative features of the obtained results.

2. A dislocation and its electrostatic analogue as a combined line defect in a piezoelectric medium 2.1. Three definitions of a dislocation As a purely topological line defect, a dislocation in a piezoelectric medium is defined in a completely the same way as that in any other type of solid. There are at least three independent but completely equivalent definitions of a dislocation. The idea of the first definition [50] was based on the Burgers circuit in a crystal lattice. One can form a circuit in a crystal making equal numbers of steps from atom to atom on the opposite sides of the path (say, n steps down, m steps to the right, n steps up and m steps to the left). In a perfect crystal the last step would bring one to the initial point. However, if the crystal contains a dislocation and the circuit links the latter, the circuit will fail to close. The vector going from the start point of the circuit to its end is defined as the Burgers vector b, which is a basic characteristic of a dislocation.

52


Ch. 72

The second definition [51] is just a continuum variant of the first one. The line integral of the elastic displacements over the closed contour γ , which links a dislocation, must fail to vanish, defining the value of the same Burgers vector, bi =

dui .

(1)

γ

In eq. (1) the direction of integration is supposed to be related to a chosen direction of the dislocation by the rule of a right-hand screw. Thus, a change of the dislocation direction leads to a change of the sign of the Burgers vector. This reflects the physical difference between dislocations of different signs (e.g., right-hand and left-hand screw dislocations or two edge dislocations with extraplanes situated on the opposite sides from the dislocation line). The rather paradoxical appearance of eq. (1) just indicates that the vector field of displacements u(r) in a crystal with a dislocation must be discontinuous or multivalued. This reflects the fundamental new quality which a crystal acquires after a topological transformation creating a dislocation. The atoms of such a crystal cannot be brought into one-toone correspondence with the atoms in the initial perfect crystal. Accordingly, it is principally impossible to decompose the displacement field u(r) into elastic and inelastic parts. However, it can be done for the total distortion ui,j [52,53]: ui,j = βj i + βj0i .

(2)

Here the first term describes the elastic distortion determined by the local change in interplane distances and lattice rotations, and the second term, known as the plastic distortion, characterizes the transformation creating the dislocation. In these terms eq. (1) reduces to [53,54] bi =

βj i dxj ,

(3)

γ

or by (2), bi = −

γ

βj0i dxj .

(4)

The latter equation directly leads to the third definition of a dislocation. This definition describes the specific transformation procedure necessary for creating a dislocation and simultaneously removing the multivalued character of the displacement field u(r). Let us make the cut over a smooth, however arbitrary, surface S bounded by the dislocation line C and impose a jump of the displacement vector u at this surface by the value of the Burgers vector b (see Fig. 1): −

+

uS − uS = b.

(5)

§2.2


53

Fig. 1. The dislocation line C, the surface S bounded by C and the contour γ linking C.

As a convenient alternative to the cut along S, one can introduce a singular plastic distortion at S (however retaining a multivaluedness of the u(r) field) βj0i (r) = −bi nSj δ r − rS · nS ,

(6)

where rS indicates points of the surface S and nS is the unit normal to the surface at the point rS . The positive direction of nS is related to the chosen positive direction of the dislocation line C by the right-hand screw rule. Indeed, one can easily check the validity of (4) with (6) taking into account that on γ close to the surface S r − rS · nS = s tγ · nS ,

γ

dxj = tj ds,

(7)

where tγ is the unit vector tangent to the contour γ at the point s (Fig. 1). So −

γ

dxj βj0i

= bi

ε −ε

ds nS · tγ δ nS · tγ s = bi .

(8)

We stress that the above exercise is absolutely insensitive to the choice of the surface S. In other words, the surface transformation described related to the third definition of a dislocation indeed creates the same line defect as that defined by conditions (1). On the other hand, this means that the plastic deformation is not a function of the body state, but depends on particular ways of formation or motion of dislocations [55]. We shall not go further in this brief introduction to the theory of dislocations. One can find details in many available books (see, e.g., [56–60]).

2.2. Electrostatic analogue of a dislocation It is customary to make use of the analogy between dislocation stress fields in elastostatics and magnetic induction fields created by line electric currents in magnetostatics [54]. Indeed, these line sources and their fields are mathematically very similar. However, for

54


Ch. 72

dielectric media considered in this work it seems to be more relevant to look for electrostatic analogies with dislocations, the more so that an electromechanical coupling occurring in piezoelectrics creates the very specific situation, when elastic and electric fields do not exist without each other. On the other hand, as we shall see, it is really possible to introduce a very close electrostatic analogue of a dislocation. The idea is based on the Dirichlet problem: to find the electrostatic field distribution in a dielectric medium excited by a given constant jump of potential ϕ prescribed at some internal non-closed surface S, finite or semi-infinite. The latter requirement is very essential, being equivalent to the necessary existence of a line bounding the surface. Indeed, the electric potential ϕ is defined to the accuracy of an arbitrary constant. So, if in some closed domain of the body the potential ϕ differs from that in the surrounding medium by any constant ϕ, this means just one thing: those potentials are counted from different reference points. Such jumps of the potential certainly cannot excite any electric fields. The situation, however, is radically changed when the surface S is not closed but is bounded by some line C (see the same Fig. 1). It is rather evident that in this case we acquire an electrical singularity along the line C. And in complete analogy with a dislocation the electric field excited in the medium will be sensitive only to the position of the line C, but not to the form of the surface S. Indeed, let us add to the surface S any other surface S ′ bounded by the same line C. Consider the closed domain between S and S ′ . It is clear that by changing its potential by the constant ϕ we do not change any physical fields, however simultaneously the Dirichlet jump ϕ is removed from the surface S to S ′ . Thus, the electrostatic source introduced is a line defect, and its similarity to a dislocation is even closer than one could expect. This is just the case when it is very convenient to look for the electric field E counting it from the “external” source field E0 produced by the same jump of potential ϕ at the surface S in vacuum: E = E0 + E′ .

(9)

The form of the field E0 can be found immediately, E0 (r) = −ϕnS δ r − rS · nS ,

(10)

which is very similar to the source term (6) in the theory of dislocations. In fact, we deal here with an electric capacitor of infinitesimal thickness. In the vicinity of its edge an additional correcting electric field E′ must arise. It is evident that in contrast to the field (10) singular throughout the cut-surface S, the distribution E′ (r) should be regular everywhere except of the “edge” line C. That is why in this case the potential ϕ(r) should be introduced for the depolarization field E′ = −∇ϕ rather than for the total field E. Comparing the relation −ϕ,l = El − El0 with similar eq. (2), one can see that in this scheme the electric field El is a counterpart of the elastic distortion βlk , and the depolarization field Ei′ – to the total distortion uk,l . However, as follows from eqs (6) and (10), beyond the arbitrary cutsurface S those pairs of electric fields (E and E′ ) and distortions (∇ ⊗ u and β) coalesce into one field E = E′ and one distortion ∇ ⊗ u = β. We stress that in a complete space including the cut, only the fields ∇ ⊗ u and E′ (but not β and E) are irrotational.

§2.3


55

2.3. Combined electro-elastic line defect – a 4D dislocation Thus, with a double jump at the same surface S of both the displacement, u = b eq. (5), and the electric potential, ϕ, we obtain along the line C bounding S the combined electroelastic line defect, which can be conveniently described as a 4D dislocation characterized by the “Burgers” 4-vector b(4) =

b ϕ

(11)

.

Let us formally introduce the corresponding four-dimensional “displacements” UK =

uk , K = k = 1, 2, 3; ϕ, K = 4.

(12)

It is essential that the space remains 3-dimensional, so the corresponding “elastic and plas0 , and “stresses” tic distortions”, UlK and UlK iJ are represented as non-square 3 × 4 matrices

0

βlk , K = k = 1, 2, 3; K = k = 1, 2, 3; βlk , 0 . (13) UlK = UlK = −El , K = 4; El0 , K = 4;

σij , J = j = 1, 2, 3; iJ = (14) Di , J = 4. Let us define also the extended “elastic moduli” of a piezoelectric CiJ Kl = cij kl

for J, K = j, k = 1, 2, 3,

= elij

for J = j = 1, 2, 3, K = 4,

= eikl

for J = 4, K = k = 1, 2, 3,

= −εil

for J = 4, K = 4.

(15)

It is easily checked that this tensor retains some symmetry CiJ Kl = ClKJ i .

(16)

With the above notation the standard constitutive equations σij = cij kl βkl − ekij Ek ,

(17)

Di = eikl βlk + εik Ek ,

(18)

take the appearance of the modified Hooke’s law iJ = CiJ Kl UlK ,

(19)

56


Ch. 72

and the equations of equilibrium, div σ = 0,

(20)

div D = 0,

(21)

are combined into iJ,i = 0.

(22)

Substituting here (19) with 0 , UlK = UK,l − UlK

(23)

one finally obtains the extended 4D equation of equilibrium 0 . CiJ Kl UK,li = CiJ Kl UlK,i

(24)

Thus, we have come to rather compact equation (24) describing coupled fields in an arbitrary piezoelectric medium excited by the combined electro-elastic line defect, which will be called below a generalized line defect or a 4D dislocation. We remind that this defect is formed by the transformation which is described by the generalized plastic distortion 0 } defined by U0 = {UlK U0 (r) = −nS ⊗ b(4) δ r − rS · nS .

(25)

This differs from corresponding eq. (6) for an ordinary dislocation only by the replacement of the Burgers vector b by the 4-vector b(4) , eq. (11). Here we must stress that even when ϕ = 0, i.e. when our line defect is reduced to an ordinary dislocation, b(4) = (b, 0)t , the formalism considered remains four dimensional due to a 4D character of coupled electroelastic fields in piezoelectrics.

3. Fundamental properties of 4D dislocations of an arbitrary shape 3.1. Generalized Green’s tensor for the problem considered Eq. (24) has a form CiJ Kl UK,li + FJ = 0

(26)

similar to the standard equation of elastostatics with FJ being the 4D “force” (J = 1, 2, 3, 4) 0 . FJ = −CiJ Kl UlK,i

(27)

§3.1


57

Accordingly, it is natural to construct for its solution a Green’s function analogous to that used in the theory of elasticity. The corresponding Green tensor, GKM (R) (K, M = 1, 2, 3, 4), describing electro-elastic fields in piezoelectrics must satisfy the equation CiJ Kl GKM,li + δMJ δ(R) = 0.

(28)

After a Fourier transformation of this equation with GKM (R) =

1 (2π)3

d3 k GKM (k)eik·R

(29)

and δ(R) =

1 (2π)3

d3 k eik·R ,

(30)

one obtains −(ki CiJ Kl kl )GKM (k) + δMJ = 0.

(31)

Let us introduce the unit vector κ = k/k

(32)

and the 4 × 4 matrix (κκ) with the components (κκ)J K = κi CiJ Kl κl .

(33)

In these terms the Fourier amplitude GKM (k) given by (31) reads GKM (k) =

1 (κκ)−1 KM . 2 k

(34)

This is a homogeneous function of k of degree −2. For any such function the Fourier integral (29) is identically reduced to a line integral along the unit circle in the plane orthogonal to R:

3

d k GKM (k)e

ik·R

π = R

2π

dϕ GKM (nϕ ),

(35)

0

where nϕ is a unit vector scanning the circle of integration and remaining normal to R (see Fig. 2). Thus, combining (29) with (34) and (35) we finally obtain the extended form for the Green’s tensor G(R) describing electro-elastic fields in piezoelectrics: G(R) =

1 8π 2 R

0

2π

dϕ (nϕ nϕ )−1 .

(36)

58


Ch. 72

Fig. 2. The contour of integration |nϕ |= 1 in the plane nϕ · R = 0.

Here the 4 × 4 symmetric matrix (nϕ nϕ ) is defined in the same way as (33). Formally, the expression (36) has precisely the same appearance, which is known for Green’s tensor in the theory of elasticity [61,62]. However, one should remember that in pure elastic case both G(R) and (nϕ nϕ ) are 3×3 matrices. And our extension of the formalism is reduced to an increase of their dimension by means of replacement of the elastic stiffness tensor cij kl by the combined tensor CiJ Kl (15) including all the material constants of a piezoelectric body. With a known Green’s tensor a solution of the initial eq. (26) for an arbitrary source vector F(r) is found immediately:

UK (r) = d3 r′ GKM (r − r′ )FM (r′ ) =

1 8π 2

d3 r′

FM (r′ ) |r − r′ |

2π

0

dϕ (nϕ nϕ )−1 MK ,

(37)

where the unit vector nϕ belongs to the plane orthogonal to the vector R = r − r′ . In particular, eq. (37) gives the general solution for electro-elastic “displacement” fields U = (u, ϕ)t excited by arbitrary distributions of the force f(r) and charge q(r) densities F = f(4) = (f, −q)t in a piezoelectric medium of unrestricted anisotropy. It will be useful also for further applications to present here an explicit form of the ordinary Green’s function for isotropic elasticity [30]: Gkm (R) =

τk τm + (3 − 4ν)δkm , 16πμ(1 − ν)r

(38)

where μ and ν are the shear modulus and the Poisson ratio, respectively, and τ is the unit vector along R: τ = R/R.

(39)

3.2. “Displacement” and “distortion” fields of a generalized line defect Now we are ready to derive the “displacement” field excited by our 4D dislocation line C of an arbitrary shape situated in an infinite anisotropic piezoelectric medium free of body

§3.2


59

forces or charges. One should just substitute into (37) the corresponding source “force” given by (27). The result is

0 UK (r) = − d3 r′ GKM (r − r′ )CiMN l UlN,i (r′ ), (40) 0 (r′ ) is defined by eq. (25). Let us make use here of the identity where UlN

0 (r′ ) = GKM (r − r′ )UlN,i

∂ 0 0 (r′ ), (r′ ) + GKM,i (r − r′ )UlN GKM (r − r′ )UlN ′ ∂xi

(41)

where it was taken into account that ∂ ∂ GKM (r − r′ ) = − GKM (r − r′ ). ′ ∂xi ∂xi

(42)

The volume integral in (40) with the first term in (41) by means of the Gauss theorem is immediately reduced to the integral over the infinite surface of our unbounded body, where 0 must vanish. Thus, eq. (40) takes the form both GKM and UlN UK (r) = −

0 (r′ ), d3 r′ GKM,i (r − r′ )CiMN l UlN

(43)

which with (25) is automatically transformed into the integral over the surface S resting on the dislocation line (Fig. 1),

(4) (44) UK (r) = CiMN l bN dSl GKM,i r − rS . S

Here by definition d Sl = dS nSl .

(45)

By transforming eq. (44), one can also find a representation of the vector field U(r) in the form of a line integral along the contour C coinciding with the dislocation line (Fig. 1). The corresponding result for an elastically isotropic medium based on eq. (38) is given by the formula b (b × R) · ξ C b × ξC 1 1 u(r) = − dl − − ∇ dl , (46) 4π 4π C R 8π(1 − ν) R C which is known as the Burgers solution [50]. In (46) ξ C is the unit tangent vector to the line C (Fig. 1) at the point r′ = rC belonging to C, R = r − r′ ,

R = |r − r′ |,

(47)

60


Ch. 72

and is the solid angle subtended at r by the surface S resting on the line C, =

S

dS

nS · R R3

(48)

(here r′ = rS belongs S). By definition this solid angle depends only on the configuration of the line C bounding the surface S being insensitive to the shape of S, which shows again that a dislocation is a line defect. On the other hand, (r) is a discontinuous function across the surface S, changing by 4π on linking the curve C. Accordingly the solution (46) automatically satisfies eq. (1),

γ

dui = −

bi 4π

γ

d = bi .

(49)

An extension of the Burgers solution (46) for elastic media of an arbitrary anisotropy has been given by Indenbom and Orlov [63] (see also [64]). In contrast to the isotropic expression (46), its absolutely nontrivial extension [63] has proved to be practically unknown to the scientific community, at least until the publication [64]. We shall not reproduce it here, but instead directly present the result of its further generalization, valid for the 4D dislocation in an arbitrary piezoelectric medium considered in this work. The derivation is very similar to that used in [63,64]. Thus, the extension of (46) in our case reads b(4) 1 U(r) = − + 4π 8π 2

1 dl R C

2π

0

dϕ (nϕ nϕ )−1 nϕ , ξ C × nϕ b(4) ,

(50)

where the 4 × 4 matrix (nϕ , ξ C × nϕ ) of the type (c, d) ≡ (cd) is defined by a convolution similar to (33) (c, d)KM ≡ (cd)KM = ci CiKMj dj .

(51)

As already was stated, the solid angle (48) is a discontinuous function having a jump 4π at crossing the surface S. Accordingly, the total distortion UK,l does have a structure similar to (2), 0 UK,l = UlK + UlK ,

(52)

0 are given by (13), (25). In spite of (52), the “elastic distortion” can be where UlK and UlK found by a differentiation of eq. (50) or (44) if the cut along the surface S is assumed,

r − rS = 0,

(53)

providing a nonsingular behavior of the function U(r). Here we must recall that the surface S is arbitrary. So, if we are interested in some particular point r, which happens to lie on S, it is always possible to deform S so that r would be beyond the cut.

§3.2


61

In evaluating the elastic distortion UlK it is more convenient to return to the initial eq. (44), which leads to

(4) (54) UlK (r) = CiMNp bN dSp GKM,il r − rS . S

It is easily checked that the above surface integral is equivalent to the line integral along the dislocation providing an extension of the Mura representation [65] for pure elasticity, (4) dlj GKM,i r − rC , UlK (r) = −CiMNp bN εjpl (55) C

where

dlj = ξjC dl

(56)

and εij k is the permutation tensor defined by εij k =

if ij k = 123, 231 or 312; −1 if ij k = 132, 213 or 321; 0 if ij k contain any repeated indices.

1

(57)

This tensor is connected with the Kronecker delta by the identity εij k εimn = δj m δkn − δj n δkm .

(58)

The equivalence between (54) and (55) is proved using the Stokes theorem

εij k dSi A...,j = A... dlk , S

which by (58) is transformed into

dSj A...,k − dSk A...,j = εij k A... dli . S

(59)

C

(60)

C

S

Indeed, combining (54) with (60) and taking into account that at the surface S by eqs (28), (53) CiMNp GKM,ip = −δN K δ r − rS = 0,

(61)

one immediately obtains eq. (55). By a similar reasoning there can also be found the alternative representation to (55) extending the corresponding result of Indenbom and Orlov [47]: (4) dlj xr − xrC GKM,is r − rC . (62) UlK (r) = CiMNp bN εjpr C

62


Ch. 72

Starting from an expression of this type, Indenbom and Orlov [31] have derived their famous formula expressing the elastic distortion of an arbitrary dislocation loop in terms of elastic fields of straight dislocations going from the observation point r to the points rC at the dislocation line C. Brown [66] has independently found an analogous formula for the particular case of a plane dislocation loop. We shall again omit derivations, which are rather cumbersome, and present only the ready result: the extended Indenbom–Orlov formula describing the general case of an arbitrary piezoelectric medium, UlK (r) = −

1 2

C

dl ξpC ξqC

∂2 UlK ξ C , r − rC . ∂xp ∂xq

(63)

Here UlK (ξ C , r − rC ) is the electro-elastic “distortion” field created at the point ξ C by the straight 4D dislocation having the same Burgers vector b(4) as the loop C and directed along r − rC . Multiplying both sides of (63) by CiJ Kl , eq. (19), we arrive at the same relation between “stresses” iJ (14) for the dislocation loop and for the appropriate set of straight dislocations. Thus, eq. (63) directly shows how 3D fields of an arbitrary dislocation can be constructed from much more simple 2D fields of straight dislocations. This provides a useful algorithm for computing complex dislocation fields. Willis [67] and Steeds and Willis [68] have already obtained important results concerning a realization of this program for pure elasticity.

3.3. Energy of a 4D dislocation and its interaction with external fields Consider an unbounded piezoelectric medium excited by the 4D dislocation loop (Fig. 1) and by some other sources. It will be convenient for further calculations to introduce an imaginary cut along an arbitrary surface S resting on the dislocation line C. As we know, such a cut excluding the surface S from the body does not change any physical results, however it provides a single-valued behavior of the “displacement” 4-vector U = (u, ϕ)t and excludes the “plastic distortion” U0 = (β 0 , E0 )t , eqs (13), (25), which vanishes everywhere beyond the surface S. This allows putting in all equations UiK = UK,i ,

E′ = E.

(64)

As was shown by Eshelby [69], the elastic energy of the body deformed under isothermal conditions is thermodynamically described by the Gibbs free energy potential F = U − T S.

(65)

For a piezoelectric medium we should take for the same purpose the Gibbs electric enthalpy (β, E) = F − D · E.

(66)

§3.3


63

Its expansion in powers of β and E counted from the ground state is 1 1 (β, E) = cij kl βij βkl − ekij Ek βij − εkl Ek El . 2 2

(67)

With constitutive equations (17), (18), definitions (13)1 , (14) and relations (64) we obtain 1 1 = (σij ui,j + Dk ϕ,k ) = iJ UJ,i . 2 2

(68)

So, the internal energy of a piezoelectric body, deformed under isothermal conditions, is W=

1 2

dV iJ UJ,i .

(69)

V

Here V is supposed to be the whole space, except the space in the cut. Let us now divide the electro-elastic fields U(r) and (r) each into two parts related to the dislocation considered and to the other sources of any origin, except for body forces and body charges. We shall mark the first by the superscript d, and the second – by the superscript o: U = Ud + Uo ,

(70)

= + .

(71)

d

o

Substituting eqs (70) and (71) into (69) one obtains 1 W= 2

dV

V

d d UJ,i iJ

1 + 2

V

dV

o o UJ,i iJ

1 + 2

V

d o d o UJ,i UJ,i + iJ . dV iJ

(72)

Here the first term describes the self-energy of a 4D dislocation W

self

1 = 2

V

d d . UJ,i dV iJ

(73)

The second term is the analogous self-energy of the other sources, and the third term – the interaction energy between the dislocation and the external field. In view of (16), (19) and (64)1 , the last interaction energy reads W int =

V

o d dV iJ UJ,i .

(74)

The integral (73) is divergent close to the dislocation line, therefore for the calculation of a dislocation self-energy it is customary to introduce a cut-off corresponding to the radius of the dislocation core of order of the Burgers vector b. In [30,64,70] one can find very rigorous consideration of this problem for a pure elastic anisotropic medium. In contrast,

64


Ch. 72

the interaction energy (74) is a well-defined quantity and can be further transformed. Taking into account that in the absence of body forces and body charges the divergence of o o = 0, eq. (22), the integrand in (74) can be reduced to vanishes, iJ,i o d o d d o o UJ ,i − iJ,i = iJ UJ ,i . UJ,i UJd = iJ iJ

(75)

Hence, by Gauss’ theorem, the integral (74) is transformed to a surface integral over both sides of the cut, S + and S − , the first with the outward normal –nS , and the second with the outward normal nS ,

(4) d int o o W = dSi iJ UJ = bJ . (76) dSi iJ S + +S −

S=S −

In (76) we have taken into account that by the third definition of a dislocation, eq. (5), − + U rS − U rS = b(4) .

(77)

With eq. (76) we are prepared to find an extension of the familiar Peach–Koehler formula [71] determining the force on the dislocation from the external fields. Of course, a force in the given case should be treated in a thermodynamic sense. It has nothing to do with any interatomic forces; rather it is similar to a force on a line of charge from an electric field or on a conductor with an electric current from a magnetic field. In our case the distributed force F per unit length of the dislocation from the other sources of electro-elastic fields is determined by the variation of the interaction energy δW int due to a virtual displacement δxp in the configuration of the dislocation line C. By definition [30], δW

int

=−

ds Fp δxp .

(78)

C

On the other hand, the same variation of (76) gives (4)

δW int = εipq bJ

C

ds ξqC δxp iJ ,

(79)

where for brevity we have omitted the superscript o at iJ . Combining the last two equations with recognition that the variation δxp is arbitrary, one finds (4)

Fp = εipq iJ bJ ξqC ,

(80)

which differs from the initial Peach–Koehler formula by a natural replacement (4)

σij bj → iJ bJ = σij bj + Di ϕ,

(81)

where σij and Di are external fields of stress and electric displacement. We can see that due to electro-elastic coupling a nonvanishing force on a dislocation may arise due to

§4


65

external electric field E even in externally nondeformed piezoelectric media (β o = 0) and even when we deal with an ordinary 3D dislocation (ϕ = 0). Indeed, in the latter case by eqs (11), (14) and (17) (4)

iJ bJ = σij bj = −ekij Ek bj ,

(82)

Fp = −εipq ξqC Ek ekij bj .

(83)

and

As is well known, dislocations in crystals can move conservatively only in so-called slip planes, containing their Burgers vectors. If n is the normal to such a plane, then n × ξ C is the direction of a possible glide of dislocation. So, as a rule, the most physical interest is concerned with the projection of F (80) on n × ξ C , (4) (4) F = F · n × ξ C = −εipq εprs iJ bJ ξqC ξsC nr = ni iJ bJ .

(84)

The scalar quantity F (84) is an extended resolved force. It is customary also to introduce the so-called resolved shear stress τ = F /b. When the external force on a dislocation with the Burgers vector b is provided by the electric field E, eq. (84) with (82) is simplified to Fel = −Ek ekij ni bj .

(84a)

4. Electro-elastic fields of straight 4D dislocations In this subsection we shall consider the very important particular case of straight dislocations in infinite piezoelectric bodies. The fields of straight dislocations are themselves of essential physical interest. However, as we have seen, the basic characteristics of fields excited by curvilinear dislocations of an arbitrary shape can be expressed in terms of corresponding fields of straight dislocations with the same Burgers vector which increases the significance of the forthcoming material.

4.1. The integral representations Consider an infinite piezoelectric medium with a straight 4D dislocation having the direction ξ C = t (z-axis), the Burgers vector b(4) , the cut surface S (xz-plane) with the normal n (y-axis) and the in-plane normal m to the dislocation line (x-axis), see Fig. 3. In this case the contour C of the integration in eq. (55) is reduced to the z-axis (−∞ < z < ∞) with rC = zt, dlj = tj dz, so that one obtains (4) UlK (r) = −CqMNp bN εjpl tj

∞

−∞

dz GKM,q (r − zt).

(85)

66

Ch. 72


Fig. 3. Straight dislocation C in the coordinate system {x, y, z} and some vector geometry.

Substituting here (29) and (34) we have 1 (2π)3

(4) UlK (r) = −iCqMNp bN εjpl tj

d

3

kq k eik·r 2 k

(κκ)−1 KM

∞

−∞

dz exp(−ikz z). (86)

By definition, the last integral in (86) is 2πδ(kz ), therefore the Fourier integral is reduced to a double integral in the plane {m, n}. We shall evaluate this integral in the polar coordinate system (kρ , ϕ) counting the angle ϕ from the vector n (Fig. 3). In this system κ ≡ k/k = n(ϕ) ≡ nϕ (Fig. 3), k · r = kρ κ · ρ, ρ = r − zt. Taking into account that [64]

0

∞

dkρ exp(ikρ nϕ · ρ) = i/nϕ · ρ,

(87)

one can easily derive (4)

UlK (ρ) =

CqMNp bN εjpl tj (2π)2

2π

dϕ

0

nq (ϕ)[(nϕ nϕ )−1 ]KM , nϕ · ρ

(88)

where the integral is understood in a principal value sense. Eq. (88) represents the extension of the Barnett–Swanger formula [72]. Let us slightly rearrange the latter result. Noting that by definition εjpl tj = mp (ϕ)nl (ϕ) − ml (ϕ)np (ϕ),

(89)

§4.1


67

and making use of the identity (nϕ nϕ )−1 KM (nϕ nϕ )MN = δKN ,

(90)

one arrives at the other form of (88): (4)

UlK (ρ) =

bN (2π)2

2π

dϕ 0

{nq (ϕ)CqMNp [(nϕ nϕ )−1 ]KM nl (ϕ) − δN K δlp }mp (ϕ) . nϕ · ρ

(91)

With (91) the corresponding generalized stress given by eq. (20) reads (4)

iJ (ρ) =

bN (2π)2

2π

dϕ

∗ CiJ Np (nϕ )mp (ϕ)

0

nϕ · ρ

(92)

,

where we have introduced the plane tensor of our extended “elasticity” (19) ∗ −1 CiJ n C − CiJ Np . Np (n) = CiJ Kl nl (nn) KM q qMNp

(93)

∗ ni CiJ Np (n) = 0,

(94)

Eqs (91) and (92) are an extension of the Indenbom–Alshits formulae [73] initially obtained for pure elasticity. In ordinary elasticity the tensor analogous to (93) has quite a series of useful properties ∗ [53,64]. Some of them hold also for piezoelectrics. For instance, the projection of CiJ Np (n) on n from both sides identically vanishes, ∗ CiJ Np (n)np = 0.

This tensor plays the main role in the theory of internal elastic fields excited by 1D distributions of sources described by a plastic distortion tensor dependent on only one coordinate, β 0 = β 0 (r · n). In the latter case the corresponding stress field is given by [53] as 0 σij (r · n) = Cij∗ np (n)βpn (r · n).

(95)

Let us prove that a similar solution is valid also for the generalized elasticity of piezoelectrics. Consider an infinite piezoelectric medium with a given 1D distribution of distortions U0 (r · n). In terms of eq. (27) this is equivalent to the body forces 0 FM (r · n) = −CqMNp UpN,q (r · n) = −nq CqMNp

∂ U 0 (r · n). ∂(r · n) pN

(96)

The generalized stress distribution determined by eq. (19) in the given case is 0 iJ (r · n) = CiJ Kl UK,l (r · n) − UlK (r · n) .

(97)

68

Ch. 72


For the 1D field UK (r · n) considered the Green function is also one-dimensional, GKM (r · n), so that we can look for UK,l (r · n) in the form UK,l (r · n) =

∞

−∞

ds GKM,l (r · n − s)FM (s).

(98)

Let us express the function GKM,l (r · n − s) in (98) as a Fourier integral GKM,l (r · n − s) =

inl 2π

∞ −∞

dk kGKM (k)eik(r·n−s) ,

(99)

where the Fourier image of GKM (r · n) is found similarly to (34), 1 [(nn)−1 ]KM . k2

GKM (k) =

(100)

Substituting (99) and (100) into (98) and noting that

∞

−∞

0 (k)k, ds FM (s)e−iks = FM (k) = −inq CqMNp UpN

(101)

one easily finds 0 UK,l (r · n) = nl (nn)−1 KM nq CqMNp UpN (r · n).

(102)

∗ 0 iJ (r · n) = CiJ Np (n)UpN (r · n),

(103)

Combining eqs (102) and (97) we finally obtain

∗ where the tensor CiJ Np (n) is defined by (93). With (102) and (103) one could derive eqs (91) and (92) much more easily basing on the Radon transformations

iJ (ρ) =

0

2π

dϕ iJ (ρ · n),

0 (ρ) = UpN

2π 0

0 (ρ · n), dϕ UpN

(104)

bearing in mind that r · n = ρ · n, (4) 0 UpN (ρ) = −bN np H (ρ

b(4) np · m)δ(ρ · n) = N 2 (2π)

0

2π

dϕ , (ρ · nϕ )(n · mϕ )

and making use of the identities np = np (ϕ)(n · nϕ ) + mp (ϕ)(n · mϕ ) and (94)2 .

(105)

§4.2


69

4.2. The matrix representations The expressions found for dislocation fields, eqs (88), (91) and (92), contain in the denominators of the integrands the scalar product ρ · nϕ , which provides a singularity at integration. Therefore, as was already noted, all above integrals are understood in the principal value sense. This is not very convenient for numerical calculations and can be eliminated [74,75]. Up to now our coordinate system was independent of the observation point ρ. Let us abandon this invariance and choose the orientation of the vector m in Fig. 3 not in the S plane but along the vector ρ, so that ρ = ρm,

ρ · nϕ = ρm · nϕ = −ρ sin ϕ.

(106)

Substituting into (88) together with (106) the relations nq (ϕ) = −mq sin ϕ + nq cos ϕ,

(107)

εjpl tj = mp nl − ml np ,

(108)

we have UlK (ρ) =

(4) 2π bN −1 n (mm) − m (nm) dϕ (nϕ nϕ )MK l NM l NM (2π)2 ρ 0

2π cos ϕ (nϕ nϕ )−1 dϕ − nl (mn)N M − ml (nn)N M MK . sin ϕ 0

(109)

Now let us derive a useful identity starting from the trivial (nnϕ ) ≡ (nnϕ ).

(110)

Replace in the left-hand side of (110) nϕ by −m sin ϕ + n cos ϕ and in the right-hand side n by nϕ cos ϕ + mϕ sin ϕ. After simple rearrangements the identity becomes (nn)

cos ϕ cos ϕ ≡ (nm) + (mϕ nϕ ) + (nϕ nϕ ) . sin ϕ sin ϕ

(111)

Multiplying both sides of (111) by (nn)−1 from the left and by (nϕ nϕ )−1 from the right, one finally obtains (nϕ nϕ )−1

cos ϕ cos ϕ ≡ (nn)−1 + (nn)−1 (nm)(nϕ nϕ )−1 + (nn)−1 (mϕ nϕ )(nϕ nϕ )−1 . sin ϕ sin ϕ (112)

Noting that the principal value integration of the first term vanishes, P

2π

dϕ 0

cos ϕ = 0, sin ϕ

(113)

70


Ch. 72

we see that the last integral in (109) is no longer singular. Let us introduce two 4 × 4 matrices

2π

2π 1 1 Q=− dϕ (nϕ nϕ )−1 , S=− dϕ (nϕ nϕ )−1 (nϕ mϕ ). (114) 2π 0 2π 0 In these terms eq. (109) takes the form

UlK =

(4) bN −ml SN K + nl (mn)(nn)−1 S + (mn)(nn)−1 (nm) − (mm) Q N K . 2πρ (115)

This representation looks rather bulky compared with eq. (88), however it has the important advantage being much more suitable for computing. Let us calculate now the self-energy (73) of a straight 4D dislocation per its unit length. It is clear that the bulk integral in (73) considered per unit length is reduced to the integration over the plane {m, n}. It is convenient to evaluate the integral in the polar coordinates {ρ, ψ} (Fig. 3)

1 self W (116) = ρ dρ dψ iJ (ρ)UiJ (ρ), 2 where for brevity we omitted the superscripts d and replaced UJ,i by UiJ bearing in mind that we do not introduce any cut along the S surface. In this case one can substitute into 0 and make use of identical vanishing of the first integral due to the (116) UiJ = UJ,i − UiJ Colonnetti theorem [53]. Taking into account the logarithmic divergence in (116) both at ρ → 0 and at ρ → ∞, the integration will be accomplished in the ring between ρ = r0 and ρ = R, where r0 is of order of the lattice parameter and R of order of the crystal size: W self = −

1 2

R

ρ dρ r0

0

2π

0 (ρ). dψ iJ (ρ)UiJ

(117)

Combining (117) with (92) and (105)1 one obtains W self =

(4) (4) 2π bJ bN R ∗ dϕ mi (ϕ)CiJ ln Np (nϕ )mp (ϕ)I, r0 8π 2 0

(118)

where I is the integral I = sin ϕ

0

2π

dψ

δ(sin ψ)H (cos ψ) = −1. sin(ψ − ϕ)

(119)

Summarizing, the self-energy of a straight 4D dislocation per unit length is given by R self (120) = E0 ln W r0

§4.3

71


with the prelogarithmic energy factor E0 defined by the convolution E0 =

1 (4) (4) b BI J bJ 4π I

(121)

where the 4 × 4 matrix B is introduced according to BJ N = −

1 2π

2π

0

∗ dϕ mi (ϕ)CiJ Np (nϕ )mp (ϕ).

(122)

In view of (94) the latter matrix can be presented in an invariant form similar to (114) B=−

1 2π

2π

0

dϕ (mϕ nϕ )(nϕ nϕ )−1 (nϕ mϕ ) − (mϕ mϕ ) .

(123)

Thus, as we have seen, all basic characteristics of electro-elastic fields of a straight dislocation can be expressed in terms of three matrices Q, S and B, eqs (114), (123). They depend only on the orientation of the dislocation line in the given piezoelectric medium. In the ordinary theory of elasticity the applied method is known as the matrix (or integral) formalism [30,64]. Eqs (114), (115) and (120)–(123) extend the most important results of this theory for the case of piezoelectric media. 4.3. The Stroh-like algebraic representations In this subsection we shall discuss an alternative approach to a description of 2D electroelastic fields excited by straight-line sources. This approach is based on the Stroh-like formalism for piezoelectrics introduced by Jergan [44] and Barnett and Lothe [45]. It reduces the problem to a purely algebraic procedure of finding eigenvalues and eigenvectors of some 8×8 matrix. As will be shown, it allows considering in a unified manner the 8D general line source, which consists of the above 4D straight dislocation, the line of body forces f(ρ) = fδ(ρ · n)δ(ρ · m),

(124)

and the line of body charges q(ρ) = qδ(ρ · n)δ(ρ · m),

(125)

all combined in the same straight line. Thus, the equilibrium equation (22) is transformed into iJ,i = −fJ(4) δ(ρ · n)δ(ρ · m),

(126)

where f (4) is the generalized 4D force f (4) = (f, −q)t .

(127)

72


Ch. 72

Let us introduce the “stress” 0 (ρ) = m ⊗ f(4) δ(ρ · n)H (ρ · m).

(128)

It is easily checked that in these terms eq. (126) is reduced to (see Fig. 3) ( + 0 )xJ,x + ( + 0 )yJ,y = 0.

(129)

Eq. (129) allows us to introduce a stress function 4-vector V(x, y), V(x, y) =

J(x, y) , D(x, y)

(130)

so that + 0 xJ = VJ,y ,

+ 0 yJ = −VJ,x .

(131)

0 ), −VJ,x = (nm)J K UK,x + (nn)J K (UK,y − UyK

(132)

0 0 , ) + xJ VJ,x = (mm)J K UK,x + (mn)J K (UK,y − UyK

(133)

With this substitution, eq. (129) is identically satisfied and the constitutive equation (19) with (23) is transformed to the system

where by eqs (105)1 and (128) (4)

0 = −bK H (x)δ(y), UyK

(4)

0 = fJ H (x)δ(y). xJ

(134)

After some rearrangements in eqs (132) and (133), this system can be transformed to the single eight-dimensional equation: (I∂y − N∂x )η(x, y) = −gH (x)δ(y),

(135)

where η(x, y) is the unknown eight-component vector function t η(x, y) = U(x, y), V(x, y) ,

(136)

g = [b, ϕ, −f, q]t ,

(137)

g is the strength of the general line defect,

I is the unit matrix and N is the 8 × 8 constant matrix: (nn)−1 (nn)−1 (nm) , N=− (mn)(nn)−1 (nm) − (mm) (mn)(nn)−1

(138)

§4.3

73


depending only on the orientation of the frame {m, n} and on the material constants CiJ Kl (see eq. (51)). The matrix N is an extension [44,45] of the 6 × 6 Stroh matrix initially introduced [25,26] for purely elastic media. Note that eq. (135) is equivalent to a system of eight differential equations of the first order, in contrast to the standard system of four differential equations of the second order given by eq. (24). As a result, the right-hand side of (135) contains the Heaviside function H (x), instead of the delta-function δ(x) in (126). The next natural step in our analysis is a Fourier transformation of the unknown function η(x, y) and the Heaviside function in the right-hand side of eq. (135) with respect to x:

∞

1 2πi

η(x, y) = H (x) =

dk exp(ikx)η(k, y),

−∞ ∞ −∞

1 η(k, y) = 2π

dk exp(ikx). k

∞

dx exp(−ikx)η(x, y), (139)

−∞

(140)

In the Fourier representation of H (x), as usual, for the sake of convergence in (140), k was given an infinitesimally small negative imaginary part k = Re k − iε,

ε → +0.

(141)

At x > 0 the integration in (140) can be reduced to the integration over the closed path CR shown in Fig. 4a. Calculating the residue at the only pole k = 0 one easily obtains H (x > 0) = 1. Similarly, at x < 0 the integration path in (140) can be deformed into the contour shown in Fig. 4b with no poles inside which leads to the result H (x < 0) = 0. Naturally, (141) will be implied in all further equations. After the Fourier transformation eq. (135) takes the form (I∂y − ikN)η(k, y) = −g(2πik)−1 δ(y).

(142)

Fig. 4. Paths of integration for evaluating the integrals in terms of residues in the complex (k) plane.

74


Ch. 72

We stress that the matrix N in (142) is independent of k (see eq. (128)). It is convenient at this stage to introduce eigenvectors ξ α and eigenvalues pα of the matrix N: Nξ α = pα ξ α .

(143)

According to [45,47,48], they form four pairs of complex conjugates. With an appropriate choice of numbering, ξ α+4 = ξ ∗α ,

pα+4 = pα∗ ,

α = 1, . . . , 4;

Impα > 0 for α = 1, . . . , 4;

(144)

Im pα < 0 for α = 5, . . . , 8.

(145)

The Stroh eigenvectors ξ α are orthogonal and complete [61,63,64]: ξ α · Tξ β = δ αβ ,

(146)

8

(147)

α=1

ξ α ⊗ Tξ α = I,

where the 8 × 8 matrix T has the following block form: 0 I T= . I 0

(148)

Now let us return to eq. (142). We shall look for a solution of this equation in a customary form of the sum of a general solution obeying the corresponding homogeneous equation, (I∂y − ikN)η(k, y) = 0,

(149)

and a partial solution satisfying the initial equation (142). The formal solution of the homogeneous eq. (149) is written as η(k, y) = exp[ikNy]X,

(150)

where X is an arbitrary eight-component vector independent of y (but not of k). Calculating the scalar product of this vector with eq. (147) from the right and substituting the result into (150), one obtains the general solution of eq. (149) ηgen (k, y) =

8

α=1

(X · Tξ α ) exp[ikNy]ξ α =

8

cα ξ α exp[ikpα y]

(151)

α=1

with a set of unknown coefficients cα , α = 1, . . . , 8, replacing the unknown 8-vector X. Adding to (151) the partial solution of eq. (142) 8

H (y) ηpart (k, y) = − ξ α (ξ α · Tg) exp[ikpα y] 2πik α=1

(152)

§4.3


75

and transforming the result to x-space, by eq. (139) with (141), we have 8

1 ∞ dk η(x, y) = ξ α exp ik(x + pα y) bα (k) − (ξ α · Tg)H (y) . 2πi −∞ k

(153)

α=1

Here the 8 unknown coefficients bα have appeared from the substitution to (151) cα = bα [2πik]−1 exp(ikpα y).

(154)

Let us find these coefficients from the requirement of the convergence of the integral in (153). The crucial role belongs here to the sign of the product Im(pα )ky. Bearing in mind (145), we can conclude that the function bα (k) must be constructed in such a way as to exclude from the integration the terms relating to ky < 0 for α = 1, . . . , 4, and to ky > 0 for α = 5, . . . , 8. One can easily check that the only variant satisfying the above criterion is provided by the choice bα (k) = (ξ α · Tg)H (−k Im pα ).

(155)

Substitution of (155) into (153) leads to 8

sgn(y) η(x, y) = − (ξ α ⊗ Tξ α )g 2πi α=1

∞ −∞

dk ik(x+pα y) e H (ky Im pα ), k

(156)

where we have made use of the identity H (a) − H (−b) = sgn(a)H (ab) = sgn(b)H (ab).

(157)

Now one can see that the integrand in (156) is exponentially small at k → ±∞ for all numbers α and independently of a sign of y. However, there is some problem with the integration along the path (141) close to the pole k = 0. Fortunately, as shows a rigorous analysis, this pole contributes to η(x, y) (156) just a large constant term ∝ ln ε. Such a contribution is unimportant, because, as we know, all components of the function η(x, y) = [U(x, y), V(x, y)]t are defined to the accuracy of an arbitrary constant. With this remark, after the integration in (156) we obtain η(x, y) =

8 1 sgn(Im pα )[ξ α ⊗ Tξ α ]g ln(x + pα y), 2πi

(158)

α=1

which is very similar to the classic result of the theory of elasticity [25,27,64]. It is convenient along with decomposition (136) to do the same with the eigenvectors ξ α , ξα =

Uα Vα

.

(159)

76


Ch. 72

In these terms the “displacement” 4-vector U(ρ) of our generalized dislocation is U(ρ) =

8 1 sgn(Im pα )Uα Vα · b(4) ln(m · ρ + pα n · ρ), 2πi

(160)

α=1

where we have returned to an invariant presentation. In order to eliminate the multivaluedness of the logarithm let us again introduce the cut along the surface S in Fig. 3. In this case the total and “elastic” distortions coincide, so that UK,l (ρ) = UlK (ρ) =

8 ml + pα nl 1 . sgn(Im pα )UKα Vα · b(4) 2πi m · ρ + pα n · ρ

(161)

α=1

The corresponding “stress” field is iJ (ρ) =

8 CiJ Kl (ml + pα nl ) 1 . sgn(Im pα )UKα Vα · b(4) 2πi m · ρ + pα n · ρ

(162)

α=1

The identical equivalence between eqs (161), (162) and (91), (92), respectively, can be proved quite similarly as has been done in [64] for ordinary elasticity.

5. Some concluding remarks Thus, an extension of the basic results of the anisotropic theory of dislocations formally reduces to almost trivial replacements of some three-dimensional vectors and matrices (apart from the space distance characteristics) by their four-dimensional analogue. However, this visual simplicity of the results obtained should not screen a fundamental transformation of dislocation fields in piezoelectrics due to a new quality arising in such media because of the electroelastic coupling occurring there. We stress that the important field transformations mentioned are not at all provided by an extension of a dislocation itself into a four-dimensional line defect. The edition to an ordinary dislocation of the electrostatic line defect of the “strength” ϕ just makes the theory presented to be more “symmetric” with respect to both coupling fields involved, elastic and electric. On the other hand, the electrostatic line defect introduced is not at all a purely theoretical object. As was noticed by J.P. Hirth [76], these and some other electrostatic defects might arise in domain structures of ferroelectrics. However their consistent physical analysis will require separate studies, which are definitely beyond the framework of this paper. Anyway, as is easily seen, even at ϕ = 0, when we deal with an ordinary dislocation described in our formalism by the Burgers vector b(4) = (b, 0)t , this dislocation certainly retains in its field both elastic and electric components. As is known, a piezoelectric coupling in practice usually is not very strong. It is customary to characterize a relative effectiveness of such coupling by the dimensionless electromechanical coefficient κ 2 = e2 /εc, where e, ε and c are typical values of the components


77

of piezoelectric, dielectric and elastic moduli involved (ordinarily even for good piezoelectrics κ 2 ∼ 0, 1). Accordingly, electrically induced additions to elastic distortions in such crystals are limited to corrections of the same order of 10%. However, electric fields excited by dislocations in piezoelectrics cannot be considered as corrections, because they arise on a zero background. And it is not widely understood that close to a dislocation line its electric field is not at all weak. Indeed, considering κ 2 as a small parameter one can estimate from eqs (54), (55) that the magnitudes of the electric field E and the distortion β in the vicinity of a dislocation are related to each other as e E ∼ β. ε

(163)

√ Substituting here e = κ εc and β ∼ b/2πρ one obtains E∼

√ bκ c/ε . 2πρ

(164)

The corresponding estimate of the maximum electric field Em close to the dislocation core at ρ ∼ b for typical magnitudes c ∼ 103 MPa = 1010 dyn/cm2 , ε ∼ 1 and κ 2 ∼ 0, 1 is given by Em ∼ 108 V/m = 1 MV/cm.

(165)

Of course, at the distances from a dislocation, say ρ ∼ 10 µm, the electric field reduces to a much more modest magnitude E ∼ 10 V/cm. However estimate (165) of the amplitude Em in the dislocation core makes a strong impression and might be very important for any phenomena on a microscopic scale, like dislocation pinning, interaction of crossing dislocations, diffusion of charged point defects in the dislocation field etc. In particular, the last effect may be completely responsible for an aging pace of freshly introduced dislocations, i.e. for their pinning by charged point defects.

Acknowledgements The authors are grateful to Prof. F.R.N. Nabarro, Prof. J.P. Hirth and Prof. A. Every for helpful comments. The paper was supported by the Polish–Japanese Institute of Information Technology, Warsaw (Research Grant No. PJ/MKT/02/2005). V.A. was also supported by the Polish Grant #4T07A02327 and by the Kielce University of Technology (Poland).

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M.Y. Chung and T.C.T. Ting, J. Appl. Mech. 62 (1995) 423. M.Y. Chung and T.C.T. Ting, Int. J. Solids Struct. 33 (1996) 3343. H.O.K. Kirchner and V.I. Alshits, Phil. Mag. A 74 (1996) 861. M.L. Dunn and H.A. Wienecke, Int. J. Solids Struct. 33 (1996) 4571.

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J.P. Nowacki and V.I. Alshits Ding Haojiang, Chenbuo and Liangjian, Int. J. Solids Struct. 34 (1997) 3041. Ding Haojiang, Chi Yuwei and Guo Fenglin, Int. J. Solids Struct. 36 (1999) 2613. Q.-H. Qin, Arch. Appl. Mech. 69 (1999) 133. Pin Lu, M.J. Tan and K.M. Liew, Int. J. Solids Struct. 37 (2000) 1065. Y. Mikata, Int. J. Engng. Sci. 38 (2000) 605. E. Pan and F.G. Yuan, Int. J. Engrg. Sci. 38 (2000) 1939. J.P. Nowacki, V.I. Alshits and A. Radowicz, Int. J. Appl. Electromagnetics and Mechanics 12 (2000) 177. J.P. Nowacki, V.I. Alshits and A. Radowicz, J. Tech. Phys. 43 (2002) 133. J.P. Nowacki, V.I. Alshits and A. Radowicz, Int. J. Engng. Sci. 40 (2002) 2057. J.P. Nowacki, V.I. Alshits and A. Radowicz, Intern. Journ. Appl. Electromagnetics and Mechanics 14 (2001/2002) 429. L. Merten, Phys. Kondens. Mater. 2 (1964) 53. L. Merten, Phys. Kondens. Mater. 2 (1964) 66. L. Merten, Z. Naturforsch. 192 (1964) 788. L. Merten, Z. Naturforsch. 192 (1964) 1161. A.M. Kosevich, L.A. Pastur and E.P. Feldman, Sov. Phys.-Crystallogr. 12 (1968) 797. G. Saada, Phys. Stat. Sol. (b) 44 (1971) 717. G. Faivre and G. Saada, Phys. Stat. Sol. (b) 52 (1972) 127. J.D. Eshelby, W.T. Read and W. Shockley, Acta Metall. 1 (1953) 251. I.S. Smirnova, Sov. Phys. – Solid State 15 (1974) 1543. A.R. Huston and L.R. Walker, J. Appl. Phys. 50 (1979) 6247. A.N. Stroh, Phil. Mag. 3 (1958) 625. A.N. Stroh, J. Math. Phys. 41 (1962) 77. K. Malen, Phys. Stat. Sol. (b) 44 (1971) 661. C. Teodosiu, Elastic Models of Crystal Defects (Springer, Berlin, 1982). D.M. Barnett and J. Lothe, Phys. Norv. 7 (1973) 13. D. Bacon, D.M. Barnett, and R.O. Scattergood, Prog. Mater. Sci. 23 (1979) 51. V.L. Indenbom and S.S. Orlov, Sov. Phys.-JETP 6 (1967) 274. J. Lothe, Phys. Norv. 2 (1967) 153. J. Lothe, in: Fundamental Aspects of Dislocation Theory, Vol. 317, Spec. Publ., eds J.A. Simmons, R. De Wit and R. Bullough (Nat. Bur. Stand., Washington, 1970) p. 11. V.L. Indenbom and J. Lothe (eds), Elastic Strain Fields and Dislocation Mobility (North-Holland, Amsterdam, 1992). D.M. Barnett and J. Lothe, J. Phys. F 4 (1974) 1618. D.M. Barnett, in: Structure and Deformation of Boundaries, eds K. Subramyan and M.A. Imam (Metallurgical Society, London, 1986) p. 31. K.-C. Wu and Y.-T. Chiu, Int. J. Solids Structure 32 (1995) 543. V.I. Alshits and H.O.K. Kirchner, Phil. Mag. A72 (1995) 1431, 1445. V.I. Alshits, H.O.K. Kirchner and G.A. Maugin, Math. and Mech. Sol. 6 (2001) 481. F.R.N. Nabarro and E.J. Kostlan, J. Appl. Phys. 49 (1978) 5445. W.C. Moss and W.G. Hoover, J. Appl. Phys. 49 (1978) 5449. L. Stagni and R. Lizzio, Int. J. Engng Sci. 24 (1986) 471. M.Yu. Gutkin, A.E. Romanov, Phys. Stat. Sol. A 129 (1992) 363. E. Jergan, Diploma thesis (University of Trondheim, Norwegian Institute of Technology, 1970) (unpublished). D.M. Barnett and J. Lothe, Phys. Stat. Sol. (b) 67 (1975) 105. V.I. Alshits, A.N. Darinskii and J. Lothe, Wave Motion 16 (1992) 265. J. Lothe and D.M. Barnett, J. Appl. Phys. 47 (5) (1976) 1799. J. Lothe and D.M. Barnett, Physica Norvegica 8 (4) (1977) 239. V.I. Alshits, A.N. Darinskii, J. Lothe and V.N. Lyubimov, Wave Motion 19 (1994) 113. J.M. Burgers, Proc. K. Ned. Akad. Wet. 42 (1939) 293. F.R.N. Nabarro, Advances in Physics 1 (1952) 269. E. Kröner, Kontinuumstheorie der Verzetzungen und Eigenspannungen (Springer, Berlin, 1956).


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[53] V.L. Indenbom, in: the book [34], Chap. 1, p. 1. [54] R. de Wit, Solid St. Phys. 10 (1960) 249. [55] A.M. Kosevich, in: Dislocations in Solids, Vol. 1, ed. F.R.N. Nabarro (North-Holland, Amsterdam, 1979) p. 33. [56] W.T. Read, Dislocations in Crystals (McGraw-Hill, New York, 1953). [57] A.H. Cottrell, Dislocations and Plastic Flow in Crystals (Oxford University Press, Oxford, 1953). [58] J. Friedel, Les Dislocations (Gauthiers-Villars, Paris, 1956). [59] F.R.N. Nabarro, Theory of Crystal Dislocations (Oxford University Press, Oxford, 1967). [60] J. Hirth and J. Lothe, Theory of Dislocations, 2nd ed. (Wiley, New York, 1982). [61] J.L. Synge, The Hypercircle in Mathematical Physics: A Method for the Approximate Solution of Boundary-Value Problems (Cambridge University Press, Cambridge, 1957). [62] D.M. Barnett, Phys. Stat. Sol. (b) 49 (1972) 741. [63] V.L. Indenbom and S.S. Orlov, in: Proc. Kharkov Conf. on Dislocation Dynamics (Physical-Technical Institute of Low Temperatures, Academy of Sciences of the USSR, Kharkov, 1968) p. 406. [64] J. Lothe, in the book [34], Chap. 4, p. 269. [65] T. Mura, Phil. Mag. 8 (1963) 843. [66] L.M. Brown, Phil. Mag. 15 (1967) 363. [67] J.R. Willis, Phil. Mag. 21 (1970) 931. [68] J.W. Steeds and J.R. Willis, in: Dislocations in Solids, Vol. 1, ed. F.R.N. Nabarro (North-Holland, Amsterdam, 1979) p. 145. [69] J.D. Eshelby, Solid St. Phys. 3 (1956) 79. [70] S.D. Gavazza and D.M. Barnett, Scripta Metall. 9 (1975) 1263. [71] M. Peach and J.S. Koehler, Phys. Rev. 80 (1950) 434. [72] D.M. Barnett and L.A. Swanger, Phys. Stat. Sol. (b) 48 (1971) 419. [73] V.L. Indenbom and V.I. Alshits, Phys. Stat. Sol. (b) 63 (1974) K125. [74] D.M. Barnett and J. Lothe, Phys. Norv. 7 (1973) 13. [75] R.J. Asaro, J.P. Hirth, D.M. Barnett and J. Lothe, Phys. Stat. Sol. (b) 60 (1973) 261. [76] J.P. Hirth, private communication.


CHAPTER 73

Statistical and Dynamical Approaches to Collective Behavior of Dislocations G. ANANTHAKRISHNA* Materials Research Centre and Centre for Condensed Matter Theory, Indian Institute of Science, Bangalore-560012, India Dedicated to the memory of F. R. N. Nabarro, a poineer in theory of dislocations

* E-mail: [email protected], fax: 91 80 2360 7316



Contents 1. General introduction 83 2. Collective effects of dislocations 86 2.1. Near stationary patterns 87 2.2. Propagative patterns 89 3. Intermittent collective effects and their characterization 93 3.1. Introduction 93 3.2. Characterization of intermittent collective behaviour 94 4. Recent advances in statistical description of dislocation dynamics 105 4.1. Introduction to statistical and stochastic approaches 105 4.2. Early statistical models of dislocation dynamics 109 4.3. Recent statistical approaches to collective dislocation dynamics 116 4.4. Distribution-theoretic approach to collective effects of dislocations 119 4.5. Fluctuation induced patterns 127 5. Dynamical approach to collective behavior of dislocations 136 5.1. General introduction 136 5.2. Introduction to nonlinear time series analysis 137 5.3. Multifractal analysis 141 5.4. Characterization of experimental stress–strain series 143 5.5. Discussion 155 6. Dynamical approach to modeling the Portevin–Le Chatelier effect 157 6.1. Early models for dynamic strain aging 157 6.2. General introduction to dynamical models of the Portevin–Le Chatelier effect 159 6.3. Introduction to a fully dynamical approach to the Portevin–Le Chatelier effect 160 6.4. Slow–fast dynamics in the Portevin–Le Chatelier effect 169 6.5. Ananthakrishna’s model 169 6.6. Comparison with experiments 180 6.7. Dynamic strain aging model for the Portevin–Le Chatelier effect 198 6.8. A multiscale model for the Portevin–Le Chatelier effect 208 7. Discussion and outlook 215 Acknowledgements 217 References 217

1. General introduction In the spirit of all statistical theories, one should expect that macroscopic measurable quantities that define plastic deformations, namely, stress, strain, strain rate, etc., should be derivable from the micro-mechanics of the constitutive elements of the system, namely dislocations and other defects that contribute to the flow. Considering the fact that a large body of knowledge has accumulated on the mechanics of dislocations, and their interactions, it might come as a surprise that developing a first principles theoretical framework of plastic deformation, one that is an equivalent of statistical mechanics of condensed matter systems, has remained elusive till date. At a naive level, one should anticipate that the basic premise of any statistical description stated above should be applicable to the ensemble of dislocations. This is only true when dislocation density is low, where one can ignore correlations between dislocations. However, at relatively high densities, large number of dislocations can move coherently. Advances in describing such collective effects have been rather slow. The reasons for the inability to formulate a theoretical framework for plastic flow are not difficult to trace. We list them below. – The first major source of difficulty is that plastic deformation is a highly dissipative irreversible process. When the system is not far from equilibrium, it may be possible to adopt nonequilibrium statistical mechanical extensions to a limited extent. However, in the process of driving the system out of equilibrium, new states of order are created. An additional difficulty is that nearly 90% of the work done on the system goes in the form of heat [1,2]. Unfortunately, even in the context of condensed matter physics, there is no accepted framework of equilibrium or nonequilibrium statistical mechanics applicable to systems driven away from equilibrium where the levels of dissipation are high. Moreover, dislocations are defects that form a part of the crystalline environment and the heat generated goes in to the crystalline complement of the sub-system of the ensemble of dislocations. There is no straightforward way of accounting for this lost energy. – The highly dissipative nature of plastic deformation coupled with the fact that different patterns are seen in far-from-equilibrium situation also implies that nonlinearities play a fundamental role, at least in the case of various spatio-temporal dislocations structure [3–5]. One generic approach for addressing patterns emerging in driven system is through a description of identifiable collective modes of the system using approach in nonlinear dynamics. In the past few decades, the area of nonlinear dynamics has developed into a specialized subject and there is no accepted framework that integrates this discipline into statistical mechanics of plasticity. In this respect, though stochastic approach cannot be regarded as a first principles approach, it can be used for integrating nonlinearities. – In plasticity, one needs to deal with thermal as well as athermal activated dislocation processes. Stochastic methods are well suited for handling athermal fluctuations as there

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is no necessity for satisfying a consistency criterion between fluctuations and dissipation, though the strength of the fluctuating force has to be determined from other considerations. In the case of the Brownian motion describable in terms of a Langevin equation, the velocity of the particle is considered to evolve slowly under the influence of the average effective force acting on the particle. The effect of the rapid collisions of the particle with molecules of the bath is then treated as a fluctuating force. However, fluctuations and dissipation experienced by the particle due to viscous medium are not independent and are expected to obey the so-called ‘fluctuation-dissipation’ theorem. One prerequisite for this decomposition is that the order parameter variable(s) should have a substantially slow evolution compared to the ‘fast degrees of freedom’ of the heat bath molecules whose effect is lumped as a fluctuating noise. One other related direction that could be useful is the Ginzburg–Landau approach. This approach has been exploited very successfully in many problems relating to pattern formation in condensed matter physics in situation where a free-energy functional can be defined. (A good example is the spinodal decomposition.) This approach may prove to be useful in the case of plasticity, particularly when one can justify the introduction of a ‘free energy like function’ on physical grounds. In particular, such ‘potential like’ functions can be derived by reductive perturbative technique from a set of nonlinear coupled differential equations. – In general, plastic deformation is inhomogeneous, particularly at high strains. This is reflected in the strain localization that has been observed at several length scales [6]. The associated time scales are quite small and thus a simple spatial average cannot be taken to represent the correct average. This also raises questions about coarse graining length scales.1 – Any averaging procedure has to take into consideration the inhomogeneous nature of deformation that will eventually set in. This becomes even more clear when one considers different types of dislocation patterns, each having its own characteristic length and time scales. In some cases, the duration of the development of the dislocation pattern could be rather long like in the case of the formation of cell structure observed using Transmission Electron Microscopy (TEM) deformed in multislip conditions, or the strikingly regular structures formed during cyclic deformation with low plastic strain amplitudes [7–12]. Or the pattern could emerge on a rather fast time scale as in the case of the Portevin–Le Chatelier effect [13] where both fast and slow time scales are involved simultaneously. Often, the associated length scales could range over several orders. Further, these time scales and length scales themselves evolve during the transition from the micro to meso to macro scales which is at the root of development of collective behavior involving a large number of correlated events. Thus, one needs to have an appropriate framework for carrying out averaging over the range of length and time scales. – Finally, dislocations are line defects with a tensorial character. This leads to certain technical complications that are not encountered when the constituent elements are point objects. 1 The usual way coarse graining is carried out is to start from a description at some scale and first calculate the correlation length and use this length scale to obtain coarse grained description. Such an approach is possible even when a hierarchy of rate equations are essential for description of the system as in the case of clustering of vacancies [14,15]. However, even today, there is no equivalent approach in the context of dislocation dynamics.

§1

Statistical and dynamical approaches to collective behavior of dislocations

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The initial optimism that collective behavior of dislocations can be described on the basis of properties of individual dislocations and their interaction evaporated by mid 1960. In the last three decades, some progress has been made in dealing with the collective behavior of dislocations. The above comments also indicate that different groups have focused on addressing issues related to specific examples of dislocation patterns. These approaches can be broadly classified as statistical and dynamical approaches. In either case, any correct framework should have the ability to bridge all the length scales observed in each of these patterns. Among the statistical approaches, three distinct approaches have been followed. The simplest, and in fact the earliest approach undertaken, is to set up phenomenological equations of motion for the probability distribution of a set of variables representing the ‘state’ of the ensemble of dislocations [16–18]. In parallel, a Langevin approach that recognises inherent fluctuations for local quantities over and above their mean was also suggested. The magnitudes of fluctuations are determined without explicitly stating the underlying distribution. More recently, Langevin type of approach has been followed to understand mesoscopic patterns [5]. The third one follows the standard route adopted in nonequilibrium statistical mechanics, namely the distribution function-theoretic approach [5]. One element that is common to different approaches, at least at the conceptual level, is the necessity to use coarse grained dislocation densities, usually, one each for the different kinds of dislocations types needed for a physical description. (In some cases such densities can be obtained by integrating out irrelevant variables.) Such equations are often set in the form of coupled nonlinear equations. In stochastic approaches, noise is introduced as necessary fluctuations resulting from the environment. The dynamical systems approach in physical sciences has exploded since the observation by Lorenz that climatic changes could well be unpredictable. This approach has been very successful in describing phenomenon which hitherto were beyond linear theories, with applications ranging from physics, chemistry, biology to geology [19–21], and to materials science as well [3]. In the case of plasticity, a dynamical description starts with a set of coupled nonlinear partial differential equations for the coarse grained densities of dislocations which are thought to be some sort of collective modes responsible for the eventual emerging patterns. This is followed by setting up equations of motion based on physical inputs on dislocation mechanisms that are responsible for the observed patterns. Usually, such equations contain a set of parameters that control the nature of solutions called drive parameters, for example applied strain rate. In such a nonlinear set of equations, the nature of the solutions changes when one of the control parameters is varied at some particular value, i.e. there is a point of bifurcation. Varieties of bifurcations have been classified in the literature some of which are relevant to dislocation patterns. These necessarily involve selection of certain wave vectors which determine the nature of the pattern. The set of differential equations often supports a sequence of bifurcations eventually leading to chaotic solutions in certain domains of the parameter space. The geometrical nature of the orbit in the phase space, called a strange attractor, exhibits self-similarity property. When spatial degrees of freedom are included, the dynamics is termed spatio-temporal chaos. There are standard methods of characterizing the bifurcation sequences and ways of characterizing chaotic state through correlation dimension and Lyapunov exponents. There are other techniques of dynamical systems that can be adopted for studies on spatio-temporal aspects of

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dislocation patterns. Two such methods that need to be mentioned here is the reductive perturbative technique and slow manifold approach. One important technique developed in the context of dynamical systems is the analysis of time series obtained from experiments [20,22]. Detailed analysis of the stress–time series has shown that there is a substantial amount of hidden information contained in the stress–strain curves [3]. In the context of the Portevin–Le Chatelier effect, there are different types of serrations in the range of strain rates where the effect is seen whose wave forms have been stated in qualitative terms. The importance of different type of serrations is that they have been identified with the band types occurring in regions of the applied strain rates. However, these serrations were not quantified till recently. Dynamical approach to the analysis of stress–time series help us to quantify the nature of serration in different regimes of strain rate. This coupled with direct observations of the band types have given insights that hitherto were not possible. The purpose of this article is to review the recent advances on both stochastic and nonlinear dynamical approaches which have given a semblance of coherence in the description of collective effects of dislocations. Much of this has been possible due to recent advances in improved experimental techniques and import of new concepts and methods from physics literature. As there have been several review articles on the general topic of collective behavior of dislocations [3–5,23,24], we confine our attention to dealing with topics that can be presented with a general perspective set out in the introduction. We first briefly summarize essential features that form the statistical or the dynamical signatures of the collective phenomenon under consideration. The first part contains the statistical approach. We begin by collecting known methodologies in the condensed matter literature that have been successful in describing collective phenomenon pointing out merits and demerits when applied to collective behavior of dislocations. Then, we briefly outline early statistical approaches to dislocation dynamics and follow it up with the recent trends that include discussion on fluctuations in plastic flow followed by the Langevin and Fokker–Planck approaches, and the distribution-theoretic approach to collective behavior of dislocations in two dimensions. The second part is devoted to a dynamical approach. The focus is largely on the Portevin–Le Chatelier effect. One important question that has received some attention is how to unravel hidden information in the stress–time series. Even though this has been discussed in our last review [3], the recent progress will be summarized. This will be followed by a few recent models that attempt to describe the complex spatio-temporal features of the Portevin–Le Chatelier effect. We show that the application of new concepts such as fractals, self-affinity, multifractals, percolation, etc. has given an understanding that would not have been possible otherwise.

2. Collective effects of dislocations It is well recognized that plastic deformation is a highly dissipative irreversible nonequilibrium process where nonlinearities play a fundamental role. Thus, the emergence of collective behavior of dislocations is a very distinct possibility. At low levels of deformation, by a continuity argument one expects homogeneous deformation. However, under suitable conditions of deformation and choice of materials, spatial and spatio-temporal patterns have

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been observed. At a microscopic level, these patterns arise due to the collective behavior of dislocations. They can be broadly classified on the basis of the associated time and length scales. For example, the cell structure observed in f.c.c. metals under multi-slip conditions and the persistent slip bands observed in cyclic deformation are examples of patterns that develop over long time scales [3]. On the other hand, a type of propagative bands referred to as the Lüders bands [25,26] observed in uniaxial tension tests are characterized by short time scales [3]. Here, a single band propagates along the sample like a solitary wave at a constant stress level corresponding to the lower yield point. This is an example where the pattern developed over a short time also propagates. Yet another and even more complex spatio-temporal pattern is observed during tension tests of dilute metallic alloys in a certain range of strain rates and temperatures. This phenomenon has come to be known as the Portevin–Le Chatelier (PLC) effect [13]. Here a uniform deformation mode becomes unstable leading to a spatially and temporally inhomogeneous state. The temporal aspect manifests itself in the form of serrations on the stress–strain curves of the sample [3,4], and the associated localized plastic deformation bands are visible to the naked eye. Experimental observation on strained single and polycrystals have demonstrated that even in situations where the macroscopic deformation is considered uniform, plastic activity is heterogeneous from the beginning and dislocation activity is localized to a small volume. For example, in the case of Lüders band which is normally considered as inhomogeneous deformation, slip line observations [6] have demonstrated that the deformation is actually heterogeneous, having multiple length scales. The Lüders front consists of slip band bundles of dimension typically 0.1 mm that can be resolved under better resolution [6]. At the next level, these consist of slip bands of about few microns arranged in nearly regular manner which themselves consist of slip lines of nanometer length scale. More recent observation on slip line morphology [27,28] have shown that the heterogeneity extends to nanoscales with groups of dislocations moving coherently. For details we refer the reader to several detailed reviews on this subject [6] (see also the most recent Refs [3,4]). Here, we focus on the salient features of these patterns as a background material from a general perspective of this article, namely, stochastic and dynamical features that suggest the directions pursued in recent years.

2.1. Near stationary patterns Among the dislocation patterns formed on long time scales, two types of patterns can be distinguished; those formed in single slip involving only one active Burgers vector such as the persistent slip bands (PSB) and those under multislip conditions such as cells. The basic distinction in the patterns arises due to the differences in dislocation mechanisms. In the former case, long range interactions appear to dominate the dipolar or the multipolar structures while dislocation intersections play an important role in the latter. Three-dimensional cell structures are most commonly observed patterns in multislip deformation conditions. Cell structure refers to a type of heterogeneous pattern with regions of high density of dislocations called the cell walls separated by cell interiors of low density. Indeed, while the cell walls form only about 10% of the total volume, most dislocations are trapped in these walls. In single crystals of f.c.c. metals oriented in multiple-slip

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conditions as well as polycrystals, cell structures are seen in a range of temperatures. In single crystals, dislocation patterns begin to form at the end of stage II, but are seen prominently in stage III. This is suggestive of the role played by screw dislocations. In b.c.c. metals, cell structure is seen above the transition temperature. Very interesting organized structures have been observed when f.c.c. and b.c.c. metals and alloys are subjected to low amplitude (∼10−2 ) cyclic deformation at moderate temperatures (less than 0.5Tm ) well summarized in Refs [7–10,12,29]. When single crystals of f.c.c. metals such as Cu, Ni and Ag oriented for single slip are subjected to cyclic deformation, several different types of patterns emerge of which matrix structure and PSBs are well studied and characterized. At low and moderate temperatures and for small strain amplitudes, the matrix structure emerges first followed by the persistent slip bands. Each of these structures corresponds to a well defined stage in the cyclic stress–strain curve. The matrix structure consists of dense regions of dipolar veins of irregular shape. When the plastic strain amplitude is increased, one sees periodic arrays of thin dipolar veins. The PSBs are thin lamellae of width ∼1–2 µm parallel to the primary glide plane (and per that might traverse the entire sample. As can pendicular to the primary Burgers vector b) be seen from Fig. 1, these lamellae are arranged in a ladder like structure and consist of a dense set of dislocations of primarily bi-/multipolar edge character separated by channels typically ten times the width of the walls. The walls have high density of dislocations ∼1015 m−2 while the channels have at least two orders less dislocation density of mainly screw character. These patterns basically arise from the unmixing of screw and edge dislocations. As screw dislocations annihilate more easily than the edges, the edge dislocations segregate into dipolar veins separated by a channel of relatively low density for the screw channels to shuttle.

Fig. 1. Persistent slip bands with a characteristic ladder like structure embedded in a matrix structure in copper subjected to cyclic deformation to strain amplitude 4.1 × 10−3 . (Courtesy of U. Holtzworth. After M. Zaiser et al. [30].)

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These patterns can have two characteristic length scales, one that corresponds to the size of the dense dislocation regions and the other that of the dislocation poor regions. For these patterns, the characteristic size decreases with an increase in stress levels. This has come to be known as the similitude principle. Some empirical scaling laws have been noted that reflect universal features of the patterns. The linear dimension of the cells is found to be inversely proportional to the flow stress or the square root of the dislocation density given by τ b = αbρ −1/2 = K . G r

(1)

Here b is the magnitude of the Burgers vector and G the shear modulus, r is the linear size of the pattern, α ∼ 0.35 and K is a constant. For the PSBs, K ≈ 3–4. In monotonic deformation, K ≈ 10. A similar value is reported for polycrystalline materials at various temperatures and various deformation conditions [31].

2.2. Propagative patterns 2.2.1. The Lüders band For the sake of completeness, we briefly mention the characteristic features of the Lüders band. The Lüders phenomenon [25,26] refers to a single band of plastic deformation traveling along the specimen, observed when tensile samples are loaded at a constant cross-head velocity. The band formation is attributed to the unpinning of initially locked dislocations with no further repinning occurring during propagation (as in the case of the PLC effect) as the time scale for diffusion of the impurity atoms is much larger than the total duration of the experiment. The band is found to be oriented at a well defined angle with respect to the specimen axis, typically 50–55◦ which can be related to the orientation of the grains. The band nucleation usually occurs at one grip following the drop in stress from the upper yield point to the lower yield point. The plastic zone moves at a constant lower yield stress level. A clear demarcated front separates the plastically deformed part of the sample from the undeformed one into which it propagates until the specimen is uniformly deformed at the so-called Lüders strain. Thereafter, the plastic deformation proceeds uniformly with positive strain hardening. The Lüder phenomenon is a strain softening instability. Rather complex dislocation processes, almost invariably involving cross-slip, take place during strain softening. 2.2.2. The Portevin–Le Chatelier effect In contrast to the above, the Portevin–Le Chatelier effect is a strain rate softening instability. The phenomenon was first recognized by Savart [32] but the first detailed study was undertaken by Le Chatelier in 1909 on mild steel specimens [13], subsequently in Duralumin by Portevin and Le Chatelier in 1923, hence the name Portevin–Le Chatelier effect [13]. When a specimen of a dilute alloy (such as Al or Cu alloy, or mild steel) is strained in uniaxial loading, the mechanical response is sometimes not smooth but discontinuous. In constant applied strain rate tests, the stress vs. strain (or time, which is proportional to

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Fig. 2. Stress–time curves for Al-5at%Mg alloy at T = 300 K showing change over from type C → type B → type A serrations with increasing strain rate. (a) Type C, ǫ˙a = 5 × 10−6 s−1 . (b) Type B, ǫ˙a = 5 × 10−4 s−1 . (c) Type A, ǫ˙a = 5 × 10−3 s−1 (after Ref. [43]).

strain) curves exhibit a succession of stress drops and reloading sequences. Each stress drop corresponds to the nucleation, and sometimes the propagation along the specimen gauge length, of a band of localized deformation. A characteristic feature of the effect is that it occurs within a well-defined range of strain rates and temperatures. The stress–strain curve shows serrations that correspond to the nucleation of the PLC bands. In polycrystals, these serrations and the associated bands are classified into three generic types shown in Fig. 2. On increasing the applied strain rate or decreasing the temperature, one first finds the type C band, identified with randomly nucleated static bands with large characteristic stress drops. The serrations are quite regular. Then the type B ‘hopping’ bands are seen with each band forming ahead of the previous one in a spatially correlated way. The serrations are more irregular with amplitudes that are smaller than that for the type C. Finally, one observes the continuously propagating type A bands associated with small stress drops. Given an alloy one can construct a phase diagram in the variable ǫ˙ and 1/T where T is temperature [33]. These different types of PLC bands are believed to represent distinct correlated states of dislocations in the bands. From a dynamical point of view, this jerky or stick-slip kind of behavior is related to the discontinuous motion of dislocations, namely, the pinning (stick) and unpinning (slip) of dislocations. The well accepted classical explanation of the PLC effect is via the dynamic strain aging (DSA) concept first introduced by Cottrell [34,35] and later extended by others [3,36–40]. In the current picture of the dynamic strain aging, solute atoms diffuse by either volume or pipe diffusion, to the mobile dislocations arrested temporarily at

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obstacles during their waiting time. Thus, the longer the dislocations are arrested, larger will be the stress required to unpin them. As a result, when the contribution from aging is large enough, the critical stress to move a dislocation increases with increasing waiting time or decreasing imposed strain rate. When these dislocations are unpinned, they move at large speeds till they are arrested again. At high strain rates (or low temperatures), the time available for solute atoms to diffuse to the dislocations to age them decreases and hence the stress required to unpin them decreases. Thus, in a range of strain rates and temperatures where these two time scales are of the same order of magnitude, the PLC instability manifests. The competition between the slow rate of aging and sudden unpinning of the dislocations, at the macroscopic level translates into a negative strain rate sensitivity (SRS) of the flow stress as a function of strain rate. This is the basic instability mechanism used in most phenomenological models [3]. Penning, in his landmark paper [36], was the first who recognized negative strain rate sensitivity as a condition for repeated yielding. Even though negative SRS is a function of macroscopic variables such as stress, strain, and strain rate, these variables are actually treated as local variables. As the concept of dynamic strain aging is the basic physical mechanism adopted directly or indirectly in modeling the PLC effect and diffusion of solute atoms is central to this, the nature and the role played by diffusion has been experimentally investigated. These studies show that three different mechanisms, namely pipe diffusion (at low temperatures), in the stacking fault ribbon between the partial dislocations (at intermediate temperatures) and bulk diffusion at high temperatures, have been identified in Cu-Mn and Cu-Al alloys [41,42]. The disappearance of the PLC effect is then attributed to bulk diffusion. Although the band nucleation and propagation is an important feature of the PLC effect, there are only few measurements of the relevant features [43] and the most recent one is by Neuhäuser’s group using a laser scanning extensometer technique [44]. As the latter is by far the exhaustive one, we shall describe these results in some detail. One advantage of this technique is that it is a noncontact method apart from allowing for simultaneous measurement of propagation properties and the local strain. This should be contrasted with the method adopted by Chihab et al. [43] (shown in Fig. 2) which uses special illumination technique for observation of bands, which however is not suitable for measurement of plastic strains within the bands. In the laser extensometer technique, white reflector markings (of width 1 mm) are applied on the sample surface along the length of the sample such that one white marking and unlacquered segment constitutes an extensometer (of 2 mm gauge). These markings are detected from the reflected laser beam which is scanned by a rotating prism. The local strains can be detected by a time delay technique from the edges of the same markings. The method allows for recording up to 22 neighboring extensometers. The observations, apart from confirming several known facts about different types of bands, also gives quantitative information about the highly correlated propagating nature of the type A band as can be seen in Fig. 3. For Cu15%Al, the band velocity is typically 105 µm/s at 323 K. The local strain jump when the band passes is about 0.4%. The discontinuous propagation of type B bands can also be clearly observed in their measurements where stepwise strain jumps are seen. For the type C bands, there is no spatial or temporal correlation as in earlier measurements. In addition, these authors have investigated the influence of various parameters such as the grain size, temperature, strain rate and sample thickness on the band velocity. They

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Fig. 3. A plot of the location of band as a function of time for type A PLC band. After P. Hähner et al. [44].

find that the band width wb , band strain ǫp and the velocity cb all increase with strain rate having a power law dependence with an exponent α. While the exponent values for the first two are nearly 0.2, that of the velocity is 0.8. They also find that the grain size has practically no effect on the band velocity or the band width. Interestingly, while band velocity and band strain are independent of the specimen thickness, band width increases with the specimen thickness. 2.2.3. Length scales and spatial coupling Coming to the associated length scales, we first note that when plastic deformation is uniform, the associated length scale is of the order of the sample dimensions and thus is large compared to any internal material length scale. Thus, conventional constitutive formulations of plastic flow that do not include internal length scales are sufficient to describe the deformation process. However, when strain localization occurs due to collective effects, gradients of plastic strain may appear. When the plastic slip activity localizes into deformation bands, the characteristic length scale of the deformation is of the order of the width of the active glide bands. In a number of situations, the latter becomes small enough to be of the order of some internal length scale originating in the material structure, such as the grain size in polycrystals. In such conditions, internal length scales must be used to account for detailed structure of the strain field. One way to include information of length scales and time scales is to include them in the constitutive laws by assuming that the flow stress depends on strain or strain gradients. The fact that slip propagates suggests the existence of a strong spatial coupling between neighboring elements which also provides a characteristic length scale. Therefore, gradients and length scales have to be incorporated into the formulations in order to account for the propagation phenomena. However, the physical origin of the coupling is still controversial which we shall discuss briefly. For a detailed discussion of the spatial coupling, see the recent review Ref. [3]. There are many mechanisms that have been suggested in the literature which may be classified into two distinct groups. Those that originate either from mechanical effects re-

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lated to the loading mode or triaxiality of the stress field, or from intrinsic material properties, at the scale of dislocation mechanisms or the intergranular compatibility stresses. For instance, necking is a typical plastic strain localization process. For plastic materials, it is given by the Bridgman [45] form which connects the radius of the neck to the radius of curvature of the neck. A simple expansion gives a diffusive type of spatial coupling. In contrast, spatial coupling by material effects may originate either from the microstructure, i.e. from dislocation mechanisms in a single crystal, or from the mesostructure, more specifically from compatibility mechanisms between grains in a polycrystal. The microstructural mechanisms can be of two types: the spreading out of dislocation glide from an active region can be triggered either by double cross-slip or by long range elastic stresses stemming from dislocation interactions [44]. Several authors have suggested diffusive type coupling arising from cross-slip [40,46–49]. Aifantis [46,47] was the first to use a diffusive coupling to model the spatial inhomogeneity. He suggested that the internal stress could be modified to take the form σint = σ0 + θ ǫ + c∂x2 ǫ. Later, [48], it was argued that three different mechanisms such as intergranular incompatibility of stresses within the polycrystal, nonlocal strain hardening and coupling of adjacent slip planes by double cross-slip can lead to such a diffusive coupling. Kubin and Estrin [40,49] suggested that the diffusive term could be introduced by modify∂2ǫ ing the Orowan equation to include a second order strain gradient term ǫ˙ = bρg V + D ∂x 2. Here, the pseudo diffusion coefficient is supposed to describe the exchange of mobile screw dislocations between adjacent glide planes via a double cross-slip mechanism. Hähner et al. [44] argue that long range internal stresses lead to a similar coupling. Ananthakrishna [50,51] incorporated the effect due to long range back stress arising from the immobile dislocations into the cross-slip term. However, once these basic length scales and time scales corresponding to a particular problem are introduced, a velocity selection mechanism still needs to be considered.

3. Intermittent collective effects and their characterization 3.1. Introduction Observation of slip lines is a classical subject of determining the slip systems. Long before dislocations were postulated, surface marking observed during deformation of a material were noted [52]. Slip lines are steps produced when dislocations emerge on the crystal surface during plastic deformation. They are generally oriented along the intersection of the close packed crystallographic planes with the surface, the slip height being maximum along the direction of the Burgers vector. There is a vast amount of literature on slip line patterns (see the review by Neuhäuser [6]). These studies, particularly on f.c.c. metals and alloys, reveal that there is a fine structure which can be resolved using light and electron microscopic techniques. A fine slip line with a step height of a few nm typically has 15 to 20 dislocations. However, the structure of slip terraces observed on the surface is far more complex. One major feature that emerges is that there is clustering of dislocations at finer and finer levels that can be resolved using techniques that give increasing levels of resolution such as light microscopy, electron microscopy (EM) and atomic force microscopy.

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For instance, each slip band observed in light microscopy at a finer scale (EM) is seen to consists of clusters of fine slip lines. In a heterogeneous slip situation, these slip bands are found to be arranged in a near regular manner to form slip-band bundles. The step wise growth of slip lines clearly suggests that several dislocation groups move coherently and that collective effects play an important role in the formation of slip lines and slip bands. Thus, it appears that the motion of a dislocation segment triggers the motion of other dislocations. Obviously, the localized yet collective behavior of dislocations at finer and finer scales cannot be understood on the basis of the dynamics of individual dislocations and their interactions. Moreover, the hierarchy of length scales resolved by electron micrographs of neutron irradiated copper [6,53] is suggestive of a fractal structure. Neuhäuser [6] estimates the fractal dimension to be about 0.7. Later, Kleiser and Boˇcek [54], in their study on slip line structure of Cu report a fractal dimension of 0.5 in the scale 0.06 to 2 µm. The distribution of slip heights and distances reflect the activity of dislocations, their sources and degree of heterogeneity [6]. Based on this, the estimated physical quantities of interest at these local scale are found to be several orders of magnitude larger than the corresponding quantities on the macroscale. For instance, the local strain rates are about six to seven orders of magnitude larger than the applied strain rates. Similarly, the local dislocation density is of similar orders of magnitude higher than the average dislocation density. The active slip volume has been estimated to be of the order of 10−10 m3 which suggest that the activity is limited to a very small volume. Till date, the collective behavior of dislocations at these scales is yet to be understood. One feature of the slip line morphology is that the extent of heterogeneity is generally higher for f.c.c. metals and their alloys compared to the b.c.c. alloys, apart from being sensitive to other factors like stacking fault energy, short range order, etc. [27,28]. In order to understand the mechanisms leading to the surface roughness, several different methods of complementary nature have been used recently. For instance, in situ optical microscopy, in situ TEM of dislocation motion in thin single crystal foils, and TEM of dislocation configurations after deformation, and atomic force microscopy (AFM) [27,28] have been used to understand the mechanisms leading to heterogeneity. The better resolution offered by AFM has helped to obtain more quantitative information of the degree of heterogeneity. (See Fig. 4.) For instance the distribution of slip heights and distances at micrometer resolution has been obtained. The slip height distribution is found to be lognormal at micrometer levels. Such studies can also provide information of whether dislocation motion is correlated or random, or whether they are smooth or jerky [28]. These investigations show that while the structure of slip bundles of two different metals (or alloys) could be similar at meso and macroscales, they can be different at the micrometer level. More importantly, these studies also help to identify the possible mechanisms causing heterogeneity of the slip lines. For instance, in CuAl, short range order together with low stacking fault energy produces a high degree of heterogeneous slip. This also implies strong fluctuations in internal stresses [55]. 3.2. Characterization of intermittent collective behaviour The fine structures of slip lines formed when Fe3 Al alloy samples are subjected to compression tests have been examined using both optical technique and AFM over a wide range

§3.2 Statistical and dynamical approaches to collective behavior of dislocations

Fig. 4. Homogeneous and heterogeneous slip in Fe3 Al. (a) Successive optical video frames of localized cluster of slip bands. (b) SFM micrograph of a typical section. (c) SFM image of localized slip band with homogeneous continuation in both sides. (d) Step profile of (c). After A. Brinck et al. [28].

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scales (0.1 to 50 µm). Heterogeneity of plastic flow has been analysed using a wavelet transform [56]. The wavelet transform is able to reproduce the internal structure of slip heights at various scales of length. These and the earlier studies [28] suggest that the slip height morphology could be self-affine rather than self-similar. The concept of self-affine geometry is different from that of a fractal in that the scale factors are different in different directions. For example, the height y(x), has a different scale factor compared to the distance along the x-direction. (See eq. (2) below.) This is best explained by considering the Brownian motion. The probability of finding a particle in the interval z and z + dz at a 2 time t is given by P (z, t) = exp√−z /2t . This Gaussian distribution has the scaling property 2πt

λ1/2 P (λ1/2 z, λt) = P (z, t). If the surface is self-affine then a more general scaling relation λH P (λH z, λt) = P (z, t) is satisfied, where H is called the Hurst exponent which is a measure of the roughness of the surface. For the Brownian motion, H = 1/2. The self-affine geometry is quite commonly observed in many branches of science from simple example of spreading of an ink drop placed on a porous paper to complex growth processes. Other well studied examples are surface morphology of fractured surfaces [57], interface growth problems such as growth of thin films by molecular beam epitaxy [58,59], growth of thin films by vapor deposition [60] and bacterial growth in biology. A detailed quantitative characterization of the self-affine nature of slip morphology has been reported recently [61]. Using AFM and scanning white-light interferometry, the range of length scales covered extend from 10 nm to 2 mm. A one-dimensional profile of the step height y along a chosen direction x has been used for the analysis. Polycrystalline samples of Cu were deformed at strain rate 1.5 × 10−3 s−1 to different levels of strain ranging from 2.5% to 23% with a view to obtain the dependence of the roughness exponent on strain. At a working level, there are various methods of estimating the roughness exponent which are generally classified as intrinsic and extrinsic [62]. One of the simplest quantifiers of self-affine nature determines how the height y at x is correlated to the height at x + L given by y(x) − y(x + L) ∼ LH .

(2)

Here the average . . . is over all values of x. A simple check of comparing surface profiles at two different scales shows that the profiles are statistically similar except for scale factors. Fig. 5(a) shows log-log plot of |y(x) − y(x + L)| as a function of L. The scaling regimes in the range 0.05 to 5 µm corresponding to AFM, and 0.5 to 100 µm corresponding to scanning white-light interferometry (SWL) ranges are clear from the figure. Noting that the values of the Hurst exponents obtained from these scaling regimes can be seen to be similar, the combined range of length scales can be seen to extend over four orders of magnitude. A plot of the Hurst exponent for various strain values is shown in Fig. 5(b) which shows that the Hurst exponent2 has a tendency to saturate to a value of 0.75 with increas2 To give a physical feel for what the Hurst exponent signifies in terms of correlation, we note that the usual Brownian motion corresponds to no correlation as it is generated using independent increments. Then, H > 1/2 corresponds to positive correlation which means that if the height of the profile has increased in the past, it has a tendency to increase in future as well. On the other hand, if H < 1/2, there is a negative correlation and the trend of the future profiles is opposite to the past and hence is more oscillatory.

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(a)

(b) Fig. 5. (a) A log-log plot of height difference as a function of distance. Here the curves correspond to (1a) ǫ = 9.6% using AFM, (1b) ǫ = 9.6% using SWL, (2a) ǫ = 17.8% using AFM and (2b) ǫ = 17.8% using SWL. (b) Dependence of the Hurst exponent as a function of strain. After M. Zaiser et al. [61].

ing strain beyond 10%. The corresponding fractal dimension3 DF = 2 − H is also shown in Fig. 5(b). The scaling property of self-affine surfaces also means that the probability distribution P (yL ) of the step height difference yL = y(x) − y(x + L) should obey the scaling relation P ((L/L0 )H y/y0 ) = P (y/y0 ). (Appropriate values of y0 and L0 are used for normalization.) This means that one should find a data collapse of the height 3 A word of caution here is worthwhile. There is no unique relation between fractal dimension and the Hurst exponent of a self-affine record. See Mandelbrot [67,68]. To appreciate this consider the conventional definition of fractal dimension, say the box counting method that uses box of size bd where d is the embedding dimension. Then, if we rescale b → λb, then N (λ) ∼ λ−DF . If we use this definition to cover a fractal Brownian record with a Hurst exponent H , we get different results that depend on the scale used. In the limit of small box size, one gets an DF that satisfies DF = 2 − H , but if one chooses a coarser box, it would not satisfy this expression. This is basically due to the fact that when one covers a distance b along the x-axis, the height or the so-called ‘range’ changes by bH , i.e. for z(t), z(bt) ∼ bH z(t). Thus, if one uses a box of size b × b to cover the set, then as long as bt ≪ |z|, we get N (b × b) ∼ b−2+H ∼ b−DF . See Ref. [69].

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Fig. 6. Collapse of height difference distribution function for various L values for strain 9.6%. After M. Zaiser et al. [61].

difference distribution for a range of values of L which the authors verify [61]. They do this by plotting the cumulative distribution normalized to unit variance, for a fixed stain value 9.6%, but for L in the range 50 nm to 100 µm as shown in Fig. 6. The analysis has other implications. For instance, the estimated fluctuations in strain y/L at the lowest length scale (∼50 nm) is about the same order as the mean, while at the upper end strain fluctuations are about 15% of the average strain. The study also raises doubts about the choice of an appropriate length scale for coarse graining in any theoretical framework. As the slip lines are caused by the emerging groups of dislocations, these investigations suggest that correlated motion of dislocations must extend to within the bulk material as well. Such a collective motion of dislocations can be captured by acoustic emission technique as it is a nondestructive technique that is sensitive to the microstructural changes taking place inside the system. The method is widely employed in the detection of earthquakes of small magnitudes [63], understanding and mapping nucleation events of fracture in seismology [64], and in the study of martensitic transformation [65]. Acoustic emission (AE) is generally attributed to the sudden release of stored strain energy, although the details of the mechanism are generally system specific. In the context of plasticity, the abrupt dislocation motion produces acoustic waves which can be recorded by piezoelectric transducers. Recently, Miguel et al. [66] have analysed the statistics of the AE signals obtained from uniaxial compressive creep of ice crystals. The transparent nature of ice has been used to eliminate the possibility of the AE signals arising from microcracks. In this case, dislocation glide occurs on the basal planes along preferred slip directions. The recorded acoustic signals show intermittent AE activity with a power law distribution for the energy bursts P (E) ∼ E −α extending over several orders of magnitude. It is clear from Fig. 7 that the data for a range of applied stress values from 0.03 MPa to 0.086 MPa fall on the same power law with an exponent α = 1.6 which might suggest insensitivity of the exponent to

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Fig. 7. Plot of the distribution of energy bursts. After Miguel et al. [66].

stress values. This must be contrasted with the dependence of fractal dimension on strain [61]. Power law distributions of acoustic emission are observed in unusually large number of situations as varied as volcanic activity [70], microfracturing process [71] and peeling of an adhesive tape [72]. One framework that attempts to provide some explanation for the ubiquity of power laws is the concept of self-organized criticality (SOC) introduced by Bak and coworkers [73]. The present situation appears to add one more example to the long list of a dissipative system driven to a critical state. In order to understand the intermittent behavior, these authors have also carried out 2-d simulations using N -edge dislocations gliding along the x-direction in the overdamped limit. Initially 1500 dislocations with equal number of Burgers vectors of both signs are introduced and allowed to relax to metastable configuration. A constant shear stress is then applied and the evolution of the configuration is monitored. Dislocation multiplication and annihilation processes are included by providing appropriate local rules that are implemented. Initially, the system relaxes slowly till it reaches a constant strain rate corresponding to a stationary creep situation that depends on the applied stress. Most dislocations are captured into metastable configurations of cell walls. But, once in a while one does find that some dislocations break free there after moving at large velocities only to be captured into another region of walls retaining the overall metastable structure. The distribution of energy associated with the abrupt movement of groups of dislocations has been calculated and shown to follow a power law with an exponent 1.8. Considering that these simulations are carried out in 2-d, the exponent value can be considered close to the experimental value. It is tempting to think that the dislocation activity in the bulk also forms a fractal set in view of the intermittent AE activity in the bulk coupled with the fractal nature of slip terraces. Further, considering the fractal dimension being 0.5 of one-dimensional slip terraces obtained by two different methods [54,61], one should expect that the fractal dimension

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corresponding to the intermittent collective motion of dislocations in bulk should be 2.5. Even though these studies only refer to spatial correlations with no dynamics involved, it appears likely that the intermittent dislocation activity is correlated both in space and in time. Extending their earlier studies on ice [66], Weiss and Marsan [74], use AE technique again, to investigate the clustering in space and time by using multiple-transducers. These transducers help them to map the hypocenters of acoustic waves generated by dislocation avalanches. The AE sources are located based on the arrival times much the same way as is done in study of fracture of rocks samples in seismology [64,75]. Thus, the method has the ability to provide the locations in three dimensions along with the time of occurrence of the event. In this experiment also ice is used. The creep of ice exhibits the primary, secondary and tertiary branches. Most of the signals are recorded in the tertiary branch. These authors first verify the power law distribution of the energies of the AE signals. (The exponent value is 1.6, same as in the earlier experiment [66].) The spatial correlation is studied using the correlation integral which is the number of points within a distance r defined by C(r) =

2 r − Ri − Rj , N(N − 1)

(3)

i<j

where is the step function, N is the number of spatial points considered and Ri is the coordinates of the AE sources. If the collection of the source points has a scale free structure, then the correlation integral is expected to scale as C(r) ∼ r ν , where ν is correlation dimension which is a lower bound of fractal dimension [76]. A log-log plot of C(r) as a function of r is shown in Fig. 8. It is clear that there is a scaling regime of nearly two orders in r with an exponent of 2.5 confirming the scale invariant structure of dislocation avalanches. This also suggests that the system is driven to the edge of criticality. The possibility of clustering of events both in space and time can be checked by computing the conditional probability of two events being separated by a distance r given that they are separated in time by n events, denoted by P (r|n). Here the index corresponding to the nth event is used rather than time. If the events are uncorrelated, this should be equal to P (r) which is the derivative of C(r). On the other hand, P (r|n) − P (r) should increase with decreases in n if an event triggers another event. A plot of P (r|n) is shown in Fig. 9 for n = 1–5 and n = 76–80. Weiss and Marsan [74] show that the likelihood of the occurrence of subsequent events increases following the occurrence of an event within its neighborhood. Thus, the analysis shows that closer the events are in space, closer they are in time as well, demonstrating the clustering property of correlated dislocation events both in space and time. 3.2.1. Characterization of PSBs and cellular patterns X-rays had been a major tool for extracting information about internal stresses till the advent of TEM. Since then, TEM has been extensively used to study both elementary dislocation reactions as well as dislocation patterns. These structures are recognized by the differences in the contrast patterns arising from entangled dislocation density. One of the limitations of TEM is that the samples must be thin enough for observation. For this reason, questions have been raised about the possibility of the changes in dislocation

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Fig. 8. A log-log plot of correlation integral of hypocentre locations. After J. Weiss and D. Marsan [74].

Fig. 9. Plot of the conditional probability that two events are separated in space by a distance r given that the events are separated in time by n (thick lines). Thin lines show the error envelope of standard deviation obtained from P (r) occurring by chance. Note the deviation of P (r|n) from P (r) for distances smaller than 20 mm. After J. Weiss and D. Marsan [74].

arrangements due to the very process of thinning. For instance, annealing of dislocation to the boundaries could occur. Moreover, as in situ observation of the deformation process is difficult, samples have to be unloaded for observation, which also rises questions about

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possible relaxation of dislocation networks occurring under unloading. These questions have been addressed quite early [77,78]. For instance, Essmann showed that by irradiating the sample, dislocation configurations can be locked [77]. TEM images have since been used for statistical characterization of dislocation patterns. Usually, TEM micrographs are preprocessed to ensure that artifacts of the procedure followed do not affect the results. These images are partitioned into boxes of size l after thresholding, which essentially converts the micrograph into black and white areas corresponding to dislocation rich and poor regions. Then, the number of black (or white) pixels in a box of dimension l of the digitized image are used for further analysis. In cases where dislocation pattern exhibits a range of length scales, or in situations where the range of length scales for the study is large, combining images of neighboring regions of the pattern would be desirable. Then, it is necessary to match the intensities (normalize the intensities) across two different micrographs of adjoining regions. This method has been effective in statistical characterization of scale free dislocation patterns [30,79]. The simplest statistical feature of any pattern is average size. In many dislocation patterns like the PSBs or the cell structure, one can easily identify the dislocation rich and depleted regions. However, boundaries may not have any particular shape as in the case of cellular patterns. Then, one can use some linear dimension (like square root of the area or the linear dimension along some chosen direction) as a measure of the size. Stereological information can be used to get the statistics in three dimension. Simple statistical measures such as the average do not require sophisticated methods even when one does not have good statistics. Thus, averages of cell size, mean channel size of the PSBs or average misorientation in cells were reported fairly early. However, with improved image processing techniques, micrographs of TEM can be used to obtain the distribution of the wall thickness and the channel sizes. The distributions of the PSBs and matrix structure are found to be peaked around the mean with that for the former being sharper than that for the latter. One other statistical quantity that can be easily calculated which yields useful information is the spatial autocorrelation function defined by C(R) =

n(r )n(r ′ ) − n(r )2 , n(r )2 − n(r )2

(4)

where n(r ) is the density (or intensity, black or white pixel at r ) of the micrograph. In general, if two spatial points are weakly correlated, the C(R) decreases starting from unity. (Note that the function is defined such that at R = r − r′ = 0, it is unity and goes to zero for large R as n(r )n(r ′ ) → n(r )2 .) In the case of the PSBs, say, for those obtained from single crystals of Cu deformed under cyclic deformation to saturation levels at room temperature, C(R) shows an oscillatory trend reflecting the periodic nature of the PSBs. The distance between successive peaks in C(R) is ∼1.7 µm which is close to the value one can obtain from a simple measurement of distance between the walls using a scale. In addition, the length scale over which the C(R) decays to half its value starting from unity reflects the thickness of the walls assuming the initial starting point is at the wall. In contrast, the matrix structure C(x) shows only one dominant hump which decays thereafter due to statistical dispersion [5]. In this case, the mean distance is about 2.4 µm.

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Fig. 10. Transmission electron micrograph of cell structure obtained from deforming Cu to 68.2 MPa oriented along [100] direction. After M. Zaiser and P. Hähner [80].

For the case of cell structure in Cu single crystals deformed at room temperature shown in Fig. 10, a visual inspection already suggests that there is no characteristic length scale, and cells of all sizes are likely to be present. In this case, it has been shown that the autocorrelation function decays slowly with r [5]. In fact, the distribution function is peaked at the origin and can be described a power law distribution P (r) ∼ r −α suggesting the selfsimilar geometry of the cell structure. In such a case, the average size is not representative of the structure. Indeed this might be one reason for large deviations in the mean cell size reported in the literature. Gil Sevillano and co-workers [81] were the first to recognize the possibility of fractal geometry of the cell structure based on the measurement of the fractal dimension of the microstructure of cold worked Cu. (This was in fact motivated by modeling dislocation lines gliding through a planar array of point obstacles. Their results suggest that gliding dislocation line has a fractal structure.) More recently, Hähner et al. [30,79] have carried out a detailed analysis of the fractal characteristics of dislocation cell structure. They consider cell structure in f.c.c. crystals deformed in tension in high symmetry oriented single crystals of Cu (100 and 111 directions). The TEM micrographs are preprocessed and converted to bit maps which are used for further analysis. Extensive investigations have been carried out on Cu single crystals oriented along [100] deformed at room temperature at a strain rate ∼5 × 10−5 s−1 to varying stress levels up to 75.6 MPa. There are several equivalent dimensions that can be defined for a fractal object [69], such as the box dimension DB , mass dimension DM , Minkowski dimension DMB and the gap dimension DG . The box counting method is the simplest wherein the set is covered with a box of length l and the number of boxes of size l containing at least one black pixel is counted. For the

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Fig. 11. Size spectrum of dislocation-free regions of the cell pattern in a [100]-oriented f.c.c. Cu-1.4%Mn single crystal deformed to τext = 58.2 MPa reduced stress. After M. Zaiser and P. Hähner [80].

sake of obtaining consistency in the estimation of the dimension, these authors calculate all these dimensions and show that estimates of all these different ways of calculating the dimensions give consistent answer for Cu single crystal deformed to 75.6 MPa; the dimension is DF = 1.79 ± 0.04. The scaling regime is typically about one and half orders and improves if a larger image is used as in the case of the CuMn alloy is shown in Fig. 11. The analysis also shows that the fractal dimension is insensitive to the orientation though the cells are elongated along the direction of the tensile axis. However, the fractal dimension appears to depend on the flow stress in a nonlinear way for small values of the flow stress but depends linearly for large values. Gil Sevillano et al. have reported fractal cell patterning of cold rolled Cu polycrystals up to strains of the order of unity beyond which they find well defined large cell size. The observation of the dependence of fractal dimension on flow stress has been confirmed by Székely et al. [82]. The fractal dimension of the cellular pattern in 3-d is just DF + 1. 3.2.2. In situ dynamics of dislocation patterns Conventionally, two complimentary tools used for studying dislocation patterns are line profile analysis of X-ray diffraction patterns and electron microscopy. While EM provides detailed information of dislocation structure, it is essentially static as well. Moreover, the patterns observed in EM are not representative of the true bulk situation as the method needs thin samples whose dimensions are much smaller than the grain or the pattern size that is being studied. This often leads to stress relaxation due to migration of dislocations to the surfaces. On the other hand X-ray line profile can, in principle, give information of the bulk dynamics, but the results are averages over grains and orientations. As a result, the basic questions of how dislocation patterns form and at what point of deformation, have remained unanswered. Recently, Jacobsen et al. [83] have designed a novel method of X-ray diffraction that gives insight into the underlying dynamics leading to dislocation patterns. The experiment essentially consists of a diffraction set up with two area detectors A and B that have resolution on a grain and subgrain level, respectively. (The authors use subgrain to the cell interior which is not a standard terminology. We shall use cell interior.)

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The source is a highly penetrating synchrotron beam with a narrow energy and angular spread with dimension smaller than the cell dimension. The detector A has a resolution of ∼0.02 Å−1 and that of B much smaller ∼0.0005 Å−1 . The orientation of the lattice can also be measured. Using this, the dynamics of individual subgrains are followed during an in situ tensile deformation of thin films of 99.99% pure Cu. This tomographic methods has a capability of illuminating a volume of 8000 µm3 of the sample. The pattern formed is similar to the one shown in Fig. 10. The samples are strained 3% to 4% linear dimension in steps of 0.04%. The method relies on obtaining a three-dimensional reciprocal space map originating from a cell interior volume using the detector B by rotating around the x-axis. An analysis of the data shows that the reflections obtained from the high resolution map comprise a set of individual peaks and thus the cell interiors are essentially dislocation free. Further, these regions are separated by dislocation rich regions. The estimated cell interior volume fraction is consistent with that obtained by TEM. Under increasing stress, the cell sizes decrease. The changes in the pattern are usually thought of to be through cell splitting. However, the dislocation free regions witness very interesting intermittent dynamics. The peaks appear and disappear during the course of deformation. The interpretation these authors provide is that dislocation free regions emerge out of sea of dislocation rich regions and vanish during the course of the deformation. As for the question of onset of the pattern, they state that the cell formation is initiated early in the plastic deformation. Finally, the pattern does not change in time when the strain is held fixed. While the intermittent dynamics already forces theorists to rethink about the approach that needs to be taken, more detailed experiments are necessary to throw more light on the dynamics.

4. Recent advances in statistical description of dislocation dynamics 4.1. Introduction to statistical and stochastic approaches Plastic deformation cannot be an exception to the principles of statistical physics that all macroscopic laws should be derivable starting from the microscopic laws governing the constituent elements. The obvious method for computing the macroscopic averages is to solve the equations of motion, which however is impracticable as the number of dislocations is huge. The subject of statistical physics addresses this connection between the microscopic and macroscopic laws without having to solve a large number of equations. The question is how much of this formalism that is well developed can be borrowed for description of plastic flow? In statistical mechanics, it is usually possible to associate a microscopic variable with a macroscopic measured property, for example – energy, magnetization etc. This is not always true; a simple example would be entropy. While these quantities are highly fluctuating at the microscopic level, the equivalent macroscopic quantities are very much reproducible under repeated measurements given identical macroscopic conditions. This means that nature performs some kind of averaging that smoothens these fluctuations. This principle of averaging is at the root of establishing a connection between microscopic and macroscopic

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quantities. Let ( q , p) ≡ ( q1 . . . qN , p1 . . . pN ) to be a point in the phase space corresponding to a microstate of the system with ( qi , pi ) representing the coordinate and momentum of the ith particle. Consider some macroscopic quantity M(t) where t refers to the time of measurement and let m( q , p, t) represent the corresponding microscopic quantity which necessarily depends on ( q , p). For simplicity, consider a system which can be described by classical mechanics. The appropriate weight factor that accounts for different microstates of the system is supplied by the phase space distribution function f ( q , p). Then,

m(p, q, t) = M(t) = dp d q m( q , p, t)f ( q , p). (5) Thus M(t) is a linear functional of m( q , p, x, t). If m =1, then we get

dp d q f ( q , p) = 1.

(6)

Since the microstate should be somewhere in the phase space, we have f ( q , p) 0.

(7)

Then, the interpretation of f ( q , p) is clear. It refers to the probability of finding the system at ( q , p) in the phase space. Now if we are interested in time dependent properties, a natural starting point is the Hamiltonian of the entire collection of the particles. Given this, we know that the time evolution of a point in the phase space is governed by the laws of mechanics, namely, Hamilton’s equations of motion ∂ H ( q , p) d qi = ; dt ∂ pi

∂H ( q , p) dpi =− , dt ∂ qi

i = 1 to N.

(8)

Then, the time development of m can be thought of as arising from the changes in the distribution function f ( q , p) resulting from the evolution of the ( q , p) (or through the dependence of m on the evolution of ( q , p)). The change in f ( q , p) is obtained by taking the Poisson bracket of f ( q , p) with H (p, q). Now let us examine the possibility of adopting this approach to plastic flow. We first note that dislocations, carriers of plastic flow, are line defects in the crystalline medium, the latter only playing a passive role. Thus, the equations of motion of the atoms or molecules are clearly not directly relevant. But, dislocations are not thermodynamics defects. This immediately implies that the Hamiltonian picture, even if one can define such a quantity for the system of dislocations, has to be abandoned. Instead, we can start with the equations of motion of dislocations and their interactions. As stated earlier, the interaction of dislocations, both through the long ranged stress fields and the short ranged interactions, is mediated by the crystal, which by itself is passive. Moreover, the number of dislocations is not conserved and thus, instead of probability density, one needs to consider a distribution function of the coordinates, velocities and other attributes like the Burger vectors and line elements of all the dislocations. Finally, in this case, it appears that all macroscopic measurable quantities do not necessarily have their microscopic equivalent, although, in some

§4.1


107

cases it may be possible like strain rate, strain etc. However, there are other local variables that are unique to plasticity which are important. Despite the limitations expressed above, the most natural approach would be to use a joint distribution for all the dislocations. Indeed, this is the approach adopted by Groma and his collaborators [82,84–86]. However, within the above framework, there is no consistent way of including the nonconservative nature of dislocation multiplication and annihilation. As mentioned in the introduction, an alternate method is the stochastic approach, often adopted for analysing the effect of fluctuations in driven systems [87–89]. This method is applicable to equilibrium situations as well, but is not restricted to thermal fluctuations. In this sense, the stochastic approach can be adopted as several dislocation processes are athermal. But the price one pays is that it is phenomenological. In the absence of a more fundamental theoretical framework, the phenomenological approach can prove to be advantageous as the methodology imposes no limitation in dealing with nonconservation of dislocations. There are several ways of treating a problem depending on the physical situation. One standard method is to regard the underlying process to be Markovian and to set up a master equation in the state space of the system with transitions between the allowed states. The transition matrix between the states is set up based on physical considerations. Indeed, this approach has proved useful in nonconservative situations such as birth and death processes [87]. Another stochastic method that is useful is the Langevin approach in which the variable under question is assumed to evolve in a stochastic manner, i.e. the variable is subject to fluctuations over and above the mean. To understand the physical origin of the fluctuating nature of the variable, consider the prototype situation of the motion of a Brownian particle in a fluid. This erratic motion of the Brownian particle arises due to the collisions of the particle with the molecules of the fluid. Indeed, the thermalization of the particle is through these collisions as the molecules themselves are in thermal equilibrium corresponding to a temperature T . The effect of the rapid collisions of the particle with molecules of the bath is then treated as a fluctuating force. On the other hand, random collisions driving the motion of particle produces the viscous drag when averaged over time. For this reason, the dissipation experienced by the particle due to viscous medium and the fluctuations are not independent and have to obey the so-called fluctuation-dissipation relation. The above discussion also means that the mean velocity of the particle evolves slowly under the influence of the average effective force acting on the particle. Thus, for a general situation, one prerequisite for using a Langevin approach is that the order parameter variable(s) should have a substantially slow evolution compared to the random noise. In fact, the usual assumption of white noise statistics is meant to mimic these rapid uncorrelated collisions. It may be worthwhile to state here that in the case of a Brownian particle, one can derive the Langevin equation starting from the total Hamiltonian of the system consisting of the particle and the set of heat bath oscillators by eliminating the fast degrees of freedom corresponding to heat bath oscillators. However, such an exercise is not possible in the case of plastic flow. As mentioned earlier, both thermal and athermal fluctuations arise in plastic deformation. However, during plastic deformation, as nearly 90% of the work done on the system is lost in the form of heat, there is no compelling reason to respect the fluctuation-dissipation relation. In view of this and the discussion above, the Langevin

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approach is well suited for dealing with athermal fluctuations as long as the separation between the fast and the slow time scales can be substantiated. As we shall argue, this condition appears to be well satisfied. The origin of the noise is attributed to the significantly fast time scales of a large number of degrees of freedom. However, the strength of the noise itself may remain a parameter or must be determined by other considerations. There could be several microscopic dislocation mechanisms that can contribute to the noise and it may not be easy to model its statistical properties starting from microscopic mechanisms. Let us briefly outline the Langevin approach. We assume that the mean equation for the variable under question is known from some consideration and the statistical properties of the noise are given. Then, the Langevin equation has a form X˙ = F (X, t) + f (X)ξ(t).

(9)

Here, the statistical properties of the ‘driven random variable’ X are to be determined once the statistical properties of the ‘driving noise’ ξ are given and the nature of the calculus to be used is also supplied. Usually, one uses white noise with a zero mean assumption, namely ξ(t) = 0;

ξ(t1 )ξ(t2 ) = 2Dξ δ(t1 − t2 ).

(10)

The mean equation itself can be nonlinear in the order parameter variable X. Here, Dξ is the strength of the noise, not necessarily of thermal origin. In eq. (9), the noise is multiplicative, meaning that the magnitude of fluctuations depends on the value of the state variable through the function f (X). The noise need not be white, it can be coloured.4 When f (X) = 1, the noise is additive, namely, the magnitude of fluctuations is independent of the value of the state variable. It is also clear that when the equation of motion is nonlinear, one will have to deal with a hierarchy of moment equations and hence, it is usually difficult to obtain a closed form solution for the probability distribution of the random variable X. In principle, the same procedure can be adopted in the case of a set of coupled dynamical equations, i.e. noise components could be added to the coupled set of order parameters equations. In such cases, the nature of the solution can change abruptly as a control parameter crosses a critical bifurcation value. Then, near the point of bifurcation, fluctuations exhibit features observed near a critical point. This means that one will find critical slowing down and related phenomenon [90]. The above approach can be extended when explicit spatial coupling needs to be considered. A variant of this approach is the Ginzburg–Landau approach where the equation for the order parameter is assumed to be derivable from a free energy functional. In this case, the equation of motion reads ∂F (X, t) + ξ(t). X˙ = − ∂X

(11)

4 It is possible to drop the white noise restriction and use correlated noise, also called colored noise. See Ref. [88].

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109

In condensed matter situations, even when the system is driven away from equilibrium, one can envisage a free energy functional on the basis of multiplicity of the allowed steady states. However, a rigorous derivation of the free energy is not always possible. Recently, this approach has been taken by Onuki [91] to study dislocation patterns. This approach, though not popular in plasticity, may find use in future.

4.2. Early statistical models of dislocation dynamics Fluctuations inherent in equilibrium systems are of thermal origin. However, in driven systems, particularly systems far-from-equilibrium, fluctuations can be both thermal as well as athermal. This is particularly so in the case of plastic deformation. Athermal fluctuations are seen in other systems as well, one that is well studied is granular materials [92]. Apart from standard reasons that can be ascribed, contributions to fluctuations in the case of plastic deformation can arise due to several dislocation mechanisms themselves. For instance, multiplication of dislocations depends on spatial location, and similarly short range interaction of dislocations such as annihilation, formations of locks etc., also fluctuate. Thus, in principle, local quantities such as strain rate, local stress, etc. fluctuate. Studies on the importance of fluctuations in the context of a simpler mechanical system (compared to plasticity) where the mechanical response is controlled by point defects, have been demonstrated in a number of situations like the Snoek relaxation [93], Gorsky relaxation [94], etc. This was a theme that was pursued well at the Kalpakkam school in the 80s [95]. The inevitability of statistical description of dislocations has been recognized for a long time [16,17,96–99]. Early efforts were not so much concerned about addressing the collective behavior of dislocations, but with a recognition that several physical quantities in plastic deformation, such as the internal stress, strain rate etc. are inherently fluctuating.5 Of these, the work of Lagneborg and his collaborators [17], that of Stout [99], and Ananthakrishna and Sahoo [16] were directed at formulating a statistical approach to dislocation dynamics. While these approaches are quite different, the measured macroscopic quantities are treated as averages over an appropriate distribution function. The work of Stout recognizes that local strain rate is a fluctuating variable. He derives a dislocation balance equation in the presence of dislocation multiplication and loss that has an integro-differential form similar to the Boltzmann equation. The interaction of the subset of dislocations with all other subsets is bilinear in the distribution function with a kernel that represents the nature of dislocation-dislocation interaction. The full form of the kernel obviously depends on the various dislocation mechanisms which have not been addressed in the paper. The author obtains a solution for the distribution function in the absence of dislocation multiplication and annihilation. The approach illustrates that a full fledged description of dislocation dynamics even for a homogeneous case is far from easy, but it does address some of the issues relevant for a statistical description. Lagneborg and collaborators [17] recognize the necessity of defining immobile and mobile dislocation densities and their evolution during plastic flow. The strain rate is con5 While some of these early attempts summarized here already have the seeds of a stochastic framework, they have largely gone unnoticed.

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sidered as being controlled by the rate at which dislocation segments surmount the local obstacles by stress-assisted thermal activation. They also recognize that the waiting time is much larger than the flight time. According to this picture, mobile dislocation segments are those that have segment length l > αGb/σ with those less than this value being considered immobile. Here σ is the applied stress and α is the strength of the obstacle. Then, ∞ αGb/σ ρm = αGb/σ lφ(l) dl and ρim = 0 lφ(l) dl. The distribution of line lengths is determined by ¯ − σint (l)) b ∂φ(l, t) Ha − bA(σ . = −νD exp − ∂t l kT

(12)

Here, l is the link length, νD is the Debye frequency, Ha is the total activation enthalpy of controlling barrier, A¯ is true activation area and σint is critical internal stress necessary to overcome the elastic interaction. This equation needs modification as the total dislocation density evolves with strain which is then accounted for. The resulting evolution equation is solved numerically and applied to a few cases with good agreement with experiments. The work has been later extended also. 4.2.1. Stochastic description for dislocation dynamics: A Fokker–Planck like equation The work of Ananthakrishna and Sahoo [16] also addresses the homogeneous deformation with a view to explain certain experimental observations on some simple materials like LiF. Careful experiments by Gilman and Johnston [100,101] demonstrated that the velocity of dislocations in strained LiF crystals decreases by a factor proportional to the total density of dislocations in the sample (as formation of dipoles is the cause of linear hardening). This means that there is a feed back mechanism that opposes the applied stress, i.e. a back stress proportional to the total density of dislocations. While the origin of this back stress was physically well known, a proper understanding based on statistical approach was lacking till then.6 A natural approach to this issue is clearly a statistical description of the dynamics of dislocations. Here, we deal with this formulation in some detail for two reasons. First, the methodology illustrates a stochastic approach to dislocation dynamics even for a simple homogeneous situation is already sufficiently complicated that involves a nonlocal integropartial differential equation. Second, this approach forms the basis for building a model for the Portevin–Le Chatelier effect which has succeeded in explaining several generic features of the PLC effect. Starting from a distribution function-theoretic approach, one can derive a creep law applicable to materials like LiF. Using velocity of dislocations as a random variable, a Fokker–Planck like equation for the distribution of dislocation velocities is written down. Consider all segments of dislocations having a velocity between v and v + v. Then the equation of motion for the distribution function is set up based on the known dislocation mechanisms valid for LiF [100,101]. In this case, only dipoles are formed. Other mechanisms considered are multiplication of dislocations and immobilization due to solute atoms or other pinning centers. Then, the continuity equation for the distribution 6 Clearly this feature can only arise due to modifications occurring in the distribution function during deformation. This is perhaps one of the few examples where the mathematical origin of the back stress has been demonstrated explicitely. See also [102] for the square root law dependence of the back stress.

§4.2


function ρ(v, t) of dislocation line segments having a velocity v is given by ∂ρ(v, t) Q ∂ρ B0 v bσa ∂ ρ(v, t) + +S = − ∂t ∂v m m m ∂v ∂ρ(v, t) ∂ (βv − f )ρ(v, t) + Dv + S, = ∂v ∂v

111

(13)

Here b is the Burgers vector, B0 is the drag force, bσa is the force per unit length, m the mass per unit length, and Dv (or equivalently Q) is the diffusion constant in the velocity space. The source term has the following contributions:

∞ ρ(v − v ′ )ρ(v ′ ) dv ′ S = (−α + θ v)ρ(v, t) − μ +h

∞

−∞

−∞

ρ(v − v ′ )

∂ρ(v ′ ) ′ dv . ∂v ′

(14)

The first term corresponds to a loss term due to the immobilization of dislocations due to solute atoms or other pinning centers, the second term is the production of dislocations due to cross-glide, the third and the fourth terms correspond to the interaction of dislocations resulting in either the annihilation of dislocations or formation of dipoles. Consider two dislocation segments with velocities v ′ and v ′′ involved in the interaction process. Then, the probability should be proportional to the product of the distribution functions for these two subsets, i.e. ρ(v ′ , t) and ρ(v ′′ , t). As the interaction leads to a reduction in the velocity of the combined entity (which in this case refers to the possibility of forming a dipole), one can write v = v ′ + v ′′ − v. The contribution to the rate of change of the distribution function is then given by integrating over all values of either of the velocities subject to the above constraint. This results in the last two terms in eq. (14) after expanding and retaining the leading order terms. For more details we refer the reader to the original paper [16]. In reality, one should construct equations of motion for the distribution functions of the positively and negatively signed dislocations ρ ± (v, t). Setting up such equations poses no additional problems at all (these would have a form similar to those derived by Groma and coworkers). However, obtaining the solutions for such a set of coupled integro-partial differential equations would be very difficult (in fact, one should introduce one more equation for the dipole population as well). For this reason, the distinction between the positive and negatively signed dislocations is ignored, i.e. ρ + (v, t) = ρ − (−v, t). The above equation can be solved using natural boundary conditions, i.e. ρ(v, t), vρ(v, t), ∂ρ(v,t) ∂v → 0 as v → ±∞. The total number of dislocations, N , is not conserved and is given by ρ(v, t) dv = N(t). This approach unifies several phenomenological relations introduced in the literature in different contexts. The following relations follow immediately. dN = (−α + θ V )N − μN 2 , dt dV + (β + μN)V = f − hN + θ v 2 − V 2 , dt

(15) (16)

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where V is the average velocity v. The second relation, with some approximation determined by the relative magnitudes of the various parameters, gives an expression for the back stress in the steady state: V =B−

h N = b(σa − σint ). β

(17)

Here B = f/β. In this case, the back stress is linear in the density as measured in experiments [100,101]. These two equations are adequate to obtain a creep law. This creep law is similar to that obtained by Webster [103]. The adjustable parameters are h and μ with the values of other parameters being known from experiments (for instance θ ∼ 30 cm−1 , B = f/β ∼ 3.03 × 10−3 cm s−1 ). The value of h/β has been fixed using the data from the velocity reduction due to hardening as measured in experiments [100,101]. Other parameters are obtained from the fit with the creep curve). The calculated creep curve fits the experiments very well as is clear from Fig. 12 (for more details see Ref. [16]). Although eq. (13) appears to be complicated, one can obtain exact expressions for the first four cumulates given by k1 = v = V =

h (f − hN) = B − N, β β

(q − hNV ) h k2 = v 2 − V 2 = = 2Dv − N V , β β h k3 = − N v 2 , β

(18) (19) (20)

Fig. 12. Calculated creep (continuous line) and experimental (•) creep curve. The influence of the parameter h/β is also shown. N0 = 7.5 × 104 cm−2 , Ns = 8.4 × 106 cm−2 , μ = 0, α/βθ = 0.02.

§4.2


h k4 = − N v 3 . β

113

(21)

Apart from the fact that the nature of the distribution is clearly non-Gaussian, the distribution function has interesting features. As the third cumulant is negative, the leading edge is sharper than the trailing edge. Further, as k4 is also negative, it has a platikurtic nature. From the four relations, it appears plausible that the central moments kn , n > 4, may have the same behavior as k3 and k4 , i.e. kn = −(h/β)Nv n−1 . If we assume this, we can obtain closed form expressions for the characteristic function of the distribution [16] which satisfies nearly the same equation as that for eq. (13). One interesting aspect of this distribution function is that it becomes sharper with time, i.e. k2 decreases as a function of time, a feature that mimics increasing hardness with time. The above description in terms of a distribution function also provides a natural tool for quantifying mobile dislocation density. Formally writing ρ(v, t) = ρim (v, t) + ρm (v, t) where the subscripts stand for the immobile and the mobile components respectively, we obtain NV = Nim Vim + Nm Vm , where vm = Vm and vim = Vim are the average velocities of the mobile and immobile components respectively. Since Vim is expected to be small, we have, NV ≈ Nm Vm . By reformulating the above equations, we can account for the production of immobile population consisting of dipoles and dislocations arrested at pinning points (see [104] for details), we obtain N˙ m = θ Vm Nm − μNm2 − αNm ,

N˙ im = kμNm2 + αNm − μNm Nim ,

(22) (23)

where Nim is the total density of dipoles, and k is a constant representing the fraction of dipoles formed in the term representing the collision integrals in eq. (14). (This is equivalent to hNs /f . The value of k that fits the creep data is k = 0.978 which is close to hNs /f used in Fig. 12.) The last term in eq. (23) is introduced to account for the release of a dislocation from a dipole due to its collision with a mobile dislocation. These two equations can be rewritten in terms of the mobile fraction which in turn can be used to obtain the creep curve, which fits the data equally well as the earlier description (in terms of N and V ). Thus, the two descriptions are equivalent. In the later picture, the entire time dependence of the flow is controlled by the mobile dislocation density Nm with Vm changing only weakly with time except near the point of inflection [104]. The procedure adapted is not restrictive and can be generalized to other more complicated situations such as the forest hardening if the governing mechanisms are known. For instance, the hardening law 1/2 in the case of Si like materials has been derived and has a back stress of the form Nim . Interestingly, the creep curve in this case can be cast into a universal curve independent of the applied stress and temperature [102,105], a feature observed in experiments. The above model is only suitable for simple creep situations. Now consider generalizing this model to a more complicated situations like the PLC effect for which other dislocation populations are involved, for instance the Cottrell density. For the PLC effect, dynamic strain aging [35] is an important modification that needs to be included. In addition, we take a less restrictive interpretation that the interaction of two mobile dislocations leads to immobile dislocations which is assumed to represent the formation of forest dislocations.

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However, the rate of formation of the immobile population should in reality be Nim Nm instead of Nm Nim . Yet for simplicity, we shall retain the latter. While this is an over simplification, it is adequate to capture the essential feature of the instability mechanism in the PLC effect as we shall show in the next section. Eqs (22) and (23) then be modified to N˙ m = θ Vm Nm − μNm2 − μNm Nim + λNim − αNm ,

t N˙ im = kμNm2 − μNim Nm − λNim + K(t − t ′ )Nm (t ′ ) dt ′ .

(24) (25)

−∞

The above equation for the immobile component includes an important generalization, namely, the inclusion of dislocations with clouds of solute atoms [35]. The last term in eq. (25) represents the process of immobilization of mobile dislocations due to solute atoms. To complete the modeling for the PLC case, we have included thermal or athermal reactivation of immobile dislocation, i.e. −λNim which is an incoming contribution to N˙ m . Now considering the nature of the last t term in eq. (25), we can define an equation for Cottrell type of dislocations using Nc = −∞ dt ′ Nm (t ′ )K(t − t ′ ). Using a single time scale α ′ to represent the rate of slowing of mobile dislocation due to the aggregation of solute atoms, we can use α exp α ′ t. Then, differentiation of Nc gives N˙ c = αNm − α ′ Nc .

(26)

From eq. (26), it is clear that α refers to the concentration of the solute atoms which participate in slowing down the dislocations.7 Then, we have N˙ m = θ Vm Nm − μNm2 − αNm − μNm Nim − λNim ,

N˙ im =

μNm2

′

− μNm Nim − α Nc − λNim ,

N˙ c = αNm − α ′ Nc .

(27) (28) (29)

This completes the model equations for the PLC effect which will be dealt with in detail in the next section. 4.2.2. Langevin approach to fluctuations during yield drop Most experiments measure only the average response of a specimen that is being deformed. In view of the fact that fluctuations should be inherent during deformation, a natural question arises – what is the role of these fluctuations and how do we determine the influence even when the deformation is homogeneous? Here, we recall that it is well known that the maximum rate of production of dislocations coincides with the yield drop. Naturally, this must manifest itself in some correlation function. In addition, one also knows that the motion of dislocations is jerky, being pinned and unpinned as the threshold stress for unpinning is reached. The latter manifests itself in the form of acoustic emission [106]. Experiments that monitor the acoustic activity during deformation show a pronounced 7 Here we have presented details of modeling these equations as they have not been explained in the original reference [107,108].

§4.2


115

maximum in the AE signal amplitude that coincides with the yield drop. As AE signals have a high degree of fluctuations, the feature may be taken as a signature of the enhanced levels of fluctuations. The pinning-unpinning of dislocations in the presence of noise with the existence of a threshold for unpinning, falls into the class of interesting systems studied in stochastic processes. These studies show that enhanced levels of fluctuations should be seen in systems that go from an unstable state to a stable state [109]. Following this approach, one can study the nature of fluctuations during a yield drop. Indeed, such a study has been carried out by Valsakumar et al. [18]. Consider materials like GeSi which display a sharp yield drop [102,105]. For such materials, the equation of motion for dislocation density coupled to the machine equation is given by dN = V N − b1 N 3/2 , dt dσa = K[˙ǫa − bV N]. dt

(30) (31)

Here, V is the velocity of dislocations, taken to be V = V0 (σa − hN 1/2 ) which is valid for materials like GeSi. One standard method of investigating fluctuations is through the Langevin approach by assuming that there are fluctuations over and above the mean. The noise term, in principle, can be additive or multiplicative. In the context of plasticity, the physical interpretation of the above two equations is that they are to be regarded as local variables, in particular, the noise term for the stress arises from local fluctuations in the internal stress field. To illustrate the enhanced levels of fluctuations inherent to such equations, it is adequate to use an additive noise. As such, eqs (30), (31) are not suitable for investigating fluctuations as these equations involve nonanalytic functions of the density variable [87]. Therefore, we first transform these two equations by using l = N 1/2 and then use additive noise terms for each of these variables.8 The statistical properties of the noise are taken to be white with zero mean and no cross-correlation between the two noise components in the two equations. Conventionally, the strength of the noise is taken to have an inverse square root dependence on the system size which in this case was 5 × 10−7 [87]. It may be possible to estimate the strength of the noise terms along the lines suggested by Hähner [110]. Note that these coupled set of Langevin equations give rise to a hierarchy of moment equations. Thus, some decoupling scheme needs to be effected. The analysis is carried out using both Gaussian decoupling and Monte Carlo methods. The averages are carried out over a large number of hysteries. One possible interpretation for these different hysteries is that they represent fluctuations sampled at various spatial locations of the sample. The results of the Gaussian decoupling agree very well with those obtained using Monte Carlo method. For details we refer the reader to the original paper [18]. Here we present the principal results and discuss the implications. The average values of σa and l or equivalently N , are well reproduced (i.e. their behavior is the same as that obtained from eqs (30), 8 Choice of improper random variables is known to lead to considerable technical problems. This transformation avoids such a possibility. See Ref. [87].

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Fig. 13. Anomalous stress fluctuations during an yield drop.

(31)) which is as should be expected. However, fluctuations grow rapidly reaching a maximum, coinciding with the time at which the yield occurs. In addition, the magnitude of fluctuations at the yield point is almost four orders larger than the initial value. This is unusual as large fluctuations are not usually found in near-stationary systems or near-equilibrium situations. However, we note that this is a general result, i.e. anomalous fluctuations manifest in all systems which pass from an unstable state to a relatively stable state [109]. Thus, such fluctuations should manifest in situations where dislocations are unpinning as they reach the threshold stress.9 Fig. 13 shows σa2 − σa 2 . Such anomalous fluctuations should be seen in scattering experiments. Indeed the attenuation of ultrasonic waves during a yield drop experiments does show a marked behavior. Our interpretation is that this is a consequence of anomalous fluctuations arising from rapid multiplication of dislocations around the upper yield point and not merely due to the change in link length. This method can in principle be used to provide a definition of an yield point in view of the fact that there is no agreed value that should be used as an offset strain. In this method, one can choose the value of strain at which the fluctuations exhibit a maximum. 4.3. Recent statistical approaches to collective dislocation dynamics 4.3.1. A stochastic approach to dislocation dynamics The Langevin equation discussed in the last section was introduced more from physical considerations. Let us now examine the experimental evidence in support of separation of time scales in plastic flow to see if it is meaningful to take a Langevin approach to 9 In spirit, this is somewhat similar to idea proposed by Hähner [110] with an important difference, namely that in his formalism, the magnitude of fluctuations is controlled by strain rate sensitivity.

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dislocation dynamics. As pointed out in the Section 3.1, early studies on f.c.c. metals have shown that slip in these materials occurs in a localized manner (with a typical active slip volume ∼10−10 m3 ) with several dislocations moving coherently in the form of avalanches [6]. The local shear strain rate attributable were found to be nearly six orders larger than the applied strain rate. Similarly, the duration between the occurrence of the slip-line bundles exceeds six order of magnitude. Thus, there appears to be a clear separation of time scales between the duration of occurrence of the slip-lines and the time scale associated with the applied strain rate. This suggests that there are internal processes that occur at considerably faster time scale than the macroscopic measurable quantities such as the mean shear strain rate, shear stress etc. This, as we have seen, is a prerequisite for describing the evolution of the system as a Langevin process. One other feature of dislocation motion that is important from the stochastic point of view is the intermittent jerky motion of dislocations during which they undergo repeated pinning and unpinning, the latter occurring when the threshold for unpinning is reached. Once unpinned, dislocation segments move in bursts and get arrested at subsequent locations. This actually implies that, for most of the time, dislocations are at rest. As discussed in the previous section, one should expect anomalous fluctuations when dislocations are on the verge of the threshold [18,109]. (Note that at any given time there will be a certain proportion of dislocation configurations which will be near the threshold of unpinning.) In other words, the entire process of pinning and unpinning is a typical unstable system and such systems are subject to anomalous fluctuations [18,109]. Thus, the jerky nature of dislocation motion is suggestive of a high degree of fluctuations in the velocities of dislocations. Indeed, numerical simulations [111] show that there is a large dispersion in the distribution of dislocation velocities. There is also evidence from the numerical integration of the equations of motion wherein dislocation interaction is mediated by long range stress fields. This shows high degree of fluctuations in the internal stress [112]. As pointed out earlier, the magnitude of fluctuations is not determined. The first attempt to estimate this is due to Hähner [110]. In the following, we briefly review this approach. The basic idea underlying this approach is that dislocation interactions at the microscopic length scales lead to fluctuations in the internal stress. Again, the physical basis for this can be found in the separation of the relevant time scales; the time scale corresponding to the macroscopic evolution of dislocation structures is much longer than the short time scale over which the mobile dislocation segments interact with the local internal stress field, and which in turn is determined by the local dislocation configurations. To begin with, we recall that the effective shear resolved stress τeff acting on a gliding dislocation is equal to the balance of external applied shear resolved stress τext and the internal shear resolved stress τint , i.e. τeff = τext − τint .

(32)

During glide, a dislocation interacts with the surrounding dislocation configurations and consequently experiences a highly fluctuating environment both in space and time. Thus, δτeff = −δτint .

(33)

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To understand the influence of these fluctuations, consider the following average for a dislocation gliding in a slip system with a velocity v that also fluctuates. Then, vτint =

1 t

0

t

dt vτint =

1 x

x 0

dx τint = 0.

(34)

The specific assumption here is that the time average . . . is taken over a sufficiently long time such that the gliding dislocation samples sufficiently large number of dislocation configurations. Note that while the spatial average of τint is zero, the temporal average τint = 0 as the dislocation spends most of the time at points of high back stress. The above equation can be generalized to mesoscopic length scales with the assumption that the average work done by the internal stress field on moving dislocations is zero (in the absence of dislocation multiplication and inertia) or equivalently the work done by external stress on the system being completely dissipated into heat. Then for the plastic shear rate, we have γ˙ τint = 0

or γ˙ τeff = γ˙ τext .

(35)

This assumes that the correlation length ξ of the fluctuations is much larger than the dislocation spacing. Using eqs (32), (33), (35) we get δ γ˙ δτeff = γ˙ τeff − γ˙ τeff = γ˙ τint .

(36)

The autocorrelation γ˙ τeff is then obtained using the definition of strain rate sensitivity S=

∂τeff , ∂lnγ˙

(37)

which gives

and

τint γ˙ 2 , (δ γ˙ )2 = S (δτeff )2 = Sτint .

(38)

(39)

As S can be regarded as the response function, eqs (38), (39) may be interpreted as “fluctuation-dissipation relations of plastic flow” as they connect local fluctuations in strain rate, effective stress and mechanical dissipation [110,113,114]. Clearly, the strain rate sensitivity plays an important role in these relations which does help to rationalize the difference in the mechanical responses of the weakly rate-sensitive f.c.c. materials from that of the strongly rate-sensitive b.c.c. materials. For b.c.c. materials deformed below the transition temperature, S is large which implies high stress fluctuations and low strain-rate fluctuations; the latter can be interpreted as homogeneous flow. In contrast, for the weakly

§4.4


119

rate-sensitive f.c.c. materials, strain-rate fluctuations are large which implies inhomoge2 neous flow. (For instance, one can estimate δγ˙γ˙2 = τSint to range from 10 to 1000 for the f.c.c. materials which can be interpreted as collective dislocations movement.) It is useful to translate these results into velocity fluctuations. When the velocities of dislocations are limited by electron and phonon drag, we can use v = bτeff /B, where B is the drag coefficient. Using this, we get the equivalent of eq. (38) to be (δv)2 b2 ρτint = . Bγ˙ v2

(40)

Here, we have used the Orowan equation. An order of magnitude estimate of this can be found by using the values of local dislocation density, strain rate and B. Using some typical numbers for B ∼ 10−5 Pa s, ρ = 1012 m−2 , b = 2.5 × 10−10 m and strain rate γ˙ = 2 10−3 s−1 , along with τint ≈ τext ∼ 5 MPa, one gets (δv) ∼ 107 . This can be compared v2 with the results of slip-line cinematography on α-brass by Neuhäuser [6] which show that the local shear strain rates are six to seven orders of magnitude higher than the applied strain rate in the range which can be regarded as an estimate of the relative fluctuations of local to global velocities. Direct measurement of dislocation velocities are difficult in experiments, but one can estimate these quantities in simulations. For example, one can compare the above prediction with the recent simulations by Devincre and Kubin [111] to check if the prediction of eq. (40) is validated. In their simulations, they use B = 5 × 10−5 Pa s, and γ˙ = 50 s−1 , the flow stress was 6 MPa. Using their figures gives a factor 56 which compares well with that obtained from eq. (40).

4.4. Distribution-theoretic approach to collective effects of dislocations 4.4.1. Introduction In plasticity, defects contributing to plastic flow are dislocations (and point defects which we shall ignore here). Thus, at the microscopic level, the most natural starting point is the motion of dislocations under the influence of external forces and other mutually interacting dislocations. As the macroscopic quantities are averages over these equations of motion, an averaging procedure that eventually connects the microscopic properties to the macroscopic properties can be obtained in principle. From a statistical mechanics point of view, an appropriate quantity for averaging is the distribution function of N dislocations. As stated in the introduction, while the general approach is known in the context of condensed matter physics, in the case of plasticity, two issues complicate attempts to set up the equivalent statistical description. The first major complication arises from the fact that dislocations are line defects described by the associated Burgers vectors and dislocation lines, even though this can be dealt with in principle [24,115]. (The long ranged nature of the interaction between dislocations in itself does not pose serious problems except in analytical calculations.) The second difficulty arises from the fact that plasticity is inherently a highly dissipative process wherein a large part of the work done on the system (consisting of the

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ensemble of dislocations and the crystalline medium in which they are embedded) is lost in the form of heat (see for example [1,2]). This in turn implies that the framework that is sought to be developed should be able to deal with this. Here it is pertinent to point out that there are other important manifestations of dissipation in the general context that remain to be included. These issues relate to modeling of dislocations processes such as annihilation, entanglement, etc., within the scope of the statistical description.10 Finally, we note that dislocation pattern formation is very distinct from pattern formation in condensed matter physics. In the latter case, collective modes correspond to the system as a whole while patterns arising during plastic deformation are patterns of the subsystem of dislocations embedded in a larger crystalline medium. In this respect, information about the crystalline nature of the whole system enters only in an indirect way, for example, glide planes, dissipation, etc. As we shall see, we will consider the ensemble of dislocations without any direct reference to the crystalline structure of the medium of which dislocations are an integral part. 4.4.2. Statistical framework for parallel straight dislocations in two dimensions To start with, following Groma [84], a major simplification can be achieved if one limits the description to two dimensions. Consider an ensemble of N dislocations parallel to the z-direction intersecting the xy plane at points ri . This amounts to describing dislocations as point particles and hence allows one to take recourse to the standard distribution function approach corresponding to N point particles of nonequilibrium statistical mechanics [116]. For simplicity all dislocations are assumed to have the same magnitude of the Burgers vector, b. The starting point is the over damped limit of the equation of motion for a dislocation in the xy plane, i.e. vi = si Mb

li τext

+

j =i

l i lj sj τind ri

− rj

.

(41)

Here, si refers to the sign of the Burgers vector (+ or −), li denotes the slip plane of the li lj dislocation and M its mobility. τind (ri − rj ) is the shear stress on the dislocation at ri in li the slip plane li by a dislocation at rj in the slip plane lj , and τext is the external shear stress resolved along the slip plane. Then τind is given by τind =

x(x 2 − y 2 ) Gb . 2π(1 − ν) (x 2 + y 2 )2

(42)

In the discussion that follows we drop the slip line index for sake of simplicity and also assume that we consider glide of dislocations and not their climb. In principle, the above equations of motions can be solved and the averages computed. However, here we start with the distribution function for the ensemble of N dislocations defined by FN (r1 , . . . , ri , . . . , rN ) referring to the joint probability distribution of N dislocations at positions r1 , . . . , rN . We note that the N particle distribution function is symmetric under exchange of any two particles, i.e. FN (r1 , . . . , ri , . . . , rj , . . . , rN ) = 10 We note that these are much easier to include within the framework of reaction-diffusion approach.

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121

FN (r1 , . . . , rj , . . . , ri , . . . , rN ) a property that will be useful in computing averages. The N particle distribution function is too complicated and for many practical calculations, one does not need the FN , and we can define the reduced distribution functions for one particle, two particles, etc., by ρ1 (r1 ) = ρ2 (r1 , r2 ) = ρ3 (r1 , r2 , r3 ) =

N i

δ(r1 − qi )FN ( q1 , . . . , qN ) d q1 . . . d qN ,

N N i

j =i

δ(r1 − qi )δ(r2 − qj )FN ( ql , . . . , qN ) d q1 . . . d qN ,

N N N i

j =i k=i,j

(43)

(44)

δ(r1 − qi )δ(r2 − qj )δ(r3 − qk )

×FN ( ql , . . . , qN ) d q1 . . . d qN .

(45)

Note that FN being symmetric under exchange of particles, N equivalent terms are generated in eq. (44), N (N − 1) in eq. (45), etc. Alternately, the ‘reduced distributions’ can be defined by integrating out other variables. Then, ρ1 (r1 ) = N

FN (r1 , . . . , rN ) dr2 . . . d qN ,

ρ2 (r1 , r2 ) = N(N − 1) ... ... ... N! ρk (r1 , r2 , . . . , rk ) = (N − k)!

FN (rl , r2 , r3 , . . . , rN ) dr3 . . . drN , FN (rl , . . . , rk , rk+1 , . . . , rN ) drk+1 . . . drN .

(46) (47)

(48)

The phase space volume is not conserved in the case of plastic deformation. However, the equations of motion derived are based on the assumption that the number of dislocations is conserved. This condition can be relaxed later resorting to phenomenology. Multiplying eq. (41) by FN and integrating over rk+1 to rN , we get the equation of motion for the joint reduced distribution function of k dislocations ρk (r1 , . . . , rk ) that takes the form k k ∂ρk + ▽i si ρk (r1 , . . . , rk ) τext + sj τind (ri − rj ) ∂t j =i,sj

i

+ (N − k) ▽i

k

i=1,sj

drk+1 ρk+1 (r1 , . . . , rk+1 )

× si τext + si sk+1 τind (ri − rk+1 ) . sk+1

(49)

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Here we have absorbed the factor Mb by rescaling the time variable. From this we obtain

∂ρ1 (r1 ) = − ▽1 s1 τext ρ1 (r1 ) − ▽1 dr2 ρ2 (r1 , r2 ) s1 s2 τind (r1 − r2 ). (50) ∂t s 2

Similarly, ∂ρ2 (r1 , r2 ) = − ▽1 s1 τext + s2 τind (r1 − r2 ) ρ2 (r1 , r2 ) ∂t

− ▽1 s1 s3 dr3 ρ3 (r1 , r2 , r3 )τind (r1 − r3 ) s3

+ other terms under the exchange of subscripts 1 and 2.

(51)

Thus, the equation of motion for the k-particle distribution function involves the k + 1 particle distribution function. This is the conventional BBGKY hierarchy well known in nonequilibrium statistical physics. Clearly, solution of these set of equations is equivalent to solving for the N particle distribution function itself, which is impractical. Truncation schemes for the solution of this hierarchy are available in the literature which can be used for further studies. These equations obey an interesting scaling property [117]. Eqs (49), (50), (51) are invariant under the scale transformation 1

ρ → cρ ρ,

σext → cρ2 σext ρ,

−1

r → cρ 2 r,

t → cρ−1 t,

(52)

with the assumption that ρk scales as kth power of ρ1 . These class of solutions have been called as similitude solutions corresponding to different values of external stress. These can be parametrized by using the length scale corresponding to the mean dislocation density

1 ρ= ρ1 (r1 ) d3 r, V 1

i.e. all lengths are scaled with respect to ρ − 2 , external stress is scaled proportional to 1 ρ 2 and time with respect to ρ −1 . The scaling relations in eq. (52) are consistent with the Taylor relationship which states that the flow stress scales inversely with the square root of the mean dislocation density, and the law of similitude that relates the length scale corresponding to dislocation pattern to the inverse of the flow stress. When there are no correlations between particles, the k particle density can be decomposed as the product of k single particle densities, i.e. ρk (r1 , . . . , rk ) =

k

ρi (ri ).

(53)

i=1

However, in most cases where there is interaction between particles, usually short ranged in most other systems, correlations are induced and hence in general the above expression is

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123

not valid. In particular, in the case of dislocations, the interaction is long ranged and hence one expects correlations to be important. The information about the extent of correlation is contained in the higher order distribution functions. The simplest, for example, is the distribution function ρ2 (r1 , r2 ) of two dislocations. The deviation from the product of single dislocation distribution functions gives the extent of the correlation. One usually writes ρ2 (r1 , r2 ) = ρ1 (r1 )ρ1 (r2 ) + g2 (r1 , r2 ).

(54)

In the above expression, g(r1 , r2 ) = ρ2 (r1 , r2 ) − ρ1 (r1 )ρ1 (r2 ) is a measure of the correlation. As |r1 − r2 | → ∞, g(r1 , r2 ) → 0, the distance beyond which g(r1 , r2 ) is insignificant gives the correlation length. In a similar way all higher order distribution functions can be expressed in ‘cluster representation’ form. For example, ρ3 (r1 , r2 , r3 ) = ρ1 (r1 )ρ1 (r2 )ρ1 (r3 ) + ρ1 (r1 )g2 (r2 , r3 ) + ρ1 (r2 )g2 (r1 , r3 ) + ρ1 (r3 )g2 (r1 , r2 ) + g3 (r1 , r2 , r3 ).

(55)

Here, g3 (r1 , r2 , r3 ) is the irreducible part of the correlation between the three dislocations. Physically, it is clear that when the separation between any two dislocations is large, they should not be correlated. Thus, one has ρ3 (r1 , r2 , r3 ) → ρ2 (r1 , r2 )ρ1 (r3 )

as |r1 − r3 | → ∞ and |r3 − r2 | → ∞.

Similar boundary conditions ρ3 (r1 , r2 , r3 ) → ρ2 (r1 , r3 )ρ1 (r2 ) hold when the separation between other coordinates become large. The above forms are standard decompositions used in stochastic processes and statistical physics as well [89,116]. Now consider rewriting eq. (50) for dislocations with opposite Burgers vector. Then, we have ∂ρ+ (r1 ) = − ▽1 τext ρ+ (r1 ) ∂t

− ▽1 dr2 ρ++ (r1 , r2 ) − ρ+− (r1 , r2 ) τind (r1 − r2 ) ,

(56)

and ∂ρ− (r1 ) = ▽1 τext ρ− (r1 ) ∂t

− ▽1 dr2 ρ−− (r1 , r2 ) − ρ−+ (r1 , r2 ) τind (r1 − r2 ) ,

(57)

where + and − denote the sign of the Burgers vector. At a working level, this hierarchy of BBGKY equations need to be truncated at some order k for these equations to be of any practical value. Moreover, purely from physical

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considerations, one expects that the first few correlations are adequate. Using eq. (54) in eqs (56), (57) we get ∂ρ+ (r1 ) = − ▽1 τext ρ+ (r1 ) ∂t

− ▽1 dr2 ρ+ (r1 )ρ+ (r2 ) − ρ+ (r1 )ρ− (r2 ) τind (r1 − r2 ) − ▽1

and

dr2 g++ (r1 , r2 ) − g+− (r1 , r2 ) τind (r1 − r2 ) ,

(58)

∂ρ− (r1 ) = ▽1 τext ρ− (r1 ) + ▽1 dr2 ρ− (r1 )ρ+ (r2 ) − ρ− (r1 )ρ− (r2 ) τind (r1 − r2 ) ∂t

+ ▽1 dr2 g−+ (r1 , r2 ) − g−− (r1 , r2 ) τind (r1 − r2 ) , (59)

Dropping g(r1 , r2 ) a compact set of equations that have physical meaning can be written down in terms of the total dislocation density be ρ(r ) = ρ+ (r ) + ρ− (r ) and the difference in densities of opposite sign k(r ) = ρ+ (r ) − ρ− (r ). Then, one can define the mean internal stress field by (mf )

τint (r , t) =

k(r2 , t)τind (r − r2 ) dr2 .

(60)

Then, using the definitions of ρ and k in eqs (58), (59) we get ∂ρ(r ) (mf ) + ▽τext k(r , t) + ▽k(r , t)τint (r , t) = 0, ∂t ∂k(r ) (mf ) + ▽τext ρ(r , t) + ▽ρ(r )τint (r , t) = 0. ∂t

(61) (62)

The first equation is the evolution of the total density while the second equation represents the conservation of Burgers vector. However, the growth of both depend on the mean internal field which in turn depends on the difference in dislocation density of opposite Burgers vector. Further, k can be related to ensemble averaged Nye–Kröner tensor αij [118]. Thus, the mean stress field given by eq. (60) satisfies the equations 2 χ = τ (r ) =

∂k(r ) Gb , ∂y 1 − ν

∂ 2 χ(r ) , ∂y∂x

where χ is the stress function, G the shear modulus and ν the Poisson’s ratio.

(63) (64)

§4.4


(a)

125

(b)

Fig. 14. (a) A plot of dislocation density obtained with dislocation multiplication and periodic applied stress of high frequency. (b) The corresponding autocorrelation function. After I. Groma and P. Balough [85].

The simplest way to check the possibility of the emergence of new states of order is to investigate the stability of the homogeneous solutions of these equations. This shows that the real parts of the eigenvalues vanish for wave vectors parallel to the x- or y-directions. This means that periodic perturbations along these directions are marginally stable. However, if there is a source term S(ρ) in eq. (61) such that ∂S/∂ρ > 0, then homogeneous solutions are no longer stable. 4.4.3. Pattern formation As these equations were derived with the assumption of no sources and sinks, such terms can only be introduced in a phenomenological way. From physical considerations, multiplication of dislocations will depend on stress, total density and other parameters not mf included in the formalism, i.e. S = S(ρ, τext , τint , . . .) which can only be introduced in eq. (61) to retain conservation of Burgers vector. Groma and Balough [85] suggest a term of the form S(ρ, τ, . . .) = Cτ 2 ρ for the production of dislocations on the basis that part of the local work done goes in the production of dislocations. To account for annihilation of dislocations, a loss term proportional to b2 G2 ρ 2 can also be included. Numerical results correspond to the solution of eqs (61), (62) (with source and sink terms) on a grid of 128 × 128 with periodic boundary conditions. The initial distribution of the total dislocation density has been chosen to be random such that Burgers density k is nearly zero. The internal stress is evaluated through eq. (60) using fast Fourier transform. Numerical investigations of these equations have been carried out for different conditions such as no source term, source term but no sink term, source term with and without external load, etc. [85]. The influence of both constant and periodic stress has been investigated. The initial distribution of the Burgers density k has been taken to be small compared to ρ. For the case when periodic stress is used, the autocorrelation function C(r ) =

ρ(r − r′ )ρ(r ′ ) dr ′ ,

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shows a definite (average) periodicity with a length scale that is a fraction of the system size. The peaks are located along the diagonal. With increasing frequency of the periodic applied stress, the length scale of this periodicity changes with the appearance of additional peaks in the autocorrelation function. This also suggests that the pattern results from a superposition of several wave vectors. As an example, Fig. 14(a) shows the dislocation density map obtained under periodic external load [85] at high frequency. However, the strong dependence of the patterns on frequency is inconsistent with experiments. The figure shows nearly regular arrangement of dislocations which is considered to be similar to the matrix structure observed in cyclic deformation. (Compare Fig. 1 in Section 2.) However, patterns visually similar to Fig. 14 have been reported by the authors under various conditions. For example similar patterns whose peaks in the correlation function appear along the positive diagonal have been reported with no external stress. Thus, the physical origin of such patterns is not clear. One possible explanation is that the pattern is entropic driven. This will become more apparent in the case where there is no applied stress but dislocation multiplication is present. Under these conditions, the system allows only relaxation processes which are determined by the time scales of internal relaxation in the absence of dislocation multiplication and the time scale of production of dislocation. Once dislocations with opposite Burgers vectors are locked in a ‘dipole’ configuration,11 these cannot be easily destroyed as there is no applied stress. Thus, as the eventual observed patterns have lower entropy than the initial uniform distribution, the process is entropy driven. Finally, the approach can be easily extended to the study of patterns in a double slip configuration [86]. Groma and Bako consider 220 parallel edge dislocations under constant external stress with no dislocation multiplication. The obtained pattern is inhomogeneous and anisotropic which the authors refer to as cell-like configuration. However, these cell walls are predominantly made up of dislocations belonging to the same slip system and aligned in the other slip direction. The pattern is scale free with a fractal dimension of 1.85. 4.4.4. The effect of correlations In all situations where a hierarchy of equations is involved, the most natural question to be answered is at what stage the truncation should be closed so that adequate correlations are taken into account. As the self-consistent field equations ignores correlation completely, the lowest order distribution that needs to be considered is the three particle function which will have to be represented in terms of the lower ones. An easy way to determine the two point correlation function is to carry out simulations of N dislocations similar to that in Ref. [119]. Such studies show that the scaled correlation function is actually short ranged with nearly 1/r singularity for small distances and an exponentially decaying form for large values. (See, for example, Fig. 8 in Ref. [5].) The numerical simulations of eq. (42) shows that the internal stress field fluctuates around a mean value and these can be considered as delta correlated [120]. Thus one can write (mf )

τint (r ) = τint (r ) + δτint (r , t), 11 Note that within this formalism, there is no natural way of defining a dipole. The closest entity is k.

(65)

§4.5


127

where δτint (r , t) is the fluctuating term with a zero mean. (Here τint (r ) refers to internal stress at a point r due to all other dislocations given by τint (r ) = N r − ri ).) It i=1 τint ( has been shown that the correlation τint (r )τint (r ′ ) is actually determined by the pair correlation function g(r , r′ ). The stress correlation function has been shown to fall off in an exponential manner. (See Ref. [5] for more details.) Further, using methods similar to those in random walk problems [121], the first leading (nontrivial) contribution to the distribution function of the internal stress P (τint ) comes from the pair correlation function [112] or the pair distribution function ρ2 (r1 , r2 ). The probability distribution of the internal stress on a given slip plane has been shown to decay asymptotically as Cρ(r )τ −3 . The half width of P (τ ) is determined by the correlation properties of dislocation ensemble [112]. Finally, some progress has been made in using the third order reduced distribution function [5,117]. For example, Kirkwood approximation which expresses higher order distribution functions in terms of products of three different possible ρ2 has been used [117]. However, a lot more needs to be done in using higher order correlations.12

4.5. Fluctuation induced patterns 4.5.1. Introduction Plastic deformation of metals can give rise to different types of patterns such as the matrix structure, PSB and the cell patterns. Of these, cell patterns are formed in single crystals of f.c.c. metals oriented in multiple-slip geometry. As mentioned in the introduction, cell structures are seen in f.c.c. polycrystals in a range of temperatures. In single crystals, dislocation patterns begin to form at the end of stage II, but are seen prominently in stage III. This is suggestive of the role played by screw dislocations. The formation of dislocation patterns requires going from microscopic length scales to macroscopic length scales. Thus an appropriate theory requires bridging several levels of length scales. The most obvious method to pursue is to consider a collection of individual dislocations and their interactions and hope to carry out some averaging procedure to obtain the relevant quantities. This however is impossible as the number of dislocations that one needs to consider is enormous. There are a few approaches proposed in the literatures that attempt to explain the formation of dislocation patterns. One of the early models attempts to use thermodynamic approach wherein the formations of cell structure is assumed to emerge as a consequence of the second law of thermodynamics [122,123]. In this picture, these patterns are considered as resulting from energy minimization. For example, the total energy arising from the pattern is assumed to consist of a checkerboard pattern of cells with alternating sense 12 There are deeper questions on the validity of using the higher order distribution functions for describing pattern formation itself. This stems from the observation that for an inhomogeneous case, higher order functions cannot be functions of the difference in the coordinates. This can be seen as follows. In the case of homogeneous systems ρk (r1 , . . . , rk ) = ρk (r1 − rj , r2 − rj , . . . , rk − rj ). This implies that ρ1 (r ) = constant. However, for inhomogeneous situations this is not valid. This also means that ρ1 = ρ1 (r ). Thus, in principle, only single particle density can be used for describing patterns. This also points to using dynamical or Langevin type of description for ρ1 (r ). Indeed a Langevin type of description has been reported in Ref. [112]. In addition, there are issues about lack of stationarity that need deeper consideration.

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of orientation and the line energy. This gives an explanation for the scaling relation expressed by eq. (1). However, as these patterns are formed under highly nonequilibrium conditions, a thermodynamic approach is unlikely to hold. Following an approach similar to that in phase separation problems, Holt has proposed a model which again appears to be inapplicable as the situation is not a near equilibrium one [124]. As the above situation is representative of a highly nonlinear dissipative structure, a nonlinear dynamical approach has been taken by Walgraef and Aifantis [125]. The model is based on different types of densities. One other model is due to Kratochvil [126]. Finally one should mention the model due to Differt and Essmann [127] which also follows a dynamical approach. Some of these models have been discussed in detail in an earlier review and hence will not be discussed here [3]. However mention must be made here of another recent dynamical approach to understand the PSBs by Rashkeev et al. [128]. They begin with the Walgraef–Aifantis model and derive a Ginzburg–Landau type equation for the amplitude of the soft mode in the neighborhood of the bifurcation point. They obtain patterns that change slowly close to Eckhaus’s instability [129]. The two-dimensional pattern resembles the ladder structure. 4.5.2. Fluctuation induced dislocation patterns The approach taken here is based on stochastic dynamics which recognizes that there is a clear separation of time scales corresponding to microscopic and macroscopic length scales. In the context of pattern formation, the time scale over which these patterns are formed is much longer than the time scale over which gliding dislocations interact with other dislocation configurations.13 On the other hand, the fast microscopic degrees of freedom can be taken to give rise to ‘a background noise’ of the internal stress [110]. Based on the formalism provided earlier, one can estimate the correlation length and time that will be relevant for the problem.14 This can be defined as the length scale over which two dislocations interact strongly enough to be able to glide in a correlated manner when their interaction stress exceeds the stress fluctuations exerted by the effective medium consisting of all dislocations. This is given by ξcorr =

Gb Gb . = √ 2 4π Sτint 4π δτeff

(66)

Assuming that dislocations move in groups as observed in experiments [6], they should be expected to experience fluctuations in the effective internal stress which will be eventually limited by their annihilation or arrest. This time scale is a measure of the correlation time tcorr given by tcorr =

bρm L , γ˙

(67)

13 Conventionally such an effective stochastic dynamics is obtained by eliminating the fast degrees of freedom.

This is far from easy. 14 The magnitude of the correlation length ξ corr so obtained can in principle provide the scale over which the coarse graining operation can be preformed.

§4.5


129

where L is the slip line length. This time scale is much shorter than the time scale of evolution of the pattern and hence the noise can be regarded as δ correlated, i.e.

τint δ γ˙ (t) δ γ˙ (0) = 2 γ˙ 2 tcorr δ(t). S

(68)

It should be noted that the strength of the fluctuations is inversely proportional to the strain rate sensitivity which determines the magnitude of intermittency. The above stochastic basis for inherent fluctuations in plastically deforming materials has been used for modeling patterns like cell structure in tensile situation or the matrix and the persistent band structures observed in cyclic deformation of f.c.c. materials. The model due to Hähner considers these patterns as noise induced transitions [130]. While these are spatio-temporal patterns, a much simpler problem of obtaining the probability distribution of the patterns is attempted. To keep the model simple, a single order parameter is envisaged that can be identified with the immobile dislocation density (dipoles or dislocation tangles) or the total. The general form of the stochastic differential equation for dislocation patterning can be written as ∂ρ = F (ρ)γ˙ , ∂t

(69)

where F (ρ) describes dislocation multiplication and dynamic recovery. Within the framework discussed earlier, as γ˙ is a fluctuating quantity, one can write γ˙ = γ˙ + δ γ˙ , where δ γ˙ is assumed to have a white noise property given by eq. (68). Then, a general stochastic evolution equation for the slow order parameter like variable ρ can be written as ∂ρ = F (ρ)γ˙ + g(ρ)δ γ˙ . ∂t

(70)

In general F (ρ) is different from g(ρ) whose functional form needs to be modeled for the relevant physical situation. Eq. (70) is a Langevin equation with a multiplicative noise. In physical terms this means that the noise amplification depends on the state variable whose magnitude is described by some function of ρ, here, g(ρ). In terms of scaled nondimensional quantities, one can write ∂ρ = f (ρ) + Dm g(ρ)η, ∂t

(71)

where Dm is strength of the noise which is taken to satisfy eq. (68). The scaled noise variable then satisfies η(t)η(t ′ ) = δ(t − t ′ ). (72) The solution of such a multiplicative stochastic differential equation is straightforward once the type of calculus that should be used is specified.15 Basically, this amounts to

15 Clearly, the arguments provided for multiplicative nature of the noise do not appear to be specific to this particular case and thus raises questions of validity of this assumption.

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specifying when (in time) the noise component is added. Most physicists tend to use Stratonovich calculus wherein the noise component is added at the mean value of t and t + t where t is the time increment for the time evolution. For the Ito calculus, the noise component is added at t. The multiplicative nature of the stochastic differential equation also implies that there is a noise induced drift term. In the case of an additive noise, the drift term is entirely determined by the deterministic part. This is better illustrated by writing down the Fokker– Planck (FP) equation corresponding to eq. (71) which is given by [87,88] ∂P (ρ, t) ∂ =− ∂t ∂ρ

f (ρ) +

2 Dm ∂ D2 ∂ 2 g(ρ) g(ρ) P + m 2 g 2 (ρ)P , 2 ∂ρ 2 ∂ρ

(73)

where we have used the Stratonovich calculus and the second term on the right-hand side is the noise induced drift term stated above. In most physical problems one has an additive noise in eq. (71), i.e. g(ρ) = 1, and the second term on the right hand side of eq. (73) will be absent and the drift term is controlled by f (ρ) alone. The FP equation is usually solved with the initial condition on the conditional probability distribution P ρ, t + t ′ |ρ0 , t ′ = P ρ, t|ρ0 , 0 , (74)

with appropriate boundary conditions. In Ito’s calculus, the drift term in eq. (73) would be absent. Using the boundary condition that the probability current vanishes at the boundaries, one can easily obtain the stationary probability distribution given by

ρ ′ 2 N ′ f (ρ ) Ps (ρ) = , (75) dρ exp 2 g(ρ)ν Dm g 2 (ρ ′ ) 0 where ν = 1 for the Stratonovich and ν = 2 for the Ito calculus.16 4.5.3. Fractal cell patterns Now we consider the specific case of cell formation observed in single crystals of f.c.c. deformed in symmetrical multislip conditions where the patterns are statistically isotropic. While the original model actually derives the order parameter equation starting from the evolution equations for both the mobile and immobile dislocation densities, here, it would be adequate to use only one density given by ∂ ρ˜ = 1 − ρ˜ + D˜ m Da + ρ˜ η, ∂ t˜

(76)

where ρ˜ is an appropriately scaled nondimensional total density given by ρ = c1 ρ˜ =

Aτeff γ˙ B2 γ˙

2

ρ, ˜

16 According to van Kampen, ‘the physicist’s conclusion is that whenever his ramification leads him to the conviction that a situation should be described by eq. (71), he must have made a logical mistake – unless his consideration also tells him how to interpret it’.

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and t = c2 t˜ =

Aτeff γ˙ t˜. (B2 γ˙ )2

A and B2 are rate constants for dislocation multiplication and recovery in the original set of equations [114]. In the above equation, the first two terms on the right-hand side represent the deterministic evolution. The third term contains both additive noise with a strength Da and as well as multiplicative noise with a strength Dm . Thus, the nature of the stationary distribution will depend on the strength of each of these terms. In terms of original variables, the noise term η is given by δ γ˙ =

Da =

τeff + S , τext

(78)

Dm =

(79)

2tcorr δ γ˙ 2 η, c2

(77)

with

and tcorr τint . c2 S

Thus, for this case, f (ρ) ˜ =1−

ρ˜

and g(ρ) ˜ = Da +

Using this in eq. (75) we get

ρ. ˜

4(1+2Da )/D 2 −1 4 4 Da (Da + 1) m Ps (ρ) . ˜ = N Da + ρ˜ exp − 2 ρ˜ + 2 Dm Dm (Da + ρ) ˜

(80)

(81)

Here N is the normalization constant. The nature of the stationary distribution changes as the strengths of the additive and multiplicative noise Da and Dm are varied as can be seen by examining the condition for the extremum of the distribution given by f=

2 ∂g Dm g . 2 ∂ ρ˜

Two extrema

ρ˜ ± exist if

2 2 Dm < Dmc = 4 1 + 2Da − 2 Da (Da + 1) .

(82)

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Here, + and − sign correspond to the maximum and minimum values respectively. From this expression, it is clear that the critical value of the multiplicative noise Dmc decreases monotonically with the noise intensity of the additive noise Da . When Dm < Dmc , the stationary probability distribution peaks at the deterministic value, ρ˜ = 1 (determined by Aτ γ˙ 2 f (ρ) ˜ = 0). (Equivalently, in unscaled dislocation density ρ = ( B2eff γ˙ ) . See [114].) This corresponds to a statistically homogeneous situation where the structure is characterized by a well defined scale. However, for Dm > Dmc , the peak shifts toward ρ˜ = 0 and the stationary probability distribution decreases monotonically from ρ˜ = 0. Thus in this model, there is a transition from a peaked nature of the stationary probability distribution to a power law type as we increase the intensity of the multiplicative noise. Physically this means that a transition occurs from a pattern with a well defined scale to a scale-free power law. The trends of the model appear to be consistent with experimental observations. The model predicts that dislocation cell formation is favored if strain rate fluctuations are large. For instance, an estimate of the order of magnitude of fluctuations in f.c.c. metals is given by τint /S ∼ 400 which is large. This suggests that the cell structure is readily seen in f.c.c. metals. In contrast, for the b.c.c. metals, for temperatures below the transition temperature, fluctuations are small. The interpretation is that a cell structure is not easily formed. As mentioned, for multiplicative noise intensity Dm greater than Dmc , the peaked nature of the distribution is lost. If one is interested in the specific feature of the above model, namely a crossover from a peaked nature to a power law type of distribution, it would be adequate to use only the multiplicative noise. Thus, setting Da = 0, in eq. (81) gives ˜ = N ρ˜ Ps (ρ)

2) − 12 (1−4/Dm

ρ˜ 4 −α . exp − 2 ρ˜ = N ρ˜ exp − 4 2 Dm Dm

(83)

Here again, the magnitude of the noise intensity determines the nature of the distribution. 2 < 4, the probability distribution peaks around the deterministic value of ρ˜ = 1 For Dm corresponding to statistically homogeneous pattern with a well defined scale. In physical 2 > 4, one gets a power space, the corresponding length scale is ρ ˜ −1/2 . In contrast, for Dm law that is singular at the origin, but is well behaved for large ρ. ˜ A plot of the distribution is shown in Fig. 15. It would be useful to impose the physical condition that the upper limit of ρ˜ is set by its value in the cell walls, ρ˜w . Then, for the power law case, in the physical space, the minimum length scale is set by ρ˜w −1/2 . Yet, as the distribution peaks at low values of ρ, ˜ low density dislocation regions are much more abundant. ˜ exhibits a power law form, it can be used to describe fractal structures. InAs Ps (ρ) deed fractal analysis of TEM micrographs of the cell structure observed in pure Cu and CuMn alloys deformed under symmetrical multislip conditions has been discussed in Section 3.2.1 [79,30]. These studies show that the size distribution of the cells obeys a power law scaling corresponding to self-similar ‘pore fractals’. We recall here that for tensile deformation of Cu single crystals along a [100] or [111] direction, the dimension DF was found to be in the range 1.5 to 1.8 [30]. To map the theoretical parameter with the measured fractal dimension, consider the normalized distribution given by ˜ ∼ ρ˜w−α+1 ρ˜ −α , Ps (ρ)

(84)

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Fig. 15. The probability distribution for various values of Dm showing a crossover from peaked nature to a power law scaling type distribution.

with α = 21 (1 − D42 ). Noting that the mean dislocations spacing is proportional to the cell m ˜ we have Ps (r) ∼ r 3−2α . Then the corresponding cumulative probabilsize, i.e. r ∝ 1/ ρ, ity distribution P (r < R) ∼ R −(2−2α) can then be compared with the measured cumulative probability distribution in the analysis of the fractal structure of the cells that obeys P (r < R) ∼ R −Df which gives 2 . Df = 2(1 − α) = 1 + 4/Dm

This correspondence is meant for the two-dimensional micrographs analysed. However, 2 ), in reality, the cells are three-dimensional. Thus, the corresponding DF (3) = (2 + 4/Dm where the argument refers to three dimensions. Finally, in the real physical situation, the In this case, the highest dislocation density is in the scaling is seen in a range [rmin , rmax ]. walls of the cells and hence, rmin ∼ 1/ ρ˜ w . Indeed, in the analysis of micrographs, a lower cut off is in fact seen which can be identified. The physical interpretation of the power law distribution is clear. Noting that ρ˜ → 0 implies regions of low density of dislocations, identifying the interior of the cells with holes, holes of all sizes occur in a statistically homogeneous way in the physical space, i.e. a scale free fractal is seen. More importantly, in the spirit of this model, the spongy nature of the pattern arises due to fluctuations intrinsic to some materials. The fractal structure should therefore be considered as a noise induced transition [130]. As argued earlier, these fluctuations are large in the case of f.c.c. metals (in contrast to the b.c.c. metals) for a range of strain rates and temperatures. Thus, the model predicts fractal structure in these f.c.c. metals. In case of solid solutions, S can be varied over a wide range by changing strain rate and temperature. Beyond the regime of dynamic strain aging, solution hardening can lead to increased value of S implying to low levels of noise. From the formula relating the fractal dimension with noise, this would imply higher value of DF . Indeed, DF is higher for CuMn compared to pure Cu.

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4.5.4. Patterns induced in fatigue This approach can be extended to other types of dislocation patterns, observed under cyclic deformation [10,29], namely the matrix structure and the persistent slip bands. These patterns basically arise from the unmixing of screw and edge dislocations. As screw dislocations annihilate more easily than the edge, the edge dislocations segregate into dipolar veins separated by a channel of relatively low density for the screw channels to shuttle. Thus one can take dipole density as the order parameter variable. The model formulated is based on some assumed mechanisms. In this model, dipole density is the order parameter. The generation rate is taken to be constant. One mechanism by which the dipole population is depleted can arise by annihilation of one dislocation in the dipole when a mobile edge dislocation approaches closer than a critical distance. Thus, loss rate of dipole populations is proportional to the dipolar density. Usually dipoles are not mobile. However, they can be displaced by strong stress gradients generated by a passing edge dislocation. In fact, this is the basis for Kratochvil model [131] where dipoles are swept from one region of space and deposited in another region by mobile screw dislocations. The mechanism has been verified in numerical simulations as well [3, 111]. In this model, this process is however assumed to be random. This rate is taken to be proportional to ρ(1 − ρ) where ρ is probability of finding regions that have dipoles and 1 − ρ the probability of dipoles being absent. Combining these rates, we get ∂ρ = k − ρ + Dm ρ(1 − ρ)η. ∂t

(85)

Here ρ is the scaled dipolar density whose bounds are zero to unity. As mentioned earlier, the matrix structure is formed at low strain amplitudes and the PSB at higher amplitudes. Within the context of the noise √ induced transitions, it is then natural to assume that the strength of the noise Dm ∝ k γ where γ is plastic strain amplitude. In this case, Hähner assumes Ito calculus to be valid [113] for this situation. This gives the associated Fokker–Planck equation D2 ∂ 2 ∂P (ρ, t) ∂ (k − ρ)P + m 2 ρ 2 (1 − ρ)2 P . =− ∂ρ 2 ∂ρ ∂ t˜

(86)

Here all variables are in scaled form. The stationary distribution is readily obtained by using the fact that the current vanishes, Ps (ρ) = Nρ

2(1−2k)

−2

2 Dm

+1

2(1−2k)

2

(1 − ρ)

2 Dm

−1

k 2 2k − 1 − . exp 2 1−ρ ρ(1 − ρ) Dm

(87)

Clearly the distribution vanishes at ρ = 0 and ρ = 1 as it should. Further, it is symmetric with respect to ρ = 1/2 and k = 1/2. For small noise Dm < 2, the stationary probability distribution exhibits a maximum at ρ = 1/2 with some dispersion (Fig. 16). This corresponds to a homogeneous distribution 2 = 2 (for k = 1/2) the of the dipoles in space. As the noise strength is increased beyond Dm

§4.5


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Fig. 16. Top: The bimodal symmetric distribution represents a matrix like structure. Bottom: Dipole density 2 = 10 corresponds to PSB like for low dipole generation rate k = 0.3 for different values of noise strength. Dm situation.

distribution becomes bimodal, symmetrically located around ρ = 1/2. Thus, the dislocation dipole poor and rich regions are equal. This corresponds to a matrix structure wherein the volume fractions of the veins and channels are equal. For 0 < k < 1/2, varying the intensity of the noise causes a crossover from a peaked 2 → 8 as k → 0. distribution to a bimodal one (Fig. 16). The critical value approaches Dmc More importantly, for this case, the height of the peak for ρ < 1/2 becomes larger that for ρ > 1/2. This feature can be interpreted as these regions being smaller in physical extent compared to dipole poor regions. The peak of the dipole rich regions occurs at 2 and that of channels at ρ = k/D 2 . For k → 0, D 2 is large and thus, ρw ≈ 1 − 1/Dm ch m m ρw → 1 and ρch → 0. Finally, it can be easily verified that Stratonovich calculus also gives similar results except that the critical noise strength would be different.

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5. Dynamical approach to collective behavior of dislocations 5.1. General introduction The second part of this review is devoted to the description of the collective behavior of dislocations that manifests over a relatively short time scales. We shall focus entirely on the rich spatio-temporal features of the PLC effect that have defied proper understanding for a long time. However, this phenomenon has been recognized as some kind of a dynamical instability almost right from the introduction of the concept of dynamic strain aging. Thus, it is no surprise that substantial progress has been made using dynamical approaches, particularly so, since the introduction of a dynamical model due to Ananthakrishna and coworkers [107,108]. As mentioned earlier, this is one of the few instances in plasticity that requires proper description of the fast and slow time scales. However, even within known approaches in physics, order parameter fields are identified with slow variables. Thus, a proper description of the PLC effect requires special methods that have been developed in the field of dynamical systems. We will postpone a review of these methods to the section on modeling. Here, we begin with the progress made in the application of another dynamical technique that has been used to get insight to seemingly irregular serrations in the PLC domain. One of the easiest quantities to monitor in any spatially extended system is some kind of spatial average, for instance the stress. This represents the average over the dislocation activity in the entire sample. One expects that even the strongly intermittent nature of collective behavior of groups should be reflected in the nature of stress fluctuations as the deformation proceeds. The stress however, is an average of a large number of strain rate bursts occurring at different spatial locations which in turn result from the collective behavior of groups of dislocations. It has been long believed that different types of stress serrations observed in the entire range of strain rates and temperatures should contain information about the nature of the bands. However, until recently, only qualitative geometrical forms have been used to label type C, B and A serrations [132]. This wealth of spatio-temporal features has long defied a proper understanding. If one wants to extract more information from the stress–strain curves, a radically different approach appears necessary as such questions appear to depend on determining the individual events that make up this average which clearly has no unique answer. Two different techniques that attempt to recover the hidden order in the stress serrations, namely time-series analysis and a multifractal approach for characterization of the time series have been introduced into plasticity theory by Ananthakrishna and coworkers [50,133–137]. The advances made in this direction were triggered off by the prediction made by Ananthakrishna’s model (AK model) that these serrations were a result of deterministic dynamics of a few collective modes of dislocations. The model suggests that the erratic nature of serrations observed over a range of strain rates and temperatures has a hidden order which, in principle, can be unfolded. The method of inverting irregular scalar time series to obtain the underlying order is a well developed subject in nonlinear dynamics. Central to this approach is the premise that the apparent randomness in the stress–strain curves is a projection from a higher-dimensional deterministic dynamics to one dimension and that it is possible to recover the dynamics by a reconstruction process. Even though the methodology has been discussed in our last

§5.2


137

review, we shall, for the sake of completeness, briefly summarize the relevant background material here, as these methods are not commonly used in metallurgical literature. For details see Ref. [3].

5.2. Introduction to nonlinear time series analysis Till the development of chaos theory, all irregular time sequences observed in nature or in the laboratory in varied fields were thought to have originated from stochastic processes in the system. It is now known that a coupled set of few (at least three) nonlinear differential equations can exhibit strongly irregular time evolution. Such systems have been termed as chaotic. With the development of theory of dynamical systems and discovery of chaotic behavior in deterministic systems, attempts were made to identify the chaotic behavior in experiments. Subsequently, a sub-branch of nonlinear dynamics, namely nonlinear time series analysis has developed to deal with characterization of time series. The subject has grown so rapidly that several review articles and monographs already exist in the literature [20,22,138,139]. Recently developed methods of nonlinear time series analysis give sufficient tools to understand the underlying dynamics of irregular time series. In the following, we discuss the tools necessary to identify and characterize a time series from a deterministic system. 5.2.1. Characterization of chaotic behavior The most important attribute of a chaotic system is the lack of predictability arising out of sensitivity to initial conditions. No matter how small the indeterminacy in the initial conditions, it very soon explodes at an exponential rate thereby rendering the prediction of the future of the closeness of the two orbits impossible. This also means that there is a loss of memory of the initial state. In the case of experimental time series, the indeterminism of initial conditions (or measurement noise) arises due to a variety of reasons. For instance, lack of control over external or internal parameters is one possibility. Another source is the fact that the measuring instrument has its own vibrational frequencies and is coupled to the system. There may also be limitation in the accuracy of measurement both in terms of the frequency of measurement and the precision of the data that is recorded. There are several, qualitative as well as quantitative, methods of distinguishing a chaotic signal from a stochastic one. Some of them are rather simplistic but are quite useful in giving a clue about the possibility of chaos. Such methods are easy to implement or to visualize in terms of plots. Some others give quantitative estimates and one needs to carry out several complementary methods till one is certain that the signal is chaotic. One usually begins with the visual inspection of the data. A plot of the data as a function of time gives hints of the possible presence or absence of stationarity, drifts, systematically varying amplitudes or time scales, rare events, the extent of noise, etc. Enlarged views of different portions of time series can also give an idea of the internal structure of the time series. Tests for stationarity are easy, provided the given time series is much longer than the longest characteristic time scale of the system under consideration. The usual method adopted is to calculate the running variance of the time series. For a stationary

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time series the running variance should remain constant within the acceptable error limits for the entire time series. Stationarity can also be tested by calculating statistical quantities like correlations, etc., for different sections of the time series. The autocorrelation function c(τ ) gives information about how much the variable x at a time τ is correlated to its value at a later time. This is defined as c(τ ) =

1 xi xi+τ − xi 2 2 σ

(88)

where τ is in units of the sampling time t, · represents the average values and σ 2 the variance of the time sequence. If the time series is periodic, then c(τ ) will be periodic in τ . For chaotic systems, it is known that c(τ ) decays rapidly. Obviously, c(0) = 1, and for some τ = τ ∗ , c(τ ) = 0. Thus at τ ∗ , xi and xi+τ become statistically independent, so would be the vectors (in the d-dimensional embedded space) defined by the delay time τ ∗ . The rapid decay of the correlation function implies that there is a loss of memory of the initial state in a short time. The oscillatory nature is suggestive of a large number of frequencies. The power spectrum is useful in identifying periodic or quasiperiodic time series, since they give sharp peaks at the dominant frequencies and their harmonics. The associated power spectrum of a chaotic time series shows a broad band spectrum of frequencies. In contrast, in the case of white noise, the autocorrelation will be δ-correlated and the oscillatory nature will be absent. However, autocorrelations are not characteristic enough to distinguish chaotic signals from stochastic ones. (See Figs 22 (a) to (d) in Ref. [3].) In most experiments, only one variable can be monitored and the PLC instability is no exception. However, one knows that the dynamics of the system is controlled by many degrees of freedom. Therefore, one attempts to reconstruct the original attractor by embedding the scalar time series in a higher-dimensional space and in the process obtains information about the number of degrees of freedom required for a dynamical description of the system. Such embedding procedures have been built on a sound mathematical framework. Whitney [140] has shown that in general a D-dimensional manifold can always be embedded in a phase-space of dimension d = 2D + 1. The extension of this theorem to strange attractors is due to Takens [141] and Mane [142]. They have shown that the multivariate dynamics of the attractor can be unfolded from the scalar time series in a higher-dimensional space under quite general conditions. Recently it has been shown that the number of degrees of freedom of a dynamical system is given by the integer value of embedding dimension d > DF [143]. Packard et al. [144] have demonstrated numerically that in case of chaotic systems, the multi-dimensional phase space can be reconstructed from a single scalar time series. Following Ruelle [145], they suggested the method of delay coordinates for the reconstruction of the phase-space. According to Takens [141], the d-dimensional attractor can be reconstructed using time delay coordinates. That is, ξi = {xi , xi+τ , xi+2τ , . . . , xi+(d−1)τ }, the delay time τ can be chosen arbitrarily for an infinitely long and noise free data. The reconstructed attractor has the same dynamical invariants such as correlation dimension and Lyapunov spectrum as that of the original attractor.

§5.2


139

5.2.2. Correlation dimension Grassberger and Procaccia [76,146] introduced the correlation dimension as a variant of fractal dimension in the context of a strange attractor. In the d-dimensional embedded space, one can construct a quantity called the correlation integral, C(r) that represents the fraction of the number of pairs of the embedding vectors (ξ (i), ξ (j )) whose distance is less than r. If the underlying dynamics is of deterministic chaotic origin, then the reconstructed attractor would be a fractal object. Then, the correlation integral C(r), in the limit of small r behaves as C(r) ∼ r ν where the exponent ν is termed as the correlation dimension. This has been shown to be a lower bound to the fractal dimension DF [76,146]. Theiler [147] pointed out that the correlation between points on the reference trajectory (which neither diverges nor shrinks), can lead to wrong conclusions in the calculation of correlation dimension. This problem can be rectified by ignoring points on the reference trajectory. This is done by using a window w such that on either side of the reference point, w points on the reference trajectory will be ignored. Usually, the window chosen is a few times the autocorrelation time. The correlation integral C(r) is then defined by C(r) =

NT NT 2 r − ξ (i) − ξ (j ) , (NT − w)(NT − w − 1)

(89)

i=1 j =i+w

where (·) is the Heaviside step function and NT = N − (d − 1)τ , τ being the delay time. Correlation dimension can be obtained from the convergence of slopes of double logarithmic plot of C(r) versus r for various d. In practice, the scaling regime is seen for intermediate values of r. Most experimental signals from chaotic systems are corrupted by random noise. The usual sources of noise are either the ‘machine’ noise or that associated with limitation in the precision and frequency of measurement. Given such a time series, it is therefore necessary to quantify the extent of the signals arising from intrinsic nonlinearity as against that due to random noise. There are sophisticated methods which allow elimination of the noise component [22]. One method which will be discussed below is called the singular value decomposition method [148]. This method, in addition to curing the time series of noise, also provides an estimate of the dimension of the attractor. There is an additional complication in experimental signals from those mentioned above, namely, they are invariably short. This requires modifications of the methods developed for ideal systems. Here we present quantitative estimates of realistic time series which allows for unambiguous identification of chaos. 5.2.3. Singular value decomposition Given the reconstructed d-dimensional vectors ξi = {xi , xi+τ , . . . , xi+(d−1)τ }, with a delay-time τ , the trajectory matrix of the system is given by 1 AT = √ ξ1 , ξ2 , . . . , ξNT . NT

(90)

It is clear that the trajectory matrix A is a NT × d matrix. To obtain the principal components of the matrix, we note that any NT × d matrix whose number of rows NT is greater

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than or equal to its number of columns d, can be written as a product of a NT × d columnorthogonal matrix U, a d × d diagonal matrix W with positive or zero elements (singular values), and the transpose of a d × d orthogonal matrix V, i.e. A = U.W.(V)T , where the diagonal values of W give the eigenvalues of the matrix A. Eigenvalues obtained by this method are conventionally ordered in a decreasing order. The plot of normalized singular values as a function of its index is known as singular value spectrum. If some of the principal values say, (q + 1, . . . , d) are zero, then the trajectory remains confined to the subspace spanned by the basis (1, . . . , q). In practice the presence of noise prevents these eigenvalues from taking zero value exactly. However, there may be a set of eigenvalues which are small compared to the largest one. Then an estimate of the embedding dimension is obtained by the sharp decrease in the eigenvalue to a certain low level. Then, one can obtain the cured trajectory matrix by retaining only the first q components and back rotating these components. Often, this method is used as a noise reduction technique. More importantly, this method can be used for visualization of the attractor by plotting the first few principal values. We illustrate this method in this section. The normalized spectrum decreases exponentially and saturates to a ‘floor level’ corresponding to the noise level. This method has been used earlier in the context of stress–time series from PLC effect [51,135]. Finally, the most important attribute of a chaotic time series is the existence of a positive Lyapunov exponent. Lyapunov exponents provide a quantitative estimate of the extent of local divergence or contraction of the nearby orbits arising from sensitivity to initial conditions. For a d-dimensional embedding space there are d Lyapunov exponents. Eckmann et al. [138] and Sano and Sawada [149] proposed methods of estimation of the Lyapunov spectrum from time series. (A host of other methods have since been introduced [139].) Both methods rely on the construction of a sequence of (d, d) tangent matrices Ti which map the small difference vector ξ (j ) − ξ (i) to ξ (j + k) − ξ (i + k), i.e. ξ (j + k) − ξ (i + k) = Ti . ξ (j ) − ξ (i) , (91)

where k is the evolution time in units of time step t. Then, the tangent matrices are re-orthogonalized using the QR decomposition, where Q is an orthogonal matrix and R is an upper triangle matrix with positive elements. Then, the Lyapunov exponents λl are given by p−1 1 ln(Rj )ll , λl = ktp j =0

l = 1, 2, . . . , d,

(92)

where p is the number of available matrices and kt is the propagation time. Since the effect of randomness will be more in case of very small difference vectors, the method can be modified by choosing the orbits in a shell rather than a sphere [150]. There are other checks that are carried out using surrogate data analysis [151]. In this analysis, the basic idea is to generate an ensemble of random data sets with the same spectral properties as that of the original data. Then, compute a statistic for these computer generated surrogate data sets as well as for the given time series. If the value of statistic computed for the original data is significantly different from that computed for the surrogate data sets, then the original data is not from a stochastic process, or in other words

§5.3


141

nonlinearity is detected in the data. One usual method of generating surrogate data is by the phase-randomized Fourier transform method. One can use the correlation dimension or any other invariant as a discriminating statistic. One should expect that the positive Lyapunov exponent is a better discriminator. However, Theiler [151] points out that this could be problematic due to inherently unrealistic values obtained for surrogate data as well. In view of this, we have analyzed the problems encountered in calculating the Lyapunov exponents for surrogate data sets. We propose improvements for the existing algorithm for the calculation of Lyapunov exponents from time series, and the method works well even in presence of noise up to ∼15% [136,152]. The largest Lyapunov exponent calculated using this method works well as a discriminating statistic in surrogate data analysis.

5.3. Multifractal analysis There are a number of examples in nonlinear systems which present us a variety of complex fractal objects and strange sets. Apart from the chaotic attractor discussed above, examples include the configurations of Ising spins at a critical point [153], region of high vorticity in fully developed turbulence [154], percolating clusters and their backbones [155], and diffusion-limited aggregates [156] to name a few. The highly nonuniform probability distributions arising from the nonlinearity of these systems often possess rich scaling properties including that of self-similarity. Consider the strange attractor shown in Fig. 17. We see that the attractor is not uniformly dense and the orbit visits some portions more often than others. This nonuniform distribution of the density results in varying local fractal dimensions over the attractor. The study of the long-term dynamical behavior of a physical system can then be attempted by the characterization of the fractal properties of a measure that can be associated with the nonuniform distribution. This forms the basis of the

Fig. 17. 3-dimensional projection of the strange attractors of Ananthakrishna’s model.

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multifractal formalism. A multifractal is an interwoven structure of infinitely many fractals. They are characterized by a large number of dimensions and a related spectrum of scaling indices. The multifractal formalism describes the statistical properties of singular measures associated with the nonuniform distribution in terms of their singularity spectrum [157,158] or their corresponding generalized dimensions [159]. Here, we briefly present a method proposed by Chhabra et al. [160] for determining the singularity spectrum of a multifractal directly from an experimental data. Consider a chaotic time series {xi }, i = 1, . . . , K. These points belong to a trajectory in d-dimensional space. The trajectory does not fill the d-dimensional space even when K → ∞, because the trajectory lies on a strange attractor of dimension D, D < d. Let the attractor be divided into boxes of size l and let Mi be the number of times the trajectory visits the ith box. Defining Pi = limK→∞ (Mi /K), we can define an exponent (singularity strength) αi given by Pi (l) ∼ l αi .

(93)

If we count the number of boxes N (α) where the probability Pi has singularity strength between α and α + dα, then f (α) can be loosely defined [158] as the fractal dimension of the set of boxes with singularity strength α given by N (α) ∼ l −f (α) .

(94)

This formalism leads to the description of a multifractal measure in terms of interwoven sets of Hausdorff dimensions f (α) possessing singularity strength α. On the other hand, the generalized dimensions Dq , which correspond to scaling exponents for the qth moments of the measure, provide an alternative description of the singular measure. (Indeed, the original formalism in terms of Dq ’s can be traced back to Renyi [161].) We can define a series of exponents parametrized by real number q, according to q Pi (l) ∼ l (q−1)Dq . (95) i

In the limit of the box size going to zero we get the conventional definition q ln i Pi (l) 1 lim . Dq = (q − 1) l→0 ln(l)

(96)

A complete knowledge of the set of dimensions Dq is equivalent to a complete physical characterization of the multifractal. The generalized Renyi dimensions are the exponents that characterize the nonuniformity of the measure; positive q’s accentuate the denser regions and negative q’s accentuate the rarer ones. In contrast to the complicated geometrical interpretation of Dq ’s, f (α) provides a precise and intuitive description of the multifractal measure in terms of interwoven sets with differing singularity strengths α. From eqs (93), (94), (95) and (96), we can write, Dq =

1 qα(q) − f α(q) . q −1

(97)

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Thus, if we know f (α), and the spectrum of α values, we can find Dq . Alternatively, given Dq , we can find α(q) since α(q) =

d (q − 1)Dq , dq

(98)

and, knowing α, f (α) can be obtained from eq. (97). When f (α) and Dq are smooth functions of α and q, they are simply related through a Legendre transformation τ (q) = (q − 1)Dq . The method of directly computing f (α) from a time series is focused on the fact that f (α) is simply the dimension of the measure-theoretic support of a particular measure (see Ref. [160] and the references therein). The prescription for computing f (α) is as follows. Cover the experimental set (attractor) with boxes of size l and compute the probability in each of these boxes Pi (l). Then construct a one-parameter family of normalized measures μ(q) where the probabilities in the boxes of size l are [Pi (l)]q . μi (q, l) = q j [Pj (l)]

(99)

As in the definition of generalized dimensions, (eq. (96)), the parameter q here provides a microscope for exploring different regions of the singular measure. The Hausdorff dimension of the measure-theoretic support of μ(q) is given by f (q) = lim

l→0

N

i=1 μi (q, l) ln μi (q, l)

ln l

.

(100)

Here N refers to the number of boxes or intervals of length l required to cover the set. The average value of the singularity strength αi = ln(Pi )/ ln(l) with respect to μ(q) can be computed by evaluating α(q) = lim

l→0

N

i=1 μi (q, l) ln Pi (q, l)

ln l

.

(101)

Eqs (100) and (101) provide a definition of the singularity spectrum which can be used to compute the f (α) curve directly from a given experimental data. To illustrate this method, Chhabra et al. [160] use the analytically solvable example of the two-scale Cantor measure. They compare their results with the analytical results obtained by Halsey et al. [158] to evaluate the accuracy of the method.

5.4. Characterization of experimental stress–strain series Several investigations have been reported in the literature that verify the existence of a chaotic regime of stress fluctuations [133–137,162]. Subsequent investigations on single crystals of Cu10%Al that combine dynamical and statistical methods have demonstrated

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the existence of two dynamical regimes; the chaotic dynamics detected at low and medium strain rates, and power law state of stress drops occurring at high strain rate [136]. As chaos is characterized by only few degrees of freedom while the power law state is scale free and hence a system with an infinite number of degrees of freedom, these are dynamically distinct states. Studies in Refs [33,134–136,162,163] taken together suggest that two distinct dynamical regimes encountered in single crystals might also be found in polycrystals. Earlier studies on polycrystals reported only chaotic dynamics [135]. In what follows, we summarize the results of studies on the time series and multifractal analyses on two sets of stress–time series, one from a single crystal and another from a polycrystal. This provides a basis for understanding the differences between them as also to see if the crossover in dynamics reported in single crystals is universal. Finally, multifractal analysis which inherently combines the virtues of both statistical and dynamical approaches, is used for analysing the region of crossover. 5.4.1. Single crystal data The analysis of the stress–time series shown in Fig. 18 has been reported in Ref. [136]. The correlation dimension is ν ≈ 2.3 for the lowest and medium strain rate time series. This means that the number of degrees of freedom required for the description of the dynamics of the system is given by the minimum integer larger than ν + 1, which is four [143], consistent with that used in the original model. An independent check of the number degrees of freedom can be obtained by calculating the singular values of the time series. For example, the normalized eigenvalues (with respect to the largest) are shown in Fig. 19(a). The relative strength of the fourth eigenvalue drops more than two orders of magnitude compared to the first and changes very little beyond the fourth. This gives an estimate of the dimension of the experimental attractor to be four which is consistent with that obtained from the correlation dimension.17 Then, for the visualization of the experimental attractor, one can use the dominant eigenvalues to reconstruct the nature of the attractor. Here, the experimental attractor is reconstructed using the first three principal directions of the subspace Ci ; i = 1 to 3. Instead of using C1 , C2 and C3 , Fig. 19(b) shows the reconstructed experimental attractor for the time series for ǫ˙a = 1.7 × 10−5 s−1 in the space of specifically chosen directions C1 –C2 , C3 and C1 . This choice has been used to permit comparison with the attractor obtained from the model. Correspondingly, the Lyapunov exponents for the first two stress–time series (Fig. 18(a), (b)) turn out to be positive. In contrast, the time series for highest (Fig. 18(c)) does not show any converged value of correlation dimension nor a positive Lyapunov exponent. Instead the distribution of stress drops shows a power law [136]. Thus, for single crystals, there is a clear evidence of (low-dimensional) chaotic dynamics for low and mid range of strain rates while at high strain rates, one finds a (infinite-dimensional) power law state. 5.4.2. Analysis of polycrystal data As stated in the introduction, earlier studies on polycrystals of AlMg have confirmed chaotic dynamics [135]. However, as these studies were restricted to a limited range of 17 Usually, for a time series from model systems, one finds a floor level below which the eigenvalues saturate. This is taken as the dimension of the actual attractor. However, in real situations, as in the present case, the eigenvalues do not saturate due to the presence of noise.

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Fig. 18. (a,b,c) Experimental stress–strain curves for ǫ˙a = 3.3 × 10−6 , 1.7 × 10−5 and ǫ˙a = 8.3 × 10−5 s−1 [136].

strain rates, the existence of power law distribution of stress drops was not verified. Thus, one would like to know if the power law states are seen at high strain rates in polycrystals as well. Here we discuss the results of such an analysis on the stress–time series obtained for a range of strain rate in Al2.5%Mg polycrystals of different microstructures. The study concentrates on regions of strain rates with only type B and type A bands, as they are expected to show interesting dynamical behavior in contrast to the type C bands which are uncorrelated. The motivation here is not merely to identify the different dynamical regimes, but their correlations with type B and A bands in polycrystalline samples as well. Apart from the dynamical and statistical methods, multifractal analysis of the stress–time series particularly in the region of crossover in dynamics will be discussed. The motivation for using this method stems from a conceptual similarity between the present transition from hopping to propagating bands in the present case and the Anderson transition in disordered systems [164]. In the Anderson model [165], wave functions are localized when the energy, E, is below the mobility edge Ec and extended for E > Ec . In the neigborhood of Ec , the

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(a)

(b)

Fig. 19. (a) Normalized eigenvalues of singular value decomposition. (b) Reconstructed experimental attractor from the time series corresponding to ǫ˙a = 1.7 × 10−5 s−1 shown in Fig. 18(a) [50,51].

states have been shown to exhibit a multifractal character [165]. In the PLC effect, the hopping type B bands are essentially localized, whereas type A bands are delocalized in the sense that they are propagating. The analysis attempts to quantify this crossover from localized to delocalized nature of the bands using multifractal analysis. The dynamical regimes associated with types B and A bands are identified as chaotic and power law type respectively, and the range of multifractality exhibits a sharp peak in the transition region. Three different types of tensile samples of an Al-2.5%Mg alloy were cut out of a polycrystalline cold-rolled sheet, parallel to the rolling direction. The first set, labeled r, consisted of specimens extracted from the as received samples. The grains were anisotropic in shape with an aspect ratio of 5. A second set, a, was annealed for 4 h at 593 K and water-quenched leading to less anisotropic grains. The third group, aa, was subjected to an additional 3 h annealing at 733 K before quenching. This leads to nearly equiaxed struc-

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Fig. 20. Stress–time series for the as received samples r at three strain rates: (a) 5.56 × 10−6 s−1 , r1 (c) 2.8 × 10−4 s−1 , r4 and (e) 5.56 × 10−3 s−1 , r7 . (b), (d) and (f) are the corresponding plots of |dσ/dt| [137,166].

ture. Tests were carried out at room temperature for eight values of ǫ˙a = V /L in the range of type B and A bands, from 5.56 × 10−6 s−1 to 1.4 × 10−2 s−1 . The stress was recorded at a sampling rate of 20 Hz – 200 Hz. Three typical stress–time curves, σ (t) (corrected for the drift due to strain hardening18 ) are shown in Fig. 20 together with |dσ/dt|. In the region of strain rates 5.6 × 10−6 s−1 ǫ˙a 1.4 × 10−4 s−1 , the data sets typically contain 10 000 to 12 000 points. The band patterns are of type B. First consider the dynamical analysis of the stress time series. For this range, as the results are similar for the different data sets r, a and aa, the results will be illustrated with the data file at ǫ˙a = 5.6 × 10−6 s−1 for the r1 sample. Fig. 21(top) shows the log-log plot of C(r) for d = 15 to 18 using τ = 8. The slopes are seen to converge to a value ν ∼ 4.6 for d = 17 and 18 in the range −4.9 < ln r < −3.2. The correlation dimension has been calculated for the range of strain rates 5.6 × 10−6 s−1 to 1.4 × 10−4 s−1 which turns out to be independent of the strain rate.19 The Lyapunov spectrum has been computed using 18 Serious mistakes are committed in using these kind of techniques which assume stationarity. Usually, the serrations override a generally increasing level of stress. As these methods are applicable to only stationary situation, and the drift due to hardening gives rise to nonstationarity, the hardening contribution needs to be subtracted. Moreover, PLC instability is a strain rate softening instability and hardening does not play a role in triggering the instability. 19 A word of caution is warranted here. The pitfalls of carrying out time series analysis are numerous. For instance, in many cases, when embedding is carried out to high dimensions, an impression of convergence of slope at large dimensions, typically 7 or 8 is created. A better way to check for convergence of slopes for successive d log C(r) values of d is to plot d log r as a function of log r for various embedding dimensions. If one finds an overlap

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Fig. 21. Top: Correlation integral C(r) using d = 15 to 18 (top–down), τ = 8 for ǫ˙a = 5.56 × 10−6 s−1 for the r1 sample. The curves corresponding to d = 15 to 17 have been displaced with respect to d = 18 by a constant amount. Dashed lines are guide to the eye. Bottom: Lyapunov exponents vs. shell size ǫs for applied strain rate ǫ˙a = 5.56 × 10−6 s−1 ; embedding dimension d = 5 [137,166].

Eckmann’s algorithm [138], suitably modified for short noisy time series. In the modified algorithm, enough number of neighbors of a vector ξi contained in a shell of size ǫs centered on ξi is sampled so that the statistics of the uncorrelated noise corrupting the original signal averages out [136,152]. The dynamics is considered to be deterministic if a fair range of shell size ǫs can be found where stable values (i.e. constant in that range) are found for both a positive as well as a zero exponent [136]. The Lyapunov spectrum shown in Fig. 21(bottom) has been calculated using d = 5. Stable positive and zero Lyapunov exponents are seen in the range 6% < ǫs < 12%. The Lyapunov dimension DKY is obtained from the spectrum by using the Kaplan–Yorke conjecture which relates the dij

λ

i mension of the attractor to the Lyapunov spectrum through the relation DKY = j + |λi=1 j +1 | j j +1 such that 1 λi > 0 and 1 λi < 0. The value of DKY obtained from the Lyapunov

of curves for the slopes for some successive d values in a range of log r, then only can one claim the existence of finite correlation dimension. See for details Refs [135,152]. Moreover, the higher the value of the correlation dimension one suspects, the longer time series is essential. In addition, one should carry out the Lyapunov spectrum calculation and check if the DKY so obtained agrees with that obtained using the correlation dimension. It is also necessary to carry out surrogate data analysis as well.

§5.4


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Fig. 22. Distributions |D(ψ)| for the data sets at (a) ǫ˙a = 5.56 × 10−6 s−1 , (b) 1.4 × 10−3 s−1 and (c) 5.56 × 10−3 s−1 [166].

spectrum gives DKY ≈ 4.6 which is close to the dimension obtained using the correlation integral. This is adequate to conclude that the time series is of chaotic origin. Similar results have been obtained with other samples in this region of ǫ˙a , with values of ν and DKY in the range 4.4 to 4.6. In contrast, the correlation dimension did not converge, nor a positive Lyapunov exponent could be found for the data sets at higher ǫ˙a , implying that the dynamics is no more chaotic in this region of strain rate. For these cases, only the largest Lyapunov exponent could be calculated as the data contain only 3000 to 5000 points. Now consider analysing the statistics of the stress drops. Again, the results are illustrated using the set r1 . Instead of dealing with the distribution of stress drops as was done for the single crystals, the quantity |dσ/dt| which reflects the bursts of plastic activity (see Fig. 20 right panel) is used. (Actually, the finite difference approximant ψi (ti ) is used for statistical and multifractal analysis.) Let ψ denote the amplitude of the bursts and t their durations, then the corresponding distributions D(ψ) of ψ , and D(t) of t are easily obtained. Plots of D(ψ) are shown in Fig. 22 for typical values of ǫ˙a . Peaked distributions are seen in the chaotic regime (5.6 × 10−6 s−1 to 1.4 × 10−4 s−1 ) indicating the existence of characteristic values. However, for higher strain rates, on the basis of single crystal results, one should expect to find a power law state,20 which is clear from Fig. 22(c) for ǫ˙a = 5.56 × 10−3 s−1 . The distribution becomes broader and asymmetrical in the mid-region of ψ with increasing strain rate ǫ˙a , eventually leading to power law distributions (Fig. 22(c)). For the data set at ǫ˙a = 5.6 × 10−3 s−1 , the distribution has the form D(ψ) ∼ ψ −a over one order of magnitude in ψ with a ∼ 1.5. Similarly, a power law distribution D(t) ∼ t −b for the duration of the bursts is found with b ∼ 3.2 and for the conditional average ψc ∼ t x with x ∼ 4.2. As expected, these exponents that characterize the power law dynamics satisfy the scaling relation x(a − 1) + 1 = b. Similar results are also found at the highest strain rate ǫ˙a = 1.4 × 10−2 s−1 . Thus, the region of power law dynamics extends from ǫ˙a = 5.6 × 10−3 s−1 onward, coinciding with the region of type A bands. 20 In contrast to self-organized criticality type dynamics where power law statistics arises when spatially extended slowly driven systems evolve naturally to a critical state, here the power law arises at the upper end of the strain rate. However, we shall ignore this distinction and continue to use power law dynamics and SOC interchangeably.

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(a)

(b)

Fig. 23. (a) Plot of

i μi ln pi vs. ln δt for the r6 file. The slope gives α. (b) Plot of r6 file. The slope gives f (α).

i μi ln μi vs. ln δt for the

As mentioned in the introduction, multifractal analysis is eminently suited if the object considered has a broad distribution of ‘length’ scales. In the present context, it is clear that the extent of propagation increases as we increase the applied strain rate beyond the type B bands. This is again similar to the transition region between localized and delocalized states in the Anderson transition [165]. In this case also, there is a broad distribution of time scales in the region separating chaos and power law states (Fig. 22(c)). Thus, one can anticipate that multifractal analysis will quantify this heterogeneity. Let N = N (δt) be the number of time intervals δt with m points, required to cover [0, K]. Then, Kthe normalized ψ / amplitude of the bursts in the ith interval δt, is pi (δt) = m k=1 im+k j =1 ψj , usually called the probability measure. The nonuniformity of the measure is captured by the range of multifractality θ = αmax − αmin , where αmin and αmax are the extreme values of α. Both α and f (α) can be directly calculated [160] using the measure given in eq. (99). α and f (α) can be directly calculated through eqs (100), (101). This canonical method has been shown to be suitablefor the analysis of short experimental data sets [160]. Plots of i μi ln pi vs. ln δt and i μi ln μi vs. ln δt for q = −5,0

§5.4


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Fig. 24. Multifractal spectrum (α, f (α)) for the applied strain rate ǫ˙a = 1.4 × 10−3 s−1 for the file r6 , q ∈ [−5, +5].

Fig. 25. The multifractal range α vs. applied strain rate ǫ˙a . Regions of chaotic type B and power law type A bands are marked [166].

and 5 are shown in Fig. 23(a), (b). Note that the scaling range extends over two orders of magnitude. The spectrum (α, f (α)) is shown in Fig. 24 for ǫ˙a = 1.4 × 10−3 s−1 . The dependence of the multifractal range α = αmax − αmin on ǫ˙a is summarized in Fig. 25 for different types of microstructures. Also displayed are the different dynamical regimes together with the band types in Fig. 25. As can be seen α has relatively low

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values at both low and high strain rates. The small value of α in the chaotic regime is due to the sharp peak in D(ψ) (Fig. 22(a)), while that in the SOC region is due to the scaling nature of the distribution (Fig. 22(c)). In contrast, a sharp peak observed at intermediate ǫ˙a clearly signals a transition in the nature of dynamics from chaotic B-type to scale invariant power law stress drop statistics for the A-type bands. 5.4.3. The PLC dynamics as a dynamical critical point phenomenon Finally the stress–time series of the PLC effect has also been analysed from the view point of critical point phenomenon. Recall that a second order transition such as magnetic, liquidgas or superconducting transition, is a continuous transition wherein the order parameter, say the magnetization or the density of the fluid, goes to zero from a finite value as we approach the critical temperature Tc from below. The order parameter is zero in the high temperature phase above Tc . In the neighborhood of the transition, several measurable quantities like the magnetization and susceptibility diverge as we approach the critical point as a power of T − Tc . The correlation length also diverges at this point. These divergences are a reflection of growing fluctuations as we approach the critical point. For instance, as we approach Tc , fluctuations in the density of the fluid grow which also have a tendency to live longer and longer. In fact, these fluctuations are visible to the naked eye as critical opalescence. The physical extents of these fluctuations grow and extend to the size of the sample as we approach the critical point. (See Fig. 1.6 of Ref. [167].) The long lived nature of fluctuations has been referred to as critical slowing down which can again be described by some dynamic scaling exponents [90]. D’Anna and Nori [168] attempt to regard the PLC effect as a dynamical critical phenomenon. In order to obtain long time series, they carry out compression tests on polycrystalline samples of Al4%Mg at 18◦ C for various compression speeds. They define ǫ = − ln(1 − l/ l0 ) as a measure of the extent of deformation. Instead of the stress– strain curves that were used in dynamical analysis, they consider − dσ dt (as in Section 5.4.2) which essentially represents bursts of plastic strain rate. This allows them to identify the waiting time tw,n between successive bursts n and n + 1 (after eliminating the small drifts). A plot of − dσ dt for various values of compression speeds is shown in Fig. 26. The first burst appears at ǫ = 16%. Initially, the intervals between bursts decrease with increasing deformation. However, for large deformation, tw,n again tends to increase beyond 35%. The waiting time appear to be random for ǫ < 35%, (perhaps chaotic, which they do not verify) beyond which the waiting time appears to be more regular. They make a special mention that they do not find a power law statistics for the bursts sizes in reference to the early studies in Ref. [163]. However, the magnitudes of the stress drops obey power law and not − dσ dt (one does not imply the other). Only much later was it shown [137,166] that bursts also show a power law distribution, but in a different material. They identify two critical values of ǫc1 and ǫc2 , one at low values and another at high values respectively (for different velocities) with respect to which tw,n diverges, i.e. tw,n = A1 (ǫn − ǫc1 )−β1 for the low values and tw,n = A2 (ǫc2 − ǫn )−β2 for high values. The values of ǫc1 and β1 are found to be 0.13 and 0.58, respectively. Similarly, ǫc2 and β2 are found to be 1.1 and 1.8. Fig. 27 shows a plot of tw,n as a function of ǫn . The plot shows large fluctuations and critical slowing down in the number of bursts per unit time in the jerky

§5.4 Statistical and dynamical approaches to collective behavior of dislocations

Fig. 26. Typical plots of −dσ/dt verses time for deformation velocity v = 0.18 µm s−1 for various values of deformation ǫ. Note that on an average, the waiting time decreases with increasing values of ǫ. After D’Anna and Nori [168].

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Fig. 27. The waiting time tw,n between successive bursts as a function of ǫn . The power law is shown by the dashed line. Large fluctuations and critical slowing down are clear from the plot. After D’Anna and Nori [168].

phase close to the critical values of ǫc1,2 . The parameter (ǫ − ǫc1,2 ) takes the role of the reduced temperature in equilibrium systems. The influence of the compressional speed on the jerky regime has also been investigated. Fixing the deformation parameter at 25% and increasing the speed v decreases the average tw , with the average being taken over 30 events around this value of ǫ. The dependence of the average waiting time is described by tw = Bv −α . D’Anna and Nori find that the exponent is 0.8, the same for two different values of ǫ = 25% and 60%. These results can be explained by noting that fast drive rates and high deformation are equivalent in the PLC effect in the following sense. At slow speeds, there is sufficient time for relaxation to allow the loading-unloading cycles to complete, which involves a large number of channels to be activated and deactivated. In contrast, at high speeds, a few percolating type channels dominate the dynamics. The similarity of the PLC dynamics with flux line lattices has also been pointed out [169]. From the above discussion, it appears that there are few differences in the nature of PLC dynamics observed during compression in comparison with that in tension tests. The existence of lower and upper critical values of the deformation parameter is similar to the existence of critical strains in the usual PLC effect under tension. However, the analysis carried out by our group so far has not shown any evidence of critical slowing features, particularly at higher deformation rates. This might be entirely due to the fact that we never reach such high deformation levels. The other difference is the presence of power law

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statistics in tension tests which is not observed in these compression tests. A comparison between Fig. 20 for Al-Mg polycrystals and Fig. 26 might provide a hint. While there are bursts of all sizes in Fig. 20, the burst in Fig. 26 is abrupt, starting from the baseline. This might suggest that the plastic component is not easily relaxed in a compression test. Note that in Fig. 20(e), the plastic component is evident even in the loading part. More work is necessary to understand these differences.

5.5. Discussion Distinct dynamical regimes and spatio-temporal patterns are displayed in the above analysis. Chaos is associated with type B hopping bands at low applied strain rates and power law dynamics is present at high ǫ˙a in the domain of type A propagating bands. The crossover from chaotic to power law dynamics is clearly signaled by a burst in multifractality. Such a diversity in dynamics poses a challenge for modeling the PLC effect. What needs to be understood is why the spatial correlations between bands and the complexity of the dynamics increase with increasing ǫ˙a . The above studies on Al-Mg polycrystals establish a correspondence between the distribution of plastic activity bursts, the band types and the types of dynamics. While the existence of the two distinct dynamical regimes, one at low and mid range of strain rates and another at high strain rates, has been reported earlier for Cu-Al single crystals [136], the studies on polycrystals establish a connection of these dynamical regimes with band types as well. However, it must be stressed that the existence of the two dynamical regimes themselves is universal. In polycrystals, the chaotic regime is associated with hopping type B bands. The chaotic regime is observed at low strain rates in all microstructures and invades higher strain rates as the material is microstructurally hardened. As chaotic behavior is conveniently described by the evolution of a small number of coupled degrees of freedom including dislocation densities, the sensitivity to microstructure is no surprise. In the upper range of strain rates corresponding to type B bands, asymmetrical distributions of bursts of plastic activity and an increased level of multifractality signal the extent of the transition region between chaos and SOC. In this region of crossover, increased levels of heterogeneity hint at mixture of chaotic and power law dynamics. While the range of chaotic dynamics is sensitive to microstructure, the range of power law dynamics and type A propagating band is not structure-sensitive which is understandable considering the scale-free correlations associated with SOC. Multifractality provides a measure of the heterogeneity of the dynamics in this region, in the form of increased levels of the order parameter α in the region of crossover between the high and low strain rate regimes. Short of resorting to full scale modeling, some physical explanation about the sequence of dynamical changes observed with the increase in strain rate appears possible based on the relevant time or length scales. The characteristic loading time tl between two serrations varies, as the reloading sequences are not purely elastic. However, tl also shows a general decreasing trend with the loading rate. One picture that can be assumed is that the spatial coupling originates in the elastic internal stresses arising from geometric incompatibilities

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between differently strained regions, both in single and polycrystals. Let the plastic relaxation and recovery of these internal stresses occur with a characteristic time tr . Then if sufficient time is allowed, it would lead to a decreasing intensity of the spatial coupling. Further, one can define a correlation length scale lp , which is essentially the distance over which internal stresses in the strained regions can contribute to the activation of slip in the undeformed ones. This length scale decreases with increasing relaxation of the spatial coupling. In simple terms the lack of spatial correlation for the type C can be seen to arise from the fact that, at low strain rate, the loading time is much larger than the relaxation time (tl ≫ tr). Thus, the strain gradients produced by a previous band are relaxed below the level of internal stress fluctuations, which implies that the correlation length scale is small. As the applied strain rate increases, less time is available for plastic relaxation to occur. In this case, the nucleation of a band is favored within the finite correlation distance lp of the previous band as tl ∼ tr . This leads to the hopping nature of the band. However, while the spatial correlation is nonzero, its range of influence is limited to the extent that band initiation occurs ahead of it. Considering the fact that two successive events are correlated yet separated in time, one does not expect a large number of degrees of freedom in the dynamics. The occurrence of chaos seems natural, but clearly requires considerable dynamical information to model this feature (see next section). At high strain rates, there is hardly any time for plastic relaxation to occur during the reloading time. This also means that the amplitude of the stress drops progressively becomes small. Thus, the stress is always close to the critical value for the onset of a plastic burst, as expected in a critically poised system. New bands form before plastic relaxation is complete, which results in a recurrence of partial relaxation events. Hence, a hierarchical distribution of plastic activity bursts occurs, leading to the power-law distributions without a characteristic value associated with SOC dynamics. Thus, the type A bands propagate continuously due to a large correlation length scale lp . The crossover regime in the dynamics is expected when the reloading time tl is less than the plastic relaxation time tp . The main conclusion from this qualitative discussion is that a valuable spatio-temporal model for the PLC effect must include not only the local internal stresses, which are usually modeled via a Laplacian term, but also a time-dependent mechanism for their relaxation. The difference in the increased value of the correlation dimension of the polycrystal from that of single crystals needs some interpretation. This is likely to be due to a transition between planar, crystallographic single slip in single crystals to the three-dimensional and multislip nature of polycrystal plasticity. In the domain of power law statistics of stress drops that coincides with the region of propagating bands, we note that the values of exponents are also higher in the case of polycrystals compared to single crystals. One may speculate that, in polycrystals, this feature is the result of a smaller scale in the dislocation microstructure. The above methods of analysis of the deformation tests on bulk samples can be used to gain insight into collective defect properties. However, the notion that dislocations exhibit chaotic or power law dynamics may apply as well to macroscopically uniform plastic flow. This conjecture can be checked by monitoring variables more sensitive to local bursts of dislocation activity than the applied stress or the average strain. One such method could

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be an appropriate acoustic emission technique. For instance, acoustic emission data from ice crystals deformed in creep have recently been analyzed, leading to the identification of power-law distributions and possibly SOC dynamics [66,74]. The above approach illustrates the power of modern methods of analysis prevalent in dynamical systems. It gives a level of understanding that has not been possible till now. Clearly the above analysis gives a hint of the interplay of length scales and time sales that lead to the complex spatio-temporal features of the PLC dynamics. A full understanding of these results will have to wait for in depth modeling based on dynamics which will be considered in the next section.21

6. Dynamical approach to modeling the Portevin–Le Chatelier effect 6.1. Early models for dynamic strain aging Historically, the earliest explanation for the PLC effect was put forward by Le Chatelier before the advent of dislocation theory. He suggested that plastic deformation induces a phase transformation which hardens the alloy and this transformation requires some time to relax resulting in serrations. The current understanding of the PLC effect is based on dynamic strain aging [34,35], and its improvements [37,38,173–175]. In this picture, the motion of dislocations is recognized to be jerky with dislocations aging during the period when they are temporarily arrested at obstacles (in contrast to the original suggestion that dislocations drag the solute atmosphere [35]). One critical feature that has attracted some attention is the existence of critical strain before the onset of the PLC instability. The possible causes have been attributed to enhanced solute diffusivity due to deformation induced vacancies [37,38] and increased mobile dislocation density. Both these mechanisms are considered to contribute for substitutional alloys while vacancies make no contribution for the interstitial alloys [174]. A generalization of the Cottrell and Bilby model [34] for including saturation effects of solute concentration due to aging and which includes the dependence on the aging time tw , is given by

C0 tw 2/3 . C = Cs 1 − exp − Cs tw0 Here Cs and C0 are the saturation and the bulk values respectively of the concentration and tw0 is the time constant associated with diffusion of solute that depends on solute 21 Recently Barat et al. [170] reported results of a scaling analysis using standard deviation (SD) and diffusion entropy (DE) analysis [171] of the stress–time series. They have considered Al-Mg alloys deformed over a range of strain rates where the Portevin–Le Chatelier effect is seen. The authors claim that the overall dynamics of the PLC effect follows a Levy-walk property in contrast to the chaotic or self-organized critical dynamics reported earlier [136]. This conclusion can be easily shown to be an artifact of applying a methodology applicable to stationary time series [171] to the raw PLC time series which is nonstationary due to the drift in the time series arising from hardening term. Even a small amount of drift added to a stationary Gaussian noise gives rise to a Hurst exponent unity for SD while it should give H = 1/2 for stationary Gaussian noise. As their analysis uses the raw PLC time series with hardening contribution, it always leads to a Hurst exponent value close to unity [172]. Moreover, the analysis does not distinguish different types of serrations found with increasing strain rate.

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mobility, solute concentration and binding energy to dislocations [175]. Kubin and Estrin [176,177] addressed the problem of the existence of critical strain(s) without having to resort to vacancy aided enhancement of diffusion. They demonstrated that by considering the dependence of incremental strain on the mobile and forest densities, it is possible to obtain estimates of critical strain values. They assume that the strain rate sensitivity S can be expressed as a sum of two additive contributions, S0 from the activation of dislocation motion in the absence of DSA and Sa , from DSA. S0 is considered as a constant for the sake of simplicity. Sa depends on the increase in stress due to aging of dislocations pinned at obstacles. The aging kinetics of Louat [175] translates to the dependence of stress τage (T , tw ) on the waiting time tw given by τage = τage0 1 − exp −(/γ˙ tw0 )2/3 ,

(102)

where is the elementary incremental strain produced when all mobile dislocations perform a successful thermally activated step through forest obstacles, τage0 is the maximum value of the stress produced due to aging. τage depends on and γ˙ through tw = /γ˙ . Then the contribution from DSA to the total SRS is Sage (γ ) =

∂τage . ∂ ln γ˙ γ

Using the condition for instability S(γ ) = 0 leads to the transcendental equation X exp(−X ) = A where X = (/γ˙ tN0 )2/3 and A = 3S0 /2τage0 . This function has a maximum at X = 1. For a given temperature and strain rate, the range of instability is then limited to X1 < X < X2 . The critical strain rate γ˙c (γ ) should have a similar functional form as (γ ). There3/2 fore, the condition for the appearance of serrated flow is given by X1 τage0 γ˙ (γ ) 3/2 X2 τage0 γ˙ . In general, and tw0 evolve with strain. This also implies that one might find the stability boundary might be lot more complex. That is, one finds that there are more than two critical strains implying that there is a region of mid strain values where the PLC effect may not be seen as in Cu-3.3%Sn, for example [178]. The possible scenarios such as those with no critical strain to those with four critical strains have been discussed in detail. These depend on the ratios of X1 /X2 as discussed in Ref. [177]. These authors support the possible types of changes in by setting up coupled rate equation for the mobile and forest densities. This prediction of the model has been verified by experiments [179–181]. In the above pictures, the system is assumed to respond instantaneously to any changes in deformation condition. To account for the ensuing transient effects, McCormic [182] suggested that the effective aging time is governed by a first order relaxational kinetics with a time constant tDSA = /γ˙ . However, as the time constant goes to zero, the effective aging time goes to infinity and strain rate also goes to infinity which is unphysical. Hähner [183,184] has generalized this approach by introducing a waiting time distribution,

∞ G(tw ) − Va τeff , (103) γ˙ = ν dtw f (tw , t) exp − kT 0 where ν is an attempt frequency and G(tw ) = G∞ − G exp[−(tw /tw0 )2/3 ]. The waiting time distribution is fixed by noting that it is the product of the probability of a dislocation

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being activated in a time t − tw and the probability of not being activated in the interval (t − tw , t). This leads to a Volterra type of integral equation of the second kind. The procedure determines the tDSA introduced by McCormic [182]. It also shows that in the absence of DSA, the retardation effects vanish. Further, at high strain rates when DSA becomes inefficient, tDSA ∼ γ˙02 while at very low strain rates where dislocations get aged completely tDSA ∼ γ˙qs , where γ˙qs is the quasi-steady state strain rate (see Ref. [183]). 6.2. General introduction to dynamical models of the Portevin–Le Chatelier effect Clearly, in all these kinds of models, there is no dynamics that is so intrinsic to the PLC effect, nor is there any spatial aspect. While, at the macroscopic level, the negative SRS signals the onset of the instability, it is not adequate to describe different serrations, a signature of space averaged stress, found in different regimes of strain rates, much less the associated nature of band types described earlier. More importantly, several deeper questions remain unanswered. For instance, what is the mechanism that induces collective motion of dislocations once the sample is subjected to the regime of strain rates or temperatures in the instability domain? Outside this domain, as there is no coherence in dislocation motion, given an ensemble of dislocations, how do dislocations know when to move coherently and when not to? Another question is about the origin of negative strain rate sensitivity starting from individual dislocations. Is the negative SRS the cause of this correlated motion of dislocations or given some other prerequisites, does the negative SRS effect emerge naturally? For instance in terms of dynamics, jumps between stable manifolds occur when separated by an unstable manifold and these themselves are self-generated once the equation of motion are given. Some of these issues themselves lead to fundamental questions about the necessity for using coarse grained variables. More importantly, the results of the previous section suggest that dynamical techniques should play a natural role in modeling the PLC effect. However, most models that attempt to describe the spatio-temporal features use the negative SRS or the concept of DSA in one form or the other. Some of these models have been dealt with in the recent reviews [3,4]. The purpose of this section is to review the advances made in the last couple of years. Three different types of approaches, all of which can be broadly classified as dynamical, will be discussed. As we shall see, each model has its own merits and limitations by virtue of the philosophy adopted for modeling the PLC effect. This also leads to a broad classification of the models. The first model that we consider here, introduced by Ananthakrishna and coworkers [107,108], can be thought of as a fully dynamical model in the sense that the starting point is populations of different types of dislocations and possible dislocation mechanisms. In a certain sense, the model is technically complicated as the number of degrees of freedom is four, but uses idealized dislocation mechanisms. The second model introduced by Hähner [44,185], though it uses dynamical approach, adopts the DSA framework, but uses only two variables making it tractable. Both these however work in one space dimension while in reality, the PLC effect is observed in three-dimensional samples. The third model [186] is a mixed one that uses polycrystalline plasticity along with negative SRS as an input, but works in three dimensions and hence makes it easier to appreciate the results. The last two models, however, do not use dislocation populations.

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6.3. Introduction to a fully dynamical approach to the Portevin–Le Chatelier effect Describing spatio-temporal features of the Portevin–Le Chatelier effect is a particularly difficult task for a variety of reasons. First, these spatio-temporal structures emerge from the collective behavior of dislocations in real materials and therefore describing pattern formation in such realistic conditions is far more difficult than ‘ideal’ nonequilibrium physics situations. Second, though properties of individual dislocations and their interactions have been known for a long time, there is no accepted framework to describe the collective properties of dislocations. Some of these difficulties have been exposed in the efforts that have been pursued in the last three decades. Another important reason that makes the PLC effect much harder to describe among pattern forming systems is that it involves both fast and slow collective modes of dislocations. This requires specific techniques of nonlinear dynamics for further analysis. Further, these time scales themselves evolve as a function of strain rate and temperature which in turn leads to different types of serrations. At low strain rates, the existence of both the fast time scale corresponding to stress drops and the slow time scale corresponding to the loading time scale are clearly discernible. This also means the bistable nature of stress is reflected in the upper and the lower envelope of the stress values. However, at high strain rate, as internal (plastic) relaxation is not complete, a clear demarcation of the slow and fast time scales becomes difficult. This along with the corresponding length scales (band widths), which also evolve, points to an extremely complex underlying dynamics. It may be pointed out that slow–fast dynamics and the negative flow rate characteristic is common to many stick-slip systems such as frictional sliding [187], fault dynamics [188] and peeling of an adhesive tape [189]. The inherent nonlinearity and the presence of multiple time scales demands the use of tools and concepts of nonlinear dynamics for a proper understanding of this phenomenon. The first dynamical approach was undertaken in early 80s by Ananthakrishna and coworkers [107,108], which by its very nature affords a natural basis for the description of time dependent aspects of the PLC effect. Further, it also allows for explicit inclusion and interplay of different time scales inherent in the dynamics of dislocations [108]. Despite the simplicity of the model, many generic features of the PLC effect such as the existence of a window of strain rates and temperatures within which it occurs, etc., are correctly reproduced. More importantly, the negative SRS has been shown to emerge naturally in the model, as a result of nonlinear interaction of the participating defects [108,190]. Due to the dynamical nature of the model, one prediction that is unique to this model is the existence of the chaotic stress drops in a certain range of temperatures and strain rates [191,192]. It is this that triggered a series of experiments to verify this prediction [133–135,162] by using nonlinear time series methods [20,22]. Moreover, the fact that the number of degrees of freedom estimated from the experimental time series turn out to be same as in the model can be taken as a support for the three collective modes (plus the experimental condition) used in the model. However the recent efforts to extend this analysis of the time series to entire range of strain rates of the PLC effect throws up further questions about the possible underlying mechanisms leading to the intriguing crossover from a chaotic state at low and medium strain rates to a power law state at high strain rates [136,137,166]. As the crossover is

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observed in both single and polycrystals, the crossover appears to be insensitive to the microstructure. However, at a fundamental level, the chaotic state is dynamically distinct from the power law state. The former involves a small number of degrees of freedom characterized by the self-similarity of the attractor and sensitivity to initial conditions [20] while the latter is an infinite-dimensional state reminiscent of self-organized criticality [73, 193]. Due to this basic difference in the nature of the dynamics, most systems exhibit either of these states. From the point of view of the PLC effect, these studies also demonstrate that the nature of the dynamics in a given strain rate regime is correlated with the nature of the band type. As shown in the last section, the chaotic state has been identified with the type B bands and the scaling regime at high strain rate with the propagating type A bands [137]. Thus, clarifying the dynamics operating in these different regions can provide insight into the PLC effect. All these new insights not only confirm deterministic dynamics of the PLC effect, they also pose additional challenge for modeling the detected crossover in the dynamics and other features of the PLC effect. The dynamics of the crossover as a function of strain rate is unusual in a number of ways. First, the PLC effect is one of the two rare instances where such an intriguing crossover phenomenon is seen, the other being in hydrodynamic turbulence [194]. Second, the power law, both in the PLC effect and in turbulence, arises at high drive rates [194,195]. Here it must be stated that even within the context of dynamical systems, there are very few models that can capture this crossover. Thus, modeling such a crossover poses a challenging problem even in the area of dynamical systems, more so for the PLC effect as such a model should also recover all the features of the PLC effect in addition to explaining the crossover feature. The issues raised in the above discussion require very specific theoretical techniques. In particular, singular perturbation and slow manifold techniques are important tools which help us to analyse the AK model that incorporates multiple time scales inherent to PLC dynamics. The next section is meant to provide the background material to the reader. 6.3.1. Background material on multiple time scale dynamical systems The time evolution of an autonomous (no explicit time dependence) dynamical system can be formally represented by a differential equation of the general form d x = F( x ; ζ ), dt

(104)

where x is a vector in n-dimensional phase space, Rn , and ζ is the control parameter. The function F, in general, is a nonlinear operator acting on points in the phase space. The changes in the state of the physical system can be described by the passage of a point in the phase space (state space) as time proceeds. Let x0 be the position of the point at time t = 0 and xt its position at time t. Thus, the history of the system represented by x as a function of time is described geometrically by a curve which is a mapping from x0 to xt . This forms a trajectory in the phase space of the continuous time dynamical system. Dynamical systems may be conservative or dissipative. As plastic deformation is a highly dissipative system, we will be concerned only with dissipative dynamical systems for which the volume in phase space shrinks with time to lower-dimensional equilibrium sets like point attractors, periodic orbits, etc. This condition for dissipation is

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∇.F(x; ζ ) < 0. This means that for different initial conditions, after sufficiently long time, the motion reduces to equilibrium sets whose dimension is necessarily smaller than the phase space dimension of the dynamical system. The long term behavior is essentially governed by the properties of these equilibrium sets. These equilibrium sets can be of zero dimension, one dimension (limit cycle), a torus (two-dimensions) or strange sets which have fractal dimension. If A is an equilibrium set, the basin of attraction of A is the set of initial conditions x0 , such that all trajectories approach A as t → ∞ (stable equilibrium) or as t → −∞ (unstable equilibrium). More than one attractor with different basins of attraction can coexist for a fixed value of a control parameter. For convenience, let us assume the existence of a point attractor or an equilibrium fixed point at x = 0 for the dynamical system given by eq. (104). The stability of the fixed point can be determined by linearizing around the fixed point x = 0 described by x˙ = Dx F(0; ζ ) x

(105)

and finding its eigenvalues. A typical situation is that the eigenvalues are nondegenerate given by λi (i = 1, . . . , n) which may occur in complex conjugate pairs also. The qualitative behavior of the solutions of this linear system depends on the sign of the eigenvalues. An eigenvalue corresponds to a mode of the dynamical system, and depending on whether Re(λi ) is greater, less than or equal to zero, the modes are said to be unstable, stable or center, respectively. If Re(λi ) < 0 (Re(λi ) > 0), then as t → ∞ (t → −∞), xi′ decays (grows) with the time scale of λ−1 i . The eigenvectors of similar nature are grouped depending on the sign of the corresponding eigenvalues to form stable (E s ), unstable (E u ) and center (E c ) eigen-subspaces. E s = span e | e ∈ Eλ and Real(λ) < 0 , E u = span e | e ∈ Eλ and Real(λ) > 0 , E c = span e | e ∈ Eλ and Real(λ) = 0 .

These subspaces span the phase space, Rn = E s ⊕ E u ⊕ E c and they are invariant under the flow given by eq. (104). That is, if x(0) belongs to E s , then as t → ±∞ trajectory remains within the subspace E s . Similar arguments hold for other subspaces also. The extension of the above classification to nonlinear dynamical systems is provided through Hartman–Grobman theorem. 6.3.2. Slow–fast dynamical systems A family of dynamical systems involving multiple time scales of the interacting modes is generically called slow–fast dynamical systems. For the sake of illustration, consider a simple example of slow–fast dynamics in linearized dynamical system (around the fixed point) as given in Fig. 28. For the case of a two-dimensional dynamics, assume the existence of an attracting equilibrium point with two real eigenvalues λ1 and λ2 such that |λ2 | ≫ |λ1 |. The corresponding eigen-directions (e2 and e1 ) are shown as thick lines in the figure. The slow–fast nature of the dynamical system manifests through the difference in the magnitude of the eigenvalues. That is, the time scale of evolution along any one of

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Fig. 28. Schematic diagram of a slow–fast vector field. e1 and e2 correspond to slow and fast directions, respectively. The size of the arrow indicates the magnitude of the eigenvalue. The thin line is the generalization of the slow direction for a nonlinear system.

the eigen-directions is much larger than the other eigen-direction. In the long time limit, the dynamics is essentially controlled by the slow direction alone, i.e. the fast mode, with eigenvector e2 , being enslaved by the slow mode, e1 (see Fig. 28). A variety of physical systems exhibit characteristic features of multiple time scale dynamics. Slow–fast behavior in these dynamical systems arises typically whenever some parameter in the evolution equations takes relatively small values compared to other parameters of the system. Examples include the PLC effect, autocatalytic chemical reactions and many stick-slip systems. For instance, the autocatalytic reaction rate typically is much larger than the other rates of the reaction involved. In stick-slip systems, the loading part corresponds to the slow time scale while unloading part to the fast. This leads to multiple time scales of evolution of different species which results in a slow–fast dynamical system. For example, consider a autonomous dynamical system of the form dx = f (x, y, ε), dt dy = εg(x, y, ε). dt

(106) (107)

This dynamical system is called a slow–fast dynamical system, because of the small parameter ε multiplying one of the velocity components. To see the nature of the dynamics in the long time limit, transform the time variable as τ = εt. Then, eqs (106) and (107) become, ε

dx = f (x, y, ε), dτ dy = g(x, y, ε). dτ

(108) (109)

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In the limit ε → 0, to be consistent, f (x, y, ε) → 0, i.e. the values of x and y should be such that within the short time scale of ε −1 , f (x, y, ε) should average to zero. This also implies that the evolution is controlled only by the slow subsystem variable y. In a similar way, a transient time dynamics can also be analyzed by scaling the time variable correspondingly as τ = t/ε. At ε = 0, we obtain the fast sub system x˙ = f (x, y, 0), where y acts as a parameter. The passage from ε = 0 to ε = 0 is characterized by the inclusion of y into the dynamics of the system. The simplification in analysing the dynamics of a multiple time scales system can be illustrated by considering the last example. It is clear that the fast variable x is slaved by the slow variable y in a very short time. Thus, in the long time limit, the dynamics is controlled only by the equation for the slow variable. Hence, the almost stationary state of the fast variable is taken as dx/dt = 0 and this defines a slow curve (S), f (x, y, ε) = 0. Once the dynamics is reduced to the slow manifold, the motion of the system on this manifold can be treated as a part of a new coordinate system. To see this, define x = x − S.

(110)

Substituting the above in eq. (106), and using Taylor series expansion we find, x˙ =

df (S, y; ε)x + (x)2 . dx

(111)

However, the nature of the slow curve is determined by the sign of df dx (S, y; ε). An example of the slow curve is given in Fig. 29. Except in close neighborhood of the slow curve, x varies rapidly. For ε small, x˙ = f (x, y) is relatively large, so the solution jumps vertically up towards the slow curve where f ′ (x) < 0. Thus, points on the curve S are attracting. Along the slow curve the dynamics is controlled only by the evolution of the slow variable. An important feature of these dynamical systems of the above form is that the magnitude of the fast variable (x) changes relatively instantaneously to ‘almost’ stationary state value and further dynamics is controlled only by the slow variable. Hence, in the long

Fig. 29. Slow dynamics curve in a two-dimensional system. Slow curve with stable and unstable pieces given by continuous and dashed lines respectively. Fast directions are indicated by the double arrow heads.

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time limit, the behavior of the dynamical system can be approximated by the behavior of the slow variable with implied constraint of stationarity of the fast variable. Indeed, even the simple nullcline method of studying limit cycle solutions makes use of the separation of fast and slow variables. As we shall see, in the case of the two PLC models that will be considered, the principle is used without stating the underlying mathematical set up. Note that multiple time scales manifest themselves in the dynamics primarily in two forms: as fast transients evolving towards an equilibrium state and/or as a part of a steady state dynamics as in the case of oscillatory evolution. In either of these cases, closed form solutions available are not easy to obtain and numerical integration is inevitably preferred to study the nature of the dynamics of the system. However, in many of the nonlinear dynamical systems owing to the coupled nature of the set of differential equations, the dynamics are too involved and thus numerical integration does not provide insight into the nature of the dynamics. Then, effective mathematical techniques of analyzing these systems involve investigating the dynamics in different domains of time scales where analytical techniques can be applied [196]. A variety of techniques and approximations are available for analysis based on this approach. In many cases, the reduction process itself often provides insight into the dynamical nature of the system. We shall discuss some of the bases of these techniques. The multidimensional, multiple time scale counterpart of eq. (104) can be written as d x = f( x , y, ε), dt d y = εg( x , y, ε), dt

(112) (113)

where x, y are n-, m-dimensional vectors in Rn and Rm respectively. ε is m-dimensional vector of small valued real numbers. ε → 0 leads to a fast subsystem surface spanned by the fast variables x˙ = f ( x , y, 0) = 0. This topological surface in Rn represents a trivial invariant manifold (slow manifold) of the full system. 6.3.3. Slow manifold analysis In the slow–fast dynamical systems, the explicit presence of multiple time scales through small parameters provides a convenient setting for simplification of the asymptotic dynamics. The approach adopted depends on the separation of the time scales and generally relies on the investigation of the behavior of the trajectory in the phase space [197]. Basically, the technique accomplishes dimensional reduction, i.e. provides a smaller dimensional version of the original dynamical system which retains the essential dynamics of the original dynamical system. These analyses are best suited for the nonlinear slow– fast dynamical systems wherein all trajectories are attracted, in the long time limit, to a subspace of Rn forming an invariant manifold. The confinement of the trajectories to a subspace also implies that the effective dimension of the dynamics is reduced which facilitates analytical approach also. These methodologies have been widely discussed in the chemical dynamics literature [198,199]. A standard procedure is to use the singular perturbation technique or its variants [200]. In this approach, the solutions of the fast variable are determined by singular perturbation

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technique. Substituting these solutions into the original dynamical system gives a lowerdimensional dynamical system which retains the dynamics of the enslaved fast variable. The goal of this methodology is to obtain an approximate algebraic expression S( x , y) from eqs (112) and (113) such that x − S( x , y) δ,

(114)

x = S0 + ǫS1 + ǫ 2 S2 + · · · .

(115)

for large enough time, where δ is an arbitrarily small number. The expression x − S( x , y) = 0, therefore describes the inertial or slow manifold. Using singular perturbation methods or its variants, a solution for the fast variable can be constructed as a series expansion in the small parameter. Then, a formal algebraic approximation of an inertial manifold takes the form

This relation describes the inertial manifold with desired accuracy and when solved for, x can be used for the elimination of the fast variable from the system of equations. The zeroth order approximation of the slow manifold S0 and the resulting approximation to equations d y = εg(S0 , y), dt

(116)

are simply equations resulting from the quasi-steady state approximation to the fast variable. The higher order terms −εS1 − ε 2 S2 − · · ·

(117)

describe the error in the steady state of the fast variable induced by the application of quasi-steady state approximation. The approximations from the singular perturbations and the quasi steady state approximation agree well when the parameter of the system involve small values. The faster the ‘fast variable’, the fewer are the correction terms in the slow manifold expansion. The structure of the slow manifold is determined by the algebraic equation while the vector fields are described in a geometrical way. The connection is revealed through the singular perturbation techniques. That is, if the small parameter (singular perturbation parameter) is set to zero, the motion of the system is subject to an algebraic constraint and is confined to some geometrical structure which approximates the slow manifold of the system. The quasi-steady state approximation for the slow manifold can also be viewed as the result of a first order truncation of the series for the slow manifold structure. The inclusion of successive terms leads to fine structure in the topological manifold as folds. Hence the perturbative series corresponds to a hierarchy of approximate representations for the ODEs for the slow time evolution. This can also be seen as a perturbation description of the global flow structure and invariant slow manifold. Indeed, as we shall see, the reductive perturbative method that will be employed to understand the dynamics of the Ananthakrishna model falls in this category except that the

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slow and fast time scales are the corresponding modes near the Hopf bifurcation. This method permits analytical expressions for the amplitude and the phase of the limit cycle solutions in the neighborhood of Hopf bifurcation in analysing the AK model which is a typical example of multiple scale dynamical system. Indeed the inclusion of successive time scales leads to high order Ginzburg–Landau type of equation. 6.3.4. Connection to relaxation oscillations In general, the slow manifold analysis helps to distinguish and study the initial transient dynamics and a final slow approach to an asymptotic stationary state of the multiple time scale dynamical system. However, in such systems the asymptotic state of interest could be periodic/aperiodic oscillations. In such a situation, the oscillations typically take the form of relaxation oscillations with large amplitude jumps as well as slow evolution of the fast variable within a single oscillatory period. In other words, relaxational nature is characterized by the stick and slip regions wherein the velocities of the fast variables are small and large, respectively. With respect to the inertial manifold structure, the stick region is a part of the phase space wherein the fast dynamics is enslaved by the slow variables and the slip region is the fast ‘transient’ dynamics. In other words, the structure of the relaxation oscillation in the phase space is brought out in this approach. Hence, the inertial manifold provides a natural setting for the analysis of relaxation oscillations. A standard set of equations for relaxation oscillation originally analyzed by van der Pol [201] is dx = ε y − f (x) , dt dy 1 = − x, dt ε

(118) (119)

where f (x) = −x + x 3 /3. The trajectories satisfy the equation dy x =− 2. y − f (x) dx ε

(120)

When ε is large, the RHS is small which suggests that the trajectories may be approximated piecewise by y = const. and y = f (x). The approximate picture of the limit cycle is given in Fig. 30. The slow manifold structure of this set of equations is given by dx x3 =0→y=x− . dt 3

(121)

The zeroth order inertial or slow manifold is given by y = f (x) which is a classical S or N shaped inertial manifold structure for relaxation oscillations. This represents the entrainment of the fast variable x by the slow variable y so that the motion of this reduced system has one degree of freedom and is constrained to the neighborhood of the slow manifold S. This slow manifold structure is also illustrated in Fig. 30. The pseudo-equilibrium points and the fold curve are shown as A and B in the figure. The segment with dashed line in

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Fig. 30. Relaxation oscillations in the N-shaped slow manifold.

Fig. 30 is unstable for the fast dynamics of the system. For this case also, an analysis on the lines of singular perturbation analysis can be performed which leads to inertial manifold structure with series expansion in the small parameter ς = 1/ε. Depending on the structure of the slow manifold and the set of equations, a variety of relaxation oscillations is realizable. The relaxation oscillations need not be confined to monoperiodicity either. Rinzel [202] presented a classification of the types of multiperiodic relaxation oscillations with spiking behavior of the fast variable based on the phenomenology of relaxation oscillations and the topology of the solution structure of the fast subsystem. Bertram [203] has elaborated on this classification to include newer bursting models that did not fit in the original scene. 6.3.5. Sticky slow manifolds and canards There exists a variety of intriguing properties of the slow–fast systems that can be used to obtain better insight into the dynamics of the PLC effect. A well studied phenomenon is the delayed bifurcations [204] wherein the bifurcation point of the solutions depends on the rate of the change of the control parameter. This might have relevance in the case of PLC effect where the comparatively slow evolution of the microstructure due to hardening rate can delay the onset of the instability or lead to extinction. In slow–fast systems, since the slow variable acts as a parameter, similar effects have been noted and studied. Another facet of slow–fast dynamical systems is canard solutions which are solutions that stick to the unstable manifold. Indeed, the type A regime of strain rates belong to this category as we shall see. Canard solutions have been first studied by using nonstandard analysis [205]. Eckhaus provided a ‘standard’ analysis also using asymptotic expansion techniques for these [206]. The transition of the oscillatory solution born out of a Hopf bifurcation to a canard solution is formally described in [207] with the help of asymptotic expansions. The sticking of trajectories on the unstable part of the inertial manifold has also been widely studied in the literature on mathematical biology [208]. Here, the bursting solutions are seen to be delayed beyond the critical point and have been termed as ‘delayed bursting’.

§6.4


169

6.4. Slow–fast dynamics in the Portevin–Le Chatelier effect The phenomenon of the PLC effect is an example of slow–fast dynamical systems. In experiments only stress can be measured, which typically has a saw tooth character. However, it is difficult to guess the kind of behavior of dislocations which leads to the macroscopic stress–strain behavior of these relaxational oscillations. From the dynamical point of view, one may infer that the stick-slip behavior is related to the known physical mechanism of dislocation movements, namely the pinning (stick/slow) and depinning (slip/fast) of dislocations. This forms the basis of the analysis of the dynamical behavior of the PLC effect. The dynamical model due to Ananthakrishna reflects this microscopic slow–fast character of dislocations in its structure. In this model, the slow–fast character manifests through small parameters which are related to the rate constants of creation of locks, diffusivity of the solute atoms and the elastic constant of the system. From the relative magnitudes of the rate constants, the mobile dislocation density ρm can be seen to act as a fast subsystem in comparison to the other dislocation populations. Some of these concepts are used to get an insight into the dynamics of the PLC effect. For instance, we shall use the slow manifold structure of the Ananthakrishna model for the PLC effect to show the connection between the negative strain rate sensitivity of the flow stress and different parts of the slow manifold. Indeed, the regions of fast transients dictated by the fast mobile dislocation population correspond to the unstable branch of the negative SRS while the sticky regions of the slow manifold where the fast mobile dislocation variable is enslaved by the slow immobile population corresponds to the increasing branch of the SRS. The most striking and insightful application of the structure of the slow manifold is the visualization of the dislocation configurations in various regimes of strain rate. As we shall show, the approach gives insight into the dynamics of different types of bands. 6.5. Ananthakrishna’s model The fully dynamical nature of Ananthakrishna’s model and its prediction of chaotic stress drops at intermediate strain rates as found in experiments, makes it most suitable for studying this crossover by including spatial degrees of freedom. In this review, we shall deal with several aspects of the PLC effect within the scope of Ananthakrishna’s model. As some of these methods are not commonly used in metallurgical literature, we illustrate the power of these methods by applying them to the AK model. First we consider a method of describing the bistable nature of stress, particularly at low strain rates. At low strain rates, the serrations are nearly regular, it suggests that the system makes a transition from a monostable (no serrations) to a bistable (upper and lower stress values). The simplest equivalent of this feature in dynamics is the limit cycle solution. Thus, we first discuss the existence of limit cycle solutions in the model equations by using the nullcline method. Second, at a more technical level, this also implies the existence of a free energy like function in the neighborhood of the bifurcation point. We show that it is possible to derive a free energy like function in the neighborhood of the Hopf bifurcation point which demonstrates the bistable nature of stress–strain curves. As bistability is related to the negative SRS, from a dynamical point of view, this can be tackled by analysing the stable and unstable manifolds of the

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system of equations. This allows us to provide a dynamical interpretation of the negative strain rate sensitivity of the flow stress based on the structure of the slow manifold. Finally, we consider the extension of the AK model to include spatial degrees of freedom and show that the above mentioned crossover detected in the analysis of experimental time series can be explained fully [51]. The model also exhibits the three types of bands. Significantly, the slow manifold approach helps to visualize dislocation configurations in the different band regimes of strain rates. Finally, by projecting the model onto the fast manifold, we show that one can obtain an analytical expression for the velocity of the bands. In the dynamical model due to Ananthakrishna and coworkers [107,108], the well separated time scales mentioned in the DSA are mimicked by three types of dislocations, namely, the fast mobile, immobile and the ‘decorated’ Cottrell type dislocations. The basic idea of the model is that all the qualitative features of the PLC effect emerge from the nonlinear interaction of these few dislocation populations, assumed to represent the collective degrees of freedom of the system. The model has been extended to study strain bursts in fatigue and has also been studied in detail [209–212]. As we have outlined the construction of the model in Section 4.2.1 here, following the notation in Ref. [190], we shall briefly outline the model in the scaled variables. (The starting point is the set of eqs (27)–(29), that can be written in scaled form.) The model consists of densities of mobile, immobile, and Cottrell’s type dislocations denoted by ρm (x, t), ρim (x, t) and ρc (x, t) respectively, in the scaled form. The evolution equations are: 2 m D ∂ (φeff (x)ρm ) ∂ρm 2 m = −b0 ρm − ρm ρim + ρim − aρm + φeff ρm + , ∂t ρim ∂x 2 ∂ρim 2 = b0 b0 ρ m − ρm ρim − ρim + aρc , ∂t ∂ρc = c(ρm − ρc ). ∂t

(122) (123) (124)

The model includes the following dislocation mechanisms: immobilization of two mobile 2 ), the annihilation of a mobile dislocadislocations due to the formation of locks (b0 ρm tion with an immobile one (ρm ρim ), the remobilization of the immobile dislocation due to stress or thermal activation (ρim ). It also includes the immobilization of mobile dislocations due to solute atoms (aρm ). Once a mobile dislocation starts acquiring solute atoms we regard it as a Cottrell’s type dislocation ρc . As they progressively acquire more solute atoms, they eventually stop, then they are considered as immobile dislocations ρim . Alternately, the aggregation of solute atoms can be regarded as the definition of ρc , i.e. t ρc = −∞ dt ′ ρm (t ′ )K(t − t ′ ), where K(t) is an appropriate kernel. For the sake of simplicity, this kernel is modeled through a single time scale, K(t) = ce−ct . The convoluted nature of the integral physically implies that the mobile dislocations to which solute atoms aggregate earlier will be aged more than those which acquire solute atoms later (see Ref. [190]). The fifth term in eq. (122) represents the rate of multiplication of dislocations due to crossm, slip. This depends on the velocity of the mobile dislocations taken to be Vm (φ) = φeff 1/2

where φeff = (φ − hρim ) is the scaled effective stress, φ the scaled stress, m the velocity exponent and h a work hardening parameter.

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The nature of the spatial coupling in the PLC effect has been a matter of much debate [3]. Several mechanisms have been suggested as a source of spatial coupling [3]. Within the scope of our model, cross-slip is a natural source of spatial coupling, as dislocations generated due to cross-slip at a point spread over to the neighboring elements. It −1 is this that gives rise to the last term in eq. (122) [51]. Note the factor ρim which models the fact that cross-slip spreads only into regions of minimum back stress. This also take care of the long range nature of dislocation correlations. Finally, a, b0 and c are the scaled rate constants referring, respectively, to the concentration of solute atoms slowing down the mobile dislocations, the thermal and athermal reactivation of immobile dislocations, and the rate at which the solute atoms are gathering around the mobile dislocations. These equations are coupled to the machine equation that provides a feedback mechanism

dφ(t) 1 l m ρm (x, t)φeff (x, t) dx , = d ǫ˙ − dt l 0

(125)

where ǫ˙ is the scaled applied strain rate, d the scaled effective modulus of the machine and the sample, and l the dimensionless length of the sample (we reserve ǫ˙a for the unscaled strain rate). We also note here that there is a feed back mechanism between the machine equation (125) and eq. (122). The model exhibits rich dynamics and has been studied in detail [213,214]. 6.5.1. Limit cycle solutions To begin with, we shall consider the simplest case of the instability seen under creep conditions where there is no feed back mechanism [215–218]. In this case, one observes a series of stress jumps during the course of deformation. To discuss this case, we set the spatial term in eq. (122) to zero and take φeff = φ. We first scale out the stress variable by ′ = ρ /φ m . The defining τ = φ m t which requires redefining only the immobile density ρim im ′ ′ ′ new parameters a , b0 , c are related to the old ones through a scale factor φ m , but to keep the notation simple we shall retain the same symbols with the explicit understanding just stated. Then we have 2 ′ ′ ρ˙m = (1 − a)ρm − b0 ρm − ρm ρim + ρim , 2 ′ ′ ρ˙′ im = b0 kb0 ρm − ρm ρim − ρim + aρc ,

ρ˙c = c(ρm − ρc ).

(126) (127) (128)

Eqs (126)–(128) are coupled set of nonlinear equations which support limit cycle solutions for a range of parameters a, b0 and c. The instability domain of the parameters can be calculated and approximate closed form solutions for the limit cycles obtained. 6.5.2. Approximate solution using nullcline method The fixed points are easily calculated by setting the RHS of eqs (126)–(128) to zero which ′ gives ρm,a = ρc,a = [1 − 2a + ((1 − 2a)2 + 8b0 )1/2 ]/4b0 and ρim,a = 0.5. A simple cal-

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Fig. 31. The instability region in the c–a parameter space keeping b = 10−4 . The bold line refers to ccritical plotted as a function of a [219].

culation of the linearized stability matrix shows that there is a domain of instability for the parameters√a, b0 and c. The range of values are 0 < b0 < 10−2 , 0 < c < 0.25, and 0.33 < a < 1/ 2. For a fixed value of b0 = 0.0004 which is at the lower end, the instability domain is shown in Fig. 31. From the range of the parameters, it is clear that the three time scales are in principle different from each other. In particular, ρm is a fast variable, and thus can be eliminated adiabatically. This aspect becomes transparent if the above equations are written in terms of variables which are deviations from the steady state. Defining new variables which are deviations from the steady state by X = ρm − ρm,a ,

′ ′ Y = ρim − ρim,a

and Z = ρc − ρc,a ,

(129)

eqs (126)–(128) take the form X˙ = − αX + ϒY + b0 X 2 + X Y , Y˙ = −b0 ŴX + Y − aZ − b0 X 2 + X Y , Z˙ = c(X − Z),

where ′ α = a + 2b0 ρm,a + ρim,a − 1, ′ Ŵ = ρim,a − 2b0 ρm,a ,

ϒ = ρm,a − 1,

= ρm,a + 1.

(130) (131) (132)

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We refer to eqs (130)–(132) as the full model. Now, we can rescale the time-like variable by τ ′ = b0 τ , then we have b0

dX = − αX + ϒY + b0 X 2 + X Y , ′ dτ dY = − ŴX + Y − aZ − b0 X 2 + X Y , ′ dτ dZ c = (X − Z). dτ ′ b0

(133) (134) (135)

In the limit of small b0 , |X | → ∞, unless the left hand side vanishes identically. Thus, we can eliminate X in favor of the other two variables and obtain: dY = −b0 aX + 2(ρm,a + X )Y − aZ , dτ dZ = c[X − Z], dτ

and

(136) (137)

where X=

1 1 −(α + Y) + (α + Y)2 − 4b0 ϒY 2 . 2b0

(138)

(The other root for X is unphysical since it corresponds to negative dislocation density.) It must be emphasized that the procedure of adiabatic elimination is quite effective as a ≫ b0 , but becomes more exact as the value of the parameter b0 gets smaller. We call these reduced set of eqs (136)–(137) as the reduced model. Approximate analytical expressions for the nullclines have been derived [219] and limit cycle solutions have been obtained. For details, we refer the reader to Ref. [219]. This method consists of first finding the nullclines N1 (Y) and N2 (Y), respectively, from the conditions Y˙ = 0 and Z˙ = 0: N1 (Y) = X (Y) + and N2 (Y) = X (Y).

2 ρm,a + X (Y) Y, a

(139)

(140)

Nullclines are then plotted on the Y–Z phase plane for a given set of parameter values (a and b0 ). For details, see [219]. If these parameter values fall within the instability region the nullclines intersect in the region of negative slope. Fig. 32 shows the two nullclines for a = 0.53 and b0 = 0.004. This intersecting set consists of two types of branches, CD and AB with positive slope (shown by dashed lines) and another branch with negative slope. Jumps occur at points D and B to the other branches as shown. While CD and AB are the slow branches, the

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Fig. 32. Phase plane portraits: The nullclines N1 (Y) ( broken line) and N2 (Y) √ (full curve) are shown (a) to (d) for b0 = 0.0001. For a < 1/3, the steady state is stable (a), for 1/3 < a < 1/ 2, the steady √ state is unstable leading to limit cycle solutions, (b) and (c). (d) The steady state is again stable for a > 1/ 2 [219].

jumps DA and BC are assumed to be instantaneous. Then, the slow branches are integrated over to yield the limit cycle solution. Using the analytical expressions for the nullclines, expressions for the mobile density can also be obtained. Fig. 33 shows plots of the mobile density calculated by using analytical expressions, the full set and the reduced set of equations. The solution obtained using adiabatic approximation is quite close to the exact solution. One can also obtain the period of the limit cycle. Using the Orowan equation, various features of the instability such as the amplitude and frequency of the jumps on the stress–strain curve are obtained and then compared with known experimental results. The qualitative features match well with the experimental results [215–218]. 6.5.3. Reductive perturbative approach As mentioned in the introduction, in the regime of low strain rates (type C), two distinct levels of stress are discernible that correspond to the top and bottom of the serrations. We first attempt to understand the origin of this bistability in this model as no input of negative SRS feature has been incorporated. We shall show that this arises due to Hopf bifurcation

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Fig. 33. Comparison of the waveforms obtained by various methods for b = c = 10−4 and a = 0.63. The full curve corresponds to approximate closed form solution. The dotted curve corresponds to the numerical solution of all the equations and dashed curve numerical solution using eqs (136), (137) [219].

of the system of equations. Here again we consider the creep case where the scaled stress is held constant and ignore the spatial inhomogeneous deformation. We begin with a brief outline of the reductive perturbative approach to problems of formation of new states of order in far-from-equilibrium situations. Transitions occurring in these systems are quite analogous to equilibrium phase transitions. The general idea is to construct a potential-like function for the order-parameter-like variable in the neighborhood of the critical value of the drive parameter. This would permit the use of the methods developed in equilibrium phase transition for further analysis. Below the point of Hopf bifurcation of the system where the fixed point is still stable, a pair of complex conjugate eigenvalues with real negative parts and another real negative eigenvalue exist for the linearized system of equations around the steady state. As we approach the critical bifurcation value from below, the real part of the pair of complex conjugate eigenvalues approaches zero from the negative side, and hence the corresponding eigen-directions have a slow time scale. As we increase the drive parameter further, the real parts of the complex conjugate eigen-values become positive and this region of the drive parameter is unstable. Thus, while the two eigenvectors corresponding to the pair of complex conjugate eigenvalues are the slow modes, the eigenvector corresponding to the real negative eigenvalue is a fast (and decaying) mode. For this reason, the slow modes determine the formation of new states of order. The reductive perturbative method is a method where the slow enslaving dynamics is extracted in a systematic way [220–224]. The method involves first finding the critical eigenvectors corresponding to the bifurcation point and expressing the general solution as a linear combination of these vectors. The effect of the nonlinearity is handled progressively using the multiple-scale method. The equation governing the complex order parameter takes the form of the Stuart–Landau equation, and corresponds to the time-dependent Ginzburg–Landau equation for a homogeneous medium. On the other hand, the asymptotic solution, which is a limit cycle, collapses to the subspace spanned by

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the slow modes with no trace of the fast mode. There are number of equivalent methods including reduction to the center manifold [224]. Eqs (130)−(132) will be solved using the reductive perturbative method.22 Writing these equations as a matrix equation where the nonlinear part appears separately from the linear part, we obtain dR = LR + N, dτ

(141)

where R =

X Y Z

L=

−α −b0 Ŵ c

(142)

, −ϒ 0 −b0 ab0 0 −c

,

(143)

and the nonlinear part N is given by N =

−b0 X 2 − X Y 2 b0 (b0 X − X Y) . 0

(144)

As we are interested in a series expansion around the point of Hopf bifurcation, we consider the stability of the fixed point as a function of the parameter c. This is done by finding the value of c = c0 at which the real part of the complex eigenvalue of the stability of the matrix L vanishes. Since c is nonnegative, we obtain a unique c0 for the allowed pair of a and b0 values within the instability region. Fig. 34 shows a three-dimensional plot of the instability region involving all the three parameters of the model. To obtain an approximate analytical solution of eq. (141), we follow a reductive perturbative approach similar to that used in Refs [222] and [223]. We choose c = c0 (1 − ǫ) with 0 < ǫ ≪ 1, and write the matrix L as a sum of two matrices, L = L0 + ǫL1 , where L0 is the matrix L evaluated for c = c0 , and L1 ≡

0 0 −c0

0 0 0 0 0 c0

.

(145)

and λ0 = T ,

(146)

The eigenvalues of L0 are λ1,−1 = ±iω

22 The same procedure has been applied to the reduced set of eqs (136), (137). However, the results are less accurate as these equations ignore one more time scale.

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Fig. 34. The instability region in the a, b0 and c parameter space. It is bounded by three surfaces, namely, the c0 surface shown by a series of curved lines, and c = 0 plane and b0 = 0 plane [209].

where ω2 = P , where P and T are sum of the eigenvalues, and sum of the product of the pairs of eigenvalues evaluated at c = c0 . Taking the solution for R as a growth out of the critical eigen-modes, we express it as a linear combination of these eigen-modes: ) = eiωτ r1 + 0 eλ0 τ r0 + ∗ e−iωτ r1∗ = R(τ

−1 j =1

j eλj τ rj ,

(147)

where rj ’s are right eigenvectors defined by L0 rj = λj rj with r−1 = r1∗ . We also introduce left eigenvectors, sjT , defined by sjT L0 = λj sjT , where T stands for the transpose. Substituting this expression for R in the matrix equation, eq. (141), and multiplying both sides of the equation by one of the left eigenvectors, we obtain an equation governing the corresponding amplitude: eλj τ

dj =ǫ μj k k eλk τ + gj lm l m e(λl +λm )τ . dτ k

(148)

l,m,ml

Expressions for the coefficients μj k and gj lm are given in Ref. [209]. We express j as a power series expansion in ǫ 1/2 : (1)

(2)

(3)

j = ǫ 1/2 ψj + ǫψj + ǫ 3/2 ψj + · · · ,

(149)

and introduce multiple time scales such that d ∂ ∂ ∂ + ǫ2 + ···, = +ǫ dτ ∂τ ∂τ1 ∂τ2

(150)

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where τ1 = ǫτ , τ2 = ǫ 2 τ, . . . . Substituting these expressions for j and d/dτ into the equation for the amplitudes, eq. (147), we successively solve by equating terms of the same order in powers of ǫ. First, terms of O(ǫ 1/2 ) give (1)

∂ψj

∂τ

= 0,

(151)

(1)

implying that ψj

is constant in the time scale of τ . O(ǫ) terms give the equation

(2)

∂ψj

∂τ

=

(1)

(1)

gj kl ψk ψl e(λk +λl −λj )τ ,

(152)

k,l,lk

which, upon integration, gives (2)

ψj eλj τ =

(1)

(1)

hj kl ψk ψl e(λk +λl )τ ,

(153)

k,l,lk

where hj kl = gj kl /(λk + λl − λj ). O(ǫ 3/2 ) terms give the equation ∂ψj(3) ∂τ

+

∂ψj(1) ∂τ1

=

(1)

μj k ψk e(λk −λj )τ

+

k

k,l,lk

(1) (2) (2) (1) gj kl ψk ψl + ψk ψl e(λk +λl −λj )τ ,

(154)

where gj kl have been given in Ref. [209]. Using the compatibility condition, we match terms that are varying on a slow time scale found on both sides of the equality, and extract the slow dynamics (1)

∂ψj

∂τ1

(1) 2 (1) (1) = μjj ψj + ηj ψ1 ψj .

(155)

An expression for ηj is given in [209]. (The subscript j = 1 is left out from 1 for the sake of brevity.) To O(ǫ 1/2 ), = ǫ 1/2 ψ (1) and, thus, eq. (155) takes the form of a cubic Stuart–Landau equation: d ∂F = ǫμ + η||2 = − . dτ ∂

(156) 2

4

− η|| is free energy like Note that μ and η are complex coefficients and F = − ǫμ|| 2 4 iτ function. is the complex order parameter given by = ||e . Both the amplitude and the frequency are easily determined.

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Fig. 35. A plot of the bifurcation diagram in the a–b0 plane. The instability region is bounded by nearly parabolic curve and b0 = 0 line. The unshaded (shaded) region is the supercritical (subcritical) bifurcation. The light and dark shaded regions refer to the regions where the quintic and septic amplitude equations, respectively, hold [209].

This solution exists provided η is negative since μ is positive. η is found to be negative over a major part of the instability region in the b0 –a plane, as shown in Fig. 35 (the unshaded region). In this case, since the amplitude of the order parameter grows continuously in proportion to ǫ 1/2 , the transition is continuous (a second-order-type transition) corresponding to supercritical bifurcation. However, there is a relatively small portion of the instability region, shown in the same figure in different shades, where η is found to be positive implying that the transition is discontinuous corresponding to a supercritical bifurcation. In this regime, one has to go to quintic or even higher orders in the amplitude equation to obtain an expression for the order parameter. In fact, an order parameter equation up to septic order has been derived [209] that covers the whole parameter space (Fig. 35). The solutions obtained from the above equation agree well with the numerical solution obtained by solving eqs (122)–(124). (See Ref. [209] for details.) A similar approach can be undertaken for solving eqs (136), (137) where the fast variable X is eliminated adiabatically. In this case, the solutions obtained do not match very well with the numerical solutions obtained by solving all the three equations, as the fast time scale is fully eliminated, i.e. the singular parameter has been set equal to zero. See [210] and also see the remarks in Section 6.3.1 on singular perturbative technique. The above analysis shows that the bistable nature of amplitudes of various densities arises due to the forward Hopf bifurcation. In the parameter space of a–b0 (see Fig. 35), large regions correspond to supercritical bifurcation where the amplitude grows smoothly. However, for low values of b0 , there is region of a values for which the bifurcation is subcritical. For such parameter values, the bistable nature of densities are abrupt across the transition. It is this region that would be relevant to the experimental situation. One more comment that may be relevant to the dislocation community. The present exercise shows that a free energy like potential function can be derived in the neighborhood of the

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bifurcation point.23 However, as the derivation is valid only in the vicinity of the bifurcation point, the nature of solution obtained from this analysis cannot be pushed deep into the limit of instability. Finally, in principle, a similar analysis can be carried out for the full set of four variables with applied strain rate as the drive parameter. However, the present analysis already shows that there would be stress drops due to the bistable nature of the densities. 6.5.4. Numerical solution of the model We first note that the spatial dependence of ρim and ρc arises only through that of ρm . We solve the above set of equations by discretizing the specimen length into N equal parts. Then, ρm (j, t), ρim (j, t), ρc (j, t), j = 1, . . . , N , and φ(t) are solved. The widely differing time scales [50,190,213] call for appropriate care in the numerical solutions. The initial values of the dislocation densities are so chosen that they mimic the values in real samples. As for the boundary conditions, we note that the sample is strained at the grips. This means that there is a high density of immobile dislocations at the ends of the sample. We simulate this by employing two orders of magnitude higher values for ρim (j, t) at the end points j = 1, and N than the rest of the sample. Further, as bands cannot propagate into the grips, we use ρm (j, t) = ρc (j, t) = 0 at j = 1 and N . As in the original model without spatial degrees of freedom, the PLC state is reached through a Hopf bifurcation and is terminated by a reverse Hopf bifurcation (with the other parameters kept in the instability domain). The number of complex conjugate roots are 2N , the negative ones are N and one zero exponent. The boundary of ǫ˙ is approximately in the range 10 to 1000 for a = 0.8, b0 = 0.0005, c = 0.08, d = 0.00006, m = 3.0, h = 0 with D = 0.5, beyond which a uniform steady state exits. For further numerical and analytical work, the model equations can be projected onto slow and fast subspaces.

6.6. Comparison with experiments Before discussing the detailed analysis of the results of the extended AK model, we begin by comparing the nature of serrations obtained in the model in different regimes of strain rates with that in experiments, as this is the simplest feature that is readily calculated. Plots of two experimental stress–strain curves from CuAl single crystals corresponding to the chaotic and power law regimes of applied strain rates are shown in Fig. 36(a), (b). The stress–time series in the intermediate and high strain rate regimes from the model are shown in Fig. 36(c), (d). The similarity of the experimental time series with that of the model in the respective regimes is clear. At the next level, one can compare the dynamical features of the experimental time series with that of the model. Recall that the analysis of the stress–time series given in Fig. 36(a), (b) has been reported in the last section [136] to be chaotic with a correlation dimension 23 The possibility of free energy like function has been proposed based on physical considerations in [225]. However, the procedure given above provides a formal mathematical basis without recourse to physical arguments. Note also that while the approach is natural to systems driven out of equilibrium, providing physical basis in these situations is harder.

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Fig. 36. (a) and (b): Experimental stress–time series: (a) chaotic state at strain rates ǫ˙a = 1.7 × 10−5 s−1 and (b) power law state at ǫ˙a = 8.3 × 10−5 s−1 . (c) and (d): Stress–time series from the model at (c) ǫ˙ = 120, (d) ǫ˙ = 280.

of ν = 2.3 along with the existence of a positive Lyapunov exponent. A plot of the reconstructed attractor has been shown in Fig. 19. This can be compared with the strange attractor obtained from the model in the space of ρm , ρim and ρc (at an arbitrary spatial location, here j = 50 and N = 100) shown in Fig. 37 for ǫ˙ = 120 corresponding to the mid chaotic region (see below). There is a striking resemblance. Note the similarity with the experimental attractor particularly about the linear portion in the phase space (arrow in Fig. 19(b)). This direction can be identified with the loading direction in Fig. 36(a). Note that the identification of the loading direction is consistent with the absence of growth of mobile density.

Fig. 37. Attractor from the model for N = 100, j = 50. Compare with the experimental attractor shown in Fig. 19, [50,51].

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On the other hand, as shown in the last section, in contrast to the experimental time series at low and medium strain rates (Fig. 36(a)), the time series at the highest strain rate (Fig. 36(b)) is not chaotic. Instead it shows a power law distribution of stress drops [136]. The model time series at high strain rates beyond ǫ˙ ∼ 280, shown in Fig. 36(d) also shows no inherent scale in the magnitudes of the stress drops [136,137]. Further, most stress drops are generally small. Indeed, these small stress drops override a generally increasing level of the stress that leads to a larger yield drop. This feature is also clear in the experimental time series at the highest strain rate (Fig. 36(b)). In experiments, this large yield drop follows a series of small ones and is associated with the band having reached the end of the specimen. Even in the model the larger yield drops are associated with the band reaching the end of the sample as we shall show later. We have analysed the distributions for the stress drop magnitudes φ and their durations t. The distribution of stress drop magnitudes, D(φ), shows a power law D(φ) ∼ φ −α . This is shown in Fig. 38(a) (◦) along with the experimental points (•) corresponding to ǫ˙a = 8.3 × 10−5 s−1 . Clearly both experimental and theoretical points show a scaling behavior with an exponent value α ≈ 1.1. The distribution of the durations of the drops D(t) ∼ t −β also shows a power law with an exponent (a)

(b)

Fig. 38. (a) Distributions of the stress drops from the model (◦), from experiments (•) for ǫ˙ = 280 and ρ¯m (♦) from the model. Solid lines are guide to the eye. (b) Largest Lyapunov exponent as a function of strain rate, [50,51].

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value β ≈ 1.3. The conditional average of φ denoted by φc for a given value of t behaves as φc ∼ t x with x ≈ 1.55. The exponent values satisfy the scaling relation β = x(α − 1) + 1 quite well (note there is a slight hint D(φ) of a hump at large sizes). Some insight into the cause of this could be obtained by studying the growth of mobile dislocation density during this period. One can also look at the scaling behavior of the total density in the sample at a given time using ρ¯m (t) = ρm (x, t) dx. Let ρ¯m (t) denote the increase in ρ¯m (t) occurring during the intervals of the stress drops. Then, one should ex−γ pect that the statistics of ρ¯m also to exhibit a power law, i.e. D(ρ¯m ) ∼ ρ¯m , with the same exponent value as α. A plot of D(ρ¯m ) for N = 300 is shown in Fig. 38(a) (♦). The extent of the power law regime is nearly two orders with γ ≈ 1.1, same as α. The large bump at high values corresponds to the small hump in stress drop statistics. This arises partly due to the effect of finite size of the system as in many models and partly due to the band reaching the edges [188]. The physical cause is due to high levels of stress at the grips. Noting that dislocation bands cannot propagate into regions of high stresses, it is clear that the edges cause distortions in the otherwise smoothly propagating bands leading to large changes in ρ¯m (t). (See Fig. 2(c) in Ref. [226].) Typically, the influence of the edges is felt by the band when it is 20 sites away. (Increasing N from 100 to 300 increases the scaling regime by half a decade and the peak of the bump reduces from 700 to 500, thus indicating the influence of the finite size of the system.) To understand and characterize the dynamics of this crossover, a natural tool is to study the distribution of Lyapunov exponents as a function of the applied strain rate in the entire interval where the PLC effect is seen. For the present purpose, it is adequate to consider the behavior of the largest Lyapunov exponent. A rough idea of the changes in the dynamics of the system can be obtained by studying the dependence of the largest Lyapunov exponent (LLE) as a function of the strain rate. Fig. 38(a) shows that the LLE becomes positive around ǫ˙ ≈ 35 reaching a maximum at ǫ˙ = 120, practically vanishing beyond 250. Periodic states are observed in the interval 10 < ǫ˙ < 35. Thus the region of chaos is restricted to 35 < ǫ˙ < 250. In the region ǫ˙ 250, the dispersion in the value of the LLE is ∼5 × 10−4 which is the same order as the mean. Thus, the LLE can be taken to vanish beyond ǫ˙ = 250. The power law distribution of stress drops is found precisely in this region of vanishingly small values of LLE and is similar to experimental situation. 6.6.1. Negative strain rate sensitivity The PLC effect is an example of slow–fast dynamical system which is clear from the sawtooth character of the stress profile at low strain rates. In general, such wave forms are a reflection of a slow time scale corresponding to increasing load and fast time scale corresponding to an abrupt fall in the stress. The general mechanism in all such relaxation oscillations is stick and slip. In the case of the PLC effect, the collective pinning and unpinning of dislocations leads to the negative strain rate behavior of the flow stress. The fact that the negative SRS branch cannot be measured in a strict sense is well recognized, but the presence of this branch clearly shows up in the dynamics of the PLC effect. Even so, early formulations and the way experimental measurements have been carried out have given rise to considerable confusion. Here we discuss briefly the concept of negative SRS and working methods adopted in the literature, and also provide a dynamical interpretation

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of negative SRS wherein a clear connection will be established with the slow manifold of the AK model. Penning [36] was the first to recognize the necessity of negative SRS in the PLC effect. Theories of dynamic strain aging assume that the interaction of dislocations with solute atoms when averaged over the specimen dimension can be represented by a constitutive relation connecting stress, strain, and strain rate which is conventionally written as [36] σ = hǫ + F (˙ǫ ).

(157)

The basic assumption inherent in eq. (157) is that stress can be split into a sum of two functions, one is a function of ǫ alone and the other is a function of ǫ˙ alone. Then, the SRS is defined as ∂σ dσ (158) = ǫ˙ . S= ∂ ln ǫ˙ ǫ d˙ǫ

Clearly, this definition uses ǫ, σ and ǫ˙ as state variables even though these variables are history dependent. At a working level, however, strain is fixed at a small nominal value and the flow stress at that value is used to obtain the SRS [227]. There are a few attempts to ‘measure’ the unstable branch as a function of the strain rate [227], even though there is a full recognition of the limitations of such a measurement [228]. Following the decomposition of Penning [36], Kubin et al. use Al-5%Mg alloy with the test carried out at 300 K. By fixing the strain at ǫ = 8 × 10−2 , they measure the mean of the upper stress values of the serrations. This is taken to represent the unstable branch (for more details see Ref. [228]). Our approach to understand the origin of negative SRS is based on the relaxation oscillations inherent to the dynamics of the PLC effect. In this picture one concludes the existence of the unstable branch on the basis of strain bursts, but one never records any points in this region. The basis for the negative SRS uses the relaxational nature of stress arising in the model. 6.6.2. Slow manifold analysis Within the scope of the AK model, while the negative SRS was numerically determined quite early [108], the analytical approach was only recently provided by Rajesh and Ananthakrishna [190] in terms of the structure of the slow manifold of the model [213]. The methodology of slow manifold analysis is basically a dimensional reduction procedure that provides a smaller-dimensional version of the original dynamical system that retains the essential dynamics. This is best suited for analysis of nonlinear slow–fast dynamical systems wherein all trajectories are attracted, in the long time limit, to a subspace of Rn which forms an invariant manifold. The geometry of the slow manifold of the original model has been analyzed in detail [190,213]. The analysis shows that the relaxational nature of the PLC effect arises from the atypical bent nature of the manifold. Here we recall some relevant results on the slow manifold of the original model (D = 0). (We shall later extend the ideas to the situation when the spatial degrees of freedom are switched on.) Slow manifold expresses the fast

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variable in terms of the slow variables. At a working level this is done by setting the derivative of the fast variable to zero [190,213] 2 + ρm δ + ρim = 0, ρ˙m = g(ρm , φ) = −b0 ρm

(159)

where δ = φ m − ρim − a. The variable δ, as we shall see, has all the features of an effective stress and thus plays an important physical role [213], particularly in studying the pinningunpinning of dislocations. We note that δ is a combination of the two slow variables φ and ρim both of which take small positive values. Hence, δ takes on small positive and negative values. Using eq. (159), we get two solutions 1/2 ρm = δ + δ 2 + 4b0 ρim /2b0 ,

(160)

one for δ < 0 and another δ > 0. For the region δ < 0, as b0 is small ∼10−4 , we get ρm /ρim ≈ −1/δ which takes on small values. On the other hand, when δ > 0, ρm ≈ δ/b0 which is large. These two regions physically correspond to different parts of the slow manifold. A plot of the slow manifold in the δ–ρm plane is shown in Fig. 39(a). For the sake of illustration, we have plotted a monoperiodic trajectory describing the changes in the mobile density during a loading-unloading cycle. The inset in Fig. 39(a) shows ρm (t) and φ(t). As can be seen, there is a region where the ratio of the mobile to immobile density is small ρm /ρim ≈ −1/δ and negative which is marked as S2 . As ρm is small, S2 can be identified with the ‘pinned state of dislocations’. In contrast, for positive values of δ, as ρm is large, we refer to it as the ‘unpinned state of dislocations’. These two pieces S2 and S1 are separated by δ = 0, which we refer to as the fold line [190,213]. For completeness, the corresponding plot of the slow manifold in the (ρm , ρim , φ) space is shown in Fig. 39(b), along with the trajectory and the corresponding symbols. In this space, one can see that δ = φ m − ρim − a = 0 is a line that separates the pieces S2 and S1 of the slow manifold, and hence the name fold line. The fixed point of the system of equations (122)–(125) lies close to the point D on S1 , which in the PLC region is an unstable saddle focus. Thus, there is an unstable manifold that separates the two stable pieces of the slow manifold S1 and S2 . Note that beyond the PLC region the fixed point becomes stable meaning we have no serrations. We shall map the various regions in Fig. 39(b) with that in Fig. 39(a) as the latter will be convenient for studying dislocation configurations later. In Fig. 39(b), the trajectory enters S2 at A and leaves at B. For this part of trajectory, the value of δ (in Fig. 39(a)) decreases from zero to a maximum negative value and reverts to zero value. Beyond this point as the trajectory leaves S2 in Fig. 39(b), δ becomes positive. The corresponding points are marked on both the figures. In addition, in the inset of Fig. 39(a), we have shown the correspondence with ρm . The segment AB in Fig. 39(b) can be identified with the flat region of ρm (t) in the inset of Fig. 39(a) with the same symbol. As the trajectory crosses δ = 0, ∂g/∂ρm becomes positive and it accelerates into the shaded region (Fig. 39(a)) rapidly till it reaches ρm = δ/2b0 (point C). Thereafter it settles down quickly on S1 decreasing to D rapidly till it reenters S2 again at A. The burst in ρm (inset in Fig. 39(a)) corresponds to the segment BCDA in Figs 39(a) and (b). The nature of trajectories for higher strain rate remain essentially the same, but are chaotic.

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(a)

(b)

Fig. 39. (a) Bent slow manifold S1 and S2 (thick lines) with a simple trajectory for ǫ˙ = 200 and m = 3. Inset: ρm (dotted curve) and φ (solid line). (b) The same trajectory in the (φ, ρim , ρm ) space [213].

6.6.3. Connection to negative SRS The above discussion on the slow manifold of the model demonstrates how to separate out different dislocation mechanisms that contribute to the time development of the variables. This should therefore help us to set up a correspondence between the stress–strain rate space and the slow manifold (Figs 39(a), (b)). Having identified the regions of the slow manifold with the pinned and unpinned states of dislocations, we now consider the variation one cycle of stress when dislocations are pinned and are unpinned. First consider eq. (125) for D = 0 which reduces to φ˙ = d[˙ǫ − ǫ˙p ],

(161)

where ǫ˙p = φ m ρm defines the plastic strain rate. We now calculate strain rate sensitivity numerically using eqs (122)–(125). This is shown in Fig. 40 (◦). From our earlier dis-

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Fig. 40. The empty circles show the phase space projection of stress vs. strain rate ǫ˙p . The dotted line represents the negative strain rate sensitivity region. The thick lines are analytical approximations to corresponding regions (Ref. [213]).

cussion, we know that when the trajectory is on S2 , ρm is nearly constant and small in magnitude. As this implies a pinned configuration, according to eqs (122)–(125), φ should increase monotonically and hence the segment AB on S2 in Fig. 39(b) corresponds to the rising branch AB in Fig. 40. For this branch, one can easily see that the (mean) value of S ∼ 3.5 using eq. (158). Further, at the point where the trajectory leaves S2 part of the slow manifold, namely B in Fig. 39(b), the value of δ approaches zero (Fig. 39(a)), and correspondingly φ reaches its maximum value. Once the trajectory leaves S2 and jumps to S1 , ρm increases abruptly outside S2 (see BC in the inset of Fig. 39(a)), δ = 0 line separating the pinned state from the unpinned state. Thus, δ = 0 physically corresponds to the value of the effective stress at which dislocations are unpinned. This evidently corresponds to the strain rate jump from B to C in Fig. 40. Note that the slope ∂φ/∂ ǫ˙p for this portion of the orbit is negative and quite small unlike the zero value when an abrupt jump is assumed across the stable branches. Further, we know from Fig. 39, once the trajectory reaches S1 , the value of ρm decreases rapidly resulting in the decrease of ǫ˙p . Thus, the region CD in Fig. 40 corresponds to CD segment of the trajectory on S1 in Fig. 39(a) (or inset of 39(a)). For this branch, one can quickly check that the strain rate sensitivity is positive, having a mean value (∼1.5) which is a factor of 2 less than that for the branch AB, implying that the nature of dissipation is quite different from that operating on AB. This is consistent with known facts about the two branches as mentioned in the introduction. Combining this with the fact that ρ˙m is decreasing, the branch CD in Fig. 40 can be identified with the slowing down of the mobile dislocations. The above picture can be made more concrete by actually calculating stress as a function of strain rate. This can be done analytically as approximate equations of motion for each part of the slow manifold AB on S2 , BC outside the slow manifold, CD on S1 , and unstable DA corresponding to the jump between the two pieces of the slow manifold, are known.

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Now consider the equation for ǫ˙p given by d˙ǫp = ρm mφ m−1 φ˙ + φ m ρ˙m , dt

(162)

which on using eqs (122) and (125) gives ǫ˙p d˙ǫp = dφ

mǫ˙ d φ

+ δ − ǫ˙p2 md φ + d(˙ǫ − ǫ˙p )

b0 m φ m + ρim φ

.

(163)

Note that in the slow manifold description, all slow variables appear as parameters. However, since SRS describes the dependence of the slow variable φ as a function of the (derived) fast variable ǫ˙p , we will consider the other two variables ρim or δ or both as parameters. Our interest here is to obtain approximate expressions for ǫ˙p (φ) on different branches. To do this, we use typical values of δ and ρim for the interval under question. As stated earlier, the trajectory has different dynamics in different regions of the slow manifold. These are (a) on S2 where ρ˙m is nearly zero for the entire time spent by the trajectory on S2 , (b) just outside S2 where ρ˙m ∼ ρm δ, (c) on S1 where ρm ∼ δ b0 for ǫ˙p > ǫ˙ , and (d) when the trajectory jumps from S1 to S2 . Approximate solutions obtained for these cases are shown in the φ–˙ǫp plot by solid lines. (For details see [213].) The approximate solutions shown by the bold lines are quite close to the numerically exact result shown in Fig. 40 by ◦. 6.6.4. Types of bands As we have shown, the spatially extended AK model is able to capture two important dynamical features of the PLC effect detected in the analysis of experimental time series and also provides a basis for understanding the dynamics.24 First, the model explains the crossover from chaotic regime found in mid range of strain rates to the power-law regime of stress drops observed at high strain rate in experimental time series. Now we address the important characteristic feature of the PLC, namely, the nature of bands in the regions of strain rates. Most models of dislocations bands use diffusive coupling although the physical mechanism of the term is different in different situations [3]. An important feature of the spatial coupling in the model is that it accounts for spreading of dislocations into regions of low −1 ). The term also determines the back stress once dislocations are unpinned (the factor ρim length scale over which dislocations spread into the neighboring elements. Thus, while dislocation pinning and unpinning gives a heterogeneity in space (in principle), regions of low ρim are favored for dislocation multiplication and spreading into neighboring regions. Further, this type of spatial term couples length scale to time scale in a dynamical way as ρim itself evolves in time and hence the associated time scale. Indeed, multiplication m ), and hence this rate itself changes dynamically of dislocations depends on stress (i.e. φeff 24 The first attempt to simulate band properties and the associated serrations is due to Neelakantan and Venkataraman [229]. The authors use analog electronic simulations that use the Penning model to obtain several features of the PLC effect. Indeed they recognize that the nature of σ –˙ǫ curve evolves with strain. They include noise effects as well. The nature of serrations for different strain rates are quite close to what is observed in experiments.

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Fig. 41. Spatially uncorrelated bands at ǫ˙ = 40 [51,230].

leading to changes in the time scale of internal relaxation as a function of ǫ˙ . We expect this to lead to changes in spatial correlation as strain rate is increased. We begin with the numerical results [226]. For ǫ˙ < 10 and ǫ˙ > 2000, we get homogeneous steady state solutions for all the dislocation densities, ρm , ρim and ρc . In these ranges of strain rates, φ takes the fixed point values asymptotically. In the region where interesting dynamics of chaotic and power law states are observed, the nature of the dislocation bands can be broadly classified into three different types occurring at low, intermediate and high strain rates described below. For strain rates, 30 ǫ˙ < 70, we get uncorrelated static dislocation bands. The features of these bands are illustrated for a typical value, say for ǫ˙ = 40. A plot of ρm (j, t) is given in Fig. 41. Dislocation bands of finite width nucleate randomly in space and they remain static till another band is nucleated at another spatially uncorrelated site. The associated stress–time curves which are nearly regular have large characteristic stress drops. The distribution of these stress drops is found to be peaked as in experiments at low strain rates [137]. At slightly higher values of strain rates, 70 ǫ˙ < 180 we find that new bands nucleate ahead of the earlier ones, giving a visual impression of hopping bands. This can be clearly seen from Fig. 42 where a plot of ρm (j, t) is given for ǫ˙ = 130. However, this hopping motion does not continue till the other boundary. They stop midway and another set of hopping bands reappear in the neighborhood. Often, nucleation occurs at more than one location. Stress–time plots in this regime have a form similar to Fig. 36(c) with the average amplitude of the stress drops being smaller than that for the localized nonhopping bands at low strain rates as seen in experiments. These stress drops also have a nearly symmetric peaked distribution as in the previous case but slightly skewed to the right similar to those observed in experiments [137]. As the strain rate is increased further, the extent of propagation increases. Concomitantly, the magnitudes of the stress drops decrease. We see continuously propagating bands even at ǫ˙ = 240 as can be seen from Fig. 43. One can see dislocation bands nucleating from one end of the sample (j = 0, t = 25, 50 and 75) and propagating continuously to the

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Fig. 42. Hopping type bands at ǫ˙ = 130 (arrow shows one such band) [51,230].

Fig. 43. Fully propagating bands at ǫ˙ = 240 [51,230].

other end. Often, we see a band nucleating at a point, branching out and propagating only partially towards both the ends. We also find reflection after reaching the end of the sample. (See Fig. 2(c) in [226].) Unlike the present case which exhibits rather uniform values of ρm , we usually find irregularities as the band reaches the edges. The stress strain curves in this region of strain rates exhibit scale free feature in the amplitude of the stress drops (Fig. 36(d)) with a large number of small drops. As can be seen from Fig. 36(d), the mean stress level of these small amplitude stress drops increases until a large yield drop is seen. This large stress drop corresponds to bands having reached the end of the specimen. 6.6.5. Visualization of dislocation configurations The above analysis of the dynamics presented in Section 6.6 and the numerical results in the last section show that the chaotic regime of strain rates corresponding to hopping type bands (as also the randomly nucleated uncorrelated bands at low strain rates), and the fully

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propagating to the power law regime of stress drops. This was precisely the correlation that was established in Ref. [137] in the experimental studies on Al-Mg alloys. Now, the basic question here is: What is it in terms of dislocation configuration that is so different between these regimes of strain rates? Is there a way of visualizing the dislocation configuration? The techniques of slow manifold introduced in Section 6.6.2 (see also Section 6.3.2) provides such a tool as we shall show. The extension of slow manifold approach when the spatial degrees of freedom are included does not pose any complication as the slow manifold is defined at each spatial point. In this case, a convenient set of variables for visualization of dislocations is (ρm (x), δ(x), x). Moreover, the variable δ(x) has been identified with the unpinning threshold. Here, we recall that we have shown that the model successfully reproduces the crossover from a low-dimensional chaotic state at medium and low strain rates to a power law state of stress drops seen at high strain rates [51]. The former corresponds to type C and B bands and latter to the propagating type A band [137]. If the difference in the dynamics in these regimes of strain rates is the underlying cause for the different types of bands seen with increasing strain rate, this should be reflected in the configuration of dislocations in the respective regimes. Thus, our aim is to investigate the nature of typical spatial configurations in the chaotic and the power law regimes of stress drops and study the changes as we increase the strain rate. For simplicity, we shall use h = 0 for which we have φeff = φ. (It is straightforward to extend the arguments to the case when h = 0.) Then, the plastic strain rate ǫ˙p (t) is given by 1 ǫ˙p (t) = φ (t) l m

0

l

ρm (x, t) dx = φ m (t)ρ¯m (t),

(164)

where ρ¯m (t) is the mean mobile density (= j ρm (j, t)/N in the discretized form). With the inclusion of spatial degrees of freedom, the yield drop is controlled by the spatial average ρ¯m (t) rather than by individual values of ρm (j ). Further, we note that the configuration of dislocations changes during one loading-unloading cycle. However, one should expect that configurations will be representative for a given strain rate. Further, we know that the drastic changes occur during a yield drop when ρ¯m (t) grows rapidly. Thus, we focus our attention on the spatial configurations on the slow manifold at the onset and at the end of typical yield drops. First consider the configuration seen just before and after the yield drop when the strain rate is in the chaotic regime. In this regime, the stress drop magnitudes are large which implies that the change in mobile density is large. Figs 44(a), (b) show the configuration of dislocations before and after the yield drop for a typical value of ǫ˙ = 120. It is clear that both at the onset and at the end of a typical large yield drop, the δ(j ) values, which reflect the state of the system – whether pinned or unpinned – is negative and correspondingly the mobile densities ρm (j )’s are small, i.e. most dislocations are in a strongly pinned state. (Recall that δ signifies how close the spatial elements are to the unpinning threshold.) The arrows show the increase in ρm (j ) at the end of the yield drop. We have checked that this picture is a general feature for all yield drops in the chaotic regime of strain rates. Now consider dislocation configuration in the power law regime of stress drops at high strain rates, say, ǫ˙ = 280, at the onset and at the end of an yield drop shown in Figs 44(c), (d)

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Fig. 44. Dislocation configurations on the slow manifold at the onset and at the end of an yield drop: (a) and (b) for ǫ˙ = 120 corresponding to hopping type band (chaotic regime), and (c) and (d) for ǫ˙ = 280 corresponding to propagating type band (scaling regime) [51,230].

respectively. In contrast to the chaotic regime, in the scaling regime, most dislocations are clearly seen to be at the threshold of unpinning with δ(j ) ≈ 0, both at the onset and the end of the yield drop. This also implies that they remain close to this threshold all through the process of a stress drop. We have verified that the edge-of-unpinning picture is valid in the entire power law regime of stress drops. Further, as a function of strain rate, we find that the number of spatial elements reaching the threshold of unpinning δ = 0 during a yield drop increases as we approach the scaling regime. Thus, the slow manifold technique affords a way of visualizing dislocation configurations in different regions of strain rates. The interesting aspect that emerges is that while the power law statistic of stress drops corresponding to propagating bands already suggests that it is in the critical state, configurations of dislocations are at the edge of unpinning provides a visual confirmation or criticality. 6.6.6. Band velocity Now let us consider the possibility of calculating the velocity of the propagating bands in the high strain rate limit. Eqs (122)–(125) constitute a coupled set of integro-partial differential equations, and hence cannot be dealt with in their present form. To reduce these equations to a manageable form that is suitable for further analysis, we first study how the changes in the location of the fixed point occur with applied strain rate. We first note that the fixed point is an unstable saddle focus that lies on the S2 part of the slow manifold. To see how the location of this changes with applied strain rate, we have plotted

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Fig. 45. Top: Eigenvalue spectrum of the fixed point of the model. ωr and ωi refer to the real and imaginary components of the complex pair of eigenvalues, λ1,2 are the real eigenvalues. Bottom: Slow manifold showing a trajectory for the space independent model near the reverse Hopf bifurcation point, at ǫ˙ = 90, m = 2. • is a fixed point of eqs (1)–(3) and (6) [51].

the eigenvalue of the fixed point as a function of ǫ˙a shown in Fig. 45 (top). As can be seen, the region of instability is bounded by a forward Hopf bifurcation followed by a reverse Hopf bifurcation. From the figure, it is clear that for high strain rates, the presence of the latter gives rise to a finite rate of softening of the eigenvalue of the fixed point [51,213]. This implies that the amplitude of the PLC serrations becomes increasingly small as we go towards the upper critical threshold of the PLC boundary. A phase plot in the ρm –δ variables (i.e. the projection of the slow manifold) is shown in Fig. 45 (bottom). In this case, the orbit spirals around the unstable fixed point several times before touching the S2 part of the slow manifold. Note that when the orbit visits the region of S2 , the mobile density mobile density is very small and remains so for a fairly long duration till it escapes from S2 during which most dislocations are in the pinned state or equivalently, immobile density is relatively large. In contrast, for the present case of high strain rates, the mobile density is going through a series of oscillatory changes as the orbits execute several turns around the fixed point during which the immobile density changes marginally. Further, as the orbit executes one turn, there is one small yield drop. However, the orbit executes several turns around the fixed point, each turn leading to larger loop sizes, i.e. larger values

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of ρm and consequently to successively larger stress levels than the earlier one, before briefly visiting S2 . This feature of small stress drops overriding a general increasing level is clear in Fig. 36(d). It must be noted that during this period, the stress level shows small fluctuations around some mean value and therefore can be considered as not varying much. Under these conditions the dynamics is entirely controlled by the spiralling motion around the fixed point. Thus, the entire dynamics is essentially described by the fast variable, namely the mobile dislocation density; the other two variables ρim and φ can be taken to be parameters. Such a situation is described by the transient dynamics dictated entirely by equation of the fast variable (the so-called layer problem [231]) and we are therefore justified in using only the evolution equation of the fast variable in terms of the slow manifold parameter δ = φ m − ρim − a. Since the trajectory rarely visits the S2 part of the slow manifold, we restrict the calculations to δ small and positive. The physical picture of a propagating solution is that as the orbit at a site makes one turn around the fixed point (i.e. δ small but positive), around the value of the applied strain rate, the front advances by a certain distance along the specimen like the motion of a screw. The equation for the mobile dislocation density ρm in terms of δ is ∂ρm ∂ 2 ρm 2 + δρm + ρim + D ′ , = −bo ρm ∂t ∂x 2

(165)

where D ′ = Dφ m /ρim . Since the slow variables ρim and φ are treated as parameters, this has the form of the Fisher–Kolmogorov equation [232] for propagating fronts, which has been well studied. This equation can be reduced to the standard form 2 ∂Z ′∂ Z . = Z(1 − Z) + D ∂t ′ ∂x 2

(166)

This is obtained by first transforming ρm = X − ρim /δ, dropping the term 2b0 ρim /δ compared to δ in the linear term in X, and then using Z = Xδ/b0 and t ′ = tδ. It is clear that Z = 0 is unstable and Z = 1 is stable. As the form of our equation has been reduced to the standard form, all other results carry through, including the nonlinear analysis. Using ′ ′ the form for propagating front Z = Z0 eωt −kx , the marginal velocity is calculated using v ∗ = Re ω(k ∗ )/ Re k ∗ = dω/dk|k=k ∗ and Im dω/dk|k=k ∗ = 0, giving the velocity of the bands v ∗ = 2 [232,233]. In terms of the variables in eq. (165), the marginal velocity is √ v ∗ = 2 Dδ.

(167)

In order to relate this to the applied strain rate, we note that for a fixed value of the strain rate (where propagating bands are seen), the average level of stress drop is essentially constant. Thus, from eq. (125), we see that in this regime of high strain rates, the applied l strain rate ǫ˙ is essentially balanced by the plastic strain rate (1/ l) 0 φ m ρm (x, t) ≡ ǫ˙p . Then, using φ m = ǫ˙ /ρ¯m , and using δ = φ m − a − ρim , we get v=2

ǫ˙ D ǫ˙ − a − ρim . ρ¯m ρim ρ¯m

(168)

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∗ , the fixed point value. Thus, for all practical purposes, At high applied strain rate ρ¯m ∼ ρ¯m we can assume ρ¯m as a constant. From the above equation, we see that the velocity of the propagating bands is proportional to the applied strain rate. This result is similar to the −1 which also appears to be result obtained recently by Hähner et al. [44]. Further, v ∝ ρ¯m consistent with an old experimental result. (See Fig. 7 of Ref. [234] which appears to fit v ρ¯m = constant.) This result needs further experimental support. Note also that the velocity is inversely proportional to the square root of the immobile density. We have numerically calculated the velocity of the continuously propagating bands at high strain rates from the model which confirms the linear dependence of the band velocity on applied strain rate. In the region of strain rates ǫ˙ = 220 to 280 (corresponding to unscaled strain rate values 10−4 –1.5 × 10−5 s−1 ), we find that the unscaled values of the band velocity increases from 100 to 130 µm/s. These values are consistent with the experimental values reported by Hähner et al. [44].

6.6.7. Dynamical interpretation of band types Recall that earlier analysis [137] has demonstrated that type B bands are associated with the chaotic regime and type A with the power law state. This can be compared with the results from the model. The bands seen in the AK model are clearly correlated with the two distinct dynamical regimes investigated. The hopping type bands belong to the chaotic regime, a result consistent with the recent studies on Cu-Al polycrystals [137]. On the other hand, the propagating bands are seen in the power law regime of stress drops [50], again consistent with these studies [136,137]. Curiously, however, even the uncorrelated bands predicted by the model also belong to the chaotic regime. This needs further experimental confirmation. We now explain these results based on the dynamics of the model. We first note that each spatial element is described by three dislocation densities (eqs (122)–(125)). Consider one of these elements being close to the unpinning threshold, i.e. δ = 0. The analysis also shows that ρim is out of phase with ρm [190,213]. When the orbit is about to leave S2 , i.e. when ρm (x) is at the verge of a sharp increase, ρim assumes the highest value locally. However, −1 the extent of the spatial coupling is determined by ρim . But the magnitude of ρim itself decreases with the applied strain rate, being large at low strain rates [190,213]. Thus, the spatial extent of this is small at low ǫ˙ and large at high ǫ˙ . On the other hand, the growth and decay of ρm (x) with x occurs over a short time scale which is typically of the order of the correlation time, τc , measured by using the autocorrelation of φ(t). Beyond this time, the memory of its initial state is lost. To see how the correlation length scale and time scale change as we increase applied strain rate, consider an initial state when a band is formed at some location. Before the memory of this initial state decays; if a new band is not created, we get an uncorrelated band. On the other hand, if a new band is created before the memory of the initial state decays, there are two possibilities. If another band is created just before the correlation decays substantially by that time, we get a hopping type band. If however, even before the burst of ρm (j ) decreases beyond its peak value, new sources of creation of ρm occur, then we end up seeing a propagating band. An analysis of the correlation time shows that it increases with the applied strain rate. Concomitantly, ρim decreases with ǫ˙ which implies that the spatial correlation increases. Under these conditions, only partial plastic relaxation is possible at high values of ǫ. ˙ More importantly, the fact that both in

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experiments as well as in the model a power law exists for the stress-drops implies that there is no characteristic time scale. Indeed, the power spectrum of φ(t) also exhibits a power law. Thus, the correlation time actually diverges. Thus not only the length scale but the correlation time also diverges. This discussion clarifies the dynamic interplay of time scales and length scales. Moreover, as the spatial coupling term allows the spreading of dislocations only into regions of low ρim or low back stress, the propensity for continuous propagation of the band is enhanced when ρim is small. In addition, we find that higher values of ρim in the wake of the band which favors propagation into regions of smaller immobile density thus determining the direction of propagation also. 6.6.8. Summary and conclusions To summarize, we have shown that established methods of analysis in the area of dynamical systems can be gainfully employed to get a good insight into the PLC effect. Moreover, these methods offer dynamical interpretation of most generic features of the PLC effect within the context of the AK model. The first question we have addressed is the mathematical mechanism leading to the bistable nature of dislocation densities which translates to two levels of stress values seen at low strain rates. Two distinct methods have been used. The first one uses a simple nullcline method wherein the approximate analytical form of the nullclines have been used to obtain not just the wave forms of the mobile density, but the steps on the creep curve. The second approach uses the reductive perturbative approach which is essentially a method that successively obtains contributions from multiple time scales. In dynamical approaches, the order parameter fields are the slow modes. The reductive perturbative approach used is a method wherein fast modes are enslaved by the slow modes in a systematic manner. The resulting equations are for the order parameter variable which in the present context are the amplitude and phase of the limit cycle solutions. This also means that one associates a free energy like function in the neighborhood of the bifurcation point. In principle, this free energy function can be used as a starting point for a possible Ginzburg–Landau theory mentioned in the introduction. The next question that we have addressed is the origin of negative strain rate sensitivity of the flow stress, which in general terms also means bistable strain rate values. The idea is that different parts of phase space can be essentially described by approximate dynamics wherein a few dislocation mechanisms play a dominant role, well captured by an analysis of the slow manifold. In the case of the AK model, the slow manifold has two distinct branches, one where the dislocations are in the pinned state and another where they are in the unpinned state. The two stable branches are separated by an unstable manifold. Approximate equations of motions are first obtained from the original complicated coupled equations for these branches. Using these, each branch of the SRS is then calculated. The interpretation is that the increasing nature of stress with strain rate and the decreasing trend of stress at high strain rate correspond to the stable parts of the manifold S2 (pinned state of the system) and S1 (slowing down of mobile population before pinning) respectively. These two stable branches of SRS are separated by an unstable branch – which results from the unpinning of dislocations and hence corresponds to the unstable manifold of the system of equations. Another important advantage of the slow manifold description is that it is particularly useful in giving a geometrical picture of the spatial configurations of dislocations. The

§6.6


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crucial parameter that describes the state of dislocations is δ which has all features of effective stress of unpinning dislocations. This method also helps us to visualize dislocation configurations in different regimes of strain rates, for example, randomly nucleated type of bands, hopping type and the propagating type. The first two have been identified with the chaotic state and the type A with the power law regime of stress drops [51,137]. The study shows that the configurations of dislocations that are largely in the pinned state in low and medium strain rates (chaotic domain) are pushed to the threshold of unpinning as we increase strain rate (power law stress drop regime). Thus, the propagating type of band corresponds to a critical state of unpinning. Finally, the complementary dynamical method offers a dynamical reason for the smallness of the yield drops in the high strain rate region of band A [190,213]. Indeed, in this regime of strain rates the existence of a reverse Hopf bifurcation at high strain rates [51, 190,213] leads to softening of the eigenvalues (as a function of the applied strain rate), and the orbits are mostly restricted to the region around the saddle node fixed point located on the S1 part of the manifold. Note also that there is a dynamic feedback between the stress determined by eq. (125) and the production of dislocations in eq. (122) which provides an explanation for the slowing down of the plastic relaxation. This sets up a competition between the time scale of internal relaxation and the time scale determined by the applied strain rate (essentially the Deborah number). While the time scale for the internal relaxation is increasing, that due to the applied strain rate is decreasing. It must be emphasized that the long range interaction which appears to govern the collective effects stems from the dif−1 factor. This gives long range correlations that increases fusive coupling modulated by ρim with strain rate. More importantly, the correlation time scale also goes to infinity as is clear from the power law distribution of stress drops both in the model and in experiments. The dependence of the band velocity on ǫ˙ and ρ is consistent with experiment. Regarding the spatial features seen in the model, we stress that all these features emerge purely due to dynamical reasons without any recourse to using the negative strain rate sensitivity feature or DSA as an input, as is the case in most models [33,37,44,163,235]. Even the recently introduced poly-crystalline plasticity model which reproduces the crossover behavior also uses the negative SRS as an input [186]. The dynamical approach followed here clearly exposes how the slowing down of the plastic relaxation occurs due to a feedback mechanism of dislocation multiplication and applied strain rate as we reach the power law regime of stress drops. However, it should be stated that while the three different types of bands have features of the uncorrelated type C, hopping type B and the propagating type A bands found in poly-crystalline materials, there is no element of poly-crystallinity in the model in its present form. In poly-crystals, other types of coupling terms do arise which have also been modeled by diffusive type terms [3]. One way of including the presence of grain boundaries within the natural setting of the model is to recognize that cross-slip will be hindered near the grain boundaries which also leads to a term similar to the present diffusive term. Such a term can account for the back stress arising from the incompatibility of grains. As the forms of these terms are similar, the basic results are unlikely to change although one should expect a competition between the terms operating within a crystal and those at the grain boundaries. Finally, it should be stressed that the model uses simplified dislocation mechanisms. For instance, all the dislocation reaction terms have been simplified as quadratic or bi-quadratic terms in the dislocation densities. In reality one should

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1/2

have terms like ρm ρim . This, however, is a necessary simplification if one wants to use dynamical systems techniques. It is also possible to introduce additional variables, for instance, one can introduce an equation for the solute concentration itself. However, these tend to obscure the physics of the PLC effect. In summary, the AK model, though an idealized one, is sufficiently complicated at a technical level. This also means sophisticated dynamical techniques need to be used to understand the physics underlying the model. Though simplistic, the model appears to capture most generic results of the PLC effect. It is also the only model that uses dislocation densities and hence can be extended to two and three dimensions. 6.7. Dynamic strain aging model for the Portevin–Le Chatelier effect The concept of dynamic strain aging is the basic physical mechanism adopted directly or indirectly in modeling the PLC effect. Hähner et al. [44,185,236] formulate a PLC model based on DSA framework with a view to capture the observed features in their experiments [44]. In this model, as in many earlier models, aging occurs when mobile dislocations are temporarily arrested at obstacles like forest dislocations which are unpinned by thermal activation. The plastic strain rate is then expressed through an Arrhenius relation G0 + G σeff ǫ˙p = ν exp − , (169) + kT S0 where ν is an attempt frequency, an elementary strain increment when the mobile dislocations are activated once, G0 an activation enthalpy in the absence of DSA and S0 is the instantaneous strain rate sensitivity. The additional activation enthalpy G reflects the change arising from the aging process. Thus, G is proportional to the solute concentration accumulated on dislocations. In this model, G is taken to be a dynamical variable as it increases with waiting time of dislocations at obstacles and hence decreases with the imposed time scales as well. Howewer, dislocation densities are not included in line equations. The driving force on mobile dislocations is given by σeff (ǫp , ǫ˙p , G) = σext − σint (ǫp ),

(170)

where σext is the applied stress which can be a function of time as in the case of stress rate experiments and σint is the athermal back stress which depends on strain ǫp . In this model, σint = hǫp where h is the work hardening coefficient. ext The instantaneous strain rate sensitivity is defined by S0 = ∂∂σ ln ǫ˙p ǫ , G which is always positive, represents the response in the absence of strain hardening and aging. However, ∂σext dG ∂σext , (171) S∞ = = S0 + ∂ ln ǫ˙p ǫ ∂G ǫ d ln ǫ˙

is the response of the material observed after all the internal degrees of freedom has been fully relaxed to a new steady state. The PLC effect arises due to negative values of S∞ .

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To model the evolution of G, we note that the aging process contributes to an increase in G and unpinning of dislocations to a decrease in G as the solute content at the core of dislocations decreases. These two contributions are written in the following form G˙ = η

G∞ G

(1−n)/n

(G∞ − G) −

ǫ˙p G.

(172)

Here η is the aging rate which is proportional to the solute mobility, n an exponent and G∞ is the asymptotic value of G. The first term is the aging contribution and the second term the loss due to unpinning of dislocations and hence G relaxes at a rate proportional to the plastic strain rate ǫ˙p . For short times, G grows as a power law. (The value of n is typically 1/3 as measured in several experiments [41,42,181,237,238].) Instead of working with the driving force σeff , a generalized driving force defined by σeff ν G0 f ≡ exp − exp , η kT S0

(173)

is more convenient to deal with. The idea is to write the evolution equation for the generalized driving force f coupled to that of G. We can have σ˙ eff ηh G 2 f , f− exp − S0 S0 kT (1−n) n G∞ G G. ∂t G = η (G∞ − G) − ηf exp − G kT ∂t f =

(174)

(175)

For the constant strain rate case these equations must be coupled to σ˙ ext = Eeff

1 ǫ˙a − l

0

l

G f , dx η exp − kT

(176)

where the applied strain is given by ǫ˙a = v/ l with v being the imposed cross-head velocity, l is the length of the sample and Eeff is the combined modulus of the sample and the machine. Eqs (174), (175) are written in the form of scaled variables f˙ = σf ˙ − θf 2 exp(−g),

(177)

g˙ = (g/g∞ )

(178)

−m

(g∞ − g) − f g exp(−g),

where m = (1−n)/n, g = G/kT , g∞ = G∞ /kT , σ˙ = σ˙ ext /ηS0 , θ = h/S0 and t˜ = ηt where the over dot now refers to differentiation with respect to the scaled time. 6.7.1. Spatial coupling The topic of spatial coupling in plasticity has been a source of much debate [3,4]. Several mechanisms have been suggested in the literature [3,4]. The presence of a gradient type of spatial coupling was suggested by Ananthakrishna and Valsakumar in the context of their

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model [108] and a Laplacian type of coupling was suggested by Aifantis [47,239]. Several mechanisms have been discussed as a source of spatial coupling. For instance, Bridgman factor, compatibility stresses across the grains, and double cross-slip have been mentioned in the literature [3]. In the present context, in principle, diffusion of solute atoms could contribute to spatial coupling which however can be estimated to be too small to produce the characteristic length scale [44] relevant in this case. The authors argue that correlated dislocation glide due to internal stresses is the source of coupling in the present model. Using this, a diffusion type of coupling is introduced into the equation for g. Noting that even for m = 1, eq. (178) is nonlinear, further analysis is limited to m = 1 case which simplifies to g˙ = g ′′ + g∞ − g − f exp[−g]g,

(179)

where the prime denotes the differentiation with respect to scaled spatial coordinate x˜ = √ (η/Dg )x with Dg = βg gmin (G/S0 )˙ǫb2 s 2 , where s is the specimen thickness. One can easily show that the asymptotic SRS is given by ǫ˙p g∞ . (180) S∞ = S0 1 − [1 + ǫ˙p /(η)]2 η Investigation of the stability of eqs (178), (179) with respect to the fixed point g∞ σ˙ g∞ f0 = exp , g0 = , θ 1 + σ˙ /θ 1 + σ˙ /θ

(181)

√ is straight forward. The eigenvalues are given by λ± = κ ± κ 2 − ω2 with κ = 12 ( σθ˙ (gs − 1 − θ ) − 1) and ω2 = σ˙ ( σθ˙ + 1). For a Hopf bifurcation, κ 0 and κ 2 ω2 . This gives the unstable region √ [g∞ − 2 − θ − g∞ (g∞ − 4 − 2θ )] σ˙ < 2(1 + θ ) θ √ [g∞ − 2 − θ + g∞ (g∞ − 4 − 2θ )] . < 2(1 + θ )

(182)

The authors state that stress rate and strain rate can be mapped one-to-one through ǫ˙ = σ˙ ext / h which is a steady state deformation condition. The instability domain is shown in Fig. 46 for different values of the hardening rate. As can be seen, increased hardening rates decrease the instability boundary. Also, in the absence of a hardening term (θ˙ = 0), one can easily check that g∞ < 4 for instability to occur. The authors state that this is a subcritical bifurcation state. One can have a clear idea of the structure of the limit cycles by plotting the two nullclines given by f˙ = 0 and g˙ = 0. The nullclines given by g∞ σ˙ (g) (f ) − 1 exp(g), (183) fnull = exp(g), and fnull = θ g

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Fig. 46. The PLC instability region for various values of the hardening parameter θ . After Hähner et al. [44].

Fig. 47. Nullclines corresponding to f˙ = 0 and g˙ = 0 (dashed lines) along with limit cycle solutions. (a) Low value of θ = 0.01. (b) High value of θ = 0.5 and g∞ = 6. After Hähner et al. [44].

intersect at the fixed point in the unstable regime. Numerical solution of the limit cycle for two different values of the hardening coefficient (σ˙ = θ = 10−2 , g∞ = 6 and σ˙ = θ = 0.5, g∞ = 6) show qualitatively different types of solutions as can be seen from Fig. 47(a), (b). For low hardening rate, the solutions stick to the stable branches of the nullcline determined by g˙ = 0 and jump only at the point of instability, while for high values of θ , the limit cycle stays away from the stable branches for all times. 6.7.2. Band properties Now consider the spatial aspects of the band formation and propagation. Concentrating on type A band, we first assume that f = −cf ′ and g˙ = −cg ′ where c = (Dg η)1/2 c0 . Using this in eqs (178), (179) we get, cf ′ − θ exp[−g]f 2 = 0,

(184)

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g ′′ + cg ′ − f exp[−g]g + g∞ − g = 0.

Ch. 73

(185)

As mentioned earlier, the nature of the limit cycle solutions is sensitive to the work hardening rate θ . As the period of the limit cycle corresponds to the nucleation and propagation of the band, it is necessary to understand the influence of θ . Since the plastic strain rate is ǫ˙p = ηf exp(−g), the peak strain rate in the band is decided by the relative minimum of f exp(−g) or g ′ = f/f ′ . Using this in eqs (184), (185), we get the band strain rate ǫ˙b to be ∗ + g ′′ g∞ − gmin ǫ˙b , = ∗ −θ η gmin

(186)

∗ is the residual DSA enthalpy realized at the peak strain rate value (see Fig. 47). where gmin ∗ and hence the effect As can be seen from the above equation, θ cannot be larger than gmin ∗ of work hardening can be understood by comparing θ with gmin . For large g∞ , this gives gmin ≈ g∞ (g∞ − 1) exp[−(g∞ − 1)]. The values of gmin ranges from 0.2 for g∞ = 6 to 10−6 for g∞ = 20. Then the transition from weak to strong hardening regime is given by

θ=

h g∞ (g∞ − 1) exp −(g∞ − 1) . S0

(187)

Further, the switching curve approximation holds if DSA is not strong, i.e., 4 < g∞ < 8. 6.7.3. Band characteristics From a physical point of view, in the case of continuous propagating band type A, the entire local strain rate contained within the band is transported continuously. Further, the fact that the band is localized implies that the total deformation in the entire sample is concentrated in the band width, i.e. l ǫ˙a is accommodated in the band. Thus, one should have l ǫ˙a = ǫb cb and l ǫ˙a = ǫ˙b wb . Here ǫb and ωb are band strain and width respectively These band parameters can be calculated analytically. For this, we note that from eq. (185), we have c=

+∞

dx˜ fgg ′ exp[−g] I1 = . +∞ ′ 2 I2 −∞ dx˜ (g )

−∞

(188)

This can be integrated by parts and using eq. (184), we get θ I1 = c

+∞

−∞

2 dx˜ (1 + g) f exp[−g] .

(189)

This integral can be evaluated by noting that ǫ˙p is a sharply peaked function which can be approximated by a delta function. For weak hardening rate θ ≪ 1, we get θ ǫ˙b 2 I1 = (1 + gmin ) w, c η √ where w = (η/Dg )wb is the dimensionless band width.

(190)

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For evaluating I2 , we note that for low θ , the orbit sticks to stable branches of the characteristic curve g˙ = 0 and jumps between branches abruptly at the extremum values of the nullcline, i.e. f = fmax and f = fmin . This also means that one can use approximate expressions on these branches to obtain 1 1 I2 = (gmax − gmin )2 ≈ 2(gmax − gmin )2 . + (191) wI wIII Using these approximations, one gets the following expressions for the band velocity, band width and band strain given by √ Dg 1/4 (1 + gmin )/2 √ l ǫ˙a θ √ , cb = η gmax − gmin wb Dg l ǫ˙a gmin wb = 2 + , η g∞ − gmin η

ǫb =

l ǫ˙a . cb

(192)

(193) (194)

It is evident that the band velocity cb can have a nonlinear dependence on the applied strain rate through the dependence of band width on ǫ˙a . As expected, cb vanishes as D → 0. Within the scope of this model, band plastic strain diverges as D → 0 [44,185]. Further, cb depends on square root of the hardening rate which implies that higher the hardening rate, larger is the velocity and lower the band strain. These results have been verified numerically [44,185]. For the strong hardening case, the limit cycle solutions no longer stick to the stable branches of the nullclines corresponding to g˙ = 0 and thus the above results do not hold. However, it is possible to derive equations for this regime as well. These are given by θ l ǫ˙a , 2 ∗ 1 + gmin 2 Dg wb = ∗ )2 , ∗ θ η (gmax − gmin cb =

ǫb =

l ǫ˙a =2 . cb θ

(195) (196) (197)

In contrast to the weak hardening case, we note that both cb and ǫb do not depend on D, but the bandwidth does. This is consistent with the experimental observation that bandwidth is affected by specimen thickness which can be rationalized recognizing that Dg ∝ s 2 . 6.7.4. Numerical results A detailed numerical investigation of the model has been carried out [44,185]. Here, we briefly summarize salient numerical results of the model. First consider the constant stress

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rate case. The parameter values chosen are σ˙ /θ = 1 [σ˙ = 0.01, or σ˙ ext = 10−3 MPa/s, θ = 0.01 or h = 1000 MPa], η = 0.1 s−1 , S0 = 1 MPa, = 10−5 and g(0) = g∞ = 6. As the nature of the solutions obtained without the spatial coupling term already gives a good insight into the dynamics, we first consider this. Starting from the initial values, f (0) = 10−3 and g(0) = 6, the system of equations goes into an oscillatory state. While g goes through a rapid variation, f increases slowly, corresponding to the region where the orbit sticks to the nullcline corresponding to g˙ = 0. Thus, f is slow while g is a fast variable. Correspondingly, stress increases in step with the plastic strain. Eqs (184), (185) have been studied in detail for a range of values of the parameters, in particular for constant strain rate conditions ranging from ǫ˙ = 1 × 10−7 s−1 to 6 × 10−6 s−1 . Other parameters are held at E = 105 MPa, Dg = 107 m2 /s, l = 0.1 m. A parabolic hardening coefficient decreasing linearly with space averaged plastic strain is used (h = 103 MPa). During the course of deformation, the hardening parameter θ changes (a)

(b)

Fig. 48. (a) Plots of f and g as a function of time for strain rate ǫ˙a = 10−6 . (b) Plot of f and g on an expanded scale showing one single event. After E. Rizzi and P. Hähner [185].

§6.7


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from 0.01 to 0.002 (h ranging from 1000 to 200). With the initial condition that the PLC band is nucleated at one end due to a perturbation at one end, a type A band that propagates along the length and gets reflected repetitively at the ends is seen. However, a more natural initial condition corresponds to solving these equations with random initial values for g and f over the set of all blocks that range from 1 to 30 corresponding to local variation of 10% yield stress. Fig. 48(a) shows the variation of f and g for a mid block as a function of time along with an enlarged plot of a single event (Fig. 48(b)) for ǫ˙a = 10−6 . It is evident that while f does not change significantly, g changes abruptly at specific intervals. The time evolution of the plastic strain rate and plastic strain at the same block are shown along with a blow up of one event in Fig. 49. While the local plastic strain rate is quite substantial, the corresponding stress fluctuations on the stress–strain curve are small. The initial part of the stress–strain (corresponding to slightly irregular be-

(a)

(b)

Fig. 49. (a) Plots of plastic strain rate and plastic strain as a function of time for strain rate ǫ˙a = 10−6 . (b) An expanded plot for one strain rate burst. After E. Rizzi and P. Hähner [185].

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haviour of f and g in Fig. 48(a)) corresponds to type C and B bands which eventually settles down to a sawtooth type of waveform of the stress that correspond to type A band. The nature of band movement obtained from the model can be compared with that seen in experiments by an appropriate choice of the strain rate. For instance, Fig. 50(a) shows a space–time plot of the band in course of time for ǫ˙a = 2 × 10−6 s−1 . This can be compared with space–time plot seen in experiments shown in Fig. 50(b). As can be seen, in both cases, different types of propagating modes such as multiple band propagation with band collisions, zig-zag propagation with reflections at one end and parallel propagation are seen. Detailed simulations also show that the band velocity and width are function of hardening rate. The scaled band velocity cb ∝ θ 0.54 obtained from simulations is shown in Fig. 51(a) with the dependence of band width for applied strain rate ǫ˙a = 10−6 . This is in good agreement with the square root dependence on θ predicted by eq. (193). The authors have also numerically verified the analytical results in eqs (193), (194) by plotting

(a)

(b) Fig. 50. (a) Space–time plot of band propagation for ǫ˙a = 2 × 10−6 s−1 obtained from the model. (b) Experimental space–time plot of band propagation. After Hähner et al. [44].

§6.7


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√ cb (wb /θ ) as a function of strain rate for various values of the hardening parameter θ shown in Fig. 51(b). The authors also note that the experimental results in Cu-15%Al actually correspond to the strong hardening case. Further the power law dependence of the band parameters observed in experiments is shown to be in qualitative agreement with the results of their model by noting that the pseudo-diffusion constant depends linearly on the band strain rate. The stability of type A band is also discussed. For details, we refer the reader to Refs [44,185]. While the agreement with experiment appears to validate the model, a few observations are necessary. One positive feature of the model is that there are only two modes and thus many of the results are easy to appreciate. However, the model suffers from the same drawback as the AK model in that only one spatial dimension is considered and there is no element of polycrystallinity. In addition, some results that are consistent with experimental observations are a consequence of phenomenology. For instance, the dependence of band velocity on thickness is due to the dependence of the pseudo-diffussion

(a)

(b) Fig. 51. √ (a) Band speed and band width as a function of hardening coefficient. (b) Rescaled band velocity cb (wb /θ) as a function of strain rate for different hardening coefficients. After Hähner et al. [44].

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constant on S. In addition, while considering band properties, a parabolic hardening is used, and the spatial variation of the hardening is also put in by hand. Similar dependence of pseudo-diffusion coefficients on the thickness and strain rate can arise in other situations for example, the Bridgman factor. Finally, as long as band strain is localized as in any nonlinear model, the dependence of w should follow.

6.8. A multiscale model for the Portevin–Le Chatelier effect 6.8.1. Introduction Although the fully dynamical model due to Ananthakrishna and the DSA based model due to Hähner discussed in the last two sections are very different from each other in terms of the physical basis as also the details, they address the complex dynamics of the PLC effect from a dynamical point of view. This points to some convergence of views that deterministic dynamics plays an important role in the description of the PLC effect. This view is also supported by the results of time series analysis discussed in Section 5.2. However, both work in one dimension and use a diffusive coupling, although the nature of the suggested coupling in the two models are quite different. Moreover, there is no consensus on the nature of spatial coupling or the associated length scale that should be used. An alternate approach for introducing spatial coupling is through strain gradient plasticity theories which involve a gradient length scale in its constitutive form through a Laplacian [240,241]. More recently, further extension has been carried out by using constitutive relations in three-dimensional finite element (FE) framework which show propagative type of bands [235]. A recent publication [186] addresses the complex spatio-temporal evolution of the PLC dynamics by embedding a polycrystalline model in a three-dimensional finite element framework. In this framework, spatial coupling arises in a natural way through the local variation in material anisotropy arising from spatial variation in crystallographic texture. The basic idea is that during deformation, grains begin to flow at different stress levels depending on the orientation of the lattice: some grains are “soft” and some are relatively “hard”. This renders a natural spatial variation in deformation response as the local anisotropic viscoplastic response varies throughout the specimen. As a consequence, the usual kinematic decomposition of deformation into elastic and plastic parts, combined with a statement of equilibrium and solution of the boundary value problem leads to development of long-range stresses. This capability to derive a property variation from a physical basis allows for a spatial coupling without any ad hoc gradient constitutive formulation. Further the formulation attempts to use a multiscale approach as we shall see. 6.8.2. Constitutive model An elasto-plastic formulation of the model begins with the single crystal kinematics described using the deformation gradient F = Fe Fp = RUFp ,

(198)

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where Fp is the plastic shearing along the crystallographic systems and Fe stems from the rotation R and elastic stretch of the crystal denoted by the right stretch tensor U. The ˙ −1 is given by velocity gradient tensor L = FF ˙ −1 RT + Fe Lp F−1 ˙ T + RUU L = RR e .

(199)

The plastic velocity gradient Lp = F˙ p Fp−1 arises from the shear rates γ˙s on all the active s s s s systems. Thus, Lp = M s=1 γ˙s b0 ⊗ n0 where b0 and n0 are unit vectors directed along the slip direction and the normal to the slip plane respectively. The slip system shear rate is related to the resolved shear stress through n τs (200) γ˙s = γ˙0 sgn(τs ), τ0

where τs is given by τs = (bs0 ⊗ ns0 ) : 2μǫ ′ , where ǫ ′ is the deviatoric component of the strain tensor ǫ and μ is the Lamé constant. The parameter n is taken to be 20 in the calculation basically to ensure the desired slip activity. A proper choice of reference strain rate and reference stress is also necessary. Dynamic strain aging, a necessary component of the PLC effect, is introduced through the reference stress or glide resistance (τ0 ). This resistance is given by Kubin and Estrin [176,177] 2/3 , τ0 (γ˙0 , τǫ ) = τa + τǫ + S0 ln(γ˙0 ) + f0 1 − exp −γ˙ ∗ /γ˙0

(201)

where τa is the athermal stress characterizing rate independent interactions of dislocations with long range barriers, and τǫ describes strain hardening and recovery effects. S0 is the SRS associated with the activation of overcoming the short range obstacles in the absence of DSA. f0 is the saturation level of the stress arising from aging and γ˙ ∗ is a reference strain. The evolution of strain hardening component of the reference stress with accumulated slip Ŵ (sum of all accumulated slips in all the allowed slip systems) is given by τǫ τλ dτǫ , (202) = θ0 1 − + dŴ τǫs τǫ dτλ 1−1/r = cτλ . dŴ

(203)

Here τλ is an additional stress variable used to describe deviation from Al-Mg alloys from an exact Vo´ce behavior with saturation stress τǫs = τǫs (γ˙0 ) [242]. The model also includes plastic relaxation. (For details about the FE approach and the numerical scheme, we refer the readers to Ref. [186].) Each material point is assigned a polycrystal comprising of N crystals all experiencing the same velocity gradient L using a Taylor type of relationship [243]. The macroscopic Cauchy stress tensor is an average over all grains. The behavior of the material at some point x is obtained as the response of an aggregate of N crystals in a representative volume around x . While different crystal stress tensors emerge due to initial crystallographic

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textures, only the macroscopic tensor satisfies the equilibrium and boundary conditions. A fully-implicit elastoviscoplastic finite element formulation is used to solve the boundary value problem. A very short time step is necessary to capture the rapid transients concomitant with localization of plastic flow. A mesh containing 1440 elements in a 5 × 8 × 36 arrangement is used to represent a tensile specimen, with velocity boundary conditions imposed at the ends of the sample. 6.8.3. Results While the model uses a multiscale approach combining polycrystalline plasticity with FE technique, the guiding principle is that the complex dynamics is controlled by deterministic dynamics. This is reflected in the quantities that are measured and comparisons made with earlier work. Apart from the nature of the band types obtained in different regimes of strain rates, most results are focused on dynamical and statistical features that characterize the PLC dynamics as illustrated in several previous publications [50,108,133–137,166,191]. Different band types are observed by changing the applied velocity. At high strain rates 10−3 s−1 which is close to the upper limit of negative SRS, propagative type A bands are seen. A typical sequence of snap shots of the band corresponding to eight grains per element is shown in Fig. 52. Fig. 53 shows the corresponding stress–strain curves for two different cases of simulation; one and eight random crystallographic orientation assigned to each element. More regular serrations are observed for the eight grains/element (which can be attributed to larger spatial coupling). When the band reaches the end of the specimen, one sees a large stress drop as in experiments as well as in the AK model. The average flow stress is higher in this case compared to one grain/element case as can be seen from Fig. 53. The higher value of the flow stress can be attributed to hardening arising from shorter wavelength in the spatial variation of the Taylor factor. Now consider the analysis of the statistical features of the stress drops amplitudes and durations. Recall that an analysis of experimental stress–strain curves for the region of band A showed a power law distribution of stress drops and their durations [33,136,137]. As in the experimental stress–strain curves, there is a hardening factor that contributes to the general increases in the stress level as well as the changes in the amplitudes of serrations. This systematic variation of stress is scaled out by a moving average method. The distribution of stress drops P (σ ) of the stress drop amplitudes σ and the distribution of associated time intervals P (t) of the durations t, both exhibit power law with exponents α = 1.35 and β = 1.92. Further, σ ∼ t x with x = 2.85. As mentioned, these exponents have to obey a scaling relation β = 1 + x(α − 1) which they do. Now consider the possibility of type C bands which are seen at the lower end of the strain rates. In experiments, these are seen in specimens hardened by cold rolling without any annealing. Appropriate information is transferred to the mesh used for simulation (see for details Ref. [186]). Basically, the rolling and normal directions were aligned along the longest and shortest dimensions of the tensile sample. The rate of work hardening is decreased to account for the hardening of the prepared sample by neglecting τλ . In this case, the serrations are quite regular as in experiments and the distribution of the magnitudes of the stress drops turns out to be nearly Gaussian. The bands in this case are static and are found to be nucleated randomly both in space and time. The lack of correlation is found to

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Fig. 52. Type A propagating bands for applied strain rate ǫ˙a = 10−3 s−1 for eight grains/element. The sequence of images shows the band getting reflected at the left end of the specimen with band reorientation after reflection. The images are shown at intervals of 0.05 s. After S. Kok et al. [186].

Fig. 53. Stress–strain curves for one grain and eight grains per element. The inset shows the serrations obtained from experiments in Al-2.5%Mg alloy at a strain rate 1.4 × 10−4 s−1 . After S. Kok et al. [186].

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Fig. 54. Location of successive type B bands for applied strain rate ǫ˙a = 2×10−4 s−1 . Intermediate images show a local increase in plastic strain rate prior to band formation followed by a local decrease upon disappearance of the band. After S. Kok et al. [186].

be the result of the small time scale of plastic relaxation time tpl (0.05 s) compared to the average reloading time which is of the order of 2.0 s in this case. The type B bands are found in the intermediate range of strain rates. In this case, simulations have been carried out on a mesh of 2 × 6 × 36 grid with total of 648 elements. Sixteen random crystallographic orientations are used per element. Fig. 54 shows a sequence of images of the bands along the length of the sample. Each band is static but a new band is created ahead of it that gives the impression of hopping type propagation. The snap shots of the images shown in the figure are not at regular intervals but at appropriate times to show the development of plastic activity intermediate to the band formation. At the location of the bands, one regularly observes a local increase in plastic strain rate prior to the band formation followed by a decrease after the disappearance of the band. The statistical distribution of the stress drops and durations are skewed to the right as in the case of the experimental stress drop distribution [137,166]. As demonstrated earlier in experiments on Al-Mg [137,166], as also predicted by the AK model, the intermediate regimes of strain rate should exhibit chaotic stress drops. To see if the model also exhibits the region of chaos in range of type B bands, a single crystallographic direction was assigned to each element to reduce the computational time as long stress–time series are required for the analysis. The methodology used is the same as for experimental time series outlined in Section 5.2. A given time series is embedded in higher dimension and the corresponding metric invariants like the correlation dimension

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Fig. 55. Correlation integral obtained from experiments and that from simulations with the correlation dimension ν = 4.15 from experiment and 4.26 from simulations. See Fig. 21 (top). After S. Kok et al. [186].

Fig. 56. Type B bands: (Left, Right) The reconstructed strange attractor obtained from experiments and simulations respectively. As in Ref. [50], the linear portion of the phase plot refers to the loading direction. After S. Kok et al. [186].

and Lyapunov spectrum are calculated. A log-log plot of correlation integral is shown in Fig. 55 for embedding dimension d = 13 to 16. The slope converges to ν = 4.26 which is close to the value obtained for the type B band experimental serrations. (Compare with Fig. 21 (top) of Section 5.4.2.) The spectrum of Lyapunov exponents have also been calculated and the Kaplan–Yorke dimension DKY obtained from the spectrum turns out to be 4.2 for the model attractor consistent with the value obtained for the correlation dimension. Using singular value decomposition outlined in Section 5.2.3, the reconstructed attractors corresponding to experimental time series of Al-Mg alloy and that of the model are shown in Fig. 56(a), (b). It is clear that the experimental attractor and the model attractor look very similar. 6.8.4. Conclusions The emphasis of the model is to include various ingredients that operate at different length scales in order to achieve a good description of the complex spatio-temporal dynamics. At

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the finest scale, the Kubin–Estrin model [176,177] is used to describe the effect of dynamic strain aging on glide resistance. Going up in scale, the usual Taylor-type assumption is used where each (lattice) orientation associated with an element quadrature point is assigned the same velocity gradient and the macroscopic Cauchy stress becomes the average over the same set of orientations. The grain size dependence introduced by increasing the number of grains per Gauss point has practically no effect on the localization process itself. This might suggest that spatial coupling is not controlled by intergranular stresses consistent with experimental observation in Ref. [44]. Instead, the coupling is due to stress gradients arising from incompatibilities between differently strained areas in the localization region. All the important features of the three types of bands are well reproduced in three dimensions without introducing a length scale for spatial coupling. Further, the associated dynamical regimes are well reproduced and are consistent with those found in experiments [137,166] as also with the predicted regimes [51,108,191,226,230]. Fig. 57 shows the distribution of plastic strain rate offset from the threshold value for all the three types of bands. The type A is closest to the threshold while type C is farthest from the threshold. This result is somewhat similar to the state of dislocations described in the AK model [51,230]. It may be recalled that the slow manifold analysis of the AK model showed that in the regime of low and medium strain rates corresponding to random

Fig. 57. Distribution of plastic strain rate offset from threshold value for localized deformation events for type A, type B and type C bands. For the propagating type A bands, the strain rate value remain close to the threshold value while it is more for the type B and the highest for the type C. After S. Kok et al. [186].

§7


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and hopping type bands, most dislocations are in the pinned state while at high strain rate where propagating bands are seen, dislocations are on the verge of unpinning [51,230]. The explanation for the three different types of bands is similar to that provided in earlier reports, namely a competition between the times scales of plastic relaxation and the reloading time scale [136,137,166]. The study of the model is not complete as it does not address the dependence of concentration, which in principle can be addressed. Temperature dependent properties are also not addressed.

7. Discussion and outlook With the material presented in the review, we now reexamine whether we are in a better position to understand the collective effects of dislocations and whether we have clear answers to the several issues raised in the introduction. To appreciate this we note that substantial gains in understanding dislocation patterns have been possible due to good interplay between theory and experiment. Two distinct types of advances have been reported. The first one is a result of improved experimental techniques both in terms of the introduction of new experimental techniques as well as improved accuracy of measurement. Further, the import of new theoretical concepts and techniques has lead to new experiments. There has been a healthy interaction between theory and experiment that has led to positive gains on both fronts. For instance, earlier results have been interpreted from the new perspective of concepts borrowed from physics and used to formulate new approaches to understand the collective behavior of dislocations. As an example of this, one can mention the results on slip-line morphology that have been known for some time [6]. These results clearly suggest that there is a separation of time scales which is a prerequisite for using Langevin dynamics approach. This has been exploited by Hähner for constructing models for explaining some collective effects [110,113,114]. In the same spirit, micrographs obtained from the earlier experiments by Essmann (see Ref. [30]) have been reanalysed in the light of fractal concept. This has also helped the authors to construct a Langevin dynamics based model to explain the scale invariance of the pattern [79,30]. Another example is the application of the concept of self-affinity for characterization of slip-line morphology [61]. Here, it appears that experiments were designed with this in mind. A third kind of progress has been achieved through an interplay of theory and experiments at successive stages. Attempts to verify theoretical prediction based on the AK model that stress serrations are chaotic [191,192] have led to new experiments, which in turn have led to subsequent modeling efforts. While early attempts [133–135] were directed at verifying the prediction of chaotic stress drops, subsequent efforts to characterize serrations in the entire range of the PLC instability [136,137,166] triggered further modeling efforts [51,226,230]. These efforts have given insights that would not have been possible but for the new ideas borrowed from dynamical systems. The analysis clearly shows that the apparently random looking stress serrations of the PLC instability have a hidden order that results from the deterministic dynamics of a few collective degrees of freedom of dislocations. Apart from verifying the prediction, the method confirms that the minimum number of collective modes required for a description of the dynamics of the PLC effect are only few [107,108]. This also meant that the approach to modeling the PLC effect

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should be from a dynamical angle, which is the approach originally taken. However, the subsequent analysis of the PLC stress–time series showed that there is a crossover in dynamics from chaotic to power law state of stress drops. This intriguing crossover from a low-dimensional dynamics to a high-dimensional power law state of stress drops needed further modeling efforts. This in turn required new dynamical techniques such as the slow manifold approach to understand the underlying dynamics in the model. The recent efforts in modeling the cooperative behavior of dislocations reviewed here can be classified as being stochastic and dynamical approaches. Even the early efforts used stochastic methods using both Fokker–Planck and Langevin approaches, though for a simple homogeneous situation. These studies showed that even the homogeneous case was sufficiently complicated. In addition, they point to the fact that there was a recognition of the importance of including the fluctuating nature of internal variables [16–18,99]. More recent approaches involve the Langevin dynamics and distribution function-theoretic approaches [84–86,110,113,114]. Note that these models deal with patterns that evolve slowly like the PSB, cells and matrix structure. In contrast, all models for propagative instabilities such as the PLC effect are dynamical. This might raise the question whether certain types of patterns have intrinsically stochastic or dynamical character? If so, how do we recognize what their signatures are? While a definite answer may be difficult, from a theoretical point of view, slowly evolving patterns are more amenable for stochastic modeling as the order parameter variables evolve slowly while the local internal variables are rapidly fluctuating. However, this in itself does not give any hint of whether the noise is additive or multiplicative. According to van Kampen [87], it is very difficult to argue unambiguously that the noise should be multiplicative. Usually, in most situations, noise is taken to be a additive. All this does not mean that the slowly evolving patterns cannot be modeled using dynamical approach. Indeed, they can be as long as there is a hint from experiments that changing one of the drive parameters leads to abrupt changes from a homogeneous to inhomogeneous state, keeping all other conditions fixed. It must, however, be mentioned that there have been earlier attempts by Walgraef and Aifantis [125] and the Kratochvil group [131] to explain the matrix and PSB structure which are dynamic in nature. More recently, Glazov et al. [211] attempt to explain the features of PSB as an Eckhaus instability [129]. These authors derive a Ginzburg–Landau equation using the Walgraef–Aifantis model and show that the Eckhaus instability offers the correct platform for the particularly slow growth of the wavelength changing pattern. In contrast, the PLC instability has enough indicators from experiments and theory that a natural approach would be dynamical. As plastic deformation is a highly nonlinear process, nonlinearities also appear in a statistical description as well. For example, in the Langevin description, they appear in the deterministic part and they can also appear in the noise coefficient. In the distributiontheoretic approach, while the equation for the decoupled joint probabilities are linear in the densities, the infinite hierarchy makes it highly nonlinear. This is also more obvious when the hierarchy of equations is truncated using any decoupling scheme. The two distinct stochastic approaches to modeling while appearing distinct are obviously related. In fact, by assuming that local internal stress is fluctuating, the results of Langevin approach have been shown to be similar to those of the distribution-theoretic approach. As mentioned earlier, distribution-theoretic approach appears to have an inherent limitation in describing


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patterns as translational invariance is violated. In this sense, the Langevin approach that uses local dislocation density is suitable. As for propagative instabilities, there is convergence of views that the general framework should be dynamical. Details of models would be different depending on what is sought to be described and what levels of sophistication are desired. This is what is reflected in the three models described here. From a technical level, the AK model requires sophisticated techniques to interpret the model. However, all generic features of the PLC effect arise purely due to the inherent dynamics of the interacting populations of dislocations. The DSA based model is amenable to simple techniques and results are easy to appreciate which however does not include dislocation densities. One major criticism against both these models is that they work in one space dimension and introduce spatial coupling based on physical arguments. However, as the question of spatial coupling is still a matter of debate, the crystal plasticity based model overcomes these two limitations. Even if most results are numerical, the dynamical as well as band properties are appealing, even though, the model does not include dislocation densities. While reasonable progress has been achieved, a coherent framework for describing pattern formation in plastic deformation is nowhere in sight. Deeper issues as to how to account for the high level of dissipation when most energy input is lost in the form of heat, coupled with the far-from-equilibrium nature of the patterns, remain open. In conclusion, it should be mentioned that even within a condensed matter description, there is no agreed framework for describing these kinds of situations.

Acknowledgements It is a great pleasure to thank several of my collaborators. I wish to acknowledge my earliest collaborator Dr. D. Sahoo in developing the model for the PLC effect. Much of this work on the PLC effect would not have been possible but for numerous collaborations with Drs M.C. Valsakumar, K.P.N. Murthy and several of my former students, Prof. M. Bekele, Drs S. Noronha, S. Rajesh and M.S. Bharathi. I also acknowledge fruitful collaborations with Dr. L.P. Kubin, M. Lebyodkin, S. Kok and Profs C. Frassengeas and A.J. Beaudoin. I would like to acknowledge the help from Dr. B. Ashok for reading the manuscript and K.G. Lobo in the preparation of the manuscript. The author wishes to acknowledge the award of Raja Ramanna Fellowship and BRNS grant No. 2005/37/16.

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CHAPTER 74

Topological Modelling of Martensitic Transformations R.C. POND, X. MA, Y.W. CHAI Department of Engineering (Materials Science and Engineering), University of Liverpool, Liverpool L69 3BX, UK and

J.P. HIRTH 114 E. Ramsey Canyon Road, Hereford, AZ 85615, USA



Contents 1. Introduction 227 2. Topological constraints for diffusionless processes 229 2.1. Virtual frontal migration of terraces 230 2.2. Topological parameters defining interfacial line-defects 232 2.3. Conservative and non-conservative motion of interfacial defects 237 2.4. Sessile and glissile defect intersections 241 2.5. Summary of constraints for martensitic transformations 242 3. Transformation crystallography 243 3.1. Coherency strains 243 3.2. Interfacial structure 246 3.3. Orientation relationship 248 3.4. Habit plane inclination 250 4. Transformation displacement 250 4.1. Transformation strains caused by disconnections 251 4.2. Accommodation strains caused by LID 252 4.3. Large strain effects 254 5. A comparison of the topological and phenomenological models of martensitic transformations 5.1. Structural differences 254 5.2. Magnitude of the differences between the TM and PTMC models 255 6. Closing remarks 259 6.1. Summary of the topological model 259 6.2. Topological modelling of other transformations 260 Acknowledgements 260 References 260

254

1. Introduction The properties of many technologically important materials arise through martensitic transformations; examples are the hardenability of steels [1], the toughening of ceramics [2] and the shape-memory and super-elasticity of alloys [3,4]. Since martensitic transformations can occur rapidly even at low temperatures, it is inferred that they are diffusionless processes [5]. Moreover, a transformation product exhibits a well-defined habit plane and orientation relationship (OR) with the parent phase, and produces a homogeneous deformation [6]. In the 1950s, Wechsler, Lieberman and Read [7] and Bowles and MacKenzie [8] formulated the phenomenological theory of martensitic crystallography (PTMC), which predicted these crystallographic features for a range of transformations in a manner consistent with contemporary experimental observations. At the heart of this theory is the proposition that the parent-martensite interface is an invariant plane of the shape transformation, so that no long-range strain field penetrates into the adjacent crystals. No description of the transformation mechanism or explicit demonstration of its diffusionless nature is embodied in the PTMC. The objective of the present work is to develop a physical model of martensitic transformations, inspired by recent studies using transmission electron microscopy (TEM) and advances in defect theory, and examine its competence regarding the prediction of transformation crystallography and insight into the transformation mechanism. Recent TEM investigations by several workers [9–13] show the structure of martensitic interfaces to comprise coherent terraces reticulated by arrays of localised interfacial line defects. Two types of defects are found in these interfaces, namely defects causing lattice-invariant deformation (LID), such as slip or twinning dislocations, and transformation dislocations [5]. The former are well understood, and their roles in crystal interiors and interfaces have been elucidated [14]. However, transformation dislocations exhibit step character as well as dislocation character, and the formal development of the theory taking both these topological properties into account in an integrated manner has only been presented recently [15,16]. To emphasise the key relevance of this understanding to the present work, our model is referred to as the topological model (TM), and transformation dislocations are referred to as disconnections to distinguish them from defects without step character [17]. Fig. 1 shows three high-resolution TEM images of interfaces in ferrous alloys consistent with being coherent terrace segments viewed edge-on and disconnections viewed end-on. To construct a feasible model of a particular parent-martensite interface using the TM, one must identify both the terrace and the array of dislocations and disconnections that remove any long-range coherency strains; thus, one can quantitatively determine the habit plane inclination and OR. One must also demonstrate that lateral disconnection motion along the terraces is diffusionless. The identification of candidate terraces and defects is highly constrained topologically by the requirement of zero long-range diffusion. Moreover, there are further physical constraints relating to tolerable coherency strains, reason-

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(a)

(b)

(c) Fig. 1. TEM micrographs of the habit plane in (a) Fe-Cr-C [9], (b) Fe-Ni-Mn [11], and (c) Fe-Ni-Mn [12]; the coherent {111}γ /{110}α terraces are seen edge-on and the disconnections end-on. Parts (a) and (c) are reproduced with the permission of Taylor and Francis Ltd. http://www.tandf.co.uk/journals.

§2

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able mobility of disconnections and available modes of LID, as enumerated in Section 2. The topological constraints derive from the ‘rules’ governing the glide and climb of disconnections, which are more complex than the corresponding case of dislocations in the bulk because of the juxtaposition of crystals with different densities. In addition, when a disconnection is shown to be glissile on a given terrace, it must also be shown that its intersections with dislocations are not sessile. In Section 3, the structure of parent-martensite interfaces is discussed, and illustrated by analysis of examples in the orthorhombic to monoclinic transformation in ZrO2 , and the bcc to hcp transformation in Ti alloys; the terrace/defect structures are determined, as well as the habit plane and OR. Section 4 is an account of the transformation mechanism and displacement, and Section 5 is a comparison of the TM and PTMC approaches. The final section is a summary of the TM for martensitic transformations, and a short discussion of its application to other types of transformations involving interfacial line defects.

2. Topological constraints for diffusionless processes The objective of this section is to demonstrate that identification of a viable terrace plane and the superposed disconnection and dislocation arrays is highly constrained if the migration process is to be diffusionless. Importantly, the extent of diffusion accompanying migration of an interface is not determined solely by the structure of the interface, but also depends on the migration mechanism. All mechanistic models for martensite transformations imagine that the process proceeds by the lateral motion of the disconnections shown in Fig. 2. In Christian’s nomenclature [22] this motion is orderly, or military, involving

Fig. 2. Schematic illustration of a parent-martensite interface showing the terrace segments and defect arrays, similar to the suggestions of several previous workers [18–21]. Coherently strained terraces are reticulated by arrays of disconnections (b, h) with spacing λD and crystal slip or twinning dislocations (b) in the (lower) martensite crystal. The terrace and habit (primed) coordinate frames are shown and the line directions of the disconnections, ξ d , and dislocations, ξ LID , are parallel to x and close to y ′ respectively.

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motion of an atom from a given site in the matrix to a known site in the martensite. The motion is analogous to that for growth of a singular surface from the vapour by a step mechanism as shown long ago by Gibbs [23]. The free energy barrier for direct motion by atom jumps across the interface in a stochastic, or civilian, process is prohibitive. Nevertheless a virtual frontal migration process is useful in calculating the diffusional constraints associated with transformation. If the disconnection array in Fig. 2 were to move laterally by the disconnection spacing λD , the interface would advance normal to the terrace plane by the distance h. On the other hand, if the interface were incoherent, with no vestige of a terrace structure, true frontal migration might be possible; however, such interfaces have not been observed for martensite. We now demonstrate that semi-coherent interfaces like that in Fig. 2 can migrate in a diffusionless manner by lateral disconnection motion whereas the virtual frontal migration needs long-range diffusion. Thus, while we concentrate on the consideration of the military process here, it is helpful to begin this section by quantifying the diffusional flux associated with the civilian process; the results provide a constraint to possible martensitic transformations. The arguments presented relate to the ‘single interface’ configuration [6] wherein the interface separates two semi-infinite crystals; thus, any expansion/contraction perpendicular to the interface is unconstrained. Such a configuration would be exhibited near the middle of the broad face of a martensitic plate and is the basis for all mechanistic models for the habit plane.

2.1. Virtual frontal migration of terraces Consider the (100)o /(100)m terrace plane between orthorhombic and monoclinic ZrO2 crystals depicted schematically in Fig. 3(a); here the two crystals viewed along [010]o / [010]m exhibit their natural lattice parameters [10] so they have misfit parallel to [001]o /[001]m , designated y in the terrace coordinate frame, but are coherent parallel to the viewing direction, x. (The monoclinic form can be viewed as being a distorted fluorite structure [10]; the orthorhombic form can be viewed as a reconstruction from two monoclinic unit cells. Thus, the double-monoclinic cell comprises four (002)m atomic planes parallel to the terrace and the single-orthorhombic cell comprises four (004)o atomic planes.) Let Xn (o) and Xn (m) be the number of atoms per unit volume in the two (natural) crystals; when multiple species, A, B, . . . are present (i.e. Zr and O for the current example), one uses Xn (o)A and Xn (o)B and so on. A fully coherent terrace is shown in Fig. 3(b) where the misfit has been removed by homogeneous uniaxial strains, e(o)yy and e(m)yy , of the two crystals; the strains can be partitioned between the crystals in any chosen manner. After homogeneous straining, the atomic densities are modified to Xc (o)A = (1 + e(o)yy )−1 Xn (o)A , and Xc (m)A = (1 + e(m)yy )−1 Xn (m)A , and so forth for other species [24]. Consider frontal migration of the incommensurate interface, Fig. 3(a), by a virtual civilian process. This is a hypothetical process since any interface like that in Fig. 3(a) would relax to either a coherent interface, Fig. 3(b), or one containing misfit compensating defects [25]. Imagine that a segment with unit area moves unit distance, n, into the orthorhombic crystal. This volume of parent material is replaced by a volume of product phase that differs

§2.1

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(a)

(b)

(c)

Fig. 3. Schematic view of the (100)o /(100)m terrace plane between orthorhombic and monoclinic ZrO2 . (a) The interface is incoherent and the crystals exhibit their natural lattice parameters. The heights of the doublemonoclinic and single-orthorhombic cells are designated d(m) and d(o) respectively. (b) The interface is coherent following uniaxial strains of the two crystals. (c) Corresponding cells in the coherently strained crystals, showing their common area at the interface; the single-monoclinic unit cells are indicated.

by the factor d(m)d(o)−1 , where d(m) and d(o) represent the relevant interplanar spacings of the terrace planes in the two crystals as indicated in Fig. 3(a) (in this case these spacings correspond to four (002)m and four (004)o atomic plane spacings respectively). Thus, the number of A atoms, NA , of the parent that are added (positive values) or removed (negative values) in this process per unit area of interface per unit distance migrated by the interface into the orthorhombic parent is equal to NA = Xn (o)A − d(m)d(o)−1 Xn (m)A . (2.1)

Similar expressions can be written for the other species present. Inspection of Fig. 3(a) shows that the virtual frontal migration of incommensurate terraces would, in general, require finite fluxes, NA , NB etc. However, zero diffusional flux can arise in certain instances, and migration of the coherent interface in Fig. 3(b) is an ex-

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ample. Consider corresponding cells in the two coherently strained crystals sharing an area of terrace plane common to the two cells, designated gc (o) = gc (m), as shown hatched in Fig. 3(c); the choice of cells is not unique, and is discussed further later. Let the number of atoms of species A in these coherent ‘o’ and ‘m’ cells be Z(o)A and Z(m)A , so Xc (o)A and Xc (m)A can be expressed as {Z(o)A (gc (o)d(o))−1 } and {Z(m)A (gc (m)d(m))−1 } respectively; substituting these expressions into eq. (2.1), one obtains −1 −1 − Z(m)A gc (m)d(o) , NA = Z(o)A gc (o)d(o)

(2.2)

and similarly for all species present. However, the number of atoms of each species in the two corresponding coherent cells must be the same, i.e. Z(o)A = Z(m)A etc., and hence expression (2.2) confirms that NA , NB , . . . = 0. One could now determine the diffusional flux arising when the semi-coherent interface in Fig. 2 migrates frontally by finding the flux associated with the virtual climb of the disconnections and dislocations embedded in the terraces, and this is treated after a discussion of interfacial line-defects. Even though this is not a physically realizable motion, it is equivalent in initial and final states to motion created by glide of disconnections along the terrace and is a convenient way to calculate the net flux.

2.2. Topological parameters defining interfacial line-defects The first issue to be considered is the definition of b and h for disconnections and dislocations in interfaces; these quantities have a critical bearing on interfacial structure, diffusional flux accompanying migration, and shape transformation. The parameters b and h for a defect are defined with respect to a chosen reference structure for which, by definition, the defect content is zero. In the case of defects in single crystals, the obvious reference is the perfect crystal itself, but, for bicrystals, the choice is less straightforward. In the present work it is convenient to use both natural and coherent reference bicrystal structures, and their use for identifying the total defect content of the habit plane is described below. The next issue reviewed is conservative versus non-conservative motion of interfacial defects. ‘Glide’ and ‘climb’ behaviour of interfacial defects is unlike that of dislocations in single crystals because of density and/or chemical composition differences between the adjacent crystals. The topological theory of interfacial defects [15,16] enables the Burgers vector, b, and step height, h, of defects to be defined. According to the theory, these parameters are determined by the symmetries of the adjacent crystals, their lattice parameters and orientation relationship. As an illustration, we consider admissible disconnections and dislocations in an interface resembling the parent-martensite interface in Ti. Unit cells of the bcc (β) and hcp (α) crystals exhibiting their room temperature lattice parameters (a(β) = 0.3280 nm, a(α) = 0.2951 nm and c(α) = 0.4679 nm) and mutually disposed with the Burgers OR, are depicted in Fig. 4(a). This is the natural reference bicrystal, and the misfit along the ¯ ¯ ¯ and 0.87% parallel to (21¯ 1)/(1 100) terrace plane is evident (3.8% parallel to [111]/[1120] ¯ [110]/[0001]). The coherent reference is depicted in Fig. 4(b), where the β and α phases have been homogeneously strained with opposite senses so that the misfit on the terrace

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(a)

(b)

(c)

Fig. 4. (a) Schematic illustration showing the ‘natural’ reference bicrystal formed by β and α crystals with the ¯ ¯ Burgers OR and interface parallel to (21¯ 1)/(1 100); the crystals misfit in the x and y directions defining the interface plane. (b) The ‘coherent’ reference state; the lattice sites are depicted in black, α, and white, β, forming a dichromatic pattern. The Burgers vectors of two admissible defects are shown. The sense vector ξ points out of the page. (c) Corresponding cells in the coherently strained crystals, with hatching to show their common area at the interface.

plane is removed. The β crystal has been expanded parallel to y and the α crystal compressed to produce periodicity in this direction, and similarly along x. These strains change the symmetries and densities of the β and α phases as discussed above for ZrO2 , thereby creating a terrace; corresponding cells are depicted in Fig. 4(c).

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Only the lattice sites are illustrated in Fig. 4(b), and these are shown interpenetrating to form a dichromatic pattern, so-called because the α crystal is considered to be black and the β white [26]. Admissible defects [15] have Burgers vectors given by b = t(β) − t∗ (α),

(2.3)

where t(β) and t∗ (α) are translation vectors of the β and α crystals respectively in the relevant reference state, the asterisk indicating that the latter has been expressed in the bcc coordinate frame. The set of potential Burgers vectors defined by expression (2.3) is represented graphically by vectors joining black sites to white ones in a dichromatic pattern, and two examples are indicated in Fig. 4(b). The formation of a candidate disconnection in the coherent reference state is shown ¯ ¯ layer are shown. schematically in Fig. 5(a); for clarity only the atoms in one (011)/(000 2) According to expression (2.3), its b is given by the difference between tc (β) = [100] and ¯ this corresponds to the Volterra operation required to form equivalent tc (α) = 1/3[21¯ 10]; terrace structures on either side of the crystal surface steps shown in Fig. 5(a). In the terrace plane coordinate frame, xyz, indicated in Fig. 5, b has components bx = 0, by = 0.0483 nm and bz = 0.0123 nm. These are the appropriate coordinates for b because they correspond to the local configuration near a dislocation in an equilibrated interface. The figure also reveals that this disconnection has step character because the location of the terrace plane is different on either side of the defect. To define the height of this interfacial defect, the heights of the two crystal surface steps in the reference state are considered first; if n is the unit normal to the terrace plane, these are h(β) = n · t(β), and h(α) = n · t∗ (α) respectively.

(a) Fig. 5. (a) Schematic illustration of the formation of a disconnection; the b of the defect with respect to the coherent reference state is the Volterra operation required to create equivalent interfaces on either side of the crystals’ surface steps. The ‘overlap’ step height of the defect is the smaller of the two surface steps, h(α) in this case. The sense vector ξ points out of the page. (b) Schematic diagram showing the atomic shear and shuffle motions as the defect moves to the right. (c) Schematic illustration of a disconnection with single atomic layer step height separating interfaces with different atomic configurations.

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(b)

(c) Fig. 5. (Continued).

The step height of the interfacial defect is defined to be the smaller of h(β) and h(α), i.e. h(α) in this case. This height is referred to as the ‘overlap’ step height, h, and is shown later to relate to the amount of material transferred from one crystal to the other when a disconnection moves laterally. Also, the z component of b is given by bz = h(β) − h(α). All the atomic sites for this disconnection are shown in Fig. 5(b); if the defect moves to the right, a fraction of the black atoms are sheared into their corresponding white sites, and the remainder must shuffle [27] to the correct positions as indicated. Strictly speaking this could modify the concept of a military transformation, which would apply to the sheared sites in Fig. 5(b). In high symmetry crystals there can be alternate paths for the shuffle motions, which could require thermal activation. Hence a degree of stochastic behaviour could enter the overall process. From the viewpoint of symmetry theory [26], the smallest h

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Fig. 6. Schematic illustration of the formation of a disconnection in the coherent (100)o /(100)m terrace plane in ZrO2 . The sense vector ξ points out of the page.

¯ atomic planes high in for a possible disconnection in this Ti example is actually two (21¯ 1) ¯ the white crystal and one (1100) corrugated atomic plane in the black (α-Ti has a double atom basis associated with each lattice point), as illustrated Fig. 5(b). A disconnection with h equal to a single atomic layer in each crystal, Fig. 5(c), separates different interface structures on either side, and therefore this configuration is excluded. In the case illustrated, the nearest neighbour distances of atoms at the interface remain unchanged but the second nearest neighbour configurations are different, and consequently the interfacial energies are different. Motion of this defect would be glissile and would not require shuffles. As a second illustration we consider a disconnection in the interface between orthorhombic and monoclinic crystals of ZrO2 discussed earlier. Fig. 6 shows the formation of a disconnection in the coherent terrace with bx = 0, by = 0.16 nm, bz = −0.002 nm, and h = h(o) = 1.014 nm; note that, in this case, bz is negative and smaller than for Ti. Similar to the Ti case, only one quarter of the atoms will be sheared into the correct positions so extensive shuffling must accompany disconnection motion, which is likely to reduce the mobility of the defect. Crystal dislocations correspond to special cases of defects defined by expression (2.3) where either t(β) or t∗ (α) is the null vector. The formation of a candidate dislocation with ¯ is depicted schematically in Fig. 7. Although the α tc (β) being null and tc (α) = 1/3[2110] ¯ phase exhibits a surface step on (1100), none is present on the β surface so the overlap step height is zero in such cases, implying that no material would be transferred from one crystal to the other by lateral motion. This distinction between crystal dislocations (h = 0) and disconnections (h is finite) turns out to be important in our model of martensitic processes, as is discussed further later. A twinning dislocation that reaches an interface displays the same topological features as a crystal dislocation, except that a stacking-fault extends into one crystal in this case. In the nomenclature of the PTMC, crystal slip and twinning defects are referred to as being lattice-invariant defects, or LID.

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Fig. 7. Schematic illustration of the formation of a crystal dislocation at the interface; the b of the defect is ¯ 1/3[2110], and the ‘overlap’ step height is zero. A Volterra displacement by b will create equivalent interfaces on either side of the perturbation. The sense vector ξ points out of the page.

2.3. Conservative and non-conservative motion of interfacial defects When unit length of dislocation in a single crystal moves a distance represented by the vector m, the number of substitutional atoms, N , which must diffuse to (positive values) or from (negative values) the core of the defect, is given by N = b · (ξ × m)X,

(2.4)

where X is the number of atoms per unit volume. If more than one species, A, B, . . . , is present, one uses NA and XA and so on. The sense vector ξ and m define the plane of motion of the defect, and expression (2.4) embodies the condition for conservative motion, or glide, since N is zero if b lies in this plane. Non-zero values of N , of course, imply that the motion is by climb. Expression (2.4) can be readily adapted to the case of interfacial defects by substituting for b from eq. (2.3), and bearing in mind that the number of atoms per unit volume may be different in the two phases. Thus, for defects in the Ti interfaces considered above, NA = t(β) · (ξ × m)X(β)A − t∗ (α) · (ξ × m)X(α)A .

(2.5)

Similar expressions can be written for each of the two species in the ZrO2 case. In expression (2.5), the line direction is taken to lie in the interfacial plane, and the translation vectors and densities are defined with respect to the relevant reference state. This expression also encompasses LID when one of the translation vectors is the null vector and the

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other represents slip or twinning. Using this expression, one can determine the diffusional flow accompanying lateral motion, where m is parallel to the terrace, and frontal motion where m is parallel to n. 2.3.1. Lateral motion of disconnections Consider the leftward lateral motion across the coherent terrace of the disconnection illustrated in Fig. 5(a). In this case, ξ is parallel to x, and m is parallel to −y, so ξ × m is anti-parallel to n. Substituting into expression (2.5), and using tc (β) · n = h(β), and tc *(α) · n = h(α), one finds that the atomic flux per unit length of defect per unit distance moved is equal to NA = h(α)Xc (α)A − h(β)Xc (β)A .

(2.6)

Furthermore, since we may re-express Xc (α)A as {Z(α)A (gc (α)h(α))−1 } and similarly for Xc (β)A , it follows that NA = 0 in this case and the disconnection motion is conservative. The b of this disconnection is not parallel to the (terrace) plane of motion, but the defect is, nevertheless, ‘gliding’ along the interface. In other words, lateral motion of this disconnection is glissile because material is thereby being exchanged between corresponding cells containing the same number of atoms. Thus, glissile disconnections glide across terraces without the necessity of climb, even though a component of the Burgers vector is normal to the terrace plane. The disconnection shown in Fig. 6 is also glissile, and its lateral motion causes exchange of material between the cells depicted in Fig. 3(c). Of course, not all disconnections on a terrace are glissile; a schematic illustration of the formation of a sessile disconnection in ZrO2 is shown in Fig. 8.

Fig. 8. Schematic illustration of the formation of a sessile disconnection in the coherent (100)o /(100)m terrace plane in ZrO2 . The sense vector ξ points out of the page.

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In summary, lateral motion of a disconnection along a terrace is only glissile when the terrace planes of the two crystals have the same projected atom density per unit area and are continuous through the disconnection’s step, even though the terrace planes in the adjacent crystals may have different inter-planar spacings. To illustrate this point, Fig. 9 shows the schematic formation of a disconnection similar to Fig. 5(a), except that Xc (α)A has been doubled by a compressive strain parallel to z; this retains the glissile nature of the disconnection, and reduces h(α) to half its original value. The resulting disconnection can move conservatively, despite the enhanced magnitude of bz since the terrace planes remain ¯ atomic planes in the white continuous through the disconnection’s step (i.e. two (21¯ 1) ¯ crystal abut each (1100) corrugated atomic plane in the black crystal across the step). Of course, the shear and shuffle motions are not the same as for the defect in Fig. 5(b). Similarly, for the disconnection depicted in Fig. 6, four (004)o planes abut four (002)m planes of equal atomic density across the step. On the other hand, four (004)o planes in the orthorhombic ZrO2 crystal terminate at the step in Fig. 8, so this defect is sessile. An alternative and useful view into the glide versus climb behaviour of disconnections is obtained from the argument advanced by Hirth and Pond [16] who expressed eq. (2.5) explicitly in terms of h and bz . Because h(β) − h(α) = bz , and, for a defect where h = h(α) (as for Fig. 5(a)), the incremental flux per unit length of defect, δNA , accompanying motion by the distance δy, is equal to δNA = hXcA + bz Xc (β)A δy,

(2.7)

Fig. 9. Schematic illustration of a glissile disconnection. This defect is similar to the disconnection depicted in Fig. 5(a) except that the density of the α crystal has been doubled by compression parallel to z. The sense vector ξ points out of the page.

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Fig. 10. Schematic illustration of the overlap step and dislocation contributions to the atomic flux accompanying lateral motion of a disconnection. The flux due to motion of the step portion, Ns , is associated with transfer of material from one crystal to the other, whereas that due to motion of the dislocation portion, Nd , involves β matter only in this case. (For simplicity, the disconnection is depicted before the Volterra operation to close the gap has been carried out.) These two fluxes are quantified by the two terms in expression (2.7).

where XcA = Xc (β)A − Xc (α)A . Expression (2.7) emphasizes the distinction between the glide/climb behaviour of crystal defects and disconnections along interfaces; in the former case (slip or twinning dislocations), h = 0 so climb occurs if bz is finite. On the other hand, for formal purposes, the step and dislocation portions of a disconnection can be treated separately, and a nonzero flux may arise as a consequence of either component. Motion of the step portion causes transfer of material from one crystal to the other, and the magnitude of the flux depends on h and the difference between the densities of the two phases, XcA . Motion of the dislocation portion does not involve transfer of material, and the magnitude of the flux is determined by bz and the density of one of the crystals, Xc (β)A in this case. These two contributions to δNA are depicted schematically in Fig. 10. 2.3.2. Virtual frontal motion of disconnections Expression (2.1) shows that the hypothetical frontal migration of an incommensurate terrace like those in Figs 3(a) and 4(a) generally involves a finite diffusional flux. If such an interface were to relax forming a misfit-relieved semi-coherent structure as in Fig. 2, the diffusional flux accompanying migration would be unchanged. This finite flux would be associated with the climb of the disconnections in the direction normal to the interface because the intervening coherent segments can migrate conservatively. One can establish the flux magnitude by using eq. (2.5) as follows for the ZrO2 case. With ξ parallel to x and m parallel to n, the term tc (o) · (ξ × m) is equal to – tc (o)y , and similarly for the second term. Thus, the flux per unit area of interface per unit distance moved into the parent crystal is equal to −1 NA = − tc (o)y Xc (o)A − d(m)d(o)−1 tc∗ (m)y Xc (m)A λD ,

(2.8)

where λ−1 D is the number of disconnections per unit length in the y direction. This expression conceives the hypothetical frontal motion of a disconnection as the coupled climb of

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two crystal dislocations, one in each crystal, and also takes into account the fact that the monoclinic dislocation moves d(m)d(o)−1 further than the orthorhombic one. Expressions (2.1) and (2.8) lead to identical values of flux in selected cases.

2.4. Sessile and glissile defect intersections Consider two disconnections, (ba , ha ) and (bb , hb ) intersecting as depicted schematically in Fig. 11 where the dislocation components are represented by bold lines. Unlike intersecting defects in single crystals, the present defects are constrained to lie in the interface, and so cannot pass through each other to form jogs. Here the defects mutually step each other, thereby creating additional lengths of dislocation; defect ‘a’ is increased in length by hb , and, conversely, defect ‘b’ is increased by ha . These additional lengths are represented in Fig. 10 by the vertical double-thickness bold line. Motion of this intersection segment requires an incremental flux, δN i , given by δN i = hb X bxa δy + bya δx + ha X bxb δy + byb δx ,

(2.9)

where X represents the density of a particular species in one of the two crystals in the relevant reference state, depending on the sign of bza for the first bracket and bzb for the second. In general, expression (2.9) indicates that δN i will be finite, implying that disconnection intersections are sessile. Consequently, there will be an osmotic resistance to the motion of the intersecting arrays that could be overcome only at very large thermodynamic driving forces. Hence, the accommodation of the terrace misfit in parent-martensite interfaces by intersecting arrays of disconnections is unlikely. However, intersections need not be sessile when one of the arrays comprises LID since h = 0 for such defects. This is the justification for adopting the interface model shown in Fig. 2 with intersecting arrays of disconnections and dislocations. Thus, if defect ‘a’ is a disconnection and ‘b’ is a crystal dislocation, the first bracketed term in eq. (2.9) is zero, and it has been demonstrated elsewhere [28] that the intersection segment can be glissile as long as the crystal defect in question can reach the terrace plane by glide. These arguments provide physical insight into the role of LID

Fig. 11. Schematic illustration of the intersection between two disconnections. The dislocation portions are depicted as bold lines. These defects mutually step each other, and additional lengths of dislocation segments arise at the intersection, indicated by the double thickness bold vertical line.

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Fig. 12. Schematic illustration of a sessile intersection between a crystal dislocation (b, 0) and a disconnection (b, h). The line direction of the dislocation is parallel to y and b is parallel to x, so motion of the disconnection in the y direction would require climb of the vertical edge segment of the dislocation on the disconnection’s step.

in martensite formation, and are consistent with the suggestion of Christian [22] relating to glissile interfaces. If an inappropriate mode of LID is activated during a transformation, leading to sessile intersections, the transformation kinetics are likely to be diffusion controlled. For example, if the misfit parallel to x in Fig. 4(a) were accommodated by dislocations from the α crystal with ξ parallel to y and b = [0001] [29], sessile intersections with the orthogonal set of disconnections would arise, as depicted schematically in Fig. 12. On the other hand, if ¯ the activated LID is {1101} twinning, with b inclined to the terrace plane, as is actually observed in Ti [30], the intersections will be glissile. In the course of a transformation, interfaces made sessile by intersections as above may still move without requiring diffusion. In many cases the thermodynamic driving forces for martensite transformations are large. Under such circumstances the sessile points may move athermally, leaving a trail of the appropriate point defects in their wake. The critical spacing of sessile points, above which athermal motion is possible, is given by eq. (16-16) in [14].

2.5. Summary of constraints for martensitic transformations In the foregoing, the diffusional flux associated with the migration of an interphase interface between crystals with the same chemical composition but different structure has been considered. For the case of epitaxial crystals, the frontal migration of an incommensurate interface by a civilian process generally requires diffusion. However, if such structures relax into a semi-coherent form, with coherent terrace segments and a superimposed array of disconnections and dislocations to accommodate the coherency strains, diffusionless migration is possible by the military process of lateral disconnection motion across the terraces. Such interfaces exhibit properties consistent with the characteristic features observed in a martensitic transformation. However, for diffusionless transformations, the terraces, disconnections and dislocations invoked in this model are subject to several topological constraints as enumerated below. 1. The terraces must be commensurate. In stiff engineering materials, this condition can be met by modest strains to produce coherent terraces.

§3


243

2. The active disconnections must be glissile, which arises when the terrace planes of the adjacent crystals are continuous through the disconnections’ steps. 3. Where an array of mobile disconnections and a nonparallel array of LID are required to accommodate the coherency strains at a terrace, the intersections between these arrays must also be glissile. The modes of LID that satisfy this criterion can be deduced from expression (2.9); slip or twinning dislocations that can reach the terrace plane by glide form glissile intersections with mobile disconnections. In addition, the intrinsic mobility of disconnections needs to be sufficiently large for a feasible transformation to occur. Atomistic simulations of disconnection motion in twin boundaries indicate that their mobility is generally greater if |b| and h have small magnitudes [31]. Defects with large |b| tend to be unstable with respect to decomposition, and those with large h tend to have narrow cores and their motion requires extensive atomic shuffling. On these grounds, the mobility of disconnections in Ti, Fig. 5, is expected to be greater than those in ZrO2 , Fig. 6. The corresponding cells contain only two atoms in the former case, and half of these are sheared into the correct position by disconnection motion, whereas the cells in the latter case contain 8 Zr and sixteen O atoms, with only one quarter being correctly placed by shear, so extensive shuffling is also required in this case. In-situ TEM observations of disconnection motion in ZrO2 [10] confirm the sluggish motion of the defects illustrated in Fig. 6. In the following sections, these conditions are met for all disconnections, which are therefore glissile.

3. Transformation crystallography The objective of this section is to formulate expressions for the transformation crystallography arising in the present model, specifically the microscopic and macroscopic interface structure and the orientation relationship. As depicted in Fig. 2, the microscopic interface structure comprises coherent terrace segments with a superimposed disconnectiondislocation array to accommodate coherency strains. Since disconnections exhibit step character, the macroscopic interface inclination, or habit plane, differs from the terrace plane by an extent that depends on the disconnection step height, h, spacing, λD , and line direction ξ d . The latter two quantities are determined by the contribution of the disconnections to misfit accommodation on the terrace. Thus, this section commences with a discussion of terrace misfit, and subsequently considers its accommodation by disconnections and LID. In general, it is found that such defect arrays remove long-range strains from the coherent terrace reference state but also introduce ancillary tilt and/or twist deviations. These angular deviations modify the short-range strain field associated with misfit relief, and contribute to the overall OR of the unperturbed parent and martensite crystals taken at long-range from the habit plane.

3.1. Coherency strains In stiff engineering materials like ZrO2 and Ti, candidate terrace planes are anticipated to exhibit relatively small coherency strains [5,25] as depicted in Fig. 13 which shows a plan

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(a)

(b) Fig. 13. Scale drawing of the terrace planes in (a) ZrO2 and (b) Ti. Full lines represent the martensite and dashed lines the parent crystals, and bold lines depict the coherent state after equal and opposite straining, exx and eyy , of each phase.

view of (a) the (100)o /(100)m interface between orthorhombic and monoclinic ZrO2 , and ¯ ¯ (b) the (21¯ 1)/(1 100) interface between β and α-Ti. In the terrace plane coordinate frame, x, y, z, the parent and martensite crystals of ZrO2 are naturally coherent parallel to x but misfit parallel to y, whereas misfit arises in both directions for Ti. In the figure, the parent and martensite crystals are equally and oppositely strained to form the coherent reference states illustrated. The total coherency strains εyy have similar magnitudes, namely 3.84% and 3.80%, whereas the strains εxx are zero and 0.86% in ZrO2 and Ti, respectively.

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Fig. 14. Schematic illustration showing the coherency interfacial dislocation density of the ‘coherent’ reference.

In the present context, one can represent these elastic coherency strains as arising from a continuous array of infinitesimal coherency-producing dislocations as described by Olson and Cohen [32]. Without loss of rigour, one can imagine these to be discretized on atom rows as in Fig. 14. The defect content, bc , of a coherent interface with respect to the natural reference can be calculated from the Frank–Bilby equation [33] as follows bc =

−1 c n

− cM −1 n v,

(3.1)

where c n is the homogeneous lattice deformation producing the coherently strained parent lattice from its natural state, cM n is the corresponding deformation of the martensite lattice, and v is a probe vector in the coherent terrace. These matrices can be expressed as (I + E(π)ij ), and (I + E(μ)ij ), where E(π)ij is given by E(π)ij =

e(π)xx 0 0

0 e(π)yy 0

0 0 , 0

(3.2)

and similarly for E(μ)ij . The probe vector, v, must lie in the terrace plane with sign consistent with the RH/FS convention [14], and, if the dislocation line direction points outward from the diagram, an admissible probe intersecting these defects is v = [010]. The matrix in expression (3.1) becomes

(1 + e(π)xx )−1 − (1 + e(μ)xx )−1 0 0

(1 + e(π)yy

)−1

0 − (1 + e(μ)yy )−1 0

0 0 , 0 (3.3)

so the dislocation content intersected by v is bcy = [0, (1 + e(π)yy )−1 − (1 + e(μ)yy )−1 , 0] per unit length. For the Ti example, this content can be visualised, as indicated in Fig. 14, to comprise one interfacial dislocation per period with Burgers vector obtained from the

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¯ (defined in the natural reference state) substitution of t(π) = 12 [111] and t(μ) = 31 [1120] into expression (2.3). Per unit length, |bcy | is equal to the total misfit strain parallel to y, hereafter written as εyy . In Fig. 13, e(π)yy and e(μ)yy are taken to be equal and opposite with magnitude eyy , so, to first order, εyy = 2eyy in this case; we note that the sign of εyy is negative here corresponding to compression of α with respect to β. These defects are not disconnections, but interfacial coherency dislocations since they do not exhibit step character, Fig. 14. A similar procedure can be followed for the defect content producing coherency in the x direction.

3.2. Interfacial structure Having represented the elastic coherency strain by an array of defects, one can proceed to relieve this strain plastically by introducing appropriate admissible defects as defined in the coherent reference state. For example, if the misfit strain εxx is zero, as is the case for ZrO2 , an array of the disconnections shown in Fig. 7 with ξ d parallel to x could relieve the strain εyy . However, the step character of these defects causes the habit plane to become inclined by the angle θ from the terrace plane, as depicted schematically in Fig. 15. Thus, the habit plane is not known unless this spacing λD is determined, and thus a probe vector, v, cannot be defined a priori for use in the Frank–Bilby equation. Similarly, Bollmann’s formulation for interfacial defect content [25] cannot be used since the orientation of the interfacial plane is taken to be an independent variable. In an alternative method due to Pond et al. [34] the dislocation portion of each disconnection is considered to be located at the mid-point of its step. Moreover, the step of each defect is taken to be orientated

Fig. 15. Schematic illustration showing the disconnection and ‘coherency-producing’ defect content of a parentmartensite interface, with Burgers vector components resolved in the terrace (upper) and habit plane (lower) frames. The terrace plane is inclined at an angle θ to the horizontal habit plane.

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perpendicular to the terrace, so that adjacent defects are separated by the distance λD along the terrace. This is an excellent approximation when λD > 4l, where l is the width of the actual step projected onto the terrace plane; for shorter terrace lengths it would be necessary to consider the distribution of the dislocation content along the step. Each disconnection carries with it both the shear εyz associated with by and the normal strain εzz associated with bz , as depicted in Fig. 15. For simplicity, the defect content producing coherency, bcy , is shown as a single symbol on each terrace. There will, of course, be local strains near the disconnections, but these will become uniform at a distance about λD from the interface according to St. Venant’s principle, and thereafter will exactly cancel the coherency strain. In order that no long-range strains arise, the coherency misfit must be accommodated along the habit plane by the relevant component of the disconnections’ b, i.e. expressed in the primed coordinate frame of the habit plane, b′cy = −b′y .

(3.4)

Thus, the disconnection spacing is adjusted until this condition is satisfied. Since each disconnection exhibits the step height h, this configuration will also determine the inclination, θ , of the habit plane with respect to the terrace plane. Thus, by re-expressing b′cy and b′y , θ can be found in terms of the coherency strain and the disconnections’ b. The ‘coherency-producing’ defect content per terrace segment is λD bcy = λD εyy . Hence, using λD = L cos θ = h cos θ/ sin θ , we obtain ′ = εyy h cos2 θ (sin θ )−1 . bcy

(3.5)

Also, by′ = by cos θ + bz sin θ.

(3.6)

Hence, by substituting (3.5) and (3.6) into expression (3.4) we obtain −εyy = by tan θ + bz tan2 θ h−1 ,

(3.7)

which can be solved explicitly for θ . The calculated ideal disconnection spacing may or may not be an integral number of lattice spacings along the terraces. If the Peierls barrier for disconnection motion has been eliminated by thermal fluctuations, the disconnections can assume an irrational, uniform spacing. If the Peierls barrier is present and traps the disconnections, then the disconnection spacings to remove misfit may vary periodically. Sets of disconnections with spacing na are separated by others with spacing (n + 1)a. If this does not satisfy the zero strain condition, an added perturbation with a longer wavelength is added and so forth. The disconnection/terrace structures of the habit planes determined by the procedures outlined above for the ZrO2 and Ti cases are summarised in Fig. 16, which shows that the inclination of the habit plane with respect to the terrace plane orientation in the parent crystal is similar in the two cases, primarily because the ratio by / h is similar for both materials. However, the spacing of the disconnections is about four times greater for the

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(a)

(b) Fig. 16. Schematic illustration of the terrace/disconnection structures of habit planes in (a) ZrO2 and (b) Ti. The Burgers vector components of the disconnections, their step heights, and the coherency strains are shown.

ceramic because by and h are approximately four times larger than their counterparts in the metal. These schematic illustrations closely resemble the interface structures observed using TEM, Fig. 17; for ZrO2 the disconnections seen in (a) are those depicted in Fig. 6. In Ti-10wt%Mo, which has the same lattice parameters as Ti, ‘double’ disconnections (2b, 2h) are observed in Fig. 17(b) rather than the single defects depicted in Fig. 5(a), but this does not affect the habit plane orientation since λD also doubles. The stability of ‘double’ disconnections perhaps relates to the energy of their cores relative to the terraces. In the relatively low magnification image Fig. 17, obtained by means of diffraction contrast, the short-range strain field of the coherent terraces is evident, consistent with the topological model. The procedure for finding the LID spacing, λl , on the terrace is more straightforward since these defects do not exhibit steps. In the case of ZrO2 , no LID is required, while for Ti the spacing λl is equal to bx /εxx . If twinning is the active LID mode for Ti, bx represents the x component of the Burgers vector, bt , of the twinning dislocation and the spacing λl can be alternatively expressed in terms of the twinning fraction [6]. The intersections of the disconnection and LID arrays are glissile in this case because bt is inclined to the terrace plane. No LID is seen in the image of the alloy, Fig. 17, because the viewing direction is along x. 3.3. Orientation relationship Fig. 15 shows that there may be a residual defect content in the habit plane due to the ′ ; these non-cancelling components constitute a tilt wall causing a components bz′ and bcz rigid-body rotation, ϕ, of the martensite with respect to the parent about x with magnitude ′ φ = 2 sin−1 bz′ + bcz /2L . (3.8)

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(a)

(b) Fig. 17. Transmission electron micrographs of (a) ZrO2 , courtesy of Chen and Chiao [10], and (b) Ti-10wt%Mo [34]. The habit planes in both cases exhibit a terrace/disconnection structure.

Using L = h/ sin θ,

(3.9)

′ bcz = −bcy sin θ = −εyy h cos θ,

(3.10)

bz′ = bz cos θ − by sin θ,

(3.11)

and

we obtain the following expression: φ = 2 sin−1 bz cos θ − by sin θ − εyy h cos θ sin θ/2h .

(3.12)

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For the ceramic, φ is only −0.035◦ , whereas that for Ti is 0.528◦ ; this arises mainly because bz is considerably smaller, and has the opposite sign, in ZrO2 compared with Ti. In the Ti case, one must also consider the tilt introduced by the LID; for a full treatment it is necessary to take into account that the line direction of the LID, ξ l , is close to y ′ rather than parallel to y, see Fig. 2. However, when θ is small, this difference can be neglected, and an additional rotation of −0.263◦ can be taken to arise about y. The rotations about x due to the disconnections and about y due to the LID must be combined to obtain the overall angular deviation from the Burgers orientation reference.

3.4. Habit plane inclination In the earlier version of the TM [34] the analysis of the habit plane assumes that the tilt, ϕ, occurs in the martensite so the habit plane is inclined from the terrace plane orientation in the martensite by θ − ϕ, while that in the matrix is inclined by θ . However, the rotation ϕ ′ components as in Fig. 15, which together produce a tilt wall. This arises from the bz′ and bcz is shown directly in the TM and is implicit in the PTMC in that the habit plane is known to have a terrace plane-disconnection structure. Actually, the rotation of such a tilt wall is partitioned equally (in the present isotropic elastic approximation) between the two phases [35]. Thus, a portion φ/2 appears in each of the adjoining phases. The OR is unchanged from the previous analysis, still given by ϕ. However, the habit plane in the martensite is inclined to the terrace plane in that crystal by θ − ϕ/2, and by θ + ϕ/2 with respect to the terrace plane in the matrix. Often, ϕ is small and this correction to the habit plane orientation is also small.

4. Transformation displacement By means of optical or scanning electron microscopy, one can reveal the displacement accompanying a transformation by observation of the re-orientation of fiducial lines across parent-martensite interfaces [36], as shown in Fig. 18 for Cu-16.9Zn-7.7Al (wt%). The objective of this section is to quantify such displacements in terms of the topological model where they are envisioned as the sum of elastic coherency strains and plastic strains due to defect arrays superimposed on the terrace. The distortion tensor, dij , is given by ∂uj ∂uj ∂ui 1 ∂ui 1 ∂ui dij = + . = εij + φij = + − ∂xj 2 ∂xj ∂xi 2 ∂xj ∂xi

(4.1)

Here, the ui are displacements and the φij are rotations. For engineering strains, Ŵij , with off-diagonal components equal to twice εij , the general form of the strain matrix for the cases illustrated previously, expressed using the terrace plane coordinate frame, is given by Ŵxy =

γxx γyx 0

γxy γyy 0

γxz γyz γzz

.

(4.2)

§4.1


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Fig. 18. Scanning electron micrograph of a fiducial marker on Cu-16.9Zn-7.7Al (wt%); the marker appears to be continuous but re-oriented as it crosses the surface traces of habit planes. The spatial resolution is too low to reveal any inhomogeneous displacement in the vicinity of the interface.

As explained in the foregoing, the misfit at the habit plane is removed by appropriate arrays of disconnections and LID, subject to the constraint that glissile motion remains possible. Thus, using the habit plane frame, the transformation strain when all misfit is relieved is given by ′ Ŵxy

=

0 0 0 0 0 0

′ γxz ′ γyz ′ γzz

,

(4.3)

where Ŵ ′ = RŴR−1 and R is the matrix representing the rotation of the habit plane frame with respect to the terrace plane as produced by the transformation shear. The components of Ŵ, R and the rigid body rotation, , can be deduced directly from the model, as shown in the following two sub-sections for disconnections and LID separately. Finally, the superimposition of strains from the two sets of defects is considered and it is demonstrated that, in general, more than one mode of LID may be necessary for Ŵ ′ to attain the ideal form of expression (4.3).

4.1. Transformation strains caused by disconnections Fig. 19 depicts a parent-martensite interface viewed along the x, x ′ direction parallel to the disconnection lines. Initially the interface is located at the lower position, Fig. 19(a), ′ , has been accommodated plastically by the disconnecand the elastic coherency strain, γyy tion array. Beyond the short-range inhomogeneous displacement field of the interface, the ′ ) two crystals adopt their natural lattice parameters. The residual defect content (bz′ and bcz causing the rigid-body rotation about the x-axis, φ = φyz , is indicated by the dislocation symbols perpendicular to the habit plane. In this schematic picture the rigid body rotation φ/2 associated with the partitioning of φyz , as discussed in Section 3.4, is suppressed for

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Fig. 19. Schematic illustration of the displacements produced by disconnections during migration of the interface from position (a) to (b). The residual defects along the interface act as a tilt wall, and synchronous motion of disconnections along the terraces effects the lattice deformation.

clarity. Now imagine that the interface moves to the location depicted in Fig. 19(b) by synchronous lateral motion of the disconnections leftward along the terraces, as represented by the dislocation symbols on the left hand side of the figure. The displacement thereby induced in the transformed material, Ŵ ′ , can be determined from the components of Ŵ as follows. Initially, the side faces of the parent crystal are vertical in the diagram, and become inclined after the homogeneous strains, γyz = h−1 by and γzz = h−1 bz , accompanying the transformation. In the case illustrated γxz = 0 and γxy = 0 since bx = 0 and γyy is obtained from expression (3.3). The matrix R required to re-express these strain components in the habit frame corresponds to a rotation by θ + φyz /2, where the former term is given by expression (3.7) and the latter by eq. (3.12). The matrix representing the total rigid-body rotation, , corresponds to a rotation about x by φyz , and is independent of the frame chosen.

4.2. Accommodation strains caused by LID Several workers have used TEM to study accommodation of coherency strain by LID; for example, Hammond and Kelly [30] investigated twinning and Sandvick and Wayman [37] examined slip. The simplest case of LID occurs when the disconnections have the optimum edge dislocation arrangement of Section 3 and there is a glide system available with dislocations that intersect the terrace plane along the y axis and has b = (bx , 0, bz ). The presence of bz permits intersections of the LID defects and the disconnections to be glissile and the absence of by allows efficient compensation of the coherency strain γxx by essentially

§4.2


253

Fig. 20. Schematic illustration of the strains produced by LID during migration of the interface. The residual defects along the interface act as a tilt wall, and glide of the LID shears the martensite. The line direction of the dislocations, ξ , is out of the page.

the one slip system alone. If the LID defects are very closely spaced, other systems may be required as discussed later. Fig. 20 is the schematic view along the y, y ′ direction, showing ′ , the residthe LID array which plastically accommodates the elastic coherency strains γxx ual defect content producing φxz , and the traces of the slip (twinning) planes separated by the distance q. Glide of the LID through the martensite in the wake of the moving interface causes the shear strains indicated on the side faces and the accumulation of dislocations at the interface. The components bx accommodate the coherency strain γxx when the dislocations have the appropriate spacing λL , while the components bz are in a tilt wall array and produce a rigid body rotation φL . As with the tilt produced by the disconnections this tilt is partitioned between the parent and martensite as ±φL /2, although this partitioning is not depicted in the figure, where the total rotation is shown in the martensite crystal. If the interface advances in the z direction the displacement components of the transformed material are γxz = q −1 bx and γxx = bx /λL . The rotation ϕL = ϕLxz = 2 sin(bz /2λL ) is carried forward with the interface, and the rotation R corresponds to a rotation about y by ϕL /2. For the case of ZrO2 , no LID is involved because the parent and martensite crystals are naturally coherent parallel to x, as depicted in Fig. 13(a). However, for the TiMo alloy, the principal coherency strains parallel to the x and y axes are both finite, and the former ¯ α twinning which intersects the terrace plane along y. For can be accommodated by (1101) simplicity in analysis, we assume the twinning dislocations to form an array of individual defects rather than coalescing into twins (the results for the two cases are the same except for some local strain fields in the actual twinning case). For twinning dislocations with b ¯ α , λL = 7.16 nm and the rigid body rotation ϕL is 0.263◦ [38]. parallel to [1102]

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4.3. Large strain effects Throughout, we have treated an idealized situation where the transformation disconnections have the ideal edge orientation and where the strains are sufficiently small that the LID strain is simple and the LID rotation can be superposed on the rotation accompanying the transformation strain. For larger strains or when there are a limited number of possible LID systems, as for low symmetry structures, other effects become important. Firstly, the transformation disconnections may have mixed screw/edge character of the dislocation component. These disconnections must form before the LID since the transformation provides the driving force for LID formation. However the disconnections are glissile and may move into mixed dislocation configurations to help in accommodating the total strain. Then only the smaller edge component would remove misfit in the transformation strain direction y ′ , while the screw component would provide an accommodation strain component ′ . γxz Second, the accommodation strain requires essentially a plain strain tensile/compression plastic deformation for higher symmetry systems or a combination of such normal strains and an in-interface shear for lower symmetry systems. In general, just as for arbitrary deformation of a crystal, the von Mises requirement for five independent deformation systems applies. For the transformations, these would include the disconnections plus four LID systems, either twinning or slip. For strains below 3 to 5 percent, crystal plasticity studies show that other complications can be present. These include such effects as unusual slip systems [39] or the suppression of an expected system by latent hardening. One consequence would be that the operative LID systems could vary locally for a given macroscopic martensite plate or among a group of plates. The associated rotations and hence habit planes could also vary. Finally, the LID in the simple treatment is selected to remove misfit strains on the terrace plane. However, the LID defects appear at an interface that is rotated from the terrace plane by the angle θ , Fig. 2. Thus while the terrace misfit strains are removed by the LID systems, the kinks in the LID defects where the defects rise from one terrace to an adjoining one still have uncompensated strain fields. Also the rotation axis associated with the LID is no longer orthogonal to that of the disconnections. The LID can no longer be simply superposed on that of the disconnections but can only be superposed on the interface that is already rotated by θ , i.e. on embedded coordinates. The formal treatment of these effects entails a very lengthy mathematical development and is not presented here. In most cases, however, the strains are sufficiently small, and the defects therefore are sufficiently widely spaced, that these more complicated effects can be neglected.

5. A comparison of the topological and phenomenological models of martensitic transformations 5.1. Structural differences The habit plane, orientation relationship and displacement for the transformations in ZrO2 and Ti as predicted by the topological model differ from those obtained using the PTMC,

§5.2


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and it is important to explore the origin of these discrepancies. To initiate this discussion, we consider the notion that the habit plane is an ‘invariant-plane’ as envisaged in the PTMC. In general, there is no plane of exact matching between two perfect crystals, even when a rigid-body rotation is permitted, although an invariant-line may be found in certain circumstances; using the PTMC algorithm, one finds this invariant-line for the transformation RB [5–8]. Matching of further vectors at the habit plane requires the martensite phase to be sheared by a single mode of LID, represented by P, and these defects lie parallel to the invariant-line in the habit plane. Thus conceived, the habit plane is an ‘invariant-plane’ of the total shape deformation RBP. In fact, invoking LID in this manner is equivalent to the notion of misfit accommodation by defects in the habit plane direction perpendicular to the invariant-line. In the topological model, no invariant-line is sought, but one set of disconnections and one (or more) set of LID accommodate the terrace coherency strain, leading to a misfit-relieved habit plane, as represented by expression (4.3). Moreover, such a misfit-relieved habit plane leads to a transformation displacement of the same nature as the ‘invariant-plane’ displacement of the PTMC, and is therefore consistent with experimental observations of fiducial lines. Hence, the key structural differences between the two approaches are the requirement of an invariant-line in the PTMC, compared with an array of disconnections in the topological model, and partitioning of the tilt as explained in Section 3.4.

5.2. Magnitude of the differences between the TM and PTMC models The case of ZrO2 is especially straightforward in this context because one principal strain of the lattice deformation is zero, so no LID need be invoked; finding the invariant-line, vi , in the yz plane and the requisite rigid-body rotation is straight forward. The Ti case is, strictly, 3-D, but it is instructive here to suppress the small misfit parallel to x so that it becomes 2-D. A graphical illustration of the PTMC procedure for determining vi is to find the intersection between the unit circle and the deformation ellipse representing the pure deformation B [5,6,40], as shown in Fig. 21 for the two cases of interest here. The rigid-body rotation R is that needed to rotate Bvi back into coincidence with vi . For ZrO2 , Fig. 21(a), the inclination of vi to the horizontal is 13.529◦ and the rigid-body rotation, φP , with respect to the reference, Fig. 3(b), is a clockwise rotation (of the monoclinic crystal) by 0.035◦ , while for Ti, Fig. 21(b), these are 11.364◦ and 0.528◦ (of α) anticlockwise with respect to the reference shown in Fig. 4(b). The invariantline for Ti is illustrated in Fig. 21(c), showing the matching of vectors at the interface. This picture is to be contrasted with the terrace-disconnection configuration predicted by the topological method, Section 3, and the high-resolution images of actual interfaces, Fig. 17. We now investigate further the extent of discrepancy between the predictions of the two approaches. Pond and co-workers [34] have considered the habit plane orientations according to the two theories, and demonstrated that these are congruent when the inter-planar spacing of the terrace planes is identical in both crystals. In this circumstance, bz = 0, because no misfit is present on the disconnection steps, and, by substitution of expression

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(a)

(b)

(c) Fig. 21. Schematic illustration of parent and product lattices viewed along x, of the unit circle and deformation ellipses for (a) ZrO2 and (b) Ti; the principal axes, invariant-lines and undistorted but rotated lines are shown. (c) Invariant-line interface for Ti assuming coherency normal to the page.

§5.2


257

(3.7) into (3.12), one finds that φyz is also zero. Introducing appropriately spaced disconnections to accommodate the terrace misfit, one determines a misfit-relieved habit plane with the same inclination as obtained using the invariant-line method of the PTMC. However, when bz is finite, discrepancies arise, and, to illustrate this, the magnitude of bz can be varied artificially by systematically changing the terrace plane spacing in the parent, while holding the strain εyy constant. The habit plane inclination according to the PTMC and TM for ZrO2 and Ti are depicted in Fig. 22 as a function of bz / h(m), where h(m) represents the terrace plane spacing of the martensite. The actual values of bz / h(m) for ZrO2 and Ti are −0.0026 and 0.048 (indicated by arrows on Fig. 22) respectively, and the discrepancies in PTMC and TM habit plane inclinations are −0.006◦ and 0.503◦ for the two materials. More generally, Fig. 22(a) shows that the values of ωP and θ are in close agreement when bz < 0, but the PTMC prediction is higher than the TM value for bz > 0; to first order, θ = ωP − ϕP in the latter region. This arises primarily as a consequence of the way the tilt is distributed between the crystals [35]; in effect, the tilt resides in the product for both models when bz < 0, whereas, for bz > 0, it resides in the parent for the PTMC and the product for the TM. Thus, in the latter region a difference close to ϕ arises. The actual partitioning of the tilt can be incorporated into the TM, as described in Section 3.4; Fig. 22(b) shows the predicted habit plane inclinations for the case of symmetrically distributed tilt. Physically, it is unrealistic for any rotation to be partitioned to the parent. Compatibility can lead to the rotation being partitioned to the product near a plate tip, where displacements in the parent are suppressed. In the PTMC there are no relaxations. The theory essentially gives an invariant plane (or line) where two rigid half crystals can be brought together without further strain of either crystal. Yet for the bz > 0 case, this constrained case has the consequence that rotations are partitioned to the parent. Physically, when the interface relaxes to a dislocation/disconnection structure, the rotation is partitioned to the product [35]. Hence agreement of the two theories to first order in the prediction of habit planes can be achieved if the PTMC result is modified by subtracting φ from the habit planes of both phases in the bz > 0 case. For the partitioned rotations, this modified result would also agree with the TM. There would always be some differences between the modified theories to second-order. These differences essentially arise from second-order nonlinear effects such as the use of the median lattice to define bz . Alternatively, one could convert the TM result to the PTMC result by essentially adding a small increment to the tilt wall (to make the PTMC bz′ equal to the TM bz ) and adding an increment to by to remove the contribution of bz to by′ . Pond et al. [41] give an example showing that indeed both components must be modified to make the two solutions coincident. The difficulties with this from a mechanistic viewpoint are that the addition of these components entails the addition of a specific number of extrinsic dislocations to the interface and that the partitioning of the rotation achieves the agreement between the theories in a simple physically consistent manner.

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(a)

(b) Fig. 22. Graph showing the variation of habit plane inclination to the terrace plane as a function of bz / h(m) for ZrO2 and Ti according to the PTMC and TM; (a) no partitioning, and (b) symmetrical partitioning of the tilt in the TM.

§6


259

6. Closing remarks 6.1. Summary of the topological model The topological model provides a physical understanding of the structure of martensitic interfaces and the diffusionless nature of transformations consistent with experimental observations including recent high-resolution transmission electron microscopy studies. Martensitic interfaces are modelled as coherent terraces with superimposed disconnections and LID that relieve any coherency strains. In addition, lateral motion of disconnections across the terraces propagates the transformation without long-range diffusion. Such conservative motion imposes severe topological constraints on the nature of admissible terraces; terrace coherency is revealed as a key feature in this context, and feasible terraces are furthermore restricted to those with modest coherency strains. Moreover, admissible disconnections are limited to those that can move conservatively on the terraces and the active LID must not form sessile intersections with them. In this model, the habit plane deviates from the terrace plane because each disconnection exhibits step character, and, for minimal long-range strain, the angle between the two planes can be found. A small rigid-body rotation of the adjacent crystals away from that of the ideal terrace configuration arises as an ancillary consequence of the defect arrays present. Due to the discrete crystallographic nature of the Burgers vectors of the disconnections and LID, uncompensated components perpendicular to the habit plane arise and lead to the formation of tilt walls. These rotations can be determined for the equilibrium configuration and the tilts are partitioned between the adjacent crystals. The transformation displacement can be expressed in terms of the strains introduced by the disconnection and LID arrays; if the misfit at the habit plane is fully relieved, fiducial markers across the transformation front will be continuous but re-oriented. In view of the topological constraints imposed by the need for a conservative transformation mechanism, it is not surprising that a relatively small number of crystalline materials exhibit martensitic behaviour even where the thermodynamic driving force is appreciable. Two transformations, orthorhombic to monoclinic ZrO2 and bcc to hcp Ti have been discussed in some depth in Section 3; the predicted interface structures, Fig. 16, show close agreement with TEM observations, Fig. 17; in particular, micrographs clearly display coherent terrace/disconnection structures. In Fig. 17(a) the disconnections in ZrO2 are sufficiently well separated that the strain-fields emanating from the coherent terraces can be seen. In these cases, the coherency strains parallel to the terraces and the dilatation normal to the terraces are relatively small. Where these strains are larger, complete relief of misfit may require the activation of multiple LID systems, as concluded in Section 4, and this may inhibit transformation except at large driving forces. In a recent study [42], the topological model was applied to the cubic to monoclinic transformation in a PuGa alloy. In this case, the dilatation normal to the proposed terrace plane is large, yet the agreement between the present theory and the experimentally determined habit plane is close. Preliminary application of the TM to ferrous alloys [43] shows that a variety of disconnection/LID networks can accommodate terrace coherency. When the LID mode is twinning, habit planes close to {259} are predicted, similarly to the PTMC, and in good agreement with experimental observations. When the LID mode is slip, TM solutions close to {575} and {121} are

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obtained consistent with experimental observations, but which are not consistent with the PTMC.

6.2. Topological modelling of other transformations A promising feature of the topological model is that it provides a comprehensive framework inter-relating martensitic transformations to other types of transformations occurring by dislocation processes. Mechanical twinning [44] can be regarded as a special instance where the same constraints apply although the structural ‘transformation’ is a rotation in this instance. The case of diffusive-displacive transformations (see [45] for a review), where the crystallographic features of martensite are exhibited but where diffusion occurs concurrently, is another related transformation. For martensitic transformations, the number of substitutional atomic sites and the atomic species must be conserved since no change of composition occurs. However, for diffusive-displacive transformations, the former may be conserved while the latter is not if the diffusional flux of, say, species A to the boundary is equal and opposite to the flux of species B away from the interface or if ordering is involved in the transformation. The formation of lamellar γ -TiAl and α2 -Ti3 Al from a disordered parent has been explained in these terms [46]. The transformation kinetics of diffusive-displacive transformations would obviously be diffusion controlled. Another instance of diffusion control would be where the mode of LID activated gives rise to sessile intersections with the disconnection array. In this way, processes like some Widmanstatten transformations, which exhibit crystallographic features very similar to martensitic ones but progress with much slower kinetics (see Ref. [25] for example), can be understood. Finally, many diffusional transformations are known to occur through a disconnection mechanism and the present framework provides the means for quantitative analysis [47–50].

Acknowledgements The authors express their gratitude to the late Professor J.W. Christian for encouragement and stimulation, and to Professor P. Kelly, Dr. T. Nixon and Dr. S. Celotto for valuable discussions.

References [1] [2] [3] [4] [5] [6]

G.B. Olson and W.S. Owen (eds), Martensite (ASM International, USA, 1992). Y.-M. Chiang, D. Birnie and W.D. Kingery, Physical Ceramics (Wiley, New York, 1997). K. Otsuka and C.M. Wayman, Shape Memory Materials (Cambridge University Press, Cambridge, 1998). K. Bhattacharya, Microstructure of Martensite (Oxford University Press Inc., New York, 2003). J.W. Christian, The Theory of Transformations in Metals and Alloys (Pergamon Press, Oxford, 2002). C.M. Wayman, Introduction to the Crystallography of Martensite Transformations (Macmillan, New York, 1964). [7] M.S. Wechsler, D.S. Lieberman and T.A. Read, Trans. AIME 197 (1953) 1503. [8] J.S. Bowles and J.K. MacKenzie, Acta Metall. 2 (1954) 129, 138, 224.

Topological modelling of martensitic transformations [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

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G.J. Mahon, J.M. Howe and S. Mahajan, Phil. Mag. Letts. 59 (1989) 273. I.-W. Chen and Y.-H. Chiao, Acta Metall. 33 (1985) 1827. T. Moritani, N. Miyajima, T. Furuhara and T. Maki, Scripta Mater. 47 (2002) 193. K. Ogawa and S. Kajiwara, Phil. Mag. 84 (2004) 2919. R.C. Pond and S. Celotto, Int. Mat. Revs. 48 (2003) 225. J.P. Hirth and J. Lothe, Theory of Dislocations (McGraw-Hill, New York, 1968). R.C. Pond, Line defects in interfaces, in: Dislocations in Solids, Vol. 8, ed. F.R.N. Nabarro (North-Holland, Amsterdam, 1989) p. 1. J.P. Hirth and R.C.Pond, Acta Mater. 44 (1996) 4749. J.P. Hirth, J. Phys. Chem. Sol. 55 (1994) 985. M.G. Hall, H.I. Aaronson and K.R. Kinsman, Surface Sci. 31 (1972) 257. A.L. Roitburd, Solid State Physics 33 (1976) 317. U. Dahmen, Scripta Metall. 21 (1987) 1029. D.A. Smith, Scripta Metall. 21 (1987) 1009. J.W. Christian, Metall. Mater. Trans. 25A (1994) 1821. J.W. Gibbs, Scientific Papers, Vol. 1 (Dover, New York, 1961) p. 325. R.C. Pond and X. Ma, Z. für Metal. 96 (2005) 1124. J.M. Howe, Interfaces in Materials (Wiley, USA, 1997). R.C. Pond and D.S. Vlachavas, Proc. Roy. Soc. London, A 386 (1983) 95. A.G. Crocker, Phil. Mag. 7 (1962) 1901. R.C. Pond and F. Sarrazit, Interface Sci. 4 (1996) 99. J.P. Hirth, T. Nixon and R.C. Pond, in: Boundaries and Interfaces: The David A. Smith Symposium, eds R.C. Pond, W.A.T. Clark, A.H. King and D.B. Williams (TMS, Warrendale, 1998) p. 63. C. Hammond and P.M. Kelly, Acta Metall. 17 (1969) 869. R.C. Pond, A. Serra and D.J. Bacon, Acta Mater. 47 (1999) 1441. G.B. Olson and M. Cohen, Acta Metall. 27 (1979) 1907. B.A. Bilby, R. Bullough and E. Smith, Proc. Roy. Soc. 231A (1955) 263. R.C. Pond, S. Celotto and J.P. Hirth, Acta Mater. 51 (2003) 5385. J.P. Hirth, R.C. Pond and J. Lothe, Acta Mater. 54 (2006) 4237. D.P. Dunne and C.M. Wayman, Acta Met. 18 (1970) 981. B.P.J. Sandvik and C.M. Wayman, Metall. Trans. 14A (1983) 835. Y.W. Chai, Ph.D. thesis (University of Liverpool, 2005). P. Franciosi, M. Berveiller and A. Zaoui, Acta Metall. 28 (1980) 273 S.Q. Xiao and J.M. Howe, Acta Mater. 48 (2000) 3253. R.C. Pond, X. Ma and J.P. Hirth, in: Solid to Solid Phase Transformations in Inorganic Materials, eds J.M. Howe, D.E. Laughlin, J.K. Lee, D.J. Srolovitz and U. Dahmen (TMS, Warrendale, 2006) p. 19. J.P. Hirth, J.N. Mitchell, D.S. Schwartz and T.E. Mitchell, Acta Mater. 54 (2006) 1911. X. Ma and R.C. Pond, J. Nucl. Mat., in press. J.W. Christian and S. Mahajan, Prog. Mater. Sci. 39 (1995) 1. B.C. Muddle, J.F. Nie and G.R. Hugo, Metall. Mater. Trans. 25A (1994) 1841. R.C. Pond, P. Shang, T.T. Cheng and M. Aindow, Acta Mater. 48 (2000) 1047. T. Furuhara, J.M. Howe and H.I. Aaronson, Acta Metall. Mater. 39 (1991) 2873. C. Laird and H.I. Aaronson, Acta Metall. 15 (1967) 73. Y. Mou and H.I. Aaronson, Acta Metall. Mater. 42 (1994) 2159. A. Garg, Y.C. Chang and J.M. Howe, Acta Metall. Mater. 41 (1993) 235.


CHAPTER 75

Dislocations and Twinning in Face Centred Cubic Crystals MAREK NIEWCZAS Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7



Contents 1. 2. 3. 4. 5. 6. 7.

Introduction 265 Crystallography of deformation twinning 267 Twinning elements in the fcc lattice 269 Description of twinning transformation – lattice correspondence 270 Twinning transformation of the perfect lattice – geometry of the transformation of dislocations Geometry of plastic deformation of single crystals – crystallographic notation 278 Methodology of TEM analysis 280 7.1. Twin-matrix orientation relationship 280 7.2. Defect identification 284 8. TEM observations of the dislocation substructure prior to twinning 285 8.1. Primary dislocations 287 8.2. Lomer–Cottrell network 289 8.3. Faulted dipole 291 8.4. Stacking fault tetrahedra (SFT) and small defect clusters 293 9. Twinning transformations of dislocations 295 9.1. Geometry of twinning transformation in a dislocated lattice 295 9.2. Transformation of primary dislocations 299 9.3. Transformation of Lomer–Cottrell (LC) dislocations 309 9.4. Transformation of faulted dipoles 326 9.5. Transformation of Stacking Fault Tetrahedra 339 10. New elements of the twinned structure 343 10.1. Extrinsic stacking faults 344 10.2. Frank dislocations 346 10.3. Secondary stacking faults 356 10.4. Fine debris 357 11. The strength of twinned crystal 359 12. Summary 360 Acknowledgments 361 References 361

275

1. Introduction When face centred cubic (fcc) metals are subjected to external stress a certain amount of stress is accommodated by elastic flow after which the material deforms plastically by the formation and glide of line defects, dislocations. Plastic deformation of fcc metals has been extensively studied both experimentally and theoretically, references to reviews of this field are given [1,2]. It is assumed that the reader is acquainted with the properties of dislocations, the part they play in plastic deformation and the main features of the dislocation substructure at various stages of deformation. Dislocation glide is easiest on close packed planes in close packed directions. Suitably oriented fcc metal crystals deform by predominantly primary glide, that is glide on the most highly stressed {111} plane in the most highly stressed 110 direction. Internal stresses due to primary dislocations are relieved by secondary slip, appropriate combinations of glide on other {111}110 systems. Secondary dislocation density steadily increases until it overtakes that of primary dislocations. The result of this activity is an ever denser and more complicated network of defects such that increasingly larger stresses are required to deform the specimen, i.e. the specimen work hardens. At very high stresses, such as those encountered at room temperature in crystals deformed into late stage three of work hardening, or during deformation at low temperatures, another mode of deformation comes into play, namely, deformation twinning or mechanical twinning. Deformation twinning has long been recognized as an important mode of plastic deformation not only in many metals and alloys [3–11] but also in intermetallic compounds [12], semiconductors [13], minerals [14,15] and crystalline polymers [16]. The early work on twinning was reviewed by Hall [17] and Cahn [18,19] while Narita and Takamura [20] and Christian and Mahajan [21] have covered more recent work, the latter article is a comprehensive treatment of deformation twinning in fcc, body-centred cubic (bcc) and hexagonal close-packed (hcp) metals and alloys. Perfect dislocations can dissociate giving rise to partial dislocations and their associated stacking faults. Mechanical twinning is accomplished by certain well-defined movements of partial dislocations. On the atomic scale, there is a homogeneous shearing of the crystal lattice, individual atoms move by a fraction of an interatomic spacing generating a new lattice in twin orientation to the parent lattice. Eventually much of the crystal is converted to twin orientation. For room temperature deformation of fcc metals and alloys twinning usually occurs only late in the deformation, in Stage 3, when the substructure is heavily dislocated. The deformation substructure has been extensively studied and the densities and distribution of dislocations and their Burgers vectors are quite well-documented. The formation of macroscopic twins in the crystal requires that thousands of layers of the parent crystal are swept by twinning dislocations as they advance in a highly organized fashion of coordinated atomic displacements. In its progress through the lattice a twinning dislocation

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must interact with every element of the pre-existing substructure which is then transformed to twin configuration. Some important physical aspects of twinning deformation in dislocated material were noted during the 1950s and 1960s. For example, Thompson and Millard [22] studied twin formation in dislocated hcp crystals and carried out one of the first qualitative analyses of the transformation of dislocations by twinning. Sleeswyk and Verbraak [23] dealt with the problem of incorporation of slip dislocations into mechanical twins in the bcc lattice. They provided the first quantitative analysis of the transformation of the Burgers vectors of slip dislocations by 111{112} twinning using matrix algebra and proposed geometrical models describing the behavior of dislocations during this process. Quoting from Sleeswyk and Verbraak: If the crystal containing a dislocation is sheared homogeneously by uniform slip of successive layers of atoms past each other, the original Burgers circuit will, in general, suffer a shear deformation too. Unless the original Burgers vector happens to lie in one of the layers of atoms that describes the shear movement, its direction, and often its magnitude, will be changed. Such a homogeneous shear is known to occur during mechanical twinning, it is, therefore, to be expected that the Burgers vector of a dislocation incorporated in a twin has its magnitude and direction changed.

Later the matrix approach was used to analyze the transformation of slip dislocations during 103{301} twinning in tetragonal β-tin [24] and in hcp metals [25]. The situation in which a new phase is created by transformation of a pre-existing defect structure renewed interest in the phenomenon of martensitic transformation. Analysis of the transformation of glide dislocations in austenite during martensitic transformation was carried out by Bernshteyn and Shtremel [26]. These authors maintained that of twelve different a2 110 Burgers vectors of glissile dislocations in austenite, eight are converted to a 2 111 and four become a100 after transformation. Shyne and Nix [27] argued that the increased strength of martensite in steels arises at least partially from the transformation of glide dislocations to sessile products. The geometry of twinning, and the occurrence of twinning in various systems has been well studied, but relatively little attention has been given to how twinning dislocations can pass through a dislocation substructure of the density and complexity of that found in highly deformed metals and what happens when they do so. Transformation of dislocations in real materials by twinning remained somewhat underinvestigated, probably because the resolution obtainable with earlier electron microscopes was not sufficient to resolve features found in heavily deformed material. In the late 1980s Basinski became interested in these processes with reference to the behavior of Cu-Al alloy single crystals, which exhibit twinning after advanced deformation by glide (e.g., Szczerba and Korbel [28]). He raised the question of what happens to primary dislocations after twinning is completed in these systems. To study this problem, Basinski’s group [29] chose Cu-Al single crystals oriented for single glide with tensile axes between 8◦ and 10◦ from [110]. These crystals first deform mainly by primary slip but the fraction of glide on the conjugate system steadily increases with increasing deformation. When the tensile axis of the sample has rotated well past the 100–111 symmetry boundary, twinning deformation is induced on the conjugate plane [29]. This work provided the first direct evidence of the transformation of primary a a 2 110 dislocations to sessile 2 200 dislocations in the twin. Studies of the transformation

§2

Dislocations and twinning in face centred cubic crystals

267

of the dislocation substructure in Cu-8%Al single crystals of the same orientation carried out by Korbel [30] have also indicated that such a transformation is possible. The present work is in essence a continuation of the work of Basinski and coworkers [29] and is focused on studies of the defect structure inherited by a twin from the parent crystal during the mechanical twinning process. The experimental methods are essentially those employed in the previous work, the specimens were crystals initially oriented for single glide with a range of compositions from high purity copper to Cu-8%Al. After Stages 1 and 2 of plastic deformation the specimens enter Stage 3 and eventually deform mainly by conjugate twinning. Twinning has been observed at high stresses for temperatures in the range 4.2 K to 300 K. The work is presented in 12 sections. The first sections deal with the theoretical framework for analysis of twinning transformations in a perfect lattice. This includes an overview of the correspondence matrix, which compactly describes how crystallographic planes and directions are transformed by twinning. The transformation in fcc crystals is then discussed purely geometrically in terms of the Thompson tetrahedron, which shows how Burgers vectors and glide planes are related before and after twinning. Later (Section 6) an account is given of the experimental method and materials used throughout the work to examine deformation behavior both with and without twinning. Section 7 outlines the application of transmission electron microscopic (TEM) techniques to the analysis of defect structures in both parent (untwinned, matrix) and twinned material. The very useful stereographic technique, which illustrates the relationship between the parent and twinned lattices, is explained. The method by which various lattice defects are analyzed and identified is also outlined. Having discussed the techniques used, this section concludes with a description of the main features of the dislocation substructure found in plastically deformed parent crystals just before the onset of twinning: it is through this substructure that twinning dislocations must travel. The core topic to be considered here, analysis of the nature of the dislocation substructure in twinned material, is discussed in Section 9. This section begins with a summary of established facts concerning the transformation of primary dislocations by twinning. Then follows an outline of the transformation of other elements of the defect structure such as Lomer–Cottrell dislocations, faulted dipoles, stacking fault tetrahedra, and fine dislocation debris, drawing from information garnered from rigorous TEM studies and by geometrical modeling techniques. Throughout this article terms such as parent lattice, matrix material, and untwinned material, are used interchangeably in referring to material in which no twinning has occurred. Crystallographic indices in the parent lattice are usually written with subscript (or superscript) M, those in the twinned lattice are written with subscript (or superscript) T.

2. Crystallography of deformation twinning A twin is a structure whose crystal lattice is a mirror image of that of its parent, the plane about which they are oriented is the mirror plane. Twins are known to be formed during solidification from liquid or vapor phase (growth twins), during solid-state phase transformation (transformation twins), during recrystallization of cold worked material (annealing twins) and during plastic deformation under externally applied stress (deformation twins).

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Fig. 1. Crystallographic elements of twinning. K1 is the first undistorted plane, the twinning plane. K2 is the second undistorted plane, it rotates to KT 2 in the twin. η1 is the twinning direction, η2 is the conjugate twinning direction which becomes η2T lying on KT 2 in the twin lattice. S is the shear plane containing η1 and the K1 and K2 plane normals. The twinning shear is given by s = 2 tan α, where α is half the angle between η2 and η2T .

Deformation twins are produced as a result of a homogeneous shear in the parent lattice such that the product has the same crystal structure as the parent and the two crystal lattices are oriented according to a well-defined crystallographic relationship. The crystallographic relationship between these two lattices may be described by four interdependent twinning elements K1 , K2 , η1 , and η2 , which constitute a twinning mode [17,31,32]. Fig. 1 shows the well-known diagram describing all twinning elements. K1 is the plane on which twinning shear occurs, it is variously referred to as the twinning plane, the habit plane, the composition plane, or the invariant plane of the twinning shear. This plane is unaltered by the transformation so that any direction in it is unaffected by a twinning shear, the plane is therefore also known as the first undistorted plane. The twinning shear proceeds on the twinning plane along the η1 direction, known as the twinning direction or shear direction. The conjugate to the twinning plane is the K2 plane, also known as the reciprocal twinning plane, the second invariant plane, or the second undistorted plane. This plane is labeled the KT2 plane in the twin, but as in the case of the K1 plane, it preserves its crystallographic identity in the twinned lattice. The K2 plane contains the η2 direction, defined as the conjugate twinning direction or conjugate shear direction. This direction is transformed to η2T in the KT2 plane in the twin, as shown in Fig. 1. The plane containing η1 and η2 is defined as the plane of shear, S, in Fig. 1 [17,32–34]. Alternatively the plane of shear can be defined as the plane containing η1 and the normals to the K1 and K2 planes. Also, η2 is the direction of intersection of the K2 plane and the plane of shear S [21]. When twinning occurs, the material above the K1 plane is sheared by an amount defined by the twinning shear, s. The magnitude of the twinning shear determines the position of the KT2 plane and η2T in the twin lattice. The twinning shear, s, is obtained from the relation s = 2 tan α, where α is half angle between η2 and η2T (Fig. 1). Analysis of the crystallographic relationship between the parent and the twinned lattices shows that two types of twin relationships can exist. In type I twins, a twin lattice is produced by reflection of the parent lattice in the K1 plane or by rotating it 180◦ around the normal to the K1 plane. In this case the K1 plane and the η2 direction must be rational, while the remaining elements are irrational. Type II twins result when parent and twin lattices are related by rotation of 180◦ around the η1 direction or by reflection in the plane

§3


269

normal to η1 . In the case of type II twins, the K2 plane and η1 direction must be rational and, as before, the remaining elements are irrational. In most cases of twinning in metals and alloys all twinning elements are rational. According to a crystallographic theory of deformation twinning developed by Bilby and Crocker [31] these twins, termed compound twins, represent a degenerate case of a spectrum of general orientation relationships between parent and twin in various lattice systems. The twinning mode of a structure is unambiguously defined when K1 and η2 , or K2 and η1 or, alternatively, K1 , η1 and s are specified. In practice however description of a twinning mode is given by providing all twinning elements K1 , K2 , η1 , and η2 , together with the amount of associated twinning shear (e.g., [32]). Corresponding to a given twinning mode there exists a reciprocal twinning mode K′1 , K′2 , η1′ , η2′ with the same magnitude of shear, which satisfies the following relationships: K′1 = K2 , K′2 = K1 , η1′ = η2 , η2′ = η1 . The twinning mode of bcc crystals is the reciprocal of that of fcc crystals [31,32]. Twinning deformation produces a new lattice and results in a macroscopic change of shape of the parent crystal. Bilby and Crocker (1965) [31] define a twinning shear as any shear that restores the lattice or a superlattice in a new orientation. This general definition of twinning leads to additional possible twinning modes in crystalline lattices in which no restriction on the rationality of the twinning elements is imposed. This means that three or four twinning elements may be irrational, leading to non-conventional twins with no obvious orientation relationship.

3. Twinning elements in the fcc lattice Twinning shear in an fcc lattice is accomplished by a displacement a6 112 applied successively to {111} layers on the twinning plane. In practice this occurs by passing Heidenriech–Shockley partial dislocation with an a6 112 Burgers vector over every {111} plane above the twinning plane. The schematics of deformation twinning are shown in Fig. 2a in which part of the block of atoms between the two arrows is sheared by application of an a6 112 shear. The change of packing sequence in the sheared lattice, which leads to the formation a of portion of crystal in twin orientation, can easily be appreciated by filling up the lattice sites in the upper, sheared part of the crystal model with atoms, as shown in Fig. 2b. The close-packed plane onto which the initial twinning shear was applied in the parent crystal (marked by an arrow in Fig. 2a and indexed as layer B in Fig. 2b) becomes the mirror plane between the two lattices. If the same block of atoms is then sheared by a2 110, equivalent to passing an ordinary (perfect) dislocation on every {111} plane, as shown in Fig. 2c, the original orientation, that of the lower (parent) portion of the crystal, is restored and no new lattice is formed (Fig. 2d). All twinning elements in fcc crystals may be conveniently represented by a Thompson tetrahedron distorted by application of a twinning shear [29,35]. Fig. 3a shows an undistorted tetrahedron ABCD whose faces are {111} close-packed fcc lattice planes. Fig. 3b represents the situation where a twinning shear occurs on plane EFG, the K1 plane, along the η1 direction parallel to 112. Every {111} layer above the composition plane EFG in Fig. 3b is successively sheared by an a6 112 vector, resulting in distortion of the upper part of the ABCD tetrahedron giving the new configuration ABCEFGD’. The second

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(a)

(b)

(c)

(d)

Fig. 2. a) Representation of the distortions of a fcc lattice due to application of a6 112 and a2 110 shears. (a) an a 112 twinning shear is applied to every {111} layer of the block of atoms between the two black arrows, which 6 leads to configuration (b) a portion of the lattice is in twin orientation with respect to the original crystal. (c) An a 110 shear is applied to the lattice between the two arrows yielding structure (d), the model has been filled 2 with atoms to illustrate that the untwinned structure is restored.

undistorted plane, K2 , which in the original ABCD tetrahedron is represented by the BCD plane (Fig. 3a) or BCGF (Fig. 3b) is rotated to the {111} plane FGD’ in the twinned lattice (KT2 plane). Similarly, the η2 direction is rotated to η2T in the KT2 plane. From the above representation it is clear that in fcc crystals the K1 , K2 and η1 , η2 twinning elements are of {111} and 112 types respectively. Also, it is clear that the plane of shear which contains the η1 and η2 directions must be a {110} plane. The magnitude of the twinning shear √1 , 2 in an fcc lattice is given by the ratio of the length of the Burgers vector of a Heidenreich– Shockley dislocation, a6 112, to the length of an a3 111 vector or the spacing of {111} planes. In fcc crystals there are four close-packed {111} planes each containing three 112 directions, thus there are twelve independent twinning systems analogous to that discussed above.

4. Description of twinning transformation – lattice correspondence It is of interest here to develop a formalism which describes the transformation of crystallographic planes and directions produced by a twinning shear. As early as 1899 Mügge [36]

§4


(a)

271

(b)

Fig. 3. a) The Thompson tetrahedron, ABCD, whose faces represent close-packed planes in a fcc lattice. b) The distorted Thompson tetrahedron, ABCEFGD’ after application of an a6 112 twinning shear sequentially to every {111} layer above the composition plane, EFG. The twinning shear occurs on the K1 (twinning, EFG) plane in T the η1 = 112 direction. K2 , the second undistorted plane, BCGF, rotates to KT 2 , FGD’ in the twin. η2 and η2 are the conjugate twinning directions in the matrix and twin respectively.

studied twinning in various noble metals and derived a set of equations describing the transformation of Miller indices of crystallographic planes and directions by fcc twinning. The more recent theory developed by Bilby and Crocker [31] describes the relationship between parent and twin lattices and allows prediction of the majority of operating twinning modes for type I and type II twinning in various crystal systems. This theory was not able however to describe anomalous relations in unusual twins formed in crystal systems which do not satisfy classical twin orientation relationships, e.g., during double twinning in magnesium [33,34]), or twins in crystalline mercury [37,38]. In a series of papers by Bevis and Crocker [39–41] the theory was generalized to incorporate additional twinning modes. A key point in the theory of deformation twinning is the concept of the correspondence matrix, which provides a link between the indices of a vector in the parent and the twinned lattice. For a comprehensive treatment of this concept together with the fundamentals of the general theory of lattice transformations the reader is referred to the book by Christian [32], the work of Christian and Crocker [42] and more recently the work of Christian and Mahajan [21]. The focus of this article is type I twinning transformation in fcc crystals, which is accomplished by the homogeneous deformation of a crystal lattice described by a deformation gradient matrix in the form [43]: 1 + sm1 n1 S= sm2 n1 sm3 n1

sm1 n2 1 + sm2 n2 sm3 n2

sm1 n3 sm2 n3 1 + sm3 n3

(4.1)

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where s = √1 is the amount shear, m is the twinning direction (m = 112), and n is 2 the normal to the twinning plane K1 (n = 111). The deformation matrix S represents a physical quantity, a tensor of the second rank which is subject to the transformation law, thus when the axes are changed, the nine coefficients Sij of this tensor in the old axis system transform to the nine coefficients Sij′ in the new axis system and vice versa [44]. The determinant of the deformation gradient matrix S is 1, i.e. there is no volume change associated with twinning deformation. Any vector uM = [u1 , u2 , u3 ] defined with respect to the basic orthogonal system e1 = [100]M , e2 = [010]M , e3 = [001]M in the parent lattice is sheared to the vector vM = [v1 , v2 , v3 ] in the same lattice, according to the linear transformation relation: vM = SuM

(4.2)

where: uM denotes the column matrix of vector uM , vM denotes the column matrix of vector vM and S is a 3 × 3 matrix built from the components of the second rank deformation tensor expressed with reference to the parent axis system (4.1). The transformation given by the above equation is termed affine: collinear points remain collinear after transformation and coplanar lines remain coplanar after transformation. If both uM and vM remain fixed in space, one can express their coordinates with respect to the new twin lattice, e1 = [100]T , e2 = [010]T , e3 = [001]T using the law of transformation of vector components [44]: u′M = AuM ,

(4.3.1)

v′M = AvM

(4.3.2)

where: uM , vM denote the column matrices of vectors uM and vM in the parent lattice, u′M and v′M denote the column matrices of these vectors in the twin lattice, and A is a 3 × 3 matrix which represents the coordinate transformation of the crystal axis system ei from the parent to the twin lattice. The first subscript of the components aij of the matrix A in eq. (4.4) refers to the twin axes and the second to the parent axes, aij being the cosine of the angle between ei in the twin and ej in the parent [43]. As is well known, any arbitrarily chosen set of coordinate systems can be represented by a transformation matrix (eq. (4.4)) whose components define the cosine of an angle between appropriate axes in the new and old systems. a11 Aij = a21 a31

a12 a22 a32

a13 a23 . a33

(4.4)

The difficulty in defining the above transformation matrix for the parent-twin coordinate transformation arises from the fact that the mutual orientation of the 100 crystal axes in the two lattices is not obvious. Therefore one chooses a set of axes associated with the matrix-twin interface which allows easy representation of the orientation relationship of the two lattices. Christian [32] showed that an appropriate selection of orthogonal axes

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Fig. 4. Diagrammatic representation of the relationship between coordinate systems in the parent crystal and twinned lattice together with the sequence of transformations which relates a matrix vector, uM , to its product vector, vT , in the twin. A is the transformation matrix relating crystal axes systems eiM and eiT in the parent and twin. X and Y are transformation matrices relating the coordinate system xi in the parent with crystal axis systems eiM and eiT in the parent and twin respectively. R is the rotation matrix for the coordinate systems xi and yi associated with the interface.

in the parent lattice is such that x1 is parallel to the η1 direction, x3 is normal to the K1 plane and x2 is their cross product (Fig. 4). In type I twins the two lattices are related by a 180◦ rotation about the normal to the K1 plane, which determines the axis selection in the twinned lattice. It follows that x1 in the parent crystal is parallel to −y1 in the twin, x2 is parallel to −y2 in the twin, and x3 is parallel to y3 in the twin (Fig. 4). This selection of axes, xi and yi , enables easy and unambiguous derivation of the relationship between planes and directions in the parent and the twinned lattices. Let vT be a new vector in the twin lattice obtained by transforming vector uM of the parent crystal by twinning represented by matrix S in eq. (4.1) (Fig. 4). The purpose is to derive a relation between components of the vector uM in the parent crystal and the shear product vector, vT , in the twin. This is achieved by using the following series of transformations, shown in Fig. 4. First we transform vector uM to vector vM in the matrix by the twinning shear described by eq. (4.2), i.e.: vM = SuM . Next we transform vector vM to the xi coordinate system associated with the twin-matrix interface in the parent lattice, extended on η1 , K1 ⊗ η1 and K1 directions, as discussed previously. The transformation matrix between crystal axis system ei and coordinate system xi in the parent is defined by the transformation matrix X. We thus obtain the intermediate (operational) vector vINT from the transformation: vINT = XvM .

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This intermediate vector vINT , is now rotated by 180◦ around x3 (Fig. 4), an operation which transforms the parent crystal to twin orientation. The matrix which describes this transformation is a rotation matrix defined here as R. The components of the new vector vROT obtained in this transformation are given by the relation: vROT = RvINT . In the last step the components of the vROT vector are expressed in terms of the twin lattice. If the transformation matrix between the “interfacial” xi axis system and the twin system ei is given by Y, the vectors vT and vROT are related by: vT = YvROT . This series of transformation relations leads to the final expression for vT in terms of uM in the form: vT = YRXSuM .

(4.5)

It can easily be shown that matrices X and Y are mutually orthogonal, so that matrix Y is equal both to the reciprocal of matrix X and to the transpose of the matrix, eq. (4.5) can be rewritten as follows: vT = X−1 RXSuM .

(4.6)

The product of matrix multiplication X−1 RX is called the reindexation matrix, denoted by U in eq. (4.7). U = X−1 RX.

(4.7)

The reindexation matrix is obtained as an orthogonal similarity transformation of the operator matrix R, which relates two lattices by rotation of 180◦ around the normal to the K1 plane. The reindexation matrix allows “reindexing” of any vector or plane of the parent lattice to its corresponding representation in the twin lattice. For example, components of vectors uT and vT in Fig. 4 are related to vectors vM and uM via the reindexation matrix such that uT = UuM and vT = UvM , and reindexation of a plane with normal n is given by the relation nT = nM U−1 . Eq. (4.6) can be rewritten in shorter form by using the reindexation matrix (eq. (4.8)) vT = USuM .

(4.8)

Also the matrix product US can be replaced by matrix C, the correspondence matrix, which now directly relates vector vT in the twinned lattice to the original vector uM of the parent crystal. The correspondence matrix represents an essential concept in predicting the physical transformation of lattice planes and directions resulting from the twinning process. This matrix combines the effect on any vector of deformation by a twinning shear and

§5


275

reindexation of its components in the new twin crystal lattice. It can easily be shown that any plane normal, nM , or vector, vM , in the parent crystal structure will be transformed into a plane normal, nT , or vector, vT , in the twinned lattice according to relations (4.9) and (4.10) [29]: vT = CvM ,

(4.9)

nT = nM C−1 .

(4.10)

We derived the correspondence matrix C, which provides a direct relation between the two lattices, using the parameters K1 , η1 , and s, all of which are basic elements for the construction of both the deformation gradient tensor in eq. (4.1) and the reindexation matrix in eq. (4.7). Alternatively, the correspondence matrix can be derived using K1 and η2 , as shown by Cahn [45]. Due to the high symmetry of the cubic lattice, the twin-parent relationship in cubic crystals is represented by one variant of the correspondence matrix, which is a degenerate case of the more general Bevis–Crocker theory of deformation twinning.

5. Twinning transformation of the perfect lattice – geometry of the transformation of dislocations ¯ plane in the [121] direction. The twinning shear that is discussed here occurs on the (111) All matrices needed to calculate the correspondence matrix are given below [29]. S is the deformation gradient matrix, R is the rotation matrix, U and C are the reindexation and correspondence matrices respectively, and X and X−1 are those matrices required to calculate the reindexation matrix U as in eq. (4.7). 7 −1 1 2 4 2 , 1 −1 7

(5.1)

−1 0 0 R = 0 −1 0 , 0 0 1

(5.2)

1 S= 6

1 X= 2

X

−1

2 4 2 −3 0 3 , 2 −2 2 2 −3 2 4 0 −2 , 2 3 2

(5.4)

−1 −2 2 −2 −1 −2 , 2 −2 −1

(5.5)

1 = 2

1 U= 3

(5.3)

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Table 1 Twinning transformation of Burgers vectors and lattice planes in fcc crystals (after Basinski at al. [29]) Matrix dislocation

Twin dislocation

vectors

plane

vectors

plane

a/2[011] a/2[101] ¯ a/2[110]

¯ (111)

¯ a/2[01¯ 1] ¯ a/2[020] ¯ a/2[011]

¯ (200)

a/2[011] a/2[110] ¯ a/2[101]

¯ K1 (111)

¯ a/2[01¯ 1] ¯ a/2[1¯ 10] ¯ a/2[101]

¯ KT 1 (111)

¯ a/2[011] ¯ a/2[110] ¯ a/2[101]

K2 (111)

¯ a/2[110] ¯ a/2[011] ¯ a/2[101]

¯¯¯ KT 2 (111)

¯ a/2[011] a/2[101] a/2[110]

¯ (11¯ 1)

¯ a/2[110] ¯ a/2[020] ¯ a/2[1¯ 10]

(002)

¯ a/2[011] a/2[110] ¯ a/2[101]

(100) (001) (010)

¯ a/2[110] ¯ a/2[1¯ 10] ¯ a/2[101]

¯ (1¯ 11) ¯ (11¯ 1) ¯ 1) ¯ (10

¯ a/3[111]

¯ K1 (111)

¯ a/6[141]

a/3[111]

K2 (111)

¯ a/6[1¯ 4¯ 1]

¯¯¯ KT 2 (111)

a/6[011] ¯ a/6[011] ¯ a/6[101] a/6[101] ¯ a/6[110] a/6[110]

(100) (100) (010) (010) (001) (001)

¯ a/6[01¯ 1] ¯ a/6[110] ¯ a/6[101] ¯ a/6[020] ¯ a/6[011] ¯ a/6[1¯ 10]

¯ (1¯ 11) ¯ (1¯ 11) ¯ 1) ¯ (10 ¯ 1) ¯ (10 ¯ (11¯ 1) ¯ (11¯ 1)

¯ K1 (111) K2 (111) ¯ (111) ¯ (11¯ 1)

1 C= 2

−1 −1 1 −2 0 −2 . 1 −1 −1

¯ KT 1 (111)

¯ KT 1 (111) T ¯ ¯ ¯ K2 (111) ¯ (200) (002)

(5.6)

Using the correspondence matrix method outlined above, Basinski and co-workers [29] analyzed the transformation of all crystallographic planes and directions associated with slip dislocations during type I twinning in fcc crystals. Szczerba [46] extended this analysis to other planes and directions for both twinning and supertwinning in fcc and bcc lattices. Table 1 lists the transformations of all possible slip systems in an fcc lattice and includes directions and planes associated with other defects to be discussed later in this work. The geometrical features of the transformation of ordinary dislocations in an fcc crystal lattice inferred from Table 1 are best depicted with the help of the Thompson tetrahedron

§5


(a)

277

(b)

Fig. 5. Perspective view of the distorted Thompson tetrahedron showing physical transformation of {111} closepacked planes and 110 close-packed directions in a fcc lattice.

in which the edges represent Burgers vectors of perfect dislocations and the faces represent {111} glide planes. Operation of the correspondence matrix on the Thompson tetrahedron transforms it to a new configuration consisting of two {100} and two {111} phases. This distorted configuration illustrates the fundamental aspects of the physical transformation of a parent lattice by twinning and provides a convenient geometrical model for analysis of the transformation of Burgers vectors and glide planes in fcc crystals. Fig. 5a shows the model previously introduced in Fig. 3b in perspective view on the {111}M ABFE plane, where the lower portion, ABCEFG, is untwinned (parent) material and the upper portion, EFGD’ has undergone twin transformation. Evidently the ABFE glide plane transforms to {100}T , EFD’ in the twin. In Fig. 5b, another view of the model shows that CAEG, the second {111}M plane of the parent lattice, transforms to {100}T , GED’ in the twin. In summary, inspection of this model clearly shows that two of the four {111}M glide planes in the fcc lattice transform to {100}T planes in the twin; the other two retain their closepacked structure, BCGF (K2 ) is rotated to {111}T , FGD’ in the twin while K1 is unchanged, hence its name undistorted or invariant plane. The model in Fig. 5 also facilitates analysis of the transformations of glide directions, i.e. Burgers vectors, of dislocations. AE, which represents 110M , is transformed to 100T , ED’ in the twin. BF, representing 110M is transformed to 110T (FD’), and 110M (CG) is transformed to 110T , GD’ in the twin. The remaining 110M directions are unchanged. These transformations of directions and planes can of course be inferred from Table 1, which provides a rigorous quantitative analysis including both the direction and length of the Burgers vectors of the dislocations and their glide planes. An important consequence of the twinning transformation in fcc materials, which can be inferred from the above analysis, is that certain components of the dislocation substructure which were mobile in the matrix become sessile in the twin, they acquire 100T Burgers

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vectors and {100}T planes in the twin, configurations which are unique to twinning. On the other hand, defects associated with {100}M planes are sessile in the parent crystal, such as for example Lomer dislocations [47] or Lomer–Cottrell (stair-rod) dislocations [48], may become either glissile products associated with {111}T planes or sessile dislocations associated with {110}T planes in the twin (Table 1, Szczerba [46]). Table 1 shows that dislocations lying on the K1 or K2 plane in the matrix and whose Burgers vectors thread the plane, including a3 111M sessile dislocation loops to be discussed later, transform to other defects with Burgers vector a6 141T deposited on K1 = KT1 and KT2 planes in the twin. Finally, Table 1 shows the transformation of stacking fault tetrahedra, a feature commonly observed in deformed materials, which consist of stair-rod dislocations bounded by stacking faults on {111}M planes. The transformation of the Burgers vectors of stairrod dislocations follows the principles previously discussed for transformation of perfect dislocations however, stacking faults on two {111}M planes of the tetrahedron transform to {100}T planes in the twin. Some sophisticated dislocation mechanism must be involved if such defects are to be fully incorporated into the twin. The next part of this work deals with analysis of the dislocation substructure in the twin which was inherited from the parent crystal. There is also be discussion about whether the theoretically predicted transformations actually take place in real materials. We shall also attempt to provide physically-based modeling of dislocation mechanisms which might be involved in the twinning transformation.

6. Geometry of plastic deformation of single crystals – crystallographic notation The crystallographic notation used here in describing the geometry of plastic flow in Cualloy single crystals is that described by several authors [20,49–52]. In this convention the ¯ (111), ¯ (11¯ 1) ¯ and (111) respectively. primary, conjugate, cross and critical planes are (111), The tensile axis of the crystals is [541], and the mutually perpendicular faces are parallel ¯ and (11¯ 1). ¯ The tensile deformation of fcc crystals with similar crystallographic to (123) orientation has been thoroughly studied [53–59]. During tensile deformation the single ¯ crystals deform by dislocation glide mainly on the primary slip system (111)[101] M and the tensile axis rotates towards the glide direction, [101]M . Stresses due to primary dislocations are relieved by a particular set of secondary shears [58]. The density of secondary dislocations gradually increases until it overtakes that of primary dislocations, the flow stress correlates well with secondary dislocation density. If the flow stress in deformed samples becomes sufficiently high, for example during deformation at very low temperature, twinning will occur. Blewitt, Coltman, Redmann [60] were the first to recognize this phenomenon in copper deformed at 4.2 K. The necessary condition for twin nucleation requires that the resolved shear stress on the twinning system exceed the critical stress for twinning nucleation, which is of the order of γb , where γ is the stacking fault energy and b is the Burgers vector of the twinning dislocation. In the case of single crystals oriented for single glide one can adjust the deformation parameters, i.e. the flow stress via the deformation temperature or the strain rate, and the stacking fault energy via alloying, so that after the tensile axis of the sample rotates past the [100]–[111] symmetry line (overshoot) the

§6


279

¯ conjugate twinning system, (111)[121] M operates. Conjugate twinning occurs when overshoot begins in high purity copper single crystals deformed at 4.2 K whereas at 78 K the crystals neck because the resolved shear stress on the twinning system is not high enough to activate twinning dislocations [59–61,63]. For the same reason, Cu-6%Al single crystals twin on the conjugate plane when deformed at 78 K, whereas they neck at 295 K [57]. The exact orientation of the tensile axis of the crystals at the onset of twinning may vary in different alloy systems [53,54]. The amount of overshoot appears to be determined by the amount of slip achieved by the dominant and secondary systems, which in turn is probably affected by the internal stress distribution due to the development of a specific type of dislocation substructure. X-ray orientation measurements show that crystals of Cu8%Al have overshot the symmetry boundary by about 2◦ at the onset of twinning [59]. The occurrence of twinning during tensile deformation is associated with a noticeable acoustic effect. Fig. 6 shows stress strain data for a high purity copper single crystal deformed at 4.2 K, Cu-4%Al single crystal deformed at 78 K and a Cu-8%Al single crystal deformed at room temperature. The onset of twinning occurs at about 95% of strain in Cu, after 98% of strain in Cu-4%Al and after about 73% of strain in Cu-8%Al [57,59–61,64]. Twinning propagates through the sample as a Lüders-like front in the form of bursts giving rise to audible clicks and producing load instability.

Fig. 6. Tensile stress-strain characteristics of Cu, Cu-4%Al and Cu-8%Al single crystals oriented for single glide deformed at 4.2 K, 78 K and 300 K respectively. Twinning is marked by abrupt load drops which occur at high strain when the tensile axis has rotated into the conjugate triangle after about 95%, 98% and 73% of strain for Cu, Cu-4%Al and Cu-8%Al crystals, respectively.

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Fig. 7. Optical micrographs of the twinning transformation in Cu-8%Al deformed at room temperature showing propagation of the twinning front from the onset of twinning in (a) to the later stages when there is a change of shape and reduction in cross-sectional area of the sample, as in (f).

Fig. 7 shows a sequence of macroscopic observations of a Cu-8%Al single crystal at different stages of twinning at room temperature. The propagation of the twinning front and the transformation of the sample, resulting in change of shape and reduction of the cross sectional area, are clearly visible. The twinned and untwinned regions are easily recognized in single crystals and detailed examination of slip markings on both faces of the crystal permits precise determination of the orientation of the sample. This is crucial in TEM studies during sample sectioning and subsequent preparation of thin foils.

7. Methodology of TEM analysis 7.1. Twin-matrix orientation relationship We now discuss the framework for the TEM analysis of the dislocation substructures which form in the matrix and in the twin. One of the most important steps in this analysis is clarification of the orientation relationship between parent and twin: one needs this information

§7.1


281

to select the diffracting conditions in the two lattices and to interpret the diffraction contrast properties of the defects. Unlike polycrystals, single crystals offer a model material system where diffracting conditions can be controlled unambiguously in relation to both the parent and twin lattices. Figs 8a and b show a stereographic projection on the (100)M plane of the parent lattice and the corresponding stereographic projection of the twin crystal, these represent the unique parent-twin orientation relationship for the case of conjugate twinning in fcc crystals [29]. The tensile axis and traces of the crystal faces of the virgin crystal are ¯ M and (11¯ 1) ¯ M , respectively, in Fig. 8a. As mentioned earrepresented by [541]M , (123) ¯ M , (111) ¯ M, lier, the primary, conjugate, cross and critical glide planes are defined as (111) ¯ ¯ (111)M and (111)M respectively in the parent lattice. One can monitor the orientation of the crystal, and therefore the position of the primary and other glide planes, either by taking X-ray diffraction photographs or by examining the slip traces produced by dislocations emerging at the lateral surfaces of deformed crystals. Accurate determination of the crystal orientation is necessary when cutting appropriately oriented slices from deformed crystals for preparation of TEM foil specimens. The dislocation substructure in both the parent and twin can be studied in various cross-sections each of which offers access to a different set of operating reflections and therefore to a different part of the total picture. For example, ¯ M , the primary glide plane of the parent crystal, i.e. with foil in foils cut parallel to (111) ¯ M or [1¯ 11] ¯ M , by appropriately tilting the foil one can put the conjugate normal either [111] ¯ M , critical (111)M or cross (11¯ 1) ¯ M plane into diffracting position, so that six differ(111) ent 111M operating reflections are available to analyze the dislocation network. There are also three different {220}M planes and therefore six different 220M reflections available in this foil orientation. The relative position of the traces of all glide planes in a given crosssection, which is carefully controlled at the time of sample sectioning and later during foil preparation, determines uniquely the foil normal and removes ambiguity both in terms of the type and the sense of the operating reflections. The corresponding orientation of the twin lattice is determined from the stereographic projection in Fig. 8b. This twin projection is uniquely defined with reference to the parent lattice through the reindexation matrix, U, given in eq. (5.5) and is obtained by reindexing the features of the stereographic projection of the parent following the principles discussed in Section 4. For twinning occurring on a plane other than the conjugate plane, the parent-twin relationship would be represented by a different reindexation matrix and a different pair of stereographic projections. ¯ T , [010]M is reindexed For conjugate twinning the [100]M direction is reindexed to [1¯ 22] ¯ T , and [001]M is reindexed to [22¯ 1] ¯ T . Reconstruction of the entire stereographic to [2¯ 1¯ 2] projection based on three known directions is straightforward. The relative orientation of two lattices in any arbitrary sample cross-section can be determined by superimposing the ¯ M of the matrix two stereographic projections. For example, for a foil cut parallel to (111) ¯ M , the corresponding foil normal in the twinned orientation is having foil normal [111] ¯ T . This orientation offers four low index planes in the twin unambiguously identified [5¯ 11] ¯ 1} ¯ T , {002}T , {020} ¯ T and {022}T , each of which can readily be set in diffracting as {11 condition, thus giving enough information for analysis of the dislocation substructure in ¯ M having a foil normal this cross-section (Fig. 8b). In the case of a foil cut parallel to (11¯ 1) ¯ ¯ ¯ [111]M the corresponding foil normal in the twin is [115]T . Diffracting planes such as ¯ 1} ¯ T , {200} ¯ T , {020} ¯ T and {2¯ 20} ¯ T , are easily accessible in this cross-section (Fig. 8b). {11

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(a)

(b) Fig. 8. Stereographic projections for (a) the parent and (b) the twin lattices representing the unique parent-twin ¯ M orientation relationship for conjugate twinning in the crystals studied, after [29]. In projection (a) the (123) ¯ M crystal faces, and the plane containing the tensile axis, are shown by dashed lines, the [541] tensile and (11¯ 1) axis is also marked. Corresponding orientations of relevant planes and directions in both lattices are marked by ¯ M, ¯ M , (111) shadowed circles. The parent primary, conjugate, cross and critical glide planes are defined as (111) ¯ M and (111)M , respectively (see text for details). (11¯ 1)

§7.1


(a)

(b)

(c)

(d)

(e)

(f)

283

Fig. 9. The orientation of traces of various crystal planes in foils cut parallel to the primary, cross and critical glide ¯ M = [5¯ 11] ¯ T , [11¯ 1] ¯ M = [115] ¯ T , and planes of the parent crystal, i.e. with foil normal (FN) corresponding to [111] ¯ T respectively. The sense of the foil normal and the operating reflection can be determined by [111]M = [1¯ 5¯ 1] monitoring traces of the crystal planes in a given cross-section.

Other foil orientations offer different sets of diffracting planes that can be determined from the relations given in Fig. 8. To illustrate the importance of controlling the orientation of the foil normal when identifying the lattice defects to be discussed later in this work, Fig. 9 shows the orientation

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of the traces of various crystal planes for a given foil orientation in sections cut parallel to the primary, cross and critical glide planes of the parent lattice both in the parent and the twin. One can easily determine the type and the sense of the operating reflection when a plane is set in diffracting position by monitoring traces of the crystal planes in a given cross-section.

7.2. Defect identification Discussion of the details of electron diffraction theory that provide the foundation for the analysis of defect structures in deformed material is beyond the scope of this work and the reader is referred to such texts as Hirsch et al. [65] or Williams and Carter [66]. In the present work TEM methods, which utilize two-beam bright-field and weak-beam darkfield diffracting conditions [67], have been employed to study the nature of dislocations and other defects in both the parent and twinned material. These studies involve determination of Burgers vectors and glide planes of the dislocations by analyzing the diffraction contrast properties of a given defect under known diffracting conditions and determining its spatial orientation in the crystal lattice. To identify the Burgers vector of a dislocation in elastically anisotropic materials such as copper and copper alloys with any confidence usually requires comparison of the observed TEM images with those calculated using anisotropic elasticity [68]: this is true for both two-beam and weak-beam techniques. However, for large values of the deviation parameter “s” the intensity distribution of the diffracted beam is known to be well-approximated by the kinematic theory of electron diffraction. This facilitates identification of various crystal imperfections, and the contrast of dislocations can be quantitatively evaluated based on the value of the product g . b or g . b × u where g is the diffracting vector, b is the Burgers vector and u is the dislocation line direction [65]. For a perfect dislocation the product g . b is an integer and identification of the Burgers vector requires finding the reflection in which the dislocation becomes invisible. The invisibility criterion, g . b = 0, implies that all lattice displacement components of a defect lie in the diffracting plane. Complications may arise since the displacement field of a general dislocation will distort any diffracting plane, so that even when g . b = 0 there will be residual contrast. Therefore complete invisibility of a dislocation requires that both g . b = 0 and g . b × u = 0 are satisfied simultaneously. Image calculations suggest that a perfect dislocation will exhibit weak residual contrast and effectively is indistinguishable from the background when the parameter m = 18 (g . b × u) 0.08 [65]. Partial dislocations, for which g . b is no longer an integer, are considered invisible at g . b = 1/3 and are visible at g . b = 2/3. These simple considerations provide a crude rule of thumb for identifying various defects in the substructure of deformed material have been employed in the present work. Tables 2 and 3 show values of g . b for the parent and twin dislocations considered in Table 1 under diffracting conditions available in selected cross-sections in both lattices. It is clear from the tables that the Burgers vector of an a[100]T dislocation, which is the transformation product of a primary a2 [101]M dislocation (Section 5), can be unambigu¯ T , {002} ¯ T and {200}T diffracting planes available in [151]T ously identified using the {111} foils, i.e. parallel to the critical plane (111)M of the matrix (Table 3c). This combination

§8

285


Table 2a ¯ M g.b values for the dislocations in the parent in various reflections available in section parallel to primary (111) plane Plane/Burgers vector ‘b’ ¯ 111

111

Operating reflection ‘g’ 11¯ 1¯ 022

202

¯ 220

¯ (111) ¯ (111) ¯ (111)

a/2[011] a/2[101] ¯ a/2[110]

0 1 1

1 1 0

−1 0 1

2 2 −1

1 2 1

−1 1 2

¯ (111) ¯ (111) ¯ (111)

a/2[011] a/2[110] ¯ a/2[101]

0 0 0

1 1 0

−1 0 1

2 1 −1

1 1 0

−1 0 1

(111) (111) (111)

¯ a/2[011] ¯ a/2[110] ¯ a/2[101]

1 1 0

0 0 0

0 1 1

0 −1 −1

1 1 0

1 2 1

¯ (11¯ 1) ¯ (11¯ 1) ¯ (11¯ 1)

¯ a/2[011] a/2[101] a/2[110]

1 1 0

1 1 1

0 0 0

0 2 1

1 2 1

1 1 0

(100) (001) (010)

¯ a/2[011] a/2[110] ¯ a/2[101]

1 0 0

0 1 0

0 0 1

0 1 −1

1 1 0

1 0 1

¯ (111) (111)

¯ a/3[111] a/3[111]

1 1/3

1/3 1

1/3 −1/3

0 4/3

4/3 4/3

4/3 0

(100) (100) (010) (010) (001) (001)

a/6[011] ¯ a/6[011] ¯ a/6[101] a/6[101] ¯ a/6[110] a/6[110]

0 1/3 0 1/3 1/3 0

1/3 1/3 0 1/3 0 1/3

−1/3 0 1/3 0 1/3 0

2/3 0 −1/3 2/3 −1/3 1/3

1/3 1/3 0 2/3 1/3 1/3

−1/3 −1/3 1/3 1/3 2/3 0

¯ T . Other dislocations of g . b values is unique to a dislocation with Burgers vector a2 [020] may exhibit similar contrast properties in different operating reflections as indicated by the absolute values of g . b in Table 3, making identification of their Burgers vectors more difficult. One must follow these defects from the beginning of deformation, since information related to their evolution, their spatial distribution, their glide planes, etc., is invaluable in understanding the dislocation products resulting from the twinning transformation.

8. TEM observations of the dislocation substructure prior to twinning The nature of the dislocation substructure developed during plastic deformation of single crystals of fcc metals and alloys has been the subject of numerous experimental and theoretical studies and has been broadly documented in the literature. The morphology, type, and spatial distribution of dislocations produced during deformation is strongly dependent upon the stacking fault energy of an alloy. Lower stacking fault energy materials such as

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Table 2b ¯ M g.b values for the dislocations in the parent in various reflections available in section parallel to cross (11¯ 1) plane Plane/Burgers vector ‘b’ ¯ 111

111

Operating reflection ‘g’ ¯ 111¯ 022

202

220

¯ (111) ¯ (111) ¯ (111)

a/2[011] a/2[101] ¯ a/2[110]

0 1 1

1 1 0

0 0 0

0 1 1

1 2 1

1 1 0

¯ (111) ¯ (111) ¯ (111)

a/2[011] a/2[110] ¯ a/2[101]

0 0 0

1 1 0

0 1 1

0 −1 −1

1 1 0

1 2 1

(111) (111) (111)

¯ a/2[011] ¯ a/2[110] ¯ a/2[101]

1 1 0

0 0 0

−1 0 1

2 1 −1

1 1 0

−1 0 1

¯ (11¯ 1) ¯ (11¯ 1) ¯ (11¯ 1)

¯ a/2[011] a/2[101] a/2[110]

1 1 0

0 1 1

−1 0 1

2 1 −1

1 2 1

−1 1 2

(100) (001) (010)

¯ a/2[011] a/2[110] ¯ a/2[101]

1 0 0

0 1 0

−1 1 1

2 −1 −1

1 1 0

−1 2 1

¯ (111) (111)

¯ a/3[111] a/3[111]

1 1/3

1/3 1

−1/3 1/3

4/3 0

4/3 4/3

0 4/3

(100) (100) (010) (010) (001) (001)

a/6[011] ¯ a/6[011] ¯ a/6[101] a/6[101] ¯ a/6[110] a/6[110]

0 1/3 0 1/3 1/3 0

1/3 0 0 1/3 0 1/3

0 −1/3 1/3 0 0 1/3

0 2/3 −1/3 1/3 1/3 −1/3

1/3 1/3 0 2/3 1/3 1/3

1/3 −1/3 1/3 1/3 0 2/3

for example the Cu-8%Al alloy studied in this work, develop a more planar dislocation arrangement, whereas higher stacking fault energy systems such as pure copper develop a characteristic three dimensional structure. Qualitatively the dislocation content is similar in these systems and includes features such as primary dislocations, secondary dislocations with a variety of Burgers vectors, primary unfaulted dipoles, faulted dipoles, Lomer– Cottrell networks, stacking fault tetrahedra and trihedra, and small defect clusters [63]. For details related to the formation, spatial distribution and the configuration of these entities, the reader is referred to the review articles by Amelinckx [69] and Basinski and Basinski [1] and the references available in these works in addition to the more recent work of Niewczas [63]. In the past, TEM studies carried out on deformed materials have usually been limited to stages of deformation where defects are relatively easily distinguishable. There have been very few high resolution TEM studies of the dislocation substructure developed at high strains, thus there is little detailed information on the state and morphology of the substructure at the onset of twinning. This section outlines the deformation substructure

§8.1

287


Table 2c g.b values for the dislocations in the parent in various reflections available in section parallel to critical (111)M plane Plane/Burgers vector ‘b’ ¯ 111

11¯ 1¯

Operating reflection ‘g’ ¯ 111¯ 022

202¯

¯ 220

¯ (111) ¯ (111) ¯ (111)

a/2[011] a/2[101] ¯ a/2[110]

0 1 1

−1 0 1

0 0 0

0 1 1

−1 0 1

−1 1 2

¯ (111) ¯ (111) ¯ (111)

a/2[011] a/2[110] ¯ a/2[101]

0 0 0

−1 0 1

0 1 1

0 −1 −1

−1 1 2

−1 0 1

(111) (111) (111)

¯ a/2[011] ¯ a/2[110] ¯ a/2[101]

1 1 0

0 1 1

−1 0 1

2 1 −1

−1 0 2

1 2 1

¯ (11¯ 1) ¯ (11¯ 1) ¯ (11¯ 1)

¯ a/2[011] a/2[101] a/2[110]

1 1 0

0 0 0

−1 0 1

2 1 −1

−1 0 1

1 1 0

(100) (001) (010)

¯ a/2[011] a/2[110] ¯ a/2[101]

1 0 0

0 0 1

−1 1 1

2 −1 −1

−1 1 2

1 0 1

¯ (111) (111)

¯ a/3[111] a/3[111]

1 1/3

1/3 −1/3

−1/3 1/3

4/3 0

0 0

4/3 0

(100) (100) (010) (010) (001) (001)

a/6[011] ¯ a/6[011] ¯ a/6[101] a/6[101] ¯ a/6[110] a/6[110]

0 1/3 0 1/3 1/3 0

−1/3 0 1/3 0 1/3 0

0 −1/3 1/3 0 0 1/3

0 2/3 −1/3 1/3 1/3 −1/3

−1/3 −1/3 2/3 0 1/3 1/3

−1/3 1/3 1/3 1/3 2/3 0

that exists in single crystals of copper alloys just before the onset of twinning: this is the structure that undergoes transformation when twinning begins.

8.1. Primary dislocations Fig. 10 shows two-beam bright field observations of the dislocation substructure in a Cu8%Al single crystal deformed at room temperature in foils cut parallel to the primary glide ¯ M . Micrographs taken in three 111M reflections available in this section alplane, (111) low identification of primary dislocations with Burgers vector a2 [101]M , shown by arrows ¯ and critical g = 111 in Fig. 10a. These defects are visible both in the conjugate g = 111 ¯ By examinreflections (Fig. 10a,b) and vanish in the cross-glide plane reflection g = 11¯ 1. ¯ ing values of g . b for the set of operating reflections available in the 111M foil given in Table 2, one sees that only an a2 [101]M dislocation satisfies this combination of visibility /invisibility conditions. The primary dislocations lie in the foil plane and exhibit strong

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Table 3a ¯ T plane g.b values for the dislocations in the twin in various reflections available in section parallel to (5¯ 11) ¯ M plane in the parent corresponding to section parallel to the primary (111) Plane/Burgers vector ‘b’

Operating reflection ‘g’ ¯ 111

¯ 020

002

022

¯ 022

¯ (200) ¯ (200) ¯ (200)

¯ a/2[01¯ 1] ¯ a/2[020] ¯ a/2[011]

0 1 1

1 2 1

−1 0 1

2 −2 0

0 2 2

¯ (111) ¯ (111) ¯ (111)

¯ a/2[01¯ 1] ¯ a/2[1¯ 10] ¯ a/2[101]

0 0 0

1 1 0

−1 0 1

−2 −1 1

0 1 1

¯ (1¯ 1¯ 1) ¯ (1¯ 1¯ 1) ¯ (1¯ 1¯ 1)

¯ a/2[110] ¯ a/2[011] ¯ a/2[101]

1 1 0

1 1 0

0 1 1

−1 0 1

1 2 1

(002) (002) (002)

¯ a/2[110] ¯ a/2[020] ¯ a/2[1¯ 10]

1 1 0

1 2 1

0 0 0

−1 −2 −1

1 2 1

¯ (1¯ 11) ¯ (11¯ 1) ¯ 1) ¯ (10

¯ a/2[110] ¯ a/2[1¯ 10] ¯ a/2[101]

1 0 0

1 1 0

0 0 1

−1 −1 1

1 1 1

¯ (111) ¯ (1¯ 1¯ 1) ¯ (200)

¯ a/6[141] ¯ a/6[1¯ 4¯ 1] ¯ 1] ¯ a/3[11

1 1/3 −1

4/3 4/3 −2/3

1/3 −1/3 −2/3

−1/3 −5/3 0

5/3 1 −4/3

¯ (1¯ 11) ¯ (1¯ 11) ¯ 1) ¯ (10 ¯ 1) ¯ (10 ¯ (11¯ 1) ¯ (11¯ 1)

¯ a/6[01¯ 1] ¯ a/6[110] ¯ a/6[101] ¯ a/6[020] ¯ a/6[011] ¯ a/6[1¯ 10]

0 1/3 0 1/3 1/3 0

1/3 1/3 0 2/3 1/3 1/3

−1/3 0 1 0 1/3 0

2/3 −1/3 1 −2/3 0 1/3

0 1/3 1 2/3 2/3 1/3

contrast, which is constant along the length of the dislocation line. These dislocations are collected into a complicated planar arrangement, as shown in Fig. 10. In the micrograph of Fig. 10d the direction of the primary Burgers vector coincides approximately with the trace ¯ M , the primary dislocations exhibit a range of orientations of the cross-glide plane, (11¯ 1) and there are many dislocations of pure screw or 30 degree character. Isolated line segments between nodes formed with other elements of this highly dislocated structure may extend to 2–3 micrometers. These observations also reveal the existence of short dislocation segments, shown by arrows in Fig. 10b, aligned along the intersection of the primary and conjugate planes. These defects, discussed in more detail later, are Lomer–Cottrell ¯ M Lomer dislocations formed by the interaction of networks and are identified as a2 [011] dislocations of the primary and conjugate systems, they are visible in the g = 111 and ¯ operating reflections in Fig. 10. The micrographs also show dislocations belongg = 111 ing to the primary glide system but with Burgers vector a2 [011]M coplanar with the primary Burgers vector, an example of this type of defect is shown by an arrow in Fig. 10c. These

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Table 3b ¯ T plane, g.b values for the dislocations in the twin in various reflections available in section parallel to (115) ¯ M plane in the parent corresponding to section parallel to the cross (11¯ 1) Plane/Burgers vector ‘b’ ¯ 111

¯ 020

Operating reflection ‘g’ ¯ 200 220

¯ 220

¯ (200) ¯ (200) ¯ (200)

¯ a/2[01¯ 1] ¯ a/2[020] ¯ a/2[011]

0 1 1

1 2 1

0 0 0

−1 −2 −1

1 2 1

¯ (111) ¯ (111) ¯ (111)

¯ a/2[01¯ 1] ¯ a/2[1¯ 10] ¯ a/2[101]

0 0 0

1 1 0

0 1 −1

−1 −2 −1

1 0 −1

¯ (1¯ 1¯ 1) ¯ (1¯ 1¯ 1) ¯ (1¯ 1¯ 1)

¯ a/2[110] ¯ a/2[011] ¯ a/2[101]

1 1 0

1 1 0

−1 0 −1

0 −1 −1

2 1 −1

(002) (002) (002)

¯ a/2[110] ¯ a/2[020] ¯ a/2[1¯ 10]

1 1 0

1 2 1

−1 0 1

0 −2 −2

2 2 0

¯ (1¯ 11) ¯ (11¯ 1) ¯ 1) ¯ (10

¯ a/2[110] ¯ a/2[1¯ 10] ¯ a/2[101]

1 0 0

1 1 0

−1 1 −1

0 −2 −1

2 0 −1

¯ (111) ¯ (1¯ 1¯ 1) ¯ (200)

¯ a/6[141] ¯ a/6[1¯ 4¯ 1] ¯ 1] ¯ a/3[11

1 1/3 1

4/3 4/3 −2/3

−1/3 1/3 2/3

−1 −5/3 0

5/3 1 −4/3

¯ (1¯ 11) ¯ (1¯ 11) ¯ 1) ¯ (10 ¯ 1) ¯ (10 ¯ (11¯ 1) ¯ (11¯ 1)

¯ a/6[01¯ 1] ¯ a/6[110] ¯ a/6[101] ¯ a/6[020] ¯ a/6[011] ¯ a/6[1¯ 10]

0 1/3 0 1/3 1/3 0

1/3 1/3 0 2/3 1/3 1/3

0 −1/3 −1/3 0 0 1/3

−1/3 0 −1/3 −2/3 −1/3 −2/3

1/3 2/3 −1/3 2/3 1/3 0

dislocations are visible both in g = 111 (Fig. 10b) and g = 11¯ 1¯ (Fig. 10c) and vanish in ¯ (Fig. 10a). g = 111 8.2. Lomer–Cottrell network Fig. 11 shows an example of a Lomer–Cottrell network in a Cu-4%Al single crystal de¯ M . The pronounced formed at 78 K: the foil was parallel to the primary glide plane (111) network of short dislocation segments aligned along 110M , directions is characteristic of these networks. There are two families of Lomer–Cottrell dislocations formed by the interaction of primary dislocations with those of the conjugate and critical slip systems. Independent analysis, not shown here, revealed that these dislocations exist in a low energy configuration formed when the primary and inclined planes meet at an acute angle with constituent dislocations bounding an intrinsic stacking fault. This configuration of

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Table 3c ¯ T plane, g.b values for the dislocations in the twin in various reflections available in section parallel to (1¯ 5¯ 1) corresponding to section parallel to the critical (111)M plane in the parent Plane/Burgers vector ‘b’ ¯ 111

002

Operating reflection ‘g’ ¯ ¯ 200 202

202

¯ (200) ¯ (200) ¯ (200)

¯ a/2[01¯ 1] ¯ a/2[020] ¯ a/2[011]

0 1 1

−1 0 1

0 0 0

−1 0 1

−1 0 1

¯ (111) ¯ (111) ¯ (111)

¯ a/2[01¯ 1] ¯ a/2[1¯ 10] ¯ a/2[101]

0 0 0

−1 0 1

0 −1 1

−1 1 2

−1 −1 0

¯ (1¯ 1¯ 1) ¯ (1¯ 1¯ 1) ¯ (1¯ 1¯ 1)

¯ a/2[110] ¯ a/2[011] ¯ a/2[101]

1 1 0

0 1 1

−1 0 1

−1 1 2

1 1 0

(002) (002) (002)

¯ a/2[110] ¯ a/2[020] ¯ a/2[1¯ 10]

1 1 0

0 0 0

−1 0 −1

−1 0 1

1 0 −1

¯ (1¯ 11) ¯ (11¯ 1) ¯ 1) ¯ (10

¯ a/2[110] ¯ a/2[1¯ 10] ¯ a/2[101]

1 0 0

0 0 1

−1 −1 1

−1 1 2

1 −1 0

¯ (111) ¯ (1¯ 1¯ 1) ¯ (200)

¯ a/6[141] ¯ a/6[1¯ 4¯ 1] ¯ 1] ¯ a/3[11

1 1/3 −1

1/3 −1/3 −2/3

−1/3 1/3 2/3

0 0 0

2/3 2/3 −4/3

¯ (1¯ 11) ¯ (1¯ 11) ¯ 1) ¯ (10 ¯ 1) ¯ (10 ¯ (11¯ 1) ¯ (11¯ 1)

¯ a/6[01¯ 1] ¯ a/6[110] ¯ a/6[101] ¯ a/6[020] ¯ a/6[011] ¯ a/6[1¯ 10]

0 1/3 0 1/3 1/3 0

−1/3 0 1/3 0 1/3 0

0 −1/3 1/3 0 0 −1/3

−1/3 −1/3 2/3 0 1/3 1/3

−1/3 1/3 0 0 1/3 −1/3

Lomer–Cottrell dislocations has been observed in all systems studied. In many places of the network L–C segments along 110M tend to curve, indicating that they may glide under the action of applied stress (e.g., Karnthaler [70]). Since it must involve rearrangement of the stacking fault, the process is expected to occur more easily in higher stacking fault energy materials deformed at higher temperatures. Thus the characteristic appearance of Lomer–Cottrell networks is rarely observed in copper deformed at room temperature to high strains but is a characteristic feature of the substructure in copper deformed at low temperature, e.g., 4.2 K. In crystals deformed to the onset of twinning, Lomer–Cottrell networks coexist with other elements of the dislocations substructure. Fig. 12 shows the substructure just before the onset of twinning in four reflections available in foils cut parallel to the primary glide plane in Cu-8%Al single crystals. A Lomer– Cottrell network is visible in the right hand side of the micrograph. There are extended stacking faults at the nodes of the network. Also, stacking fault ribbons arising from the dissociation of primary dislocations are clearly visible at the arrows in Fig. 12b. One can

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Fig. 10. Two-beam bright field observations of the dislocation substructure in primary glide plane foils from a Cu8%Al single crystal deformed at room temperature, taken in three g = 111M reflections. Primary dislocations ¯ M dislocation segments are shown by with Burgers vector a2 [101]M are shown by arrows in (a). Lomer a2 [011] arrows in (b) and (c).

follow the contrast properties of component Heidenreich–Shockley and stair-rod dislocations with the help of Table 2 using the arguments discussed in Section 6.2. 8.3. Faulted dipole The arrangement of faulted dipoles (also known as Frank dipoles) in the parent crystals is discussed with reference to Fig. 12. TEM observations of the dislocation substructure before twinning have revealed the presence of high densities of faulted dipoles in many materials. Examples of this structure, short dislocation segments in the lattice, are marked

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Fig. 11. Weak-beam observations of a Lomer–Cottrell network in a Cu-4%Al single crystal deformed at 78 K, foil cut parallel to the primary glide plane, taken in two opposite g = 111 reflections. Short LC dislocation segments aligned along 110M directions are characteristic of these networks. For better resolution, the weakbeam micrograph is printed as a negative.

FD in Figs 12a and 12d. Previous studies established that the majority of these faulted dipoles are of vacancy type and are in “Z configuration”. These structures are composed of two a6 112M Shockley dislocations of opposite sign on primary planes, two a6 110M stair-rod dislocations of opposite sign at the acute intersection of the primary and inclined plane, and three intrinsic stacking faults bounded by partial dislocations [71–77]. Faulted dipoles are derived from dissociation of narrow primary 60◦ unfaulted dipoles, which give rise to two families of dipoles lying in the primary glide plane aligned along the two 110M directions not parallel to the primary Burgers vector [72–78]. These two families of faulted dipoles are clearly visible in Fig. 12. Dipoles running along 011M , the direction of inter¯ M . These section of the primary and conjugate planes have a total Burgers vector of a3 111 ¯ defects are visible only in the conjugate plane reflection g = 111 in Fig. 12a. The second ¯ M , the intersection of the primary and critical planes, family of dipoles, aligned along 110 a have a Burgers vector of 3 111M . These dipoles are visible in g = 1¯ 1¯ 1¯ and g = 02¯ 2¯ in

§8.4


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Fig. 12. Weak-beam micrograph of the substructure in a Cu-8%Al single crystal before the onset of twinning (i.e. parent, matrix, or untwinned structure) in four reflections available in sections parallel to the primary glide ¯ M plane. Lomer–Cottrell network can be seen in the right hand side of the micrograph. Faulted dipoles, a3 [111] ¯ M , are marked by FD in (a) and (d). aligned along 011M , and a [111]M aligned along 110 3

Figs 12b and 12d. Both dipole families vanish in g = 11¯ 1¯ in Fig. 12c. Faulted dipoles in highly deformed material become very refined, ranging from small loops to dipoles rarely exceeding 1 micron in length (Fig. 12). The detailed geometry and contrast properties of faulted dipoles formed in copper single crystals have been discussed in detail elsewhere [63,64]. 8.4. Stacking fault tetrahedra (SFT) and small defect clusters Stacking fault tetrahedra (SFT) have been observed in high densities in the substructure of a range of deformed metals and alloys. Depending upon the stacking fault energy of the system, the size of these defects in highly deformed material ranges from 2–20 nm

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(a)

(b)

(c)

Fig. 13. (a) Weak beam observations of the substructure in Cu-4%Al single crystals deformed at 78 K to the onset of twinning. Stacking fault tetrahedra (SFT) are recognized as small triangles or incomplete triangles, with characteristic fringes produced by stacking faults on {111}. Short segments of Lomer–Cottrell dislocations aligned along two 011M directions are visible in various parts of the micrograph. (b) and (c) show high resolution TEM observations of small defect debris marked by arrows.

[63,79]. Because of their small size they are often not detected under strong beam diffracting conditions. SFTs can be formed by various mechanisms including both conservative and non-conservative motion of dislocations. These defects have been investigated both experimentally and theoretically by a number of authors who provided early TEM observations or dealt with theoretical considerations regarding their formation or stability [80–84]. Fig. 13a shows SFTs formed in Cu-4%Al single crystals deformed at 78 K to the onset of twinning. These defects are recognized in TEM micrographs as small closed or unclosed triangles, with characteristic fringes produced by stacking faults on {111} visible in larger SFTs. In general SFTs coexist with all other elements of the dislocation substructure. In the structure shown in Fig. 13, these defects are seen in the neighborhood of short segments of

§9


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¯ M and distributed in various Lomer–Cottrell dislocations aligned along [011]M and [110] places in the network. Their size usually does not exceed 20 nm. The figure shows that the overall density of dislocations is high. Apart from small SFTs, TEM observations also show a high density of dot-like defects of dimension below 2 nm. These are either SFTs, faulted loops, or clusters of point defects, such structures are observed in all regions of the dislocated lattice [63]. Together with other very small entities these defects contribute to the fine debris structure visible with higher resolution TEM techniques. Figs 13b and 13c are high resolution electron micrographs showing examples of these defects, marked by arrows, in Cu-4%Al single crystals deformed to the onset of twinning at 78 K.

9. Twinning transformations of dislocations 9.1. Geometry of twinning transformation in a dislocated lattice When twinning occurs, the relation between crystallographic parameters in the twinned and untwinned material is succinctly given by the reindexation matrix and the correspondence matrix (Section 5, Section 7.1). These relations apply to all twins in an fcc lattice irrespective of how they are formed, the presence of twins can thus be verified by TEM analysis of the crystallographic parameters in different regions of a specimen. The crystallographic relations show that, for example, a 100M foil normal in the matrix corresponds to a 221T foil normal in the twin, a 111M direction in the matrix will become 511T in the twin, and 110M directions which do not lie in the habit plane will become 411T in the twin, etc. (Section 7.1). In the case of the single crystals being considered here, the twinning shear is polarized, ¯ M plane twinning occurs only in the [121]M direction and not in in the conjugate (111) ¯ M direction or any other 112M direction. This determines the way in which the the [1¯ 2¯ 1] crystallographic planes and the features of the dislocation substructure are transformed by twinning. From a TEM analysis of the spatial orientation and distribution of features in the parent and twinned lattices, Basinski and co-workers [29] showed that indeed the geometry of the twinning shear follows the predictions of the correspondence matrix, whereas the mutual orientation relationship between parent and twin is described by the reindexation matrix, as discussed previously. Figs 14a and 14b are multibeam bright-field micrographs of the dislocation substructure in both parent and twin lattices in a Cu-8%Al crystal. The {111} glide planes in the parent and their transformed products in the twin are easily visible due to the high densities of dislocations associated with them. The insert, an atomistic model of the distorted Thompson tetrahedron, is included to facilitate interpretation of these micrographs. In Fig. 14a the foil was cut parallel to the parent critical plane (111)M , i.e. BCGF in the atomistic model. With appropriate tilting of the foil one can reveal features of the dislocation substructure in both ¯ M plane, on which the twinned and untwinned lattices. Apparently, the trace of the (111) ¯ M , to become the primary dislocations lay, changes across the matrix-twin interface, (111) ¯ (200)T in the twin. The atomistic model shows that indeed these traces are produced by dislocations associated with the primary ABEF plane of the parent and dislocations on

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(a) Fig. 14. Multibeam TEM observations of the dislocation substructure in both parent and twin lattices in a Cu8%Al crystal, showing the effect of the twinning shear on the spatial distribution of dislocations associated with {111}M close-packed planes in the parent. The distorted Thompson tetrahedron is included to facilitate analysis of various features of the substructure. (a) Section parallel to the parent critical plane (111)M , BCGF in the model. (b) Section parallel to the parent cross glide plane, ACEG in the model. After Basinski and co-workers [29].

EFD’ in the twin. The intersection of these two planes with the BCGF plane of the foil under given diffraction conditions produces traces approximately along AE and ED’in the parent and twin respectively, as marked in Fig. 14a. Observations from other foil orientations yields a complete set of data on the crystallographic relation between twinned and untwinned material. Fig. 14b shows multibeam bright-field TEM observations from a foil parallel to the parent cross glide plane, ACEG in the atomic model, after suitable tilting. The matrix-twin interface is oriented approximately horizontally on the micrograph and is distinguished by the presence of stacking fault fringes on the K1 plane. The parent and twin crystals, both of thickness exceeding 1 micrometer, are separated by this interface. Transformation of the primary glide plane ¯ T is obvious from the micrograph. Also, the critical plane BCFG marked ¯ M to (200) (111) by a high density of dislocations, is rotated to FGD’ in the twin. The transformation of the ¯ T in the twin is expected from the corresponcritical (111)M plane of the matrix to (1¯ 1¯ 1) dence matrix. In summary, the TEM observations confirm that the spatial arrangement of dislocations resulting from twinning is in accord with the predictions of the theory dis-

§9.1


297

(b) Fig. 14. (Continued.)

cussed in Section 5, however detailed resolution of individual defects is impossible using multibeam diffraction conditions due to blurring [65]. The complexity of the dislocation substructure resulting from the twinning transformation can be appreciated by inspecting micrographs obtained using weak-beam diffracting conditions. Fig. 15 shows the defect structure in a twinned region of a Cu-8%Al single crystal, the plane of the foil is parallel to the cross glide plane of the parent, i.e. the foil ¯ operating reflection is marked by ¯ T in the twin, the weak-beam g = 020 normal is [115] an arrow. The most notable feature which distinguishes untwinned from twinned material ¯ interface. is the presence of long planar layers of stacking faults extended along the (111) These defects are visible on the micrograph as fringes aligned along the intersection of the twin plane and the plane of the foil. The crystals are never fully converted to the twinned configuration: some matrix structure always remains [85]. The stacking faults in Fig. 15 represent layers of untransformed matrix left within the twinned material, their thickness ranges from one to a few atomic layers. Other parts of the twinned sample may contain matrix remnants of macroscopic size. In the case of the substructure shown in Fig. 15 the extent of the homogeneously twinned material separated by the interfaces is on average between 0.3 µm and 0.5 µm. Thus there are volumes of material with a high density of defects, inherited from the parent, that are available for Burgers vector analysis. The weakbeam TEM observations confirmed previous findings on the geometry of the dislocation

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¯ T Fig. 15. Weak-beam observations of the dislocation substructure in a twinned Cu-8%Al single crystal in [115] ¯ The micrograph reveals a complex, highly dislocated, dislocation substructure, the spatial dissection, g = 020. tribution of these defects is in accord with the predictions of the correspondence matrix. Arrangements of disloca¯ T , (1¯ 1¯ 1) ¯ T and in (200)T inherited from dislocations in parent crystal are marked on the micrograph. tions in (111)

network inherited from the matrix. They clearly show that there is a large density of dis¯ T and (1¯ 1¯ 1) ¯ T planes, as marked on the micrograph. There is also a locations in the (200) ¯ interfaces, and a high large number of dislocations deposited or associated with the (111) density of dot-like defects distributed throughout the volume of the twinned crystal. In the

§9.2


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upper right corner of the micrograph, extended stacking faults initiated at the interfaces are visible on planes other than the interface planes. In the homogeneous twin widely dissociated dislocations transformed from parent dislocations are visible. These defects are analyzed in detail in a later part of this work. However, in précis, the expected spatial configuration of the dislocation substructure inherited by twins from the parent crystal is completely compatible with the theoretical predictions.

9.2. Transformation of primary dislocations 9.2.1. TEM observations The dislocation substructure in the twin which originates from primary dislocations in the matrix is expected to be in the form of cube dislocations, i.e. dislocations which inherit an a ¯ 2 [020]T Burgers vector and are deposited on (100)T planes. To study the planar arrangement of these dislocations TEM foils were cut parallel to (100)T . Figs 16a and 16b show two-beam bright-field observations of two areas in a twinned Cu-8%Al crystal containing a higher and a lower density of transformed defects respectively. Both micrographs were ¯ operating reflection, i.e. with the twinning plane set in the diffracttaken with the g = 111 ing position. It is useful for the reader to compare this structure with that in Fig. 10a which shows the arrangement of primary dislocations before the onset of twinning imaged under the same diffracting conditions when the twinning plane is set “edge-on”. Fig. 16 reveals the rather complicated arrangement of transformed dislocations. The effect of the twinning shear applied to this structure is clearly visible from the directional nature of the transformed defects. Homogeneous twins separated by interfaces are on average 50 nm–100 nm thick and they cut the original dislocation lines into shorter pieces, so that the transformed dislocations appear to be more refined than the primary dislocations. In a highly dislocated structure cube dislocations exist in a spectrum of orientations, they are often contained within transformed tangles and therefore are difficult to distinguish from other elements of the substructure inherited by twin. However, observations carried out on less dislocated regions of the crystal allow these defects to be precisely identified, examples of cube dislocations are marked by arrows in Fig. 16b. These dislocations exhibit relatively strong contrast along the dislocation line, which indicates that they are aligned in the plane of the foil, i.e. in the (100)T plane and that they do not dissociate into partial dislocations. This property is discussed later. The TEM observations also reveal a high density of defects associated with interfaces oriented edge-on and marked as fine sharp lines running perpendicular to the direction of the operating reflections. These defects are discussed in more details in Section 10.2. Secondary dislocations are contained in the dense dislocation tangles visible in the micrographs: they contribute to the development of the dimensional dislocation substructure in the twin. To determine the Burgers vectors of transformed defects, Basinski and co-workers carried out weak-beam studies of the twinned substructure in foils cut parallel to the critical glide plane of the parent. As discussed in Section 6, the diffraction conditions available with this foil orientation allow unambiguous identification of cube dislocations. Fig. 17 shows weak-beam observations of the transformed substructure in three reflections available with this foil orientation. Dislocations marked by arrows in Fig. 17a and enlarged in

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Fig. 16. TEM bright-field observations of the dislocation substructure in a (100)T section from a Cu-8%Al single crystal, areas containing (a) a higher and (b) a lower density of defects. Transformed segments of primary dislocations are shown by arrows.

§9.2


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Fig. 17. Weak-beam observations of the dislocation substructure in Cu-8%Al in three reflections available in ¯ T sections parallel to the critical plane of the parent. Dislocations marked by arrows in (a) and enlarged in [1¯ 5¯ 1] ¯ T (e) and (f) are a2 [020]T cube dislocations. Dislocations marked by arrows in (b) and enlarged in (d) are a2 [01¯ 1] conjugate dislocations on the twinning plane. After Basinski et al. [29].

¯ and vanish when g = 002¯ and g = 200. Only Figs 17e and 17f are visible only with g = 111 a dislocations with Burgers vector 2 [020]T satisfy these invisibility conditions. Careful inspection of the contrast properties also shows that cube dislocations are not dissociated in their plane. Other defects, marked by arrows, are also evident in Fig. 17b, they are visible only in g = 002 and vanish in other operating reflections. The nature of the oscillatory contrast of one of these defects, visible under higher magnification in Fig. 17d, reveals that ¯ T Burgers ¯ twinning plane. These dislocations thus have an a [01¯ 1] they belong to the (111) 2 a vector (Table 3c) and are transformed from 2 [011]M conjugate dislocations in the matrix.

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9.2.2. Geometrical considerations In the case of mechanical twinning, the geometry of the transformation of the Burgers vector and the dislocation line direction is expressed concisely by the correspondence matrix. During transformation every individual atom in the parent lattice, including all atoms within the dislocation core, is sheared to its new position by the same fraction of interatomic spacing, i.e. by the Burgers vector of the twinning dislocation, the continuity of a dislocation line is therefore preserved in the twinned lattice. The sequence of events which may occur during the shearing process is shown in Fig. 18 for the same model as that in Fig. 2a. Fig. 18a shows schematically a dislocation line marked by a black arrow in 60◦ orientation with respect to the a2 [101]M Burgers vector which forms the edge of the model. During the initial intersection by the twinning dislocation a short jog equal to the length of its Burgers vector is formed on the primary dislocation (Fig. 18b). As the transformation proceeds and subsequent layers are sheared, the matrix-twin interface moves and leaves behind a transformed segment of dislocation line between nodes N1 and N2 . At some stage of twin propagation, part of the dislocation may still exist in the old matrix while a transformed portion will be incorporated into the twin (Fig. 18c). Eventually a long segment of dislocation line is deposited in the twin lattice and the dislocation acquires a new Burgers vector. The physical basis for the transformation of a Burgers vector of any dislocation by twinning is its reaction with the transforming (twinning) dislocation, in this case a Heidenrich– Shockley dislocation with Burgers vector a6 [121]M . One can predict the Burgers vector of the product using simple vector algebra, as in any dislocation reaction, by addition and/or subtraction of the Thompson vectors in an fcc lattice [86]. As discussed above, a twin-transformed dislocation must be incorporated into the twin lattice (Fig. 18c), thus its Burgers vector must be reindexed with reference to the new coordinate system. This is indicated schematically in Fig. 18d showing that addition of the Burgers vectors of a primary a2 [101]M dislocation and an a6 [121]M twinning dislocation results in formation of an a6 [424]M dislocation. This resultant Burgers vector when reindexed with respect to the ¯ T , marked on the model. Similarly, a primary dislocation a [101]M twin lattice gives a2 [020] 2 a ¯ will become 6 [141]T when expressed in twin co-ordinates, as shown in Fig. 18d. Reindexation of a vector does not change its length, the same line is merely expressed in two ways, ¯ T are the same, as are a [424]M it is easily verified that the lengths of a2 [101]M and a6 [141] 6 a ¯ and 2 [020]T . To better understand the dislocation reactions occurring during transformation of other defects to be discussed later, it is important to examine in more detail the dislocation reactions that occur at nodes N1 and N2 during transformation of primary dislocations (Fig. 18c). For dislocations meeting at a node the total Burgers vector is zero [87]. If all dislocations entering or leaving the node are signed consistently, the Frank node equation is written ni=1 bi = 0, where bi are Burgers vectors of n dislocations meeting at the dislocation node. Throughout this work we use the convention that all dislocations entering the node are positive and those leaving the node are negative. This of course will affect the way Burgers vectors are indexed and marked in the figures. In Fig. 18c, at the node N1 the primary dislocation a2 [101]M is entering the node and is positive, the twinning dislocation a 6 [121]M is also positive and the product of the dislocation reaction, i.e. the transformed dislocation which leaves the node is defined as a6 [424]M . The same dislocation entering

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(d) Fig. 18. Representation of the transformation of dislocations on a close-packed plane of the matrix. The black arrow marks a dislocation line oriented at 60◦ to the a2 [101]M Burgers vector along the edge of the model. During the intersection process the a6 [121]M twinning dislocation operates in sequence on every {111} plane ¯ T dislocation lying on transforming the a2 [101]M dislocation segment between nodes N1 and N2 into an a2 [020] a (200)T plane in the twin (a)–(c). The physical basis for the transformation of a Burgers vector is the reaction between the Burgers vector of the transforming (twinning) dislocation with that of the cut dislocation.

¯ M . The node equation at a dislocation node N1 is the node N1 would be indexed as a6 [4¯ 2¯ 4] written: a a a a ¯ [101]M + [121]M = [424]M = [020] T. 2 6 6 2

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With respect to the twin lattice this equation can be written: a ¯ a ¯ T = a [020] ¯ T = a [424]M . [141]T + [1¯ 2¯ 1] 6 6 2 6 At node N2 , the transformed dislocation is entering the node and is positive, a6 [424]M . The primary and twinning dislocations are leaving the node and, according to our convention, ¯ 1] ¯ M and a [1¯ 2¯ 1] ¯ M respectively (Fig. 16c). The node equation are therefore negative, a2 [10 6 at N2 with respect to the parent lattice can be written as follows: a a ¯ ¯ a ¯ M, [424]M = [10 1]M + [1¯ 2¯ 1] 6 2 6 and with respect to twin lattice: a ¯ a ¯ ¯ a [020]T = [14 1]T + [121]T . 2 6 6 9.2.3. Dislocation modeling The dislocation mechanisms by which long segments of primary dislocation, after interacting with twinning dislocations, acquire a new Burgers vector and are incorporated into the twin lattice will now be discussed. Various dislocation models for the nucleation and growth of twins in fcc and other structures have been reviewed by Christian and Mahajan [21]. Here we describe one particular mechanism, known as the pole mechanism, which with one unified model is able to illustrate important geometrical and dislocation aspects of the transformation process. Cottrell and Bilby (1951) [88] noticed long ago that when a forest dislocation has a component of its Burgers vector normal to the twinning plane and equal to the spacing of these planes, there is a screw-like arrangement of atoms around the dislocation line. In the present case an a2 [101]M primary Burgers vector has an a ¯ ¯ ¯ 3 [111]M component equal to the spacing of the (111) planes normal to the (111) twinning plane. The primary dislocation, which acts as a pole dislocation in this configuration, produces helical-like distortion of the twinning plane around the dislocation line, as shown in Fig. 19a. This topology facilitates the spiraling movement of the twinning dislocation around the pole dislocation during the intersection process and allows macroscopic twins to develop [21,88]. Larger numbers of closely spaced primary dislocations of different sign will produce rather complicated distortions of the twinning plane in their neighborhood. The glide planes are no longer flat as they are in the defect-free lattice, but are organized in a staircase-like arrangement giving an unlimited number of forest dislocations which may act as poles for the twinning dislocations throughout the dislocated crystal. ¯ M Consider a primary a2 [101]M dislocation dissociated into two Shockley partials a6 [211] a ¯ and 6 [112]M in the primary glide plane (111)M . The sequence of events which occurs during intersection with a twinning dislocation is shown in Figs 19b–19d [46]. The twinning dislocation propagates on the twinning plane, extends the stacking fault, and eventually interacts with the dissociated primary dislocation. Following the theory of Wessels and Nabarro [89,90], which deals with aspects of unstable glide during the intersection of forest dislocations dissociated into Shockley partials by glide dislocations, Szczerba suggested

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Fig. 19. (a) Spiral topology of the twinning plane around a forest dislocation with a Burgers vector component equal to the lattice spacing. The approximate orientation of the dislocation line is shown by a black arrow in figure (a). (b)–(d) A twinning dislocation intersecting a primary dislocation which is dissociated into Shockley partials, after Szczerba [46]. (b) At the point of intersection the primary dislocation reassociates forming a perfect dislocation. (c) As the twinning arms wind up around the primary pole the twinning partial reacts with the primary dislocation on every glide plane. The dislocation segment between the two nodes is fully incorporated into the ¯ T Burgers vector. twin lattice inheriting a new a2 [020]

[46] that as the twinning partial approaches the primary dislocation it leads to constriction of the two partials, so that the primary dislocation reassociates to form a perfect dislocation at the point of intersection and thus removes the stacking fault in the primary glide plane (Fig. 19b). This eliminates the necessity for transformation of the stacking fault and simplifies the process of incorporation of the dislocation line into the twin lattice. The formation of such a constriction is expected to be energetically favorable. Thompson [91] and Stroh [92] have analyzed the energy of an extended jog versus a contracted jog which might be formed during the intersection of non-coplanar dislocations, and have shown that

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it depends strongly upon the ratio of the stacking fault width to the length of the Burgers vector. For fcc systems considered here, the energy of an extended jog is higher than that of a contracted jog. The primary dislocation acts as a pole dislocation for the twinning partial and after being constricted its transformation proceeds in the same way as that discussed with reference to the geometrical model in Section 9.2.2 and Fig. 18. The spiral nature of the twinning plane around the primary dislocation will allow the twinning dislocation to move in such a way that one arm winds up and the other arm winds down around the pole (Fig. 19c). The small jog of length equal to the Burgers vector of the twinning dislocation formed on the primary dislocation during the intersection event shown in Fig. 18b is the first elementary segment of the dislocation line to be incorporated into the twin lattice. This jog also acquires a new Burgers vector, to be discussed later. At a certain stage during the first revolution around the pole the two arms of the twinning dislocation of opposite sign need to pass each other on two adjacent planes if their spiraling movement is to continue, as shown in Fig. 19c. Assuming there is no energy barrier for the passing process, further helical-like movement of the twinning dislocations around the pole dislocation will lead to the growth of a macroscopic twin and two matrix-twin interfaces will propagate. After every revolution, the twinning partial reacts with the primary dislocation. As the twinning arms move away, two nodes N1 and N2 are formed and move along the constricted pole dislocation in accord with the propagating interfaces. Continuation of this process leads to the development of a macroscopic twin by the pole mechanism. The dislocation segment between the two nodes is fully incorporated into the twin lattice and is described by a new a ¯ 2 [020]T Burgers vector. The equation which is satisfied at node N1 can be written with reference to the parent lattice as: a ¯ a a a a ¯ [211]M + [112]M + [121]M = [424]M = [020] T. 6 6 6 6 2 By analogy, the equation at node N2 reads: a a ¯ a a [424]M = [211] M + [112]M + [121]M . 6 6 6 6 As mentioned above with reference to Fig. 19c, during the first revolution around the pole the spiraling arms of the twinning dislocation need to pass each other on neighbouring atomic planes for twin growth to continue. In real crystals the energy barrier is too large for such a process to occur and the two arms of the twinning dislocation could not pass. The mechanism would then produce only one layer of stacking fault, whereas TEM observations of fcc materials show homogeneous twins of thickness often exceeding 1 µm, with a high density of incorporated defects. Szczerba [46] has put forward a very neat and simple mechanism which overcomes this perceived problem. He suggested that the passing stress problem can be solved by considering the interaction of primary dislocations with a wall of two Shockley dislocations. Fig. 20 shows the intersection of the primary dislocation with two twinning partials bounding stacking faults on their glide planes [46]. For simplicity a primary perfect dislocation is considered. In terms of the movement of

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Fig. 20. The intersection of a primary dislocation with two twinning partials bounding stacking faults on their glide planes [46]. The lower arm of the first (upper) Shockley partial is sweeping the same physical plane as the oppositely signed upper arm of the second (lower) Shockley partial; these two partials will locally annihilate at the jog and will restore the continuous dislocation line which propagates further on the twinning plane. The upper and lower arms of the twinning dislocation anchored at nodes N1 and N2 are now two interplanar spacings apart; the passing stress barrier between two Shockley dislocations spiralling around pole dislocation is thus reduced (after Szczerba [46]).

individual Shockley dislocations the process is analogous to that discussed above, however after the first revolution the lower arm of the first (upper) Shockley partial is sweeping the same physical plane as the oppositely-signed upper arm of the second (lower) Shockley partial. These two partials will locally annihilate at the jog and will restore the continuous dislocation line, which under the action of applied stress can propagate freely and extend the stacking fault in this plane. The two remaining arms of the twinning dislocation anchored at nodes N1 and N2 are now two interplanar spacings apart. The energy barrier for passing these two dislocations is of the order of 0.05 eV per unit length of Burgers vector which can be overcome by thermal energy [64]. If a larger number of twinning dislocations is involved in the process this will further reduce the energy barrier for the passing of the outermost dislocations in the wall. Thus it is evident that there is no geometrical difficulty and that the pole mechanism can transform long segments of primary dislocation and incorporate them into macroscopic twins. Chiu and co-workers [93] used this concept in the analysis of the transformation of a2 110 and a011 dislocations in gamma TiAl and to explain thickening of the twin in this system. As is well known, changing the sense of the Burgers vector of a primary dislocation will change the sense of spiraling around the pole, but will not affect the growth of the twin. This was discussed in detail by a number of researchers in relation to various aspects of nucleation and growth of twins by the pole mechanism [64,94–98]. Clearly, the transformation of a long dislocation segment can be effectively accomplished during intersection of this dislocation with an incoherent twinning front consisting of a large number of partial dislocations of the same sign, one on top the other, gliding ¯ M . Such twinning fronts have been observed during the onset of twinning in Cuon (111) 8%Al single crystals [64]. Inspecting the plane topology in the neighborhood of the pole dislocation more closely (Figs 21a and 21b) leads to the conclusion that as the incoherent

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Fig. 21. Diagram showing the detailed interaction of a twinning dislocation with an a2 [101]M forest dislocation. (a) Twinning dislocation marked “upper” approaches the a2 [101]M dislocation segment shown as a black line. ¯ T between the nodes. (c) A twinning dislocation (b) Reaction of two dislocations leads to the formation of a2 [020] gliding on an adjacent plane, marked “lower”, restores the continuity of the dislocation on the “lower” twinning plane by exchanging arms of two chopped upper and lower dislocations in the twinning front. (d) The twinning ¯ T. dislocation propagates further leaving behind a segment of primary dislocation transformed to a2 [020]

twin wall intersects and reacts with the primary dislocation on every glide plane along the dislocation line, two processes may occur simultaneously. There will be local reaction of the Burgers vectors of the primary and twinning dislocations, also exchange of part of the dislocation line between the chopped upper and lower dislocations in the wall, as shown schematically in Figs 21c and 21d. This restores continuity of every twinning partial in the wall, with the exception of the two outermost partials, which define the position of the matrix-twin interfaces. As a consequence the continuity of a twinning front is fully restored and it can propagate further under applied stress as a wall of n − 2 twinning dislocations, leaving behind a segment of primary dislocation incorporated in the twin lattice ¯ T Burgers vector. If the primary dislocation is not dissociated, which now has an a2 [020] as shown schematically in Fig. 22, it will be effectively transparent to the twinning front, which could then propagate easily through a large number of dislocations distributed in the

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Fig. 22. Schematics of the transformation of a primary a2 [101]M dislocation by an incoherent twinning front. Spontaneous transformation of the defect occurs by local reaction of the Burgers vectors of primary and twinning dislocations followed by an exchange of arms between the chopped upper and lower dislocations in the front. ¯ T in This restores the continuity of the twinning front leaving behind a segment of primary dislocation, a2 [020] the twin. Circles correspond to nodes N1 and N2 in previous figures.

lattice, causing spontaneous transformation of these defects. If the primary dislocation is dissociated it would be expected that the twinning wall will first cause association of two partials to form a perfect dislocation before the front can continue through the defect. Whether any of these mechanisms is likely to operate in a real lattice and what kind of atomic rearrangements occur remains to be established by appropriate atomistic simulations. However, the TEM observations discussed in Section 8.2.1 show that a high density of primary dislocations becomes incorporated into the twin lattice and that many of these ¯ T Burgers vector and (100)T glide plane, dislocations have cube configuration, i.e. a2 [020] and that they are not dissociated in the twin lattice. This suggests that the processes discussed above are likely to operate in real crystals. It will be shown latter in section 10 that ¯ T Frank dislocation by appropriate primary dislocation may also be transformed to a3 [111] reaction with the twinning partial.

9.3. Transformation of Lomer–Cottrell (LC) dislocations Correspondence matrix analysis indicates that twinning should result in the transformation of sessile Lomer dislocations in the parent into glissile dislocations in the twin [29,46]. ¯ M , a [110]M , and This is reflected in Table 1 which shows that the Burgers vectors a2 [011] 2 a ¯ 2 [101]M of Lomer dislocations lying on (100)M , (001)M and (010)M planes are trans¯ T , a [1¯ 10] ¯ T , and a [101] ¯ T lying on formed to dislocations with Burgers vectors a2 [110] 2 2 ¯ ¯ ¯ ¯ ¯ ¯ (111)T , (111)T and (101)T planes respectively. In particular, the transformation of the ¯ M to a [110] ¯ T , and a [110]M to a [1¯ 10] ¯ T are of interest because Burgers vectors a2 [011] 2 2 2 they are prominent features of the dislocation substructure in a range of copper alloy crystals, as discussed in Section 8.2. The present section deals with TEM observations of these

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dislocations in parent and twinned material in single crystals of two materials of different stacking fault energy, namely pure copper and Cu-8%Al. Dislocation modeling of these processes is also introduced. 9.3.1. TEM observations of LC networks in twinned crystals It was shown elsewhere [62,85] that the dislocation substructure of high purity copper deformed to very large strains becomes very refined, the segments of LC dislocations found in the highly dislocated substructure usually do not exceed 50 nm in length [63]. At high stresses, and with the additional thermal energy available in materials deformed at room or higher temperatures, these dislocations can glide on {100} planes, the LC networks then lose the characteristic lattice-like appearance observed after low temperature deformation, shown for example in Fig. 11. The present work was therefore carried out on single crystals deformed at very low temperatures, where this easily-identifiable structure is relatively stable and can be recognized in the presence of other defects. Fig. 23a is a weak-beam micrograph of a LC network lying on the primary glide plane in a deformed copper single crystal before the onset of twinning at 4.2 K. Fig. 23b, from a crystal deformed into the twinning region, shows the substructure transformed by twinning. LC dislocations are shown by arrows in Fig. 23a and it can be seen that in a few places these dislocations have moved out of their initial positions along 110, indicating that they may glide under high stresses on {100} planes even at very low temperatures. The primary dislocations react with secondary dislocations to form LC networks, which in this case are parallel to the foil and exhibit strong contrast along the dislocation line. There are also conjugate dislocations visible in this network that give rise to oscillating contrast because they are inclined to the foil. Both primary and secondary (conjugate) dislocations form nodes with LC segments. The observations show that more LC segments are formed at the intersection of primary planes with conjugate planes than with critical planes, at least in this volume of the crystal. As the network undergoes twinning transformation, the substructure becomes increasingly more refined and some defects are chopped by propagating twinning dislocations, so that much dislocation debris in the form of dot-like defects is left in the material, as seen in Fig. 23b. The TEM observations also reveal that, in addition to fine debris, there are dislocation segments lying along the trace of the interface, shown by arrows in Fig. 23b. The length of these entities is comparable to the length of LC dislocations formed at the intersection of primary and conjugate planes in the parent crystal (Fig. 23a), they give rise to strong fringe-like contrast not observed before. A higher magnification micrograph of one of these defects is shown by an arrow in Fig. 24. The overall configuration of the defect inherits the shape of the Lomer lock from which it was formed, it includes two arms, marked by dagger-like symbols, and a central fringe-like part approximately 30 nm in length, marked by an arrow. The fringe-like con¯ T resulting from dissocitrast properties of this defect arise form the stacking fault on (1¯ 11) ation of a perfect dislocation into Shockley partials in this plane, which, in turn, was derived from the transformation of the parent LC lock. These “unlocked” LC dislocations might ¯ T except that they are pinned by immobile glide under the action of applied stress on (1¯ 11) arms, the whole defect configuration therefore reproduces the LC shape which makes it readily identifiable in the substructure. The two dislocation arms have a2 [020]T Burgers vectors since they are transformed from primary dislocations. As they lie on (100)T , they

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(b) Fig. 23. Weak-beam observations of a Lomer–Cottrell network in a copper single crystal deformed at 4.2 K (a) ¯ the operating reflection of the twinning before the onset of twinning, and (b) into the twinning region. g = 111, plane. Section parallel to the primary glide plane. LC dislocations in the matrix and transformed LC dislocations in the twin are shown by arrows in (a) and (b) respectively. Printed as negatives for better resolution.

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Fig. 24. Higher magnification view from Fig. 23b. The transformed LC dislocation, marked by an arrow, is pinned by nodes formed from sessile arms of as-transformed primary dislocations shown by dagger symbols. Printed as negative for better resolution.

are slightly inclined to the foil in this cross-section and therefore exhibit oscillatory contrast with longer periodicity (compare Fig. 23a). Details of the transformation reactions of the LC configuration will be discussed later. To investigate further the defects transformed from LC dislocations TEM studies were carried out on Cu-8%Al crystals deformed at room temperature. This material has a lower stacking fault energy than that of pure copper, stacking fault ribbons arising from dissociation of perfect dislocations are therefore wider, thus facilitating identification of these defects in twinned material. Fig. 25 shows weak-beam observations of the twin substructure in Cu-8%Al in a section parallel to the cross-plane of the parent, which corresponds to ¯ T foil normal in the twin, the operating reflection of the twin lattice is g = 020. ¯ The a [115] matrix-twin interfaces, marked by stacking fault fringes, run from the bottom right to the top left of the micrograph. The substructure is filled with short defects of length approximately 40–50 nm represented by contrast fringes not parallel to the interface fringes and therefore associated with stacking faults on other than the twinning plane. These stacking faults are produced by the dissociation of perfect dislocations into Shockley partials, their width varies along the dislocation line and is usually less than 100 nm. From the foil orientation and the orientation of stacking fault fringes it can be determined that the dissociation ¯ T . Defects that are deposited on these planes during the of these defects occurs on (1¯ 11) twinning transformation lay on (100)M in the parent lattice. Fig. 26 shows detailed weak-beam analysis of defects visible in the lower central part of the micrograph in Fig. 25, four operating reflections are available in this foil orientation, i.e ¯ T . The dislocation shown by the arrow, dissociated into Shockley partials, foil normal [115] ¯ M , which coincides with the trace of the interface on the is aligned along [011]T = [01¯ 1] micrograph in Fig. 26a. Defect visibility depends upon the operating reflection and there¯ 1¯ reflection of the twinning plane fore provides a valuable analytical tool. In the g = 11 (Fig. 26a) the dislocation exhibits strong fringe-like contrast due to the stacking fault, it must therefore be lying on other than the twinning plane, which is set in diffracting con-

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Fig. 25. Weak-beam observations of the twin substructure in Cu-8%Al crystals. Stacking fault fringes on the ¯ T twinning plane run diagonally across the micrograph from lower right to upper left. The substructure (111) contains a large density of short segments of transformed LC dislocations of length approximately 40–50 nm ¯ T which are not parallel to the interface plane (see text for details) represented by contrast fringes on (1¯ 11) ¯ g = 020.

¯ and g = 200 twin reflections (Figs 26b and c) the dislocation is dition. In the g = 020 recognized by strong contrast arising from the stacking fault and one bounding partial dis¯ reflection the dislocation exhibits weaker contrast arising mainly location. In the g = 2¯ 20 form the stacking fault. Table 3b shows that these contrast properties are exhibited only ¯ T or a [110] ¯ T consisting of two a [12¯ 1] ¯ T and by dislocations with Burgers vector a2 [110] 2 6 a ¯ 6 [211]T Shockley partials bounding the stacking fault, no other defect shows the same visibility/invisibility properties in the set of reflections used. Hence, this defect arises from ¯ M or LC dislocation on the (100)M plane in the matrix, formed by interaca Lomer a2 [011] ¯ T dislocation tion of primary and conjugate dislocations resulting in formation of an a2 [110] a ¯¯ a ¯ in the twin lattice, which subsequently dissociates into 6 [121]T and 6 [211]T Shockley ¯ T. partials on (1¯ 11) We checked this conclusion by comparing the experimental and simulated images of ¯ T and a [12¯ 1] ¯ T Shockley partials this dislocation assuming that it dissociates into a6 [211] 6 ¯ ¯ on (111)T in the twin lattice. Fig. 27 shows the calculated images for various operating reflections available in a ¯ T foil, and, for comparison, the appropriate observed images are reproduced as small [115] inserts. Referring back to Fig. 26 or 25 we conclude that there are a number of defects

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Fig. 26. Weak-beam analysis ¯ T foils. The available in [115]

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of features in the lower central area of Fig. 25 using four operating reflections a [110] ¯ T dislocation, dissociated into a [12¯ 1] ¯ T and a [211] ¯ T Shockley partials is 2 6 6

shown by arrows in the micrographs.

in other volumes of the crystal which satisfy these contrast properties. This indicates that many LC dislocations in the parent structure can indeed be transformed by twinning into mobile dislocations in the twin lattice, as suggested by Szczerba [46]. LC dislocations resulting from interaction between dislocations of the primary and crit¯ M in the matrix, are less common than those resulting from ical systems, lying along [110] interaction with the conjugate system (Section 7 and Fig. 23) which lie along [011]M . The correspondence matrix method predicts that a2 [110]M LCs on the (001)M plane in the ¯ M direction in the matrix will be transformed to a [1¯ 10] ¯ T dislocations on the (11¯ 1) ¯ T [110] 2 ¯ plane in the [011]T direction in the twin. One can visualize this transformation by referring to the distorted Thompson tetrahedron shown for example in Fig. 3b and the insert

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¯ T dislocation dissociated into a [211] ¯ T and a [12¯ 1] ¯ T Shockley partials Fig. 27. Simulated images of an a2 [110] 6 6 ¯ ¯ on the (111)T twin plane. For comparison, the inserts are actual experimental images obtained in the operating ¯ T foil. reflections available with a [115]

in Fig. 14b, where alignments along BF, which represent LC segments in the matrix, are transformed to FD’ in the twin. In contrast to the transformation of conjugate LCs formed ¯ M , where the direction of the dislocation line is preserved, the transalong [011]M = [01¯ 1] formation of critical LCs requires substantial rotation of the dislocation line. In addition, they also inherit a new Burgers vector. Studies were carried out on various systems and indications that the transformation of these defects follows the predictions of the correspondence matrix were found only occasionally. Fig. 28 shows, at larger magnification, the area just above the marker in Fig. 15. By controlling the foil orientation one can determine the spatial orientation of a given defect, including the plane on which it is located in the twin lattice. Analysis showed that

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Fig. 28. Weak beam images of an as-transformed Lomer–Cottrell network in Cu-8%Al. The dislocation marked ¯ plane and forms a node with transformed LC dislocations marked “2” and “3” “1” is dissociated in the (111) ¯ T and (11¯ 1) ¯ T respectively. A longer faulted dipole aligned along [011]T is shown which are dissociated on (1¯ 11) ¯ ¯ T is shown by an arrow g = 020. at FD. A short faulted dipole aligned along [011]

the dislocation marked by number 1 in Fig. 28 is dissociated into two Shockley partials ¯ T . The partial dislocations themselves are not visible in the on the twinning plane (111) g = 020 operating reflection, the strong contrast comes from the stacking fault on the twin¯ T and is inherited from a ning plane. The dislocation indicated by number 2 lies on (1¯ 11) LC resulting from interaction between primary and conjugate dislocations. A dislocation ¯ T along [011] ¯ T and is most probably transformed from marked by number 3 lies on (11¯ 1) the LCs formed along the intersection of primary and critical planes in the matrix. At this stage it is unclear why these defects are observed so rarely compared to the other family of LCs formed at the intersection of primary and conjugate planes, possible reasons are discussed later.

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9.3.2. Dislocation modeling ¯ M Lomer locks along [011]M . Consider the config9.3.2.1. Transformation of a2 [011] uration of a Lomer lock formed as a result of the interaction of primary a2 [101]M and ¯ M dislocations, as shown in Fig. 29. The reaction between these two conjugate a2 [1¯ 10] ¯ M and conjugate dislocations occurs along [011]M , the intersection of the primary (111) a ¯¯ a ¯ a ¯ (111)M planes according to the equation: 2 [101]M + 2 [110]M = 2 [011]M . The Burgers vector of the resulting Lomer dislocation lies on (100)M , it is therefore sessile (Fig. 29a). The nature of the lock is not changed if one arm of the primary dislocation sweeps around the node N1 as shown in Fig. 29b. Consider now an a6 [121]M twinning dislocation ex¯ M plane which contains the line of the tending the stacking fault on the conjugate (111) Lomer dislocation (Fig. 29b). Any dislocation with both line and Burgers vector belonging to this plane is transparent to the trailing twinning dislocation and is left unchanged, i.e.

(a) ¯ M Lomer lock formed by the interacFig. 29. Schematics of the transformation of a Lomer lock. (a) An a2 [011] ¯ M dislocation. (b) The same configuration formed differently, tion of a primary a2 [101]M and conjugate a2 [1¯ 10] ¯ twinning plane. (c) After passing a conprimary arms approached by a6 [121]M twinning dislocation on a (111) ¯ M dislocation. (d) All jugate dislocation the twinning partial reacts with a Lomer dislocation to form an a6 [114] dislocations form double dislocation pole sources, two primary dislocations with parallel Burgers vectors anchored at nodes N1 and N2 ; the twinning partial can wind around the primary poles producing a macroscopic ¯ T Burgers twin. (e), (f) In the next stage of twin growth the as-transformed Lomer dislocation inherits an a2 [110] ¯ T and a [211] ¯ T Shockley partials on the (1¯ 11) ¯ T plane in the twin lattice. vector which can dissociate into a6 [12¯ 1] 6

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(e) Fig. 29. (Continued.)

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(f) Fig. 29. (Continued.)

it will preserve its Burgers vector and dislocation line direction in the twin lattice after passing the twinning dislocation [46]. Depending upon the relation between their Burgers vectors, these two dislocations will experience either repulsive of attractive interaction. During collision one might expect formation of an intermediate dislocation with Burgers vector equal to the sum of the Burgers vectors of the two colliding dislocations. In the ¯ M dislocation into the twin latparticular case of the incorporation of the conjugate a2 [1¯ 10] ¯ M Burgers tice, shown in Fig. 29c, an intermediate dislocation would acquire an a6 [2¯ 11] a a ¯¯ a a ¯¯ ¯ T . Under apvector according to the reaction: 2 [110]M + 6 [121]M = 6 [211]M = 6 [211] plied stress this intermediate dislocation would subsequently transmit a twinning Shock¯ T , so that, with respect to the twin coordinates, the dissociation reaction is: ley a6 [1¯ 2¯ 1] a ¯ a a ¯¯¯ 6 [211]T = 2 [110]T + 6 [121]T . Thus, as the twinning dislocation propagates through the ¯ M = a [110]T indexed lattice, it leaves behind unchanged conjugate dislocations a2 [1¯ 10] 2 with respect to either matrix or twin coordinates. For a conjugate dislocation of opposite sign, the transformation could involve formation of another intermediate dislocation with Burgers vector a6 [451]M ( a2 [110]M + a6 [121]M = a6 [451]M ). Similarly, this dislocation would transmit twinning dislocations while leaving the conjugate dislocation unchanged in the twin lattice. After colliding with and passing the conjugate dislocation, the twinning partial subsequently reacts with the segment of Lomer dislocation, as shown schematically ¯ M + a [121]M = a [114] ¯ M . At this stage the in Fig. 29c, according to the equation a2 [011] 6 6

§9.3


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whole configuration forms a double dislocation pole source with two primary dislocations ¯ M = a [110] ¯ T with parallel Burgers vectors anchored at nodes N1 and N2 by an a6 [114] 2 jog. This configuration is analogous to the twinning source with an extended sessile jog discussed by Basinski, Szczerba and Embury [96] and Niewczas and Saada [64]. The two ¯ M provide poles for the arms of the primary dislocation on the primary glide plane (111) ¯ M Lomer dislocations and spiraling movement of the twinning dislocation. Both a2 [011] a 2 [101]M primary dislocations have the same Burgers vector component normal to the ¯ M , equal to the spacing of these planes, so that the presence of twinning plane, i.e. a3 [111] Lomer dislocations does not change the spiraling nature of the glide planes produced by the primary dislocation arms. This may not necessarily be true for other dislocations discussed latter in Section 9.4. As was discussed in Section 9.2.2, after every revolution the arms ¯ M planes of the twinning dislocation being chopped into two branches on adjacent (111) a wind up or down around the pole and transform primary dislocations 2 [101]M into cube a ¯ a ¯ 2 [020]T dislocations. The transformed 6 [114]M Lomer dislocation can be fully incorporated into the twin lattice by such a pole mechanism provided the twinning dislocation continues its spiraling movement. After the first revolution the twinning dislocation will ¯ M jog, which could prevent it encounter a passing stress barrier arising from the a6 [114] from further movement. However, when two twinning dislocations pass the jog together, as shown schematically in Fig. 29d, the interaction energy between the jog and two Shockley partials is reduced to zero [64]. Thus, the twinning dislocations will be able to break away from the jog and continue spiraling around the primary dislocations. The same argument applies with other dislocations in the twinning plane. As the twin grows, the nodes N1 and N2 , at which the reaction between twinning and pole dislocation occurs, move along the propagating interfaces defining the boundary between transformed and untransformed ¯ T Burgers vector and a new material. The Lomer dislocation inherits both a new a2 [110] ¯ T plane in the twin lattice, as shown in Fig. 29e. To lower its energy the dislocation (1¯ 11) ¯ T bounding a stacking fault according can dissociate into two Shockley partials on (1¯ 11) ¯ T = a [12¯ 1] ¯ T + a [211] ¯ T (Fig. 29e). The final configuration of such to the reaction a2 [110] 6 6 defects will consist of a glissile dislocation dissociated into two partials anchored by two a ¯ 2 [020]T sessile dislocation arms. Configurations of this type are commonly observed experimentally in twinned copper and copper alloys (Section 9.3.1). Fig. 29 shows a diagrammatical representation of the transformed configuration of a LC lock utilizing the pole mechanism. The following reactions between Burgers vectors need to be satisfied at a given node at any stage of the process: (i) (ii) (iii) (iv)

at node N1 at node N2 at node N3 at node N4

a a a a ¯ 2 [020]T = 6 [424]M = 2 [101]M + 6 [121]M , a a a a ¯ 2 [101]M + 6 [121]M = 2 [424]M = 2 [020]T , a ¯ a a ¯ a ¯¯ 6 [121]T + 6 [211]T = 2 [110]T + 2 [020]T , a a ¯ a ¯¯ a ¯ 2 [110]T + 2 [020]T = 6 [121]T + 6 [211]T .

Geometrically, the transformation of a LC lock would proceed in a way analogous to the transformation of a Lomer lock. The difficulty arises from the fact that a LC dislocation consists of two Shockley partials on the primary and conjugate planes and a stair rod dislocation at the intersection of these two planes, all dislocations include stacking faults on the primary and the conjugate planes. Reactions occurring between twinning dislocations and

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various partial dislocations in the extended configuration of the LC lock could be complicated. In particular a difficulty arises in the transformation of stacking faults on different planes by the twinning partial. This problem is discussed in more detail later in Section 9.4. It is not clear how this dislocation progresses through the structure of extended LC network, atomistic studies are required to provide a clear picture of these processes. One possibility is that all partial dislocations in a LC lock associate to form a perfect Lomer dislocation during interaction with the twinning dislocation, subsequent transformation steps would then follow the path described above for transformation of a Lomer lock. The above discussion considers only the geometry of transformation of Lomer dislocations and neglects any interaction of the twinning dislocation with the internal stress field from existing defects, in particular stress barriers that may prevent a twinning dislocation from rebuilding a macroscopic twin by the pole mechanism. The reader is referred to the work of Niewczas and Saada [64] who discussed such processes in relation to the configuration of a twinning source similar to the one considered above, and also the work of Song and Gray [97]. Finally it can be shown that the transformation of Lomer or LC locks can be accomplished by incoherent twinning fronts moving through these defects in a way discussed in Section 9.2.2. ¯ M . As mentioned in Section 9.3.2.2. Transformation of a2 [110]T Lomer locks along [110] 9.3.1, defects transformed from the family of Lomer or LC locks formed along the intersection of the primary and critical planes have occasionally been observed in twinned material (Fig. 28). Therefore, we discuss briefly the possible mechanism involved in the transformation of these defects. The lock is formed in the reaction between a2 [101]M primary and a a a a ¯ ¯ 2 [011]M critical dislocations according to the equation: 2 [101]M + 2 [011]M = 2 [110]M [47]. The glide plane of the dislocation contains both the Burgers vector and the line direction (001)M , therefore the dislocation is sessile in the parent lattice. In contrast to the Burgers vector of either primary or critical dislocations involved in the reaction, the prod¯ M twinning plane. Thus uct a2 [110]M Lomer dislocation has no component normal to (111) this dislocation does not produce the spiral topology in the twinning plane around its line, which is necessary for operation of a pole mechanism. Some other process must therefore be involved. The incorporation of a2 [110]M Lomer dislocation into the twin lattice can be realized if an incoherent twinning front moves through the crystal. Fig. 30a shows the relaxed core ¯ M , direction of the dislocation line. Fig. 30b structure of this dislocation along the [110] ¯ M , which reveals the topology of the shows the same dislocation in projection along [11¯ 2] twinning planes in the crystal. In this orientation the dislocation line is inclined, extending ¯ layers marked by arrows in Fig. 30b. As can be seen in this figure, large between the (111) distortions of the lattice occur in the dislocation core along its line, but the twinning planes are flat and no spiral topology is developed. The Lomer dislocations act as a forest for the propagating twinning Shockley and as the Burgers vector is included in the twinning plane the configuration allows pure forest cutting of the stationary Lomer by a propagating twinning partial and no reaction between Burgers vectors occurs at the point of collision of the two dislocations. As a result of the twinning Shockley cutting through the Lomer dislocation, a jog of length a6 [121] is formed on the Lomer dislocation. Full incorporation ¯ of this dislocation into the twin lattice requires that cutting must occur on every (111)

§9.3


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¯ M , direction. Fig. 30. (a) The relaxed core structure of an a2 [110]M Lomer dislocation lying along the [110] ¯ ¯ (b) The same Lomer dislocation in projection along [112]M , revealing the topology of the twinning planes in the neighbourhood of the dislocation line. (c), (d) Schematics of the transformation of a Lomer dislocation by an incoherent front.

plane over the whole length of the dislocation line. During this process the direction of the ¯ T and the Burgers vector of the sheared dislocation dislocation line is transformed to [011] is preserved in the twin lattice. The cutting of a Lomer dislocation by an incoherent twinning front is shown schematically in Figs 30c and 30d. As the front approaches the Lomer dislocation in the forest ¯ layer (Fig. 30c). After the twinning partials position, intersection occurs on every (111) cut through a segment of the Lomer dislocation, since its Burgers vector is in the twinning plane, the continuity of the front is preserved and it propagates with the same number “n” of dislocations (Fig. 30d). The process is analogous to the case discussed in reference to the transformation of a primary dislocation shown in Fig. 22. However in the present case no dislocation reaction is taking place during collision and the twinning dislocation is not chopped into two branches on adjacent glide planes. This would require exchange of

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the arms of the twinning dislocation in the wall to restore the twinning front after cutting takes place. The front leaves behind a segment of Lomer dislocation with Burgers vector a ¯¯ 2 [110]T incorporated in the twin lattice, the dislocation acquires a new line direction and ¯ T and (11¯ 1) ¯ T respectively. plane, [011] The transformation of the Burgers vector can be predicted both by the correspondence matrix (Table 1) and by the node equations which read: a a a a ¯ T [110]M + [121]M = [451]M = [4¯ 5¯ 1] 2 6 6 6

and

a ¯¯¯ a ¯ T + a [1¯ 10] ¯ T. [451]T = [1¯ 2¯ 1] 6 6 2 ¯ T , which subseDuring intersection, a jog acquires an intermediate Burgers vector, a6 [4¯ 5¯ 1] ¯ T and leaves an a [1¯ 10] ¯ T jog in the twinned quently transmits a twinning Shockley a6 [1¯ 2¯ 1] 2 lattice. The dislocation becomes glissile after the twinning transformation and can lower its ¯ T in the twin lattice, as shown energy by dissociating into two Shockley partials on (11¯ 1) in Fig. 28 at “3”. The above exercise represents a more general example of a twinning transformation of a forest dislocation whose Burgers vector is in the twinning plane: the glissile or sessile nature of the transformed dislocation in the twin lattice is determined by its glide plane. Consider now an extended LC lock consisting of two Shockley partials and a stair rod dislocation bounding stacking faults on the primary and critical planes. The transformation of this extended defect involves interactions of the twinning partial with the component dislocations and cutting of the stacking faults, which is more difficult to model. For the transformation to proceed, we suggest that, as discussed previously, as the twinning front approaches the LC defect its component partials are constricted to reform a perfect Lomer dislocation at the point of intersection. In the absence of stacking faults, incorporation of this defect into the twin lattice occurs following the same principles as in the transformation of a perfect Lomer dislocation. As discussed in Section 9.2.2, formation of such a constriction is expected to be energetically favourable as the energy of an extended jog is higher than that of a constricted one [91,92]. Fig. 31 shows sequences of the intersection process. All transformations of dislocations occur at high stresses typically of the order of 100 MPa and larger. As two families of LC dislocations are “unlocked” in the way discussed above, they become glissile in the twin lattice and will tend to glide out of their initial positions under the influence of applied stress. Unless these transformed dislocation segments are relatively short and are pinned by immobile nodes or some other obstacles, they will glide on the close-packed planes and will be difficult to identify in the twinned structure. However, these defects can be traced back to LC dislocations in the parent lattice: one strong indication is the presence of dissociated dislocations on {111} planes not coplanar with the twinning plane. Also the presence of extended stacking faults on other than the twinning plane in the twinned lattice, visible for example in the upper part of the micrograph in Fig. 15, suggests that such transformations have taken place. However, as discussed in Section 10, such faults might also be formed to release internal stresses

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Fig. 31. Schematics of the intersection of a twinning dislocation with an extended Lomer–Cottrell dislocation ¯ M , the intersection of the primary and critical planes. formed along [110]

built up in this structure [64], so that identification of the component partial dislocations is crucial. As previously mentioned, defects originating from the interactions of primary dislocations with those of the critical system are less numerous that those derived from interactions with the conjugate system. Basinski and Basinski [58], measured the amount of glide on all slip systems in copper crystals throughout stages 1, 2 and 3 of deformation and showed there are two kinds of secondary slip. One kind occurs in response to stresses caused by the presence of primary dislocations and is the same for all crystal orientations. The second kind, which occurs in response to applied stress, gives rise to the secondary slip markings that appear on the crystal surface. However, this response is not in proportion to the resolved applied stress on the systems. For certain orientations, the initial Schmid factor for the critical system is almost equal to that of the primary system but the critical system contributes only a few percent to the total glide. Also, as had previously been deduced from stress-strain curves, the amount of slip on the conjugate system increases smoothly and it eventually becomes the dominant system. These data indicate that even for crystals having a high critical Schmid factor, the conjugate system is still the more active one, so that it is not surprising that conjugate LC locks are more numerous than those of the critical system especially in the parent crystal just before the onset of twinning. The predominance of

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conjugate LC locks also might depend on the relative stabilities of the transformed defect configurations under conditions of high stress.

9.4. Transformation of faulted dipoles Faulted dipoles are easily recognized by their characteristic Z-configuration with compo¯ M . The spatial arrangement of faulted nents along the matrix directions [011]M and [110] dipoles in the parent crystals was discussed in Section 8.3 and their contrast properties under week-beam diffraction conditions have been thoroughly analyzed [63]. Briefly, faulted dipoles comprise four component dislocations, two Shockley partials of opposite sign on the primary planes and two stair-rod dislocations of opposite sign at the acute intersection of the primary and secondary planes. Three intrinsic stacking faults determine the vacancy nature of the faulted dipoles. Two stair-rod dislocations formed along the intersection of either primary and conjugate or primary and critical planes render the structure sessile. Frank dipoles are formed by reassociation of Shockley partials and stair rod dislocations to a3 111 sessile Frank dislocation enclosing intrinsic stacking fault on a {111} plane. In this section the effect of twinning on faulted dipoles in materials of different stacking fault energies is considered and details of the resulting Burgers vectors are given. Information on the modeling of the defect transformation is also provided. 9.4.1. TEM observations Fig. 32 shows TEM observations of the twinned substructure in a copper crystal deformed to the onset of twinning at 4.2 K, the foil plane is parallel to the cross glide plane of the ¯ T in twin co-ordinates. The micrountwinned crystal lattice, the foil normal is thus [115] graph shows the very small-scale nature of the substructure. Distributed throughout the micrograph there is a high density of dot-like debris and defects with strong line-like contrast some of which are marked by arrows. These defects are short, rarely exceeding 20– 30 nm in length. They all terminate within the foil suggesting that they are short dipoles. The distorted Thompson tetrahedron shown in an insert to the figure illustrates the threedimensional orientation of the observed features. Dipoles marked by “1” and “2” represent features aligned along [011]T , EF on the tetrahedron. This direction is conserved during the twinning transformation (Table 1) so that defects which gave rise to these dipoles were aligned along the same [011]M direction in the matrix. Thus, the observed dipoles originate from faulted dipoles formed along the intersection of the primary and conjugate plane in the parent. ¯ T , FD’ in the disDipoles marked “3” and “4” represent features aligned along [011] ¯ M (Table 1), torted tetrahedron. This direction is inherited from the matrix direction [110] indicating that these dipoles originated from faulted dipoles formed along the intersection of primary and critical planes in the parent crystal, i.e. BF on the tetrahedron, and subsequently sheared to FD’ in the twin lattice. The micrograph shows that the two families of dipoles are oriented approximately perpendicular to each other, exactly as would be expected from the geometry of this projection. Close inspection of the dipole images in Fig. 32 reveals that they are not all of the same width. Dipole #1 gives wider contrast than dipole #2 or other dipoles of this family

§9.4


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¯ T foil orientation. DisloFig. 32. Weak-beam observations of the twinned substructure in a copper crystal, [115] cation dipoles marked by “1” and “2” are aligned along [011]T , and those marked “3” and “4” are aligned along ¯ T. [011]

marked by arrows. Similarly, the image of dipole #3 is wider than dipole #4 or other dipoles of its family shown by arrows. Due to the very small size of these defects the nature of the stacking faults and component dislocations could not be unambiguously determined by TEM analysis. However, tilting experiments and image simulations indicate that the dipoles are of vacancy type in Z configuration. The difference in the image width between dipoles #1 and #2 and between dipoles #3 and #4 is most likely related to a difference in their height. Image simulations indicate that contrast from dipoles having larger separation ¯ T which is is wider and shows a stronger contribution from the stacking fault on (111) invisible in narrow dipoles. Possible mechanisms leading to the formation of these dipoles is discussed later. The nature of defects inherited from faulted dipoles was also examined in Cu-4%Al single crystals deformed at 78 K. Fig. 33 shows weak-beam observations in 200 and 002 ¯ T , i.e. parallel to the critical plane twin reflections available in sections parallel to [1¯ 5¯ 1] in the parent crystal (Section 6.1). The substructure is severely refined by twinning, but a number of defects giving strong line-like contrast are distinguishable from the background of fine debris. As in the case of pure copper, these defects fall into two families. One family of dipoles visible in the 200 operating reflection in Fig. 33a, vanishes in the 002 reflection, Fig. 33b. Dipoles belonging to this family are shown by arrows in the figure: they are aligned approximately 45◦ to the trace of the interface which is marked by fringes. The

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¯ T Fig. 33. Weak-beam observations of the twin substructure in Cu-4%Al single crystals deformed at 78 K, [1¯ 5¯ 1] foil, 200 and 002 twin reflections. There are two families of dipoles, one visible in the 200 operating reflection and the other in 002.

other family of dipoles, shown by arrows in Fig. 33b, are aligned along the trace of the interface, they are visible in the 002 operating reflection but vanish in the 200 reflection (Fig. 33a). Using the arguments discussed above we find that dipoles which are visible in 002 and vanish in 002 are the transformed products of faulted dipoles formed along ¯ M . Similarly, the intersection of the primary and critical planes in the parent crystal, [110] dipoles visible in 002 and invisible in 200 (Fig. 33) are inherited from faulted dipoles formed along the intersection of the primary and conjugate plane in the parent. To explore the defects inherited from faulted dipoles in even lower stacking fault energy material such as Cu-8%Al we refer back to Fig. 28 (Section 8.3), which showed the ¯ T foil from a Cu-8%Al single crystal deformed to the dislocation substructure in a [115] onset of twinning at room temperature. The weak-beam micrograph shows a line-like defect of length >50 nm, marked FD in the lower central area of the figure. The defect is aligned along [011]T and terminates within the foil indicating that it has dipolar character. ¯ weak-beam reflection. The short-wavelength osIt exhibits strong contrast in the g = 020 cillations are due to the fact that it is inclined in the foil. As mentioned before, this defect must originate from a faulted dipole formed in the parent crystal by interaction of primary and conjugate dislocations along [011]M . Fig. 28 also showed the presence of dipoles belonging to a second family of much shorter dipoles oriented approximately perpendicular to the previous set. These dipoles are aligned ¯ T and must therefore originate from faulted dipoles formed in the parent crystal along [011] ¯ M . Closer inspection suggests along the intersection of the primary and critical planes, [110] that these dipoles are severely chopped, presumably by twinning fronts, and measurements show that these dipoles are indeed much shorter than those in the parent crystal. The observations have shown that the transformation of faulted dipoles, governed by the geometrical principles of the twinning transformation, occurs in materials of both high and low stacking fault energy, from pure copper to Cu-8%Al single crystals.

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9.4.2. Dislocation modeling 9.4.2.1. Faulted dipoles along [011]M . As has already been mentioned, if a Frank dislocation does not dissociate into Shockley partials and stair rod dislocations then the ¯ M Z-shaped faulted dipoles aligned along [011]M can be regarded as faulted loops on (111) a ¯ with an intrinsic stacking fault bounded by 3 [111]M Frank dislocations. For simplicity of ¯ M Frank loop rather than a fully modeling, in subsequent discussion we consider an a3 [111] dissociated faulted dipole, however the transition between the two structures is straightforward and was discussed in detail elsewhere [64]. ¯ M The correspondence matrix method (Table 1) predicts that a Frank dipole on (111) ¯ M extended along [011]M will be transformed to a defect on the with Burgers vector a3 [111] ¯ T and in the same direction [011]T but will have a new a [141] ¯ T Burgers same plane (111) 6 vector in the twin lattice. The Burgers vector of the dipole is equal to the interplanar spacing of the twinning planes, but as the dislocation line lies in this plane the transformation of the defect cannot occur by the pole mechanism. It can be shown that full incorporation of the faulted dipole into the twin lattice occurs only after the twinning front, consisting of a wall of twinning partials, passes through it. Consider the interaction of a twinning dislocation with a Frank loop shown in Fig. 34. For the transformation to proceed, the a6 [121] twinning dislocation and the Frank dislocation must operate on the same plane, so that the reaction occurs along the line of the Frank, as shown in the sequence of events in Figs 34a–c. The dislocation that is formed as a result

(a)

(b)

(c)

(d)

Fig. 34. Interaction of a twinning dislocation with a Frank loop. An a6 [121]M twinning dislocation reacts with ¯ T dislocation which the Frank dislocation along the line of the Frank (a)–(c) producing a new a6 [303]M = a6 [141] is incorporated in the stacking fault on the twinning plane (d).

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of this reaction inherits an a6 [303]M Burgers vector (Fig. 34d) according to the following transformation equation: a ¯ a a a ¯ [111]M + [121]M = [303]M = [141] T. 3 6 6 6 Because of the polarized nature of the twinning shear, the twinning dislocation always reacts with the same sense of Frank dislocation, reaction with opposite sign of the twinning partial or Frank dislocation cannot occur. This is determined both by the vacancy nature of ¯ M, the dipole, which determines the sense of the bounding dislocation, in this case a3 [111] and by the sense of the twinning shear (Burgers vector of the twinning dislocation, in this case a6 [121]M ) all indexed with respect to the predefined coordinate system of the matrix. Realistically, the transformation of a faulted dipole is expected to occur when an incoherent twinning front interacts with it. Fig. 35 shows possible stages of such a process. One twinning dislocation from the wall, located on the same plane as the Frank partial, interacts with the loop (Figs 35a–c) and transforms it to a new loop with Burgers vector a6 [303]M in¯ T with respect to the twin. The nature and dexed with respect to the parent lattice, or a6 [141]

(a)

(b)

(c)

(d)

Fig. 35. Cross-section of a Frank dipole showing various stages of transformation by an incoherent twinning front. (a)–(c) A twinning dislocation located on the same plane as the Frank partial interacts with the loop and ¯ T in twin indices. Each transforms it into a new loop (c) with Burgers vector a6 [303]M in parent indices or a6 [141] a [141] ¯ T dislocation subsequently dissociates into a [022] ¯ T and a [12¯ 1] ¯ T Shockley partials on a (1¯ 11) ¯ T plane 6 6 6 forming Z dipoles of vacancy type in the twin.

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integrity of the stacking fault is unchanged and after the twinning front passes, the loop and its intrinsic stacking fault are fully incorporated into the twin lattice and aligned along the same 110 direction as the parent defect. In the twin lattice the dipole can lower its energy by dissociating into partials. A number of reactions may be written for dissociation of the a ¯ a ¯ a ¯¯ a ¯ 6 [141]T dislocation (e.g., 6 [141]T = 6 [130]T + 6 [211]T ). The most probable would be a ¯ ¯ T dissociation into the higher energy 6 [022]T stair rod on the (100)T plane and an a6 [12¯ 1] ¯ ¯ Shockley partial on the (111)T plane. This is supported by TEM observations giving strong visibility conditions for these dipole components in the 020 reflection (Figs 28 and 32), strong contrast in the 002 reflection in Fig. 29, and weak residual contrast in the 200 re¯ T = a [022] ¯ T + a [12¯ 1] ¯ T , would be favoured since flection in Fig. 32. This process, a6 [141] 6 6 it leads to a decrease in energy of approximately 33%. In this case the Shockley partial and ¯ T plane, giving rise to the stair-rod dislocation bound an intrinsic stacking fault on a (1¯ 11) a vacancy-type faulted dipole in Z configuration in the twin lattice. The transformed dipole inherits its sessile nature from the originating faulted dipole in the parent, arising from the a ¯ 3 [011]T “super stair-rod” dislocation stabilizing the dipole along the intersection of the ¯ T twinning plane and the (1¯ 11) ¯ T “super stair-rod” disloca¯ T plane. It is this a [011] (111) 3 tion which differentiates the structure of this dipole from its parent. This dislocation arises simply from reaction of the star-rod dislocation of the parent dipole with the propagating partial, so that the Burgers vector of this component dislocation increases. The longer Burgers vector of the stair-rod dislocation component of the faulted dipoles in the twin should make these defects more stable, and therefore stronger obstacles for other dislocations. ¯ M . The interaction of faulted dipoles along [110] ¯ M 9.4.2.2. Faulted dipoles along [110] with twinning dislocations and their incorporation into the twin lattice deserves detailed analysis because it represents another class of twinning transformations which cannot be accomplished by the pole mechanism, and where the transformation of the stacking fault becomes important. In all cases of dislocation transformations by twinning discussed so far, the same result is obtained whether the Burgers vector of the product defect is predicted by the correspondence matrix method or by the node equation. Faulted dipoles formed along ¯ M represent a degenerate case where the two methods may lead to different product [110] defects, some discussion of this is offered. ¯ M in the parAs for other dipoles, sessile Z-type faulted dipoles formed along [110] ent crystal can be regarded as faulted loops with intrinsic stacking fault on (111)M and a 3 [111]M bounding dislocation. The correspondence matrix method predicts that a faulted ¯ M , the intersection of the primary and critical dipole on K1 (111)M aligned along [110] a planes, and having an 3 [111]M Burgers vector is transformed to a defect which will have ¯ T Burgers vector, a new [011] ¯ T direction and a new KT (111)T plane in the a new a6 [1¯ 4¯ 1] 2 twin lattice rotated with respect to the K2 plane in the matrix (Section 3, Section 5). Fig. 36 shows a diagrammatic representation of a partially transformed faulted dipole which illus¯ M ) is sheared by the trates these features. The faulted loop initially oriented along BF([110] twinning front such that part of the loop, uvxy, is still in the parent crystal (ABCEFG), and the sheared part, u′ v ′ xy, of the dipole is incorporated into the twin lattice (EFGA’B’C’) ¯ T ). The figure shows that the transformation of the loop also requires along GC ([011] shearing of the stacking fault, initially on K2 (111)M and subsequently deposited onto KT2 (111)T in the twin.

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Fig. 36. Geometrical representation of a partially transformed faulted dipole with respect to the parent and twin ¯ M ) is sheared by the twinning front such that part lattice orientations. The faulted loop oriented along BF ([110] ¯ T) of the loop, uvxy, is in the parent crystal ABCEFG and the sheared part, u′ v′ xy, of the dipole along GC’ ([011] exists in the twin lattice EFGA’B’C’. The transformation of the loop also requires shearing of the stacking fault from K2 (111)M to KT 2 (111)T in the twin.

Before considering mechanisms of transformation of this defect, we analyze individual events which occur during intersection of the faulted dipole by an advancing twinning dislocation. Fig. 37 shows the sequence of events occurring during intersection. The dipole is dissociated into component dislocations, forming the characteristic Z configuration, and ¯ M plane along its length. the twinning dislocation approaches the dipole on some (111) If the twinning partial causes constriction of the component dislocations and the stacking ¯ T and (111)T planes, annihilation of the dipole will occur along the interfaults on (111) secting plane effectively chopping the dipole into two separate loops (Figs 37a–c). Subsequent twinning dislocations operating on adjacent planes would cause further shortening of the dipole and eventually would remove it from the structure in a purely mechanistic way (Fig. 37d). It is expected that many of these defects will be chopped this way into small debris, a common feature of the twinned substructure. A Frank dislocation bounding a stacking fault on the K2 (111)M plane is oriented in the ¯ M component forest position with respect to the advancing twinning partial and has a9 [111] a ¯ of 3 [111]M Burgers vector normal to the (111)M twinning plane, which is a factor of three shorter than the spacing between these planes. Therefore in the neighborhood of a Frank dislocation the twinning planes are not flat, the dislocation causes the glide planes to assume screw topology with d(111) ¯ /3 thread length around its line. This will not allow the twinning partial to wrap around on an adjacent plane after a complete revolution, as would be the case for a primary dislocation acting as a pole. Hence, this dislocation will not be incorporated into the twin lattice by a pole mechanism as discussed in Section 9.2. ¯ M planes experience the same d ¯ /3 displacement across the stacking Twinning (111) (111) fault on (111)M . Thus the dipole itself creates an obstacle for the propagating twinning dislocation, which might get stuck at the dipole while attempting to shear it. However,

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Fig. 37. Intersection of an extended faulted dipole in Z configuration by a twinning dislocation. The twinning ¯ T and (111)T planes partial causes constriction of the component dislocations and the stacking faults on (111) and the dipole is chopped into two separate loops (a–c). Twinning dislocations operating on adjacent planes will cause further shortening of the dipole which may remove this type of defect from the structure.

TEM observations have shown that dipoles which are found in the twin lattice are isolated from other dislocations and are not associated with the interface (e.g., Figs 28, 32, 33). Thus, the mechanism operating during transformation of these defects must also account for their full incorporation into the twin lattice. If under appropriate conditions a dipole resists constriction caused by an advancing twinning partial, it may undergo transformation to a new type of dipole in the twin lattice. This requires that the bounding dislocations and the stacking faults are sheared by the advancing twinning front causing the defect to adopt a new configuration in the twin lattice. Such a transformation represents a class of phenomena not discussed before, which involve twinning transformation both of the partial dislocations with Burgers vectors normal to K2 of the matrix and the associated stacking fault enclosed in this plane. The geometry of the transformation of a stacking fault on {111}M into one on {100}T during fcc twinning has been discussed by Chiu and coworkers [93] for the transformation

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Fig. 38. Atomistic model of the structure of a narrow sessile Frank dipole in a projection which reveals the ¯ planes, marked by numbers, in the neighbourhood of the defect. The bordering a [111]M topology of (111) 3 ¯ layers. dislocations are located approximately on the fourth and twelfth (111)

of faulted dipoles in gamma-TiAl. Topologically, the transformation of the faulted dipole considered here involves similar elementary dislocation processes, however the present case leads to different resultant defects in the twin, this transformation is therefore discussed in detail below. Fig. 38 shows the structure of a narrow sessile faulted dipole in a projection which re¯ planes, marked by numbers, in the neighborhood of the defect. veals the topology of (111) ¯ layers, outThe dislocations are located on approximately the fourth and twelfth (111) side the dipole the twinning planes are almost flat thus providing no obstacle for gliding ¯ planes to the right of the fault evidently twinning partials. However, in cross section (111) are displaced with respect to those to its left, generating a discontinuity in the glide plane whose magnitude is one third of the interplanar spacing. Atomistic simulations of the structure of Frank dislocations suggest that stacking fault displacements in the relaxed dipole structure are somewhat different but these details will not change the geometrical picture discussed below. Consider a twinning partial propagating on the same plane as the jog-like Frank dislocation forming the closure of the lower end of the dipole, i.e. the fourth layer in Fig. 38. The twinning partial reacts with the Frank jog along its length producing a new dislocation whose Burgers vector is the sum of those of the Frank and the Shockley partial (Fig. 39a). The arms of the twinning dislocation can propagate further, and after winding around the long jog they annihilate on the glide plane producing a segment of Shockley partial which aligns itself on displaced layer 4 along the jog on the right hand side (Figs 38, 39b). The following equations, in matrix coordinates, are satisfied at nodes N1 and N2 : a a a [111]M + [121]M = [343]M 3 6 6

at node N1

and

§9.4


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(b)

(c)

(d)

(e)

¯ M Frank dipole by interaction with a [121]M twinning partials. (a)–(b) Fig. 39. The transformation of an a3 [111] 6 The reaction of the twinning partial with the Frank jog produces a duplet of a6 [343]M dislocation and a6 [121]M twinning partial at the end of the dipole. (c)–(d) Incorporation of the Frank dipole into the twin lattice by interaction of the dipole with the twinning partial. (e) Ladder-like structure of the transformed dipole in the twin lattice consisting of a sequence of transformed a6 [343]M and untransformed a3 [111]M segments of dislocation in series and rungs of twinning partials.

a a a [343]M = [111]M + [121]M 6 3 6

at node N2 .

This produces a duplet consisting of a transformed Frank jog and a twinning partial at ¯ M = a [111]M the lower end of the dipole, and together these become a6 [343]M + a6 [1¯ 2¯ 1] 3

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indexed with respect to matrix coordinates (Fig. 39b). The reconstructed loop continues ¯ plane number 4, leaving an its glide extending one layer of the stacking fault on (111) a ¯¯¯ a [343] jog and an [ 1 2 1] dislocation behind, incorporated in the stacking fault. Due M M 6 6 /3, the twinning dislocation cannot complete a full revto the shorter thread length, d(111) ¯ olution around the Frank pole, which would be necessary for it to be incorporated into the twin by means of a pole mechanism. Thus, for this defect to be transformed into twin configuration, another twinning dislocation must reproduce this elementary process on adjacent planes. As the next twinning partial approaches the dipole from the left hand side on the fifth layer (Figs 38a and 39c) it reacts with the Frank dislocation at nodes N3 and N4 , and part of the dislocation gets stuck in the dipole at the stacking fault between these nodes. The two outside arms of the twinning partial continue their spiraling motion around the Frank pole, they mutually annihilate on layer 5 producing a segment of twinning partial which is aligned at the fault on the right hand side on displaced layer 5 along the opposite sign partial in the dipole. The outside twinning loop is restored and expands further on its glide plane, as shown in Fig. 39c. This process will produce lateral segments of untransformed Frank dislocation between nodes N1 and N3 and between N2 and N4 , these eventually become incorporated in the twin (see also [93]). The process is repeated on every plane along the dipole length to layer 12 where the Frank dipole terminates (Figs 38 and 39d), the twinning dislocation reacts with the Frank dislocation along its length between nodes N′1 and N′2 producing an a6 [101]M dislocation at the upper end of the dipole. The following equations, in matrix coordinates, are satisfied at nodes N′1 and N′2 : a a a [101]M + [121]M = [111]M 6 6 3

at node N′1

a a a [111]M = [101]M + [121]M 3 6 6

at node N′2 .

and

As before, the outside arms of the twinning partial will wind around and annihilate producing segments of Shockley partial along the jog and restoring the outside twinning loop which then propagates further on its glide plane on layer 12. As at the lower end, this leaves the upper end of the dipole as a duplet of two dislocations consisting of transformed jog and twinning partial, which again become a6 [101]M + a6 [121]M = a3 [111]M indexed ¯ planes with respect to matrix. The twinning dislocation propagating on subsequent (111) completes the transformation of this part of the matrix into twin orientation and fully incorporates the original faulted dipole into the twin lattice. Fig. 39e shows a dipole structure in which the dislocation that borders the dipole consists of a sequence of transformed and untransformed segments, the length of the untransformed segments being approximately twice that of the transformed ones. These features of a partial dislocation dipole with Burgers vector non-coplanar with the twinning plane incorporated into the twin were also predicted by Chiu and co-workers [93] in their work on faulted dipoles in gamma-TiAl crystals. The shearing of the lattice produced by propagating twinning dislocations, which transforms faulted dipoles into twin configuration, involves simultaneous shearing of stacking faults on the K2 plane of the matrix (Figs 36 and 39). During this process, as the K2 plane rotates to KT2 in the twin, the relative position of the twinning planes across the fault is not

§9.4


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changed, the fault itself is deposited directly into the KT2 (111)T plane without changing its intrinsic nature in the twin lattice, as schematically shown in Fig. 36. Chiu and coworkers [93] proposed a formal description of such a process by introducing an additional dipole, with a Burgers vector equal to the displacement vector produced by the fault, on that fault plane where the operating twinning partial intersects the stacking fault. The resulting dipole is transformed subsequently to twin orientation and determines the type and the nature of the stacking fault in the twin. A similar situation might apply here, it would require introduction of a dipole with a Burgers vector in the K2 (111)M ¯ M . Any vector belonging to K2 retains its identity after twinplane, for example a6 [1¯ 12] ning (Section 2), which implies that the stacking fault on K2 of the matrix has the same displacement vector in the twin, thus preserving its nature. Within the twin lattice the transformed faulted dipole (Fig. 39c) consists of set of two dislocations adding up to a3 [111]M at each end of the dipole enclosing a ladder-like structure (rungs) of ± a6 [121]M twinning partials stacked on the opposite side of an intrinsic stacking fault on the KT2 (111)T plane spaced d(111) ¯ /3 apart and running along the dipole length with periodicity d(111) ¯ (see also [93]). The detailed structure of the dislocations bordering the dipole consist of sequences of transformed a6 [343]M and untransformed a3 [111]M segments of dislocation in series, the appropriate number of dislocation junctions is produced at places where dislocation reactions took place along the line. If this structure lowers its energy by annihilating pairs of ¯ M = a [111]M dislocations of opposite sign at the ends of the dipole (i.e. a6 [343]M + a6 [1¯ 2¯ 1] 3 ¯ M = 0) and a6 [101]M + a6 [121]M = a3 [111]M ) and along its length (i.e. a6 [121]M + a6 [1¯ 2¯ 1] the total Burgers vector of the defect indexed with respect to matrix coordinates becomes a a ¯¯¯ 3 [111]M , or 9 [151]T in twin co-ordinates. This indicates that in the case of a dipole built from partial Frank dislocations on the K2 plane, the Burgers vector of the defect incorporated into the twin lattice is obtained by reindexing with respect to twin coordinates rather than by operation of a correspondence matrix on the parent defect, which is equivalent to applying a sequence of deformations followed by reindexation of the parent Burgers vec¯ T in the twin. This however, would be tor, leading to a dipole with Burgers vector a6 [1¯ 4¯ 1] true only if the structure relaxes as discussed above. Possibly, the Frank dislocation bordering the dipole could be replaced by a partial dislocation whose Burgers vector is in the K2 (111)M plane and is unchanged during transformation. This can be accomplished by nucleating an appropriate loop with a Burgers vector ¯ M = a [21¯ 1] ¯ M . Thus the above argument that a Frank in K2 plane, e.g., a3 [111]M + a2 [01¯ 1] 3 a ¯¯¯ a [111] dipole is transformed to an [ 1 5 1] M T dipole which preserves the length of the 3 9 transformed dislocation would appear to be valid. This is equivalent to saying that the di¯ plane, resulting in jogs pole is transformed by a sequence of a6 [121] shears on every (111) of length equal to the Burgers vector of the twinning dislocation on the forest dislocations, which reorients this defect to its new configuration in the twin lattice without any reaction occurring between the twinning dislocation and the dipole, i.e. without changing the nature of the bordering dislocations and the enclosed stacking fault, as shown diagrammatically in Fig. 36. Chiu et al.’s observations of faulted dipoles in twinned TiAl indicate that the transformed dipoles have no fine structure of rungs, suggesting that relaxation of the internal dipole

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structure does occur [93]. This would again support the view that in our case a dipole in¯ T Burgers vector and an intrinsic stacking fault. corporated into the twin inherits a a9 [1¯ 5¯ 1] This dislocation has a {611}T glide plane so that in its transformed form the dipole is sessile, similar to its parent defect in the matrix. The transformed dipole would be expected to reduce its energy by dissociating into partial dislocations, as in the micrographs in Fig. 32. ¯ T dislocation could dissociate and decrease There is a number of ways in which an a9 [1¯ 5¯ 1] its total energy. Further decomposition of the dipole may produce either a planar configuration on (111)T or a Z or S configuration in the twin lattice. Which of these reactions actually takes place remains to be established. However, TEM data strongly indicate that non-coplanar configurations are more favourable (e.g., Fig. 32). TEM studies show that the transformation of faulted dipoles occurs when the twinning front passes through them. Fig. 40 shows a high resolution micrograph of parent and twin crystals taken in the [011]M , projection along the intersection of primary and conjugate planes, i.e. along AB in Fig. 36. The micrograph captures the intersection by the twinning

Fig. 40. High resolution micrograph of parent and twin crystals in Cu-4%Al in [011]M projection revealing the intersection of a3 [111]M Frank dipole by a twinning front. The figure shows that the upper part of the dipole is sheared and is incorporated into the twin lattice, whereas the lower part is still in the parent crystal.

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front of a faulted dipole initially located on the K2 plane in the matrix. In this geometry the fault plane of the dipole intersects the observation plane producing a set of partial dislocations along the dipole length, shown by arrows. Because the dipole is not viewed along its line, the Z-type configuration of the defect is not evident. The figure shows that the upper part of the dipole, visible in the center, has been sheared and is already incorporated into the twin lattice, whereas the lower part is still in the parent crystal. The micrograph illustrates the rather complicated nature of the twinning front on the atomic scale and indicates that shuffling of atoms occurs in large volumes of the crystal in the neighborhood of the defect and over a number of glide planes. Detailed analysis of this process is beyond the scope of the present work. The transformed dipoles should have strength comparable to that of their parent dislocations but because the faulted dipoles are fine they are expected to be less effective obstacles for mobile dislocations.

9.5. Transformation of Stacking Fault Tetrahedra As discussed in Section 8.4, in the present work stacking fault tetrahedra (SFT) are commonly observed in the substructure before the onset of twinning (see also [63]). These defects must interact in some way with a twinning front. SFTs and the structures that originate from them are small and their analysis by TEM provides a challenge, however, they present another group of defects which undergo twinning transformation, this type of transformation has not yet been discussed in the literature. SFTs consist of intrinsic stacking faults on four non-coplanar {111} planes bounded by a 110 stair-rod dislocations aligned along the edges of the tetrahedron [80,83,86]. All 6 of these component defects must undergo co-ordinated shearing if the SFT is to be fully incorporated into the twin lattice. One must develop a geometrical foundation with which to describe the transformation of partial dislocations, their glide planes, and the stacking faults comprising an SFT. Dislocation-based processes responsible for the incorporation of this defect into the twin lattice are discussed later in this section. Consider a stacking fault tetrahedron, ABCD, with intrinsic stacking faults on four {111}M planes and stair-rod dislocations along the 110M edges, AB, BC, CD and DA, as shown in Fig. 41a. The change in shape of this tetrahedron caused by twinning, i.e. transformation of parent {111}M planes and Burgers vectors of stair rod dislocations, was discussed in Section 5. To determine the nature of partial dislocations in the twin lattice, both the transformation of the dislocation line and the Burgers vector must be considered. All stair-rod dislocations are of pure edge character, so that, referring to the tetrahedron, in the parent lattice a dislocation aligned along AB has a Burgers vector lying along CD, similarly, a dislocation aligned along BD has a Burgers vector along AC, and so forth. These glide planes are (100)M for dislocations running along AB and CD, (010)M for those along AD and BC, and (001)M for those along AC and BD (Table 1). One can determine the transformation of these planes by examining the distortion of the cube caused by the operation of the correspondence matrix on it. As discussed in Section 5 and shown in ¯ T , (001)M to (11¯ 1) ¯ T , and (010)M to (10 ¯ 1) ¯ T . Thus, Table 1, (100)M is transformed to (1¯ 11) of six matrix stair-rod dislocations, the two aligned initially along AD and BC will become

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(a)

(b)

Fig. 41. (a) Detailed structure of a stacking fault tetrahedron in the matrix. (b) Distorted configuration obtained by operation of the correspondence matrix on the original tetrahedron.

¯ T and a [020] ¯ T Burgers vectors, their sessile in the twin lattice: they will inherit a6 [101] 6 lines are sheared along AD’ and BC respectively, and they have {110}T glide planes in the twin. The four remaining stair-rod dislocations remain also sessile, they have {111}T glide planes and a6 110T Burgers vectors in the twin lattice. The stacking faults initially on the ABD and ACD {111}M planes will be deposited on two {100}T planes, ABD’ and ACD’, bordered by dislocations aligned along the AB, AD’, BD’, AC and CD’ edges of the distorted tetrahedron. These features are represented schematically in Fig. 41b which shows the geometry of a stacking fault tetrahedron transformed by twinning. We now discuss the dislocation-based mechanism by which this type of transformation can be accomplished. In the previous section we considered transformation of faulted dipoles consisting of partial dislocations bordering a stacking fault on either the K1 or K2 plane in the matrix. We use this as a foundation from which to develop the geometry of interaction of a twinning partial with a three-dimensional stacking fault tetrahedron. The Burgers vectors of stairrod dislocations forming the base of the tetrahedron (AB, BC and CA edges in Fig. 41a) ¯ component normal to the twinning plane and equal to one third of the have an a9 111 interplanar spacing, (d(111) ¯ /3). The stair-rod dislocations aligned along the AD, BD and CD edges have Burgers vectors in the twinning plane, as a result of twinning their lines will be sheared to new positions but their Burgers vectors will remain unchanged. It is clear that the SFT does not develop plane topology, which precludes any spiraling motion of Shockley dislocations around a pole provided by any dislocation on a {111}M plane. It follows that a pole mechanism, in which a single dislocation might sweep all {111}M planes occupied by the SFT and incorporate it into the twin lattice, is not responsible for the transformation in this case. ¯ M plane of the tetraheConsider an a6 [121] twinning dislocation moving on the (111) dron (ABC in Fig. 42a). The twinning dislocation reacts with stair-rods along their lines producing new dislocations which enclose stacking fault on the base plane. The outside part of the twinning partial will continue its motion on the same plane thus extending the stacking fault and changing the stacking sequence in the parent lattice. For the SFT to be ¯ M incorporated into the twin, another twinning dislocations must operate on every (111) over the height of the defect. However, because atoms in the tetrahedron are displaced by (d(111) ¯ /3) with respect to those outside the tetrahedron across the stacking faults on inclined {111}M planes there is no continuity of the twinning plane across the defect, and it

§9.5


(a)

341

(b)

Fig. 42. Diagrammatic representation of the interaction of a twinning dislocation with a stacking fault tetrahedron. (a) An a6 [121]M twinning dislocation reacts with stair-rods along their lines producing new dislocations which enclose stacking fault on the basal plane of the SFT. The atomic planes inside the tetrahedron are displaced by d(111) ¯ /3 with respect to those outside across the stacking faults on inclined {111}M planes thus producing a discontinuity in the twinning planes across the defect. (b) The twinning dislocations will become stuck at the SFT, forming the characteristic Orowan loops.

is expected that twinning dislocations will become stuck at the SFT, forming the characteristic Orowan loop, as shown schematically in Fig. 42b. One possible mechanism by which the SFT may become fully incorporated into the twin is that the internal stress generated by the dislocation loops around it will relax, leading to mutual annihilation of opposite partials and transformation of the defect to its distorted configuration in the twin as shown in Fig. 41b. This would require that the SFT is penetrable by the twinning partial. This possibility may be inferred from the atomistic simulations of the interaction of SFTs with dislocations in copper carried out by Wirth, Bulatov and de la Rubia [99]. Fig. 43a shows the relaxed structure of a SFT in 110 projection [99]. The figure shows stacking faults formed on (111) planes. As discussed above, there is a discontinuity in the {111} glide planes across the SFT. The work of Wirth et al. [100] indicates that such a defect represents a strong obstacle to the motion of perfect dislocations, but that at a shear stress of approximately 300 MPa the SFT becomes penetrable through operation of a very interesting mechanism. A dislocation becomes dissociated into two Shockley partials bounding a stacking fault on its glide plane, at high enough stress the leading partial can penetrate the SFT and as it passes through the defect it leaves behind an extended stacking fault which is subsequently removed by the trailing partial. As a result, the SFT is sheared by the dislocation and acquires a jog with Burgers vector a2 110, the SFT preserves its integrity and remains intact. During the intersection, the leading partial passing the SFT acquires a critical angle of approximately 80 degrees (Fig. 43b). This suggests that twinning dislocations may not necessarily form loops around the defect, as discussed above, but the SFT may become penetrable with the right combination of resolved shear stress and bowing angle of the impacting twinning dislocation, leading to the formation of a small jog along the glide plane of the twinning partial. The results of Wirth et al. [99] indicate that some additional local atomic shuffling must occur in the SFT as the leading partial passes through it, rendering this defect a less resistant obstacle to the trailing partial. It is proposed here that a similar process might be expected to occur in the case of twinning when a wall of twinning partials moves through the lattice containing SFTs. With the high stresses obtaining in the twinning region and availability of an appropriate

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(a)

(b) Fig. 43. (a) The relaxed structure of a stacking fault tetrahedron in 110 projection after showing a discontinuity ¯ twinning planes across the SFT, after Wirth, Bulatov and De la Rubia [99]. (b) Atomistic simulation in the (111) of the intersection of a stacking fault tetrahedron by a dislocation dissociated into Shockley partials at the moment of intersection of the SFT by a leading Shockley dislocation. The photograph shows high energy atoms within the core structure of these defects, after Wirth, Bulatov and De la Rubia [100].

§10


343

arrangement of twinning partials, the SFT may be penetrable by the twinning front and therefore will assume its twinned configuration. Obviously, further atomistic simulations are required to determine whether this mechanism operates in the case of twin transformations. The twinned form of the SFT is a high energy configuration due to the faults on {100}T planes, it is not clear whether changes occur which would reduce this energy. It might be expected that the transformed SFT will rearrange to form a perfect tetrahedron in the twin, which may occur by, for example, a combination of short range vacancy diffusion, dislocation climb, and/or glide of transformed defects. Detailed atomistic simulations are needed to shed light on this problem. Thus, apart from the readily-defined dislocation arrangements discussed earlier, the twin structure contains a high density of very fine defect structures which comprise both small tetrahedra and dot-like defects in the form of vacancy clusters that presumably are derived from defects in the parent material.

10. New elements of the twinned structure The previous sections have given details of the observed nature and spatial arrangement of defects resulting from passage of a twinning front through the substructure characteristic of fcc material deformed heavily in tension. Geometrical considerations by means of which transformation products may be predicted were also discussed. Briefly, the twin substructure is comprised of well-defined dislocation layers on {100}T planes in the twin resulting from the transformation of dislocations lying on primary and cross {111}M planes in the matrix. The substructure contains cube dislocations originating from transformation of primary dislocations in the parent, these are easily recognized in low stacking fault energy materials since they are not dissociated into Shockley partials. They are often observed in a characteristic arrangement of arms attached to long jogs aligned along the parent-twin interface (e.g., Fig. 16). The substructure also contains dislocations dissociated on {111}T planes resulting form the transformation of Lomer– Cottrell networks which are a prominent feature of the matrix. The distribution of these features is inhomogeneous, their density varies, as in the parent crystal. There are also many dislocations resulting from transformation of other secondary matrix dislocations. These often quite irregular dislocations contribute to the development of complicated threedimensional arrangements in the twin. Other characteristic defects easily recognizable in the transformed substructure are two families of short faulted dipoles aligned along two 110T directions in the twin lattice, one of these is aligned along the matrix-twin interface. Finally the twinned structure contains a large density of fine nano-scale debris arising from severe refinement of defects by chopping. Three-dimensional defects inherited from the parent SFTs also contribute to the fine debris structure. So far, only those defects which result from twinning transformation of defects already existing in the parent substructure have been considered in any detail. There are other entities in the twinned microstructure, only briefly mentioned above, which are a direct product of the twinning process and are unrelated to any pre-existing structures. These elements of the twinned microstructure, extrinsic stacking faults, Frank dislocations, secondary stacking faults, and fine debris, will be considered here.

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10.1. Extrinsic stacking faults In the single crystals initially oriented for primary glide studied here, twin transformation is never 100% complete, some regions of untransformed matrix remain in the structure [57,59,61,62,64,85]. The volume of untransformed material varies between 30–60% depending upon the stacking fault energy of the material, the deformation conditions [57,61, 85], and deformation history [101]. After twinning stops, the sample comprises a layer-like structure consisting of twin and parent lamellae. In material with low stacking fault energy the thickness of the layers is of the order of microns, but in materials with higher stacking fault energy, such as copper deformed at low temperatures, the scale of the lamella is of the order of only 50–100 nm, as shown in Fig. 44 (see also [61,85]). The substructure of twinned crystals, in addition to various defects inherited from the matrix, contains long layers of stacking fault parallel to the twinning plane. It has been mentioned that these define the position of the interface (e.g., Figs 15, 17, 25, 26, 28, 33 etc.) but they have not been analyzed in detail. Fig. 45 shows the twinned structure in Cu-8%Al crystals imaged in the g = 020 twin reflection under complementary two-beam bright and dark-field diffracting conditions. The stacking faults of interest here are extended along the twinning plane and are inclined to the foil plane thus giving oscillatory contrast in the form of fringes. The character of these defects will be examined here and a detailed analysis of the displacement vector associated with them will be given in Section 10.3. The character of the stacking fault can be determined by examining the nature of the outer fringes in bright or dark field under known

Fig. 44. High-resolution TEM observations of the layer-like structure in copper after twinning deformation at 4.2 K. The scale of the matrix-twin layers in this area is of the order of 50–100 nm.

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Fig. 45. Complementary bright-dark field TEM images of stacking faults in Cu-8%Al single crystals deformed ¯ at room temperature. The specimen is imaged in both bright and dark-field modes using g = 020 ad g = 020 ¯ (b), (d) Dark field images, g = 020 and 020. ¯ operating reflections. (a), (c) Bright field images, g = 020 and 020. Analysis of the nature of the outermost fringes of the stacking faults in bright and dark-field allows determination of the sense of the slope of the stacking fault plane with respect to the direction of the operating reflection and the orientation of the foil normal, which in turn indicates that the fault in the central area of the micrograph is extrinsic in nature.

diffracting conditions, as first suggested by Howie [102]. In principle, one bright field or one dark field image and its associated diffraction pattern provide enough information to determine the character of the stacking fault. In practice, difficulties may arise in recognizing the type of the outer fringes on the image background (white or dark). To avoid this ambiguity, Fig. 45 shows two pairs of strong bright-dark field images using g = 020 and ¯ operating reflections. The analysis is focused on the fault in the central area of the g = 020 micrograph, which does not overlap with other faults on parallel twinning planes. The dynamical theory of electron diffraction [65] shows that under two-beam brightfield diffracting conditions a stacking fault inclined to the foil produces a symmetrical image with the same type of fringes at the top and the bottom of the foil. On the other hand, two-beam dark-field diffracting conditions give rise to an antisymmetrical image with opposite types of fringes at the top and the bottom of the foil. The dynamical theory

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also predicts that dark-field and bright-field images produce the same type of fringes at the top of the foil (white-white or dark-dark) and complementary fringes at the bottom of the foil (white-dark or dark-white) arising from anomalous absorption effects [65]. Fig. 45a reveals that in the bright-field image, the left outermost fringe of the fault at T is dark. The right outermost fringe must of course also be dark and is marked B on the micrograph. In the complementary dark-field image taken using the same operating reflection (Fig. 45b), the left outermost fringe at T is dark, whereas the right outermost fringe at B is white. Bright-dark field images taken using opposite direction operating reflections in Figs 45c and 45d, reveal that the outermost fringes of the fault under brightfield conditions are white (Fig. 45c). Under dark-field diffracting conditions (Fig. 45d), the left outermost fringe of the stacking fault appears white, at T, while the right outermost fringe is dark, at B. This set of complementary dark and bright field images allows the nature of the outer fringes and the bottom B and the top T of the foil to be identified in the micrographs. We can deduce the extrinsic nature of the stacking fault by examining these images with respect to the direction of the operating reflection and the orientation of the foil normal, a method first described by Art and coworkers [103], and later applied by Bell and Thomas [104,105] to tilting-beam microscopes. The extrinsic nature of various stacking faults observed in the substructure can be independently verified by matching the experimental images with those obtained in simulation calculations. It should be noted that there are other stacking faults in the neighborhood of the extrinsic faults considered above which exhibit different contrast properties, these defects are considered in Section 10.2. ¯ M As indicated above, the observed stacking faults are residual areas of matrix (111) planes remaining after twinning deformation is complete. In other words these are twinning planes immersed in the homogeneously twinned lattice which have not been swept by the twinning dislocation. In the twin orientation these faults represent extra planes inserted into the crystal they are thus extrinsic in nature. Several processes may be envisaged by which untransformed regions may remain after twinning stops. One possible mechanism has been discussed by Niewczas and Saada [64] in relation to the twinning source in the single crystals considered here. These authors argue that while the proposed source is capable of nucleating a a6 [121] twinning partial necessary to produce twinning deformation, macroscopic growth of the twin by a pole mechanism requires that a long layer of untransformed matrix is left in the twinned material. This layer will be visible as a wide stacking fault extended along the twinning plane and terminated by a jog which nucleated the twinning dislocation on one side of the fault. These defects may also be a consequence of the nature of the twinning planes in a highly dislocated substructure, their uneven topology and lack of continuity at various defects will prevent the advancing twinning dislocations from sweeping every glide plane so that untransformed layers will be left in the crystal. Finally, during the propagation of twinning fronts, if one or more twinning partials get stuck at some defect, layers of untransformed matrix may be left within the twinned material. This aspect will be developed in more detail in the next section. 10.2. Frank dislocations TEM observations show that many stacking faults are associated with characteristic dislocations observed in the twinned structure. Fig. 46 is a micrograph of the substructure in a

§10.2


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¯ T Fig. 46. Dark-field strong beam TEM observations of the substructure in a Cu-8%Al single crystal in a [115] ¯ M Frank dissection parallel to the cross plane of the matrix. The micrograph shows a large number of a3 [111] locations along the interface which give a characteristic oscillatory contrast when the twinning plane is in the reflecting position, after Basinski et al. [29]. The inserts show a magnified image of a Frank dislocations.

¯ operating reflection deformation twin in a single crystal of Cu-8%Al taken in the g = 111 of the twinning plane. There are large numbers of defects lying along the interface, the characteristic strong oscillatory contrast indicates that they are inclined to the foil plane. Previously, Basinski and co-workers [29] reported similar structures. The insert in the micrograph show a magnified images of two such defects. Since the twinning plane is in the reflecting position its Burgers vector must have a non-zero component perpendicular to the ¯ T twinning plane. Basinski and co-workers [29] showed that these structures are a3 [111] Frank dislocations lying along the [011]T direction, i.e. the intersection of the parent primary plane and the twinning plane. These dislocations are a prominent feature of the twin substructure and may be used as a fingerprint for identification of twinned material. Observations of the dislocation substructure in copper, which has a higher stacking fault energy than the alloy, show that the same feature can be seen here. Fig. 47 shows weakbeam analysis of the dislocation substructure in twinned copper using four reflections avail¯ T foils. There are extended stacking fault on the twinning plane running apable with [115] ¯ 1¯ proximately horizontally on the micrographs. The stacking fault is invisible in the g = 11 ¯ reflection, Fig. 47a, and shows vanishing, or residual contrast in g = 220, Fig. 47b. The interface is decorated with various dislocations giving strong dot-like contrast when the twinning plane is set in diffracting position (Fig. 47a), indicating that their Burgers vec-

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Fig. 47. Weak-beam analysis of the dislocation substructure in twinned copper using four reflections available ¯ ¯ T foils. Extended stacking faults on the twinning plane in the form of fringes are visible in 200 and 020 in [115] ¯ T Frank dislocation shown by an arrow in (a) exhibits very strong contrast in operating reflections. The a3 [111] ¯ and 220 ¯ reflections and weaker contrast in 200 and 020 ¯ reflections. Part of the dislocation line running along 111 ¯ T. the interface is shown by an arrow in (b). An arrow in (d) points to a faulted dipole along [011]

¯ plane. Some of these dislocations are visible when tor has a component out of the (111) the twinning plane is inclined to the electron beam, Figs 48b–d. The dislocation shown by an arrow in Fig. 47a exhibits typical features of an as-transformed primary dislocation. The length of the dislocation segment immersed in the twin is of the order of 100 nm, the dislocation is not dissociated into partials and is associated with a stacking fault on the ¯ operating reflection show that part twinning plane. Micrographs taken using the g = 220 of the dislocation line, shown by an arrow in Fig. 47b, extends along the interface. If the ¯ T character, originating from the transformation of a [101]M , it dislocation were of a2 [020] 2 would vanish in the 200 reflection. However, Fig. 47 shows that the defect exhibits very ¯ 1¯ and 220 ¯ reflections (Figs 47a, b) and only slightly weaker contrast strong contrast in 11 ¯ T Burgers vector ¯ in the 200 and 020 reflections (Figs 47c, d), indicating that it has a3 [111] (Table 3b). The presence of Frank dislocations in the twinned substructure has been known for some time, e.g., Venables [94] and there has been some discussion about how they are produced [29,64]. Venables [94] suggested that the Frank dislocation results from the reaction of a

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¯ 1] ¯ M dislocation, Fig. 48. Diagrammatic representation of the interaction of a twinning partial with a primary a2 [10 after Szczerba [46]. (a) The twin dislocation presses against the primary dislocation and reacts with the segment ¯ M and (111) ¯ M twinning plane. (b) Interaction of the aligned along [011]M , the intersection of primary (111) primary and twinning dislocations leads to the formation of a Frank dislocation along the kink line between nodes N1 and N2 . The two arms of Shockley dislocation can wind around a primary dislocation transforming it to Frank dislocation and build a macroscopic twin by a pole mechanism. (c) Configuration of the kink which leads ¯ M = a [020]T ) between nodes N1 and N2 . (d) The to the formation of a segment of cube dislocation ( a6 [4¯ 2¯ 4] 2 node conversion mechanism of Basinski, Szczerba, Embury [96] in which two arms of the twinning dislocation exchange their position and again can wind around the primary pole as in configuration (b). This will incorporate the whole primary dislocation line as a Frank dislocation in the twin.

twinning partial with a segment of primary dislocation aligned along [011]M according to the equation: a ¯ ¯ a a ¯ ¯ a ¯ ¯ [101]M + [121]M = [11 1]M = [11 1]T . 2 6 3 3 He argued that these reactions represent the most efficient way of stopping propagating twin boundaries and also explain why the twins are stable upon removal of the applied stress. It is appropriate now to revisit these interactions and to discuss details of the dislocation mechanism leading to the formation of Frank dislocations and how they are incorporated into the twin lattice. Fig. 48 is a diagrammatic representation of events which may occur when a twin¯ 1] ¯ M dislocation. The propagating twin dislocation ning partial interacts with primary a2 [10 presses against the primary dislocation and the segment aligned along [011]M , the intersection of primary and twinning plane, reacts with the oncoming twinning partial (Fig. 48a). Depending upon the orientation of the kink segment along [011]M with respect to the

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impacting twinning partial, two different configurations may result [46]. In the first case, shown in Fig. 48b, interaction of the primary and twinning dislocations leads to the formation of a Frank dislocation along the kink line between nodes N1 and N2 , according to the reaction suggested by Venables. In this geometry the right arm ‘R’ of the twinning partial enters the node N1 whereas the left arm ‘L’ comes out of the node N2 [46]. The reaction with the Shockley right arm occurring at node N1 is thus: a ¯ ¯ a a ¯ [101]M + [121]M + [111] M = 0. 2 6 3 The reaction involving the left Shockley arm at node N2 is: a ¯ ¯ a ¯ M + a [101]M = 0. [111]M + [1¯ 2¯ 1] 3 6 2 As discussed in Section 9.2, the Burgers vector of a primary dislocation has a component normal to the twinning plane equal to the spacing of these planes. The dislocation segment not parallel to the twinning plane acts as a pole for the twinning partial, the sense of spiraling is opposite to the previously considered a2 [101]M dislocation (compare Figs 19 ¯ 1] ¯ T jog and 48b). The two arms of Shockley dislocation anchored at the ends of the a3 [11 at nodes N1 and N2 , can wind around a primary dislocation outside the nodes extending the stacking fault in the way discussed in Section 9.2.2, but in the opposite sense. The right arm winds up the pole whereas the left arm winds down the pole as shown in Fig. 48b. During spiraling glide the twinning partial reacts with primary dislocation on every adjacent ¯ 1] ¯ T Burgers vector glide plane producing segments of sessile Frank dislocations with a3 [11 in the twin lattice and the dislocation line is deposited on a (100)T plane. On the other hand, if the configuration of the kink is as shown in Fig. 48c, the reaction of the twinning partial with the primary segment along [011]M leads to the formation of a segment of cube dislocation between nodes N1 and N2 according to the following reactions. For an ‘R’ Shockley at node N1 : a a a ¯ M = 0. [101]M + [121]M + [4¯ 2¯ 4] 2 6 6 For an ‘L’ Shockley at node N2 : a ¯ ¯ a a ¯ M = 0. [424]M + [10 1]M + [1¯ 2¯ 1] 6 2 6 However, it is evident from Fig. 48c, that as the arms of the twinning partial are wrapping in opposite directions outside the nodes, the right arm ‘R’ comes back and interacts with the primary pole at node N2 , whereas the left arm ‘L’ comes back and reacts with the primary pole at node N1 . The nature of the nodes is such that the twinning dislocation ‘R’ comes into the node N2 , whereas the Shockley partial ‘L’ comes out of node N1 [46]. In this process the two arms of the twinning dislocation have exchanged positions so that the original right twinning arm ‘R’ is now on the left side of the jog ready to wind up around the primary pole, whereas the left arm ‘L’ is on the right side of the jog and is ready to

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wind down the primary pole as in the previous configuration. This will incorporate primary poles into the twin lattice, transforming them to Frank dislocations according to Venables’ reaction. Between the nodes N1 and N2 the dislocation segment exists as an a6 [424]M dislocation plus two Heidenreich–Shockley partials, in total adding up to a Frank dislocation according to the reaction a a ¯ M + a [1¯ 2¯ 1] ¯ M = a [111] ¯ M, [424]M + [1¯ 2¯ 1] 6 6 6 6 or in twin coordinates a a a ¯ a ¯ [020]T + [121]T + [121]T = [111] T. 2 6 6 6 If this triplet is treated as a single dislocation, the node equations at N1 and N2 read: a a a ¯ ¯ M = 0 at node N1 , [111]M + [101]M + [1¯ 2¯ 1] 3 2 6

and

a ¯ ¯ a ¯ ¯ a [111]M + [10 1]M + [121]M = 0 at node N2 . 3 2 6 The above discussion shows that a twinning partial can always find a path to wind around a pole dislocation and therefore to develop a macroscopic twin utilizing a pole mechanism. From the perspective of defect transformation, the two configurations discussed above are equivalent and both lead to the incorporation of a primary dislocation into the twin lattice, in this case as a Frank dislocation. Formation of Frank dislocations will thus not depend upon the nature of the kink which is formed. This ingenious manoeuvre, known as the node conversion mechanism, proposed by Basinski, Szczerba and Embury [96] can be applied to other dislocation configurations and interactions. For example, it can be shown that the transformation of a primary dislocation of appropriate sign having a kink segment along the interface into a cube dislocation, and its further incorporation into the twin lattice, is governed by the same principles. This will be discussed later in relation to Fig. 51. Basinski, Szczerba and Embury [96] also demonstrated that the pole mechanism of Cottrell and Bilby for fcc twinning [88] can produce macroscopic twins if both arms of the winding twinning dislocation and the node conversion mechanism are taken into account. To complete the above discussion it is necessary to consider details of twin growth at the planar level by examining the paths of twinning dislocations interacting with jogs formed between nodes N1 and N2 in Fig. 48. ¯ 1] ¯ M dislocation in [011]M projection Fig. 49a shows the unrelaxed structure of an a2 [10 ¯ revealing the topology of (111)M twinning planes between nodes N1 and N2 introduced by the defect. An extra-half plane is located at layer 6, as shown in the figure. In the first step, the twinning Shockley dislocation gliding on the same lattice plane (layer 6) reacts with an a ¯ ¯ a ¯ ¯ 2 [101]M dislocation and forms an 3 [111]M Frank dislocation (Fig. 49b). During the first revolution, the arms of the twinning partial sweeping outside the nodes come back from

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(b)

(c)

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Fig. 49. Diagrammatic representation of the cross-section of a jog between nodes N1 and N2 showing twin ¯ 1] ¯ M dislocation growth in the configuration of Fig. 48b at the planar level. (a) The unrelaxed structure of an a2 [10 ¯ 1] ¯ M Frank dislocation by reaction of an a [10 ¯ 1] ¯ M located at layer ‘6’ in [011]M projection. (b) Formation of a3 [11 2 primary dislocation and an a6 [121]M twinning dislocation. (c) Configuration of the jog after the first revolution of ¯ the arms of the twinning partial. (d) Development of the macroscopic twin by twinning partials gliding on {111} layers in every subsequent revolution indicated by circled numbers in the direction shown by arrows.

the right side on layers 7 and 5. The twinning partials are likely to become stuck at the Frank jog, however if they break away they will continue their spiraling motion. During subsequent revolutions the twinning partials will always come back from the right side of the jog sweeping layers 8 and 4, 9 and 3, 10 and 2 etc., fully incorporating the Frank dislocation into the twin lattice. However, as mentioned above, if after the first revolution the arms of the twinning partial get stuck at the jog between nodes N1 and N2 , their arms outside the nodes may continue their glide and as they come back to the jog, they will leave part of their glide planes untransformed. Niewczas and Saada [64] discussed a similar case during the operation of a twinning source and argued that in most cases a wall of twinning partials will be built at the jog. These dislocations should eventually break away to release the internal stress arising from the wall thus transforming these layers into twin orientation. However, under appropriate circumstances if two Shockley partials pass the jog at the same time, the energy barrier of the interacting dislocations is reduced to zero [64]. Thus

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Fig. 50. Diagrammatic representation of the cross-section of the jog between nodes N1 and N2 showing twin growth in the configuration of Fig. 48c at the planar level. (a) Unrelaxed structure of an a2 [101]M dislocation located at layer ‘6’ in [011]M projection. (b) The twinning dislocation approaches the a2 [101]M jog from the left hand side and reacts with it producing an a6 [424]M dislocation. (c) During the first revolution, indicated by circled “1”, the Shockley partial arms after sweeping around nodes N1 and N2 come back to the jog from the right side on layers 5 and 6. (d) The twin thickens by twinning arms moving outside the jog. During subsequent revolutions shown by circled numbers, twinning partials come back to the jog from the left side on layers 4 and 7, 3 and 8, 2 and 9 etc. A triplet of dislocations immersed in the twin lattice, consisting of a6 [424]M and two Shockley partials circled in (d), is equivalent to a Frank dislocation. If twinning dislocations during any revolution get stacked at the jog, this will leave layer/layers of untransformed matrix within the twin lattice.

there is a probability that Shockley arms may continue their motion between nodes N1 and N2 , developing a homogeneous twin with Frank dislocations fully incorporated into the twin lattice. There is some experimental evidence that such a situation may exist under deformation conditions [106]. Let us now consider details of the interaction of a twinning partial with a primary dislocation containing a2 [101]M kink segments in the twinning plane (Fig. 48c). Fig. 50 shows the unrelaxed structure of an a2 [101]M dislocation in [011]M projection. Geometrically, this configuration is a 180◦ rotation of the previous configuration around the projection axis, the dislocation extra-half plane is located on layer 6 and extends towards the right side of the micrograph. Consider a twinning Shockley extending the stacking fault on layer 5. The

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dislocation approaches the a2 [101]M jog from the left hand side and reacts with it producing an a6 [424]M dislocation. The geometry of this configuration requires that during the first revolution the Shockley partial arms sweeping around nodes N1 and N2 , come back to the jog from the right side on layers 5 and 6, fully transforming these layers to twin orientation and leaving a segment of Shockley partial stuck at the jog (Fig. 50c). The twinning arms outside the jog continue their motion, wrap around poles, and during subsequent revolutions come back to the jog from the left side on layers 4 and 7, 3 and 8, 2 and 9 etc., producing a macroscopic twin. This leaves a set of dislocations consisting of a6 [424]M plus two Shockley partials in the fully-twinned lattice which together are equivalent to a Frank ¯ M + a [1¯ 2¯ 1] ¯ M = a [111] ¯ M ). However, as in the previous dislocation ( a6 [424]M + a6 [1¯ 2¯ 1] 6 3 case, if during a second revolution indicated by a circled number 2 in Fig. 50d, partial dislocations gliding on adjacent planes are stuck at the jog, this will not stop the twin from growing since the twinning arms may continue their spiraling motion outside the jog. In this way, this mechanism also will leave layers of untransformed matrix within the twin lattice. The presence of untransformed layers may be unavoidable because Shockley dislocations located on adjacent planes may not always be able to break away form the jog in individual attempts, thus leaving layers of stacking faults in otherwise fully-twinned material. TEM observations of the substructure reveal in many places arrangements of two or three straight dislocations, associated with the interface, which are of a form expected from the processes discussed above (see also [64]). This revives the old problem of dislocations passing stress barriers, which must play an important role in preventing completion of the twinning transformation of the material [21,42,107–109]. For a more complete discussion of passing stress barriers and references to other work the reader is referred to [64] and [97]. From the analysis presented here and in Section 9.2 it follows that a parent primary dislocation may be transformed either to a cube dislocation or to a Frank dislocation in the twin lattice depending upon its sign with respect to the incoming twinning partial. To illustrate this, Fig. 51 shows two configurations of primary dislocations containing kink segments of orientation such that a cube dislocation results from the operation of the pole source. This process is analogous to that discussed with reference to Fig. 48 however, due to the different nature of nodes N1 and N2 , the reaction between primary and twinning dislocations will always yield cube dislocations in the twin lattice irrespective of the nature of the kink segment. In Fig. 51a, interaction of the primary and twinning dislocations leads to the formation ¯ T along the kink line between nodes N1 and N2 . In this configuration of a6 [424]M = a2 [020] the right arm ‘R’ of the twinning partial enters node N1 whereas the left arm ‘L’ comes out of node N2 . The sense of spiraling is such that the left arm of the Shockley partial winds up and the right arm winds down the primary pole transforming it to a cube dislocation in the twin lattice (Fig. 51a). For the kink configuration shown in Fig. 51b, the reaction of the twinning partial with the primary segment along [011]M leads to the formation of a segment of Frank dislocation between the nodes. However, as in the configuration in Fig. 48c, as the arms of the twinning partial wrap around the primary poles outside the nodes, the right arm, ‘R’, winds down, comes back and interacts with the primary pole at node N2 , whereas the left arm, ‘L’, winds up, comes back to the primary pole and reacts with it at node N1 . As before, the two

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(c) Fig. 51. Diagrammatic representation of the interaction of a twinning partial with a primary a2 [101]M dislocation. ¯ T cube (a) Interaction of the primary and twinning dislocations leads to the formation of an a6 [424]M = a2 [020] dislocation along the kink line between nodes N1 and N2 . The two arms of Shockley dislocation wind around a primary dislocation building a macroscopic twin by a pole mechanism and transforming primary dislocations ¯ M to cube dislocations in the twin. (b) Kink configuration which leads to the formation of a segment of a3 [111] Frank dislocation between nodes N1 and N2 . (c) After node conversion the two arms of the twinning dislocation exchange positions and again can wind around the primary pole as in configuration (a). This incorporates the whole primary dislocation line into the twin as a cube dislocation (compare Fig. 48 and Fig. 19).

arms of the twinning dislocation have exchanged positions and the twinning arm ‘R’ which was originally on the right is now on the left side of the jog ready to wind down around the primary pole, similarly, arm ‘L’ is now on the right side of the jog, ready to wind up the primary pole thus transforming these dislocations to cube dislocations. It is seen ¯ M in Fig. 51c that between nodes N1 and N2 the dislocation segment exists as an a3 [111] Frank dislocation and two Heidenreich–Shockley partials whose sum is a cube dislocation, ¯ M + a [121]M + a [121]M = a [424]M = a [020] ¯ T ). Formation of Frank or cube ( a3 [111] 6 6 6 2 dislocations is equally probable since at the onset of twinning the dislocation substructure contains primary dislocations of both signs. To account for the large number of Frank dislocations observed at the parent-twin interfaces in Cu-8%Al crystals, Basinski and co-workers [29] suggested yet another possible mechanism for their formation. According to these authors these defects could originate ¯ T jog lying along the [011]T direction which, in order to lower its energy, disfrom a2 [020] ¯ T Frank dislocation and two a [1¯ 2¯ 1] ¯ T Shockley partials bordering an sociates into an a3 [111] 6 intrinsic stacking fault on the twinning plane. The extrinsic stacking fault associated with

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(a)

(b)

¯ T jog lying Fig. 52. Mechanism of formation of an extrinsic Frank dislocation by dissociation of (a) an a2 [020] ¯ T Frank dislocation and two a [1¯ 2¯ 1] ¯ T Shockley partials bordering an intrinsic along [011]T into (b) an a3 [111] 6 stacking fault on the twinning plane (after Basinski et al. [29]).

Frank dislocations formed in this way would arise from two overlapping intrinsic faults, as shown schematically in Fig. 52. Finally it should be mentioned that rigorous TEM analysis in different operating reflections is required to differentiate these two dislocations in the twinned structure of fcc materials, as discussed in Sections 9.2 and 10.2.

10.3. Secondary stacking faults Stacking faults on close-packed {111} planes non coplanar with the twinning plane have occasionally been observed in the structure of twinned materials studied in this work. The thickness of these defects may range from one to a few atomic layers. The present studies have shown that, as expected, lower stacking fault energy alloys are more likely to develop these structures than higher stacking fault energy materials. Fig. 53 shows TEM observations of the substructure in Cu-8%Al crystals in a section parallel to the (100)T plane in the twin, revealing stacking faults on three noncoplanar ¯ T twinning plane are, as dis{111}T planes. The stacking faults extended along the (111) cussed in two previous sections, represented by fringes running from the bottom left to the top right of the micrograph. These defects are visible in 020 and 002 reflections, Figs 53a ¯ reflections, Fig. 53c and Fig. 53d respectively. As and 53b, and vanish in 022 and 111 the electron wave crosses the stacking fault, it undergoes a phase shift. If this phase shift is equal to 2π , or an integral multiple of it, the stacking fault is invisible [65,66]. The contrast properties shown by these defects are compatible with faults associated with both ¯ T and R = ± a [121]T displacement vectors, consistent with the view that R = ± a3 [111] 6 these faults are produced either during formation of Frank dislocations or by dissociation ¯ T jogs. However, as discussed in Section 10.2, stacking faults associated with of a2 [020] a ¯ ¯ a ¯ a 3 [111]T and 2 [020]T jogs have an R = 6 [121]T displacement vector. This provides independent verification that wide faults extended along the twinning plane are direct products of twinning produced by glide of a6 [121]T twinning dislocations. Another feature visible in Fig. 53 is that of secondary stacking faults on two non¯ T planes recognized by their characteristic fringe spacing and coplanar (111)T and (111) their alignment approximately perpendicular to the trace of the twinning plane. These de-

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Fig. 53. Weak-beam TEM views of the substructure in Cu-8%Al crystals taken in various operating reflections ¯ T twinavailable with sections parallel to the (100)T plane in the twin. Stacking faults extended along the (111) ning plane are visible in 020 and 002 reflections in (a) and (b). The contrast properties of these faults show that ¯ T their displacement vector is R = a6 [121]T . Secondary stacking faults on two non-coplanar (111)T and (111) planes are recognized by fringes aligned approximately perpendicular the interface faults. These faults are produced by Shockley dislocations accumulated on the twinning plane cross-slipping onto secondary planes. The micrographs are printed as negatives for better resolution.

fects arise from the relaxation of internal stress by cross slip of Shockley partial dislocations accumulated on the twinning plane onto the secondary {111}T planes in the twin. Fig. 54 is a high resolution micrograph of a Cu-4%Al crystal in a [011]T section, it shows the formation of a secondary microtwin consisting of approximately 5–10 {111} layers. The process occurs by annihilation of opposite sign partial dislocations nucleating at two ledges, propagating along the interface, as shown in Fig. 54. It might be argued that annihilation of these dislocations occurred dynamically during twin growth as the ledges pass each other. Such a process would probably locally stop the twin from growing and would stabilize the untransformed matrix.

10.4. Fine debris TEM observations of the alloys studied reveal consistently that the internal structure of mechanical twins contains a high density of fine dislocation debris, a feature which has been

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Fig. 54. High resolution TEM observations of a Cu-4%Al crystal in a [011]T section showing a secondary microtwin formed inside the primary twin by annihilation of partials of opposite sign nucleating at two ledges propagating along the interface.

mentioned several times throughout this work but their nature has not yet been discussed. Fine dislocation debris, visible under weak-beam conditions, constitutes a major component of the dislocation substructure in twinned material. Every weak-beam image presented in this work has a background of dot-like debris distributed inhomogeneously throughout the volume of the twinned material. Niewczas provided some quantitative analysis of the evolution of debris in copper single crystals deformed at 78 K and showed that it consists of various kinds of small defects usually less than 20 nm in size [63]. These defects include primary edge dislocation dipoles and loops of predominantly vacancy type, vacancy-type faulted dipoles, stacking-fault tetrahedra with intrinsic faults, and point defect clusters also most probably of vacancy type and less than a few nm in size. Passage of a twinning front transforms the debris into new stable defects in the twin and also causes further refinement. Previous studies on high purity copper single crystals deformed at 4.2 K [60,62,85, 101,110] have shown that a large density of both recoverable and unrecoverable defects accumulates in the lattice, the latter control the flow stress of the material. The recoverable defects do not contribute to the flow stress but do contribute to the electrical resistivity induced during plastic deformation, this contribution anneals out from the structure between 78 K and 300 K [60,62,85,110,111]. Niewczas and coworkers [62] have shown that the twinned material suddenly acquires a very large concentration of recoverable defects, a phenomenon not observed during the earlier stages of deformation. This effect has been attributed to the refinement of pre-existing debris and other defects during the twinning process, the small defects become unstable and anneal by short circuit diffusion at various sinks present in the structure [62]. This process likely occurs in other materials that undergo twinning deformation.

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The difficulty in characterizing defects that are inherited from fine debris during twinning arises from two sources. First, their size is beyond the resolving power of the weakbeam technique and intensive studies with high resolution microscopy have not yet been carried out. Secondly, it is relatively easy to predict the shape of three-dimensional defects in the twin based on geometrical considerations, their equilibrium configuration in the twin lattice however is less easy to predict. There are also outstanding questions related to the detailed processes that occur during transformation of these defects, in addition to issues related to the stability of as-transformed debris in the new lattice in the presence of other defects. Details of such problems are beyond the scope of this work. Also, atomistic simulations [100] indicate for example that while a perfect SFT remains stable after being sheared by a perfect dislocation, the truncated SFT is absorbed by the same dislocation and disappears from the structure. Other debris might be expected to react similarly to interactions with twinning dislocations or twinning fronts. Obviously there is a need for systematic atomistic simulation studies to throw more light on the nature of very small defects and how they are affected by passage of a twinning front.

11. The strength of twinned crystal It was mentioned (Section 5) that certain transformations lead to an increase in the Burgers vector of the product, making these defects stronger obstacles for mobile dislocations. For ¯ T example, dislocations with primary a2 [101]M Burgers vector acquire the longer a2 [020] a ¯ a ¯ Burgers vector in the twin, also, 3 [111]M faulted dipoles acquire the longer 6 [141]T Burgers vector. Since very large numbers of dislocations undergo these transformations one would expect that a homogeneous slab of twinned material should be stronger than the material from which it originates. This contribution to strengthening arising from transformation of dislocations during twinning has been referred to in the literature as Basinski hardening (Kalidindi et al. [112]). Mechanical data reported here support the concept that twinning leads to macroscopic strengthening in the case of fcc single crystals. This is illustrated by Fig. 6, which shows an increase in flow stress even though there is a decrease in specimen cross-sections due to twinning. Niewczas and co-workers estimated that the strength of twinned copper single crystals in shear is about 1.89 times that of the parent crystal [62]. Micro-hardness measurements [29] also suggest that twinned fcc crystals are harder than untwinned ones. As we have shown, there is a spectrum of other transformations which do not necessary lead to stronger and more stable defects. At the onset of twinning, mobile dislocations gliding under high applied stress encounter a twin boundary or stacking faults extended along the twinning plane which provide strong obstacles to continued glide. This is known as the Hall–Petch mechanism of strengthening, which arises from the reduction in dislocation mean free path. In the case of the materials studied here this mechanism may be more important than contributions from defect transformations. However, as was shown by Kalidindi and co-workers in hexagonal titanium, where the scale of the twin structure is much larger than that produced in typical fcc materials, the Basinski hardening will be more effective than Hall–Petch strengthening [112]. In

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view of this, it should prove interesting to determine which defect transformations are responsible for strengthening deformation twins in hexagonal and perhaps also other crystal systems.

12. Summary In this chapter, we have documented the dislocation substructure in highly strained copper and copper alloy crystals initially oriented for single glide and have analyzed the consequences of passage through this complex structure of a twinning front. The relation between various types of dislocations in the parent and the twinned crystals and the nature of new lattice defects resulting from twinning is analyzed. Dislocation transformations can be classified into five groups depending upon the character of the parent defect and its transformation product. The first group includes transformation of dislocations which are glissile in the matrix but sessile in the twin. This group is ¯ T represented by the transformation of a2 [101]M primary dislocations to either cube a2 [020] a ¯ ¯ ¯ dislocations or to 3 [111]T Frank dislocations in the twin. Also in this class, (111)M pri¯ M cross planes which transform to {100}T in the twin, rendering dislocamary and (11¯ 1) tions lying in these planes sessile. The second group includes those dislocations which are sessile in the matrix but be¯ M and come glissile in the twin. This group of transformations is represented by a2 [011] a ¯ 2 [110]M Lomer or Lomer–Cottrell dislocations lying along [011]M and [110]M , conversion of {100}M to {111}T glide planes renders these formerly sessile dislocations mobile in the twin lattice. The next group, glissile to glissile transformations, includes all secondary dislocations having Burgers vectors in planes K1 and K2 , which are undistorted by passage of the twinning front. These dislocations are important in the development of the three-dimensional dislocation substructure in the twin lattice. Next are the sessile to sessile transformations represented by faulted dipoles and their products. These defects have Burgers vectors normal to the K1 and K2 planes, they may or may not inherit similar structures in the twin lattice. Finally, a fifth group of transformations has been identified here, that of three-dimensional structures such as stacking fault tetrahedra and small defect clusters. In addition to the high density of transformed dislocations visible using conventional TEM methods, the twin substructure contains another smaller-scaled component not immediately recognized against the background of defects discussed above. This component includes numerous small defects such as short sessile faulted dipoles and large densities of fine dislocation debris inherited from the parent. We showed that the pole mechanism can effectively accomplish the twinning transformation. It combines the elementary dislocation processes occurring during interaction of a twinning dislocation with a given defect with the elements of the transformation of the defect by homogeneous shear, as described by the correspondence matrix. Predictions arising from these considerations have been validated by observations of actual defects in the twinned lattice. However, the pole mechanism does suffer from the well-known passing


361

stress problem during twin growth. In some circumstances the passing stress barrier may be overcome by appropriate configurations of passing dislocations or by groups of dislocations on the twinning front. Many defects including three dimensional defects such as stacking fault tetrahedra are transformed by these incoherent twinning fronts propagating through the lattice. The discussion showed that the transformation of partial dislocations cannot be accomplished by a pole mechanism because of the shorter thread length which would prevent a twinning dislocation from revolving freely around it. The same is true for partial dipoles containing stacking faults such as Frank dipoles on the K2 plane in the matrix. A fundamental consequence of the nature of the twinning transformation in crystals containing dislocations is that the substructure acquires a high density of stacking faults along the parent-twin interface and layers of untransformed matrix, after the twinning process ceases. These features are a major component of the substructure and are characteristic of deformation twinning in fcc lattice.

Acknowledgments I wish to express my sincere gratitude to the late Professor Z.S. Basinski and to Professor M.S. Szczerba with whom I had many stimulating discussions over the years, which shaped my understanding of twinning transformations. I benefited a great deal from their knowledge, experience and patient advice. I also wish to thank Ms. Sylvia Basinski for years of encouragement, support and friendship. I am grateful to her for reading the manuscript despite sight problems, and for valuable comments and corrections, which allow me to present this article in its current form. Financial support from the Natural Sciences and Engineering Research Council of Canada is also gratefully acknowledged.

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CHAPTER 76

Elasticity, Dislocations and their Motion in Quasicrystals K. EDAGAWA Institute of Industrial Science, The University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8505, Japan and

S. TAKEUCHI Department of Materials Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510, Japan



Contents 1. Introduction 367 2. Elastic properties of quasicrystals 368 2.1. Phonon and phason degrees of freedom 368 2.2. Generalized elastic theory 376 2.3. Phonon elasticity 381 2.4. Phason elasticity 383 2.5. Phonon–phason coupling 387 3. Dislocations in quasicrystals 390 3.1. Perfect dislocations in quasicrystalline lattice 390 3.2. Displacement field and self free energy of dislocations 393 4. Dislocation motion in quasicrystals 398 4.1. Dislocation mechanism of deformation 398 4.2. Dislocation glide process 399 4.3. Dislocation climb process 405 4.4. Plastic homology 407 Appendix A: Irreducible strain components 411 Appendix B: Derivation of eqs (47) and (48) by the generalized Eshelby’s method References 413

412

1. Introduction Quasicrystals have a peculiar type of ordered structure characterized by crystallographically disallowed rotational symmetry and by quasiperiodic translational order [1–3]. Since the discovery of the quasicrystal by Shechtman et al. in 1984 [4], physical properties of quasicrystals have attracted much attention by solid-state physicists and materials scientists. The mechanical property is one of them. Due to its peculiar structural order, mechanical properties, including elastic and plastic properties, of quasicrystals exhibit characteristic features different from those of crystalline matter. Originating in the quasiperiodic translational order, quasicrystals have a special type of elastic degrees of freedom, termed as phason degrees of freedom [3,5]. Quasicrystals are accompanied by the phason elastic field in addition to the phonon (conventional) elastic field. The generalized elasticity of quasicrystals is described in terms of the two types of elastic fields [6–8]. Within linear elasticity, the elastic free energy of quasicrystals consists of three types of quadratic terms: phonon–phonon, phason–phason and phonon–phason coupling terms. Correspondingly, elasticity of quasicrystals comprises the three parts: pure phonon elasticity, pure phason elasticity and phonon–phason coupling. In relation to the generalized elasticity of quasicrystals, the concept of dislocations in quasicrystals was theoretically established [5,6,9] in an early stage of the research on quasicrystals. Experimentally, since the first demonstration by Shibuya et al. [10] in 1990 of the plastic deformation of an Al-Cu-Ru icosahedral quasicrystal at high temperatures, it has been shown that quasicrystals, both icosahedral and decagonal quasicrystals, are generally deformable at high temperatures above 0.8Tm (Tm : melting temperature), as reviewed in [11–13]. Wollgarten et al. [14,15] have shown by transmission electron microscopy (TEM) that high temperature deformation of icosahedral Al-Pd-Mn is brought by a dislocation process. Until recent years, it has been generally believed that the plasticity of quasicrystals is carried by a glide process of dislocations, and various models of the deformation mechanism have been proposed based on the dislocation glide [16–20]. However, recently, Caillard et al. [21–23] have shown by TEM that the high temperature deformation of icosahedral Al-Pd-Mn is brought not by a glide process but by a pure climb process. In this article, we review the present understandings of characteristic features of elasticity, dislocations and their motion in quasicrystals. This article is organized as follows. In Section 2, we review elastic properties of quasicrystals. Here, we show in Section 2.1 that quasicrystals have the phason elastic degrees of freedom in addition to the phonon (conventional) elastic degrees of freedom. In Section 2.2, we review the generalized elastic theory constructed by incorporating the two types of elastic degrees of freedom. Here, explicit formulae are presented for the two most important classes of quasicrystals, that is, icosahedral and decagonal quasicrystals. Sections 2.3, 2.4 and 2.5 are devoted to the review of experimental and theoretical works related, respectively, to the three parts of elasticity of quasicrystals, that is, pure phonon elasticity, pure phason elasticity and phonon–phason

368

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Ch. 76

coupling. In Section 3, we present characteristic features of dislocations in quasicrystals. In Section 3.1, we show the definition of a perfect dislocation in quasicrystals and review which types of dislocations have been experimentally observed. In Section 3.2, we demonstrate a calculation of displacement field and self free energy of a dislocation. Using a general formula of the self free energy, relative stability among dislocations with various Burgers vectors is discussed. Section 4 is devoted to dislocation motion. After reviewing briefly the experimental results about dislocation mechanism of deformation of quasicrystals in Section 4.1, we discuss detailed mechanisms of dislocation glide and dislocation climb in Section 4.2 and 4.3, respectively. Finally, in Section 4.4, we show a homologous nature in the temperature dependence of the yield stress for icosahedral quasicrystals and discuss it in terms of microscopic deformation mechanism common to icosahedral quasicrystals.

2. Elastic properties of quasicrystals 2.1. Phonon and phason degrees of freedom In general, the diffraction intensity function I (q) of solid is given by I (q) ≡ |S(q)|2 , S(q) =

ρ(r) exp(−2πiq · r) dr.

(1)

Here, ρ(r) is the atomic-density function in real space. The function I (q) observed experimentally for the quasicrystal has the following characteristics [1–3]: (1) It consists of δ-functions. (2) The number of basis vectors necessary for indexing the positions of the δ-functions exceeds the number of dimensions. (3) It shows a rotational symmetry forbidden in conventional crystallography. Conversely, we define the quasicrystal as the material with a ρ(r) which gives a diffraction intensity function I (q) satisfying these conditions. The condition (1) implies that this material has a kind of long-range translational order. The conditions (2) and (3) indicate that the order is not periodic. The translational order defined by the conditions (1) and (2) is called quasiperiodicity. According to the definition, every d-dimensional quasiperiodic atomic-density function ρ(r) can be expressed as ρ(r) =

mi ∈I

ρm1 ,...,mN exp[2πiGm1 ,...,mN · r],

(2)

§2.1

Elasticity, dislocations and their motion in quasicrystals

369

where ρm1 ,...,mN is the Fourier component associated with the reciprocal vector

Gm1 ,...,mN =

N

mn qn .

n=1

Here, {qn } (n = 1, 2, . . . , N ) (N > d) are the reciprocal basis vectors. Now, let us define an N -dimensional function ρ h (x1 , . . . , xN ) as h

ρ (x1 , . . . , xN ) =

mi ∈I

N mn xn . ρm1 ,...,mN exp 2πi

(3)

n=1

This function is periodic in every xn (n = 1, 2, . . . , N ) with the period of unity. Comparing eqs (2) and (3), we find ρ(r) = ρ h (x1 , . . . , xN ),

(4)

where xn = qn · r (n = 1, 2, . . . , N ).

(5)

This indicates that the d-dimensional quasiperiodic function ρ(r) can be described as a d-dimensional section of an N -dimensional periodic function. As an example, Fig. 1(a) presents a typical one-dimensional quasiperiodic structure known as a Fibonacci lattice, which is described as a one-dimensional section of a twodimensional periodic function (d = 1 and N = 2). Here, E and E⊥ denote the physical space and the complementary space perpendicular to it, respectively. In this case, the two-dimensional periodic structure consists of a periodic arrangement of a line segment extending in the direction of E⊥ . The line segment is called atomic surface. A point sequence is obtained on the E section, comprising an arrangement of two spacings L and S. Here, the slope of E with √ respect to the two-dimensional lattice is an irrational number τ (the golden mean: (1 + 5)/2). In this case, the resultant sequence of the two spacings becomes equivalent to the Fibonacci sequence and therefore it is called a Fibonacci lattice. The irrational slope indicates lack of periodicity in the arrangement of L and S. In the description of the Fibonacci lattice in Fig. 1(a), let us consider a translation of the two-dimensional periodic structure by a vector U with respect to the origin of the physical space E (Fig. 1(b)). The two structures on E before and after the displacement U shown in Figs 1(a) and (b) can be overlapped out to arbitrarily large finite distances by a finite translation in E . Two structures satisfying this condition are said to belong to the same local isomorphism class (LI class) [1–3]. Thus, the displacement U represents the degrees of freedom of generating a series of structures belonging to the same LI class. Obviously from the definition of the LI class, a series of structures in the same LI class

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Ch. 76

Fig. 1. A Fibonacci lattice (a), and a structure resulting from a displacement of U (b), that of u (c) and that of w (d).

are geometrically indistinguishable on any finite scale and thus they are also physically indistinguishable: they give the same diffraction intensity function I (q) and have the same energy. The vector U can be decomposed into u in E and w in E⊥ : U = u + w.

(6)

Here, u represents the degrees of freedom of d-dimensional translation in physical space, which conventional crystals also possess, and w represents (N − d) degrees of freedom characteristic of quasiperiodic system. As shown in Figs 1(c) and (d), while u results in translation of Fibonacci lattice in E , w generates a rearrangement of L and S. The two kinds of degrees of freedom are called phonon and phason degrees of freedom, and u and w phonon and phason displacements, respectively [3,5]. When these displacements vary spatially, the gradients of them yield distortion or strain. More specifically, while the gradient of u, i.e. ∇u(r) =

∂ui ∂rj

gives conventional distortion (or strain), that of w, i.e. ∇w(r) =

∂wi ∂rj

§2.1


371

Fig. 2. A Fibonacci lattice (a), and its phonon-strained (b) and phason-strained (c) structures.

yields distortion (strain) called phason distortion (strain) .1 Here, u and w are the functions of only r ∈ E and ∂r∂ i denotes a spatial derivative in E . Figs 2(b) and (c) illustrate, respectively, a phonon-strained and a phason-strained structure of the Fibonacci lattice in Fig. 2(a). In Fig. 2(b), uniform phonon strain is introduced by a compression deformation of the two-dimensional structure. On the other hand, uniform phason strain is introduced by a shear deformation of the two-dimensional structure in Fig. 2(c). It is noteworthy that the phonon and phason stain fields should have quite different dynamical properties. As described above, phason displacement results in a local rearrangement of points (atoms) such as LS ↔ SL, which is called phason flip. Examples of phason 1 In general, the gradient of displacement vector is called ‘distortion.’ In the one-dimensional case, the distor∂w tions ∂u ∂r and ∂r also represent strains in the same form. In the case of two-dimensional and three-dimensional

∂u ∂u quasicrystals, the phonon strain should be defined as a symmetrical form 21 ( ∂r i + ∂rj ) to remove the component j

∂w

i

of rigid rotation while the phason strain should have the same form ∂r i as the distortion (see Section 2.2). j

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Ch. 76


Fig. 3. Examples of phason flip in a two-dimensional Penrose lattice.

flip in a two-dimensional Penrose lattice, which is known as a typical two-dimensional decagonal quasicrystal, are shown in Fig. 3. In this case, a lattice point in the hexagon makes a transition between two (meta-) stable positions. Introduction or relaxation of phason strain requires a combination of phason flips. Generally, the phason flip is a thermally activated process and thus relaxation of phason strain whose elementary process is phason flip must proceed relatively slowly like atomic diffusion in solids [3,5,8]. This is in sharp contrast to conventional phonon strain, which can be relaxed instantaneously via displacive phonon modes. Experimentally, thermally-induced phason flips have been investigated by neutron scattering [24–27], by Moessbauer spectroscopy [28], by NMR [29, 30] and specific heat measurements [31,32]. By in-situ high-temperature high-resolution transmission electron microscopy, direct observations of thermally-induced phason flips [33–35] and those of the phason-strain relaxation process [36] have been performed. The activation enthalpy of a phason flip or collective phason flips has been estimated to be ≈1 eV in an Al-Pd-Mn icosahedral quasicrystal by radiotracer diffusion experiments [37]. In general, to decompose properly the N total degrees of freedom into d phonon and (N − d) phason degrees of freedom, or equivalently, to embed properly a given ddimensional quasiperiodic structure into an N -dimensional hypercrystal, the information on the point group symmetry of the system is needed. Below, the method of embedding is briefly reviewed for the two most important classes of quasicrystals: icosahedral and decagonal quasicrystals [3,38–41]. To describe the structure of a three-dimensional icosahedral quasicrystal, we use a sixdimensional hypercubic lattice spanned by di (i = 1, . . . , 6): di = Mij ej ,

− 1 M = a 2τ 2 + 2 2

τ ⎢τ ⎢ ⎢1 ⎢ ⎢0 ⎣ 0 1 ⎡

1 0 −1 0 0 τ τ 1 τ −1 0 −τ

1 1 −τ 0 0 −τ

−τ τ 0 1 1 0

⎤ 0 0 ⎥ ⎥ 1 ⎥ ⎥, −τ ⎥ ⎦ τ −1

(7)

§2.1


373

√ where τ is the golden mean (= (1 + 5)/2) and a is the lattice constant. ei (i = 1, . . . , 6) are the orthonormal unit vectors of the six-dimensional space. The icosahedral point group Y is generated by a fivefold rotation C5 and a threefold rotation C3 . The lattice spanned by di (i = 1, . . . , 6) is invariant under these operations. The action of these operations on di (i = 1, . . . , 6) is given as 1 ⎢0 ⎢ ⎢0 Ŵ(C5 ) = ⎢ ⎢0 ⎣ 0 0 ⎡ 0 ⎢1 ⎢ ⎢0 Ŵ(C3 ) = ⎢ ⎢0 ⎣ 0 0 ⎡

0 0 0 0 0 1

0 1 1 0 0 0

0 0 0 0 0 0

0 0 0 1 0 0

⎤ 0 0⎥ ⎥ 0⎥ ⎥, 0⎥ ⎦ 1 0

0 0 0 0 0 0 0 0 0 0 0 1 0 −1 0 0 0 0 −1 0 1 0 0 0

⎤ 1 0⎥ ⎥ 0⎥ ⎥. 0⎥ ⎦ 0 0

(8)

The character table for the icosahedral group Y is presented in Table 1. Here, the irreducible representations of the group Y are a one-dimensional Ŵ 1 , two distinct three-dimensional Ŵ 2 and Ŵ 3 , a four-dimensional Ŵ 4 , and a five-dimensional Ŵ 5 . The six-dimensional representation in eq. (8) is reducible and can be decomposed into the sum of the irreducible representations as Ŵ = Ŵ 2 + Ŵ 3 . The subspace spanned by e1 , e2 and e3 , and the subspace spanned by e4 , e5 and e6 are the eigenspaces of the representations Ŵ 2 and Ŵ 3 , and correspond to the physical space E and its complementary space E⊥ , respec tively. Shown in Fig. 4(a) are the projections di (i = 1, . . . , 6) and d⊥ i (i = 1, . . . , 6) of the basis vectors di (i = 1, . . . , 6) onto E and E⊥ , respectively. Because the sixdimensional hypercubic lattice is invariant under the inversion Ci , it has the maximum icosahedral point group symmetry Yh = Y × Ci . By placing atomic surfaces spreading in E⊥ (in this case they are three-dimensional objects) at appropriate positions in the six-dimensional lattice, a three-dimensional icosahedral quasicrystalline structure with the symmetry Y or Yh is obtained as a section on E . Here, to preserve the symmetry, each of the atomic surfaces must satisfy the site symmetry of its position in the corresponding symmetry group. Similarly to the case illustrated in Fig. 1, a six-dimensional Table 1 Character table for the icosahedral group Y Y

E

12C5

12C52

20C3

15C2

Ŵ1

1 3 3 4 5

1 τ −τ −1 −1 0

1 −τ −1 τ −1 0

1 0 0 1 −1

1 −1 −1 0 1

Ŵ2 Ŵ3 Ŵ4 Ŵ5

374


Ch. 76

(a)

(b)

Fig. 4. (a) The projections di (i = 1, . . . , 6) and d⊥ i (i = 1, . . . , 6) of the basis vectors di (i = 1, . . . , 6) in eq. (7) for icosahedral quasicrystals onto E and E⊥ , respectively; (b) the projections di (i = 1, . . . , 5) and d⊥ i (i = 1, . . . , 5) of the basis vectors di (i = 1, . . . , 5) in eq. (9) for decagonal quasicrystals onto E1 , E2 and E⊥ , respectively. Here, di (i = 1, . . . , 4) have no components of E2 while d5 has the component of E2 only.

displacement U can be defined, which can be decomposed into u ∈ E and w ∈ E⊥ , representing the three-dimensional phonon and the three-dimensional phason displacements. It is noteworthy that the hypercubic lattice is not a unique choice for embedding icosahedral quasicrystalline structure. In fact, we can use any lattice spanned by the basis vectors of the form d′i = di + cd⊥ i (c: an arbitrary scale factor) instead of the hypercubic lattice ⊥ spanned by di = di + di in embedding any given structure. In other words, we can determine arbitrarily the length scale in E⊥ . This fact should always be born in mind in discussing the magnitudes of the physical-property parameters related to the length scale in E⊥ such as phason displacement, phason strain, phason elastic constants, etc. (see Sections 2.4, 3.1 and 3.2). To describe a three-dimensional decagonal quasicrystal, we use a five-dimensional lattice spanned by di (i = 1, . . . , 5): di = Mij ej ,

§2.1

375


⎡ a √ (c1 − 1) ⎢ 5 ⎢ a ⎢ √ (c2 − 1) ⎢ 5 ⎢ ⎢ M = ⎢ √a (c − 1) 3 ⎢ ⎢ 5 ⎢ a ⎢ √ (c − 1) 4 ⎣ 5 0

a √ s1 5 a √ s2 5 a √ s3 5 a √ s4 5 0

a √ (c2 − 1) 5 a √ (c4 − 1) 5 a √ (c1 − 1) 5 a √ (c3 − 1) 5 0

a √ s2 5 a √ s4 5 a √ s1 5 a √ s3 5 0

0

⎤

⎥ ⎥ 0⎥ ⎥ ⎥ ⎥ , 0⎥ ⎥ ⎥ ⎥ 0⎥ ⎦

(9)

c

where a and c are the lattice constants, and ci = cos(2πi/5) and si = sin(2πi/5). ei (i = 1, . . . , 5) are the orthonormal unit vectors of the five-dimensional space. The character table for the decagonal group C10v is shown in Table 2. The generators of this group are a tenfold rotation C10 and a mirror σv . The lattice spanned by di (i = 1, . . . , 5) is invariant under these operations. The action of these operations on di (i = 1, . . . , 5) is given by 0 ⎢ 0 ⎢ Ŵ(C10 ) = ⎢ 0 ⎣ −1 0 ⎡

1 1 1 1 0

⎤ −1 0 0 0 −1 0 ⎥ ⎥ 0 0 0⎥, ⎦ 0 0 0 0 0 1

0 ⎢0 ⎢ Ŵ(σv ) = ⎢ 0 ⎣ 1 0 ⎡

0 0 1 0 0

0 1 0 0 0

1 0 0 0 0

⎤ 0 0⎥ ⎥ 0⎥. ⎦ 0 1

(10)

This five-dimensional representation is reducible: Ŵ = Ŵ 5 + Ŵ 7 + Ŵ 1 . The subspace E1 spanned by e1 and e2 , the subspace E⊥ spanned by e3 and e4 , and the subspace E2 along e5 are the eigenspaces of the representations Ŵ 5 , Ŵ 7 and Ŵ 1 , respectively. E1 is parallel to the quasiperiodic plane and E2 is along the tenfold periodic directions in the physical

space E = E1 + E2 . In Fig. 4(b), the projections di (i = 1, . . . , 5) and d⊥ i (i = 1, . . . , 5) of the basis vectors di (i = 1, . . . , 5) onto each space are illustrated. Because the lattice spanned by di (i = 1, . . . , 5) in eq. (9) is invariant under the inversion Ci , it has the maximum decagonal point group symmetry D10h = C10v × Ci . There are seven point groups in the decagonal system: C10 , C5h , C10h , C10v , D10 , D5h and D10h . By placing atomic surfaces spreading in E⊥ at appropriate positions in the five-dimensional lattice, a threeTable 2 Character table for the decagonal group C10v C10v

E

2C10

2C5

Ŵ1

1 1 1 1 2 2 2 2

1 1 −1 −1 τ τ −1 1−τ −τ

1 1 1 1 τ −1 −τ −τ τ −1

Ŵ2 Ŵ3 Ŵ4 Ŵ5 Ŵ6 Ŵ7 Ŵ8

3 2C10

1 1 −1 −1 1−τ −τ τ τ −1

2C52

C2

5σv

5σd

1 1 1 1 −τ τ −1 τ −1 −τ

1 1 −1 −1 −2 2 −2 2

1 −1 1 −1 0 0 0 0

1 −1 −1 1 0 0 0 0

376


Ch. 76

dimensional decagonal quasicrystalline structure with any of those decagonal point group symmetries can be constructed. Here, to preserve the symmetry, each atomic surface must satisfy the site symmetry of its position in the given point group. A five-dimensional displacement U can be defined, which can be decomposed into u ∈ E and w ∈ E⊥ , representing the three-dimensional phonon and the two-dimensional phason displacements. As in the case of icosahedral quasicrystal, we can use any five-dimensional lattice spanned by the basis vectors of the form d′i = di + cd⊥ i (c: an arbitrary scale factor) in embedding any given structure.

2.2. Generalized elastic theory As in the preceding section, in quasicrystals there exist two types of elastic degrees of freedom, that is, phonon and phason degrees of freedom. In view of this fact, a generalized elastic theory can be formulated for the quasicrystals [6–9,42–45]. In this section, we review the theory and present the explicit formulae for the icosahedral and decagonal quasicrystals. i The spatial variation of u, i.e. ∇u(r) = ∂u ∂rj yields phonon (conventional) distortion while ∂wi ∂rj

yields phason distortion. Here, u and w are the functions of

∂ ∂ri denotes

a spatial derivative in E . The phonon strain should be defined

that of w, i.e. ∇w(r) =

only r ∈ E and as a symmetrical form

∂uj 1 ∂ui uij ≡ + 2 ∂rj ∂ri to remove the component of the rigid rotation which does not change the elastic free energy. i On the other hand, the phason strain should be defined as wij ≡ ∂w ∂rj . The elastic free energy density f is given as a function of the phonon strain uij and the phason strain wij . The function f can be expanded into the Taylor series in the vicinity of uij = 0 and wij = 0. In the regime with |uij |, |wij | ≪ 1, we would omit the third and higher order terms, leaving only the quadratic terms (linear elasticity): f = fu−u + fw−w + fu−w

1 w−w 1 u−w = Ciju−u kl uij ukl + Cij kl wij wkl + Cij kl uij wkl , 2 2

(11)

where fu−u , fw−w and fu−w are a pure phonon, pure phason and phonon–phason coupling w−w u−w terms, respectively, and Ciju−u kl , Cij kl and Cij kl are the corresponding second-order elastic constant tensors. The explicit form of f depends on the symmetry of the system and we derive those for icosahedral [6,9,42–45] and decagonal [46,47] quasicrystals below. For the icosahedral quasicrystal with the point group Y , the six components uij transform under (Ŵ 2 × Ŵ 2 )sym. = Ŵ 1 + Ŵ 5 , corresponding to dilatation and shear, respectively (see Table 1). The nine components wij transform under Ŵ 2 × Ŵ 3 = Ŵ 4 + Ŵ 5 . Now let u1 and u5 be the irreducible components of the phonon strain for Ŵ 1 and Ŵ 5 , respectively,

§2.2


377

and w4 and w5 be those of the phason strain for Ŵ 4 and Ŵ 5 , respectively. These irreducible strain components are given explicitly in Appendix A. Then, we find that there exist five quadratic combinations of strains that transform as scalars, i.e. under Ŵ 1 : u1 · u1 , u5 · u5 , w4 · w4 , w5 · w5 and u5 · w5 . This fact indicates that the elastic free energy density f can be rewritten as f = fu−u + fw−w + fu−w ,

1 1 fu−u = k1 u1 · u1 + k2 u5 · u5 , 2 2 1 1 fw−w = k3 w4 · w4 + k4 w5 · w5 , 2 2

fu−w = k5 u5 · w5 .

(12)

Here, ki (i = 1, . . . , 5) are scalar constants representing the elastic constants. In many works done so far on the elasticity of icosahedral quasicrystals, a different set of elastic constants have been used: λ and μ (Lame constants) for pure phonon elasticity; K1 and K2 for pure phason elasticity; and K3 for phonon–phason coupling [48–57]. They are related to ki (i = 1, . . . , 5) as [44,58] 1 1 λ = (k1 − k2 ), μ = k2 , 3 2 1 1 K1 = (4k3 + 5k4 ), K2 = (k3 − k4 ), 9 3

1 K3 = − √ k5 . 6

(13)

Inserting eq. (13) into eq. (12) leads to f = fu−u + fw−w + fu−w ,

1 fu−u = λu2ii + μuij uij , 2

2 4 1 1 2 − wij wij + τ w12 + τ −1 w21 fw−w = K1 wij wij + K2 wkk 2 2 3 + cyclic permutations , fu−w = K3 w11 u11 + τ −1 u22 − τ u33 + 2u23 τ −1 w23 − τ w32 + cyclic permutations .

(14)

It should be noted that the equations presented in [34,35] can be obtained from eq. (14) by the substitution of wij by wj i and the coordinate transformation (x → y, y → −x, z → z) [44,58]. Comparing eq. (14) with eq. (11), we finally obtain the elastic constant tensors, as follows. The pure phonon elastic constant tensor Ciju−u kl is identical to that for conventional isotropic solids: Ciju−u (15) kl = λδij δkl + μ δik δj l + δil δj k ,

378

Ch. 76


where δij is the Kronecker delta. The pure phason elastic constant tensor Cijw−w kl and the u−w phonon–phason coupling tensor Cij kl are shown in Tables 3 and 4, respectively, using a 9 × 9 representation. Because both uij and wij are centrosymmetrical, i.e. invariant under the action of the inversion operation, elastic properties should possess an intrinsic centrosymmetry. This fact indicates that all the point groups belonging to the same Laue class have exactly the same elastic properties. Thus, the above results derived for the group Y also apply to the group Yh = Y × Ci . Table 3 w−w The phason elastic constant tensor Cij kl for icosahedral quasicrystals with the point K

group Y or Yh . A–D are defined as: A = K1 − 32 , B = K1 + K2 (τ − 13 ), C = K1 + K2 ( 23 − τ ), D = K2 w−w Cij kl

kl = 11

22

33

23

31

12

32

13

21

ij = 11

A

D

D

0

0

0

0

0

0

22

D

A

D

0

0

0

0

0

0

33

D

D

A

0

0

0

0

0

0

23

0

0

0

B

0

0

D

0

0

31

0

0

0

0

B

0

0

D

0

12

0

0

0

0

0

B

0

0

D

32

0

0

0

D

0

0

C

0

0

13

0

0

0

0

D

0

0

C

0

21

0

0

0

0

0

D

0

0

C

Table 4 u−w The phonon–phason coupling constant tensor Cij kl for icosahedral quasicrystals with the point group Y or Yh . A–C are defined as: A = K3 , B = −τ K3 , C = τ −1 K3 u−w Cij kl

kl = 11

22

33

23

31

12

32

13

21

ij = 11

A

B

C

0

0

0

0

0

0

22

C

A

B

0

0

0

0

0

0

33

B

C

A

0

0

0

0

0

0

23

0

0

0

C

0

0

B

0

0

31

0

0

0

0

C

0

0

B

0

12

0

0

0

0

0

C

0

0

B

32

0

0

0

C

0

0

B

0

0

13

0

0

0

0

C

0

0

B

0

21

0

0

0

0

0

C

0

0

B

§2.2


379

For three-dimensional decagonal quasicrystals, uij is a 3 × 3 symmetrical tensor and wij is a 2 × 3 tensor, respectively. For the point group C10v , the six components of uij and wij transform under {(Ŵ 5 + Ŵ 1 ) × (Ŵ 5 + Ŵ 1 )}sym. = 2Ŵ 1 + Ŵ 5 + Ŵ 6 and (Ŵ 5 + Ŵ 1 ) × Ŵ 7 = Ŵ 6 + Ŵ 7 + Ŵ 8 , respectively (see Table 2). Let u1 and u′1 be the two irreducible components of the phonon strain for Ŵ 1 , and u5 and u6 be those for Ŵ 5 and Ŵ 6 , respectively. Let w6 , w7 and w8 be those of the phason strain for Ŵ 6 , Ŵ 7 and Ŵ 8 , respectively. These irreducible strain components are given explicitly in Appendix A. Then, we find the following quadratic combinations of strains that transform as scalars, constituting the elastic free energy density f : f = fu−u + fw−w + fu−w ,

1 1 1 1 fu−u = k1 u1 · u1 + k2 u1 · u′1 + k3 u′1 · u′1 + k4 u5 · u5 + k5 u6 · u6 , 2 2 2 2 1 1 1 fw−w = k6 w6 · w6 + k7 w7 · w7 + k8 w8 · w8 , 2 2 2 fu−w = k9 u6 · w6 .

(16)

Here, ki (i = 1, . . . , 9) are scalar constants representing the elastic constants. Eq. (16) can be written explicitly as: f = fu−u + fw−w + fu−w , 1 1 fu−u = c11 u211 + u222 + c12 u11 u22 + c13 (u11 + u22 )u33 + c33 u233 2 2 2 2 2 + 2c44 u23 + u31 + (c11 − c12 )u12 ,

1 2 2 2 2 fw−w = K1 w11 + w22 + w12 + w21 2 1 2 2 + K2 (w11 w22 − w21 w12 ) + K3 w13 , + w23 2 fu−w = K4 (u11 − u22 )(w11 + w22 ) + 2u12 (w21 − w12 ) ,

(17)

where a different set of elastic constants are used, which are:

c11 = k1 + k5 , c12 = k1 − k5 , c13 = k2 , c33 = k3 , c44 = K1 = k6 + k8 , K2 = k6 − k8 , K3 = k7 , K4 = k9 .

k4 , 4 (18)

Comparing eq. (17) with eq. (11), we obtain the three elastic constant tensors that are presented in Tables 5–7. We note that the form of the phonon elastic constant tensor is identical to that for the hexagonal crystals. The point groups D10 , D5h and D10h belong to the same Laue class as that of C10v and therefore the above results also apply to all of them. In the decagonal system, there is another Laue class to which the point groups C10 , C5h w−w and C10h belong. Similar analysis reveals that Ciju−u kl and Cij kl for the latter Laue class are

380

Ch. 76

K. Edagawa and S. Takeuchi Table 5 u−u The phonon elastic constant tensor Cij kl for decagonal quasicrystals with the point group C10 , C5h , C10h , D10 , D5h , C10v or D10h . There are five independent elastic constants: c11 , c12 , c13 , c33 and c −c c44 . c66 is given by c66 = 11 2 22 u−u Cij kl

kl = 11

22

33

23

31

12

ij = 11

c11

c12

c13

0

0

0

22

c12

c11

c13

0

0

0

33

c13

c13

c33

0

0

0

23

0

0

0

c44

0

0

31

0

0

0

0

c44

0

12

0

0

0

0

0

c66

Table 6 w−w The phason elastic constant tensor Cij kl for decagonal quasicrystals with the point group C10 , C5h , C10h , D10 , D5h , C10v or D10h w−w Cij kl

kl = 11

22

23

12

13

21

ij = 11

K1

K2

0

0

0

0

22

K2

K1

0

0

0

0

23

0

0

K3

0

0

0

12

0

0

0

K1

0

−K2

13

0

0

0

0

K3

0

21

0

0

0

−K2

0

K1

Table 7 u−w The phonon–phason coupling constant tensor Cij kl for decagonal quasicrystals with the point group D10 , D5h , C10v , or D10h u−w Cij kl

kl = 11

22

23

12

13

21

ij = 11

K4

K4

0

0

0

0

22

−K4

−K4

0

0

0

0

33

0

0

0

0

0

0

23

0

0

0

0

0

0

31

0

0

0

0

0

0

12

0

0

0

−K4

0

K4

identical to those for the former and that Ciju−w kl for the latter is slightly different; there are two independent elastic constants K4 and K5 in this tensor, as shown in Table 8.

§2.3


381

Table 8 u−w The phonon–phason coupling constant tensor Cij kl for decagonal quasicrystals with the point group C10 , C5h or C10h u−w Cij kl

kl = 11

22

23

12

13

21

ij = 11

K4

K4

0

K5

0

−K5

22

−K4

−K4

0

−K5

0

K5

33

0

0

0

0

0

0

23

0

0

0

0

0

0

31

0

0

0

0

0

0

12

K5

K5

0

K4

0

−K4

We define the phason stress σijw ≡

σiju ≡

∂f ∂uij

∂f ∂wij

, in addition to the conventional phonon stress

. From eq. (11), these stresses can be written as

u−w σiju = Ciju−u kl ukl + Cij kl wkl ,

w−w σijw = Ciju−w kl ukl + Cij kl wkl ,

(19)

which represents a generalized Hooke’s law. Introducing a phason body-force density fiw as well as the conventional phonon body-force density fiu , we obtain a generalized form of the elastic equations of balance: ∂ u σ + fiu = 0, ∂rj ij

∂ w σ + fiw = 0. ∂rj ij

(20)

Inserting eq. (19) into (20) leads to Ciju−u kl

∂ 2 wk ∂ 2 uk + Ciju−w + fiu = 0, kl ∂rj ∂rl ∂rj ∂rl

Ciju−w kl

∂ 2 wk ∂ 2 uk + Cijw−w + fiw = 0, kl ∂rj ∂rl ∂rj ∂rl

(21) 2

∂ukl u−u ∂ uk where we take into consideration the relation Ciju−u kl ∂rj = Cij kl ∂rj ∂rl .

2.3. Phonon elasticity As shown in eq. (11), the elastic free energy density f of quasicrystals has the three quadratic terms, which represent a pure phonon elasticity, a pure phason elasticity and phonon–phason coupling, respectively. As described in Section 2.1, relaxation of phason strain proceeds diffusively with thermally activated phason flips [3,5,8], indicating that at low temperatures the phason mode should be frozen, i.e. dtd wij (t) = 0. Even at high

382

Ch. 76


Table 9 Phonon elastic constants of various icosahedral quasicrystals: λ and μ are Lame constants; G = μ, B = 3λ+2μ 3 λ and ν = 2(λ+μ) are shear modulus, bulk modulus and Poisson’s ratio, respectively. The unit of λ, μ and B is GPa Alloy system

λ

μ(G)

Al-Li-Cu Al-Li-Cu Al-Cu-Fe Al-Cu-Fe-Ru Al-Pd-Mn Al-Pd-Mn Ti-Zr-Ni Cd-Yb Zn-Mg-Y

30 30.4 59.1 48.4 74.9 74.2 85.5 35.28 33.0

35 40.9 68.1 57.9 72.4 70.4 38.3 25.28 46.5

B 53 57.7 104 87.0 123 121 111 52.13 64.0

ν

Ref.

0.23 0.213 0.232 0.228 0.254 0.256 0.345 0.2913 0.208

[60] [62] [65] [65] [65] [66] [67] [64] [68]

temperatures, it can be regarded as being effectively frozen under applied stress oscillating with a much shorter period than the relaxation time of phason strain. If additionally the residual phason strain is zero, i.e. wij (t) = 0 = const., the elasticity of quasicrystals consists entirely of the phonon elasticity. As shown in eq. (15), icosahedral quasicrystals should behave as an isotropic elastic body.2 Actually, isotropic elasticity has been observed experimentally for icosahedral Al-Li-Cu [60–62], Al-Pd-Mn [63] and Cd-Yb [64]. Using various ultrasonic methods, the elastic constants have been measured for icosahedral quasicrystals of Al-Li-Cu [60,62], Al-Cu-Fe [65], Al-Cu-Fe-Ru [65], Al-Pd-Mn [65,66], Ti-Zr-Ni [67], Cd-Yb [64] and Mg-Zn-Y [68]. The evaluated elastic constants at λ room temperature are summarized in Table 9, where G = μ, B = 3λ+2μ and ν = 2(λ+μ) 3 are shear modulus, bulk modulus and Poisson’s ratio, respectively. First, we note that the bulk moduli differ considerably for different alloy systems. We find that they correlate well with the melting temperature, which is generally observed for conventional crystals. With the exception of Ti-Zr-Ni, the main feature characteristic of the quasicrystals is that the 3(1−2ν) Poisson’s ratio is relatively small, or equivalently, that the ratio G B = 2(1+ν) is relatively large. For example, the Poisson’s ratios of the elemental metals of Al, Cu and Pd are about 0.35, 0.37 and 0.39, respectively, at room temperature. Except for Ti-Zr-Ni, the Poisson’s ratios of the quasicrystals in Table 9 are much smaller than those values and close to those of typical covalent crystals such as Si (0.22) and Ge (0.21). Tanaka et al. [65] have pointed out that the small ν or equivalently the large G/B ratio is indicative of the existence of directional atomic bonds. The bulk modulus B is a measure of the resistance against the deformation in which atomic bond lengths change with keeping bond angles unchanged. In contrast, the shear modulus G is a measure of the resistance against the deformation in which bond angles change without changing bond lengths. Therefore, the G/B ratio should become large if the atomic bonds are directional, as in covalent crystals. In relation to this fact, covalent nature of the atomic bonds has been revealed experimentally in some quasicrystals and their crystal approximants [69–71]. 2 In the third-order elastic constants described as a sixth-rank tensor, anisotropy is theoretically expected and a sign of anisotropy has actually been observed in the study of nonlinear propagation of acoustic waves in an Al-Pd-Mn icosahedral quasicrystal [59].

§2.4

383


Table 10 Phonon elastic constants of an Al-Ni-Co decagonal quasicrystal [72]: c11 , c33 , c44 , c12 and c13 are the five independent elastic constants of this system; B and G are the bulk and shear moduli calculated as Voigt averages of the moduli cij ; ν is the Poisson’s ratio calculated from the B and G values. The unit is GPa, except for ν Alloy system

c11

c33

c44

c12

c13

B

G

ν

Al-Ni-Co

234.33

232.22

70.19

57.41

66.63

120.25

79.78

0.228

As shown in Table 5, there are five independent phonon elastic constants c11 , c33 , c44 , c12 and c13 in the decagonal quasicrystals and the form of the elastic constant tensor is identical to that of the hexagonal crystal. Isotropic elasticity is expected in the basal plane, which has been confirmed experimentally for an Al-Ni-Co decagonal quasicrystal by Chernikov et al. [72]. The same group has reported the values of a complete set of elastic moduli, which are presented in Table 10. Here, a striking feature to be noted is that the deviation from complete elastic isotropy is rather small. The compressional anisotropy and shear anisotropy, measured by the ratios c33 /c11 and c44 /c66 = 2c44 /(c11 − c12 ), respectively, are 0.99 and 0.79, indicating a high degree of elastic isotropy. Inelastic neutron scattering measurements have been done for a decagonal Al-Ni-Co [73] and considerably isotropic phonon dispersion relations have been obtained, which is consistent with the result of the elastic constant measurement. So far, many structural models have been proposed for decagonal quasicrystals, as reviewed by Ranganathan et al. [74]. Most of them have a layered structure, for which one would expect anisotropic elastic properties. Actually, phonon dispersion relations have been calculated for one of such a layered structure model, exhibiting quite large anisotropy [75]. Dmitrienko [76] has suggested the possibility of both icosahedral and decagonal quasicrystals having common local atomic ordering with icosahedral symmetry. The highly isotropic elastic properties may be attributable to such an isotropic local atomic arrangement. In Table 10, the bulk modulus B and shear modulus G are calculated as Voigt averages of the moduli cij , and the Poisson’s ratio ν is obtained from the B and G values. As in icosahedral quasicrystal, Poisson’s ratio is small and the ratio G/B is large, indicating the existence of directional atomic bonds also in decagonal quasicrystals.

2.4. Phason elasticity Phason elasticity is closely related to the problem of what is the physical origin of quasicrystalline structural order and has attracted much attention since the discovery of the quasicrystal. Despite years of study, there remain two plausible competing models that explain the realization of quasicrystalline long-range order. One is a perfectly ordered model [1–3,77]. In this model, the existence of local matching rules that force the formation of the quasicrystalline order is assumed. When atomic interactions are in accordance with the local matching rules, the quasicrystal is formed as an energetically favorable structure. Introduction of phason strain induces local atomic rearrangements and thus brings matching rule violations, which increase the elastic free energy by increasing the energetic part of the free energy. The other model is a random tiling [48], in which the quasicrystal is assumed

384


Ch. 76

to be stabilized by a configurational entropy originating in many nearly degenerate ways of packing structural units. Such a configurational entropy decreases with increasing average phason strain and thus the phason strain contributes to the elastic free energy f through its entropic part. Theoretically, quasicrystals in both models show δ-functional diffraction peaks and projection of the structure along a symmetry axis in both states preserves perfect quasicrystalline order unless the projection thickness is too small [48,78]. These facts make it difficult to distinguish between the two states by a simple diffraction experiment or by high-resolution transmission electron microscopy. In early studies [48,79,80], it was pointed out that in the perfectly ordered model the phasonic elastic free energy fw−w should show a linear dependence on the phason strain, that is, fw−w ∝ |∇w(r)|, instead of a quadratic form given in eq. (11). This arises from the conjecture that the number of the matching rule violation is in proportion to |∇w(r)| in the vicinity of |∇w(r)| = 0 and that each such defect independently costs a constant energy. In this case, the radius of the curvature of the energy surface is arbitrarily small at |∇w(r)| = 0 and therefore introduction of phason strain must be strongly suppressed especially at low temperatures. Such a state is called a state of locked phason or simply a locked state. In contrast, the elastic free energy in random tiling quasicrystals is shown to have a standard quadratic form [48,79,81]. The state with such a quadratic form of elastic free energy is called a state of unlocked phason or an unlocked state. Monte Carlo simulations using various tiling models have shown that a locked state is indeed realized as a ground state and that the locked state undergoes a transition to an unlocked state at some temperature. In general, such a transition occurs at 0 K in two-dimensional systems and at some finite temperature in three-dimensional systems [79–81]. Due to all these studies, it may be widely believed that the form of the phason strain dependence of elastic free energy can distinguish between the energetically stabilized perfectly ordered model and the entropically stabilized random tiling model, and that a quadratic form experimentally evidenced by phasonic diffuse scattering (described below) is in favor of the latter model. However, based on the calculations using model quasicrystals with interatomic interactions, Koschella et al. [82] and Trebin et al. [83] have recently claimed that even energetically stabilized quasicrystals should have a quadratic form of elastic free energy, indicating that the phasonic diffuse scattering does not immediately justify the random tiling model. The quadratic form of phasonic elastic free energy generates phason fluctuations. Such phason fluctuations should induce diffuse scattering around Bragg peaks in diffraction spectra just as phonon fluctuations generate conventional thermal diffuse scattering [48, 84]. Phasonic diffuse scattering has indeed been observed experimentally for quasicrystals of various alloy systems, which evidences the quadratic form of phasonic elastic free energy. The shape and the intensity of the diffuse scattering changes depending on the shape of the basin of the energy surface, which is characterized by the elastic constants. Therefore, in principle, by examining the phasonic diffuse scattering one can evaluate the phason elastic constants. The evaluation of the phason elastic constants by diffuse scattering measurements has been reported for icosahedral Al-Pd-Mn [51–55], Zn-Mg-Sc [57] and Al-Cu-Fe [56]. In [54] and [57], absolute-scale measurements of the diffuse scattering intensity have been reported for the Al-Pd-Mn and Zn-Mg-Sc systems, respectively. From the results of the absolute-scale measurements, the phasonic elastic constants K1 and K2 have been eval-

§2.4

385


Table 11 Phason elastic constants of various icosahedral quasicrystals evaluated by diffuse scattering measurements of X-ray, neutron and electron Alloy system

Al-Pd-Mn Al-Pd-Mn Al-Pd-Mn Al-Pd-Mn Al-Pd-Mn Zn-Mg-Sc Al-Cu-Fe

Source

X-ray neutron neutron neutron electron X-ray X-ray

Meas. temp. R.T. R.T. 1043 K R.T. R.T. R.T. R.T.

K1 kB T ′

K2 kB T ′

(atom−1 )

(atom−1 )

0.06 0.1 0.13 – – 0.5 –

−0.031 −0.052 −0.052 – – 0.075 –

K1 (MPa)

K2 (MPa)

43 72 125 – – 300 –

−22 −37 −50 – – −45 –

K2 K1

−0.52 −0.52 −0.4 −0.5 −0.5 −0.15 0.5

Ref.

[54] [54] [54] [51] [73] [57] [56]

uated on the assumption that the phonon–phason coupling K3 is negligible (see eq. (14), and Tables 3, 4). On the other hand, in [51,55,56], only the shapes of the diffuse scatterings have been analyzed and the ratios K2 /K1 have been evaluated on the assumption of K3 = 0. The evaluated values of the elastic constants are summarized in Table 11. In general, the absolute-scale measurements first give the elastic constants in the form of kBKTi ′ , where kB denotes the Boltzmann constant and T ′ the temperature at which the phason fluctuation equilibrates. To evaluate K1 and K2 , we assume T ′ = 773 K [54] for the room temperature measurements and T ′ = 1043 K for the measurements at 1043 K in Table 11. We notice that the phason elastic constants in Table 11 is by about three orders of magnitude smaller than the phonon elastic constants in Table 9. It should be noted, however, that the magnitudes of the phason elastic constants change depending on the arbitrarily chosen scale factor for E⊥ , as pointed out in Section 2.1 and that the ratio K⊥ /K (K⊥ = K1 , K2 ; K = λ, μ) by itself has no physical meaning. For the icosahedral quasicrystalline state to be thermodynamically stable, the elastic free energy density f in eq. (14) should be positive definite, imposing on the elastic constants the following condition [49]: 5 K1 + K2 > 0, 3

4 μ K1 − K2 > 3K32 . 3

(22)

This indicates that K1 should always be positive. Dividing the inequalities by K1 leads to K2 3 >− , K1 5

9K32 K2 3 < − . K1 4 4μK1

(23)

In addition to this thermodynamic stability, the system should satisfy another stability condition, which is of the hydrodynamic stability [49,85]. This condition is for the relaxation rate of phason fluctuation to be finite, which requires [49] 3K32 K2 3 < − , K1 4 (λ + 2μ)K1

18K32 K2 a < − K1 b bμK1

386


√ √ a = 27 − 9 5 ≈ 6.9, b = 15 5 − 27 ≈ 6.5 , K32 K2 3 >− + . K1 4 3(λ + 2μ)K1

Ch. 76

(24)

It should be noted that the hydrodynamic stability condition in eq. (24) is less restrictive than the thermodynamic one in eq. (23). If we assume K3 = 0, eqs (23) and (24) reduce to 3 K2 3 < − < 5 K1 4

(25)

3 K2 3 − < < , 4 K1 4

(26)

and

respectively. As in Table 11, the Al-Pd-Mn system has been studied most intensively. For this system, the following facts should be noted: (1) K1 and K2 are positive and negative, respectively, |K2 | ′ 1| (2) the values k|K ′ and k T ′ are smaller at room temperature (or at T ≈ 773 K) than BT B ′ those at T (≈ T ) = 1043 K, ′ 1| (3) the decrease in k|K ′ with the temperature change from T = 1043 to 773 K is more BT 2| significant than the decrease in k|K ′, BT (4) as a result, the ratio K2 /K1 also changes slightly; it changes from −0.4 to −0.5.

First, the signs and the values of K1 and K2 in (1) and (4) are consistent with the stability conditions. Generally speaking, |K1 | |K2 | ′ −1 , ∝ T kB T ′ kB T ′

is expected when the phasonic part of the elastic free energy f w−w consists entirely of the energetic contribution. On the other hand, if it consists entirely of the entropic contribution, as in the case of an ideal random tiling, |K1 | |K2 | ′ 0 , ∝ T kB T ′ kB T ′

is expected. The fact (2) implies |K1 | |K2 | ′ k , ∝ T kB T ′ kB T ′

(k > 0),

which is closer to the latter. The facts (3) and (4) indicate that the system is close to the instability points K2 /K1 = −0.6 (thermodynamic instability) in eq. (25) and K2 /K1 =

§2.5


387

−0.75 (hydrodynamic instability) in eq. (26) and that it approaches to them with decreasing temperature. Ishii [85] and Widom [49] have shown that the hydrodynamic instability at K2 /K1 = −0.75 leads to the symmetry breaking from Yh to D3d . In relation to this result, the formation of a modulated icosahedral phase with the modulation along a threefold direction has been reported in an Al-Pd-Mn alloy with the composition close to that of the icosahedral quasicrystal [86]. For Zn-Mg-Sc, K1 is considerably larger than that for the Al-Pd-Mn system while K2 is roughly the same, resulting in K2 /K1 = −0.15 that is far from instability points. For Al-Cu-Fe [56], a positive ratio of K2 /K1 = 0.5 has been reported, which is close to another instability point K2 /K1 = 0.75 that leads to the symmetry breaking to D5d [49,56,85]. Apart from the alloy systems listed in Table 11, for Al-Pd-Re [87] qualitatively different shapes of diffuse scatterings from those for Al-Pd-Mn have been reported, although the K2 /K1 ratio has not been evaluated quantitatively. The intensity and the shape of diffuse scatterings for Cd-Yb [87] have been shown to be similar to those for Al-Pd-Mn. For a Al-Ni-Co decagonal quasicrystal, anisotropic diffuse scattering has been observed in synchrotron X-ray diffraction measurements [88]. It has been shown that the observed diffuse scattering can be attributed partly to the phasonic origin, although no quantitative evaluation of K2 /K1 have been reported.

2.5. Phonon–phason coupling In contrast to pure phonon and pure phason elasticities described in the preceding sections, the number of studies so far done on the phonon–phason coupling is limited and we still have little knowledge about it. Henley [48] has discussed the microscopic origin of the phonon–phason coupling by using a toy atomic model of a quasicrystal, in which the atoms interact with a given set of pair potentials. In general, not all neighbor pairs can have optimal distances simultaneously; this results in frustration: some linkages are under compression and others under tension. The introduction of phason strain causes a statistical imbalance between the frequencies of the appearance of various symmetry-related linkages under compression or tension, resulting in the introduction of a global phonon strain. Such a mechanism implies that the phonon–phason coupling strength can be evaluated by measuring the magnitude of the phonon strain spontaneously introduced in crystal approximants to quasicrystals. Fig. 5(a) illustrates schematically an elastic free energy density f of a quasicrystal as a function of phonon strain u and phason strain w, in the vicinity of u = w = 0 (see eq. (11)). Here, the principal axes of the quadratic surface of f (u, w) do not coincide with the u and w axes, indicating a non-zero phonon–phason coupling. In the quasicrystalline state characterized by w = 0, the minimum of f lies at u = 0, as shown in the curve 1 in Fig. 2(b). In general, each crystal approximant structure is characterized by a nonzero uniform phason strain with a fixed value w0 . Now, let us consider an atomistic model quasicrystal with a certain set of atomic interactions. We can construct any crystal approximant structure by introducing the uniform phason strain w0 characterizing the crystal approximant. Then, the dependence of the elastic energy of the crystal approximant structure on the phonon strain u should be given as

388


Ch. 76

Fig. 5. A schematic illustration of an elastic free energy density f of a quasicrystal as a function of phonon strain u and phason strain w, in the vicinity of u = w = 0 (a). The cuts along the line w = 0 and w = w0 in (a) are drawn as the curves 1 and 2 in (b), respectively. The curve 3 in (b) represents the elastic free energy of the approximant phase, which is assumed to be identical to the curve 2 except for the constant term.

f (u, w0 ) at T = 0, as shown in the curve 2 in Fig. 5(b). Therefore, if we relax the structure while keeping the global phason strain of w = w0 , a spontaneous deformation u0 should take place, where u = u0 gives the minimum of f (u, w0 ). From the quantity u0 , we can evaluate the phonon–phason coupling constant because u0 is proportional to the coupling constant. By use of such a method, the coupling constants in a model decagonal quasicrystal and in model icosahedral quasicrystals have been evaluated by Koschella et al. [82], and Zhu and Henley [50], respectively. Koschella et al. [82] have used a two-dimensional binary tiling model with decagonal symmetry, where the atoms interact with pair potentials of the Lennard-Jones type. The coupling constant evaluated is about 3% of the shear modulus. Zhu and Henley [50] have calculated the phonon–phason coupling constants for realistic models of icosahedral Al-Mn and Al-Cu-Li, which represent the two major classes of quasicrystals. The results show that (1) the sign of the coupling constant K3 in eq. (14) is positive for Al-Mn and negative for Al-Cu-Li and (2) |K3 | is 0.003–0.02μ for Al-Mn and 0.02–0.1μ for Al-Cu-Li, where μ is the shear modulus. Here, it would be interesting to examine whether the magnitude of K3 is large enough to effect the elastic instability of the quasicrystal discussed in the preceding section. Comparing eqs (23) and (24) with eqs (25) and (26), we notice that K2 K32 , 3 ≪1 (λ + 2μ)K1 μK1

§2.5


389

is the condition for K3 to be neglected in the discussion of the elastic instability. The followings are crude estimations: for Al-Mn, we assume |K3 | = 0.01μ and adopt K1 = 70 MPa and μ = 70 GPa that are for Al-Pd-Mn in Tables 9 and 11, respectively, leading to K32 μK1

≈ 0.1; for Al-Cu-Li, we use |K3 | = 0.05μ and adopt K1 = 300 MPa and μ = 40 GPa K2

that are for Zn-Mg-Sc and Al-Li-Cu, respectively, giving μK31 ≈ 0.3. These values suggest that K3 values are not small enough to be completely neglected. We may be allowed to apply the above method to real crystal approximants. Here, it should be noted that we have to make the following assumption to apply the method to the real systems. When we evaluate experimentally the phonon strain introduced in a given crystal approximant phase, the phase should be in the stable state. This indicates that the elastic free energy f of this state must be lower than that of the corresponding quasicrystalline state characterized by u = w = 0. In fact, the phase transition from the quasicrystalline state to the crystal approximant state is generally induced by the decrease in the phason elastic constants, i.e. phason softening and also by the effect of higher-order terms in the expansion of f [49,85]. This fact indicates that the functional form of f in the vicinity of u = w = 0 should be modified considerably on the transition. Nevertheless, we may be allowed to assume that such a change in the functional form of f occurs in its w-dependence only, considering the nature of the quasicrystal-to-approximant phase transition. On this assumption, the u-dependence of the elastic free energy g(u) of the approximant phase should be identical to the u-dependence of the elastic free energy f of the quasicrystalline phase at w = w0 , i.e. f (u, w0 ) except for the constant term, as shown in the curve 3 in Fig. 5(b). We find in eq. (12) that the phonon and phason strain components u5 and w5 couple in icosahedral quasicrystals. This indicates that (1) only the crystal approximants with w5 = 0 induces phonon strain, and that (2) the induced phonon strain has only the u5 component, i.e. pure shear strain. The fact (1) poses a rather severe limitation in choosing the crystal approximant for the experiment. Most of the crystal approximants so far found are of the cubic system, which have only the w4 component, i.e. w5 = 0. There have been only two non-cubic approximants known so far, which have good structural quality suitable for the present experiments: one is an orthorhombic approximant in the Mg-Ga-Al-Zn system [89] and the other is a rhombohedral one in the Al-Cu-Fe system [90–92]. Recently, Edagawa [93] has measured the phonon strain induced in the crystal approximant of the 32 - 12 - 21 orthorhombic type in the Mg-Ga-Al-Zn system and estimated the phonon–phason coupling constant K3 , as follows. In the composition Mg39.5 Ga16.4 Al4.1 Zn40.0 , a metastable icosahedral phase produced by melt-spinning underwent a transition to the stable 32 - 21 - 12 orthorhombic approximant phase on heating. By X-ray diffraction measurements, the quasilattice parameter aq of the icosahedral phase and the lattice parameters a, b and c of the orthorhombic approximant have been evaluated. Without induced phonon strain, the lattice constants of the 23 - 21 - 12 orthorhombic approximant should be − 1 a0 = 2τ 4 aq 1 + τ 2 2

and b0 = c0 = a0 /τ.

390


Ch. 76

Comparing (a, b, c) with (a0 , b0 , c0 ), the induced phonon strain components, u11 = (a − a0 )/a0 , etc., were evaluated. Finally, the coupling constant K3 has been estimated to be ≈−0.03μ. Using μ = 46 GPa for a Zn-Mg-Y in Table 9, we obtain K3 = −1.4 GPa. In a following paper [94], it has been pointed out that an unnecessary constraint was imposed in the analysis of the data in [93]. The correction by removing this constraint has been shown to give a slightly larger values of K3 = (−0.04 ± 0.01)μ and K3 = −1.8 ± 0.4 GPa. The icosahedral Mg-Ga-Al-Zn is of the Frank–Kasper type and should be compared with the results of the calculation for Al-Cu-Li by Zhu and Henley [50]. The negative sign of K3 for the Mg-Ga-Al-Zn system is in agreement with the calculation. The |K3 | value is also in good agreement with the calculation, although the calculation has yielded a considerably large range of values. Recently, Edagawa and So [94] have performed the measurements of the phonon strain induced in a rhombohedral approximant in the Al-Cu-Fe system [90–92]. They have reported a small positive K3 value of about 0.004μ for this system.

3. Dislocations in quasicrystals 3.1. Perfect dislocations in quasicrystalline lattice Because of lack of periodicity, we cannot introduce a perfect dislocation into a quasicrystal simply by applying the conventional definition of crystalline dislocation. However, a perfect dislocation can also be defined in quasicrystal by extending the definition of crystalline dislocation [5,6,9]; a perfect dislocation in quasicrystal is a line defect in the physical space E , satisfying dU = B ∈ LR , (27) C

for any loop C surrounding the dislocation line. Here, U denotes a high-dimensional displacement vector (see Fig. 1), B a high-dimensional Burgers vector and LR a set of highdimensional lattice translational vectors. Using eq. (6), eq. (27) can be decomposed into the two equations: du = b , C

C

dw = b⊥ ,

(28)

where b and b⊥ are the E and E⊥ components of B, i.e. B = b + b⊥ .

(29)

Eq. (28) indicates that in the presence of dislocations the displacement fields u(r) and w(r) are multi-valued. The dislocations defined here generally have the following properties:

§3.1


391

Fig. 6. A perfect dislocation introduced in a two-dimensional Penrose lattice. The Burgers vector b indicated by ¯ an arrow is [0110], where the indices are referred to the basis vectors di (i = 1, . . . , 4) in eq. (9) (see Fig. 4(b)).

(1) It is not accompanied by any planar fault because B is a lattice translational vector in the high-dimensional space. (2) The magnitude of distortion or strain is in inverse proportion to the distance from the dislocation center. These properties are common to conventional crystalline dislocations. However, the perfect dislocation in quasicrystal is different from crystalline perfect dislocation in that the former is inevitably accompanied by the phason strain because every high-dimensional lattice vector necessarily has a phason component. Fig. 6 presents an example of a perfect dislocation in a quasicrystal. Here, a perfect ¯ dislocation with the Burgers vector [0110] is introduced in a two-dimensional Penrose lattice that is known as a typical example of a two-dimensional decagonal quasicrystal. Here, the indices of the Burgers vector are referred to the basis vectors di (i = 1, . . . , 4) in eq. (9) (Fig. 4(b)). In this case, the dislocation is a point defect because of the twodimensional system. Here, as a simple example, the displacement vector is given by U(r, θ ) =

θ B, 2π

or equivalently, u(r, θ ) =

θ b , 2π

(30)

392


w(r, θ ) =

θ b⊥ , 2π

Ch. 76

(31)

where r is the distance from the dislocation core at the center and θ is the polar angle measured about the core. If you observe carefully the figure at grazing angle, you will find distortions and jogs in lattice planes in the vicinity of the dislocation core. Here, the distortions of lattice planes correspond to the phonon distortion or strain. On the other hand, the jogs in the lattice planes correspond to the phason one. In real quasicrystals, the displacement fields around dislocation should be calculated on the basis of the generalized elastic theory described in Section 2.2 and must be more complex than the form of eq. (31). The displacement fields around real dislocations will be discussed in more detail in the next subsection. Since dislocations in quasicrystals are accompanied by a strain field as well as those in conventional crystals, they can be observed as a diffraction contrast image by transmission electron microscopy. For dislocations in conventional crystals, the direction of the Burgers vector b can be determined by use of the invisibility condition g · b = 0 [95], where g is a reciprocal vector used for imaging. For dislocations in quasicrystals, this condition is generalized to [96,97] G · B = g · b + g⊥ · b⊥ = 0,

(32)

where G = g + g⊥ .

(33)

Here, G is a high-dimensional reciprocal vector, and g and g⊥ are its E and E⊥ components, respectively. There are two cases in which this invisibility condition is satisfied. One is the case g · b = g⊥ · b⊥ = 0,

(34)

which is called ‘strong’ invisibility condition. Because g⊥ · b⊥ = 0 is automatically satisfied when g · b = 0 is satisfied, the strong invisibility condition is written just as g · b = 0. This condition is the same as the invisibility condition for conventional crystals. The other case is g · b = −g⊥ · b⊥ = 0.

(35)

This condition is called ‘weak’ invisibility condition, which is characteristic of dislocations in quasicrystals. By use of the convergent-beam electron diffraction (CBED) technique, not only the direction but also the magnitude of the Burgers vector can be determined [98]. In the defocus CBED pattern, zeroth-order Laue zone line splits when it crosses a dislocation line, where the number n of splitting is given by [99] n = G · B = g · b + g⊥ · b⊥ .

(36)

§3.2


393

Using such methods, the Burgers vector determination has been attempted for icosahedral [99–102] and decagonal [103–108] quasicrystals. Rosenfelt et al. [102] have analyzed in detail dislocations in deformed and undeformed samples of icosahedral Al-Pd-Mn. It has been found that about 90% of the dislocations in both deformed and undeformed samples are the perfect dislocations with Burgers vector pointing along twofold direction. The rest are partial dislocations with Burgers vector pointing along fivefold, threefold and pseudo¯ 11, ¯ twofold directions. Four types of twofold Burgers vectors have been observed: 0011 ¯ 21, ¯ 0023 ¯ 32 ¯ and 0035 ¯ 53, ¯ where the indices are referred to the basis vectors di in 0012 eq. (7) with a = 6.45 A (see Fig. 4(a)).3 For the four Burgers vectors, the ratios |b⊥ |/|b | ¯ 21 ¯ is most frequently observed; are τ 3 , τ 5 , τ 7 and τ 9 , respectively. Of the four, 0012 ¯ 21. ¯ about 65% of the twofold Burgers vectors observed are 0012 The major slip planes are shown to be fivefold and threefold planes. For decagonal quasicrystals, the dislocations with the Burgers vector parallel to the periodic direction [103–106], in the quasiperiodic plane [106–108] and with both periodic and quasiperiodic components [106] have been reported. The Burgers vectors of the second ¯ whose |b⊥ |/|b | ratios are τ , τ −1 type so far reported are 10000, 01000, and 01110, −3 and τ , respectively. Here, the indices are refereed to the basis vectors in eq. (9) with a = 3.39 A. All of them are perfect dislocations with twofold Burgers vectors. 3.2. Displacement field and self free energy of dislocations The phonon and phason displacement fields u(r) and w(r) around a dislocation in quasicrystals should satisfy the definition of the dislocation given by eq. (28) and elastic equations of balance by eq. (21). In the absence of body forces, eq. (21) reduces to Ciju−u kl

∂ 2 wk ∂ 2 uk + Ciju−w = 0, kl ∂rj ∂rl ∂rj ∂rl

Ciju−w kl

∂ 2 uk ∂ 2 wk + Cijw−w = 0. kl ∂rj ∂rl ∂rj ∂rl

(37)

In principle, the displacement fields u(r) and w(r) around a dislocation in quasicrystals can be fully determined by eqs (28) and (37). The calculations of the fields around a dislocation in pentagonal [109], decagonal [110–113], octagonal [111], dodecagonal [111, 114] and icosahedral [58,115] quasicrystals have been reported. In the calculations, various mathematical methods have been applied, which are summarized in the review by Hu et al. [116]. Because eqs (28) and (37) have the forms of a simple generalization of the corresponding equations for conventional crystalline dislocations, the methods used for crystalline dislocations can be generalized easily for the application to the quasicrystalline dislocations. In the following, we briefly review the Eshelby’s method [117] generalized by Ding et al. [113]. By use of the generalized Eshelby’s method, we demonstrate a calculation of the displacement fields around a dislocation in an icosahedral quasicrystal and deduce self free energy of the dislocation. 3 In this case, the lattice spanned by d in eq. (7) with a = 6.45 A has a F-type superlattice order; the structure i has the translational symmetry only for the vectors R = 6i=1 ni di with 6i=1 ni = even. We confirm that 6 i=1 ni = even is satisfied for all the four Burgers vectors and therefore they are perfect dislocations.

394

Ch. 76


We consider a straight dislocation parallel to the x3 axis. We define αβ αβ αβ αβ a αβ (p) = B11 + B12 + B21 p + B22 p 2 ,

(38)

αβ u−w u−w w−w Bj l = δiα δkβ Ciju−u kl + δk(β−3) Cij kl + δi(α−3) δkβ Cij kl + δk(β−3) Cij kl .

(39)

det a αβ (p) = 0.

(40)

a αβ (pn )Aβ (n) = 0

(41)

where

Here, α and β take 1, 2, . . . or 6, and i, j , k and l take 1, 2 or 3. We solve the equation

This is a twelfth-order equation for p and we obtain twelve roots p1 , p2 , . . . and p12 . These roots are necessarily complex and consist of six conjugate pairs. We pick up p1 , . . . and p6 , one from each pair of complex conjugates. Solve the equations

to obtain Aβ (n) for each of the six roots of pn . Using Aβ (n) (n = 1, 2, . . . , 6), we determine the constants D(n) satisfying 6

(42)

6 αβ β αβ Re B21 + B22 pn A (n)D(n) = 0

(43)

Re

n=1

β

A (n)D(n) = bβ

and

n=1

where b = [b1 , b2 , b3 ] and b⊥ = [b4 , b5 , b6 ] are the Burgers vectors. Then, we finally obtain the displacement fields U1 (x1 , x2 ) u(x1 , x2 ) = U2 (x1 , x2 ) U3 (x1 , x2 )

U4 (x1 , x2 ) and w(x1 , x2 ) = U5 (x1 , x2 ) U6 (x1 , x2 )

with Uβ (x1 , x2 ) = Re

6 n=1

1 Aβ (n)D(n) ln(x1 + pn x2 ) , ±2πi

(44)

§3.2

395


where the sign ± is taken to be the same as the sign of the imaginary part of pn . Following the Foreman’s method [118], the self free energy F per unit length of the dislocation has the form R Kb2 , (45) ln F= 4π r0 where 6 αβ αβ Kb2 = bα Im ± B21 + B22 pn Aβ (n)D(n) .

(46)

n=1

Here, K is a factor called energy factor, and r0 and R are inner and outer cut-off radii, respectively. As described before, the dislocations with the Burgers vector along the twofold direction are by far the most frequently observed in icosahedral quasicrystals. As explained in more detail later, such a Burgers vector can be written as b = [0, 0, b ] and b⊥ = [0, 0, b⊥ ], where the indices are referred to e1 , e2 and e3 for b , and e4 , e5 and e6 for b⊥ , respectively (see Fig. 4(a)). As an example, we consider a simple case of a dislocation with such a Burgers vector and with the line direction along the x3 axis, i.e. the e3 direction. If we assume that the phonon–phason coupling is negligible, i.e. K3 = 0, we can deduce easily the displacement fields u(r) and w(r) and the self free energy F . The results are ⎡

⎢ u(x1 , x2 ) = ⎣

b 2π

0 0

tan−1

⎤

⎥ x2 ⎦ , x1

⎡

⎢ w(x1 , x2 ) = ⎣

b⊥ 2π

tan−1

0 0

⎤

⎥, B x2 ⎦ C x1

(47)

and F=

2 √ 2 μb + BCb⊥ 4π

ln

R , r0

(48)

where B = K1 + K2 (τ − 31 ) and C = K1 + K2 ( 32 − τ ) (see Table 3). The derivation of eqs (47) and (48) is given in Appendix B. Because K3 = 0 is assumed, u(r) is identical to the field around a screw dislocation in conventional isotropic elastic body, which has the same form as in eq. (31). On the other hand, w(r) deviates from the form in eq. (31). Assuming R 1 ≈ 1, ln 4π r0 as often assumed for conventional crystalline dislocations, we obtain from eq. (48): F ≈ μb2 +

√ 2 . BCb⊥

(49)

396

Ch. 76


More generally, the self free energy F of a dislocation in icosahedral quasicrystals can be written as 2 2 F = aμb + bK1 b⊥ ,

(50)

where a and b are positive constants of the order unity, which change depending on the Burgers vector direction and the dislocation line direction. Here, we note that K1 is always positive and that |K2 |/K1 and λ/μ are of the order unity. Now, let us discuss about relative stability among dislocations with various Burgers vectors, based on eq. (50). Here, we restrict our discussion to the dislocations with twofold Burgers vectors. In general, the Burgers vectors with b parallel to a twofold direction e3 in Fig. 4(a) can be written as ¯ 0, ¯ R1 = 0, 0, 0, 1, 1, R2 = [0, 0, 1, 0, 0, 1], B = mR1 + nR2 = 0, 0, n, m, m, ¯ n¯ .

(51)

Here, the indices are referred to the basis vectors di in eq. (7). We note that all these B belong to the translational symmetry vectors for the F-type superlattice order (see footnote 3 in Section 3.1). The same Burgers vectors can be rewritten as R1 = [0, 0, p, 0, 0, −τp],

R2 = [0, 0, τp, 0, 0, p]

B = mR1 + nR2 = 0, 0, p(m + nτ ), 0, 0, p(n − mτ ) ,

2a

, p= √ τ 6 − 2τ (52)

where the indices are referred to the basis vectors ei in Fig. 4(a). This shows that for those B, the b⊥ component is always parallel to the e6 direction in E⊥ . In other words, all those B lie on the two-dimensional subspace spanned by e3 and e6 in the six-dimensional space. More specifically, eq. (52) indicates that the Burgers vectors B form a square lattice in the e3 −e6 plane, as shown in Fig. 7. Here, the square lattice is tilted with respect to the e3 −e6 coordinates by θ = tan−1 (τ ) and the projections of B onto E and E⊥ give the b and b⊥ components, respectively. Now, we consider the problem as to which B gives the minimum self free energy F in eq. (50). For example, the lattice point P with (m, n) = (1, 3) can never be the minimum F point because there are other points, e.g., the point Q with (m, n) = (1, 2), whose |b | and |b⊥ | are simultaneously smaller than those for the point P. In contrast, the point Q can be the minimum F point because there are no other points whose |b | and |b⊥ | are simultaneously smaller than those for the point Q. The lattice points satisfying this condition are ¯ (2, 1), ¯ (1, 1), ¯ (1, 0), (0, 1), (1, 1), (1, 2), (2, 3), . . ., which those with (m, n) = . . . , (3, 2), are indicated by open circle in Fig. 7, and their inversion counterparts indicated by solid circle. Here, the open circle points can be expressed as B∗k = (Fk , Fk+1 ) (k = 0, 1, 2, . . .) and B∗k = (F−k , F −k−1 ) (k = −1, −2, . . .) . . . , where Fn (n 0) is the Fibonacci numbers defined by Fn+2 = Fn+1 + Fn with F0 = 0 and F1 = 1. The solid circle points can simply be expressed as {−B∗k |k = 0, ±1, ±2 . . .}. Combining the two groups, we obtain {±B∗k |k = 0, ±1, ±2 . . .}.

§3.2


397

Fig. 7. The two-dimensional plane spanned by e3 and e6 in Fig. 4(a). The Burgers vectors of the type eq. (51) form a square lattice on this plane. The open and solid circles represent the subsets {+B∗k |k = 0, ±1, ±2 . . .} and {−B∗k |k = 0, ±1, ±2 . . .}, respectively (see text).

The Burgers vectors {B∗k |k = 0, ±1, ±2 . . .} satisfy ∗ ∗ k b = b τ , k 0

∗ ∗ −k ∗ −(k+1) b = b τ = b τ , 0 ⊥k ⊥0

(53)

where b∗k and b∗⊥k are the E and E⊥ components of B∗k , respectively. From eq. (53), we obtain the following relations:

and

∗ ∗ ∗ ∗ b · b = b · b = const. ⊥0 0 ⊥k k ∗ b ⊥∗ k = τ −2k−1 . b

(54)

(55)

k

As shown above, the minimum F point necessarily lies in the subset {±B∗k |k = 0, ±1, ±2 . . .} (open and solid circles in Fig. 7) of all the Burgers vectors B defined by eq. (51) or (52). Then, which point in the subset {±B∗k |k = 0, ±1, ±2 . . .} is the minimum F point? Using the relation of eq. (54), we find that the self free energy F in eq. (50) becomes the minimum at ∗ b aμ ⊥∗ k ≈ . b bK1 k

(56)

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Ch. 76

For the Al-Pd-Mn system, we find μ ≈ 70 GPa and K1 ≈ 70 MPa in Tables 9 and 11, respectively. Thus, we obtain μ/K1 ≈ 1000. Because of a/b ≈ 1, eq. (56) gives ∗ b ⊥∗ k ≈ 30. b k

¯ 11, ¯ 0012 ¯ 21, ¯ 0023 ¯ 32 ¯ As described in Section 3.2, the twofold Burgers vectors 0011 ¯ ¯ and 003553 have been reported in the Al-Pd-Mn system. All of them belong to the subset {±B∗k |k = 0, ±1, ±2 . . .}; they correspond to B∗−2 , B∗−3 , B∗−4 and B∗−5 , respectively. The most frequently observed Burgers vector is B∗−3 . The ratios |b⊥ |/|b | for the four Burgers vectors are τ 3 = 4.2, τ 5 = 11, τ 7 = 29 and τ 9 = 76, respectively, agreeing roughly with the above estimation

|b∗⊥ | k

|b∗ | k

≈ 30 for the minimum F .

¯ For decagonal quasicrystals, twofold Burgers vectors 10000, 01000, and 01110 −1 −3 have been reported, whose |b⊥ |/|b | ratios are τ = 1.6, τ = 0.62 and τ = 0.23, respectively. All of them belong to the subset that can be defined for the decagonal system similarly to the subset {±B∗k |k = 0, ±1, ±2 . . .} for the icosahedral system. In contrast to the case of icosahedral quasicrystals, no measurements of the phason elastic constants have been reported for decagonal quasicrystals. The observed |b⊥ |/|b | ratios for decagonal quasicrystals may allow us to make a crude estimation of the magnitude of the phason μ elastic constants: K⊥ ≈ 0.6 2 ≈ 3μ ≈ 240 GPa.

4. Dislocation motion in quasicrystals 4.1. Dislocation mechanism of deformation In general, quasicrystals are brittle at room temperature and can only be plastically deformed at high temperatures above about 0.8Tm (Tm : melting temperature). In the high temperature range, systematic deformation experiments and transmission electron microscopic (TEM) observations of the deformation microstructures have so far been performed for quasicrystals of various alloy systems, as reviewed in [11–13]. In 1993 [14], Wollgarten et al. have revealed by transmission electron microscopy (TEM) that by plastic deformation the dislocation density increases to the order of 108 /cm2 from 106 /cm2 in the as-grown state, suggesting a dislocation mechanism for plastic deformation. In icosahedral Al-Cu-Fe, a twin-like structure and planar faults were observed after plastic deformation [119–121], but later high density dislocations and stacking faults were observed also in this alloy [122] as in icosahedral Al-Pd-Mn. The most direct evidence for the dislocation mechanism of plastic deformation has been obtained by an in-situ stretching experiment on a high temperature tensile stage in an electron microscope [15,122]. Under an applied stress, straight dislocations have been observed to move steadily and continuously, just like the dislocation glide process observed in-situ by transmission electron microscopy for bcc metals at low temperatures and for semiconducting crystals at high temperatures. Until recent years, it has been generally believed that the plasticity of quasicrystals is carried by a

§4.2


399

glide process of dislocations, and various models of the deformation mechanism have been proposed based on the dislocation glide [16–20]. However, recently, Caillard et al. [21– 23] have shown by TEM that high temperature deformation of icosahedral Al-Pd-Mn is brought not by a glide process but by a pure climb process. On the other hand, at low temperatures Mompiou et al. [23], Caillard et al. [123] and Texier et al. [124,125] have performed deformation of icosahedral Al-Pd-Mn phases under a high pressure and investigated the deformation microstructures by TEM observation. Mompiou et al. [23] and Caillard et al. [123] have shown that for single-crystals of icosahedral Al-Pd-Mn deformation at 573 K is exclusively due to dislocation climb. On the other hand, Texier et al. [124,125] have conducted deformation experiments of polycrystalline Al-Pd-Mn icosahedral quasicrystals in the temperature range between room temperature and 573 K. They have shown the following facts: (1) at room temperature the deformation is mostly brought by dislocation glide but dislocation climb events are also occasionally observed and (2) the frequency of the climb events increases as the deformation temperature is raised. Saito et al. [126] have recently performed deformation experiments of MgZn-Y icosahedral quasicrystals at low temperatures under a hydrostatic confining pressure superimposed to an applied uniaxial stress. They have found an irregular behavior in the temperature dependence of the yield stress. The origin of the irregular temperature dependence has been discussed in terms of possible transition between glide process at low temperatures and climb process at high temperatures. In the following two subsections, we discuss theoretically the two important dislocation processes in quasicrystals, i.e. glide and climb processes.

4.2. Dislocation glide process For a phason strain field to be produced or relaxed, phason jumps must occur via short range atomic diffusion, which takes place only as a thermally activated process, as mentioned in Section 2.1. Thus, in a low temperature range below half the melting point where the atomic diffusion is practically prohibited, the perfect dislocation glide is impossible and only the partial dislocation with the Burgers vector of b can glide, leaving behind the stacking fault (also called the phason fault) with the fault vector of b⊥ . Let the energy of the phason fault be Ŵ, the dragging stress necessary to move the dislocation is τd = Ŵ/b ,

(57)

where τ represents the resolved shear stress acting on the glide plane in the direction of b , and b is the strength of b . In contrast, at high enough temperature at which atomic mobility by diffusion is faster than the dislocation velocity, perfect dislocations can migrate accompanying both phonon and phason strains. Fig. 8 shows the structural change that should take place when a perfect dislocation moves in the two-dimensional Penrose lattice. Here, the patterns indicated by broken lines show the phason flips (see Fig. 3) that should take place when the perfect dislocation is translated to the left by the distance represented by an arrow. In actual quasicrystals, the quasicrystalline lattice is decorated by atomic clusters; depending on the

400


Ch. 76

Fig. 8. Structural changes taking place when a perfect dislocation moves in a two-dimensional Penrose lattice; the patterns indicated by broken lines show the phason flips (see Fig. 3) that should take place when the perfect dislocation is translated to the left by the distance represented by an arrow.

way of the decoration, different types of quasicrystals are produced such as the Frank– Kasper type, the Mackay-icosahedron type and the Cd6 Yb type for icosahedral quasicrystals. Fig. 9(a) shows an example of a realistic quasicrystalline structure containing a perfect dislocation: the atomic structure is a model decagonal quasicrystal (Burkov model [127]) viewed from the ten-fold axis [128]. The perfect lattice is composed of decagonal columnar-clusters with glue atoms among them. In contrast to the case of undecorated Penrose lattice in Fig. 8, we can see clearly a phason strain field by a distribution of imperfect cluster columns in Fig. 9(a). Fig. 9(b) shows the atomic structure for the dislocation position at the left end in the figure. By comparing Figs 9(a) and (b) we see a change of the cluster pattern as a result of the change of the phason strain field. As is evident from Figs 8 and 9, two types of the structural change should take place as a perfect dislocation translates on the glide plane; one is the re-tiling structural change outside the glide plane and the other is the intra-tile structural change. The energy release due to the latter structural change must be much higher than that due to the former type of the structural change, because without the latter relaxation severe nearest neighbor violation occurs, while without the former relaxation no nearest neighbor violation occurs and the energy increase is only due to a long range interaction. Therefore, in an intermediate temperature region where the short range diffusion can take place with the dislocation translation, a partial relaxation of the phason strain should occur only near the glide plane by healing intra-tile structural violation. Fig. 10 shows schematically the three cases of the structure change as a result of dislocation glide from the left end to the center: (a) low temperature case, (b) intermediate temperature case and (c) high temperature case.

§4.2


401

Fig. 9. An example of a realistic quasicrystalline structure containing a perfect dislocation: the atomic structure is a model decagonal quasicrystal (Burkov model [127]) viewed from the ten-fold axis [128]. The dislocation position is at the center in (a) and at the left end in (b).

Fig. 10. Three cases of the structure of a dislocation gliding from left to right in a quasicrystal: low temperature case (a), intermediate temperature case (b) and high temperature case (c).

In addition to the glide resistance due to the phason defect production, dislocations in quasicrystals should be accompanied by a Peierls potential barrier as the dislocations in crystals. The Peierls potential is the self-energy change with the change in dislocation position due to the discreteness of the lattice; the Peierls potential in crystals is periodic but that in quasicrystals quasiperiodic. For crystals, it is established theoretically [129,130] and experimentally [131,132] that the simpler the crystal structure having a larger d/b ratio (d: the glide plane spacing, b: the strength of the Burgers vector), the higher is the Peierls potential. It is naturally considered that the Peierls potential is high for dislocations in quasicrystals due to the complex atomic structures.

402


Ch. 76

Fig. 11. Energy variation with the translation of a straight dislocation in two different orientations in a realistic model quasicrystal [133]. Peierls potential is dependent on the dislocation orientation.

Fig. 12. Ŵ-surfaces in two directions in a model quasilattice [133].

Unless the phason relaxation occurs, the change of the potential energy with the translation of a dislocation comprises the phason fault energy and the Peierls potential which is superimposed on the former. Examples of the potential energy profile calculated for two different perfect dislocations in a realistic model decagonal quasicrystal [133] are presented in Fig. 11. It is seen that the Peierls potential is quite sensitive to the type of the dislocation as for crystal dislocations. For type (a) dislocation in the figure, the calculated stress to move the dislocation at 0 K is as high as 0.159G (G: the shear modulus) which consists of the phason production stress component of 0.048G and the Peierls stress component of 0.111G, the latter being twice as large as the former. The so-called Ŵ-surfaces (the energy surface of the plane fault created by a rigid displacement parallel to the glide plane of the upper half crystal (quasicrystal) with respect to the lower half) for the model quasiperiodic lattice in the two Burgers vector directions in Fig. 11 are shown in Fig. 12. It is to be noted that in multicomponent quasicrystals the Ŵ-surface energy consists not

§4.2


403

Fig. 13. (a): Kink-pair formation process; (b): potential profile for a kink-pair formation process under no applied stress (dashed curve) and under a stress higher than Ŵ/b (solid curve).

only of the phonon and phason strains but also of chemical disordering energy, as shown in the simulation of a model decagonal quasicrystal [134]. As seen in Fig. 12, due to the quasiperiodicity of the lattice, the Ŵ-surface also oscillates quasiperiodically. Essentially the same results have been obtained for other models [135,136]. Such an oscillation of the Ŵ-surface means that if a group of dislocations glide together on the same plane, the necessary phason dragging stress which is given by τd = Ŵ/bi becomes negligibly small as the number of dislocation increases [133,137]. Thus, the Peierls potential plays a dominant role in the dislocation glide rather than the phason dragging stress. At high temperatures, we would expect thermally activated dislocation glide. Because dislocations in quasicrystals are subject to high Peierls potential, they should tend to lie along a Peierls valley. This has been verified experimentally by in-situ electron microscopy observation of dislocation motion, where dislocations are observed to move keeping a straight form oriented in a particular direction [15,122]. The thermally activated glide process for such a dislocation is the kink pair formation followed by motion of kinks along the Peierls valley, as illustrated in Fig. 13(a) [138]. For the thermally activated kink-pair formation to occur a phason fault relaxation should take place to some extent; otherwise the enthalpy of the final state (kink-pair state) cannot be lower than the enthalpy of the initial state (straight form lying in a Peierls valley) under a reasonably low stress that does not lead to fracture. As mentioned previously, two kinds of phason fault should be produced with dislocation translation, the re-tiling fault and the intra-tile fault. The re-tiling fault has

404


Ch. 76

much lower energy than the intra-tile fault and furthermore the relaxation of the re-tiling requires cooperative jumps of a number of atoms leading to a much longer relaxation time than the relaxation of the intra-tile fault. For these reasons, it seems reasonable to assume that in the thermally activated kink-pair formation process and the kink migration process, the relaxation only of the intra-tile fault occurs. There are two regimes for the kink-pair formation process in crystals, one is the smooth kink regime and the other is the abrupt kink regime [139]. In the case of relatively low Peierls potential such as that in bcc metals or NaCl crystals, the kink shape is smooth, and in the case of high Peierls potential such as that in covalent crystals of Si, the kink shape is abrupt. Due to the lattice periodicity along the Peierls valley, there exists a periodic potential also for the kink translation, which is called the Peierls potential of the second kind. For the smooth kink, the Peierls potential of the second kind is generally negligible and the kink energy can well be represented by the line tension approximation of the bowing-out dislocation. The dislocation mobility is determined by the kink-pair formation enthalpy, which has been given by Celli et al. [140], and Dorn and Rajnak [141]. For the abrupt kink, the Peierls potential of the second kind is also involved in the kink-pair formation enthalpy and also in the kink migration process. The kink-pair formation in the abrupt kink case has been treated in terms of the kink diffusion theory by Hirth and Lothe [142], which has been applied to the interpretation of the dislocation mobility in semiconducting crystals. As mentioned above, the thermally activated dislocation migration should be accompanied by the intra-tile phason relaxation in the close vicinity of the glide phane, which requires thermally activated short range atomic jumps. Thus, even if we neglect the Peierls potential of the second kind, the kink-pair formation of a dislocation in quasicrystals has to be treated by a kink diffusion theory in a similar manner to the kink-pair formation in the abrupt kink regime. The overall potential profile for the kink-pair formation process under zero applied stress is depicted by a dashed line in Fig. 13(b). As the kink separation l of the kink-pair increases, the self-energy of the kink-pair increases towards twice the single kink energy Eκ . Constantly increasing component with a slope Ŵ is due to the unrelaxed phason strain outside the glide plane. The potential hills superimposed aperiodically on the curve are due to the activation barrier for the atomic jumps in the intra-tile relaxation process. In this potential profile, the thermally activated kink-pair formation can take place only under a stress higher than τd = Ŵ/b . The solid curve in Fig. 13(b) shows the enthalpy profile for the kink-pair formation under a stress higher than τd . The rate of the kink-pair formation under a stress τ ∗ = τa − τd (τ ∗ : the effective stress, τa : the applied stress) is determined by ∗ towards larger l value. the rate of the kink diffusion surmounting the potential barrier Hkp This rate can be formulated in perfect analogy to the kink-pair formation of the abrupt kink regime in crystal dislocations with the substitution of the kink migration enthalpy by the activation enthalpy of the atomic jump and the period of the Peierls potential of the second kind by the separation of the potential hills H ′ in Fig. 13(b). The frequency of the kink-par formation per unit length of a dislocation at low stress is given by ∗ + H′ Hkp τ ∗ b d , νkp = νk exp − 2kB T kB T

(58)

§4.3


405

where νk is the kink vibration frequency. The kink velocity is written as vk =

τ ∗ b dd ′2 H′ , exp − kB T kB T

(59)

where d ′ is the average distance between the phason jumps. For an infinite length of the dislocation, the equilibrium density of kinks is determined by the balance between the rate of the kink-pair formation and the rate of annihilation of positive and negative kinks, and the equilibrium kink separation is given by l¯k =

∗ √ ′ Hkp . 2d exp 2kB T

(60)

Let the dislocation segment length lying in a Peierls valley be L. Depending on the relative magnitude of L and l¯k , there are two cases for the dislocation mobility; one is the kink∗ = collision case and the other is the kink-collisionless case. For a typical value of Hkp ′ 3 eV and d = 0.5 nm, l¯ at 1000 K is calculated to be 3 cm, which is much larger than the dislocation segment length L, leading to the kink-collisionless case. Therefore, the dislocation glide velocity Vg is controlled only by the rate of kink pair formation on the segment length L, which is followed by the kink motion over the distance L, and is written as Vg =

∗ (τ ∗ ) + H ′ Hkp τ ∗ b d 2 L . νk exp − 2kB T kB T

(61)

4.3. Dislocation climb process For a dislocation confined in a deep Peierls valley, the climb process should be the jog-pair formation followed by the jog motion, in a similar manner to the kink-pair formation and the kink motion, as illustrated in Fig. 14(a) [138,143]. Unlike the kink-pair formation by glide, the climb motion occurs by vacancy absorption or emission at jogs, i.e., accompanying atomic diffusion to or from the jog sites. As a result, the intra-tile relaxation is expected to occur simultaneously with the dislocation climb. Due to unrelaxed phason defects outside close vicinity of the climbing plane, there should be a constant increase of the phason strain energy with the jog motion, as in the case of the kink motion. Thus, the potential energy profile under no applied stress for the jog-pair formation is like that depicted by a dashed line in Fig. 14(b) and that under an applied stress τ > Ŵ/b is shown by a solid line in the figure. In the case of the kink-pair formation process, the kink velocity is determined by the short range atomic diffusion to relax the intra-tile phason relaxation, whereas in the case of the jog-pair formation process, the jog velocity is controlled by the rate of vacancy absorption or emission at the jog, i.e. the self diffusion. Taking account of the fact that the formation energy at the dislocation core and the migration energy along the dislocation

406


Ch. 76

Fig. 14. (a): Jog-pair formation process; (b): potential profile for a jog-pair under no applied stress (dashed curve) and under a stress higher then Ŵ/b (solid curve).

core of a vacancy are considerably lower than those in the lattice, the jog velocity under an effective resolved stress σ ∗ for the climb is written as [142] 4πDs σ ∗ b a Hs exp , vj = 2kB T kTB ln z¯ /b

(62)

where a is the atomic distance along the dislocation line, Ds is the self-diffusion coefficient, Hs is the difference between the activation enthalpy of the self-diffusion in the bulk and that along the dislocation core and z¯ is the mean life length of a core vacancy along the dislocation line given by √ Hs z¯ = 2a exp . 2kB T

(63)

As in the case of the kink-pair formation, there are two cases for the dislocation velocity expression, the jog-collision case and the jog-collisionless case, depending on the relative magnitude of the equilibrium jog spacing l¯j = a exp(Ej /kB T ) and the dislocation segment length L lying in a Peierls valley. Since Ej is of the order of an eV, l¯j > L is generally satisfied. Thus, as in the case of the kink-pair formation, the jog-collisionless case is real-

§4.4


407

ized. For L < l¯j , the dislocation climb velocity Vc is determined by the rate of the jog-pair formation on the segment length L and is given by Vc =

∗ (σ ∗ ) − H /2 Hjp s 4πLDs va σ ∗ exp − , 2 kB T a kB T ln z¯ /b

(64)

∗ is the activation enthalpy of the jog-pair formation. where va is the atomic volume, Hjp Writing Ds = a 2 νD exp(−Hs /kB T ) (Hs : the activation enthalpy of the self-diffusion; νD : the Debye frequency) and approximating νa = a 3 , eq. (64) is rewritten as

∗ (σ ∗ ) + (H − H /2) Hjp s s 4πLa 3 σ ∗ . Vc = νD exp − kB T kB T ln z¯ /b

(65)

As described in Section 4.1, detailed TEM observations have revealed that in icosahedral Al-Pd-Mn climb process is the dominant deformation process over the glide process at high temperatures. This indicates Vc > Vg . Below, we make a crude estimation about relative magnitude of the velocities, which indeed justifies the fact Vc > Vg . From eqs (61) and (65), the ratio of Vg /Vc is given by ∗ (τ ∗ ) − H ∗ (σ ∗ )} + {H ′ − (H − H /2)} {Hkp ln z¯ /b s s Vg jp . = exp − Vc 8π kB T

(66)

For a typical case of T = 1000 K, Hs = 2 eV and Hs = Hs /2, the pre-exponential factor of eq. (66) is of the order of unity. H ′ may be comparable to the migration enthalpy of vacancy which is about Hs /2, and hence the second term of the exponent is small. Thus, ∗ and H ∗ or the the ratio Vg /Vc is determined essentially by the relative magnitude of Hkp jp relative magnitude of the kink Ej . Let τP /G = β, the kink √ energy2 Ek and the jog energy energy is given by Ek = 0.5 βGdb [139]. Ej = Gb2 h/{4π(1 − ν)} ≈ 0.1Gb2 h [142]. Assuming β = 10−1 ∼ 10−2 , Ek d = (0.5 ∼ 1.5) . Ej h

(67)

Since the kink height d (period of the Peierls potential) is generally larger than the jog height h (atomic spacing), Ek can be larger than Ej , and hence Vc can be larger than Vg .

4.4. Plastic homology Although the climb controlled deformation has been confirmed only in Al-Pd-Mn icosahedral quasicrystals, we assume that the high temperature plasticity of any icosahedral quasicrystals is governed by a common dislocation climb process. Then, we expect that some homologous relation should hold for icosahedral quasicrystals, as already established in bcc metals [144] and tetrahedrally coordinated crystals [145].

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Ch. 76

(a) Fig. 15. (a): Temperature dependence of the upper yield stress for icosahedral quasicrystals of Al-Pd-Mn [16, 146], Al-Cu-Fe [147], Cd-Yb [148], Mg-Zn-Y [149] and Al-Li-Cu [150]. (b): Normalized upper yield stress vs. normalized temperature replotted from (a).

Fig. 15(a) shows the temperature dependence of the upper yield stress for various icosahedral quasicrystals [16,146–150]. The temperature range of the plastic deformation differs largely among the icosahedral quasicrystals. In Fig. 15(b), we replotted the data to the normalized upper yield stress vs. the normalized temperature relations [143], where the upper yield stress is normalized with respect to Young’s modulus E and the temperature with respect to the material parameters of Young’s modulus times cube of the average atomic diameter a¯ divided by the Boltzmann constant, i.e. E a¯ 3 /kB . As seen in Fig. 15(b), a homologous relation holds approximately.

§4.4


409

(b) Fig. 15. (Continued).

For Hs = Hs /2, the climb velocity is written as Vc =

∗ + (3/4)H Hjp s 4πLa¯ 3 σ ∗ νD exp − . kB T kB T ln z¯ /b

(68)

Assuming Hs ≈ 3 eV, a = b = 0.3 nm, L = 10 µm and νD = 1014 s−1 , the preexponential factor at T ≈ 1000 K of eq. (68) is estimated to be 7 × 107 ms−1 at a low stress of σ ∗ = 10 MPa (≈10−4 E). Assuming the mobile dislocation density at the upper yield point is of the order of 109 cm−2 [151], the dislocation velocity at the yielding stage is 3 × 10−8 ms−1 for the usual strain rate of 10−4 s−1 . At the yielding stage at high temperature and low stress (≈10−4 E), the value of exponent of eq. (68) is estimated to be 35. The activation enthalpy of dislocation motion has been evaluated experimentally from the temperature dependence of the yield stress and the stress dependence of the activation

410


Ch. 76

Fig. 16. The relation between E a¯ 3 and the activation energy of self-diffusion Ed for cubic metals.

volume obtained by the stress relaxation experiments. The obtained values of the exponent are largely scattered: 40 [152], 60 and 43 [146], 45 [153], 65 [154], 20 [155] and 35 [148]. However, taking account of a variety of uncertainty in the experimental evaluation due to effects of work softening, recovery during the relaxation test and temperature dependent dragging stress etc., the above estimated value of 35 does not seem to be inconsistent with the experimental results. From the homologous plot in Fig. 15(b), the deformation temperature at low stress satisfies T ≈ 4.2 × 10−3 E a¯ 3 /kB . This indicates ∗ + (3/4)H Hjp s

4.2 × 10−3 E a¯ 3

= 35.

(69)


411

∗ ≈ 2E (E : the jog energy) at low stress, Since Hjp j j

3 2Ej + Hs ≈ 0.15E a¯ 3 . 4

(70)

The energy of a jog with a height h is written as [142] Ej =

Eb2 h Gb2 h = . 4π(1 − ν) 8π(1 − ν 2 )

(71)

Approximating b ≈ a and h ≈ a, ¯ and inserting ν ≈ 0.25 [62,64,65,68], one obtains Ej = 0.04E a¯ 3 .

(72)

In Fig. 16, we plot Ea 3 values against Hs for 16 cubic metals, showing a correlation Ea 3 /Hs ≈ 7.5 except two transition metals V and Nb. Since the same vacancy mechanism has been suggested for quasicrystals as for usual metals by diffusion experiments for specific atomic species in quasicrystals [156,157], we assume here that the same correlation holds also for icosahedral quasicrystals, i.e. E a¯ 3 /Hs = 7.5. Then, 3 Hs = 0.10E a¯ 3 . 4

(73)

From eqs (72) and (73), 3 2Ej + Hs = 0.08E a¯ 3 + 0.10E a¯ 3 = 0.18E a¯ 3 . 4

(74)

Taking various simplified assumptions made in deriving eq. (74), the agreement between the experimental relation of eq. (70) and the estimated relation of eq. (74) seems reasonable. The above results indicate that about a half of the activation enthalpy of the dislocation climb is due to the kink-pair energy and the other half to the activation enthalpy of self-diffusion.

Appendix A: Irreducible strain components The irreducible strain components in the icosahedral system are given below [42,58].

1 u1 = √ u11 + u22 + u33 , 3

⎡

⎢ ⎢ u5 = ⎢ ⎢ ⎣

1 √ 2 3

2 −τ u11 + τ −2 u22 + τ + τ −1 u33 ⎤ 1 −1 ⎥ u11 − τ u22 + u33 ⎥ 2 τ √ ⎥, ⎥ √2u12 ⎦ √2u23 2u31

412

Ch. 76


⎤ w11 + w22 + w33 1 ⎢ τ −1 w + τ w12 ⎥ w4 = √ ⎣ −1 21 ⎦, 3 τ w32 + τ w23 τ −1 w13 + τ w31 ⎡

⎤ ⎡ √ 3(w11 − w22 ) ⎢ √w11 2w33 ⎥ + w22 −−1 ⎥ 1 ⎢ ⎢ 2 τ w − τ w12 ⎥ . w5 = √ ⎢ √ 21 6 ⎣ 2 τ w − τ −1 w ⎥ 23 ⎦ √ 32 −1 2 τ w13 − τ w31

(A.1)

The irreducible strain components in the decagonal system are given below [47]. u′1 = u33 ,

u1 = u11 + u22 , w11 + w22 w6 = , w21 − w12

u5 =

u11 − u22 , 2u12 w11 − w22 w8 = . w21 + w12

u31 , u23

w13 w7 = , w23

u6 =

(A.2)

Appendix B: Derivation of eqs (47) and (48) by the generalized Eshelby’s method w−w By use of Ciju−u kl in eq. (15) and Cij kl in Table 3, we obtain

⎡ (λ + 2μ) + μp2

⎢ ⎢ a (p) = ⎢ ⎢ ⎣ αβ

(λ + μ)p 0 0 0 0

(λ + μ)p μ + (λ + 2μ)p2 0 0 0 0

0 0 μ + μp 2 0 0 0

0 0 0 A + Bp 2 2Dp 0

0 0 0 2Dp C + Ap 2 0

⎤ 0 0 ⎥ ⎥ 0 ⎥. ⎥ 0 ⎦ 0 B + Cp 2

(B.1)

The determinant of a αβ (p) reduces to the product of four subdeterminants: αβ a (p) = D1 · D2 · D3 · D4 , (λ + 2μ) + μp 2 (λ + μ)p , D1 = (λ + μ)p μ + (λ + 2μ)p 2 A + Bp 2 2Dp D3 = , 2Dp C + Ap 2

D4 = B + Cp 2 .

D2 = μ + μp 2 ,

(B.2)

Of them, D2 and D4 are relevant to the Burgers vectors b = [0, 0, b ] and b⊥ = [0, 0, b⊥ ], respectively. The roots of D2 · D4 = 0 are p1 = i,

p2 =

B i, C

(B.3)


413

and p7 = p1∗ and p8 = p2∗ . For p1 and p2 , eq. (41) gives, respectively, ⎡ ⎤ 0 ⎢0⎥ ⎢ ⎥ ⎢1⎥ A(1) = ⎢ ⎥ ⎢0⎥ ⎣ ⎦ 0 0

⎡ ⎤ 0 ⎢0⎥ ⎢ ⎥ ⎢0⎥ and A(2) = ⎢ ⎥ . ⎢0⎥ ⎣ ⎦ 0 1

(B.4)

Eqs (42) and (43) become, respectively, Re[D(2)] = b⊥

Re[D(1)] = b ,

(B.5)

and Re[μiD(1)] = 0,

Re

√ BCiD(2) = 0.

(B.6)

From eqs (B.5) and (B.7), we obtain simply D(1) = b ,

D(2) = b⊥ .

(B.7)

Inserting eqs (B.3), (B.4) and (B.7) into eq. (44) and into eq. (46) lead to eq. (47) and eq. (48), respectively.

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CHAPTER 77

Experimental Studies of Dislocation Core Defects JOHN C.H. SPENCE * Department of Materials Science University of Cambridge, Cambridge, UK

* E-mail: [email protected]

Permanent address: Physics, Arizona State University, Tempe, AZ 85287, USA



Contents 1. Introduction 421 2. Early work. Background theory 423 3. Recent theory. Too many possibilities? 428 4. Recent experiment. What is known 430 5. The way forward. UHV TEM, ELS and nanodiffraction 6. Summary 449 References 450

439

1. Introduction Despite half a century of effort, we still do not have an unambiguous physical model for dislocation motion at the atomic scale. This is a result of lack of information on the atomic structure of dislocation cores and their associated defects. The most important of these core defects may be classified as kinks, jogs, reconstruction defects (such as one-dimensional antiphase “domains” or defects (APD), also referred to as solitons), vacancies and foreign atoms. This chapter reviews what we know from experiment about these dislocation core defects and their effect on dislocation mobility. To place this work in context, a brief background of theoretical work on core structure and defects is included, but they are not the main theme, and the theoretical effects of core structure on dislocation mobility are fully covered in the chapter by Cai et al. in this series. The electronic structure of core defects forms a crucial part of this field: however we do not cover here the effects of core defects on optoelectronic device performance and recombination kinetics (see [1] for recent work). Because so much more is known about it than any other material, and because of its practical importance, most of the work described is based on silicon. Other materials are also discussed where work exists. For the strongly bonded semiconductors at low temperature the bond-breaking process during kink motion is directly responsible for the low mobility of dislocations, and so explains their brittleness at room temperature. (The Peierls stress is rate-controlling up to temperatures approaching the melting point.) The atomic mechanisms of kink motion are crucially affected by core defects such as impurities, kink structure, and anti-phase defects (APD) which may thus exert a controlling influence on bulk properties. More generally, kinks are likely to be involved in any first-order phase transition in condensed matter, and thus of fundamental importance in physics and materials science. While long-range elastic interactions between dislocations are important in controlling the interaction between dislocations (and store most of the energy), the effects of crystallography and dislocation core structure on dislocation mobility, effects not considered in the continuum elasticity theory, are responsible for the friction the crystal structure offers to dislocation motion (the Peierls stress), and are indicated directly by several phenomena. These signatures of atomic granularity include the failure of Schmid’s law (in which critical resolved shear stress does not depend on slip system), and of a strong temperature and orientation dependence of yield stress and dislocation velocity. Indentation experiments have shown that loops on different glide planes with the same shear stress component on their glide plane travel at different velocities, implicating other components of the stress tensor, contrary to Schmid’s law. Core structure and defects are also important in generating deep states responsible for recombination and other electronic properties in semiconductors, with life-time limiting effects on devices. Yet direct evidence of dislocation core structure is very difficult to obtain. Most of the evidence comes from high resolution transmission electron microscopy (HREM) and, less directly, from the spatially resolved

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spectroscopies. Indirect evidence comes from techniques such as electron spin resonance (summarized in Cai et al. in this series), Hall effect, deep level transient spectroscopy, etch pit measurements, X-ray topography, internal friction measurements, intermittent loading, energy-loss spectroscopy, internal friction, cathodolluminescence, EBIC, and in-situ TEM observations of dislocations in the various regimes of their motion. All of these methods have various strengths and weaknesses, involve differing assumptions, and provide us with estimates of different parameters, such as the density of deep states associated with dislocations, dislocation mobility, or kink formation and mobility energy. It has proven difficult to relate the results of the various methods to each other (see [2] for a review), and the spread of experimental measurements by different groups overall is about as great as those provided by competing theoretical treatments. Some fundamental questions which need to be answered in connection with dislocation core defects include the following: 1. What limits dislocation velocity at the atomic level for given conditions of stress and temperature – is it obstacles to kink motion, as in the theory of Celli and Thomson [3] or is it kink formation or migration energy, as in the Hirth–Lothe theory [4]? Is kink path length limited by obstacles or kink-kink collisions? Is kink generation on different partials correlated? 2. If obstacles are important, what is the nature of the obstacles? 3. What is the atomic structure of kinks and other defects on dislocations? 4. What are the detailed mechanisms of dislocation nucleation? 5. What are the atomic structures of jogs? 6. What are the atomic processes involved in the earliest stages of the ductile-brittle transition? 7. What are the atomic structures of the common complex extended line defects in intermetallic alloys? An ideal imaging experiment which provided video recordings of kink motion under controlled conditions of stress and temperature would answer many of these questions, even if the details of kink atomic structure could not be resolved. From recordings of kink velocity at known temperature and stress an Arhenius plot would yield kink migration activation energies. From the pre-factor given by the intercept, the entropy term might also be obtained (eq. (3) below). These results could be obtained for both left and right kinks on both partials of dissociated dislocations. Comparisons could be made and the stress dependence of the kink velocity determined. Measurements of kink density at low stress would yield directly the double kink formation energy. Video recordings would answer the question “Do kinks collide” [5], and so provide a direct test of the Hirth–Lothe theory in its two regimes. In addition, the observation of kink delay at obstacles and the measurement of waiting times would yield unpinning energies and provide a test of the obstacle theory of dislocation motion [3]. The proposed correlation between kink nucleation events on different partials [6] could be sought, and the oscillatory dependence of mobility on stress which has been proposed for certain stacking-fault widths could be investigated. Finally, entirely new phenomena and mechanisms are likely to be observed. All this information could be provided by an imaging method which allows the position of a kink to be localised to within about 0.5 nm. (The image may be that of the strain-field associated with the kink.)

§2

Experimental studies of dislocation core defects

423

A second class of experiments might provide much higher resolution images of stationary core defects in order to determine their structure and the structure of core reconstructions. (Currently, the resolution of the best TEMs is about one Angstrom.) Clearly such a technique, which would open up the whole field of kink dynamics to experimental observation and resolve the large discrepancies between theory and experiment in this field, is urgently needed. Such a method is outlined in Section 5. At the same time, the recent commercial availability of monochromators for electron energy-loss spectroscopy will provide new insight into the electronic structure of dislocation cores and may conveniently be integrated with another new technique of great promise – coherent nanodiffraction – this work is also reviewed below.

2. Early work. Background theory A review of work on all aspects of semiconductor dislocation cores prior to 1990 can be found in [7], while the role of cores in metal plasticity is covered by Duesbery in volume 8 of this series. Experimentally, it is found that dislocation velocity is a thermally activated process which depends exponentially on inverse temperature, with activation energy Q. Microscopically, it is generally accepted that dislocation motion occurs when thermal fluctuations throw a segment of dislocation line across into the next Peierls valley. These minimum-energy directions for dislocation lines lie along the [110] tunnels of the silicon structure. Kink-pair embryos smaller than a critical size x ∗ shrink back to annihilation, while larger kinks of a pair are driven apart by external stress. The effect of stress favors annihilation by collision of kinks of opposite sign on the trailing side, but advance on the leading side of the dislocation line. The height of the energy barriers for kink formation and migration (Fk and Wm respectively) are also affected by stress. The Hirth–Lothe theory of dislocation motion [4] considers the nucleation of these double kinks (with separation x), and their diffusion and drift under an external stress σ , by one-dimensional analogy with classical steady-state nucleation and growth theory for particles of size x. Then the equilibrium distribution of kink separations (constrained to zero kink current) per unit length of line is cc (x) =

1 exp −F (x)/kT 2 a

(1)

with dims L−2 . Here a is the reconstructed period along the line, and the free energy of kink-pair formation is [8] F (x) = 2Fk −

μb2 h2 − σ bhx, 8πx

(2)

where h is the kink height and μ the shear modulus. (The kink height is the distance between Peierls valleys, for example that labelled 0.33 nm in Fig. 13.) The second term represents the attractive kink-kink strain interaction (tending to annihilate by recombination kink pair embryos less then the critical separation x ∗ ) while the last term describes the

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external stress which drives kinks apart. The kink velocity for a dislocation with Burgers vector b is νk =

σ bhνD a 2 σ bh exp(−Wm /kT ) = Dp , kT 2kT

(3)

where νD is the Debye frequency, Dp the double kink diffusion coefficient and Wm = Um − T Sm the free energy of kink migration (similarly Fk = Uk − T Sk ). The entropy terms have been estimated [9] as Sk (90◦ ) = 0.5k and Sm (90◦ ) = 5k. The net double kink nucleation rate is J = Dp Co (x ∗ )/2x ′ , with x ′ = kT /(σ bh). The dislocation velocity VD = hλJ = 2h(J vk )1/2 if the dislocation segment length L is larger than twice the kink mean free path λ = 2a exp(+Fk /kT ), so that kinks frequently collide. Then VD depends exponentially on Q = Fk + Wm . If, however, kinks never collide (for example in a thin film of thickness L < 2λ where they are more likely to emerge at surfaces), then VD = hLJ and Q = 2Fk + Wm . (The simpler low temperature case where no thermal kinks are excited is described below (eq. (6)).) This change in Q for different dislocation segment lengths (producing a large change in their velocity–temperature dependence) has formed the basis for many experiments aimed at measuring Wm and Fk , as discussed below. In general we find a dislocation velocity vD (σ, T ) =

L 2νD abh2 σ exp −(Fk + Wm )/kT , kT L+λ

(4a)

which provides the appropriate limits for L ≫ λ (dislocation velocity independent of dislocation length L) and L ≪ λ (velocity proportional to L) using λ = 2a exp(+Fk /kT ). Historically, the observation of dislocation velocities independent of length in bulk material has been taken as support for kink collisions in the Hirth–Lothe theory, however research in the last decade has called this into question, as discussed in Section 4. Under conditions of high stress, the formation energy for a kink pair is modified according to [8] Fkp (τ ) = 2Fk − σ τ b3 h3 /2π. Experimental data have customarily been fitted to a model expression of the form m τ VD = V0 exp −Q/kT , τ0

(4b)

where m is a fitting parameter, often taken to be unity (as in the Hirth–Lothe theory above) except at low stress (see below). For silicon the common 60◦ and screw dislocations lying along [110] both show an activation energy Q of roughly 2.2 eV (decreasing at high stress). In more detail [2], the best fit to experiment is found to be τ Q(τ ) = 2.6 eV − 0.115 eV ln , (5) τ0 where Q(300 MPa) = 1.95 eV.

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Alternatives to the Hirth–Lothe theory have assumed a kink velocity limited by obstacles such as antiphase defects or point defects [3,6,10]. These “dragging points”, with spacing less than the kink mean free path, impede kink motion, and the probability per unit time that a kink surmounts an obstacle barrier is νd = ν0 exp −Ed /kT ,

where Ed is an unpinning energy and the waiting time is νd−1 . The consequences of this and resulting modifications to the above equations are briefly reviewed in Section 4. Electron microscope imaging in the early seventies showed that dislocations in tetrahedrally-bonded semiconductors are mostly dissociated (at rest and during motion [11]) into two 30◦ screw partials for the screw, and into a 30◦ screw and a 90◦ partial for the 60◦ dislocation. The connecting intrinsic ribbon of stacking fault (SF) is usually assumed to lie between the narrowly-spaced glide plane atoms, as shown in Fig. 1, rather than on the more widely spaced shuffle planes, since this allows the partials to glide. In the sphalerite structure this SF creates a thin lamella of wurtzite structure. However shuffle cores are seen in unit screw dislocations at high stress and low temperature by TEM [12] – their effect on mobility under these conditions is discussed by Cai et al. in this series. In covalent semiconductors other than silicon the activation energies of the two partials may differ widely, so that the leading and trailing partials travel at different speeds. In polar semiconductors, it was realized that the line of atoms along the partial dislocation core could belong to either species, so that, for example, in GaAs, α (As core) and β (Ga core) cores became possible, in addition to the shuffle and glide possibilities. For metals, dissociation occurs in both the fcc and bcc metals, while the complex superdislocation line and planar faults found in intermetallic alloys constitute a specialized study [13,14]. For most of the fcc metals, kink formation energies are comparable with kT even at rather low temperatures, so that the kink mechanism is not considered important, and in sufficiently pure copper, for example, the yield stress tends to zero. Here cores are wide and the Peierls barrier small, unlike the covalently-bonded semiconductors. Three-fold dissociation of screws in the bcc metals leads to failure of Schmid’s law. Oxide ceramics appear to offer excellent prospects for experimental imaging of their stationary dislocation core defects, but require very high temperatures for the study of kink dynamics. Throughout the nineteen seventies and eighties HREM images of dislocation cores were obtained by many groups using the instruments then available with resolution limited to about 0.26 nm (for reviews, see [15,16]; for early HREM work on the 30/90 in silicon, see, for example [17–19]. Representative of this period is the comprehensive three-dimensional analysis of dislocations at a grain-boundary in silicon which can be found in [20]). With sub-Angstrom resolution now possible, more accurate results are possible, as reviewed in Section 4. Dissociation has important consequences for core defects, kink structure and energetics, since dissociation favors core reconstruction. As shown in Fig. 2, reconstruction (“dimerization”) doubles the periodicity of the core of the 30◦ partial (relative to Fig. 1), so that kinks must move twice as far in each elementary jump. The resulting change in electronic structure reminds us of a one-dimensional metal-insulator Peierls transition. Reconstruction is thus found to clear the gap of electronic defect states, suggesting that core defects

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Fig. 1. Three-dimensional sketch of the 30◦ and 90◦ partial dislocations in the diamond structure. The core of an unreconstructed 30◦ partial lies along the left-hand edge of the dashed rectangle, the 90◦ partial lies at the right.

Fig. 2. This is the (111) plane of atoms lying on the dashed rectangle in Fig. 1 for silicon or sphaelerite (if shaded atoms are considered). The double-period reconstruction of the 90◦ partial is shown at right (from C. Koch, PhD, ASU, 2003).

such as kinks and anti-phase defects are chiefly responsible for any remaining deep states in the gap. These ideas, and that of the antiphase defect which must border independently nucleated reconstructions (Fig. 3), can be traced to early work by Hirsch and Jones in the late nineteen-seventies [21,22], based on ball-and-spoke modeling and simulations using atomic potentials. Understanding the photoplastic effect [23], and the effect of doping on dislocation velocity (through its effect on Fermi energy position, and the resulting charge on kinks with their associated deep levels) was an important motivation for this work. The first ab-initio density functional calculations for cores appeared soon after in 1981 [19], and these continued to grow in popularity throughout the nineteen eighties.

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Fig. 3(a). Arrangement of partials around a dislocation loop in silicon. At left 30/90 with 90◦ partial on right, both unreconstructed. Across the top 30/30 dissociated screw, reconstructed. Soliton at S. At right: 90/30 with 90◦ partial on right, both reconstructed. 30◦ partial has kink (K) and APD (soliton, labelled S). (From [16].)

Fig. 3(b). 90◦ partial in Si on (111) with LR/RL kinks which reverse direction of core reconstruction; DSD: Direction switching defect, RR-complex = LR(RL) + DSD, LL-complex = LR + DSD. (From [101].)

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3. Recent theory. Too many possibilities? Since about 1990 the sophistication of dislocation core calculations has increased dramatically, and many new ideas have emerged. A review for silicon can be found in the special issue of Scripta Mat. 45, No 11 (2001) devoted to dislocations in silicon. A considerable literature now covers kinks, anti-phase defects, oxygen atoms, dopants, vacancies and hydrogen for many semiconductors. We consider first the intrinsic core defects. In order to link the atomistic calculations to bulk properties, kinetic Monte-Carlo solutions to the relevant rate-equations have appeared [24] which predict dislocation mobility as a function of temperature and stress. This work also incorporates the oscillatory relation between mobility and stress at low stress due to correlated kink motion on different partials for certain stacking-fault widths [6,25]. This effect appears to explain the starting-stress anomaly in silicon at 10 MPa [26]. Entirely new kink, reconstruction, and APD structures have been discovered in the last decade, and dissociated kinks have been proposed. The core calculations of the last decade are now commonly based on density functional theory (DFT) rather than atomic potentials, and so can compare “total” core energies reliably for different structures (excluding the long-range elastic component, which depends on sample size). Values of Fk and Wm can also be computed more accurately now if reasonable assumptions are made concerning the saddle-point structures. While the DFT calculations solve the problem of sensitivity to choice of atomic potential, they are restricted to fewer atoms, so that the choice of boundary conditions becomes crucial, and several new schemes, in addition to simple periodic continuation of dipoles, have been tried. The chapter by Cai et al. reviews many of these calculations – for our purposes the main results are as follows. 1. A new reconstruction of the 90◦ partial in silicon has been proposed, which doubles the period (DP) along the core, as shown in Fig. 2 [27]. The energy of this differs little from that of the reconstructed single-period form, so both may co-exist, however kink motion will be different. Similar structures are predicted in germanium, diamond and GaAs, where polarity effects arise [28]. (It is difficult to compare the energies of the α and β forms in these simulations – the computational supercell must contain both in order to preserve charge neutrality and so becomes very large.) 2. It has been appreciated that members of a kink pair differ atomistically, so that distinct left and right forms must be considered [29]. Left kinks may be transformed to right kinks by passage of the APD (Fig. 3). The APD may bind to a kink (with a lubricating effect on mobility?), leading to a multitude of possible structures (e.g., 16 for the DP 90◦ ) when different reconstructions are allowed. The lowest-energy variants have yet to be accurately identified. These core defect structures may be classified by considering the symmetries allowed by the host crystal and the displacement field [24]. Dissociation of kinks into partial kinks has been proposed, providing a relationship between the SP and DP forms of the 90◦ partial [24]. 3. Ab-initio simulations have been extended to many more materials and defects, providing energies of some of the various core defects. Examples include kink energies in Si and GaAs [28,30–32], diamond [33], and both cubic and hexagonal silicon carbide [34]. (Core structures are identical in these polytypes, since the local environments are identical up to third nearest neighbors). Fig. 4 shows the results of

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Fig. 4. Formation Ef and migration Wm energies for kinks on the 90◦ partial in SiC. Results are shown as a function of configuration coordinate (kink pair width) for both a half-plane terminating in a Si atom, and a C atom [34]. Migration energy is controlling in both cases.

density-functional calculations for kink energy as a function of kink-pair separation in SiC [34]. The Si partial is found to be less mobile than the C, where weak reconstruction facilitates kink pair formation. For both cores, mobility is controlled by migration, not formation energies. The coexistence of shuffle and glide forms has been tested recently using ab-initio calculation [35]. The location of arsenic dopants at partial dislocations in silicon has been determined [36], leading to a negative core charge. For the effect of point defects on dislocation kink velocity, several theoretical models have appeared since the original obstacle model of Celli and Thomson [3]. These consider the effects of weak and strong obstacles to kink motion, the entrainment of point defects and the effect of a random force field [36–40]. In summary, it is evident that theoretical estimates of kink, reconstruction and other core defect energies for the same system by different groups and methods span a wide range (as do experimental measurements). The reasons for this in ab-initio work may be traced to differing treatments of boundary conditions or use of an insufficient number of atoms, together with the fact that in many cases there are simply too many possible candidate

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structures to search for a minimum energy configuration, or for the saddle-point configuration, needed to calculate migration energies. The “bottomless complexity” (Bulatov, 2001) of possibilities for core defect structures then suggests a statistical approach based on a symmetry classification of possible defects. But the best hope for progress now appears to lie with improved experiments on the most well-characterized samples, which we now review.

4. Recent experiment. What is known The following is a brief summary of experimental results on dislocation core defects and their effect on mobility, with emphasis on recent work. X-ray topography, TEM, and etch-pit methods have been used to measure the velocity of unit dislocations at various temperatures and stresses, leading to the value of the activation enthalpy Q given in eq. (5) above. However these experiments tell us little about core defects. Most experimental work on core defects has been aimed at determining the relative contributions to Q from kink formation, kink motion, and obstacle pining energies. (The energy barrier to kink motion along the core is sometimes referred to as a “secondary Peierls potential”.) Three methods have been used – TEM observations of dislocation motion (to determine their velocity at known stress, temperature and doping), internal friction measurements, and intermittent loading. The TEM work frequently uses either samples prepared by a special two-stage deformation process (resulting in partial dislocations quenched at non-equilibrium separations [41]), or samples mounted in a straining stage for the TEM [42]. By comparison with experiments on bulk samples, the TEM work deals with the regime of high stress and short segments, limited by the sample thickness. In the first case, the stress (about 250 MPa) is obtained from measurements of the partial separation and the known elastic constants. The sample is warmed in the microscope and video recordings made as the partials relax and move to equilibrium at a sufficiently low temperature to ensure that no new kinks are generated during motion. Then VD = 2chVk ,

(6)

with h the kink height (equal to the distance between (110) tunnels in silicon), and c their concentration, as in eq. (1). The change in stacking fault width after relaxation by warming can then be measured [43,44], giving the dislocation velocity and the kink migration energy Wm since no new kinks have formed. These changes in width cover a wide range in the work of several groups, and a summary analysis is given in [2]. The main conclusions are that in silicon for the dissociated 60◦ dislocation, the 90 degree partial is about three times more mobile than the 30◦ partial, the 30◦ partial of the screws is less mobile than that of the 60◦ dislocation, and the trailing partial in screws is less mobile than the leading. (The dislocation EPR activity has also been localised to these slower 30◦ partials in screws). Where the TEM observations can be made with near-atomic resolution [45], an additional estimate of the kink density c can be made, and video recordings obtained at near-atomic resolution during partial dislocation motion. This “forbidden reflection” lattice

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imaging technique is described in more detail in Section 5. The main results for dissociated 60 degree dislocations in silicon are: (i) A higher density of kinks (by a factor of 3) is found on the ninety degree partial, consistent with earlier mobility observations; (ii) From relaxation experiments at low temperatures which avoid kink nucleation, and with the electron beam turned off during partial dislocation motion (to avoid radiation-enhanced dislocation glide (REDG [46]), the kink velocity can be deduced from the dislocation velocity using eq. (6). From this, eq. (3) gives Wm (90◦ partial) = 1.24 ± 0.07 eV (T = 130 ◦ C, τ = 275 MPa), in reasonable agreement with the results of earlier experiments [43,44,47]. (For the 30◦ partial, Wm = 0.797 ± 0.15 eV). By making a rough estimate of the distribution of kink-pair separations, one may estimate the kink formation energy from these images using eqs (1) and (2) after extrapolation to the kink-kink saddle-point separation x = x ∗ = (μbh/8πσ )1/2 = 0.81 nm. For the 90◦ partial the value Fk = 0.73 ± 0.15 eV is obtained (T = 420 ◦ C, 250 MPa). These values give Q = Fk + Wm = 1.97 + 0.2 eV for dislocation segments much longer than the kink mean free path. Hence, since Fk < Wm , they suggest that, unlike metals, kink mobility rather than formation is the rate-limiting step. However the effect of unseen atomistic obstacles cannot be ruled out by these experiments. A second TEM approach is based on straightforward measurements of dislocation velocity (under conditions where the partials cannot be resolved in the images) for different segment lengths, temperatures and stresses. Recently, the use of strained layer (Si)/GeSi/Si heterostructures has been found useful for this purpose [46,48]. These provide dislocation motion under controlled conditions during observation at 1 nm resolution in the TEM. (Bulk measurements on similar material can be found in [49–51]). TEM movies of threading and misfit dislocation formation allow detailed modeling of the interaction of the dislocations with each other, and with point and planar faults. These samples also allow a quantitative analysis of dislocation velocity in the two regimes where kinks may or may not collide (due to surfaces or strong obstacles) [52]. These observations can provide an estimate of Wm , as in the work of George et al. [53] who directly observed the transition between the length dependant (L < λ) and the length independent (L > λ) regimes of dislocation velocity. Recent measurements of dislocation velocity can be found in Vanderschaeve et al. [42], who use a straining stage and obtain the stress from the curvature of the moving dislocations. For compound semiconductors, temperatures in a small range around Q/(30k) must be used, where the velocity is within video recording range. This range is too small to provide an Arrhenius plot. For silicon, a larger range of about 100 ◦ C may be used. REDG effects are analysed in detail, and the effect of surface image forces on velocity are clearly seen. Note that in the length-dependent regime of eq. (4a), Q may be determined from a plot of vD (L) at fixed temperature, while for the length independent regime the saturation velocity (where L ∼ λ) can be used to obtain a second value of Q. If obstacles are not considered, so that eq. (4a) applies, measurements of the two Q values in each regime give Fk and Wm directly by subtraction (see Table 1). However this analysis has traditionally given pre-exponential values widely different from theory, with the discrepancy attributed to a (large) entropy term. In TEM work on heterostructures in 1993, Maeda and Yamshita [5] observed velocities in both regimes but failed to find the expected change in Q at the saturation velocity (eq. (4a)). These results could be explained by assuming that the kink mean-free-path λ is limited by obstacles rather than kink-kink

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Table 1 Summary of measurements of kink migration (Wm ) and formation (Fk ) energies for 60◦ dislocations in silicon and germanium Si Wm (eV)

Ge Fk (eV)

1.6 1.6 1.5

0.45

1.4

0.6

1.6 1.2 1.8 1.8* 1.5 1.8 1.24 (90◦ )

0.7 1.0 0.6 0.5

1.3

Wm (eV) 1.1

0.4 0.727 (90◦ ) 0.797 (30◦ ) 0.9

Method

Ref.

IF IF IF IF TEM TEM MD, TEM MD,TEM MD, TEM IL TEM/IL TEM HREM HREM TEM

[106] [107] [108] [109,57] [110] [55] [68] [52,48] [51] [111] [112] [56] [45] [45] [42]

Fk (eV)

1.1

0.5

0.9

0.5

0.85

0.5

These results may or may not include entropy corrections or the Seeger–Schiller high-stress correction applicable to TEM work. Values of Fk can sometimes be deduced from the known sum (Fk + Wm ) (eq. (5)) and Wm value in the kink collision regime, but see [48]. Thin-film TEM measurements usually assume kinks reach surfaces before colliding, unlike the kink collisions assumed in bulk measurements (unless obstacles prevent this). Errors are typically 0.1 eV. IF = Internal Friction. MD = Misfit dislocations. IL = Intermittent Loading.

collisions and annihilation [46]. Then λ is less than the fixed distance D between obstacles, and we may use eq. (4a) with D = L ≪ λ, and the velocity becomes length independent, but Q = 2Fk + Wm . Reconciling all this most recent work on dislocation velocity in silicon is difficult, and cannot be said at this stage to provide definitive support for the obstacle theory over Hirth and Lothe’s diffusive theory, but we might summarize as follows: (i) All groups (recently [42,46,52–54]) find velocities proportional to length for short segments (L < λ), where kinks don’t collide, L is about half a micron, and Q = 2Fk + Wm . (ii) At longer lengths, where velocity is independent of length, Q is unexpectedly found to remain unchanged [5,52] giving support to an obstacle model, even for bulk samples [51]. The form of the obstacles, whose separation is about a micron, is unknown, but could be jogs. Kinks wait for annihilation at obstacles until met by a kink of opposite sign. The dislocation motion then becomes entirely controlled by nucleation of double kinks on pinned segments. It is possible that pinning centers present in these alloy samples are not present in pure silicon and germanium [55], where the diffusive theory still applies. Experiments on the temperature dependence of the saturation length in pure silicon (which isolate Fk ) are needed to resolve this issue.

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(iii) The experiments which directly observe kink motion and pinning in silicon [45] do find pinning centers with a spacing of about 10 nm (and unpinning energy 2.4 eV) in support of the obstacle theory. The value of Fk = 0.7 eV found in that work by estimating the density of critical kink nuclei from the images, suggests a much larger kink mean free path (about one micron) than the kink spacings seen in the images (about 10 nm), perhaps due to assumptions of equilibrium or surface roughness effects. The saturation length of about one micron from velocity measurements sets a lower limit of 0.5 eV for Fk . A summary of experimental measurements of Wm and Fk is given in and Table 1 for 60 degree dislocations in Si and Ge ([56] with additions). These values may or may not include entropy corrections or the Seeger–Schiller correction, which reduces Fk at high stress. Hull and Bean [48] have pointed out the importance of this factor for TEM experiments, where stresses (typically 1 GPa) are much higher than for bulk experiments (100 MPa). Note that values of Fk can, under certain conditions, be deduced from the known sum Fk + Wm (eq. (5)) in the kink collision regime. Thin-film TEM measurements usually assume that kinks reach surfaces before colliding, unlike the kink collisions assumed in bulk measurements (unless obstacles prevent this). Internal friction measurements have been used to give values of Wm = 1.11 eV and 2Fk = 2.07 eV for dislocations in germanium [57]. A recent theoretical model for the interpretation of this data is given by Antipov et al. [58]. The intermittent loading technique measures the glide distance of dislocations in response to a pulsed load, and has recently been applied to Si, Ge and SiGe alloys [40]. The interpretation of these data are difficult, however; a discussion can be found in [2]. Studies of dislocation mobility in materials other than silicon have also sometimes led to atomistic models which allow deductions to be made about core defects. SiC PiN diodes, for example, are capable of transmitting Megawatts of power and posses kilovolt breakdown voltages, however they degrade during forward biased operation due to an electronhole plasma which propagates stacking faults. The dislocation core energy levels (presumably at kinks) responsible for this plasma-enhanced kink motion [46] have recently been identified and the activation energy measured [59]. (For silicon, Werner et al. [60] were able to attribute similar recombination enhancement of glide to a reduction in Fk rather than Wm in their observations of the cathodo-plastic effect.) In SiC also, the ductile-brittle transition has been analysed in detail – a transition temperature is found below which a single leading partial dislocation is responsible for shear, and above which a dissociated pair of partials is responsible [61]. Measured values of Q are given for α, β and screw dislocations in Si, GaAs and InP at various temperatures by Sumino and Yonenaga [62]. Results for Wm and Fk are given for GaAs, InSb, InP, Ge and Si by Vanderschaeve et al. [42]. The mean-free path is much longer for GaAs and especially InSb (>5 microns) than silicon. In addition, β (e.g., Ga) cores are found to be about 350 times slower than α (e.g., As) cores. Dislocation dynamics and the Peierls stress in spinel are described by Mitchell [63]. A new more general double kink mechanism (in which kink-kink interaction dominates) has recently been proposed to account for the yield stress in intermetallics and refractory oxides (such as sapphire and spinel) [64]. The effect of various dopants and impurity atoms on dislocation mobility have been measured by several groups – for example in silicon Q is found to be reduced from 2.2 eV

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to 1.2 eV after irradiation by a hydrogen plasma [65]. The effect of oxygen has been extensively studied. In Czochralski-grown silicon the interstitial oxygen greatly improves mechanical strength by locking dislocations. At the higher processing temperatures, the oxygen diffuses to the dislocations. The unlocking stress then depends on the density of oxygens along the line, and can be measured by etch-pit methods during application of controlled stress in a three-point bend. The oxygen transport and segregation to the dislocation cores has been found to exhibit five regimes as a function of annealing time [66]. Precipitation of oxygen at the core has been identified, and the oxygen-dislocation binding enthalpy measured (0.2 eV below 650 ◦ C and 0.74 eV at higher temperatures). Kinks have been directly observed by TEM pinned at a core defect [45] with unpinning energy of 2.4 eV (obtained from the pinning time and temperature of 600 ◦ C), in rough agreement with calculations [67] and X-ray topography studies of dislocation motion in silicon containing known amounts of oxygen [68]. A detailed theory of the interaction of oxygen atoms decorating a moving dislocation [69] gives good agreement with the experimental measurements of dislocation mobility by Imai and Sumino [26]. The resolution of the best high-resolution electron microscopes (HREM) is now about one Angstrom. Work has continued using these to obtain new information on core defect structures over the past decade. For dislocation cores running normal to the surface of a thin film, and viewed by HREM along the line, the favorable occurrence of plane strain means that a simple projection of displaced atomic columns is obtained if the core is atomically straight (unkinked). In a thin film, however, the self-energy, together with image forces, will tend to rotate the partials as they approach the surfaces, and the final arrangement will depend on temperatures and deformation during sample preparation. However any aligned, unkinked portion of the core near the midplane will nevertheless provide a sharp image in projection, with material above and below providing a smooth background unresolved in projection. For certain line defects, such as stair-rods and sessile cores which form at the intersection of planar faults, a straight core can be expected to provide an intense HREM image. In this way the atomic positions and local structure of a Z-shaped faulted dislocation dipole in GaAs have been determined by comparing experimental and simulated HREM images [70]. (The dramatic improvement in resolution of HREM machines is evident by comparing this work with earlier lattice images of the same defect [71]). A detailed analysis of the methods of quantitative HREM used to compare experimental and computed images is given in [72], where 60◦ misfit dislocations in InGaAs/GaAs are studied and found to be of the shuffle type. The method of electron-lens aberration correction, using a throughfocus series, is treated in detail. Other recent examples of dislocation core imaging by HREM include: (i) The imaging of misfit dislocations at interfaces in metal-ceramics. Consistent with theory [73], these have been found to be set back slightly into the softer material [74,75]. Dissociated dislocation cores are imaged by HREM in SrTiO3 in combination with energy-loss spectroscopy by Zhang et al. [76,77]. For the a[100] dislocation, this work suggests a charged, oxygen-deficient core, a reduction in crystalfield splitting and an increase in the L3 /L2 ratio. Oxygen deficient cores are also found in the dissociated a110 grain boundary dislocations, which involve climb, and a reduced crystal field at the cores.

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Fig. 5. Non-planar dissociation of partial dislocation cores in TiAl, indicated by arrows. A [011] superlattice dislocation observed along the dislocation screw axis orientation. In contrast to three- and twofold planar dissociation observed in the low temperature range, non-planar dissociation involving (1−11), (100) and (−1−11) planes due to thermally activated cross-slip is observed here in the anomalous temperature range [81].

(ii) Several groups have attempted to observe the predicted non-planar three-fold dissociation of screw dislocations, which, in b.c.c. metals is thought to impede their mobility – see for example [78]. For this work, a projection along the line is used. For the (a/2)[111] screw in Mo, Sigle undertook a detailed comparison of HREM images with molecular dynamics simulations. The image contrast is dominated by the Eshelby twist relaxation in these thin films, yet evidence remains of a threefold dissociation. In CoTi (B2 structure), HREM imaging of an edge dislocation core shows a core spreading of about 1.2 nm. For screw dislocations the spreading on (010) was 1.8 nm, and this spreading is taken to account for the anomalous temperature dependence of the yield stress of CoTi [79]. (iii) In the intermetallic structures, many interesting arrangements have been found – for the L12 structure Ni3 Al, see [80], where superpartials are seen by HREM on cross-slip planes. As shown in Fig. 5, superlattice dislocations have been imaged at atomic resolution in TiAl by Inui et al. [81], showing non-planar dissociation. The Ti-56at%Al single crystal sample was compressed along [−152] at 800 ◦ C, in the anomalous temperature range. In Fe3 Al (DO3 structure), the superpartials bounding an antiphase boundary have been imaged. Here the HREM images were taken in an orientation with the beam normal to the dislocation line but lying in the plane of the planar fault connecting the partials. As a result, a rare image of a

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Fig. 6. A rare TEM image of a jog, seen at D in Fe3 Al, as viewed from the side (beam normal to dislocation line and in plane of stacking fault) [82].

jog can be directly imaged, as shown in Fig. 6 [82]. Superdislocations in Ni3 (Al, Ti) alloys are also studied by HREM in Kawabata et al. [13], which contains a summary of dissociated fault energies from these alloys. (The fault vector, energy, atomic dimensions and planar structure can often be obtained from these HREM images.) (iv) HREM imaging of climb can provide information unobtainable by other methods on the atomistic processes involved. Jogs are commonly seen as a constrictions in weak-beam images, where the partials are seen to come together at a point where climb occurs onto a new (111) glide plane occurs. A detailed analysis of climb in silicon bicrystals using the HREM technique can be found in [83]. Further observations of dislocation climb are analysed in [84] and [85] (in GaAs), while [86] summarizes the point defect complexes which facilitate climb. (v) The power of combining atomic-scale imaging and energy-loss spectroscopy (ELS) in the scanning transmission electron microscope (STEM) has recently been realized experimentally. The ELS absorption spectra probe empty states in the conduction band, and may be obtained from sub-nanometer diameter regions with an energy resolution of about 0.6 eV. (The commercial availability of monochromators since 2003 will soon extend these results to about 0.15 eV for routine work.) For example, in GaN, spectra in the 1–10 eV band-gap region were recently obtained from dislocation cores in good agreement with first-principles calculations [87]. Similar observations have also been made for diamond [88]. These low-loss spectra probe the relatively delocalised wavefunctions in and around the band-gap. Greater localisation (but a much weaker signal) can be obtained using transitions from in-

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Fig. 7. ADF STEM image of a dissociated 30◦ /90◦ dislocation in a Ge/Si alloy viewed along [110] in a thin film. The intrinsic stacking fault ISF is marked. Each bright (or dark) dot is a pair of atomic columns viewed in projection along [89] [110]. ELS spectra were taken as the lettered points: a) bulk, b) tension, c) compression, d) ISF, e) 30◦ core, f) 90◦ core.

ner shell states into band-gap states of the dislocations, and this approach has been applied to silicon by Batson [89] as shown in Figs 7 and 8. His STEM instrument provides both dark-field scanning transmission images at atomic resolution and energy-loss spectra from sub-nanometer regions which can be identified in the image. He derives models for the cores of a dissociated 60◦ misfit dislocation which are consistent with both the images and the spectra. The partials in a Ge0.35 Si0.65 alloy are shown in Fig. 7 – it is not possible at this 0.28 nm resolution to determine whether the bright or dark image blobs represent the pairs of atomic columns, which are seen in projection along [110], but these provide an accurate scale for the spectroscopy and a constraint on atomic models. Kinks along the cores may reduce the core image contrast. Dark regions are most heavily strained. Fig. 8 shows the ELS spectra (dots) recorded with 0.25 eV resolution from the regions identified in Fig. 7. This experimental data has been deconvoluted to remove background and the 2p1/2 –CB transition intensity, leaving transition from the inner 2p3/2 shell to the s- and d-projected local density of states (LDOS). A calculated alloy LDOS is also shown, and its fit to the data, including core exciton, lifetime and instrumental broadening effects. The fit to the bulk is excellent, with the symmetry points of the bandstructure labeled. The splitting of the L1 conduction band minimum at the intrinsic stacking fault (ISF) is caused by third-nearest neighbor interactions at the fault. The spectrum from the 30◦ core shows similar splitting and evidence of deep core states. The 90◦ degree core does not show this splitting, but does display also

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Fig. 8. Spatially-resolved inner-shell L-edge (2p3/2 ) energy-loss spectrum obtained from the four subnanometer regions indicated in Fig. 7 by STEM. The L1 band is split at the ISF into two components. This splitting is present also at the 30◦ partial core, but not at the 90◦ partial. Both dislocation cores support near-edge gap states at D1 , but the ISF does not.

evidence of deep core states. This suggests that the 30◦ core resembles the stacking fault, while the 90◦ core resembles the bulk. Using published core models and relaxed atomic potential modelling, Batson shows that these ELS spectra favor a

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kinked DP 90◦ [27] core, a variant of that described by Chelikowsky et al. [90] for the 30◦ partial and rule out shallow states at the stacking fault near the CB edge. Fast parallel detection is required for this work, in order to avoid contamination or hole-drilling from the small intense probe. For the application of this STEM ELS method to oxides (including superconductors) and their interfaces and defects, see [91] and references therein. The combination of HREM and ELS is applied to dislocation cores in strontium titanate by Zhang et al. [77].

5. The way forward. UHV TEM, ELS and nanodiffraction Given that the best resolution of current electron microscopes is about one Angstrom, it would seem that many of the questions set out at the beginning of this review could be answered by high resolution TEM imaging, if the core defects can be viewed in a favorable projection within samples with atomically smooth surfaces. In addition, the availability of monochromators now makes possible the collection of energy-loss absorption spectra with perhaps 0.1 eV energy resolution and nanometer spatial resolution. Finally, the information in nanodiffraction patterns can provide direct evidence for reconstruction. In this section we suggest experiments based on these techniques which might clear up these fundamental questions in kink dynamics. By remarkable good fortune, a thin [111] slab of fcc, diamond or sphaelerite structure material, in the transmission–diffraction geometry, generates additional “termination” Bragg reflections (with indices such as 1/3(422)), unless the thickness of the slab consists of exactly 3n double-layers (with n an integer) [92–94]. These “forbidden” reflections, which can also be generated by stacking faults, have found many uses [95,96]. Images formed with these reflections in the electron microscope therefore reveal directly the border of the SF, and hence any directional fluctuations in the dislocation cores bounding the fault. Simulations and experiments [45,93,97,98] provide the best hope for direct imaging of kinks and their dynamics in real time, and provide a unique window onto dislocation core structure which has yet to be fully exploited. The geometry, with the beam normal to the glide plane, produces an image which directly reveals kinks, however jog structure along the beam direction is lost in projection. The images can also be used to reveal reconstructions along the core (more readily for the 30◦ partial in silicon than the 90◦ [97]). There are no such “termination reflections” in the wurzite structure, however weaker kink contrast can nevertheless be expected [99]. Experimental work in this area is beset by two main difficulties – roughness on an atomic scale on the surfaces of the film, and REDG. The effects of surface roughness can be eliminated by using ultra-high vacuum electron microscopes (UHV-TEM) which allow surface cleaning. Alternatively, using video recordings, to a lesser extent roughness is minimized by subtracting images taken before and after dislocation core motion. (Since HREM images show columns of atoms in projection, the contrast changes due to kinks or additional atoms at surfaces are difficult to distinguish in projection.) One can only minimize the effects of REDG by using low-dose techniques, or by recording images before and after core motion, with the beam turned off during motion.

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Fig. 9. Experimental CBED pattern from an intrinsic stacking fault on (111) in silicon lying normal to the beam. The “termination” {−422}/3 type forbidden reflections can be clearly seen inside the bulk allowed {220} reflections. Circle shows aperture used for imaging (Philips FEG 400 ST, 120 kV. Probe size about 2 nm). The (1−11) reciprocal lattice point lies directly above (2, −4, 2)/3.

Fig. 9 shows an experimental image of these remarkably strong termination reflections, obtained from silicon. An image formed with the inner six of these “termination” or forbidden reflections in the (111) zone provides a lattice image of the SF alone, and its boundary at the dislocation core. The d-spacing for the “forbidden” planes is d422 = h = 0.33 nm, or one Peierls valley wide. These valleys run along the 011 tunnels in the diamond structure, orthogonal to (42−2)/3. The occurrence of termination reflections may be understood in several ways. A single (111) double layer of silicon atoms produces a much denser reciprocal lattice (which includes {−422}/3 reflections) than does an infinite crystal. This occurs because atoms in a single double layer are more sparsely packed that those in a[111] projection of three double layers, which overlap in projection, leading to a less dense reciprocal lattice without g = (−422)/3 type reflections. Dark-field images formed with them show single atomic-height surface steps on thin foils [92]. If the [−1, −1, −1] beam direction is taken into the foil, then a (1−11) reflection lies in the first-order Laue zone (FOLZ) directly above the (2, −4, 2)/3 ZOLZ termination reflection shown in Fig. 9. We can thus consider the (2, −4, 2)/3 ZOLZ spot to be the tail of the crystal shape-transform (or rock-

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ing curve) laid down around the (1−11) HOLZ spot, and extending down into the ZOLZ. Since the (1, −1, 1) reflection is weak in the ZOLZ, single-scattering theory can be used to give the intensity of the (1, −1, 1) beam as sin2 π · z · sg φg (z)2 = z2 σ 2 V 2 2 . g π · z · sg

(7)

The period of this function in z (giving rise to thickness fringes) is L = Sg−1 . The excitation error Sg of the (1−11) FOLZ reflection evaluated in the ZOLZ (i.e. √ at the (2, −4, 2)/3 position) is just equal to the height of the FOLZ, or |g(111)|/3 = ( 3/a)/3 = 1/(3d111 ) = S111 , where a is the silicon conventional cubic cell constant. Thus the period of thickness fringes is t0 = 3d111 = 0.94 nm, and we expect a sinusoidal intensity variation with thickness, with a period of three atomic double-layers. Termination reflections thus give “weak beam” thickness fringes with the period of the lattice in the beam direction. This approximate treatment assumes that the (2, −4, 2)/3 reflection is at the Bragg condition (rather than the [111] zone axis orientation used in experiments), and it ignores atomic structure within the 0.94 nm spacing along the beam path. Alternatively, we may explicitly evaluate the (2, −4, 2)/3 structure factor for silicon. If we choose an unconventional hexagonal unit cell for silicon with the c axis along the cubic [111] direction so that atoms have coordinates such as (1/3, 1/3, 1/3) etc., then a “forbidden” reflection g has hexagonal indices such as (11.0), and its structure factor becomes N

Vg =

47.878 e 47.878 e 2f (g) exp(−4nπi/3) fj (g) exp(−2πg · rj ) = j

(8)

n=0

for a crystal of N double layers, again at the Bragg condition for (2, −4, 2)/3 (cubic indices). Here Vg is in volts, is the cell volume and f e (g) the electron structure factor. Each term in the final sum is proportional to the scattering from one double layer, and these terms may be represented on an Argand diagram, as shown in Fig. 10. The imaginary part of Vg has been plotted horizontally and real part vertical. One side of the triangle represents the scattering from a single (111) double layer of atoms in silicon – to analyze the effects of shuffle or glide termination, half the lengths of the sides should be used. The lateral shear of each double layer in the diamond structure introduces a 120◦ phase shift, making a closed triangle ABC every three layers with zero resultant scattering. A thin crystal containing 3m double layers above an intrinsic SF and 3n below it has stacking sequence (3m) AB/ABC (3n), as shown. Below the SF the total scattering vector runs from the origin to each of the corners on the upper triangle in turn, and the intensity is never zero. If the scattering amplitude from one double layer is F (with intensity F 2 ), then the change in intensity due to the addition of a single double layer in an unfaulted crystal is either zero (on adding a B layer to an A layer) or F 2 (for addition of an A or C layer). For a faulted crystal the results depend on the depth of the fault, however all the possible cases may be obtained by starting at one corner of the lower triangle and ending at one on the upper. In particular, for the kink images, we may compare the diffracted intensity produced by a column of crystal within a ribbon of SF with that generated outside it. In the most

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Fig. 10. Argand diagram for a {−422}/3 reflection in silicon. Real part of structure factor F plotted vertically, imaginary part horizontal. The kinematic amplitude of Bragg scattering is proportional to a vector from √ the origin to one corner of the figure. A crystal with stacking sequence ABCAB/ABCA produces the amplitude 3 F shown.

favorable case, an unfaulted region ABCABCABC produces zero intensity, but a faulted crystal ABCAB/ABCA of the same thickness generates intensity 3F 2 . Multiple scattering calculations [93] confirm these single-scattering estimates. Fig. 11 shows more accurate multiple-scattering calculations for the intensity of the (−2−24)/3 reflection as a function of thickness, and we see that the addition of a C layer (shown primed in Figs 10 and 11) after the fault changes the intensity by about 4F 2 , in agreement with Fig. 10. In this most favorable case the intensity changes by a factor of four for the addition on one layer, effects which may be seen experimentally in Fig. 12. These calculations (unlike Fig. 10) correctly take account of the excitation errors for the termination reflections, and when this is done we find that the six {−422}/3 reflections are not equivalent – there are two groups of three, reflecting the three-fold (not six-fold) symmetry of a (111) slab consisting of p = 3m lay-

Fig. 11. Multiple scattering calculations (using 1000 beams) giving the thickness dependance of the termination reflection (−2−24)/3 in silicon at 100 kV, with beam along [111]. Each letter represents the addition of one atomic layer. A stacking fault occurs at ABA. Three of the six beams are equivalent.

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Fig. 12. Image from silicon projected along (111) obtained in an ultra-high vacuum (UHV) TEM which enables the preparation of samples with atomically smooth surfaces. The bright band is a dissociated dislocation. Distinct patches with differing shades of gray are regions in which the thickness changes by a single double-layer of atoms. (300 kV Hitachi UHV TEM Si from [100].)

ers. At the [111] zone axis orientation the forbidden reflection intensity nevertheless still falls to zero every three double layers. Fig. 12 shows a bright-field image of a thin silicon film recorded in the [111] orientation in a UHV TEM after following an annealing and oxygen-etch procedure to produce atomically smooth surfaces [100]. Remarkably, we find that each gray-level represents a change of one atomic double-layer in thickness on either side of the slab. (The gray-level boundaries cross where steps on either side superimpose.) As predicted, the image of a dissociated dislocation within the slab (the bright line across the image) shows clear intensity changes as it crosses surface steps, in agreement with detailed calculations [101]. Real crystals have atomically rough surfaces, and these effects must also be considered, in addition to other sources of background in the images, which limit contrast. Fig. 13 shows a similar image (but now with near-atomic spatial resolution) from a silicon sample without surface cleaning. It was formed by interference between the inner six 1/3{−224} termination reflections of Fig. 9 [45]. The bright diagonal band of regularly spaced dots is a lattice image of the double layer of atoms (spaced 0.33 nm apart laterally) which form the SF plane. Pairs of atoms in the SF appear as a single dark spot; bright spots are centered on

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Fig. 13. TEM image of dissociated 60◦ dislocation in silicon after relaxation viewed along the normal to the (111) glide plane. The bright diagonal band of regular dots are six-membered rings in the ribbon of SF separating 30◦ and 90◦ partial dislocation lines. Black lines run along cores of the two partial dislocations. Fine white line shows typical alternative boundary used to estimate error in counting kinks [45].

the six-fold rings of a single double-layer. The borders of this band of regular dots forms the partial dislocation cores, as shown. The white scale lines indicate one Peierls valley, 0.33 nm wide. The average SF width corresponds to a stress on the partials of 275 MPa. The effects of surface roughness are clearly seen outside the SF, however an approximate estimate of kink density can be made. A full analysis of the effect of surface roughness has been given [93], but the main points are as follows. TEM cross section lattice images of Si/SiO2 interfaces suggest that the roughness will be one or two double-layers. Large atomically flat surface islands produce sharp forbidden reflections unless p = 3m. As the island size becomes small compared

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Fig. 14. Similar to Fig. 13, but for an unrelaxed pair of partials spaced more closely than the equilibrium separation in material with heavier oxide coating.

with the coherence width of the electron beam these forbidden reflections broaden out into diffuse elastic scattering peaks. For a random distribution of surface vacancies there are no termination reflections. The width of the diffuse peaks increases with the depth of the surface roughness. We can understand this by recalling that the projected potential for a thin “perfect” crystal with atomically rough surfaces is not a laterally periodic function, hence its diffraction pattern contains elastic diffuse scattering. If surface islands are not seen in forbidden-reflection lattice images from unfaulted crystal, we might assume that the roughness can be modeled as random vacancies in the surface layer. Depending on the crystal thickness, these vacancies may or may not lie within the atomic column which contains a kink, thereby altering its contrast. Again, if images of dislocation cores show a much higher density of kinks on the 90◦ partial dislocation than on the 30◦ partial, we might conclude that surface roughness is not the dominant contrast effect. Fig. 14 shows an image from a different sample which is more heavily oxidized. Fig. 15 shows a simulation for kinked partials within a slab whose surfaces are atomically clean (but not reconstructed) – the structural model, which contains a variety of core defects, is shown in Fig. 3(b), and

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Fig. 15. Simulated HREM image of dissociated 60◦ dislocation in silicon projected along (111), based on the structural model shown in Fig. 3(b). The [110] direction is across the page. Kinks and various defects are shown on each partial. The surfaces of the slab are not reconstructed, as shown in the next figure. (Atomically smooth surfaces, 200 kV, Cs = 1.2 mm, Scherzer focus, 10 triple-layers thick). From [101].

could probably be deduced from such an image recorded under these conditions. Fig. 16 shows a remarkable experimental image [102] taken under similar UHV TEM conditions at atomic resolution, in the absence of dislocations. All the details of the famous 7 × 7 Si (111) surface reconstruction are visible for the top three layers of atoms, unlike STM images, which show only the adatoms in the first layer. The translational and symmetry averaging processes which have been applied to this image could not be applied to such an image showing dislocation cores. The ability to clean the surfaces of samples containing dislocations by in-situ heating insitu thus improves the quality of these images greatly. The question of whether such heating will anneal out all dislocations in a thin film then becomes crucial. Fig. 12 shows that this is not so. Our experience has been that there is always sufficient pinning of dislocations to allow a few to remain after a low temperature surface cleaning. The far less effective technique of taking the difference between images before and after dislocation motion is shown in Fig. 17 – this does clearly reveal strips of SF as narrow as a single Peierls valley, however their edges cannot be delineated with atomic precision from individual video frames. The field-emission electron gun provides the brightest particle source in all of physics, and the interaction of electrons with matter is much stronger than that of other particles. It follows that nanodiffraction patterns, in which the electron probe is focussed down to

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Fig. 16. Experimental HREM image of thin 111 silicon slab projected along [111]. The image was obtained in a UHV HREM instrument, and shows directly the 7 × 7 surface reconstruction, with atomic structure inset. Unlike STM images, this HREM image shows all the atoms in the top three layers in projection [102]. Fig. 15 shows the effect a dissociated dislocation would have within the sample on this image.

sub-nanometer dimensions, provide the strongest signal from the smallest volume of matter possible. Fig. 18 shows simulated coherent nanodiffraction patterns obtained with the electron beam aligned along the cores of 90◦ partial dislocation in silicon within a (111) slab about 60 nm thick (beam normal to slab). The patterns for single-period (SP) and double-period (DP) reconstructions of Figs 2 and 3(a) (RHS) are compared. The calculations include all multiple-scattering and thermal diffuse scattering effects [103]. The outer ring in the SP pattern is known as a “Higher-order Laue Zone” (HOLZ), and is produced by diffraction from planes whose normals are not normal to the beam. For the DP core, an additional “half-order” HOLZ ring appears due the new periodicity (twice that of the host

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Fig. 17. Difference between filtered video images of a moving 90◦ partial dislocation in silicon at 600 ◦ C, viewed along [111]. Dark strip is SF three Peierls valleys wide (0.99 nm), eliminated by passage of several kinks. Inset shows experimental SF image for scale. The dark patch is a portion of the shaded region in Fig. 4(a).

Fig. 18. Computed electron microdiffraction patterns with a STEM probe stationary along the core of a 90◦ partial dislocation in Si. The additional ring in the right-hand pattern clearly distinguishes the double-period from the single-period reconstruction. The intensity is displayed on a logarithmic scale. The probe diameter is 0.5 nm [103].


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crystal) which arises along the beam direction. The sub-nanometer diameter electron probe has concentrated practically all the diffracting energy onto the core, yet Bragg reflections from nearby perfectly crystalline material can still be indexed. Unlike the HREM experiments, all the core atoms contribute to the measured intensity, and the results are much less sensitive to instrumental parameters, which need not be measured. The diffuse scattering from the core generates sheets of scattering normal to the beam in reciprocal space from this line defect. The sheets are spaced at half the spacing of reciprocal-lattice points for the DP reconstruction. This nanodiffraction experiment [104] is probably the simplest possible which would produce clear, direct experimental evidence for core reconstruction. By varying the sample temperature, activation energies might be measured for reconstruction. Experimental difficulties are likely to include radiation damage in the form of “hole drilling” along the core, and provision for a simultaneous real-space imaging mode needed to locate the probe over the core [101].

6. Summary Despite the limitations of TEM methods (effects of surface forces, need to minimize REDG, diffusion of impurity atoms at surfaces to cores) it appears that this approach, especially the forbidden-reflection method, using samples with atomically clean surfaces in ultra-high vacuum, now offers the best hope for resolving the issues raised at the beginning of this article. (A critical discussion of this approach can be found in Jones [113]). Direct measurement of kink densities and core reconstruction should be possible by this method, if not the determination of kink structure at the atomic level. Direct evidence for reconstruction, and measurements of the associated activation energy might also be obtainable by electron microdiffraction with the beam aligned along the core. Electron energy-loss spectroscopy is a tool whose power is only now beginning to be realized, and which will prove far more powerful in the near future with the commercial availability of monochromated instruments. Understanding the role of obstacles in limiting kink and dislocation motion appears to be the most fundamental unanswered question, and this may be approached with further experiments on the temperature dependence of the saturation velocity at which dislocation velocity becomes independent of segment length for samples of different purity. The failure of Schmid’s law requires further investigation at the nanoscale level, and the role of the Escaig stress component, which exerts opposite forces on each partial, must be understood since experiments indicate that stress components other than the glide stress are important [7]. The nature of the shuffle core structures operating in unit dislocations at high stress and low temperature needs further elucidation, as does the transition from this form to the dissociated glide set at higher temperature. All these comments, which are made with silicon in mind, may be extended to the other semiconductors, while studies by HREM imaging of the crucial extension of cores in the bcc and intermetallic alloys remains in its very early stages. Very few atomic resolution images of the recently proposed disconnection defect have appeared [105] – these may prove crucial for theories of crystal growth. Understanding the atomistics of climb and of the earliest stages of the ductile-brittle transition are perhaps the most challenging problems ahead for the HREM method.

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CHAPTER 78

In situ Nanoindentation in a Transmission Electron Microscope ANDREW M. MINOR National Center for Electron Microscopy, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

ERIC A. STACH School of Materials Engineering, Purdue University, West Lafayette, IN 47907, USA and

J.W. MORRIS, JR. Department of Materials Science and Engineering, University of California, Berkeley, CA 94720, USA



Contents 1. Introduction 455 1.1. Overview 455 1.2. Background 455 2. Experimental procedure 459 3. Results and discussion 464 3.1. Silicon 464 3.2. Al thin films 470 3.3. Martensitic steel 482 3.4. Hard thin films on softer substrates 4. Conclusion 491 References 494

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1. Introduction 1.1. Overview The chapter describes the first comprehensive studies of the deformation of materials conducted with the mechanical testing technique of in situ nanoindentation in a transmission electron microscope (TEM). This technique makes it possible to observe the mechanical behavior of nano-scale volumes of solids in real-time. Conventional nanoindentation techniques have been developed over the past 20 years as a method to probe the mechanical properties of materials in the sub-micron size range [1,2] which is a typical dimension of the thin films used in integrated circuits and micro-electro-mechanical systems (MEMS). However, real-time observations of the mechanism of deformation in a material during nanoindentation have only become possible with the development of the in situ technique described herein. In typical nanoindentation experiments, the sample surface is indented with a sharp diamond pyramid, which has a tip that can be characterized by a radius of curvature on the order of 100 nm. Both elastic and plastic deformation results in the sample. The displacements imposed during nanoindentation are on the order of nanometers, and high-resolution load and displacement measurements are simultaneously recorded. Thus, it is possible to study atomistic-scale deformation processes, such as dislocation nucleation events [3,4] and pressure-induced phase transformations [5]. A multitude of behaviors have been observed with conventional indentation techniques [6], although direct observations of the associated mechanisms of deformation have been until now importantly lacking.

1.2. Background Indentation testing has been used for many years as a convenient method to characterize the hardness of a material, which is classically defined as a material’s resistance to plastic deformation [7]. The advent of the microelectronics industry necessitated the development of characterization tools more appropriate for the size and scale of the thin films and small structures associated with microelectronic devices. By the 1980s significant advances in nano-scale science and instrumentation led to the first “instrumented” indentation techniques at submeter length scales. Pethica, Hutchings and Oliver [1] first demonstrated the technique of providing continuous measurement of load and displacement during indentation with resolutions in the sub-microNewton and the sub-nanometer regimes. Since then, the technique of nano-scale hardness testing has become broadly referred to as nanoindentation.

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When considering how to probe the mechanical behavior of thin films and small (submillimeter) structures, there exist several alternative methods to nanoindentation. For example, wafer curvature [8,9] bulge testing [10,11] micro-beam bending [12], micro-tensile testing [13], micro-compression testing [14], and MEMS-actuated tests, e.g., [15] all measure the mechanical properties of thin films and small structures. Nanoindentation, however, is the only method that probes a small enough volume of material to observe discrete and localized deformation phenomena in a material that can be reasonably called defectfree. All other mechanical testing methods probe relatively large volumes and measure properties of materials in a continuum manner, where the mechanical properties are typically dominated by defects. During a nanoindentation experiment the peak stresses occur below the surface, in volumes that are small enough to be closely assumed defect-free, perfect crystal [16]. Clearly, indentation into defect-free volumes cannot be accommodated by conventional plastic flow – nucleation of defects must occur first. For example, Fig. 1 shows two examples of the most commonly observed discrete phenomena from a nanoindentation measurement. Fig. 1(a) shows load displacement data from both a 300 nm-thick Al film on Si and a polished single crystal of Al. Discrete displacement bursts can be seen in the loading portions of the curves. These displacement bursts are typically observed during the nanoindentation of metals in the sub-micron regime and are commonly referred to as either staircase-yielding [17] or “pop-ins” [4,18,19]. Fig. 1(b) is a conventional nanoindentation curve of single crystal 100 silicon, which shows a different kind of discrete behavior. While the loading of the sample is smooth, upon unloading a displacement burst is seen in the unloading direction. This discrete behavior is typically associated with the nanoindentation of semiconductors and is commonly referred to as a “pop-out” [20]. The pop-in phenomenon observed during the loading of a material at shallow depths is one of the most debated issues in the nanoindentation community. Although it seems reasonable that pop-in behavior is due to the nucleation of dislocations, only post-mortem TEM analysis has been used to correlate dislocations and load vs. displacement data. Page and coworkers [21] used TEM to investigate the post-indent microstructure around indentations where pop-ins had or had not occurred during the nanoindentation of sapphire. Post-indent TEM samples showed dislocation structures around indentations associated with discrete displacement bursts in the loading behavior, and no dislocations surrounding indents that exhibited superimposed loading and unloading (i.e. no pop-ins). Gerberich and coworkers [18] used Atomic Force Microscopy (AFM) to show that no observable surface deformation results from indentations of Fe-3wt%Si single crystals when there is no popin, and considerable surface deformation from indentations that displayed pop-in. Other studies have shown that the stress at which initial displacement bursts occur are on the same order of the critical stress needed to nucleate dislocations in previously defect-free material [3,22]. These measurements strongly support the idea that pop-ins are related to the nucleation of dislocations. However, in many studies of pop-in behavior it is argued whether or not displacement discontinuities are truly related to nucleation events. Other proposed rationales for such behavior are oxide fracture or activation of pre-existing dislocations [23]. Even though the peak stresses applied to a material during indentation occur below the surface [16], the in-plane stresses imposed at the surface are not zero. It is argued that the presence of a

§1.2

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(a)

(b) Fig. 1. (a) Load vs. displacement nanoindentation curve from a 300 nm thick Al film (on Si) and single crystal Al 100. Note the discrete displacement bursts during the loading portion of the curve. (b) Load vs. displacement nanoindentation curve from a single crystal n-type 100 Si wafer. Note the discrete displacement burst on the unloading portion of the curve.

native oxide layer can provide a defect source for the initiation of plasticity [23], or can act as a barrier to dislocations generated within the bulk. Typically, however, the presence of oxide layers is often ignored in analyses of load-displacement behavior measured during indentation.

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The pop-out phenomenon observed during nanoindentation in semiconductors has been shown to be associated with phase transformations. Numerous nanoindentation studies of semiconductors such as Si [5,20,24,25] Ge [26] and GaAs [26,27] have shown pop-out behavior followed by indirect evidence of a phase transformation at the time of the popout phenomenon. Although the evidence is indirect, the most common explanation for pop-outs is that the material undergoes a high pressure phase transformation upon loading (thus densifying), followed by a transformation back to a less-dense phase upon unloading. The transformation to a less-dense phase results in a large volume increase and therefore causes a sudden displacement burst in the direction of unloading (a pop-out). For example, it is believed that silicon transforms to high pressure metallic phases (e.g., β-Sn) during loading, as is inferred by in situ Raman spectroscopy [5] and resistivity measurements [28]. Upon unloading, the high pressure metallic phases are thought to transform to either a lower pressure metallic phase or undergo amorphization. Post mortem TEM analysis has provided evidence of both bcc-R8 and amorphous silicon phases [29], where the product phase is determined by the unloading rate and the depth of indentation. Conventional nanoindentation experiments have not been able to completely describe the discrete mechanisms of nano-scale plasticity that only nanoindentation can achieve. In a typical nanoindentation experiment, the contact areas are on the order of ∼100–1000 nm2 . When combined with applied forces on the order of milliNewtons this results in stresses on the order of GigaPascals [3,4,30]. These stresses inside the small volume of perfect crystal during a nanoindentation experiment approach the ideal strength of a material, a phenomenon not seen in any other mechanical test. Thus, nanoindentation makes it possible to directly probe the mechanical behavior of a material at its point of elastic instability. This behavior includes dislocation nucleation [3,5] and pressure induced phase transformations [5]. The ideal strength of a material is defined as the stress required to plastically deform an infinite, perfect crystal [31]. In reality, however, crystals are neither infinite nor perfect. As a result, most materials deform at stress levels well below their ideal strengths (e.g., for most metals τyield ≈ (10−2 −10−5 )τideal ). This is because a typical solid will relieve an imposed stress through mechanisms of deformation that are activated at stresses much lower than its ideal strength. If a solid is crystalline, the transformation to a stress-free state is accomplished by one of three basic mechanisms: (1) The solid deforms into a sheared replica of itself by moving dislocations, twinning, or diffusion. (2) The solid breaks into stress-free fragments. (3) The solid changes its shape by changing its crystal structure, a process that is sometimes called ‘transformation-induced plasticity’. The strength of the solid is the stress that triggers the easiest of these transformations [32]. While conventional nanoindentation tests are able to quantitatively measure the mechanical behavior of materials, the discrete deformation mechanisms that contribute to the measured behavior are rarely observed directly. Typically, the mode of deformation during a nanoindentation test is only studied ex situ, and ex post facto. To date, in situ observations of a material’s deformation behavior has only come through indirect techniques such as in situ Raman spectroscopy [5] and in situ electrical resistivity measurements. However, to truly observe the microstructural response of a material during indentation the deformation must be imaged at a resolution on the same order as the size of the defects created. A transmission electron microscope can provide sub-nanometer resolution with the abil-

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ity to image sub-surface phenomena such as the creation of dislocations and nucleation of phase transformations. The phenomenological interpretation of nanoindentation tests, and indeed the mechanical behavior of solids at their elastic limit, is a fundamental area of materials science that has lacked direct experimental observations. This chapter describes a novel method for exploring this nano-scaled regime of mechanical behavior and therefore the ideal strength behavior of solids.

2. Experimental procedure The in situ nanoindentation experiments described in this chapter were made possible through the development of a novel sample stage for a transmission electron microscope (TEM). Designed initially by Mark Wall and Uli Dahmen at the National Center for Electron Microscopy in Berkeley, CA in 1997 [33], the initial experiments were performed by Wall and Dahmen on Si with a stage designed for use in a Kratos High Voltage Microscope. The experiments described in this chapter were performed on subsequent stages designed for a Jeol 200 kV TEM and a Jeol 300 kV TEM. A picture of the indentation stage used in the Jeol 200 kV TEM is shown in Fig. 2. The stage consists of a diamond indenter attached to a metal rod that is actuated by two mechanisms. For coarse positioning, the indenter can be moved in 3 dimensions by turning screws attached to a pivot at the end of the rod. For fine positioning, including the actual indentation, the indenter is moved in 3 dimensions with a piezoelectric ceramic crystal, which expands in response to an applied voltage. All of the experiments were run under voltage-control, where the actual loading mechanism was the expansion of the piezoelectric crystal under an applied voltage. Multiple diamonds were used throughout the following experiments, but all of diamonds were 3-sided pyramid indenters with typical radii of curvature of approximately 75 nm as measured in the TEM. In order to be electrically conductive in the TEM, the diamond at the end of the rod must be doped with boron and attached with electrically conductive epoxy. A boron concentration of approximately 1020 ppm was achieved by annealing these diamonds in close proximity to boron-hydride coated wafers at 1000 ◦ C for 1 hour. The boron concentration was estimated based on thermal diffusion studies of single crystal diamond [34,35]. The diamond is mounted on the end of an Al rod that is in turn connected directly to the

Fig. 2. In situ nanoindentation stage for the Jeol 200 CX transmission electron microscope. The coarse positioning screws can be seen on the base at left, the diamond tip and sample holder are on the far right end.

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piezo-ceramic actuator, which controls its position in 3 dimensions and forces it into the edge of the sample. The in situ nanoindentation experiments posed three significant constraints on the required geometry of the samples. The first constraint, common to all TEM investigations, is that the region of the sample to be imaged must be thin enough to be electron transparent. The critical thickness is dependent on the material and the accelerating voltage of the microscope, but typically sample thicknesses of 200–300 nm were used. The second constraint, which is unique to the in situ nanoindentation experiments, is that the electron transparent part of the sample must be accessible to the diamond indenter in a direction normal to the electron beam. The third constraint is that the sample must be mechanically stable such that indentation, and not bending, results from the indenter pressing upon the thin region of the sample. Fig. 3 is a schematic showing the experimental setup in crosssection, demonstrating the constraints on the sample design. In order to fabricate samples that were electron transparent, accessible to the diamond indenter and mechanically stable, it was necessary to design unique sample geometries for the in situ nanoindentation experiments. Two different methods were used to meet these sample design constraints: (1) silicon wedge substrates were fabricated with bulk micromachining techniques and (2) focused ion beam (FIB) prepared samples were made from bulk materials. These two sample preparation techniques will be briefly described below. Using proven bulk micromachining and thin film deposition techniques, the silicon wedge substrates could be produced in large volumes (∼800 per wafer) and of repeatable quality. The final substrate design resulted in an electron-transparent area approximately 1.5 mm in length which was of a robust geometry for indentation. The microfabricated structures could then serve as a substrate for any material that can be deposited onto single crystal silicon in a thin film form. The wedge structures were achieved using silicon-based lithographic techniques including an anisotropic wet etch to give the wedges their basic geometry. KOH has been shown to be an extremely anisotropic etchant for Si [36,37], such

Fig. 3. Schematic showing the geometric requirements for an in situ nanoindentation sample. The sample must have the electron transparent portion of the sample accessible to the diamond indenter in a direction normal to the electron beam.

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that the {100} and {110} planes are etched much faster than the {111} planes resulting in structures which have walls defined by {111} planes. Further details of the silicon wedge fabrication process are described elsewhere [38]. Scanning electron micrographs of the final structures are shown in Figs 4(a) and 4(b). The final silicon wedge structure can then be used as a substrate upon which a thin film can be deposited for in situ nanoindentation. The Si plateau on the tip of the wedge can be tailored to provide the proper thickness for

Fig. 4. (a) SEM plan-view image of lithographically prepared silicon substrates. The middle bar of the structure is 1.5 mm long and the entire length is sharp enough to be electron transparent. (b) SEM image of the middle bar of the structure where it intersects with the side bar.

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Fig. 5. Scanning electron micrographs of the lithographically-prepared silicon wedge samples in cross-section. The sharp geometry, with a plateau of ∼20 nm (a), and the blunt geometry, with a plateau of ∼150 nm (b). The cross sections were prepared with a focused ion beam after a layer of Pt was deposited for protection.

electron transparency of whatever material is deposited on top. In the case of Al, a plateau 150 nm in width with a 250 nm Al film deposited on top proved to be electron transparent, while also thick enough to be indented. Fig. 5(a) is a cross-section of the sharpest wedge that was fabricated, which has a plateau of ∼20 nm. Fig. 5(b) is an example of a wedge left with a thicker plateau of ∼150 nm. A second sample preparation method was employed in order to perform in situ nanoindentation experiments on materials that could not be easily deposited on the lithographically-prepared silicon wedge substrates. This second method relied on a FEI [39] Strata 235 dual-beam Focused Ion Beam (FIB) which used a precisely-controlled 30 kV Ga ion beam to mill away material from a sample of the desired material. The FIB sample preparation method proved to expand the types of material systems which could be

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Fig. 6. Schematic of the focused ion beam (FIB) sample geometry. A trench is milled out of the sample leaving behind a window thin enough to be electron transparent. The diamond indenter can approach the thin window in a direction normal to the electron beam, as shown.

investigated through in situ nanoindentation, as described in the latter sections on TiN/MgO (001) and martensitic steel. The basic method for FIB sample preparation is to mill out a two opposing trenches on a flat piece of material which leaves behind a thin window that is electron transparent (< ∼250 nm in thickness). As is shown in Fig. 6, the electrons then travel along the trench and through the window, which is accessible by the diamond indenter in the direction normal to the surface. The most important aspect of the FIB sample preparation process is the protection of the sample from damage due to the 30 kV Ga ion beam. This protection was afforded by deposition of a sacrificial layer on the surface of the sample that could subsequently be removed prior to indentation. In typical FIB sample preparation, a thin layer of Pt, Pd, or W is deposited through decomposition of a metal-organic gas precursor inside the FIB for this purpose, but is not removed afterwards. The protective layer used for the TiN/MgO 001 samples was an evaporated Al thin film approximately 500 nm thick. The Al film was deposited at room temperature and at relatively low vacuum. Since the Al film was to be used purely as a sacrificial layer to absorb incident Ga ions during FIB sample preparation, a film of poor quality was advantageous for removal after FIB processing. Deposition in a low vacuum and at room temperature presumably resulted in a film with high impurity content and poor adhesion. After the electron-transparent window was milled out of the Al/TiN/MgO sample, the Al was removed with dilute HCl leaving behind a TiN/MgO sample with very little residual Al. In the case of the FIB prepared martensitic FeC samples, a protective layer of Pt was deposited in the FIB before milling, and was removed mechanically in situ with the nanoindenter inside the TEM prior to indentation.

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3. Results and discussion 3.1. Silicon Over fifty years of research on dislocation behavior in silicon supports the conclusion that dislocations do not generally move during conventional mechanical testing at temperatures below 450 ◦ C [40,41]. Under extreme conditions such as indentation loading, where localized stresses can approach the theoretical shear strength of the material, dislocation structures have been observed in silicon. Traditionally, these are thought to result from either block slip [42,43] or phase transformations [24,25,29,44] rather than the nucleation and propagation mechanisms associated with conventional dislocation plasticity. However, recent results have shown evidence of room temperature dislocation plasticity in silicon in the absence of phase transformations through post mortem transmission electron microscopy (TEM) of shallow indentations [45]. While indentation-induced dislocation nucleation has been widely documented for metallic systems [46–48], the high Peierls barrier associated with covalently bonded materials tends to suppress dislocation activity. Prior research suggests that the indentation of silicon causes plastic deformation through a series of phase transformations [49]. In some studies, dislocations have been observed [45,50–52] or their presence inferred [53,54], but they are usually associated with the formation of new phases. The presence of dislocations is typically observed within a newly transformed metallic phase, or described as a consequence of the phase transformation. Conclusive understanding of the dislocation formation has been importantly lacking, due to the inability to directly observe the evolution of deformation. Direct nanoscale observations of the mechanisms of deformation during the earliest stages of indentation in silicon have only recently been possible through the technique of in situ nanoindentation in a transmission electron microscope [55]. The in situ experiments described in this section were performed on 100 n-type single crystal silicon samples that were fabricated lithographically as described in Section 2. The wedge geometry allows the microfabricated silicon samples to provide for electron transparency as well as mechanical stability. Indentations were performed on two different wedge geometries, where the wedge was either terminated by a flat plateau ∼150 nm in width, or sharpened to a plateau width of approximately ∼20 nm (as shown in Fig. 5). In situ nanoindentation experiments were performed on the silicon wedge samples to peak depths ranging between 50 and 200 nm. Indentations to depths greater than 200 nm were not performed in situ due to the inherent limitations in electron transparency of the wedge geometry. During in situ indentation the deformation is observed and recorded in real time, and diffraction patterns are taken directly after unloading. It was found that plastic deformation proceeds through dislocation nucleation and propagation in the diamond cubic lattice. Fig. 7 shows a series of images taken during an indentation into the “blunt” geometry, which had a plateau of ∼150 nm at the top of the wedge. Fig. 7(a) shows the defect-free sample prior to indentation. Figs 7(b) and 7(c) show the evolution of elastic strain contours as the indenter presses into the sample – no evidence of plastic deformation is seen at this point. The elastic strain contours reveal the shape of the stress distribution in the sample (these are essentially the contours of principal stress). Figs 7(d) and 7(e)

§3.1 In situ nanoindentation in a transmission electron microscope 465

Fig. 7. Parts (a)–(f) are time series taken from a video of an in situ nanoindentation into silicon 100. The diamond indenter is in the top left corner of each frame, and the silicon sample is in the lower right. The time in seconds from the beginning of the indentation is shown in the top right corner of each frame. (a) Prior to indentation the silicon sample is defect free. In (b) and (c) the initial stage of indentation shows elastic strain contours resulting from the pressure applied by the indenter. In (d) and (e) the dislocations can be seen to nucleate, propagate and interact as the indentation proceeds. (f) After a peak depth of 54 nm the indenter is withdrawn and the residual deformed region consists of dislocations and strain contours that are frozen in the sample.

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Fig. 7. (Continued.) (g) A [011] zone axis electron diffraction pattern of the indented region directly after the in situ indentation. With the exception of slight peak broadening, the diffraction pattern is identical to similar patterns taken prior to indentation, showing only single-crystal diamond cubic silicon with no additional phases. The tails seen on the diffraction spots pointing in the 200 direction are a geometrical effect due to the wedge geometry, which exhibits a drastic change in thickness ¯ dark field condition. Note the continuous surface across the indented region, indicating that over a very short distance. Fig. 7(h) is the same indentation in a g = (022) the indentation left at least one side of the wedge intact.

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clearly show the nucleation and propagation of dislocations from the surface as deformation proceeds. This particular indentation was taken to a peak depth of 54 nm, resulting in the plastic zone shown in Fig. 7(f). The post-indentation selected area diffraction pattern of the indented region is shown in Fig. 7(g). Due to the high density of dislocations after indentation a slight broadening of the diffraction spots is observed. The broadening of diffraction spots is an expected consequence of a high density of dislocations created on multiple slip planes [56]. However, no additional diffraction spots or rings are present after indentation as compared to diffraction patterns taken prior to indentation. This indicates that no additional phases (crystalline or amorphous) have formed. Fig. 7(h) is a dark-field ¯ diffracted beam, showing that at least one edge of the plateau is condition using the (022) still continuous across the indented region. In the case of deeper indentations, significant metal-like extrusions were also formed during indention. These large extrusions are shown in Fig. 8(a), a post-indentation TEM micrograph of a 220 nm deep indentation with the corresponding diffraction pattern. This metal-like deformation is clearly seen in Fig. 8(b), which is a plan view micrograph of the indent shown in 8(a), taken with a field emission scanning electron microscope (FESEM). Previous studies [50,51,57] have also described metal-like extrusions resulting from indentation into Si. However, in all previous experimental cases these extrusions were attributed to the flow of a transformed metallic phase (such as β-Sn or bcc-R8). As shown in Fig. 8(a), the diffraction pattern taken after the indentation indicates that the extruded volume is entirely single-crystal diamond cubic silicon in the same orientation as the rest of the sample. In the case of the “sharp” wedge geometry the deformation behavior during indentation was identical to the case described above, showing dislocation plasticity during loading. Upon unloading, however, fracture occurred in several of the indentations. Fig. 10 shows post-indentation TEM micrographs of one such indentation into the sharp wedge geometry. As can be seen in Fig. 10(a), the fracture surface is smooth and non-crystallographic (within the resolution limit of the bright field TEM image). During unloading two cracks were observed to nucleate at the surface and propagate along a strain contour to the center, forming the semicircular fracture profile shown in Fig. 9(a). Fig. 9(b) shows the dramatic contrast between the crystallographic nature of the dislocations nucleated during indentation and the non-crystallographic nature of the fracture surface. It has been shown that stresses near the ideal shear strength are reached during indentation into defect-free metal systems, resulting in the nucleation of dislocations [4,21]. However, the critical shear stress for dislocation nucleation in metallic materials is far less than that for silicon, owing to the high Peierls–Nabarro barrier of the latter (which is directly related to the strong covalent bonding in silicon). During conventional indentation experiments in silicon, it is generally believed that deformation is accommodated by phase transformation and/or fracture, rather than dislocation plasticity. The phase transformations that occur during indentation have also been observed during diamond anvil cell experiments, where high hydrostatic pressures are applied [58–61]. It follows that during conventional indentation experiments in silicon, the critical pressure for phase transformation is reached before the shear stress reaches the critical value for dislocation nucleation. Presumably, this difference arises from the difference in sample geometry used in conven-

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Fig. 8. (a) Bright field TEM micrograph of an in situ indentation into 100 silicon that was taken to a depth of 220 nm. The image was taken in a kinematic condition in order to show the undeformed region surrounding the indented volume, which was heavily deformed through dislocation plasticity. The [110] zone axis electron diffraction pattern inset was taken after indentation and shows only the presence of the diamond cubic phase of silicon. This diffraction pattern is identical to diffraction patterns taken before indentation, except for a slight broadening of the diffraction spots due to the heavily dislocated region of the indentation. (b) A plan-view scanning electron micrograph of the same indentation. Relatively large, metal-like extrusions can be seen surrounding the indentation. These extrusions can also be seen above the indented volume in Fig. (a), and as shown by the inset diffraction pattern are extrusions of single-crystal diamond cubic silicon.

tional instrumented nanoindentation tests and the in situ nanoindentation tests described here. Conventional indentation experiments are conducted on samples well described as a flat half-space (e.g., a silicon wafer). In the case of the silicon wedge sample geometry, the sides of the wedge are of 111 orientation, making an angle of 70.5◦ to each other, and are separated by a flat plateau of 100 orientation. The plateau width, w, the wedge height, h, and the length of the wedge, L, have the relationship: L ≫ h ≫ w, such that the geometry is better described as a plane stress configuration than the plane strain configuration characterizing conventional indentation experiments. A plane stress configuration is defined by a lower hydrostatic component than a plane strain configuration. In the case of the in situ

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Fig. 9. (a) Bright field TEM micrograph after an indentation of the sharp wedge geometry. The fracture surface is the semi-circular border between the sample (on the left) and the vacuum (on the right). The image was taken in the [011] zone axis condition with the corresponding diffraction pattern inset. The stripes along the length of the sample are thickness contours resulting from the wedge-shaped geometry of the specimen. (b) A higher magnification of the region showing residual dislocations adjacent to the fracture surface. The smooth fracture surface indicates non-crystallographic fracture. The image was taken in the g = [311] condition.

experiments, an increase in the ratio of maximum shear stress to hydrostatic pressure could allow the shear stress to reach the critical value for dislocation nucleation before the hydrostatic stress reaches the critical value for phase transformation. Additionally, the presence of the surfaces near the contact area (and thus the plastic zone) in the in situ wedge samples would presumably make dislocation nucleation easier than in the half space geometry. Since the surfaces would provide a easy nucleation source, it is conceivable that the induced deformation can be completely accommodated by dislocation plasticity and the hydrosta-

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Fig. 10. Cross-section of an in situ nanoindentation sample. An Al film ∼250 nm thick is deposited onto a wedge-shaped perturbation on a Si substrate.

tic stresses required to nucleate phase transformations are thus never realized. However, this effect requires further study. Most importantly, efforts to control the loading and unloading rate should be undertaken, since loading rate has been shown to make a significant difference in the deformation pathway in silicon. Fracture in silicon is widely observed in indentation experiments and in most mechanical tests. In the case of the in situ experiments, fracture events often appear to be a direct result of the presence of dislocations. In particular, a plastic zone characterized by indentationinduced dislocations can lead to a residual stress field upon unloading, resulting in fracture. As described by Lawn [42], a residual tensile field arises during unloading due to the accommodation of the plastically deformed volume and the surrounding elastic volume. This field is comprised of high tensile stresses that drive fracture. Surprisingly, the path of crack propagation is non-crystallographic. Rather than cleaving along the lowest energy planes, the crack follows a strain contour associated with the residual stress field. This is energetically favorable if the stress along the strain contour is on the order of the theoretical cohesive strength of silicon.

3.2. Al thin films The mechanical behavior of Al thin films has important relevance to the microelectronics industry, where Al metallization is used throughout a typical microchip. Since the grain size of a metallic thin film typically scales with the thickness of the film, a variety of grain sizes can be studied by controlling the thickness of the film. In this section, the deformation behavior of sub-micron Al films is discussed. In the case of the polycrystalline Al films described in this section, films were deposited on the silicon wedge substrates by evaporating 99.99% pure Al at 300 ◦ C. A film thickness

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of approximately 250 nm gave the maximum thickness while still maintaining electron transparency at 200 kV. It is preferable to use the maximum possible thickness that is still electron transparent, since a thicker sample is less likely to bend during indentation. During in situ nanoindentation of the Al films, the 3-sided diamond pyramid indenter approaches the sample in a direction normal to the electron beam (Fig. 10). The indentation is made in the cap of film on the flat top of the wedge, which can be seen in Fig. 10. The interpretation of conventional nanoindentation data is not always clear. For example, since most metals form native oxides yielding under the nanoindenter may be governed by fracture of the oxide film rather than the onset of plastic deformation in the material itself [23]. It is difficult to resolve these mechanisms ex situ since the mechanisms associated with yielding are only indirectly elucidated from quantitative load vs. displacement behavior. Thus, the most significant advantage from performing in situ nanoindentation inside a transmission electron microscope (TEM) is the ability to record the deformation mechanisms in real time, avoiding the possibility of artifacts from post-indentation sample preparation. However, since the Peierls barrier in Al is extremely low and consequently the dislocation velocity is very fast, our video sampling rate of 30 frames per second is too slow to capture the movement of the individual dislocations. Hence, each video image captured during the in situ experiments presented here is essentially a quasi-static image of the equilibrium configuration of defects. Fig. 11 illustrates this point by showing a series of six images taken from a video during an in situ nanoindentation experiment. In Fig. 11(a), the diamond is approaching an Al grain that is approximately 400 nm in diameter. Figs 11(b) and 11(c) show images of the evolution of the induced stress contours during the initial stage of indentation, and correspond to purely elastic deformation in the absence of any pre-existing dislocations that could cause plasticity. Fig. 11(d) shows the first indication of plastic deformation, in which dislocations are nucleated; a set of prismatic loops is observed. Figs 11(c) and 11(d) are consecutive frames of the video, and are (1/30th) of a second apart. As can be seen, the exact location of the nucleation event is not discernable, since the evolution of the dislocation configuration has already proceeded beyond the point at which that might be possible. Figs 11(e) and 11(f) show the large increase in dislocation density achieved as deformation proceeds, and dislocations tangle and multiply. In order to study the crystallographic mechanisms of nanoindentation in detail it is more useful to use dark-field TEM. In a dark field condition the image is formed by using a strongly diffracted electron beam from one grain only. Figs 12 and 13 show examples of the information that can be obtained from dark-field TEM images. Fig. 12 includes a series of micrographs from the video record of an in situ nanoindentation of an Al grain that was oriented in approximately the [–1–11] direction. Figs 12(a) and 12(b) are micrographs taken prior to indentation in bright field, and dark field respectively. Figs 12(c) and 12(d) show the evolution of nanoindentation damage. As expected, dislocations glide along close-packed {111} planes, and multiply and tangle as the deformation proceeds. Fig. 13 is a more detailed examination of the development of dislocations near the yield point of the sample shown in Fig. 12. The top picture shows the entire Al grain at +0.6 seconds, followed by a series of pictures taken 0.1 seconds apart and cropped from the region outlined by the white box in the top picture. The most striking feature is the sequential appearance of prismatic dislocation loops on an axis approximately 45◦ from the direction of indentation, a mechanism of nanoindentation-induced deformation that has been

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Fig. 11. Time series of an Al grain showing the evolution of plastic deformation during an in situ nanoindentation in the [111] direction. The time elapsed from image (a) is given in seconds in the upper right corner of each frame. Images (b) and (c) correspond to elastic deformation only. In image (d) the nucleation of dislocations can be seen. Images (e) and (f) are characteristics of the resulting plastic deformation during deeper penetration and the pile-up of the dislocations at the grain boundaries and the substrate/film interface.

suggested by several researchers [5,62,63]. Typically, in order to determine the Burgers vector of a dislocation in a TEM it is necessary to tilt the sample to multiple diffracting conditions in multiple zone axes. In the case of in situ experiments, it is only possible to image an evolving defect in one diffracting condition at a time, and thus not usually possible to determine the exact character of a defect during an in situ experiment. However, a study by Bell and Thomas [64] in 1965 showed that the character of prismatic loops in Al could be determined through geometrical analysis alone without necessitating g.b = 0

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Fig. 12. Time series of an Al grain showing the evolution of plastic deformation during an in situ nanoindentation: Bright field (a) and dark field (b) image before indentation. The dark field condition used was [111]. The stripes are thickness contours due to the wedge shape of the specimen. (c) Dark field micrograph at t = 1.4 seconds showing dislocations on {111} planes. (d) Dark field micrograph at t = 2.8 seconds showing multiple dislocations.

tilting experiments. Bell and Thomas discovered that in fcc crystals such as Al, perfect prismatic dislocation loops lying on {111} planes projected not a loop, but a double arc image onto the surface of the foil in diffraction contrast. This double arc image was shown to correspond to the Burgers vector such that a line connecting the missing segments of the loop image lies along a 110 direction parallel to the Burgers vector of the perfect loop. As can be seen in Fig. 13, the dislocation image is in fact a double arc where the line of no-contrast can be seen to lie in a 110 direction. Thus, even without being able to tilt the sample to multiple diffraction conditions in situ, it can be determined that the dislocation image in Fig. 13 corresponds to a prismatic dislocation loop with a a/2110 Burgers vector. Since the loops are prismatic rather than shear in nature, they are presumably formed at the surface rather than nucleated in the bulk. However, the actual nucleation and migration of the loops to the site at which they appear is too rapid to capture, as is their motion to new equilibrium positions as additional prismatic loops are punched into the grain. The peak stresses (shear and hydrostatic) applied to a material during indentation occur below the surface 16. Thus, elastic contact theory suggests that nucleation should occur from the bulk, below the indenter/sample interface. However, the in-plane stresses imposed

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Fig. 13. Series of dark field transmission electron micrographs showing the appearance and evolution of prismatic dislocation loops in a previously undeformed part of the Al grain during an in situ nanoindentation. The numbers in the micrographs are the time in seconds from the start of the indentation.

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at the surface are not zero. During a nanoindentation test into an initially defect-free solid volume the initiation of the defects necessary to accommodate the induced deformation can occur at a number of locations, including the surface, the interface of the native oxide layer with the bulk material, or the interior of the grain [23]. This competition for the location of the initial defect nucleation will depend on numerous material and indentation parameters, including the surface properties of the material and the resulting defects. Although the precise nucleation site may be difficult, if not impossible, to establish from such experiments, in situ indentation experiments provide unique advantages over ex situ TEM analysis of post-indent dislocation configurations. In addition to the difficulty in preparing TEM samples after indentations have been conducted, the strong image forces exerted on the dislocations by the free surface can lead to very different structures after the sample has been unloaded. Indeed, in our studies it has been observed that the configuration of defects changes dramatically after the dislocations have had time to rearrange themselves and, possibly, anneal out of the sample. During the in situ experiments on the Al thin films, it is observed that the configuration of dislocations that result during loading is usually lost within minutes of unloading the sample. Specifically, the image effects of the sample surface almost always lead to a dramatic decrease in the number of dislocations left in the sample after the indentation. As compared with instrumented nanoindentation of a typical thin film (which is essentially infinite in two dimensions), our in situ sample design substantially increases the amount of surface interacting with a resulting defect configuration. Since the elastic fields of a dislocation can extend throughout the thickness of a thin film, it would be expected that the surface of a typical thin film under instrumented nanoindentation would also exhibit the ability to absorb dislocations. This observation suggests that post mortem TEM studies of indentation damage and microstructures taken after nanoindentation experiments should be viewed with some caution. In order to demonstrate the effect of the surface on the resulting dislocation configuration, Fig. 14 shows three post-indent images of the same experiment documented in Fig. 11. In this case, Fig. 14(a) is a bright field image taken approximately 2–3 minutes after the indentation. Fig. 14(b) is a dark field image using the same exact diffraction condition, and taken approximately 2–3 minutes after the bright field image (Fig. 14(a)). Fig. 14(c) is the same two images overlaid on top of each other, in which we can directly compare the dislocation configuration shown in the bright field condition (where the dislocations look black) and that shown in the dark field condition (where the dislocations look white). A bright field and a dark field image are not necessarily exact negatives of each other. However, in an isotropic material such as Al any defect seen in a dark field configuration would certainly have to show up in a bright field image using the same diffraction condition. In this case, it can be seen that the dislocation configurations do not match, and that the dark field condition shows dislocations that are not seen in the bright field condition (for example, the dislocation at the end of the [–1–11] arrow). Thus, what was inadvertently documented with the two images 14(a) and 14(b) is the rearrangement of the post-indent dislocation configuration that must have occurred sometime during the 2–3 minutes between the moment that the bright field and dark field images were captured. In polycrystalline metals with relatively large grain sizes (e.g., >1 micrometer), grain boundaries are thought to behave primarily as barriers to dislocation motion. Consequently,

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Fig. 14. Post-indent images from the experiment shown in Fig. 11 using the [100] diffraction condition. (a) Bright field image taken approximately 2– 3 minutes after the indentation. (b) Dark field image taken approximately 4–6 minutes after the indentation. (c) The DF image overlaid on top of the BF image, demonstrating the rearrangement of the dislocations which occurred sometime between the time the BF and DF images were taken.

a typical method for increasing the hardness of a metal is to change the composition or processing of the material in order to decrease the average grain size, thus increasing the total area of grain boundaries and increasing the barriers to dislocation motion. This basic

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premise of microstructure-property relations in metallurgy is known as Hall–Petch behavior, since the first studies to relate grain size with the strength of a material were performed by Hall [65] and Petch [66] in the early 1950s. There are other known mechanisms by which grain boundaries can influence the mechanical behavior of a polycrystalline metal. Li [67] described an alternate role for grain boundaries in 1963, when he first proposed that grain boundaries be thought of as sources for dislocations rather than only barriers to their motion. Indeed, recent reports using computational studies of nanoindentation have shown that relatively easier dislocation nucleation at grain boundaries can serve to lower the load at which plasticity is initiated [68]. As grain size decreases, the volume of the material associated with grain boundaries increases dramatically. For instance, the amount of material associated with a grain boundary changes from 0.01% for 3 micron grains, to 0.1% for 300 nm grains and to 1% for 30 nm grains [69]. For a given indenter size, the interaction of the grain boundaries increases as the grain size decreases. Thus, it should not be surprising that the deformation behavior associated with increased grain boundary interaction might involve mechanistic changes. These mechanistic changes might include grain boundary sliding [70,71], dislocation nucleation from the grain boundary [68], or even grain boundary movement. This final mechanism, grain boundary movement, has been observed in more macroscopic experiments, but is not typically mentioned as relevant behavior for small scale deformation. Winning, et al. [72] described the motion of Al tilt boundaries under imposed external stresses, and suggested the movement of the grain boundaries was achieved through the movement of dislocations that comprised the structure of the boundaries. Merkle and Thompson [73] ascribed the motion of grain boundaries in Au to a more localized phenomenon- the rearrangement of groups of atoms near a grain boundary to be incorporated into a growing grain. Whether the grain boundary motion is accomplished through dislocation motion or atomic rearrangement, there exists a driving force for a grain to grow or shrink under an inhomogeneous external stress. The stresses imposed by a nanoindenter are inhomogeneous [16], and can be expected to provide a significant driving force for the movement of grain boundaries. In fact, the movement of grain boundaries during in situ nanoindentation was observed to be considerable in Al grains under ∼400 nm in width, and these initial observations will be described below. When the contact area of the indenter is small compared to the grain size, the grain boundaries in the Al films were seen to act as barriers for dislocation motion. Fig. 11 shows one such case, and other examples have been discussed previously [74]. Fig. 15 shows a different situation, where the size of the indentation contact area is very large compared to the size of the grain, and in fact the indenter imposes directly onto the grain boundary during indentation. Fig. 15(a) is a TEM image of two Al grains taken before indentation. Fig. 15(b) is a TEM image of the same two grains after indentation, where the deformation of the grain that had been within the contact region of the indenter is extensive. Fig. 15(c) is the same two images, 15(a) and (b) overlaid on each other, showing that the grain boundary had moved extensively during the indentation. Undoubtedly, the change in equilibrium position of the grain boundary with respect to the two grains shows that the grain boundary must have participated in the deformation of the film in a substantial

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Fig. 15. (a): TEM image of two Al grains taken before indentation. (b): TEM image of the same two grains after indentation in the same diffraction condition. (c): The same two images, 15 (a) and (b), overlaid on top of each other, showing that the grain boundary had moved extensively during the indentation.

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Fig. 16. (a) The Al tilt boundary prior to indentation with the indenter approaching from the bottom of the image. Images (b), (c) and (d) are three images only 0.6 seconds apart, taken 18.2 seconds from the start of the indentation. The grain boundary can be seen to change positions dramatically even over this short time frame (1.8 seconds for the three images). Dislocations can also be seen extending from/to the grain boundaries in each grain. Image (e) shows is the final position of the grain boundary after the indenter has been removed, which is similar in location to the grain boundary’s original position. (f) A diffraction pattern of both grains, showing two strongly-diffracting planes, one from each grain.

way. Fig. 16 shows a time series taken from an indentation directly onto a ∼6◦ tilt boundary between two Al grains. Here, the surprisingly large in situ movement of the grain boundary during the indentation can be seen directly.

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Fig. 16(a) shows the indenter approaching the grain boundary from the bottom of the frame. Figs 16(b)–(d) are three images only 0.6 seconds apart, taken 18.2 seconds from the start of the indentation. The grain boundary can be seen to change positions dramatically even over this short time frame (1.8 seconds for the three images). Dislocations can also be seen extending from/to the grain boundaries in each grain. Fig. 16(e) is the final position of the grain boundary after the indenter has been removed, which is similar in location to the grain boundary’s original position. This is more evidence that ex situ characterization of nanoindentation experiments would miss important microstructural phenomena. Although the before (Fig. 16(a)) and after (Fig. 16(e)) positions of the tilt boundary are very similar, the boundary had actually moved considerably during the indentation. Fig. 16(f) is a diffraction pattern of both grains, showing two strongly diffracting planes, one from each grain. A more extreme example of grain boundary movement is shown in Figs 17 and 18. Fig. 17 includes a series of four images extracted from the videotape record of an in situ nanoindentation of a small grain approximately 150 nm in width that had formed at the cusp of a grain boundary between three larger grains. In Fig. 17(a), the indenter is approaching the small grain from the upper left corner of the video frame. Starting at Fig. 17(b) and ending at 17(d), the small grain can be observed to shrink dramatically over a very short time (0.2 seconds elapsed between these three frames). By comparing the pictures taken before and after indentations (Fig. 18), one can see that the two larger grains

Fig. 17. A series of images extracted from a video-tape record showing the stress-induced grain growth during an in situ nanoindentation on submicron-grained aluminum: (a) before indentation the indenter approaches from the upper left-corner, (b) the indenter makes contact with a small grain in the cusp of 3 larger grains. (c) The small grain in the cusp starts to shrink and is left as a small film (d) between the neighboring grains.

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laterally surrounding the small grain in the cusp grew at the expense of the shrinking grain. Fig. 18(a) and 18(c) are bright-field images before and after the indentation, respectively, showing the change in grain size of all of the grains. Fig. 18(b) and 18(d) are dark field images of the small grain in the cusp taken before and after the indentation, respectively, clearly showing the shrinkage of the small grain after the indentation. It can be noted by the consistent contrast in the small grain before and after indentation that the small grain did not simply change its shape, and that the volume of this grain was not conserved. If the volume of this grain had been conserved, then the thickness of the grain in the direction of the electron beam would have increased, resulting in a significant change in contrast as compared to the surrounding grains, which was not observed. This deformation mode can be effectively suppressed by the addition of a solute to the Al films. In situ nanoindentation experiments reported previously showed that the addition of as little as 1.8 wt% Mg to the same Al films pinned the high angle grain boundaries [75]. Nevertheless, the observation of dramatic grain boundary movement in the pure Al films suggest that grain boundary motion is a significant mechanism of deformation during nanoindentation, and in fact might play a large role in the softening of materials with submeter grain sizes. For example, similar grain boundary motion was found in nanocrystalline Al films with a grain size on the order of 20 nm [76]. Since, given Le Chatelier’s principle, all spontaneous processes that happen under load contribute to the relaxation

Fig. 18. Still images from the indentation shown in the previous figure. (a) Bright-field image before the indentation. (b) Dark-field image of the middle small grain before the indentation. (c) Bright-field image after the indentation. (d) Dark-field image of the middle small grain after the indentation.

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of the load, it is very likely that coalescence plays a significant role in the extraordinary mechanical behavior of nanocrystalline materials.

3.3. Martensitic steel Fe-C based martensitic steels are certainly among the most important high-strength materials. The common “dislocated lath martensite” has a high dislocation density in the asquenched condition. The microstructure has a characteristic form that includes at least three distinct kinds of grain boundaries [77,78]. (1) Prior austenite grain boundaries define the boundaries of the high-temperature phase that was present before the martensitic transformation. They remain because the martensitic reaction is a shear transformation that individually transforms each prior austenite grain. They are, ordinarily, high-angle boundaries that separate grains with no particular crystallographic relation. (2) The prior austenite grains contain sets of thin, parallel laths that are separated into “blocks” or packets. The laths within a block are almost identical in orientation and are separated by low-angle boundaries. (3) The block, or packet boundaries separate sets of laths that are, ordinarily, related to the parent austenite by different “Bain” variants of the Kurdumov–Sachs orientation relation that usually governs the martensitic transformation in steel. These are high-angle boundaries, but the grains that join at the interface are different K–S variants and are, hence, constrained to have one of the specific set of crystallographic orientations. In situ straining TEM studies of dislocation activity at grain boundaries in steel have observed individual dislocations to pile up or pass through boundaries [79,80]. These studies show interactions between pre-existing lattice dislocations and grain boundaries, and hence relate only to the initial stages of plastic deformation. The flow stress after yielding is strongly affected by dislocation interactions under conditions of high dislocation density, and deformation behavior in this regime must also be understood. Ohmura et al. [81,82] have shown through conventional nanoindentation techniques that grain boundary strengthening is a significant factor in martensitic steels, the relative roles of the three kinds of boundaries remain unclear. Morris and co-workers [83,84] have suggested on theoretical grounds that the prior austenite grain boundaries should dominate, while the lath and packet boundaries should be relatively unimportant. The low-angle lath boundaries are not expected to pose more than nominal barriers to dislocation motion while the high-angle block boundaries allow continuity of five of the six {110} glide planes of the bcc martensite with only small-angle deflections. Through in situ nanoindentation in TEM it was possible to study the interactions between the dense dislocation distribution in Fe-C lath martensitic steel with the different internal boundaries [85]. The sample used was a high-purity Fe-C binary martensite with a carbon content of 0.4 wt%. The specimen was austenitized at 1323 K for 900 s, ice-brine quenched and tempered at 723 K for 5400 s. Following heat treatment the specimen surface was mechanically polished and then electropolished. As described in Section 2, a FIB was

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used to prepare suitable samples for in situ nanoindentation. The electropolished surface was coated with two Pt films before FIB milling in order to avoid damage to the surface from the incident Ga+ ions. The Pt coating was first deposited with an electron beam, and the covered by ion beam-deposited Pt. The Pt film was removed mechanically prior to indentation by breaking it off in situ with the nanoindenter inside the TEM. The nanoindenter tip was maneuvered in the microscope to crack the brittle Pt films, and the Pt was then peeled off of the surface with the indenter to reveal a clean electropolished surface. Indentations were made into the surface such that a specific low angle or high angle boundary would be visible beneath the indenter, and the deformation at that boundary was followed in real time. Fig. 19 includes TEM micrographs of the Fe-C martensite before nanoindentation. Two suitably oriented boundaries were found and indented. The two boundaries are highlighted with arrowheads in Fig. 19. The first boundary chosen was the low-angle boundary shown in Fig. 19(a). The selected-area diffraction pattern (SADP) included in the figure shows that the grains on either side of the boundary have the same [111] zone axis, indicating that the grain boundary is low-angle. Both its low-angle character and its visual appearance strongly suggest that this boundary is a lath boundary [86,87]. The second boundary chosen was the high-angle boundary shown in Fig. 19(b). In this case, the SADP from the bounding grain nearest the surface shows a [100] zone axis pattern while that from the adjacent grain has a [111] zone axis pattern. Detailed calculations of the misorientation angle strongly suggest that the highlighted boundary is a high-angle block boundary [85]. Fig. 20 includes a series of video frames from the in situ TEM nanoindentation of the low-angle boundary. Fig. 20(a) shows the diamond indenter prior to indentation, as it approached the surface from the upper-right corner of the figure. The grain boundary is indicated by arrowheads. As the indenter penetrated into the grain, the deformation was accommodated by the motion of dislocations away from the indenter contact point toward the grain boundary. The low-angle boundary offered some resistance to the dislocation motion, resulting in a pile-up at the grain boundary that is shown in Fig. 20(b) (a frame taken at an indenter penetration depth of 46 nm). As the indenter penetrated further, a large number of dislocations were emitted on the far side of the grain boundary into the adjacent grain. A dense and tangled dislocation structure is seen on the far side of the grain boundary in Fig. 20(c), which was taken at an 84 nm penetration depth. When the indenter was withdrawn a high density of dislocations was retained near the far side of the boundary. The dislocation interactions with the high-angle grain boundary (block boundary) were dramatically different, as documented in Fig. 21. Fig. 21(a) shows the initial state before indentation, with the high-angle grain boundary indicated by arrows. During the early stages of the indentation dislocations can be seen to sweep across the grain from right to left. At approximately 60 nm penetration depth, a dense cloud of dislocations reached the highangle boundary, as shown in Fig. 21(b). On reaching the boundary the dislocations simply disappeared; there is no indication of a pile-up or significant penetration into the adjacent grain. In fact, as illustrated in Fig. 21(c), there is virtually no change in the dislocation configuration on the far side of the boundary as compared to Fig. 21(b). The grain beyond the boundary is essentially unaffected. The stable contrast in these images show that the diffraction condition in the vicinity of the grain boundary was the same throughout the deformation sequence shown- indicating

484 A.M. Minor et al. Fig. 19. TEM micrographs of the Fe-C martensite before nanoindentation including (a) low-angle grain boundary and (b) high-angle grain boundary. The grain boundaries are highlighted by arrows in both pictures. In (a) only one SADP is shown since the orientation on both sides of the grain boundary is nearly identical, while in (b) the orientations on either side of the boundary are different, as shown by the two inset SADPs. Ch. 78

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Fig. 20. In situ TEM micrographs of the Fe-C martensite, including a low-angle grain boundary: (a) before indentation; (b) at 46 nm penetration depth, showing dislocation pile-up at the grain boundary; (c) at 84 nm penetration depth, demonstrating dislocation emission on the far side of the grain boundary.

that the behavior observed was not a result of bending in the foil. This sequence demonstrates that the dislocations approaching the boundary were almost completely absorbed by it. In the case of a low-angle grain boundary, the dislocations induced by the indenter piled up against the boundary. As the indenter penetrated further, a critical stress appears to have been reached and a high density of dislocations was suddenly emitted on the far

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Fig. 21. In situ TEM micrographs of the Fe-C martensite, including a high-angle grain boundary: (a) before indentation; (b) at 83 nm penetration depth, showing dislocations sweeping across the grain and disappearing at the grain boundary (dislocations indicated by the arrow); (c) at 92 nm penetration depth, demonstrating no dislocation emission at the grain boundary.

side of the grain boundary into the adjacent grain. Since slip planes are essentially continuous across the lath boundary, the resistance to dislocation glide is presumably due to dislocation-dislocation interactions. In the case of the high-angle grain boundary, the numerous dislocations that were produced by the indentation were simply absorbed into the boundary, with no indication of pile-up or the transmission of strain. Presumably, the plas-

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tic strain was accommodated by deformation perpendicular to the foil. One explanation for this behavior is that the interface between the two K–S variants across the boundary is a {110}α slip plane that is common to both variants, with the consequence that slip in the boundary plane (grain boundary sliding) is relatively easy. Details of this theory can be found in the work of Ohmura et al. [85].

3.4. Hard thin films on softer substrates As described in Section 3.2, when a thin film is considerably softer and more compliant than its substrate, plastic deformation is confined to the film only. In thin film/substrate systems where the substrate is either softer or more compliant, however, the mechanical properties of the substrate will influence the indentation behavior at even shallow indentation depths. In the analysis of nanoindentation data, an elastically inhomogeneous thin film/substrate system gives rise to a so-called “substrate-effect”, where the mechanical properties of the substrate influence the mechanical properties of the film extracted from a nanoindentation experiment [88]. At significant indentation depths, the substrate effect is not only caused by differences in the elastic properties between the film and substrate but can be influenced by plastic deformation in the substrate as well. With in situ nanoindentation, the onset of plastic deformation in both a thin film and its substrate can be observed directly. TiN thin films deposited onto MgO substrates are one such system, where the Young’s modulus and hardness are nominally 480 GPa and 20 GPa for TiN, and 310 GPa and 4.2 GPa, respectively, for MgO [89]. The transition metal carbonitrides are a technologically important class of materials whose mechanical properties are still poorly understood [90]. As an example, it has been shown that within a given class of materials hardness values typically scale with the shear modulus [89]. Within the transition metal carbonitrides, however, their exist some striking discrepancies to this trend. In particular, while TiC and TiN have very similar shear moduli, the hardness of TiC is approximately 1.5 times the value of TiN. The underlying explanation for this relationship is not well understood. It is believed that the slip of dislocations does not occur at room temperature in these materials, but their precise deformation mechanisms are not well known [91]. The deformation mechanisms of the transition-metal carbonitrides are difficult to study, in part, due to their characteristically high melting temperatures [92]. TiN, for instance, has a melting temperature of 3000 ◦ C. The growth of TiN in thin film form is thus always at a temperature of less than half the homologous temperature, since high vacuum growth chambers cannot be operated at temperatures greater than ∼1000 ◦ C [93]. Because of their necessarily low homologous growth temperatures, transition-metal carbonitride thin films are typically grown with a relatively high amount of intrinsic defects and very small grain sizes [94]. High defect densities and small grain sizes make the imaging of these materials difficult when using diffraction contrast in a TEM. Because of the high defect densities in as-deposited films, any deformation-induced microstructural changes are easily obscured by the intrinsic defects. However, it is possible to grow heteroepitaxial thin films of TiN on relatively large pieces of single crystal MgO. Since the epitaxial films are free of grain

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boundaries, they are more ideal for studying the fundamental deformation mechanisms in these materials. In order to study the fundamental deformation mechanisms in TiN using in situ nanoindentation, TiN was deposited epitaxially on single crystal MgO with (100) orientation [95]. The ∼350 nm thick films were deposited by ultrahigh vacuum reactive magnetron sputtering on MgO (001) at 850 ◦ C in pure N 2 discharges maintained at a pressure of 5 mTorr (∼0.67 Pa). The details of the film deposition have been reported elsewhere [96–99]. In order to prepare the samples for the in situ nanoindentation experiments, a FIB was used to mill the samples to the proper geometry. This method, and the manner in which the sample was protected from the incident ion beam was discussed in Section 2. Fig. 22 shows the epitaxial TiN/MgO (100) sample prior to indentation. As can be seen by both the bright field (Fig. 22(a)) and dark field (Fig. 22(b)) micrographs, there exists a very high density of defects in the TiN pre-indentation microstructure. It is not known whether these defects were created during the growth of the film or ion beam milling. Both processes are known to create defects. The exact character of these defects is also not known, since their size is too small to be individually resolved with the TEM (at the sample thickness required for an in situ nanoindentation). Damage layers have been reported in numerous materials as a result of ion beam thinning [100]. The implantation of the energetic Ga+ ions (typically 30 keV) is known to cause amorphization in a layer 30 nm thick in Si [100]. No studies have been found on Ga+ ion damage to TiN specifically. During the indentation of the TiN/MgO, plastic flow in the MgO was extensive, and the TiN deformed by more limited dislocation motion. Figs 23(a) and 23(b) show bright field and dark field micrographs of the post-indentation microstructure from an indentation taken to a peak depth of approximately 100 nm. Dislocations have arranged themselves in what closely resembles a hemispherical plastic zone surrounding the point of indentation in the TiN. The post-indentation surface of the TiN at the contact interface was so heavily damaged that it simply appears black in both bright field and any attainable dark field conditions. This indicates that the black region at the point of indentation in TiN has such a high defect density that the electrons entering this region are almost completely absorbed or scattered. During the in situ indentation a large amount of dislocations were generated in the MgO substrate at the film/substrate interface. As can be seen in Fig. 24, indentation-induced dislocations propagated approximately twice as far in the MgO substrate as in the TiN film. During this relatively large indentation (the indentation depth was approximately 1/3 of the film thickness), the TiN film did not de-adhere from the MgO substrate. Rather, the TiN film was forced to accommodate the plastic deformation of the MgO substrate by bending. The extensive nature of the plastic deformation in the MgO led to a bending of the TiN film, as observed in Fig. 24. A hemispherical distribution of dislocations led to a 8◦ rotation of the crystal around the 001 axis underneath the indenter. The crystal was bent 4◦ on each side of the indentation, as evidenced by comparing pre-indent and post-indent diffraction patterns. Fig. 25 shows the post-indent hemispherical plastic zone in the TiN along with selected area diffraction patterns of the TiN film taken before and after the indentation. The circle drawn on the TEM micrograph in Fig. 25(a) shows the region of the TiN from which the diffraction information was taken. Fig. 25(c) shows the splitting of the (–220) diffraction

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Fig. 22. Bright field (a) and dark field (b) TEM mirographs of the epitaxial TiN/MgO (100) sample prior to indentation. Both images are taken in the g = [220] condition. The arrows showing the [100] direction in both images also point to the location of the indentation shown in Fig. 23.

spot into approximately two spots, with 8◦ of rotation between them (around the 001 zone axis). These two spots now represent each side of the TiN underneath the indenter, where the TiN film has formed what is effectively a diffuse grain boundary. Essentially, as the splitting of the (–220) spots show, one could now describe the TiN film as having a

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Fig. 23. Bright field (a) and dark field (b) TEM mirographs of the epitaxial TiN/MgO (100) sample taken after indentation. Both images are taken in the same g = [220] condition as Fig. 22. The arrows showing the [100] direction in both images also point to the location of the indentation. The diffuse nature of the dislocation plasticity can be seen in the MgO substrate, while the hemispherical configuration of dislocations around the indentation can be seen in the TiN film.

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Fig. 24. Lower magnification image of the same indentation shown in Figs 22 and 23. The bending of the TiN film can be seen more clearly with reference to the film position farther away from the indentation. Notice the dislocations emanating from the indentation axis in the MgO.

diffuse, 8◦ tilt boundary formed directly underneath the axis of the indentation direction. This boundary is comprised of a dislocation array formed during the indentation.

4. Conclusion This chapter presented initial observations using the novel mechanical testing technique of in situ nanoindentation in a TEM. Four classes of material systems were studied with this new technique: (1) bulk single crystal, (2) bulk polycrystalline material (3) a soft thin film on a harder substrate, and (4) a hard thin film on a softer substrate. The direct observations of each material system during indentation provided unique insight into the interpretation of ex situ nanoindentation tests, as well as to the intrinsic mechanical behavior of nanoscale volumes of solids. The in situ nanoindentation results presented in Section 3.1 provided the first direct evidence of dislocation nucleation in single crystal silicon at room temperature. In contrast to the observation of phase transformations during conventional indentation experiments, the unique geometry employed for the in situ experiments resulted in dislocation plasticity. The wedge geometry leads to a higher ratio of shear stress to hydrostatic pressure than that which is associated with conventional nanoindentation experiments, and therefore leads to a change in the mode of plastic deformation. These observations provide a crucial new insight into the failure of silicon-based microelectronic devices. If silicon is subjected to high stresses where the ratio of shear stress to hydrostatic pressure is large, room temperature nucleation and motion of dislocations is indeed possible. This observation leads to

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Fig. 25. (a) The post-indent hemispherical plastic zone in the TiN film. (b) Selected area diffraction pattern of the TiN film before the indentation. (c) Selected area diffraction pattern from the plastic zone of the indentation. The circle drawn over the image in (a) corresponds to the region from which the diffraction information is taken. In (c) the splitting of the [220] diffraction spots can be seen, indicating the bending of the TiN film 4◦ to each side.

speculation that room temperature dislocation plasticity is an important deformation mode even in ceramics, given small enough volumes. Using this in situ indentation technique, indentations into Al thin films were presented in Section 3.2. These results included the first real time observation of dislocation nucleation during nanoindentation of an initially defect-free Al grain. Plastic deformation in the Al


493

proceeded through the formation and propagation of prismatic loops punched into the material, and half-loops that emanated from the sample surface. Importantly, significant grain boundary movement during indentations near grain boundaries was observed. These observations strongly suggest that grain boundary movement could be a significant deformation mechanism in nanoscale metallic thin films. Section 3.3 demonstrated two different grain boundary-dislocation interaction phenomena within the same Fe-C martensitic sample. In the case of a low-angle lath boundary, the dislocations induced by the indenter first piled up against the boundary. But, as the indenter penetrated further a critical stress appears to have been reached and a high density of dislocations was suddenly emitted on the far side of the grain boundary into the adjacent grain. In the case of a high-angle grain boundary, the numerous dislocations that were produced by the indentation were simply absorbed into the boundary, with no indication of pile-up or the transmission of strain. Section 3.4 discussed the influence of the substrate on the indentation response of thin film/substrate systems where the films were harder than the substrate. During the deformation of epitaxial TiN on MgO (001), both the film and substrate deformed through dislocation plasticity. In the MgO substrate, dislocations nucleated at the film/substrate interface and flowed laterally. This lateral deformation in the MgO substrate was accommodated by the bending of the TiN film, as evidenced by the appearance of a diffuse tilt boundary. While substrate effects are inferred from load vs. displacement data obtained with conventional indentation techniques, the underlying mechanisms are often unknown. In situ nanoindentation in a TEM can provide the direct observations of the evolution of plasticity needed to resolve the mechanisms associated with the substrate effect. In prior work [48,102] attempts have been made to correlate load-displacement behavior with real-time images of the deformation response during in situ nanoindentation in a TEM. However, these attempts have relied on ex post facto determination of indenter displacement from sequential TEM images as well as indirect correlations between voltages applied to the piezoceramic actuator and measurements of known bending moments. This approach suffers from substantial uncertainties caused by non-linearities in the piezoceramic response, resulting in inexact data with low temporal and load resolution. Recently, a collaboration between the National Center for Electron Microscopy (NCEM) at the Lawrence Berkeley National Laboratory and Hysitron, Inc. [6] has led to a new in situ TEM nanoindentation holder design that includes a capacitive load sensor for quantitative force and displacement measurement during in situ indentations [103]. The quantitative holder has force resolution of ∼0.2 µN and displacement resolution of ∼0.5 nm. By correlating the force-displacement response of the material with direct images of the microstructural response of a material in real time, we have been able to study the initial stages of plasticity in metals [104,105], semiconductors [106] and even individual nanoparticles [107]. In conclusion, the experimental technique of in situ nanoindentation in a TEM has been shown to provide a unique capability for investigating the nanomechanical behavior of small solid volumes. This capability is essential to fully understanding the mechanisms associated with indentation phenomena and the fundamental deformation behavior of materials.

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CHAPTER 79

White Beam Microdiffraction and Dislocations Gradients G.E. ICE Metals & Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6118, USA and

R.I. BARABASH* Metals & Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6118, USA Center for Materials Processing, University of Tennessee, Knoxville, TN 37996-0750, USA

* Contact e-mail: [email protected]



Contents 1. Introduction: The need for white-beam microdiffraction 502 1.1. Ewald picture of X-ray diffraction 503 1.2. Qualitative influence of crystal defects on reciprocal space 506 2. Dislocation arrangements 507 2.1. Multiscale dislocation structure 507 2.2. Glossary of terms 508 2.3. Self-organization of dislocation arrangement 508 2.4. Geometrically necessary dislocations 509 2.5. Statistical description of randomly distributed dislocations 509 2.6. Dislocation boundaries types: IDBs and GNBs 511 2.7. Length scales in dislocation arrangements 512 3. Dislocation-induced redistribution of scattering intensity 514 3.1. Transverse and radial dependence 514 3.2. Correlation function 516 3.3. Influence of statistically stored (paired) dislocations on scattering 516 3.4. Influence of geometrically necessary (unpaired) dislocations on scattering 519 3.5. Connection between strain-gradient plasticity and transverse intensity 524 3.6. Diffraction by crystals with dislocation boundaries 526 3.7. Diffraction by multiscale structure with both GNDs/GNBs density 533 3.8. Measuring reciprocal-space intensity distributions 533 4. White beam analysis of the orientation space 534 4.1. Laue microdiffraction and related methods 534 4.2. Misorientation vector in Laue diffraction 538 4.3. Representative volume element for white beam diffraction 539 4.4. Natural axes of the Laue spot 540 4.5. Contrast factor for Laue diffraction 543 4.6. Split Laue spots 544 4.7. Distinct Laue patterns from crystals with random GNDs and with dislocation grouping within boundaries 544 4.8. Absorption correction 546 4.9. Laue Diffraction from the gradient cell–wall dislocation structure with decaying dislocation density 551 4.10. Size dependence of Laue intensity distribution 555 5. Determination of elastic strain 556 5.1. Local elastic strain 556 5.2. Ways to determine the full strain tensor and connection to reciprocal space mapping 558 5.3. Resolving plastic and elastic strain effects 559 6. Example applications of white beam diffraction 560 7. Instrumentation 590 7.1. X-ray source and reflectivity considerations 591 7.2. Microbeam monochromator 593 7.3. Achromatic focusing 594 7.4. Detector 594 7.5. Software 595

8. Concluding remarks 597 Acknowledgements 597 References 597

We describe how X-ray microdiffraction is influenced by the number, kind and organization of dislocations. Particular attention is placed on micro-Laue diffraction where polychromatic X-rays are diffracted into characteristic Laue patterns that are sensitive to the dislocation content and arrangement. Micro-Laue diffraction and related techniques are important for measurements where defect distributions need to be measured with good 2D and 3D spatial resolution. Diffraction is considered for various stages of plastic deformation. For early stages of plastic deformation with random dislocation spacing, the intensity in reciprocal space is redistributed about Laue spots with a length scale proportional to the number of dislocations within the sample volume and with a characteristic shape that depends on the kinds of dislocations and the momentum transfer vector. Geometrically necessary dislocations (GNDs) that contribute to lattice rotations cause the largest redistribution of scattered intensity. In later stages of plastic deformation strong interactions between individual dislocations cause them to organize into correlated arrangements. Here again Laue spots are broadened in proportion to the number of GNDs inside the wall and to the total number of unpaired walls, but the broadening can be discontinuous. With microdiffraction one can to quantitatively test models of dislocation organization.

1. Introduction: The need for white-beam microdiffraction The development of ultra-brilliant third-generation synchrotron X-ray sources [1–3] together with advances in X-ray optics [4] has created intense X-ray microbeams. Important applications include ultra-sensitive elemental detection by X-ray fluorescence/absorption and microdiffraction to identify phase and strain with submicrometer spatial resolution. X-ray microdiffraction is a particularly exciting application compared with alternative probes of crystalline phase, orientation and strain; X-ray microdiffraction is nondestructive with better strain resolution, competitive or superior spatial resolution in thick samples, and with the ability to probe below the sample surface. This offers promise for a fundamentally new approach toward characterizing dislocation behaviour by allowing for in-situ and 3D measurements with outstanding sensitivity to defects. Although X-ray microdiffraction has many important advantages compared to other probes, traditional monochromatic methods are often difficult for deformed and polycrystalline samples due to the need to rotate the sample [5]. Indeed, the key challenge of microdiffraction is to obtain precise crystallographic information while maintaining good spatial resolution in the sample. Precise crystallographic information is essential to characterize defect content, and good spatial resolution is essential to map the crystal/defect distribution. Below we briefly describe the Ewald picture of X-ray diffraction to contrast the relative advantages of monochromatic and polychromatic (white beam) microdiffraction and to clarify why polychromatic methods are essential for measurements in need of the best spatial resolution.

§1.1

White beam microdiffraction and dislocations gradients

503

1.1. Ewald picture of X-ray diffraction Although single atoms scatter X-rays very weakly, atoms arranged in a regular “crystalline” lattice, scatter X-rays efficiently when the X-ray wavelength and incident angle satisfy the Bragg condition; sin θ =

λ . 2dhkl

(1.1)

Here λ is the X-ray wavelength, dhkl is the spacing between the hkl Bragg planes in the grain and θ is the angle between the incident X-ray beam and the Bragg plane. Bragg’s law is essentially a requirement of constructive interference from specularly reflecting planes within the crystal [6]. Schematically this can be represented for any crystal lattice and for any beam directed onto that lattice by an “Ewald Sphere Diagram” that shows the momentum transfer for all possible X-ray scattering directions and compares these to the momentum transfer associated with particular sets of crystal planes in the sample. The distribution of possible momentum transfer for all possible scattering directions forms the surface of an “Ewald” sphere for a single wavelength. Similarly, the conditions for efficient Bragg scattering (eq. (1.1)) forms a 3D lattice called the “reciprocal” lattice that is fixed to the orientation of the real-space lattice of the crystal. A typical Ewald Sphere Diagram is illustrated in Fig. 1(a) for a face-centered-cubic metal (body-centered-cubic reciprocal lattice). An Ewald sphere for a monochromatic incident X-ray beam is also drawn. A Bragg reflection (efficient scattering) occurs when one of the reciprocal-space lattice points intersects the Ewald sphere. As illustrated in Fig. 1(a), when there is no a-priori knowledge of grain orientation, there can be only a small probability of even a single reciprocal space point (other than the (0,0,0)) intersecting the Ewald Sphere. In traditional monochromatic single-crystal methods the sample is rotated until a reciprocal lattice point intersects the sphere (Fig. 1(b)). For a particular momentum transfer, the X-rays are scattered in a specific direction. Area detectors naturally increase the likelihood of detecting a reflection. Sample rotations work well for large homogeneous single crystals, but with inhomogeneous or small-grained polycrystalline samples, rotations have severe drawbacks. For example, as illustrated in Fig. 2, during sample rotations, grains of the sample move into and out of a small beam. This means that some grains will only be illuminated over a small angular range and may not produce even a single Bragg reflection. Polychromatic (white-beam) Laue Diffraction is one method that overcomes this problem. Here an incident beam with a well-defined direction, but with a wide bandpass is directed onto the sample. Because of the wide bandpass, several combinations of wavelength and Bragg plane indices h, k, l are likely to simultaneously satisfy the Bragg condition. The advantage of Laue diffraction compared to monochromatic methods is illustrated in Fig. 3. As shown, the Ewald Sphere surface is replaced by a volume in reciprocal space between the lower and upper bounds of the X-ray beam bandpass. Reciprocal lattice points that lie in the region bounded by the two shells are efficiently scattered. Formally the resulting Laue pattern is composed of radial integrals through reciprocal space

504

G.E. Ice and R.I. Barabash

Ch. 79

Fig. 1. (a) Two dimensional representation of the Ewald Sphere for 1 Å X-rays and a FCC unit cell with a0 = 3.25 Å. There is only a small probability that the Ewald Sphere representing the possible X-ray momentum transfer will intersect a reciprocal lattice point (satisfy a Bragg condition). The unit vectors s and s0 give the direction of the diffracted and incident beams. (b) As the sample rotates, the reciprocal lattice points rotate with the real-space lattice and can be made to intersect the Ewald Sphere. Intersections are indicated with arrows. A Bragg reflection is detected if an X-ray sensitive detector is located to collect the reflected radiation.

Fig. 2. The grains illuminated by a penetrating X-ray beam change as the sample is rotated due to sphere of confusion errors (circle), alignment errors and due to the aspect ratio of the penetration length to beam crosssection.

weighted by geometric factors, by sample absorption and by the spectral density distribution [7]. Area detectors, under realistic conditions, detect multiple reflections simultaneously. Laue Diffraction is actually the oldest X-ray diffraction method and its prediction led to the 1914 Nobel Prize in Physics presented to Max von Laue. Laue diffraction is still a standard crystallographic method used to determine crystal orientation [8–10]. However, Laue diffraction is rarely used to measure crystal defects because the precision of most Laue instruments is low compared to standard monochromatic diffractometers/goniometers, and because the unit cell volume cannot be determined with a Laue measurement. Nevertheless, small X-ray beams overcome some of the limitations of standard Laue diffraction. With suitable instrumentation, the Laue method can be used to precisely determine orientation (local texture) of individual grains, their elastic strain and as described in this paper

§1.1


505

Fig. 3. Ewald sphere picture for Laue diffraction. Reciprocal space points located between the upper and lower bounds set by the maximum and minimum wavelengths are efficiently reflected. The Laue image is composed of radial integrals through reciprocal space. Each integral includes weighting from absorption effects, geometrical effects and the incident beam spectral distribution.

can be used to characterize dislocation distributions. Laue diffraction can also be extended to determine the full strain tensor in polycrystalline samples by the measurement of the wavelength/energy of one or more reflections. In addition to its other advantages, spatially-resolved two-dimensional (2D) and threedimensional (3D), characterization of local crystal structure is conveniently made with polychromatic X-ray microdiffraction. The local (subgrain) orientation, and the local streaking of Laue reflections can be used to fit models of the unpaired-dislocation density and the local elastic strain tensor. This information can be collected non-destructively and used for direct comparison to theoretical models. Practical microbeams use intense synchrotron X-ray sources and advanced X-ray focusing optics. One can measure 3D distributions of the local crystalline phase, orientation (texture) and elastic and plastic strain tensors with submicrometer resolution. The local elastic strain tensor elements can typically be determined with uncertainties less than 100 ppm. Orientations can be quantified to ∼0.01 degrees and the local unpaired dislocation-density tensor can be simultaneously characterized. The spatial resolution limit for hard X-ray polychromatic microdiffraction is c1 .

§2


507

Table 1 Classification of defects 1st kind

2nd kind

• • • •

• Dislocations • Disclinations • Planar defects − grain boundaries − subgrain boundaries − cell boundaries − stacking faults − twins • Large dislocation loops

Point defects, Coherent precipitates Small dislocation loops Defects small compare to crystal

Whether a defect is of the 1st or 2nd kind depends on the behaviour of the DWF at large distances (ρ → ∞). It should be noted that DWF has a complicated dependence on the details of the distortion fields near a defect. As a first approximation, 1 1 − cos(Quss ′ t ) 1 + (ϕst + ϕs ′ t ) . ρ→∞ f t

2W = Re T∞ ∼ = c lim

(1.2)

Here T∞ is the correlation function for defects at large distances (ρ → ∞), c is the concentration of defects, f is the structure factor of the average crystal, Q is the momentum transfer for certain (hkl) reflection, uss ′ t is the difference between displacements in two scattering cells s and s ′ caused by the defect located in the position t, and ϕst and ϕs ′ t describe structure amplitude changes of scattering cells s and s ′ caused by the defect located in the position t. We note that for dislocations, changes in structure amplitudes are small and the behavior of the 2W depends mainly of the asymptotic behavior of the displacement field created by the dislocation. With defects there are two possibilities: 2W is either finite at large distances or 2W tends to infinity at the large distances. It was shown [11,12] that if the displacements fall off faster then 1/r 3/2 , then the value 2W is finite, and the defects belong to the 1st kind. If the displacements decrease at a lower rate then 1/r 3/2 , then the value 2W → ∞, and these defects are of the 2nd kind. According to this classification, we can sort defects into the two types (see Table 1).

2. Dislocation arrangements 2.1. Multiscale dislocation structure Although dislocations have long been recognized as the principal mechanism for deformation, the organization of dislocations on various length scales was recognized much later. Key elements of multiscale dislocation structure were first proposed by Ashby [13], worked out by Mughrabi [14–16], Nabarro [17–19], Humphreys and Hatherly [20]. These elements include statistically stored dislocations, SSDs, where there are equal numbers

508


Ch. 79

of “+” and “−” Burgers vectors. SSDs are also called paired dislocations. Geometrically necessary dislocations (GNDs), also called unpaired or polar dislocations, are excess dislocations of one sign. Other elements include dislocation cells and walls. GNDs (unpaired) dislocations that cause streaking in Laue peaks and rocking curves can be organized as individual GNDs; as Incidental Dislocation Boundaries (IDBs); in Geometrically Necessary Boundaries (GNBs); in lamellar bands (LB) separated by some distance (SD) and in combinations of these organizations. At small strains, geometrically necessary boundaries (GNBs) are composed of dense dislocation walls (DDWs) or double walled microbands (MB) inclined in the direction of metal flow. At large strains they can form almost parallel lamellar bands (LB). These microbands are distinguishable mainly due to differences in local orientation as well as morphology differences of the dislocation structure within the deformation microbands. With increased plastic deformation, dislocations form more complicated hierarchical structures in different disclination configurations [21,22]. Some new mechanisms of dislocation/disclination arrangements are observed at the nanoscale [23–26]. A fractal description of dislocation arrangements is proposed for example by Zaiser and Hanner [27]. 2.2. Glossary of terms In part because of the widespread recent progress in understanding dislocation structures, there are a number of terms used to describe dislocations and their hierarchical organizational structures. We summarize the various terms and acronyms used in this paper for reference: • White-beam microdiffraction, Laue microdiffraction, polychromatic microdiffraction (PXM); • Paired ≡ redundant ≡ dipolar ≡ Statistically Stored Dislocations (SSD); • Unpaired, nonredundant, polar, geometrically necessary dislocations (GNDs), GN dislocations; • Incidental Dislocation Boundaries (IDBs); • Geometrically necessary boundaries (GNBs) ≡ dense dislocation walls (DDW); • Representative volume element (RVE); • Reciprocal space map (RSM); • High resolution X-ray diffraction (HRXRD); • Orientation imaging microscopy (OIM); • Differential aperture X-ray microscopy (DAXM).

2.3. Self-organization of dislocation arrangement In real crystals individual GNDs tend to group into walls to reduce the stored energy [14,11, 12,17,28–32]. The elastic energy per unit length E/ l of a crystal with randomly distributed dislocations may be written as:

§2.4


L E μnb2 ∼ ln l 4π b

509

(2.1)

(see Nabarro [17] and Krivoglaz [12]). Here E is the integrated elastic energy due to noninteracting dislocations, μ is the shear modulus, n is the total dislocation density, b is the modulus of the Burger’s vector and L is a characteristic length (e.g., grain size). The logarithmic factor ln(L/b) depends on the cut-off distance L from the dislocation. Correlations between dislocations screen the elastic fields at the so-called “correlation radius”: rc . In the presence of screening the integrated elastic energy decreases: E μnb2 rc . ∼ ln l 4π b

(2.2)

As a result, correlated dislocation arrangements can be more stable than random ones. This leads to the formation of cell/wall structures [14] that are organized with a high dislocation density within the cell walls and a low dislocation density within the cell interiors. Such structures are observed in a number of experiments [30,33–40]. Significant correlation may also occur between the dislocation positions within the wall. Models for dislocation boundary formation are analyzed in detail in [14,18,30–32]. 2.4. Geometrically necessary dislocations During initial stages of deformation without strain gradients, dislocations with opposite Burgers vectors usually have the same probability to form. With strain gradients, however (e.g., in the case of plastic bending or torsion of a crystal), more dislocations with a given Burgers vector direction are produced. Such dislocations cause not only random strain fluctuations in various parts of the crystal, but cause a correlated bending (or torsion) of the crystal planes. Such unpaired/polar dislocations are termed geometrically necessary dislocations GNDs. They may also appear at later stages of deformation when dislocation with opposite Burgers vector sign separate at large distances and groups of GNDs with certain Burgers direction accumulate in regions separated by distances large compared to the size of the so called “representative volume element” (RVE) which characterizes the volume probed with different diffraction techniques. This additional “cumulative” strain is so essential that even a relatively small fraction of GNDs gives rise to significant changes in scattering intensity distribution [15]. Crystallographic aspects of GND and SSD densities are discussed by [41,42]. It was pointed out that the term “geometrically necessary” is imprecise because the unpaired or “polar” portion of the dislocation population strongly depends on the representative volume over which the dislocation density is determined [17,41]. We later address this issue in the description of white beam diffraction due to GNDs. 2.5. Statistical description of randomly distributed dislocations The simplest model for dislocation organization consists of randomly distributed dislocations. Consider the set of edge dislocations shown in Fig. 5 with Burgers vectors, b, parallel

510


Ch. 79

Fig. 5. (a) Random numbers ct , describe statistical distribution of dislocation in the transverse plane: c2.4 = c3.7 = 1 (these numbers correspond to the edge of the extra plane), for other lattice sites ct = 0 in the XOY plane. Dashed regions S0 indicate the area of one dislocation in the transverse plane. (b) Total number of dislocations crossing the area S. The magnitude of the vector, τ , is the net number of dislocations having Burgers vector, b, crossing unit area normal to τ . It is also equal to the net length per unit volume, projected parallel to τ of all dislocations crossing unit area normal to τ . (c) GN portion of dislocation population crossing the area.

to the x-axis and dislocations lines, τ parallel to the z-axis. To describe the distribution of dislocations, we introduce the number ct =

&

1 0

such that, ct = 1 for a dislocation at position t, and ct = 0 for positions without dislocations [11,12]. For example for the dislocations shown in Fig. 5(a) only c3.4 = c4.7 = 1, for all other sites ct = 0. Each edge dislocation positioned on the t lattice site creates a displacement field with displacement uit of any ith scattering cell in the plane XY perpendicular to its line. The total displacement of the ith cell ui is due to the superposition of all dislocations:

§2.6


ui =

ct uit .

511

(2.3)

t

For example for an elastically-isotropic crystal the projections of the displacement vector in XOY plane are described by the following equations [17,43,44]: xy b −1 y tan ux = + , 2π x 2(1 − ν)(x 2 + y 2 ) uy = −

1 − 2ν 2 b (x 2 − y 2 ) . ln x + y 2 + 2π 2(1 − ν) 4(1 − ν)(x 2 + y 2 )

(2.4)

Here ν is Poisson’s ratio. Similar components of the displacement field for anisotropic crystals can be found in text books and can be used for simulation of scattering to compare with experiments on anisotropic sample.

2.6. Dislocation boundaries types: IDBs and GNBs Inhomogeneities of plastic deformations result in the formation of different types of dislocation boundaries. With respect to their origin and properties two main types are distinguished [30]: incidental dislocation boundaries (IDBs) and geometrically necessary boundaries (GNBs). IDBs: Incidental dislocation boundaries are assumed to form by statistical mutual trapping of glide dislocations. IDBs usually are thick, curved, boundaries with a loose distribution of dislocations within their walls. Two kinds of IDBs are distinguished: IDB-1s which cause very small misorientation (Fig. 6(a)) and IDB-2s with small misorientations through the boundary (Fig. 6(b)). IDBs result in long-range internal stresses that broaden diffraction peaks. GNBs: Under severe plastic deformation, dislocation tangles self-organize within sharp walls separating regions with low dislocation density [45,46]. Extended straight parallel boundaries have been termed “geometrically necessary boundaries”, GNBs (Fig. 7). GNBs have dense dislocation populations. They are assumed to form by a contrasting slip activity on each side of the boundary. The contrasting slip activity might be due to different activated slip systems in neighboring cell blocks or due to a different partition of the total shear among a common set of slip systems [31,32]. Morphological differences, which exist between the cell boundaries and the extended boundaries, indicate different mechanisms in the evolution of IDBs and GNBs. For example, IDBs and GNBs have different rates of misorientation and boundary spacing with strain. The presence of IDBs and GNBs redistributes diffracted intensity distributions in the near Bragg region of reciprocal space (Section 3.6) and the distinct diffraction behavior of IDBs and GNBs can be used to track their evolution during plastic deformation.

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Ch. 79

Fig. 6. Scheme of dislocation self-organization within incidental dislocation boundaries: IDBs-1 without misorientation (a) and IDBs-2 with small alternating misorientation through the boundary (b).

2.7. Length scales in dislocation arrangements The length scales usually distinguished with respect to the evolution of plastic deformation (Fig. 8) in solids are [21]: (1) Microscopic (Fig. 8(a)) with typical length scale l1 ∼ 1–30b. (2) Mesoscale (Fig. 8(b)) or length scale of dislocation substructure with the typical length scale l2 ∼ 0.1–10 µm. (3) Structural length scale (Fig. 8(c)), which coincides with the grain size in polycrystalline materials, l3 ∼ 20–200 µm. For single crystals this length scale may coincide for example with the cell-block size or size of fragments. (4) Macroscopic length with the typical length scale l4 10l3 . At this length scale the structural inhomogeneities of the plastic deformation typically are not essential (Fig. 8(c)). The recent developments of strain gradient plasticity represent an effort to bridge the gap between classical plasticity and dislocation theory [47–49]. Plastic-strain gradients appear either because of the geometry of loading or because the material is plastically inhomogeneous. Two length scales are introduced in the framework of strain-gradient plasticity. Gao et al. [48] related effective strain gradient via the density of geometrically necessary dislocations.

§2.7


513

Fig. 7. (a) Lattice rotations around alternating “+” and “−” tilt GNBs; (b) displacement field in different scattering cells at different locations in the vicinity of the GNB. Insert shows the change in the displacement field dependence on the distance from the GN boundary for finite length of the GN boundary.

Materials demonstrate strong size effects when the characteristic length scale associated with non-uniform plastic deformation is of the order of micrometers. Usually the smaller the size of the sample the stronger is the response [47,48]. Recently the influence of length scale on deformation was elegantly demonstrated during uniaxial compression of micron and nanosize gold pillars by Nix et al. [50,51]. Length scale and time scale effects on the plastic flow of FCC materials are also observed in MD simulations [52]. Strain gradient plasticity theories [47–49,53,54] have developed phenomenological descriptions of plastic strain and its gradient for mesoscale structures with dimensions in the range from 0.1 to 10 micrometers. Usually strain gradients become important when the densities of statistically stored and geometrically necessary dislocations (GNDs) are of the same order. Nix and Gao [55]

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Ch. 79

ˆ which is related to the Burgers vector introduce a so called “internal (materials) length” l, b and the mean spacing Ls between statistically stored dislocations at yielding (Fig. 8(a)). L2 lˆ = s . b

(2.5)

This length scale corresponds to the 1st structural level of the above classification (lˆ = l1 ). Internal (material) length scales govern strain gradient effects and determine the ratio between statistically stored and GNDs. However, due to inhomogeneity of plastic deformation and due to inhomogeneity of dislocation distributions, the internal length scale lˆ varies spatially and can only be determined locally for a particular SSD dislocation density [56]. Nix and Gao [55] also introduce a second length scale called the “mesoscale cell size” (lM = l2 ) which is a “resolution parameter”: lM ≈ βLs ;

Ls = β

μb . σy

(2.6)

Here σy is the yield stress, μ is the shear modulus, β is a cell size parameter which in their approach is usually taken from 1–10. For typical deformed metals they estimate the length scales fall in the ranges: lˆ ≈ 10–100 nm,

and lM ≈ 1–10 µm.

Five length scales are discussed with respect to boundary strengthening by Hansen [30]. He describes a multiscale analysis being applied to the relationship between strength and boundary parameters. This analysis suggests that the Hall–Petch relationship continue to be valid even for nanoscale grains. The influence of the different length scales on white beam diffraction is addressed in Section 4.

3. Dislocation-induced redistribution of scattering intensity 3.1. Transverse and radial dependence Dislocations change the diffraction conditions and enlarge the region of high intensity around each Bragg position; they introduce long-range distortion fields in the crystallographic lattice. The intensity distribution is sensitive to the kind of dislocations present in the crystal and to the organization or correlated structure of the dislocations [11,12,14,15,29,33,34,57–59]. Around each reciprocal lattice point corresponding to a reciprocal lattice vector site Ghkl the scattered intensity is a function of the diffraction vector Q = k2 − k1 = 4π/λ(kˆ 2 − kˆ 1 ) or of the deviation vector q = Q − Ghkl . Here k1 , k2 are wave vectors of the incident and the scattered waves with the directions defined by the unit vectors kˆ 1 and kˆ 2 . In the presence of dislocations, there are two distinct directions in reciprocal space (Fig. 9):

§3.1


515

Fig. 8. Schematic of different length scales in dislocation arrangement.

∼ I (θ/2θ ) describes the shape of the The radial direction: (q parallel to Ghkl ). I (q ) = radial intensity distribution of the reflection, and depends mainly on the dilatation part (strain) of the long-range displacement fields u = u(R) caused by lattice defects. Broadening in the radial direction can be qualitatively understood as due to the local variation of the lattice plane spacing caused by dislocations strain fields [11,12,15,57]. Transverse plane (orientation space): (q⊥ transverse to Ghkl ). I (q⊥ ) depends essentially on the misorientation part (lattice rotations) of the displacement fields u = u(R) caused by lattice defects. Broadening in the transverse plane represents a local variation in the lattice plane orientation. Several nondestructive probes with different resolution and penetration depth have been recently developed for the analysis of the local orientation. Transverse broadening can be used to characterize the local orientation distribution using white beam techniques [60,61], rocking curve analysis [11,12,15,33,37,62–67], reciprocal space mapping [68], and/or orientation mapping [69–71]. Ultrasmall-angle scattering is now developed for the analysis of the dislocation core [72]. Generally, the radial intensity distributions I (q ) and transverse distributions I (q⊥ ) have quite different characters correlating with distinct parameters of the dislocation structure. The intensity distribution I (q) of X-ray (or neutron) scattering due to dislocations can be computed from the expression [11,12,29]: I (q) = f 2

eiq· e−T .

(3.1)

i,j

Here f is the average scattering factor of the matrix atoms, = R0i − R0j is the undistorted distance vector between the lattice cells i, j , and the correlation function T is determined

516


by the following equations [11,12] T = nS0 1 − e(iQ·(uit −uj t ) = c 1 − e(iQ·(uit −uj t ) . t

Ch. 79

(3.2)

t

Here S0 is the area related to one possible dislocation position in the transverse plane, n is the total dislocation density, c = nS0 is a dimensionless quantity that indicates the frac tion of lattice sites intercepted with dislocations (Fig. 5(b) and (c)), n = λ nλ , λ numerates different slip systems. The correlation function T differs according to the dislocation arrangement. 3.2. Correlation function The difference between the displacements of two scattering lattice cells i and j can be written [11,29,59,73,74]: 1 uit − uj t ≈ (Rij ∇)uit + (Rij ∇)2 uit + · · · . 2

(3.3)

The correlation function T (eq. (3.2)), can be expanded with respect to small displacements. In general for an arbitrary distribution of paired and unpaired dislocations T has both imaginary and real parts T = Ti + TR : Ti = i ci (Rij ∇)(Ghkl uit ); TR = − ci 1 − cos (Rij ∇)(Ghkl uit ) . (3.4) t

t

The first term, Ti , is imaginary and describes lattice rotations due to geometrically necessary (unpaired) dislocations within the probed region. The real part of the correlation function, TR , represents the fluctuating strain fields from randomly distributed individual dislocations. It is independent of whether dislocations are paired or unpaired. As discussed below if all dislocations are paired, the imaginary part is equal to zero and the correlation function T is real. If all dislocations are unpaired the function T contains both real and imaginary parts. These two limiting cases give rise to distinctly different intensity distributions. 3.3. Influence of statistically stored (paired) dislocations on scattering For an equal number of random “+b” and “−b” dislocations the average deformation tensor is negligible and broadening of the diffuse scattering is induced by fluctuations in the local unit cell orientation and d spacing that tend to cancel over long distances. The correlation function for a paired dislocation population does not have an imaginary part (Ti = 0) and TR = C2 (Qb)2 ln L nλ ϕλ . (3.5) λ

§3.3


517

Example 1: One kind of statistically stored edge dislocations. Due to the character of the displacement field around edge dislocations, displacements occur only in planes perpendicular to the direction of the dislocation lines, τ . As a result, coherence is not changed along the direction, τ , and the scattered intensity in this direction is the same as for crystals without dislocations. This property provides a means to determine Burgers vectors by transmission electron microscopy [75] where dislocations can be aligned so they are out of contrast in the image mode. Perpendicular to the dislocation line, τ , the intensity distribution is Gaussian (Fig. 10). If only one set of dislocations is activated, the scattered intensity has a disk-like character around all reciprocal lattice points (Fig. 11). The plane of the disk is perpendicular to the dislocation line. The full-width-at-half maximum (FWHM) in the plane of the disk depends on the orientation of the diffraction vector, the Burgers vector and the dislocation line. In the direction normal to the disk the FWHM is of the order of 2π/L (L is the size of the grain (or probe dimensions) in that direction; the FWHM becomes δ(qz ) for an infinite crystal). For a crystal with edge dislocations the intensity disk is non isotropic and looks like an ellipsoid with an intensity distribution near Ghkl that depends on the “contrast factor” tensor β I (Q) =

1 (2π)3/2 N 2 exp − βxy qx qy . f δ(qz ) √ v 2 x,y |α|

(3.6)

−1 . Elements of this Here |α| is a determinant of the matrix being inverse to β: αxy = βxy matrix depend on the orientation of the dislocation and diffraction vector. Surfaces of equal intensity around any reciprocal lattice point of the crystal with edge straight dislocations are described by the second order equations x,y βxy qx qy . The eigenvectors and eigenvalues of the matrix β can be found. Matrix β is termed the “contrast factor”. Coefficients of this matrix determine intensity distribution in any direction of reciprocal space around each reciprocal lattice site. Contrast factor for each reflection depends on the mutual orientation of the diffraction vector and a dislocation slip system (Burgers vector and slip plane).

Example 2: One kind of statistically stored screw dislocations. For one set of paired screw dislocations running along the z-axis, the intensity distribution becomes simpler and the disk intensity follows the expression π qx2 + qy2 1 8π 2 N 2 . f δ(qz ) 2 exp − I (Q) = v σ σ2

(3.7)

Here 2

2

σ = 0.5n(Q · b) l;

l ∼ ln

√

√ πσ L ≈ ln nlL . 2Qb

(3.8)

From (3.7) it follows that the FWHM in the disk plane is proportional to the square root √ of the total dislocation density, n, the reciprocal lattice vector of the reflection, Q and (Qb). For Q ∼ perpendicular to b the scalar product tends zero and due to the exponential

518


Ch. 79

Fig. 9. In reciprocal space the radial direction (q Q) is distinct from the two transverse directions (orientation space) (q⊥ ). Dislocations enlarge the region of high intensity around each (hkl) Bragg point (blue shadow).

Fig. 10. A scattering intensity distribution perpendicular to dislocation lines falls off with a Gaussian distribution. This is illustrated by a colour map of intensity (top) or by a 3D representation of intensity (bottom).

term (3.6) the FWHM of the intensity distribution tends to zero. If the dislocation density is almost equal in all different possible slip systems, then the intensity distribution is roughly symmetric√ with a characteristic FWHM ∼ σ dependent on the total dislocation density n: FWHM ∝ n [7,11,12,28,33,57,59].

§3.4

519


Fig. 11. Real (a) and reciprocal (b) space for statistically stored dislocations. All reciprocal lattice points turn to isotropic disks with the plane transverse to dislocation lines direction.

3.4. Influence of geometrically necessary (unpaired) dislocations on scattering Geometrically necessary (GN) (unpaired) dislocations have a much more pronounced influence on scattering. GNDs are typically arranged in the following ways: • Unpaired random geometrically necessary dislocations (GNDs). • Unpaired geometrically necessary walls/boundaries (GNBs). • GNBs walls with IDBs and random dislocations between the walls. X-ray scattering near Ghkl by crystals with GNDs and GNBs exhibit a range of distinct intensity distributions in radial and transverse (orientation space) directions as summarized in Table 2. Below we discuss how the three models above influence reciprocal space intensity distributions. (a) Radial direction (q Ghkl ) The presence of GN (polar/unpaired) dislocations in real space causes macroscopic lattice rotations (Fig. 12). In reciprocal space, the intensity near reciprocal lattice sites is streaked. The orientation of the streak depends on the mutual orientation of GNDs slip system and the diffraction vector. Using eqs (3.2)–(3.4), we find that for a GN dislocation density, n+ λ belonging to slip system λ, the correlation function has both real and imaginary terms: Ti = −iC1

λ

n+ λ (Ri · bλ ) [Q × Rij ]τ λ ;

TR = C2 (Qb)2 ln L

nλ ϕλ .

(3.9)

λ

Here C1 , C2 are the contrast factors for transverse and radial directions, L is the size of the probed subgrain (or the cut-off radius at which the distortion field of the dislocation is screened by neighboring dislocations, whichever is smaller), and ϕλ is the orientation factor for each dislocation system. Ti is linear with respect to the density of polar dislocations n+ and goes to zero when n+ = 0. From the structure of eq. (3.9) it follows that unpaired dislocations have the same scattering intensity in the radial direction as SSD dislocations;

520

Ch. 79


Table 2 Intensity distribution in reciprocal space Dislocation arrangement

Radial direction (q ) FWHM rad ≡ δ2θ

Transverse plane FWHM rock ≡ δq⊥

Individual, randomly distributed, paired SSDs (Sections 2.3, 3.3)

Continuous Gaussian √ δ2θrand ∝ n tan θ

Isotropic continuous Gaussian √ δq⊥ ∝ n

Individual, randomly distributed, unpaired GNDs (Sections 2.4; 3.4)

Continuous Gaussian √ δ2θrand ∝ n tan θ

Continuous “flat-top” anisotropic √ max + δqmin ⊥ ∝ n; δq⊥ ∝ Ln Q

IDB-1s (Sections 2.6, 3.6)

Continuous Gaussian √ δ2θ rand ∝ n tan θ × (l − 0.5 ln(Di / h))/ l.

Continuous Gaussian √ Continuous Gaussian δq⊥ ∝ n

§3.4


521

Table 2 (Continued) Dislocation arrangement


Continuous Gaussian IDB-2s √ (Sections 2.6, 3.6) δ2θ rand ∝ n tan θ Not correlated × (l − 0.5 ln(Di / h))/ l.

Continuous Gaussian IDB-2s √ (Sections 2.6, 3.6) δ2θ rand ∝ n tan θ Correlated × (l − 0.5 ln(Di / h))/ l.

Transverse plane FWHM rock ≡ δq⊥ Anisotropic Gaussian √ √ δqrock = 2 π A0 L/D(bQ/ h)

√ √ δq⊥ = 2 π A0 rcor /DαQ

in the radial direction the cross product [Q × Rij ] = 0 and Ti vanishes. A random distribution of unpaired dislocations results in a Gaussian intensity profile in the radial direction with the FWHM rad ≡ δ2θ , which depends only on the total dislocations density n, δ2θ ∝

√

n.

(3.10)

If GNDs are grouped within walls, the radial intensity profile has a Lorentzian shape and I (q ) depends on the average distance D between walls (Table 2). In the intermediate case of dislocations distributed both in tilt walls and throughout the crystal, the radial intensity profile corresponds to a convolution of a Laue function for coherently scattering

522


Ch. 79



Paired GNBs

Continuous Lorenzian

(Sections 2.6; 3.6)

1 sec θ δ2θ ∝ D

Transverse plane FWHM rock ≡ δq⊥ L Q δq⊥ = 2 π Aξ ξ D

Unpaired GNBs (Sections 2.6, 3.6)

Continuous Lorenzian 1 sec θ δ2θ ∝ D

“Flat-top” anisotropic min δqmax ⊥ ∝ Lθ/D; δq⊥ ∝ 1/D

3 (bQ)2 ≫ Lh2 Discontinuous, if Dcb

regions between walls and a Gaussian function corresponding to the effect of the randomly distributed dislocation walls. (b) Transverse direction (orientation space) (q⊥ ⊥Ghkl ) In the transverse directions, the diffuse scattering becomes much broader for unpaired dislocations (T1 = 0) and is described by a more complicated function. For crystals with a random distribution of excess dislocations the FWHM ⊥ ≡ δq⊥ of the transverse intensity distribution scales linearly with the density of excess dislocations n+ and the length of the probed region, L, δq⊥ ∝ Ln+ .

(3.11)

However due to the dependence of Ti on the orientation of the unpaired dislocation population relative to the diffraction vector (Ti ∝ ([Q × Rij ]τ λ ), eq. (3.8)) dislocations with lines parallel to the diffraction vector do not influence the scattering intensity even in the transverse plane.

§3.4


523

Fig. 12. Real (a) and reciprocal (b) space for geometrically necessary (polar) dislocations. GNDs create macroscopic lattice rotation around the direction of dislocation lines in real space (a) and cause streaking of all reciprocal lattice sites in reciprocal space (b).

For crystals with tilt dislocation walls, Ti is proportional to the number of excess dislocation inside each wall and to the total number of excess walls. For unpaired geometrically necessary dislocations (GNDs) (Fig. 12), the intensity distribution relates the distortion tensor, ωij , due to GNDs, n+ , to the dislocation density and to the slip system (dislocation direction τ , and Burgers vector b). In this case the mean deformation tensor can be written in terms of the anti-symmetric Levi-Civita tensor of third rank ετ lm and a dislocation density tensor of second rank ρτ n ετ lm

∂ωmn ∂xl

= −ρτ n .

(3.12)

Here the Levi-Civita tensor has elements ε123 = ε231 = ε312 = 1; ε213 = ε132 = ε321 = −1; and all other εij k = 0. The index τ in ρτ n specifies the crystallographic direction of the dislocation line, and n indicates the Burgers vector direction. For a general set of edge dislocations with dislocation density ρ0 , and a line direction parallel to the plane normal [lmn], and Burgers vector [l ′ m′ n′ ], the tensor of dislocation density is written, ρτ n =

ρ0 ll ′ ρ0 ml ′ ρ0 nl ′

ρ0 lm′ ρ0 mm′ ρ0 nm′

ρ0 ln′ ρ0 mn′ ρ0 nn′

.

(3.13)

Example 1: Tilt deformation due to one set of edge GN dislocations. Consider for example, the set of edge GN dislocations with the density n+ E and dislocation lines running along the z-axis, τ Z = [001], and Burgers vector running along the x-axis b = [100] (Fig. 12(a)). There is only one non-zero component of the dislocation density tensor, ρZX = n+ E b. Eqs (3.12), (3.13) show that there are only two non-zero components of the mean distortion tensor: ωXY = −ωY X = bxn+ E.

(3.14)

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Ch. 79


This distortion field is antisymmetric and represents a pure rotation about the z-axis that increases with displacement in x. The correlation function in orientation space due to one set of edge GN dislocations may be written as following [73,74]: Edge

Ti

= −in+ E (RS · b) [Q × Rij ]τ .

(3.15)

Example 2: Twist deformation due to one set of screw GN dislocations. Consider a set of screw GN dislocations with density n+ S and with dislocation lines and Burgers vectors running along the z-axis. This set of screw dislocations corresponds to a dislocation density tensor with only one nonzero component: ρZZ = n+ S b with b = |b|. Lattice twist due to such a set of screw GN dislocations is described by an antisymmetric distortion tensor with the following nonzero components: b ωXY = zn+ ; 2 S

b ωXZ = yn+ ; 2 S

b ωY Z = xn+ ; 2 S

ωmn = −ωnm .

1 TiScrew = − ibn+ S RS · [Q × Rij ] − 2(τ · RS ) τ · [Q × Rij ] . 2

(3.16)

(3.17)

Example 3: Tilt/twist distortions from regions with multiple slip. In regions with GN dislocations activated in multiple slip systems inducing twist and tilt, deformation of the lattice depends both on screw and edge portion of GNDs. The GN related part of the correlation function in orientation space may be written as following:

1 Ti = −i n+ |bm | RS · [Q × Rij ] − 2(τ m · RS ) τ m · [Q × Rij ] 2 m m + n+ (R · b ) [Q × R ]τ S m ij m . m

(3.18)

m

Here m is a summation index.

3.5. Connection between strain-gradient plasticity and transverse intensity Plastic strain gradients appear either because of loading geometry or because of materials inhomogeneity. They cause gradients in the spatial arrangement and density of GN dislocations, and/or GN boundaries. Different models of GN dislocations, GNDs, can describe the plastic strain gradient. A variation in GND density alters the gradient of local orientation with depth and can redistribute the intensity around each Bragg reflection. Depending on the ratio between the absorption coefficient, the scattering geometry and the gradient of GND density with depth, the maximum of the streak intensity can be displaced. Different slip systems cause distinctly different Laue-pattern streaking. Experimental patterns are therefore sensitive to statistically stored dislocations and GNDs.

§3.5


525

In the mechanism-based strain-gradient plasticity (MSG) framework, plastic deformation is described [48,49,54,55] by a strain gradient tensor of the third rank ηlmk with effective strain gradient η=

c1 ηiik ηjj k + c2 ηij k ηij k + c3 ηij k ηkj i ,

(3.19)

and constants c1 , c2 , c3 . ηlmk is the second gradient of displacements ηlmk = ∂ 2 uk /∂xl ∂xm . The strain gradient tensor ηlmk may be written in terms of the anti-symmetric Levi-Civita tensor of third rank εilm and a dislocation density tensor of second rank ρik [76]: εilm ηlmk = −ρik = −τi bk δ(r).

(3.20)

Here uk , the k-th component of the displacement field u for any unit cell, is due to all dislocations in the crystal. The displacement R0i is due to all dislocations and is defined using random numbers ct by eq. (2.3). The Cartesian reference frame is set such that the x1 X-axis coincides with the direction of Burgers vector b and x3 Z coincides with the direction of dislocation lines of the dislocation net. With this coordinate system the crystal will be bent in the plane XOZ due to the net of GNDs. In this Cartesian reference frame the only nonzero components of the dislocation density tensor is ρ31 resulting in the nonzero components of the strain gradient η112 = −K and η211 = K, with effective strain gradient η = K = n+ b. Plain-strain bending of a crystal of curvature K usually is modelled by a network of randomly distributed geometrically necessary dislocations (GNDs) with the density n+ and Burgers vector b (Fig. 13). The curvature K = n+ b coincides with the above effective strain gradient, η. This enables us to write η in terms of the density of GNDs, η = n+ b. The average distance between randomly distributed GNDs at yielding, Ls , is related to the “material length” l in the MSG theory [48,49,55].

Fig. 13. Modelling of plain strain bending with random GNDs (a) and with GNBs (b).

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Ch. 79

The nonzero components of the strain gradient tensor ηij k can be modelled to fit experimental intensity distributions in reciprocal space and used to find the dislocation density tensor ρik from eq. (3.19). The dislocation density tensor ρik gives the sum of the Burgers vectors of the dislocations whose Burgers vector and line direction are directed parallel to xk and xi axes (k, i = 1, 2, 3), respectively. These are typically set as an initial input parameter for simulation of the experimental diffraction pattern. By simulating patterns with different GND slip systems, the set of predominant slip systems can be determined that are the best fit to the corresponding strain gradient tensor ηij k and experimental diffraction pattern. Typically models of the strain gradient structures with constant effective strain gradient are considered. In real crystals the effective strain gradient as well as the GNDs density may vary in space. Therefore we expand this consideration and include the case of varying effective strain gradients η (GNBs density, n+ ) distribution. In Section 4 polychromatic X-ray microdiffraction (PXM) is considered with respect to the following three models of spatial GNDs distribution: (a) constant, (b) linear and (c) exponential gradients of GNBs density (Fig. 22(a)). We note the following relationship for the three models: (1) With a uniform plastic deformation related to uniform density n+ 0 of geometrically + necessary boundaries (GNBs) formed by “thin” tilt GNBs (DDWs): ∂n ∂x = 0; (2) With a liner decay of GNBs density with depth: n+ (x) = n+ 0 (1 − βx),

∂n+ /∂x = −n+ 0 β;

(3) With an exponential decay of GNBs density with depth −γ x ; n+ (x) = n+ 0e

−γ x . ∂n/∂x = −γ n+ 0e

Here β and γ are the decay coefficients of the GNBs density (or effective strain gradient) along the beam path in the sample. In addition we later discuss the influence of absorption on profile shape. White beam intensity distributions are calculated and discussed for these three models in Section 4.10. 3.6. Diffraction by crystals with dislocation boundaries Here we expand our description of scattering to include the influence of dislocation boundaries on reciprocal space. We consider the characteristics of diffraction for the following simple models: (1) Noncorrelated incidental dislocation boundaries (IDBs): (a) IDB-1s with zero disorientation angle through the boundary, (b) IDB-2s with nonzero disorientation angle through the boundary, (c) noncorrelated IDBs, (d) correlated IDBs. (2) Dense dislocation walls or geometrically necessary boundaries (GNBs) formed by tilt “thin” equidistant dislocations walls with or without correlation. (3) Multiscale dislocation structures with GNBs separating regions with IDBs.

§3.6


527

For crystals with dislocation boundaries the diffracted intensity in reciprocal space depends on the details of dislocation distribution and morphology of the wall and is different for IDBs and GNBs. The intensity distribution depends on the correlation between the boundary spacing D, the spacing between dislocations within the wall h, and the morphological parameters of boundaries (Fig. 7). IDB-1s with n± ≪ n (very small misorientation) The stochastic formation of IDBs was described by Pantleon [31,32], and Hansen [30]. Glide dislocations trap each other statistically and form thick loose cell walls with randomly distributed dislocations within the wall and with almost equal numbers of “+” and “−” dislocations within the cell wall. These IDB trap gliding dislocation with some capturing probability P = Di /λ¯ determined by the IDB-1s spacing Di and the mean free pass ¯ of the mobile dislocation λ. The character of stress and strain fields surrounding IDBs-1 determines the style of the diffracted intensity. The central part of intensity distribution for IDBs-1 with almost equal numbers of “+” and “−” dislocations around any reciprocal lattice point is described by a Gaussian function both in the transverse and radial direction. If there is only one type of edge dislocations parallel to the z-axis grouped into walls with both “+” and “−” dislocations, then the intensity distribution near a reciprocal lattice point has a disk shape in the plane perpendicular to the z-axis. Along the z-axis the intensity distribution does not change compared to the nondeformed crystal. The full width at half maximum (FWHM) in the XOY plane depends on the orientation of the Burgers vector relative √ to diffraction vector √ and square root of total dislocation density FWHM IDB1 = (Qb)γ 0.5nl. Here l ≈ ln( nlL), and L is the effective size of a coherently (or semicoherently) scattering fragment. Compared to randomly distributed statistically stored dislocations (SSDs), the intensity distribution caused by IDBs-1 with very small misorientations is contracted by a factor √ γ = (l − 0.5 ln(Di / h)/ l) depending on the ratio Di / h. The FWHM of the intensity distribution depends on the total density of dislocations, ni , grouped within the IDB-1 wall. In the strained crystal IDB-1s are spread out to a varying extent owing to a possible spread in the positions of dislocation lines in the region adjacent to the boundary of a finite thickness (boundary width, W ) and owing to the fact that ratio h/D is finite. For IDB-1s the width of the wall W is larger then the characteristic distance between dislocations within the wall, h, so that the condition W > h is valid (Fig. 6). This is also in agreement with the model of Mughrabi [14] and Ungar et al. [28]. This relatively large width causes additional broadening at the tails of the intensity distribution. Usually the contribution of the transition regions adjacent to the wall is significant if the distance between the boundaries D and boundary width W are of the same order and D/ h 5 (this is also in agreement with Wilkens [57,58]). IDB-2s (with misorientations n± ≪ n) When mobile dislocations approach the dislocation wall they are partially trapped by the wall with an immobilization probability P . The GN portion of the dislocation population within IDB-2s causes misorientation between the adjacent cells. The misorientation angle through the IDB-2 according to Read and Shockley [77] is equal α = bDn+ . A pure sto-

528

Ch. 79


chastic formation of dislocations leads to a square root dependence of the average modulus of misorientations angle through an IDB on strain, γ [32]: |α| =

2bP γ. πDi

(3.21)

Due to statistical fluctuations and activation imbalance with time t (or equivalently with plastic strain, γ = γ˙ t) the disorientation angle through the boundary increases according to the equation, dα = bP j + bŴ. dt → −

← −

(3.22) − →

← −

Here j = j − j is the bias of dislocation fluxes j or j passing through the boundary from both sides. The activation parameter −1 < Ŵ < 1 describes how the slip systems on both sides of the boundary share the prescribed deformation. After a certain time, the densities of “+” and “−” dislocations within IDB-2s become significantly different, and the condition, n+ ∼ n becomes valid. These walls cause a radical change in the reciprocal space intensity distribution. The distribution still has relatively narrow Gaussian intensity distribution along the momentum transfer (radial direction) and a much broader intensity distribution in transverse direction. This change in intensity may be used as an indication of the formation of IDBs-2s. According to Saada and Bouchaud [78] even for an infinite dislocation wall with parallel identical dislocations any small uncorrelated disorder prevents cancellation of the periodic elastic field, thus allowing long range interactions to reappear as opposed to the case of a perfect equidistant dislocation wall (insert in Fig. 7(b)). Noncorrelated IDBs Diffraction by noncorrelated IDBs results in a broad intensity distribution in the rocking direction [29]. The transverse intensity distribution I (q)ξ of the rocking curve along the main axis ξ in the transverse plane (perpendicular to the reciprocal lattice momentum transfer, Ghkl ) has a shape close to a Gaussian function. The full-width at half-maximum of the rocking curve (FWHM rock ) is proportional to the square root of the number N of the boundaries within coherently (or semi coherently) scattering cell blocks of the effective size L:

1 πL L 1 FWHM rock = 2 A0 αQ, . (3.23) dR0s √ = Di V Aξ ξ Rs0 A0 Here A0 is the rocking contrast factor for the IDB. It depends on the mutual orientation between the diffraction vector, Q, rocking axes and orientation and type of the IDB (boundary plane and misorientation angle and rotation axis through the IDB). If disorientations through different IDBs and their positions are totally independent of each other, the disorientation angle across N boundaries increases proportional to the square root of the number

§3.6


529

Fig. 14. Increase of the misorientation angle (decrease of the distance between dislocations within the boundary) through the boundary (a), (b) and decrease of the distance (D2 < D1 ) between the boundaries (c), (d) broadens intensity distribution in the orientation space (e), (f).

√ N of the IDBs taking part in diffraction: |N | = |θ | N . If only one or two different types of IDBs are formed in the diffracting area the orientation space distribution might be anisotropic even in reciprocal space due to the orientation dependence of the contrast factor. It has a “long axis” direction and a “narrow axis” direction. This gives the possibility of finding the primary orientation of IDBs being activated in the area. The shape and size of the reciprocal space maps (RSM) strongly depend on two length scale parameters: (1) the distance between the walls, D, and (2) the distance between dislocations within the wall, h (Fig. 14). A decrease of the distance between dislocations within the boundary causes an increase of the misorientation angle through the boundary and broadens the intensity distribution of the RSM. A decrease of the distance (D2 < D1 ) between the boundaries also broadens the RSM. Correlated IDBs The disorientation angles in neighboring IDBs are usually correlated. Dislocations of the opposite Burgers vector corresponding to the same dislocation loop have a finite separation around twice the mean free path of the mobile dislocation 2λ. In the absence of a macroscopic plastic strain gradient, disorientations cannot be cumulative over several boundaries

530


Ch. 79

and the disorientation across N boundaries will level-off after an initial increase. The saturation value for the disorientation angle depends on the strain γ as following [31,32]: |α∞ | =

4bγ . πD

(3.24)

For such correlated paired arrangements of IDBs the rotation field of a single IDB is screened by surrounding IDBs within a distance dependent on the correlation length for IDBs to the saturation level. Unlike uncorrelated IDBs, the rocking curve FWHM rock depends on the size of the beam and saturates at the value:

FWHM rock = 2 A0

πrcor αQ. Di

(3.25)

Here rcor is the correlation length for IDBs. According to Pantleon [31,32], rcor is equal to several cells diameters and is usually much less than the size of the beam. Saturation of the rocking curve FWHM rock is observed experimentally for a deformed copper single crystal [37]. A detailed analysis of this experimental dependence allows for the determination of the correlation radius for IDBs. With coherent synchrotron microbeam measurements, the experimental determination of the ratio γ /Di is now possible. However due to the rocking contrast factor some boundaries may give very small contribution to the rocking curve FWHM and broadening. This is why multiple reflections should be analysed to determine the contrast factor and the orientation of the activated IDB. Diffraction by paired GNBs with/without correlation The elastic energy of deformed crystal with IDBs can be further decreased when dislocations inside the boundary rearrange. Dislocations of opposite sign annihilate and the distance between the remaining unpaired dislocations becomes nearly constant along the boundary (equidistant tilt walls). Such a wall forms a very stable configuration. They become much narrower then IDBs. Their formation indicates the next step in the deformation process and is accompanied by certain changes in the diffracted intensity. We call this boundary type “geometrically necessary” boundaries (GNBs). Consider cell blocks with different orientations in deformed material separated by GNBs, formed by thin tilt dislocation walls (Fig. 7(a)). Each tilt GNB provides a rotation between two neighboring dislocation cells around the direction of dislocation lines within the wall. The unit vector ω parallel to the rotation axis of each wall coincides with the unit vector τ along the dislocation lines in the case of a tilt boundary. We consider pure tilt GNBs formed by equidistant edge dislocations (so called “thin walls”). Such boundaries do not produce long-range strain, only rotations. That is an important difference between the IDBs and GNBs. As illustrated in Fig. 7, thin dislocation walls provide a rotation between two neighboring cell blocks around the direction of dislocation lines within the wall. The unit vector ω is parallel to the rotation axis of each wall and coincides with the unit vector τ along the dislocation lines in the case of a “thin” tilt wall formed by equidistant edge dislocations [29,79]. Long-range internal strains from individual dislocations within infinite “thin” wall shield each other (Fig. 7(b)). As a result “thin” walls do not produce long-range strain, only

§3.6


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rotations. The misorientation angle due to such a boundary is defined by the equation b/ h = 2 sin(/2), where b/ h ≈ for small angle boundary. If h is the distance between these dislocations in the wall, one can consider the boundary as a single defect producing a local rotation field. Walls made of a finite number of parallel dislocations according to [78] cause some long range interaction in the cell interior to reappear (compare an insert in Fig. 7(b)). However with the increase of the length of the walls this input becomes smaller and their displacement fields can be treated as infinite wall to a first approximation. We define the average size of the deformation cell D between the two nearest GNBs and write the linear density of GNBs as the number of walls per unit length, 1/D. The total den1 . We use a Cartesity of GN dislocations grouped in the thin GNB is denoted by n+ = Dh sian coordinate system with axis z parallel to the direction of dislocation lines within the wall and axis x perpendicular to the plane of GNB. For this coordinate system (Fig. 7) two non-zero components of the mean deformation tensor are equal: ωxy = −ωyx = x/D. This means that the deformation tensor again results in pure rotations about the z-axes. The rotation increases with displacement x. The number of GNBs per unit length and the length of the crystal in the beam direction determines the total measured rotation. If the planes of the GNBs are distributed randomly, the intensity distribution and correlation function in general are described by eqs (3.1) and (3.2). We emphasize that the influence of the dislocation walls on the scattering distribution is only characterized by the function T . Parameter c in eq. (3.2) is the concentration of GNBs, ρ = R0s − R0s ′ is the vector between two scattering cells s, s ′ . The relative displacements uss ′ between two cells due to the local rotation caused by the sub-boundaries (Fig. 7(b)) is then given by uss ′ rα = usrα − us ′ rα = csr [ωα , Rs − Rr ] − cs ′ r [ωα , Rs ′ − Rr ] ∼ α.

(3.26)

There are two different possible mutual orientations of the scattering cells: in the first case, both cells are on one side of the wall (cells s1, s2), while in the second case they are on different sides of the wall (cells s1, s3) as shown in Fig. 7(b). The scalar product Q · u is then given by Quss ′ r = −

b Q[ω · ρ] , h

(3.27)

if s1, s2 cells are on one side of the wall, or Quss ′ r = −

b Q ω · (Rs − Rs ′ − 2Rr ) , h

(3.28)

if s1, s3 cells are on the opposite side of the wall. In the general case, T R0s , ρ = Re T R0s , ρ + i Im T R0s , ρ .

For equal probabilities of boundaries with opposite rotation angles, Im T = 0 and |ρ · nα | R0s nα T (ρ) = + 1 − cos Q(ωα × ρ)α . Dα Dα α α

(3.29)

(3.30)

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Where Dα is the distance between the α-type walls. Such an expression for correlation function T results in a strong asymmetry of the intensity distribution I (q) in the vicinity of the reciprocal lattice point Ghkl , where hkl are conventional Miller indices in the tetragonal notation. In crystals with paired noncorrelated GNBs the intensity distribution is close to a Lorentzian shape in the radial direction I (qD ) = IDi

qDi , 2 + qDi

(3.31)

2 π 2 qD

where qD is the deviation of the diffraction vector from the exact Bragg position in the radial direction. Here the integral line width, as commonly used, is given by qDi = πς/D, (or in angular units by 2δθ = ςλ sec θ/D), where D is the average distance between walls as in Fig. 7(a). The contrast factor in the radial direction depends on the kind of GNBs and momentum transfer. It can be written as (m · nα ). (3.32) ς= α

The factor ς is determined by both the orientations of the wall and by the diffraction vector Q; m = Q/|Q|. Such dependence for radial line broadening corresponds to classical “particle-size broadening”. The transverse intensity distribution I (qξ ) corresponding, to rocking curve measurements has a dependence along the main axis ξ of the transverse plane, described by I (qξ ) =

2πf 2 ν 2 Q2 ζ

dR0s

R0s

q2ξ D D , exp − 0 2 4Rs Q Aξ ξ Aξ ξ Aηη

(3.33)

where ν is the volume of the unit cell, and f the scattering amplitude. The matrix Aηξ depends on the mutual orientations between the diffraction vector Q and the directions τ α of the dislocation lines inside the walls in the transverse plane (perpendicular to the diffraction vector). Aηξ may thus be viewed as a transverse orientation factor. The integral width in the transverse plane is then given by δq⊥ =

2πAξ ξ

L Q. D

(3.34)

In the simple case of two different sets of dislocation walls with equal Burgers vector in the crystal, the scattering contours around the points Ghkl will have the shape of thin disks considerably more extended in the transverse plane than in the radial direction. GNBs result in broad intensity distribution in the plane transverse to diffraction vector. The shape of the spot in the orientation space depends on the average orientation of the dislocation arrays and the diffraction vectors. The intensity distribution due to thin dense GNBs formed by equidistant tilt walls is distinct from the case of thick loose IDBs with

§3.7


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fluctuating distance between dislocations within the wall. In the rocking (transverse) direction the intensity distribution is close to a Gaussian shape and FWHM rock depends on the relative direction of rocking axes, diffraction vector and parameters of dislocation arrangement within the boundary.

3.7. Diffraction by multiscale structure with both GNDs/GNBs density A number of experimental studies of plastic deformation under different loading conditions show that the cell-wall structure with alternating regions of high and low dislocations density are typically formed [80,81]. The dislocation walls contain high density of dislocations, and cell interiors have very low dislocation population. The formation of cell walls can be detected in the diffraction pattern by discontinuities [68,79]. Generally, real crystals contain a hierarchy of dislocation structures. Some of the GNDs are distributed randomly, and the rest may form different non-random arrangements: incidental dislocation boundaries (IDBs) and geometrically necessary boundaries (GNBs). Dislocation boundaries evolve within a regular pattern of grain subdivision on two scales. The smaller scale is related to incidental cell boundaries – IDBs. The larger scale is related to so called, geometrically necessary dislocation boundaries (GNBs). Usually GNBs separate volume elements which deform by different slip system modes with different strain amplitudes [30–32,36,82]. Typically there are many IDBs separating ordinary dislocation cells between two cell-block boundaries formed by GNBs. The presence of boundaries splits the intensity distribution in orientation space of the crystal. We summarize the discussion above as follows: • Dislocations change the condition for X-ray or neutron diffraction and cause asymmetry in the reciprocal space of the crystals. • Statistically stored dislocations can be determined from the radial intensity distribution. • Dislocation groupings within the boundaries influences the character of the internal strain fields surrounding the boundary. • Two main boundary archetypes IDBs and GNBs are distinguished with respect to their origin, morphology and strain fields. • Structural transitions from IDBs to GNBs are accompanied by a change in the intensity profile both in the radial and transverse directions. • The correlation between boundaries results in the saturation of the functional dependence of the intensity distribution on the size of probed region. • Orientation space is sensitive to strain gradients and to the GN portion of dislocation population within the probed region. • Length scale parameters of the dislocation arrangement can be probed by diffraction. 3.8. Measuring reciprocal-space intensity distributions As described above, dislocations redistribute scattering intensity in reciprocal space, and intensity maps in reciprocal space can be used to characterize dislocation density and distribution. Standard monochromatic-beam methods exist for measuring the reciprocal-space

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intensity distribution of large isotropic crystals. Typically these require the sample to be rotated while the detector angle relative to the incident beam is also adjusted. Area detectors can remove the need for detector motions, but as deformation inhomogeneities are of particular interest, the need to rotate the sample to map reciprocal space makes it very difficult to probe small sample volumes. This problem gets progressively worse as the desired volume resolution increases. As we describe later, energy scanning methods can provide reciprocal space maps without sample rotations, and hold the promise for powerful analysis of defects in very small sample volumes. In addition, the future availability of area detectors with good energy resolution will allow for the collection of 3D reciprocal space volumes from small sample volumes. These methods however are still evolving and are based fundamentally on extensions of polychromatic (white-beam) Laue microdiffraction tools already deployed. Below we describe the motivation for and application of polychromatic microdiffraction for the study of dislocations in materials.

4. White beam analysis of the orientation space Polychromatic (white-beam) microdiffraction (PXM) is a rapid method for determining the local crystallographic orientation of small sample volumes. As a characterization tool of crystal structure it has several major advantages: (1) the sample does not need to be rotated which allows rapid analysis of polycrystalline materials; (2) the intensity distributions around a large number of different reflections can be simultaneously analyzed from the same subgrain; (3) one can measure an uncompromised full three-dimensional reciprocal space intensity distribution when necessary by scanning the incident beam energy, (4) the scattering from 3D spatially-resolved volumes in the sample can be measured using a technique called differential aperture X-ray microscopy (DAXM) and (5) PXM together with DAXM allows the characterization of local boundary orientation and 3D interior surface morphology. Because 3D surface morphologies can be determined, errors described by Winther et al. [83] that accrue from 2D EBSD analysis are avoided.

4.1. Laue microdiffraction and related methods The actual process of measuring local structure with polychromatic microdiffraction is outlined below. A broad-bandpass microbeam intercepts a sample and illuminates a small number of crystal grains. The overlapping Laue patterns from the illuminated grains and subgrain volumes are recorded by an X-ray sensitive area detector and fit to find the centers of the reflections. The angles between the reflections are compared to the angles anticipated for the crystal structure of the sample. When the n(n − 1)/2 angles between n reflections have angles that match those predicted for a particular set of reflections, the reflections are assumed to be indexed. Owing to the exquisite angular sensitivity of the measurements, once 4 or more reflections are indexed the probability for a false indexation is small. Based on the diffraction directions of the indexed reflections, the texture (orientation) and relative or absolute strain tensors are determined for the illuminated grains.

§4.1


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Fig. 15. Laue patterns from: (a) single Ni grain region in a Roll Assisted Biaxial-Textured Substrate (RABiTS) sample. The Laue pattern includes diffraction from the Ni substrate, from a buffer layer and from a high Tc superconductor layer. In this sample only the Nickel peaks are sharp indicating a local mosaic spread to the layers above the substrate; (b) GaN layers grown by cantilever epitaxy on Si substrate with streaked GaN Laue spots and more intensive and round Si Laue spots.

Because polychromatic microdiffraction produces Laue images from each subgrain intercepted by the X-ray beam, instead of too few reflections associated with monochromatic microdiffraction, the typical problem is an overlap of many Laue patterns that must be disentangled. The process of disentangling can be carried out either by the direct experimental use of differential aperture X-ray microscopy [61] or by pattern matching/absorption sensitive methods [84,85]. For example, the use of small X-ray beams (∼500 nm) immediately restricts the sample volume that can scatter X-rays into the CCD. The sample-volume, position is adjusted by moving the sample under the fixed X-ray beam. In some cases, (e.g., highly absorbing materials) the scattering volume is further limited to a near-surface region. In other cases (e.g., layered materials) the Laue patterns are distinct for different layers so the 3D position of each pattern can be inferred from the layer order. For example, as shown in Fig. 15, the overlapping Laue patterns from (a) layers in a roll assisted bi-axially textured substrate (RABiTS) [86] sample are very distinct with sharp reflections from the underlying Ni grains and more diffuse reflections from the buffer layer and high Tc films; (b) GaN layers grown by cantilever epitaxy on Si substrate with streaked GaN Laue spots and more intensive Si Laue spots (see Section 6.1). Measurements in three dimensions (3D) – differential aperture microscopy Although a number of possible methods exist to extract 3D structural information, differential aperture microscopy is a uniquely powerful approach that allows fully automated reconstruction of the subgrain Laue patterns generated along the incident beam path [61]. Differential aperture microscopy is illustrated schematically in Fig. 16(a)–(g). A smooth absorbing wire translates parallel to the sample surface. If the wire is moved a small in-

536 G.E. Ice and R.I. Barabash Fig. 16. (a) As the polychromatic X-ray beam penetrates the sample, it creates a Laue image from each subgrain it intercepts. The overlapping Laue images are recorded by an X-ray sensitive CCD. As a wire passes near the sample, it intercepts rays from the sample. The difference between images before and after a small (differential) move is due to rays that pass either near the leading or trailing edge of the wire. (b)–(g) Series of partially shadowed images taken at different wire positions. By triangulation the origin of the rays are determined on a pixel-by-pixel basis.

Ch. 79

§4.1


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Fig. 17. Total Laue pattern of the friction stir processed Ti and series of reconstructed depth resolved images taken with the step of 1 µm.

crement near the sample surface, the change in the Laue pattern arises from X-rays that pass near the leading or trailing edge of the wire. Since the position of each CCD pixel can be accurately calibrated, and since the position of the wire can similarly be accurately calibrated, the intensity changes in each pixel can be connected with the edge of the wire when the intensity changed, and extended to the incident beam intercept. By following the intensity changes in each CCD pixel, one can construct the subgrain Laue patterns for each volume element probed by the incident X-ray beam (Fig. 17). This reconstruction can be carried out automatically and greatly simplifies the problem of indexing the subgrain Laue patterns and determining the orientation and strain maps. By using DAXM, one can construct a three-dimensional map of the local crystallographic orientation and elastic strain tensor distributions in a sample. The three dimensional orientation distributions can be used to directly determine the dislocation tensor. This technique provides unique solutions for the local dislocation tensor, but is very time consuming, due the need to collect many X-ray images. The process of collecting differential aperture data can be accelerated by the use of multiple wires or by the use of coded apertures; coded aperture methods are an established tool for astronomical instruments. The extension of coded aperture methods to polychromatic microdiffraction and other tomography applications is an emerging area of active research. Nevertheless DAXM mea-

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surements of dislocation-tensor distribution are certain to remain relatively slow even in the most optimistic scenarios. An alternative method for characterizing the dislocation structure of small crystal volumes is to model the influence of defects on the near Ghkl region as outlined in Section 3. Here the probed volume is assumed to be composed of single-crystal grains with defects that distort and smear the Laue reflections. This method is orders of magnitude faster as a single Laue image provides information about the dislocation tensor in the sample volume probed by the penetrating X-ray beam. This method is also essential when the dislocation density is so high that the Laue spots are significantly smeared even from a DAXM resolved volume, or when a significant gradient in the density distribution exists. Indeed although differential aperture microscopy methods can be used to subdivide the crystallographic structure into small volumes, volume-average information is insufficient in cases where: • the length probed by the beam is small due to small grain size, thin film architecture or some other inhomogeneity/discontinuity, • depth probed by the beam is small in the materials with high absorption factor (Ir, Pt, Ta) • The plastic response of the material creates large strain gradient even within 1 µm and Laue spots are streaked even in the depth-resolved images obtained with DAXM (Fig. 17). Here streak-profile analysis can be used for characterization of the local dislocation population. An approach to characterize the dislocation structure and determine the activated slip systems by analyzing the streaked pattern by means of a multiscale hierarchical framework is described in detail in [7,74,88–90]. Below we outline the theoretical and experimental methods employed to interpret smeared Laue reflections. This interpretation is based on radial projections of the reciprocal-space distributions of the models outlined in Section 3.

4.2. Misorientation vector in Laue diffraction In Laue diffraction, the incident beam and scattered beam directions defines a line in reciprocal space [60,61,91]. The radial position along this line is determined by the wavelength of the scattered radiation. For example reciprocal lattice points (00h), (002h), (003h) etc. are scattered towards the same pixel on a CCD but lie at different positions radially in reciprocal space. An energy change in the incident radiation causes a change in the Ewald sphere diameter (Fig. 18(a)). To analyze the white beam intensity distribution from a deformed grain, we introduce unit vectors in each direction of scattering, kˆ = k/|k|. We define a special misorientation vector m near a Bragg reflection m = kˆ − kˆ hkl [73,74,92]. The misorientation vector m gives the difference between the unit vector parallel to Bragg reflection (hkl) and an arbitrary direction in its vicinity (Fig. 18(b)). Note that this is a crucial difference between the general intensity distribution in reciprocal space and the intensity distribution of a scattered polychromatic (white) beam. For example with respect to the (1,0,0) Bragg reflection, the distinct reciprocal space points (1,0,0.1) and (2,0,0.2) have the same m.

§4.3


539

The region of high intensity around each vector kˆ hkl in Laue diffraction (Fig. 18(a)) IL (m) is a function of a misorientation vector. Within this approximation, the resulting intensity distribution by the deformed crystal in the white microbeam method can be written as follows:

IL (m) ∼ A I0 (k)I (q) dk; q = |khkl |m⊥ + |khkl |mrad + kG/|khkl | . (4.1) = Here khkl is the radius of the Ewald sphere that passes through Ghkl , A is a constant, and mrad and m⊥ are components of the misorientation vector m along and perpendicular to the direction of the scattered beam kˆ hkl , respectively. The bandpass of the incident radiation defines a volume of reciprocal space that is bounded by the short and long wavelength limits of Ewald’s sphere (Fig. 18(a)). The misorientation vector m characterizes the Laue intensity IL (m) assuming the spectral distribution varies smoothly near the energy cork responding to the Bragg reflection. The second equation of (4.1) is valid when |k ≪ 1. 0| ˆ In the first approximation, the orientation vector m is perpendicular to khkl (Fig. 18(b)). Microfocusing optics [91] typically introduce a small 1 mrad∗ convergence to the incident beam. This convergence angle has a negligible effect along a streak, but can change the intensity distribution in the narrow direction of the streak. IL (m) should therefore be convoluted with the experimental angular resolution function. 4.3. Representative volume element for white beam diffraction Polychromatic X-ray microdiffraction is well suited for the analysis of the GNDs distribution and strain gradient effects at a micron and submicron level and provides a convenient criterion to determine the strain gradient length scale lˆ and the mesoscale cell size lM introduced by [48]. For geometrically necessary, GN, (polar) dislocations or dislocation walls the distortion tensor, ωij , due to GN dislocations, is related to the unpaired dislocation density, n+ , and slip system (direction τ , and Burgers vector b). Interestingly, our diffraction estimates for these parameters coincide with the values given in [48]. When the length scale lM associated with the deformation field is small compared to a materials length lˆ even on the micron level, the streaking of Laue spots usually occurs. As mentioned in the Section 2.1 at the differential (microscale) volume level, all dislocations are “geometrically necessary” [17,42]. Because the assignment “geometrically necessary dislocations” implies a length scale, the representative volume element (RVE) over which the polarity is determined should be described explicitly. For white beam analysis of a single crystal, the RVE is equal to the size of the probed region (Fig. 19). The polar portion of the dislocation population within each RVE causes streaking of the Laue spots. As the probed region moves through the sample, changes in the polar portion of the dislocation content and the subsequent lattice curvature are accompanied by changes in the PXM intensity distribution. For polycrystalline material the correlation length is at most the size of the grain or subgrain. After strong deformation the correlation length is restricted by the size of slip bands separated with large angle GNBs. In this case RVEs for several slip bands or subgrains are simultaneously probed by the beam.

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Fig. 18. (a) As the wave length of the white beam spectrum changes from λmin to λmax , the Ewald’s sphere “moves” through reciprocal lattice. (b) The misorientation vector m is the vector difference between a unit vector that points toward a position in reciprocal space and the unit vector that points toward the nearest Bragg reflection.

Fig. 19. Representative volume elements for single crystals (a) and for polycrystalline (b) samples in white beam diffraction.

4.4. Natural axes of the Laue spot To describe the intensity redistribution near a Laue spot, due to a particular set of dislocations, the mutual orientation of two planes is important (Figs 11, 12): • The plane perpendicular to the direction of dislocation line τ in real space; • The plane perpendicular to the direction of reciprocal lattice vector Ghkl . The line of intersection of these two planes naturally defines the direction of the streak axis ξ : ξ = τ × Ghkl /|τ × Ghkl |.

(4.2)

§4.4


541

Fig. 20. Natural axes of the Laue spot are distinct for different GN slip systems. In both cases the Laue intensity is calculated for the (222) reflection from a crystallite with 2500 interatomic steps along the Burgers vector direction and a total dislocation density n = 1011 cm−2 with √ √ 0.25% GN dislocations: (a) GNDs slip system with Burgers ¯ ¯ 2]/ ¯ 6 results in the streak axis running√along the ξ = [31¯ 2]/ 14; vector b = [110]/2, dislocation line τ = [11 ¯ (b) GNDs slip system with Burgers vector b = [110]/2, dislocation √ line τ = [11¯ 2]/ 6 results in the streak axis ¯ running along the ξ = [132]/ 14.

The second axis ν is perpendicular to the ξ axis and to the reciprocal lattice vector Ghkl . With this coordinate system, the intensity of a Laue spot is strongly elongated in the ξ direction (Fig. 20(a), (b)). Eq. (4.2) indicates that different GNDs slip systems result in a distinct Laue patterns. For example we show the change of the Laue pattern for 12 primary slip systems typically activated in FCC crystals. To understand the shape of experimental Laue images and to check the sensitivity of the Laue image to different possible dislocation kinds the Laue images corresponding to 12 different slip systems of the primary GNDs were simulated. The 12 most likely edge dislocation systems contribute distinct patterns to the beam spread. Fig. 20 demonstrates how the Laue patterns change for an FCC crystal with surface normal in [001] direction. The orientation of the crystal is chosen to be close to cubic axes with x,

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Fig. 21. The orientation of GNDs influences the Laue pattern character: Simulated Laue patterns for 12 predominant GN slip systems in FCC materials are distinct for different GNDs slip systems.

y and z axes running along [100], [010], [001] directions. When the slip system changes, all Laue streaks rotate simultaneously. Some typical peculiarities are observed: • The length of the streak (full width at half maximum in the ξ direction) FWHM ξ ≡ δmξ depends on the orientation and the number of GNBs in the probed volume. The orientation of the GNDs influences the character of Laue spot (Figs 20, 21). • Along certain zone lines Laue streaks for different (hkl) are almost parallel to each other (as the angle is practically constant along this line). • Along other zone lines the Laue streaks are slightly inclined relative to each other. This is due to the fact that the angle between the slip plane normal and the momentum transfer Ghkl gradually changes along the zone lines with hkl changing along 0h6. • For certain hkl no streaks are observed (or become very small). This is the case when the contrast factor (see next paragraph) is zero. • In the transverse direction ν, the FWHM ν ≡ δmν depends on the total number of all boundaries (GNBs and IDBs) per unit length 1/D and usually δmξ ≫ δmν (Table 3). • It is emphasized that in white beam diffraction the FWHM ≡ δm is a function of the misorientation vector m (rather than the reciprocal space vector in the conventional diffraction).

§4.5


543

4.5. Contrast factor for Laue diffraction The characteristic dimensions along the ξ and ν directions, FWHM ξ and FWHM ν depend primarily on two parameters, σξ and σν , which in turn depend on the diffraction geometry and number of GN and total dislocations. For a single GN slip system operating within the probed region, σξ and σν are given by: +

σξ = n bGhkl L 1 − (τ Ghkl /Ghkl );

bGhkl σν = 8(1 − ν)

nl . π

(4.3)

Here n is the total and n+ the GN dislocation density. Two extreme cases are important: • If σξ ≫ σν , then Laue spot turns to a streak and length of the streak, FWHM ξ ≡ δmξ ∼ σξ /|k0 | gives direct information about the GN portion of dislocation population in the probed region. In the narrow direction of the streak FWHM ν ≡ δmν ∼ σν /|k0 |. It gives information about the total dislocation density. • If σξ ≪ σν , the shape of the Laue spot becomes almost isotropic even for a single GND slip system (no streaks). In this case GNDs density is small enough and dislocation population mainly consists of SSD (the case of multiple slip is considered below). Here in any direction the Laue spot width FWHM ≡ δm ∼ σν /|k0 | and total dislocation density can be evaluated from the last expression. Under certain deformation conditions multiple slip occurs. For example, a deformation of copper single crystals with 001 and 111 load orientation involves multiple slip [14, 28,33]. When different dislocation systems are simultaneously activated, the correlation function (3.7) is a sum over all activated slip systems λ. The individual types of GNDs are identified by their Burgers vector b and the dislocation line direction, τ . Under multiple slip the correlation function T depends on the dislocation density tensor ρij and effective strain gradient tensor ηlmk . The element of dislocation density tensor can be written as: ρik = τi bk δ(r).

(4.4)

The magnitude of the vector τ , is the net number of dislocations having Burgers vector, b, crossing unit area normal to τ [93] (Fig. 5(b), (c)). Following the approach [48,49] in the framework of strain gradient plasticity, the total GNDs and GNBs density tensor ρij relates to the strain gradient tensor ηlmk , εilm ηlmk = −ρik .

(4.5)

Here ετ lm is the anti-symmetric Levi-Civita tensor. Under multiple slip, intensity distribution around each reciprocal lattice point and the streak direction ξ of the Laue spot depends on the aforementioned dislocation density tensor and effective strain gradient. The analysis of the dislocation substructure for multiple slip deformation with polychromatic microdiffraction includes the following steps:

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• Determine the lattice curvature tensor components from the experimental Laue pattern at each probed location. • Simulate Laue patterns using these components. • Perform a least square fit to the streak profile and find the components of the dislocation density tensor.

4.6. Split Laue spots If the angular misorientation between GNBs is sufficiently large, the reflection becomes discontinuous. To understand this dependence, we assume that along the penetration depth L the X-ray beam intersects several cell blocks with average size Dcb . Each cell block contributes to the diffraction. The number of such contributions is L/Dcb . Each boundary produces an average misorientation . When this misorientation exceeds the average FWHM cb ≡ δmcb of the Laue image for each cell block along the ξ axis, the Laue spots split (Fig. 22). If almost all dislocation walls are unpaired, the following criterion can be used: K= . (4.6) δmcb If K < 1, the intensity distribution of white beam reflection is continuous (Fig. 22(a)). If K > 1 the white beam reflection is discontinuous and the intensity profile along the streak consists of discrete spikes (Fig. 22(c)).

4.7. Distinct Laue patterns from crystals with random GNDs and with dislocation grouping within boundaries Unpaired boundary and individual dislocations can have either similar or distinctive effects on the Laue pattern. Both organizations of dislocations result in the streaking of Laue reflections. The same orientation of GNDs and GNBs correspond to the same direction of streaking. However GNDs and GNBs cause continuous and discontinuous intensity distributions along the streak, respectively. Due to local strains, individual dislocations influence the length of the streak more than the same number of dislocations in a boundary. Moreover, the FWHM in the narrow direction of the streak, δmν , is most strongly influenced by individual dislocations and can in principle be used to separate boundary dislocations from those located inside the cell. For a better separation of unpaired boundary and individual dislocations, the white X-ray microbeam intensity should be differentiated with respect to |k0 |. This can be done by a scan of the incident X-ray energy with a monochromator. Simulations of the intensity of scattering by crystals with different numbers of cell blocks help one to understand the main features of white-beam scattering from crystals with various dislocation arrangements (Fig. 23). To simplify the interpretation of the patterns, the total number of unpaired dislocations was kept constant. A total dislocation density of n+ = 1012 cm−2 was chosen. This value is typical of highly deformed crystals. In

§4.7


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Fig. 22. Criterion K for splitting of a Laue spot is illustrated by the simulated contour maps and intensity profiles along the streak (red) and perpendicular to the streak (green). If average distance between partial Laue spots exceeds the average FWHM f r of each scattering domain, the Laue spot splits: (a) K < 1; (b) K ∼ 1; (c) K > 1.

Fig. 23. As dislocations group within walls, splits in the Laue streak can be detected. Total dislocation density of n+ = 1012 cm−2 was kept constant. The portion of boundary dislocations gradually increased: 1% (a); 50% (b); 85% (c). Intensity profile along the streak direction (ξ ) is shown at the top of each figure (red lines). Contourmaps are shown in the bottom of each figure.

the first simulation only 1% of all dislocations are grouped within nine GNBs (Fig. 23(a)). For this case the walls are not well developed and the condition h ∼ = D is true. Here each cell block still contains many randomly distributed dislocations and has a large FWHM. This results in overlapping of the spikes. As dislocations are removed from the cells and added to the dislocation walls, a correlated misorientation develops between the neighboring parts of the crystal. When the dislocation walls are well developed, so that the distance between dislocations within the wall h is much shorter than the distance between the walls

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D(h ≪ D) (corresponding to the case when 50 or 85% of all dislocations are grouped within the walls), the intensity distribution becomes discontinuous (Fig. 23(b) and (c)). Such walls produce sharp rotations of the cell blocks with an abrupt rotational phase variation. With an increasingly large number of walls, the scattering from each cell block blends together. For higher resolution measurements, a larger number of separate spots can be detected within a streak as observed experimentally. This simulation illustrates that observed splitting of a white beam reflection (hkl) into spots depends on the following parameters: density of unpaired dislocations inside the wall, misorientation angle created by the dislocation sub-boundary, number of walls in the irradiated volume, size of a cell block, size of scattering volume and experimental resolution function. Based on the above analysis we can conclude: √ • If n+ L ≫ 0.1 nl, the intensity of scattering under a white microbeam is mainly influenced by the unpaired GND and GNB (related to correlated deformations of the lattice). For example this condition is valid if L = 1 µm, n = 1011 cm−2 , n+ = n. In this case δmξ ≫ δmν . • In the opposite case δmξ ∼ δmν , and the intensity distribution is almost isotropic. Here the intensity of scattering measured in a white microbeam experiment is significantly influenced by total dislocation density in any direction in orientation space (Table 3). In this case, a detailed knowledge of intensity distribution in 3-dimensional reciprocal space is necessary to adequately understand the dislocation structure. • When the observable intensity variations are measurable along the streak (K > 1), the dislocation structure must be predominantly delimited by unpaired dislocation walls forming GNBs. • For the same number of dislocations, the δmξ can differ by up to ∼50% depending on the hierarchical arrangement of n+ .

4.8. Absorption correction Eq. (4.1) describes the intensity distribution of Laue spots when the penetration depth is relatively small compared to the characteristic length of the lattice rotations gradient along the beam; here the structure may be considered uniform along the beam. In synchrotron measurements this condition sometimes is not valid and the dislocation distribution along the beam changes significantly. For large penetration depth , it becomes important to take into account attenuation of the intensity along the beam path both in and out of the crystal: I = I0 e−μx .

(4.7)

Here μ is the linear absorption coefficient, I is the transmitted intensity through a length x, and I0 is the initial intensity. To reveal the most essential features, we first describe the intensity distribution of an ideal crystal by delta-functions (neglecting the finite size of the crystal and other defects). The Laue image of a reference crystal before deformation consists of δ-functions in the symmetry positions of the reciprocal space of the crystal. The measured Laue intensity at each point of a Laue pattern is an integral of I (q) over

§4.8

Table 3 Intensity distribution at the Laue pattern Shape of the Laue spot

Narrow direction FWHM ν = δmν & Streak direction FWHM ξ = δmξ

Individual, randomly distributed, paired SSDs (Sections 2.3, 3.3)

Almost isotropic

Continuous Gaussian δmν ∝

Individual, randomly distributed, unpaired GNDs (Sections 2.4; 3.4)

Streak

Continuous Gaussian δmν ∝

√ n

√ n

Continuous Gaussian √ δmξ = δmν ∝ n

Continuous “flat-top”; δmξ ∝ Ln+


Dislocation arrangement

547

548

Table 3 (Continued) Shape of the Laue spot


IDB-1s (Sections 2.6, 3.6)

Almost isotropic

Continuous √ Gaussian δm ν ∝ n × (l − 0.5 ln(Di / h))/ l.

Continuous Gaussian √ δmξ = δqν ∝ n


Continuous Gaussian √ √ δmξ = 2 π A0 L/D(b/ h)

IDB-2s (Sections 2.6, 3.6) Not correlated



Ch. 79

§4.8


Shape of the Laue spot

Paired GNBs (Sections 2.6; 3.6)

broad in all directions spiky spot


Continuous Gaussian √ √ δmξ = 2 π A0 rcor /Dα

Spiky Lorenzian δmν ∝ 1/D

Spiky Lorenzian δmν ≈ δmξ ∝ 1/D


IDB-2s (Sections 2.6, 3.6) Correlated


549

550

Table 3 (Continued) Shape of the Laue spot


Unpaired GNBs (Sections 2.6; 3.6)

Streak with spikes

Continuous Lorenzian δmν ∝ 1/D

“Flat-top”, discontinuous, δmξ ∝ Lθ/D; if > δmcb

Gradient of GNDs density

Anisotropic streak with maximum at one end of the streak

Close to Gaussian √ δmν ∝ n

Anisotropic intensity distribution “double peaked”



Ch. 79

§4.9


551

the energy spectrum of the incident beam. In the case of a large penetration depth we may approximately extend the integration to infinity then the diffracted Laue intensity profile maybe written I (θ ) = A

∞

0

′ δ θ − θ (x) e−μ x dx.

(4.8)

The integral eq. (4.8) takes into account the absorption correction for an ideal reference crystal without dislocations. Below we show how it can qualitatively change the Laue intensity depending on the cell–wall structure arrangements.

4.9. Laue Diffraction from the gradient cell–wall dislocation structure with decaying dislocation density Inhomogeneous cell–wall structure results in varying effective strain gradients and creates depth dependent gradients of lattice misorientations. This causes asymmetry of the Laue images collected using synchrotron X-ray microbeams. The intensity distribution along the Laue streak varies depending on the dislocation density distribution. Large cumulative misorientation within one grain was recently observed by 3DXRD [71]. Large intensity gradients were also observed in the Laue microdiffraction experiments [94]. The precise shape of each Laue spot depends formally on the distribution of local orientations within the probed grain/subgrain. The intensity also depends on the incident-beam spectral distribution near the Bragg energy of the reflection, and the distribution of elastic and plastic deformation in the grain/subgrain. The initial spectral distribution can be described by, I0 (k), where (k = |k0 |). Isotropic strain gradient distribution To illustrate the main features of the intensity distribution of a Laue streak we consider three simple models of cell–wall density gradients: a) uniform; b) linear and c) exponential. In this paragraph for simplicity it is assumed that the cell interiors are free from dislocations and free from other long-range displacement fields. First we consider diffraction by a crystal with uniform plastic deformation related to constant density n+ of geometrically + necessary boundaries (GNBs) ( ∂n ∂x = 0) that form tilt-dislocation walls in the cell–wall structure. Each wall provides a local orientation shift, (Figs 7(a), 24). The uniform distribution of GNBs creates a constant gradient of macroscopic lattice rotations (along the beam) with depth x: θ (x) = θ0 + αn+ 0 x;

∂θ/∂x = αn+ .

(4.9)

Here α = αξ is the contrast factor along the streak. The expression for the scattered intensity may be written explicitly as follows I (θ ) = A

0

∞

αx −μ′ x dx; δ θ − θ0 − + e D

552


Ch. 79

Fig. 24. (a) Scheme of white beam diffraction probing polycrystalline sample; (b) gradients of GNDs in the plastically deformed grain near the surface.

D+ μ′ D + (θ − θ0 ) . I (θ ) = A exp − α α

(4.10)

I (θ ) has a single maximal value at θ = θ0 formed by diffraction from the near-surface region and decreases exponentially with the deviation from this angle. All simulations were performed using a microscopic description of the cell–wall structure and explicit distortion fields due to certain dislocation arrangement. Simulations of the intensity distribution for uniform density of GNBs with an absorption correction are shown in Fig. 24. If we neglect the weakening of the intensity with depth (μ = 0) the structure with constant dislocation GNBs density gives the “flat-top” shape of intensity profile along the Laue streak and for a symmetric (central) spot the maximal value is in the center of the spot (Fig. 25). Absorption redistributes intensity along the streak. The intensity has a single maximum displaced to one side of the streak corresponding to the near surface region. This is in agreement with eq. (4.10). The length of the streak increases with higher initial linear density of GNBs at the surface and higher misorientation angle created by each boundary. Linear decay of dislocation density with depth Consider linear decay of GNBs density with depth. n+ (x) = n+ 0 (1 − βx),

and ∂n+ /∂x = −n+ 0 β.

(4.11)

Here n+ 0 is GNBs density near the crystal surface, β is the decay coefficient of the GNBs density along the beam path in the sample. Lattice rotations and their gradient may for this case be written θ (x) = θ0 + αn+ 0 x(1 − βx);

∂θ/∂x = αn+ 0 (1 − 2βx).

(4.12)

§4.9


553

Fig. 25. Flat-top intensity profile without absorption correction for constant GNDs density.

For a linear decay of GNBs density the intensity profile even for semi-infinite crystal has two maxima of intensity due to the interplay between the decay of GNBs density and of scattering intensity with depth. Integration similar to eq. (4.10) leads to the following expression for the intensity of the Laue streak exp −μx2 (θ ) exp −μx1 (θ ) + . I= ∂θ x1 (θ ) /∂x ∂θ x2 (θ ) /∂x

(4.13)

Those maxima correspond to the scattering from the depths:

x1,2 =

αn+ 0 ±

+ 2 (αn+ 0 ) − 4αn0 β(y − θ0 ) 2αn+ 0 β

.

(4.14)

Simulations show that the decay of GNBs density with depth redistributes the streak intensity (Fig. 26(a)–(c)). The most unusual shape of the streak is revealed when μ ∼ β or μ < β. In this case, due to the interplay of the intensity decay with depth and the gradient of dislocation density, the streak is double peaked (Fig. 27(b)) instead of the central maximum for the structure without a gradient (although the density of GNBs decreases continuously with x). The first maximum corresponds to the scattering from a near surface region and the second is close to the scattering angle of the undistorted bulk crystal. This is also in agreement with a continuum description (eqs (4.13), (4.14)). A further decrease of the μ/β ratio results in the appearance of a very strong maximum in the position close to undistorted bulk material and long weak tail (Fig. 26(c)). These shapes in Laue streaks with strong maximum in the position close to undistorted crystal are observed in a number of experiments [7,60,89,94,95]. From the comparison of the above two models we see that the decaying strain gradient causes intensity redistribution and results in the peculiar intensity profile along the streak. When strain gradient changes weakly along the beam path (Fig. 26(a)) and due to the absorption, X-rays are scattered by planes with similar orientation. In the opposite cases (Fig. 26(c)) strain gradient decays essentially during the beam path. In this way, a poly-

554

Ch. 79


Fig. 26. Intensity profile along the streak for linear decay of GNDs density with depth.

chromatic microbeam probes different characteristic near surface or inner volumes along the beam path. The picture is similar to the scheme shown at Fig. 24. Exponential decay −γ x n+ (x) = n+ , 0e

−γ x ∂n/∂x = −γ n+ , 0e

∂θ −γ x = αn+ (1 − γ x). 0 e ∂x

−γ x θ (x) = θ0 + αn+ x; (4.15) 0e

(4.16)

With an exponential decay of GNBs density with depth, the bimodal distribution can be even more pronounced. Depending on the ratio between parameters, μ/γ the scattered intensity redistributes along the streak (Fig. 27). Similar to linear gradient the exponential decay of GNBs density with depth in the interval of μ ∼ γ parameters suppresses the first maximum and the bimodal intensity distribution of the streak intensity profile is observed (Fig. 29(b)). Depending on the ratio between μ and γ parameters the intensity of the streak redistributes between those two maxima. When μ < γ the intensity along the streak again redistributes (Fig. 27(c)). Its maximal value is displaced to the opposite side of the streak (comparing with Fig. 27(a)). It is close to the position for undistorted inner region of the

§4.10


555

Fig. 27. Intensity profile along the streak for exponential decay of GNDs density with depth.

crystal. This Laue streak shape with strong maximum in the position close to undistorted crystal was observed in the number of experiments [7,60,96].

4.10. Size dependence of Laue intensity distribution To illustrate the dependence of the Laue intensity distribution on penetration depth (L) defined by the beam path through the sample (L is a number of interatomic distances), we simulated Laue contour maps for different values of L while maintaining a constant total value (L × n+ = const.) of unpaired dislocations in the whole scattering region (Fig. 28). The orientation of dislocation system in this calculation corresponds to Burgers vector par¯ 2] ¯ and the gliding plane (11¯ 1). ¯ The simulations allel to [110], dislocation line parallel to [11 show that such a dislocation system results in a streaked intensity distribution (Fig. 28) of ¯ The characthe (222) Laue spot. The axis of the streak is parallel to the direction ξ = [31¯ 2]. ter of the intensity distributions along the ξ and the ν axes is markedly different as expected (Fig. 28). The intensity distribution along the ξ axis has flat-top shape with large FWHM. Along the ν axis it is very narrow and is best described with a Gaussian function that is sensitive to total dislocation density. The FWHM along the ξ directions (Fig. 28(c)) does not change

556


Ch. 79

Fig. 28. Dependence of Laue intensity distribution on the size of scattering region L: Contour maps around (222) reflection for two different values L (a); intensity distributions along the axis ξ (1) and ν (2) (b); FWHM along ξ and ν directions with ratio FWHM ξ /FWHM ν (an insert) as a function of L (c).

with L (as total number of excess dislocation along the penetration depth remains the same, L × n+ = const.), while along the ν direction (Fig. 28(c)) it decreases (as dislocation density decreases). This means that the total dislocation density can be determined from the intensity distribution along the ν direction, while the total number of unpaired (geometrically necessary) dislocations, providing local rotation along the penetration depth of the beam, can be obtained from the intensity distribution along the ξ direction.

5. Determination of elastic strain Although Laue streaking is most often caused by the presence of Krivoglaz defects of the 2nd kind, streaking can also be induced by elastic strain gradients. For example, elastically bent beams induce streaking in the Laue patterns since the local crystallographic orientation changes along the penetrating X-ray beam. Uniformly elongated tensile specimens however will not exhibit streaking due to elastic deformation, because each unit cell is identically aligned with the other unit cells of the sample. Below we outline methods used to determine the local elastic strain and indicate strategies to resolve whether streaking is due to dislocations or due to elastic strain gradients. 5.1. Local elastic strain Single crystal diffraction directly measures the average local strain tensor of the crystal through the distortion of the lattice parameters from their (unstrained) values. Consider for example a unit cell with lattice parameters aI and α I and a Cartesian coordinate system uI

§5.1


557

Fig. 29. Cartesian coordinates attached to the real-space unit cell. Note that u1 = a1 /|a1 |; u3 = a1 × a2 /|a1 ||a2 | sin α; and u2 = a3 × a1 .

which is attached to the crystal. We adopt a notation where the vector a1 is coincident with the u1 axis, a2 is in the u1 u2 plane, and u3 is perpendicular to the u1 u2 plane (see Fig. 29). Similar notations have been adopted by previous authors [97–99]. A position in the crystal can be specified either by a vector in the Cartesian coordinates vu or by a vector with unit cell coordinates v. The transformation from unit cell coordinates to Cartesian coordinates is given by vu = Av, where A=

a1 0 0

a2 cos α3 a2 sin α3 0

a3 cos α2 −a3 sin α2 cos β1 1/b3

.

(5.1)

Here the b1′ s and βI ’s and are the reciprocal lattice parameters and angles for the unit cell respectively. A vector position in a crystal can move due to rigid body translation of the crystal, or due to distortion of the crystal lattice (strain) [100]. The average strain of a measured unit cell can be determined by comparing the measured unit cell parameters to the unit cell parameters of undistorted material. Let AMeas be the matrix which converts a measured vector v into the measured crystal Cartesian coordinates. Let A0 be the matrix for an unstrained unit cell which converts v into the Cartesian reference frame of the measured (strained) crystal. To ensure that there is no rigid body translation between the measured unit cell and the unstrained unit cell, the origin of the unit cell vectors are made coincident. For convenience, we assume a1 of the unstrained unit cell lies along the u1 axis and a2 of the unstrained unit cell to lie in the u1 u2 plane. A position vector v in the unit cell coordinates is found in the measured-crystal Cartesian coordinate reference frame at position A0 v for the unstrained case and at position

558


Ch. 79

AMeas v for the strained case. With this definition, for the matrices AMeas and A0 there is a transformation matrix T which maps from unstrained to strained vectors. AMeas = TA0 .

(5.2)

The transformation matrix can include both distortion and rotation terms and for unstrained crystals T = I. From the definition of the strain tensor, εij , the strain tensor in the measured-crystal Cartesian-coordinate reference frame is given by, εij = (Tij + Tj i )/2 = Iij .

(5.3)

The operation represented in eq. (5.3) ensures that the tensor is symmetric, εij = εj i , and removes the anti-symmetric rotation terms [100]. The strain tensor in the crystal reference frame can be converted to a laboratory reference frame or to a sample reference frame by using a rotation matrix R. Sample

εij

= Rεij R−1 .

(5.4)

The strain tensor of eq. (5.3) contains both a hydrostatic strain (dilatation) and a shear (deviatoric) strain term. With the isostatic strain, /3 defined as the mean strain component along each Cartesian axis, eq. (5.3) can be written in terms of the two components, ⎛ ⎞ ⎛ ⎞ ε11 − 3 ε12 ε13 0 0 3 εij = ⎝ ε21 ε23 ⎠ + ⎝ 0 3 0 ⎠ , ε22 − 3 0 0 3 ε31 ε32 ε33 − 3 δ D + εij , εij = εij

(5.5)

where = ε11 + ε22 + ε33 . Here the first matrix represents the deviatoric strain tensor and the second diagonal matrix represents the dilatational strain tensor. With microdiffraction measurements, it is important to retain the full (9 parameters) information contained in the crystal strain tensor and local grain orientation; the stress tensor can be determined from the strain tensor and the anisotropic single crystal elastic constants (stiffness moduli) [100]. σij = Cij kl εkl . (5.6) kl

Nonlinear least-square fitting of the peaks is typically used to determine the local crystallographic orientation and strain. 5.2. Ways to determine the full strain tensor and connection to reciprocal space mapping Although the directions of four non-planar Bragg reflections determine the average deviatoric strain tensor of a crystalline grain, determination of the absolute strain tensor requires

§5.3


559

that at least one wavelength be determined [5]. This can be done by measuring the energy of one reflection with a solid-state detector [101] with a wavelength dispersive diffracted beam analyzer [102] or with an incident beam monochromator [102–104]. Although all three methods have been used, an incident beam monochromator provides better energy resolution than existing energy dispersive detectors and provides greater mechanical stability than a diffracted beam analyzer [105]. The availability of a nondispersive monochromator similarly allows for detailed mapping of reciprocal space within the microbeam probe volume. As illustrated in Fig. 3, the Ewald’s sphere cuts through reciprocal space with a radius adjusted by the X-ray energy. Assuming an average sample orientation, the intensity distribution in reciprocal space volume is directly mapped out by plotting measured detector intensity in each radial direction with the momentum transfer determined by the scattering angle and X-ray wavelength.

5.3. Resolving plastic and elastic strain effects In general both elastic and plastic strains are present in samples. Sometimes the strain field is mainly plastic (as in a nanoindented Cu crystal [7]). Sometimes the strain field is mostly elastic as in a bent silicon crystal at room temperatures [106] (analyzed in connection with the problem of sagittal focusing of high-energy synchrotron X-rays and of neutron imaging). In fact, elastic strains can be a direct result of compliance to dislocations, as described by eq. (2.4). As described above, polychromatic X-ray microdiffraction directly measures the average elastic strain within a resolved volume through the deformed unit cell parameters [5,106]. The plastic deformation tensor can also be determined, but the connection to experiment is less direct. Typically, streaking in Laue patterns is attributed to the presence of lattice rotations as a result of unpaired or “geometrically necessary” dislocations [74,92]. However, streaking can also result from gradients in the elastic strain tensor as elegantly illustrated in an experiment by Larson et al. [61] and Yang et al. [106]. In cases where both elastic and plastic deformations are present, the average local elastic contribution must be calculated point-by-point and used to correct the lattice rotations that then are used to calculate the deformation tensor. This analysis goes beyond the scope of this summary, which is restricted to cases where elastic contributions are assumed to be negligible [106]. However, the interplay between elastic and plastic strains is particularly relevant to mesoscale deformation structure and strain-gradient plasticity theories. Assuming linear superposition of plastic and elastic strain fields we may write the total mean deformation tensor p

e . ωij = ωij + ωij

(5.7)

Elastic strain gradients results both in macroscopic lattice rotation of the crystal (deviatoric strain) and to a linear gradient of lattice unit cell parameters (isostatic strain component). The elastic part can be obtained from the lattice parameters of the unit cell in different regions of the crystal and subtracted.

560


Ch. 79

6. Example applications of white beam diffraction Measurements in the following examples have been carried out at the dedicated polychromatic microdiffraction station ID-34-E at the Advanced Photon Source (APS), Argonne IL and on beamline 7.3.3 at the Advanced Light Source (ALS), Berkeley California. Although the experimental setups and technical approaches are similar, station 34-ID-E is equipped with differential aperture microscopy hardware and software and is designed to pass X-rays up to 20 keV, which favors 3D measurements. For both setups, a region of interest in the sample is identified using an optical microscope and the polychromatic X-ray beam is focused onto the sample. The local orientation at the sample/beam intersect is precisely determined by an automated indexing program. Details on the experimental setups and data collection can be found elsewhere [60,61,84,91,95,107,108]. Dimensions of the beam were ∼0.5 by 0.5 µm2 on 34-ID-E and ∼1 × 1 µm2 on 7.33. The penetration depth ranged from ∼20–100 micrometers depending on the material of the film, the beamline, and the substrate. In all the examples, the polychromatic (Laue) diffraction is sensitive to changes in local lattice orientation and allows the lattice curvature at any probed location to be determined. In the experimental setup, the microfocusing optics introduce a small 1 mrad convergence to the incident beam. This convergence angle has a negligible effect on broadening of the Laue spot (an order of magnitude smaller then the broadening due to typical lattice curvatures). We present the following example applications of polychromatic microdiffraction to materials characterization: • • • • •

Thin films I – Defects in GaN layers on patterned single-crystal substrates; Thin films II – Defect evolution in Interconnects; Welds-dislocation gradients in the weld joint of single-crystal superalloys; Near surface dislocation gradients in a Ni polycrystal during uniaxial pulling; Plastic deformation near a nanoindent.

In these examples we intend to demonstrate a range of capabilities of the PXM Laue microdiffraction for characterization of dislocation arrangements. We provide a comparison between the PXM results and other characterization methods such as OIM (EBSD), SEM, TEM, and optical microscopy. We also compare PXM results with finite element simulations in some examples. Example 1. Thin films I – analysis of lattice rotations, deviatoric strain, misfit and threading dislocations in stripped GaN layers GaN is an important electro-optical material with possible applications to high-performance light-emitting-diodes (LEDs). However, attempts to grow suitable GaN films have been frustrated by the presence of defects; the mismatch in the thermal expansion coefficients between the substrate and the nitride material structures causes formation of dislocations, boundaries and elastic strain. Dislocations and tilt boundaries are known to impair the performance of GaN-based light emitting devices [109]. To reduce defects, several novel growth techniques have been explored, most notably lateral epitaxial overgrowth (LEO) techniques such as pendeoepitaxy. In the pendeoepitaxy process, GaN wings are grown from the sides of GaN columns (Fig. 30(a)). The top of the column may or may not be covered with a mask (masked or

§6


561

¯ GaN and Fig. 30. Scheme of the sample geometry (a) together with a series of Laue images showing the (0118) ¯ (01124) SiC spots. The Laue images (b) for sample 1 correspond to different locations of the probing beam across the stripe: (A) in the gap between the wings; (B) center of the left wing; (C) left wing/column interface; (D)–(E) center of the column; (F) column/right wing interface; (G) center of the right wing; (H) in the gap between the wings. Typical PXM Laue pattern from the PE grown GaN on SiC (c).

maskless pendeoepitaxy) to inhibit GaN growth. When the LEO process is continued until the overgrowth of a mask, threading dislocations are observed to form a grain boundary along the edge of the mask [110]. In the case of maskless pendeoepitaxy [111] for which the growth starts from the sidewalls and simultaneously from the surface of unmasked GaN columns, the wing tilt depends on the c-axis strain in the columns. Despite extensive characterization, the basic structure of LEO films including such critical information as the direction of the wing tilt remains controversial. The application of polychromatic microdiffraction (PXM) provides detailed new information on the structure of these materials.

562


Ch. 79

The results of PXM Laue diffraction are compared with conventional HRXRD and finite element (FE) simulations. For these experiments, pendeoepitaxial (PE) grown GaN stripes with a rectangular cross ¯ section and oriented along [1100] were etched from 1 µm thick GaN single crystal layers grown by metalorganic vapor phase epitaxy on 0.1 µm thick AlN buffer layer deposited on 6H-SiC (0001) substrate. The subsequent overgrowth process was stopped before the wings coalesced. A schematic geometry of the stripe, with two wings on SiC substrate is shown in Fig. 30(a). Two PE-grown samples with uncoalesced wings with different GaN stripe geometries were analyzed. The GaN thickness, the column width and the wing width were 1.46 µm, 3.80 µm, and 0.98 µm for sample 1 and 3.53 µm, 3.46 µm and 1.74 µm for sample 2, as determined by cross-sectional scanning electron microscopy. FE simulations were conducted using the software FELT. Details of the growth process and FE simulations can be found elsewhere [111]. Polychromatic X-ray microdiffraction, and finite element simulations were used to characterize the distribution of strain, dislocations, sub-boundaries and crystallographic wing tilt in uncoalesced and coalesced GaN layers grown by maskless pendeoepitaxy. An important parameter was the width-to-height ratio of the etched columns of GaN from which the lateral growth of the wings occurred. The strain and tilt across the stripes increased with the width-to-height ratio. Tilt boundaries formed in the uncoalesced GaN layers at the column/wing interfaces for samples with a large ratio. Sharper tilt boundaries were observed at the interfaces formed by the coalescence of two laterally overgrown wings. The wings tilted upward during cooling to room temperature for both the uncoalesced and the coalesced GaN layers. Finite element simulations that account for extrinsic stress relaxation can explain these experimental results for uncoalesced GaN layers. Relaxation of both extrinsic and intrinsic stress components in the coalesced GaN layers contribute to the observed wing tilt and the formation of sub-boundaries. The sample-microbeam intersect was scanned perpendicular to the stripe axis, and Laue patterns were recorded at 1000 different points with a spacing of 0.25 µm. Several (usually 10–20) parallel line scans were recorded to obtain a 2D map of the local crystallographic orientation. Due to the small size of the microbeam (0.5 µm) it was possible to obtain separate images from different locations on the column and the wings. Because the high-energy X-ray beam passes through the GaN and the AlN layers, the SiC substrate contributes to the observed Laue patterns. A typical PXM Laue pattern is shown in Fig. 30(c). A small region of overall Laue pattern is shown in Fig. 30(b). In this region, two Laue reflections belonging to the SiC substrate and one reflection from the GaN film are visible. The SiC reflections do not change position with sample translation and are used as a reference to ¯ Laue reflection. determine the position of the GaN (0118) As the microbeam is moved along the positive x-axis, it first hits the left wing of the GaN film. For this part of the film, the GaN reflection appears at the right side of the SiC reflections, as shown in images B, C of Fig. 30(b). When the probe is in the center of the column, the GaN reflections are aligned with the SiC reflections, as shown (D, E of Fig. 30(b)). With further displacement towards the right wing the GaN reflections appear to the left of the SiC reflections (compare image F and G of Fig. 30(b)). The GaN reflection finally disappears when the microbeam is positioned in the gap between wings (A and H

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in Fig. 30(b)). The shift of the GaN reflections along with the geometry of the structure (Fig. 30(a)), clearly indicates that the wings are tilted upward [112]. Intensity profiles of the Laue spots are related to defects in the RVE as described in Section 4.3. The sample was oriented such that the beam penetrates into the sample at 45 degrees to the surface normal and parallel to the etched pedestals. Since there is translational invariance along the length of the pedestals, the beam therefore probes changes in the GaN crystallographic orientation with depth. Typical intensity profiles are shown in Fig. 31 for both samples. The broadening of the GaN Laue reflections at different locations of the wings and column is analyzed at corresponding positions along the PXM line scans, as shown in Fig. 31(d), (e). The broadening is generally more pronounced for sample 1 with the larger width-to-height ratio of the GaN column. For sample 1 the linescans in Fig. 31(a) show that the GaN reflection is particularly broadened at the column/wing interface where it splits into two distinct peaks – one from the wing and one from the column (here the large difference in orientation lateral to the column length dominates the broadening). The splitting profile shows clearly resolved peaks, which indicates that wings and column have their own well-defined orientation, and that a small angle tilt boundary has formed at the column/wing interface in sample 1 (Fig. 31(a)). For sample 2 the misorienation changes gradually transverse to the column. The wing tilt still shows an abrupt change at the interface but is much smaller than in sample 1 (Fig. 31(b)). A gradual change of the misorientation in the sample 2 is also demonstrated with the pole figure (Fig. 32(a)). An upward-tilt of the wings is in agreement with FE simulations. For the calculations we assumed that the GaN is under biaxial tensile stress in the c-plane due to the mismatch in thermal expansion coefficients between the GaN layer and the SiC substrate. FE simulations of strain in the Z directions (Fig. 32(d)) for the geometry corresponding to both samples also show that the wing tilt tends to increase with an increasing width-toheight ratio of the column. The change of misorienations through the GaN stripe obtained from the PXM data (Fig. 32(b)) demonstrates that the crystallographic tilt changes quite abruptly at the column/wing interface for sample 1. In sample 2, however, this change is more gradual, and extends over the whole column between the right and left wing. Sample 1 has a width-to-height ratio of the column of 2.60 and a wing tilt of 0.16◦ , whereas the corresponding numbers for sample 2 are 0.98 and 0.08◦ , respectively. This finding is also in good agreement with the results of HRXRD (Fig. 32(c)). The misorientation between the column and wings at their interface was quantified from the splitting of the GaN Laue spots. Using the measured misorientation at the column/wing interface, we derived the spacing between dislocations in the boundary using the Burgers relation for a small angle tilt boundary: ϑ = b/ h (Fig. 33). Here ϑ is the measured tilt angle at the interface, b is the Burgers vector modulus and h is the spacing between dislocations in the boundary. The tilt angle at the column/wing interface was always smaller then the total tilt between wing and column (especially for the sample 2). This means that thermal stresses were only partially relieved by the interface tilt boundary and some elastic strain remained. For the measured tilt angle at the interface between column and wing Laue spots of ϑ = ϑ1 = 0.12◦ for sample 1 and ϑ = ϑ2 = 0.04◦ for sample 2 and for b = 1 ¯ 3 1120 we obtained the distance between dislocations in the tilt interface boundary h1 = 98c for sample 1 and h2 = 294c for sample 2. Here c is the interatomic distance along the z-axis of the GaN. For a GaN layer thickness of 1.46 µm or 3.53 µm in either sample

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¯ GaN Laue spot as a function of misorientation, mx , for the column, wing Fig. 31. Line scans through the (0118) and column/wing interface for sample 1 (a) and sample 2 (b). The SiC line scan is also shown for comparison.

Fig. 32. Pole figure obtained with PXM analysis (a) and relative crystallographic orientation (b) of the GaN c-axis across the stripe with distance from the column center as determined from PXM analysis for sample 1 and sample 2. Conventional reciprocal space maps obtained with HRXRD reveals three peaks for each sample (c). They correspond to the column and two wings of the GaN stripe (c). Half of the RSM is shown for each sample (red color for sample 1 and blue color for sample 2). Finite element simulations of strain (d). The height Z = 0 corresponds to the GaN/SiC interface.

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Fig. 33. Dislocation model of the column/wing interface in the uncoalesced PE grown GaN film on SiC.

1 or sample 2 this amounts to 29 dislocations in the interface tilt boundary for sample 1 versus 23 dislocations for sample 2. The dislocation density in the boundary for GaN layers grown with maskless pendeoepitaxy is much smaller than for layers grown using a mask [110,112]. For a coalesced PE sample, the interpretation of PXM becomes more complicated. PXM analysis of the coalesced PE grown sample reveals two kinds of tilt boundaries in the layer. One is located in the region overgrown on top of the former column region with a tilt angle of 0.18◦ (Fig. 34(a)). The second sharp tilt boundary with a tilt angle of 0.5◦ is formed at the wing/wing coalescence interface (Fig. 34(b)). In the coalesced films the FE simulation accounted only for extrinsic stress relaxation due to the mismatch in the thermal expansion coefficients between the GaN and the SiC. These simulations predicted almost zero wing tilt due to compactness of a coalesced pendeostructure. However, PXM results (Fig. 34(a), (b)) showed the same features as for uncoalesced films: i.e. both wing tilt and a difference in strain between stripes and the wings were observed. If small notches were assumed to form on the bottom side of the coalescence front, the FE simulations indicated a tendency for downward tilt. However PXM measurements similar to ones shown in Fig. 30 indicated that the wing tilt in coalesced films is upward, as in the case of uncoalesced films. This means that further understanding is necessary to understand the mechanisms of stress relaxation in these materials. Periodic oscillations of local lattice misorientations related to the formation of GN dislocations in the coalesced GaN films show that a maximal density of GN dislocations is observed at the wing/wing coalescence interface (Fig. 34). These observations are consistent with the dislocation model of the coalesced film (Fig. 35).

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Fig. 34. Two types of interfaces observed in coalesced PE grown GaN films on SiC: PXM images at the wing/column interface (a) and coalescence interface (b). Oscillations of lattice rotations (c) and density of GN dislocations across several GaN stripes (d) and across one stripe. Maximal misorientations are observed at the coalescence interface (d).

Cantilever epitaxially (CE) grown GaN films are another variant on LEO approaches. The CE-grown GaN film on a Si (111) PXM reveals additional crystallographic tilt between the Si substrate and GaN surface normal (Fig. 36). We note that the two different kinds of tilts are demonstrated in these PXM data: wing tilt manifests itself by the horizontal displacement of the Laue diffraction between substrate and GaN spots in the Laue patterns at the wing regions, while the crystallographic tilt between the GaN(0001) and Si(111) planes for CE grown samples (Fig. 36 top), is observed by the vertical separation of the spots for CE samples (absent, for example, in the data of the sample grown by PE (Fig. 36 bottom) [113].

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Fig. 35. Model of dislocation arrangement based on PXM results for coalesced GaN film.

Fig. 36. Comparison between PE and CE grown samples: additional crystallographic tilt between the substrate and the GaN layer is found for cantilever grown GaN on Si (top) as compared to PE grown GaN on Si (bottom).

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Fig. 37. 3D distribution of lattice orientations with depth of the CE grown GaN on Si: Original PXM pattern contains GaN and Si Laue spots (a); partially shadowed with wire PXM pattern – the wire shadow is seen as a white band across the image (b). Region of interest in the center of the Laue pattern is shown with a dashed circle. Reconstructed Laue images (c)–(d) of the central part corresponding to the dashed region. Redistribution of intensity from the (0001) GaN spot to (111) Si Laue spot is observed with depth.

The depth dependent distribution of the lattice orientations in the CE-grown GaN layer and its Si substrate was additionally analyzed with DAXM technique (Section 4.1). The details of the DAXM technique can be found elsewhere [61]. Fig. 37 illustrates application of the DAXM for the analysis of the buried Ga/Si interface. The partially shadowed images (Fig. 37(a), (b)) are used to reconstruct the Laue patterns generated from sample volumes along the penetrating microbeam. Patterns corresponding to distinct 1 µm thick layers are recovered. A change of the intensity distribution from the (0001) GaN reflection at the surface of the film to the (111) Si reflection with depth is demonstrated. To summarize, in this example, X-ray PXM experiments give direct experimental evidence that the crystallographic tilt of the wings in GaN layers PE grown on SiC substrates is upward. The strain, stress and tilt across the stripe increased with the width-to-height ratio. For small width-to-height ratios the tilt is small, and it changes gradually in the region of the column. For larger width-to-height ratios the tilt is higher, and it changes abruptly at the wing/column interface. The evaluation of the local broadening of the reflections and their intensity profiles revealed that the density of defects was reduced and/or the strain was homogenized in the wings as compared to the column. A low-angle tilt boundary is formed at the column/wing interface in samples with larger width-to-height ratio of the

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GaN column. The estimated dislocation spacing at the column/wing interface is approximately three times smaller in sample 1 than in sample 2. Finite element simulations that accounted for extrinsic stress relaxation provided information that allowed an explanation of the experimental results for uncoalesced GaN layers. Relaxation of both extrinsic and intrinsic stress components in the coalesced GaN layers contribute to the observed wing tilt and the formation of sub-boundaries. Example 2. Thin films II – analysis of early stages of plastic deformation in interconnects Interconnects are thin metal film conductors that carry current on integrated circuits. To make faster smaller and more complex integrated circuits the physical size of all integrated circuit components has decreased. This decrease of interconnect line dimensions, and subsequent increase in current density (1 MA/cm2 ) has imposed tremendous challenges for materials and reliability of interconnects. Electromigration-induced failure in metal interconnect constitutes a major reliability problem in the semiconductor industry [114]. While the general mechanism of electromigration is understood, the effect of the atomic flow on the local metallic line microstructure is largely unknown [115]. White beam X-ray microdiffraction was used to probe microstructure in interconnects and has recently unambiguously unveiled the plastic nature of the deformation induced by mass transport during electromigration in Al(Cu) lines [107,108,116–118] even before macroscopic damage occurs. The first quantitative analysis of dislocation structure in a grain in the polycrystalline region of the interconnect line was performed by Barabash et al. [119,120] and it was shown that in that region of the interconnect line the dislocations with their lines almost parallel to the current flow direction are formed first. Recent studies of precipitation in Al(Cu) interconnects indicates that Cu is preferentially depleted from the cathode end of the line [121]. An honest analysis of alloying effects in electromigration, demonstrates that a complete understanding of why the addition of Cu results in greatly reduced electromigration has not yet been achieved. Data collection for this example was carried out in the group of N. Tamura at the X-ray microdiffraction end station on beamline 7.3.3 at the Advanced Light Source, Berkeley National Laboratory. Details on the experimental setting and data collection at this station can be found elsewhere [107,108]. In this example PXM is used to analyze and test the evolution of mesoscale structure during electromigration and to study possible methods of mitigating electromigration damage. Notably, early evidence of electromigration-induced plastic deformation is observed by white beam X-ray microdiffraction before evidence of morphologic changes. In this example a quantitative analysis of the dislocation structure is provided in several micrometer-sized Al grains in both the middle region and ends of the interconnect line during an in-situ electromigration experiment. The evolution of the dislocation structure during electromigration is highly inhomogeneous and results in the formation of randomly distributed geometrically necessary dislocations as well as geometrically necessary boundaries. The orientation of the activated slip systems and rotation axes depend on the position of the grain in the interconnect line. The origin of the observed plastic deformation is considered in view of constraints for dislocation arrangements under the applied electric field during electromigration. The coupling between plastic deformation and precipitation in the Al (0.5% wt. Cu) is observed for grains close to the anode/cathode ends of the line

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and near the middle of the line. The results of the measurement helps to clarify the dislocation structure arising from electromigration-induced plastic deformation in different regions (including the ends) of interconnect lines. It also shows the possible correlation between Cu drift, precipitation and formation of dislocations. The sample is a patterned Al (0.5% wt. Cu) line (length: 30 µm, width 4.1 µm, thickness 0.75 µm) sputter deposited on a Si wafer and buried under a glass passivation layer (0.7 µm thick). Electrical connections to the line are made through unpassivated Al (Cu) pads connected to the sample by W bias. The evolution of the diffraction pattern of three particular grains (Grain A with the size of ∼2.5 µm) situated approximately half-way between the middle of the line and the anode end and grains B and C close to the opposite ends of the line (Fig. 38) is described. A qualitative description and semi-quantitative interpretation

Fig. 38. Change of experimental streaking of (222) Laue spots for A grain in the middle of the line (a) and for B and C grains (b), (c) close to the opposite ends of the interconnect line. The sample was maintained at a constant temperature of 205 ◦ C (Spot 1) and the current density was progressively ramped up to 0.98 MA/cm2 and maintained at this value for a period of 20 hours (Spots 2 and 3). The direction of the current was then reversed for a total period of 19 hours. Spot (1) initial state before the current flow; Spot (2) 5 hours; Spot (3) 11 hours; Spot (4) 9 hours reversed current; Spot (5) 10 hours reversed current. Location of the grains is shown at the microstructure of the interconnect (d).

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of the entire data set collected for the present sample can be found elsewhere [108,118]. The local orientation maps obtained from the X-ray microdiffraction reveal that the grain structure of the line has a random in-plane orientation and a pronounced (111) fiber texture. Only one or a few grains span the width of the line (near bamboo configuration). Grain A is in a region of the line, where multiple grains are found transverse to the line, grains B and C are almost at the ends of the line. Dislocation structure was modeled from the Laue images by simulating the shape of the reflections observed in the experimental data. Custom software allows us to determine the orientation of the predominant dislocation network in each sample subgrain [74]. This method was extended to consider gradients with depth in density of randomly distributed geometrically necessary dislocations (GNDs) within the scattering domains separated by geometrically necessary dislocation boundaries (GNBs) and to include the strain gradient parameters. Before electromigration the Laue patterns for all grains show sharp reflections (Fig. 38 (a)–(c), spot 1). After electromigration pronounced streaking of the Laue reflections is observed in the majority of Al grains (Fig. 38(a)–(c), spots 4, 5). The average streaking direction for the grains in the middle of the line is approximately transverse to the length of the line (Fig. 38(a)) but almost parallel to the line near the ends (Fig. 38(b), (c)). During the first 20 hours, grain B is close to the anode end of the line (Fig. 38(b), spots 1–3), which becomes the cathode end after current reversal (Fig. 38(b), spots 4, 5), and grain C is in the opposite location. Plastic deformation away from line ends: Grain A. Grain A is located about 4 grains away from the line end (grain C). For grain A, continuous streaks are observed near all Laue spots after having the current equal 0.98 MA/cm2 for 5–10 hours. It is worth noting that different Laue streaks have distinct orientations on a CCD. However, in reciprocal space the reflections are streaked almost in the same direction (Fig. 39(a) and (b)). After 14 hours of current flow, the intensity distribution near a (222) Laue spot becomes asymmetric with strong maximum of the intensity on one side and a long weak tail. ¯ The orientation of the primary GNDs corresponds to a Burgers vector b = [110] and a ¯ dislocation line direction τ = [112] that is almost parallel to the direction of the current flow in that grain. After 6 additional hours at the same current density, the intensity distribution in grain A breaks into two distinct maxima (Fig. 39(c), frame 3) indicating that the dislocation structure partially relaxed with the formation of a GNB. The density of boundary dislocations is about 0.25 × 1010 cm−2 . The FWHM ξ along the streak direction within each maximum again increased (Fig. 39(c), frame 3). The density of unpaired individual dislocations within each scattering domain at this stage is equal to n+ = 0.3 × 1010 cm−2 (Fig. 39(c)). Then the direction of the current flow was reversed to −0.98 MA/cm2 and after 9 hours the randomly distributed portion of GNDs decreases, while the portion of dislocations grouping within the GNB increased (Fig. 39(c), frame 4). This indicates that the opposite direction of the current may “cure” some part of randomly distributed unpaired individual dislocations. The density of unpaired individual dislocations within each scattering domain slightly decreases to n+ = 0.2 × 1010 cm−2 . However, the dislocations being grouped into a sub-boundary form a very stable arrangement, which is not destroyed by the opposite direction of the current. After 10 hours with J = −0.98 MA/cm2 a second sub-boundary is formed. The streak splits into three distinct maximums (Fig. 39(c),

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Fig. 39. Experimental Laue streaks for five different reflections of the grain A after 18 hours of current flow on a CCD (a) and in reciprocal space (b). Evolution of the (222) Laue spots with time of electromigration (c). Simulated (d) and experimental (e) (222) Laue spots. Density of GNDs in the grain A with time of the electromigration (f).

frame 5). The second GNB creates misorientation of 0.4◦ . The total density of GN dislocations within the above two boundaries is equal to 0.6 × 1010 cm−2 . It should be noted that orientation of the secondary slip system in the grain A is slightly inclined to the direction of the current flow. For the grain A in the middle polycrystalline region of the interconnect line the best fit between the simulated and experimental Laue pattern (Fig. 39(d), (e) and Fig. 40) is obtained for such GNDs which dislocation lines are running almost parallel to the current flow. According to the model described by Barabash et al. [119,120] and Valek et al. [118], dislocation climb during electromigration becomes the main mechanism of stress relaxation and the source of vacancies (Fig. 41(a)). Stress relaxation results in the formation of GNDs accompanied by streaking of the Laue spots. GNDs further group within GNBs (Fig. 41(b)) and all Laue spots split. The elastic field around climbing dislocations generates an elastic field through the entire crystal (grain), which interacts with phonons in the crystal (phonon viscosity) as well as with electrons. Dislocations and dislocation walls scatter electrons off the initial direction by the deformation potential related to them. The major contribution to the interaction potential between electrons and dislocations comes from the deformation field in the matrix around the dislocation, and not from the dislocation core. Around an edge dislocation the deformation field decreases very slowly, as 1/r⊥ (r⊥ is the distance to the dislocation line), leading to a large scattering matrix element. However, scattering only occurs in the

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Fig. 40. Comparison between the simulated and experimental Laue spots for grains A, B and C. Experimental and simulated stereographic projections of different Laue streaks for grains A, B and C. Simulations were A ¯ performed: for grain A assuming two activated slip systems with Burgers vectors bA 1 = [011]/2, b2 = [110]/2 √ √ A A ¯ ¯ ¯ and dislocation lines running along τ 1 = [211]/ 6, τ 2 = [112]/ 6; for B grain assuming the slip system √ ¯ ¯ bB = [101]/2 6, and for a C grain assuming two activated slip systems with bc1.2 = [011]/2 and τ B = [121]/ √ √ C ¯¯ ¯ and τ C 1 = [211]/ 6 and τ 2 = [211]/ 6.

direction perpendicular to the dislocation line, because of translation invariance along the dislocation line. The scattering of the electrons can be described as (k0 − k)τ = 2πn/dτ _,

n = 0, ±1, . . . .

(6.1)

Here dτ is a lattice parameter along the dislocation line τ and k0 , k are wave vectors of incident and scattered electrons. There exists a strong anisotropy of scattering (Fig. 41). The probability of scattering depends on the difference (k0 − k) as well as on the direction of initial electron momentum k0 . When almost all unpaired dislocations with the density n+ are parallel (as in the case of Al-based interconnects) the anisotropy of scattering becomes

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Fig. 41. Scheme of the GNDs formation during electromigration (a) and their regrouping within GNBs (b). Favorable orientation of GNDs parallel to current flow with minimal electron scattering (c), and unfavorable (inclined orientation) resulting in the increase of frequency of the electron scattering effects (b).

important and the electrical properties of the interconnect depend strongly on the direction of the electric current relative to the orientation of the dislocation network [119,120]. Plastic response of the lattice in the near anode/cathode regions of the interconnect line. Grains B and C (Fig. 38(b), (c)) in the opposite ends of the interconnect line show a distinct behavior compared to the grain A (close to the middle of the line). Grain B in the beginning of the experiment is close to the anode end of the line. There is absolutely no increase in the FWHM ξ of the Laue spot indicating the absence of plastic deformation in this grain during this period of current flow (more over the FWHM in all directions of the spot even slightly decreases). After the current reversal the grain B becomes close to the cathode end of the line and its FWHM ξ , along the streak immediately increases demonstrating high activity of plastic deformation. The density of GNDs becomes equal n+ = 0.3 × 1010 cm−2 . The orientation of the primary GNDs corresponds to a Burgers vector b = ¯ ¯ (Fig. 40), and is almost perpendicular to [101] and a dislocation line direction τ = [121] the direction of the current flow. The next 10 hours of the current flow in the same direction resulted in the grouping of dislocations within a GNB. Grain C demonstrates the opposite behavior (Fig. 38(c)). In the beginning of the experiment grain C is located close to the cathode end of the line, and immediately since the current is on, the FWHM ξ of Laue streak starts to increase. Opposite to grain A the (222) Laue spot in grain C first broadens symmetrically (Fig. 39(b), curve 2). After 6 additional hours at the same current density, the intensity distribution becomes highly asymmetric (with intensity maximum on one side of the streak and long tail of

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decreasing intensity) and corresponds to dislocation density equal n+ = 0.4 × 1010 cm−2 (Fig. 38(c), spot 3). After current reversal the FWHM ξ , of the streak slightly decreases and intensity distribution breaks into two distinct maxima (Fig. 38(c), spot 4) indicating that the dislocation structure partially relaxed with the formation of a GNB. The continuation of the current flow in this direction (with grain C remaining close to the anode end of the line) does not increase FWHM ξ , of the streak, indicating the absence of dislocation activity in this grain during the “near anode” period of the test. To determine the orientation of the activated slip systems the Laue streaks corresponding to the predominant activated GNDs and GNBs arrangements in grains A, B and C were simulated. In Fig. 40, the experimental and simulated (313) and (224) Laue spots are shown for A grain, (331) and (222) Laue streaks for the grain B, and (224), and (222) Laue streaks for grain C. Laue streaks are shown in reciprocal space in the sample basis. Although orientation of streaks is different in different grains the simulated Laue streaks fit the experimental one in each grain. We performed simulation assuming one (grain B) or two (grains A and C) predominant GNDs slip systems. We can understand such peculiarities of plastic deformation in the “near anode/cathode” regions by taking into account that Cu is preferentially depleted from the cathode region of the interconnect [121]. Moreover the Al (0.5% wt. Cu) interconnect line in the initial state might contain small (∼1.2 nm) precipitates of tetragonal θ phase (Al2 Cu). The dissolution kinetics of this phase near the cathode end is coupling to the plastic deformation activity. Interestingly, that during the time of electromigration test there is no visible dislocation activity close to the anode end of the line (as observed in this study), while the “near cathode” grain is quickly plastically deformed. To interpret such a behavior the model of electromigration-induced Cu motion and precipitation in Al(Cu) interconnects, described by Witt et al. [121], was expanded to take into account coupling between plastic deformation and precipitation formation and dissolution in different regions of the line. The presence of small precipitates is known to strengthen the Al-based alloys and increase their resolved shear stress. Cu depletion in the near cathode region causes dissolution of precipitates. With Cu diffusion from the cathode the precipitates dissolve, critical shear decreases and plastic deformation is activated. This is accompanied with streaking and further splitting of Laue spots (Fig. 38(c)). With the reverse of current flow Cu concentration in this grain starts to increase and precipitates form again. Critical shear stress increases in this grain and plastic activity stops. The dislocation grouping in the wall is stable and is not destroyed by this process, however new dislocations do not appear. In the “near anode” grain B increase of Cu concentration retains high critical shear stress and plastic activity is suppressed. This supports the idea that Al (0.5% wt. Cu) interconnects are most reliable when Cu depletion from the cathode end is the slowest [121]. The start of plastic activity in all grains corresponds to highly anisotropic intensity distribution along the streak with the maximum of intensity concentrated on one side of the streak and long “tail” with gradually decreasing intensity (Figs 38, 39). Such intensity distribution is related to the gradient of the dislocation density with depth. This supports the model of dislocation climb from the interface into the depth of the interconnect line. We summarize the application of PXM to the problem of electromigration as follows. PXM reveals the formation of GNDs and GNBs much earlier than visible damage can be distinguished. An analysis of the orientation of the activated dislocation slip systems

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shows that in the middle of the interconnect line in the polycrystalline region the slip systems with dislocation lines almost parallel to the direction of current flow are activated first. Near the ends of the line plastic activity is coupled with the depletion of Cu from the cathode end of the line. There is practically no plastic activity in the “near anode” end of the line. A significant gradient in dislocation density with depth is observed. Coupling between the plastic activity and density of GNDs with the Cu depletion from the cathode and dissolution of precipitates during electromigration is demonstrated. Example 3. Dislocation formation and multiplication in the weld joint of single-crystal superalloys – comparison of PXM results with OIM (EBSD) and TEM During welding, the interaction of the heat source with the material results in melting, solidification, solid-state transformation, stresses, distortions and dislocations. Ultimately, welding related microstructures affect the cracking tendency and final properties of the weld and a fusion weld is typically divided into three regions with distinct mesoscale structure: (1) the base material (BM) which is largely unaffected by the welding process, (2) the fusion zone (FZ) where material melts and then recrystallizes, and (3) the heat affected zone (HAZ) in between the BM and FZ where plastic deformation and phase transformations can take place without melting. Here the PXM analysis of GN dislocation arrangements in a Ni-based superalloy single-crystal weld is described. Measurements were made on RENE-5 and CMSX-4 nickel-based single crystal superalloys. The locations of the BM, FZ and HAZ are schematically shown in relation to the fusion line (FL) (Fig. 42). Characterization of the weld was performed with polychromatic microdiffraction, TEM, optical and orientation imaging microscopy (OIM). The experimental procedure and data collection are described in detail elsewhere [60,61,91]. Of key

Fig. 42. (a) Section of the phase diagram with the alloy position (dashed line) showing the change in Al solubility in the matrix with temperature. (b) Scheme of the different zones position in the weld joint. (c) Homogeneous orientation of the BM according to OIM analysis. (d) Dendrite structure of RENE N5 in BM. (e) γ ′ -phase particles within the dendrite.

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importance, polychromatic microdiffraction analysis of the weld provides local information (0.5 × 0.5 µm2 ) and tests models of the local orientation and dislocation density distribution at different locations in the weld. Measurements were made in areas of the weld containing base metal (BM), fusion zone (FZ) and heat-affected zone (HAZ). Scans were carried out along lines perpendicular to the welding direction. The orientation at each position of the crystal is precisely determined by an automated indexing program [5]. A total of 1000 locations were probed with a grid size of 5 µm along several parallel lines. The results of PXM are discussed in comparison with respect to OIM, TEM, SEM and optical microscopy. The initial nominally single crystal base material had a dendrite microstructure because of its original casting (Fig. 42(d)). Dendrites were oriented along the (001) sample axis (±2.5◦ ). Each dendrite contained Ni3 Al cubical γ ′ -phase particles with their (001) plane aligned with matrix (001) plane (Fig. 42(e)). The base metal (BM) of the Ni single crystals weld samples are relatively easy to analyze under microdiffraction because the Laue spots are sharp. Analysis becomes more complicated in the FZ where multiple dendrites grow as a result of melting and solidification. It is well known that the contraction of molten weld metal during solidification is opposed by the surrounding metal giving rise to internal stresses [100]. These stresses may partially relax by plastic deformation. The structural state in the HAZ depends sensitively on the thermal history of the process including the temperature distribution, its gradient and the rate of cooling. The alloy location on the phase diagram is illustrated by a dashed line in Fig. 42(a). The very early stages of the weld process together with the BM structure determine the properties of the weld. Typical Laue images for several positions in the HAZ and FZ in the vicinity of FL are shown in Fig. 43. As shown in the Laue patterns (Fig. 43), the welding of the single crystalline Ni based superalloy is accompanied by local plastic deformation which relieves thermally induced elastic stresses. Microbeam-Laue diffraction reveals pronounced streaking in the Laue images from dendrites located in the HAZ (Fig. 43). The plastic response of the material in the HAZ can be described by the formation of GNDs and GNBs that appear in the material to relax the stress field induced during welding and subsequent cooling. Microstructure of the weld is shown in Fig. 44(c). The color insert shows the misorientation of each location in the weld joint relative to the base material obtained with OIM. The almost constant color across the weld indicates that the crystallographic orientation of dendrites formed within the FZ remains practically the same. Pole figures obtained with OIM and Laue PXM analysis (Fig. 45(a), (b)) find that a quasi single-crystal structure is retained across the entire weld joint. Although OIM shows virtually no misorientation across the weld, Laue PXM analysis reveals local changes in orientation ranging in the interval of ±1.6◦ . Moreover, Laue PXM analysis reveals severe plastic deformation with oscillating GN dislocation density across the weld (compare Fig. 43 and Fig. 44). This information cannot be extracted from OIM analysis because electron beam diffraction in OIM is too surface sensitive and has lower sensitivity to misorientations. In the center of the weld, Laue spots are sharp indicating that there is virtually no plastic strain (Fig. 43(a)). In the HAZ, large dendrites are observed with significant deformation (Fig. 43(e)).

578 G.E. Ice and R.I. Barabash Fig. 43. Laue patterns obtained from different probed locations in the weld joint (a)–(e) and SEM image of the region in the vicinity of the fusion line. Streaking (c)–(e) and splitting (b) of the Laue spots is observed. Ch. 79

§6 White beam microdiffraction and dislocations gradients Fig. 44. Pole figures of the weld obtained with OIM (a) and Laue microbeam (b). Three regions of dendrites with different cubic orientation of the dendrite growth axis in RENE N5 (c). OIM characterization of the central part of the weld (color insert) demonstrates three possible cubic growth directions. Orientation distribution in the weld (d). Color reference scheme of the orientations distribution (e). 579

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Fig. 45. Experimental (a) and simulated (b) Laue pattern in the HAZ of the Ni base superalloy weld joint.

Nucleation of dendrites from the FZ results in the formation of regions with different ¯ dendrite growth directions in the weld such as ([100], [100], [010]). Regions with the first two dendrite orientations are located in the left and the right sides of the FZ, and the third one with [010] dendrite direction is located in the central part of the joint (Fig. 44(c)). The intensity profiles of virtually all Laue spots in the HAZ are discontinuous indicating the presence of GNBs (Fig. 43(a)–(e) and 45(a)). For example, consider the image shown in Fig. 45(a). Here the investigated volume is comprised of at least two well-defined GNBs separating three cell blocks. Based on an analysis of experimental Laue spots, the misorientation angles between cell blocks can be determined and then used as input parameters for simulations [87]. The density of GNDs can be estimated by fitting the whole experimental Laue pattern as well as separate Laue spots to the simulated ones (Fig. 45(a), (b)). Assuming a constant dislocation density within the RVE and assuming an orientation for the activated slip system in each scattering fragment, we note that the whole 3D intensity distribution is simulated and compared to the experimental intensity. With these assumptions, dislocation density, orientation of the slip system, and GNBs misorientation angles between cell blocks can be fit. Once a good fit to a single reflection is achieved, the whole Laue pattern (Fig. 45), is recalculated. In this example, agreement between the model and the experimental data is good. A transition from one dislocation network to another is observed at various neighboring positions in the weld. In this example, the overall streaking direction of the Laue pattern fluctuates but maintains a similar qualitative behavior within most of probed locations in the HAZ. Parallel to the weld direction the diffraction Laue patterns show only minor changes. However, previous experiments indicate that the overall deformation behavior is highly correlated to the crystallographic orientation of the weld and to the thermal gradients and therefore the behavior of polycrystalline welds is not expected to retain translational invariance along

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the weld direction [74,90,122]. Lattices misorientations in the HAZ are primarily found to rotate around the axis perpendicular to the weld direction. The Laue-image scans perpendicular the weld joint show long and short-range oscillations of the length and width of the Laue spots (Fig. 46). These parameters correlate with the density of GN dislocation, nd , and with interfaces and/or boundaries appearing under gradients of stress fields, in particular thermal stresses [87,122]. Periodic dislocation structure is formed during continuous movement of melt zone in thin Ni-based superalloy sheet. Moreover oscillations in the dislocation structure formed under such conditions at both macro and micro scales are observed. Depending on the temperature, the formation of dislocations is accompanied by the partial or complete dissolution of γ ′ particles in the matrix. Finite element simulations of the thermal field in the vicinity of the weld pool show that the thermal gradient perpendicular to the fusion line is not monotonic (Fig. 47). Thermal gradients result in the formation and multiplication of dislocations. Their arrangement correlates with the temperature gradient field and with the dissolution and re-precipitation of γ ′ particles. The distribution of the dislocation density at the macroscale is due to symmetric temperature gradient perpendicular to the direction of melt zone movement. Within the above macro regions (FZ and HAZ), oscillations of dislocation density, nd , due to grouping at the micro scale were also observed. The typical length of the dislocation density oscillations is related to the dendrite size and the conditions of local melt and solidification (Fig. 46). The maximum GN density is observed near the FL and in the middle of the FZ at the coalescence interface between dendrites growing from the opposite sides of ¯ the FZ with different growth direction, [100], [100], [010], [122]. The value of nd calculated from the Laue pattern at each location across the weld joint is shown in Fig. 46. From the analysis of the dendrite size within the HAZ and FZ, we note that long range oscillations correlate with: (a) characteristic parameters of the weld joint; (b) position of the regions, where there is a change in size and direction of the dendrite growth; (c) positions of maximal/minimal values in temperature gradient in the HAZ. The length of short-range oscillations is about 15–20 µm. In addition, it correlates with typical dendrite size in different locations of the weld joint. The Laue PXM results were further confirmed with TEM analysis (Fig. 48). The dislocation density was minimal far from the FL and it increases as one approaches the FL. The maximum number of dislocations is observed in the matrix of the single crystal between the large γ ′ -phase particles and near the γ /γ ′ interface boundary (Fig. 48(a)). Dislocation lines having Z-shape lines are elongated along two directions mutually perpendicular to [100] and [010]. The dislocation types (A or B) are identified using different g operations. For edge dislocations (g · b) = 0 and g(b × τ ) = 0 (where τ is the edge dislocation line unit vector) and (g · b) = 0 for screw dislocations. They correspond to dislocations A and B in Fig. 48(a), (b), with Burgers vectors [101] and [011], respectively. We summarize this example: Comparison with OIM demonstrates that although both methods show similar orientation in the near surface region. The higher accuracy of the PXM method allows distinguishing between orientation fluctuations in different weld regions. The PXM Laue method additionally provides information about plastic deformation and GN dislocation density within the 50 µm beneath the surface. The GN dislocations arrangement correlates with the temperature gradient field and with the dissolution and re-precipitation of γ ′ particles. Long and short range oscillations in the dislocation struc-

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Fig. 46. Long and short range oscillations of the GND density across the weld (a) and short range oscillations near the FL (b). Maximal GND density corresponds to the coalescence interface of recrystallized dendrites in the middle of FZ (c) and interface between the HAZ and FZ near the FL (d).

Fig. 47. Distribution of temperature, its gradient and second derivative across the FZ of the weld.

ture are observed. The long range oscillation of the dislocation density is due to symmetric temperature gradient perpendicular to the direction of melt zone movement. Typical short range length scale of dislocation density oscillations is related to the dendrite size and the

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Fig. 48. Dislocation structure within the HAZ of the TSM75 single crystal, close to the FL. A – zigzag dislocations along the average direction [100] with Burgers vector parallel to b = [101]; B – zigzag dislocations along the average direction [010] with Burgers vector parallel to b = [011]. (a) Dislocation distribution along the γ /γ ′ boundaries; both A and B type dislocation sets are visible; g = (220); beam direction is close to (001). (b) Dislocation network in the γ matrix formed by B-type dislocations is visible; g = (020); beam direction is close to (001).

conditions of local melt and solidification [122]. TEM analysis confirms the presence of the same type of dislocations in the weld joint. Example 4. Lattice rotation gradients in polycrystalline Ni grains during in-situ uniaxial tension In this example the PXM analysis was applied to obtain the parameters of the gradient cell– wall structure in a grain of a Ni polycrystal after in situ uniaxial pulling [88]. Rotations within individual grains in a Ni (99.96% pure Ni) polycrystalline tensile sample with an average grain size of 200 µm were non-destructively determined as a function of depth and plastic strain. Laue patterns integrated over a penetration depth of about 30 µm are shown in Fig. 49 for 0%, and 15% macroscopic deformation. The initially simple Laue patterns at 0% strains (Fig. 49(a), (b)) are complicated by local deformation at 15% strain that makes indexing of the patterns virtually impossible (Fig. 49(c), (d)); the lattice rotations through the integrated depth of the gauge volume (∼30 µm) are much larger than the angular resolution required to distinguish reflection pairs. Grains with different crystallographic orientations demonstrate distinct streaking features (Fig. 49(c), (d)) to analyze such complicated Laue pattern depth resolved images were taken at different depths of a grain in the Ni polycrystalline sample for several grains.

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Fig. 49. Experimental (a) and simulated (b) Laue patterns for a grain of Ni polycrystal before deformation. Intensive streaking observed after 15% of tensile strain in grains with different crystallographic orientation is distinct (c), (d).

For example, the orientation of one grain surface normal after deformation was almost par¯ ¯ direction (Figs 50, 51). The allel to [233] with loading direction approximately in [112] plastic response of the material in each grain can be described by the formation of cell wall structure (Fig. 52) in the material to relax the stress field induced during pulling [88,89,94]. DAXM (Section 4.1) resolves the local orientations at specific depths and the depthresolved Laue images can be reconstructed and indexed to determine the local crystallographic orientation in small 3D volumes within the sample. The intensity distribution and the shape of the observed Laue streaks are essential inputs for simulations of the diffracted Laue patterns. In most cases, even depth-resolved microbeam-Laue images reveal pronounced streaking after 15% strain. Distinct streaking of the Laue spots is observed up to 25 µm below the sample surface (Figs 41, 51). For example in Fig. 41, the integrated Laue patterns gener-

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Fig. 50. Depth-integrated (a) and depth-resolved (b)–(f) images for a particular grain. One Laue spot shown with red dashed rectangle is blown up (g).

ated along the penetrating X-ray beam show two distinct sets of streaks. It is an indication that at least two regions with different slip systems and different strain gradient directions are present in the probed volume (RVE). In order to get additional information about the plastic strain gradients within such micrometer length scale steps we need to complement the results obtained with DAXM with the Laue intensity analysis of the streaks (Fig. 51). DAXM together with the streak analysis finds that in the near-surface region (within the depth interval of 0–10 µm) only one set of very-long streaks is present which is replaced by the second set of streaks at the depth of about 12–15 µm (Fig. 51(a), (b)). Laue streaks remain parallel to each other for all (hll) type Laue spots. In addition, all the Laue

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Fig. 51. Simulated (a) and experimental (b) Laue patterns corresponding to the depth of 10–20 µm. Laue spot (−355) demonstrates asymmetry of intensity distribution along the streak (c).

Fig. 52. Schematic of the cell–wall structure in the deformed grain.

spots demonstrate highly asymmetric intensity distributions along the streak with a strong maximum at one end of the streak and a long tail with gradually decreasing intensity. The intensity maxima in all spots are located at the same end of the streak (Fig. 51(c)). Accord-

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ing to Section 4.9 this kind of asymmetry in the Laue streaks indicates a strain gradient with depth. Below 30 µm, Laue images became sharp enough that rotations in the grains were fully resolved with DAXM (Fig. 41(e), (f)). However comparison of the sample orientation matrix at increasing depth shows that it continues to rotate. The results of the streak profile analysis are in agreement with and complementary to the results of the differential-aperture information. Each set of streaks is related to a certain (predominant) dislocation slip system. Using the PXM with streak analysis and fitting the experimental Laue pattern the dislocation density tensor was determined for both regions. A least square fit can be implemented to find the predominant slip systems corresponding to the measured strain tensor. This yields the distribution of the activated slip systems at different depth regions and can be inverted to simulate the Laue images corresponding to the best-fit slip system parameters (Fig. 51(a)). In this example, the near surface depth interval (0–10 µm), has geometrically necessary boundaries (GNBs) that are well modelled ¯ axis (Figs 50, 51). by two sets of edge dislocations resulting in the rotation around [110] At 15 µm depth a change in the slip system direction accompanied by the appearance of a new set of Laue streaks is observed. In the second depth region (15–25 µm) the activated slip systems change and result in the rotation around the [101] axis. The Laue pattern simulated with the experimentally-determined eigenvectors of the lattice curvature for a second region occupying the depth scale 15–30 µm is shown at Fig. 41(a). Our analysis of the dislocation density tensor shows that the dislocation population which is activated in this grain is mostly formed by edge dislocations with lines inclined approximately 45 degrees to the loading direction. The gradient in the cell–wall structure arrangement is understood as a result of the different number of constraints at different boundaries of the analyzed grain with the free surface. The number of constraints changes from zero at the surface to very rigid constraints at the opposite inner boundary of the grain in the middle of the sample [123]. The observed gradient in the cell–wall structure has a length scale which differs from surface roughness, known to vary from 12 to 80 nm in similar experiments [124]. The observed gradients in GNBs with depth result in cumulative rotations and occupy regions with a much larger length scale of about ∼15 µm. They indicate the evolution of a hierarchical mesoscopic substructure. It evolves as a result of translational-rotational movement of the grain with free surface during pulling including both shear and rotation. Different structural levels are observed in this grain. The length scale at which the change of slip system orientation takes place (at the depth of about 12–15 µm) is comparable (half size) to the length scale of fractal dimension found during tensile testing of Ni single crystal with AFM [124]. It is also in agreement with the length scale of mesoscopic slip sub-bands observed in Al single crystal strained in tension [125] and with earlier observations using optical microscopy by Honeycombe [126]. The above earlier papers demonstrate the inhomogeneities of plastic deformation based on the analysis of surface orientation maps. Gradients in the GNDs density with depth were also observed by Mughrabi [15,16,65–67] and Ungar et al. [127] with conventional XRD. Our results confirm his observation that near the surface the ratio of GNDs to SSD density increases due to a reduced number of constraints near the surface. The analysis of 3D Laue diffraction synchrotron measure-

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ments thus allowed us to measure directly inhomogeneities and orientation gradients and distribution of GNDs with depth in the grain. We conclude, that in this example, depth-related inhomogeneities of plastic deformation are observed in a Ni grain of a uniaxialled deformed polycrystal. A decaying GNDs density with depth in the near surface grain results in an asymmetric Laue streak profile with a strong intensity maximum at one end of the streak and a long weak tail. The analysis of the experimental intensity distribution along the streak profile estimates the dislocation density gradient with grain depth due to distinct characteristic positions of the strong intensity maximum at one side of the streak for all Laue spots when corresponds to the case β > μ (Fig. 26(c)). Qualitatively such a distinct asymmetry in the intensity distribution along the streak profile means that both sharp and small strain gradients are present. This is definitely measurable with the penetration length of short-wavelength X-rays. Both the large rotation field near the surface and the smaller gradient within the grain are probed simultaneously, as shown in Fig. 24. This result is in agreement with our DAXM measurements of the grain rotation. The analysis of the dislocation density tensor shows that slip systems with dislocation lines inclined at approximately 45 degrees to the loading direction are activated first. A cumulative gradient in the cell–wall structure arrangement is observed for a “free surface” grain with the density of GNBs decaying into the depth of the grain. Several lamellar bands with different predominant slip systems and effective strain gradients are formed within the first 40 µm depth of the grain. Change from one slip band to another takes place at the depths of ∼12–15 µm beneath the surface. Regions of cumulative rotations with a length scale of about 15 µm are observed. The results of microscopic description are in good agreement with predictions from the continuum approach. Example 5. Analysis of heterogeneous dislocation network near an indentation in a Cu single crystal Microbeam-Laue analysis was applied to a complicated heterogeneous dislocation network arising from plastic deformation in a nanoindented Cu single crystal. The orientation of the Cu single crystal surface normal was parallel to [111], and the upper side of the pyra¯ midal nanoindent is almost parallel to [110]. The corresponding coordinate axes around the nanoindentation are shown in Fig. 53. Laue images were taken at various positions in and near the pyramidal nanoindent crater. Some experimental images are presented together with simulated images. Both the experimental data and the model images reveal long streaks with distinctive orientations that can abruptly change direction. To understand the shape of experimental Laue images and to check the sensitivity of the Laue image to different possible orientations of dislocations, the Laue images were simulated for different model arrangements of the primary unpaired dislocations. In FCC crystals, typical edge dislocation lines are parallel to the directions 112 with Burgers vectors being parallel to the directions 110 and corresponding glide planes {111} [17,43]. The plastic response of the material subjected to indentation can be understood by formation of excess or “geometrically necessary” dislocations (GND) that appear in the material to accommodate the indentor [7,91,95,96]. Usually only a few of these dislocation systems are activated depending on the conditions of deformation. The change of orientation of

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Fig. 53. Laue images reveal which dislocations are operative around the indentation. The following dislocations systems are used to model Laue images for A, B and C positions of the indentation: (a) two activated dislocaˆ ˆ ¯ direction: 1) τˆ [211], ¯ ¯ b[101]; tion system provide total rotation around [110] b[011]; 2) τˆ [12¯ 1], (b) beam ˆ ¯ intersects five regions with different orientations of geometrically necessary dislocations: 1) τˆ [101], b[10 1]; ˆ ˆ ˆ ˆ ¯ ¯ 2], ¯ b[110]; ¯ b[110]; ¯ 2) τˆ [1¯ 21], b[101]; 3) τˆ [11 4) τˆ [11¯ 2], 5) τˆ [2¯ 11], b[011]; (c) beam intersects four ˆ ¯ 2) τˆ [11¯ 2], ¯ regions with different actual dislocation systems and different rotation axes: 1) τˆ [011], b[01 1]; ˆ ˆ ˆ ¯ 2], ¯ b[110]; ¯ Scattering by dislocation system with τˆ [101], found for b[110]; 3) τˆ [11 4) τˆ [101], b[10 1]. ˆ ¯ ¯ 1] B and C positions is equivalent to the scattering by two octahedral systems (τˆ [121] + τˆ [121]) with b[10 and equal dislocation density. To distinguish those cases, simultaneous analysis of several Laue spots is usually performed.

the primary dislocation set surrounding the nanoindentation results in a reorientation of the Laue images (Fig. 53). A simple image (uniaxial elongated streak) is obtained for the position (A). The streak can be described by two activated dislocation systems resulting ¯ direction with total dislocation density n = 2 × 1011 cm−2 in the rotation around [110] and an excess dislocation density n+ = 0.5n. Images taken at the positions (B) and (C) (Fig. 53(b), (c)) are more complicated. At these positions, the penetrating beam intersects several fragments with different rotation axes relatively to the indentation. Plastic deformation during indentation activates different slip systems of the dislocations in these fragments.

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The application of DAXM for the analysis of GNDs distribution around the nanoindent [95,96] makes it possible to observe the dramatic change in the activated dislocation networks along the beam penetration depth. With depth, the beam intersects several regions with different GND dislocation network. The transition from one dislocation network to another and corresponding Laue images around the indentation are shown in Fig. 53(b), (c). These dislocation sets form a dislocation network surrounding the whole indentation and can be used to explain the elasto-plastic response of the crystal during nanoindentation.

7. Instrumentation Specialized instrumentation is required to efficiently perform polychromatic X-ray microdiffraction. This equipment is schematically shown in Fig. 54. The actual process of measuring local structure with polychromatic microdiffraction was outlined in Section 4. A broad-bandpass microbeam intercepts a sample and illuminates a small number of crystal grains. The overlapping Laue patterns from the illuminated grains are recorded by an X-ray sensitive area detector and fit to find the reflection centers. The angles between the reflections are compared to the angles anticipated for the crystal structure of the sample. When the n(n − 1)/2 angles between n reflections have angles that match those predicted for a particular set of reflections, the reflections are assumed to be indexed. Owing to the exquisite angular sensitivity of the measurements once 4 or more reflections are indexed the probability for a false indexation is small. Finally, the texture (orientation) and relative or absolute elastic strain tensors are determined for the illuminated grains. The dislocation tensor is then determined from the 3D local orientation distribution. The small X-ray beam is focused onto the sample with non-dispersive total-externalreflection Kirkpatrick–Baez mirrors (Fig. 54). The beam can be switched between polychromatic and monochromatic modes with a special microbeam monochromator (Fig. 54).

Fig. 54. Schematic of the key elements of a 3D polychromatic X-ray microprobe. The intense white-beam source can either be passed directly to the focusing mirrors or monochromated by a special translating monochromator. The K-B mirrors focus the polychromatic radiation to a submicron spot. The sample is translated with a precision 3 axis stage and a heavy-Z wire runs near the sample surface to decode the origin of the Laue patterns along the incident X-ray beam. The X-ray Laue patterns are detected with an X-ray sensitive CCD.

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The idealized approach outlined above, is often complicated by significant deformation that smears Laue spots and makes it difficult to index reflections or by the presence of submicron grains that create thousands of Laue patterns from the penetrating beam. For this reason, a suite of tools has developed to address various complexities of real data. These tools include the hardware and software to resolve the Laue patterns generated along the beam path (DAXM) and energy scanning methods that differentiate scattering with respect to momentum transfer. Below we outline the special instrumentation and software essential for general-purpose investigations of deformation in materials.

7.1. X-ray source and reflectivity considerations As discussed previously, the central problem of microdiffraction is to simultaneously achieve good momentum-transfer and good spatial resolution. Laue methods help solve most of the spatial resolution problems by providing reciprocal-space projections without sample rotations. However, in order to make precision measurements of crystal properties such as orientation (elastic strain) at least 2 (4) reflections are required; many more reflections are useful to reduce statistical and instrumental uncertainties to achieve 100 microradian angular (100 ppm strain-tensor) resolution. However, there is a compromise between high-energy beams that provide many simultaneous reflections, low sample heating and good penetration, and low-energy beams that have better reflectivity and better near-surface sensitivity. To understand this compromise it is important to understand how reflectivity, number of reflections, penetration and sample heating scale with wavelength, bandpass and total power. Number of reflections. As described pictorially in Fig. 54 the total volume of reciprocal space sampled by a Laue instrument depends on the solid angle of the detector, its orientation with respect to the incident beam, the average wavelength of the radiation and the beam bandpass. For detectors oriented at 90 degrees to the sample, the reciprocal space volume is given by, V∼

λ . λ3 λ

(7.1)

Here is the solid angle subtended by the detector, λ is the average X-ray wavelength, and λ is the bandpass. For a simple FCC real-space lattice with unit cell dimension a0 the reciprocal-space lattice is a BCC lattice with reciprocal space unit cell dimensions 2/a0 . The total number of reflections anticipated is therefore given by, nr =

a03 λ . 4λ3 λ

(7.2)

Whereas high X-ray energy (short wavelength) increases the number of reflections collected for a given bandpass and detection geometry, higher X-ray energy drives reflections to large momentum transfer, Q, with higher-order indices and lower reflectivity. Reflectivity is further reduced by the decreasing atomic scattering factor of high-indices reflections,

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Fig. 55. The integrated intensity for typical reflections goes down by approximately two orders-of-magnitude from 10 to 20 keV for Cu and Al with a fixed 2θ angle of 90 degrees.

by geometric wavelength dependence, and by the Debye factor. For example, the reflectivity for small grains can be estimated within the kinematic approximation [6] and scales as, R ∝ λ3 f 2 e−2M .

(7.3)

Here the atomic scattering factor, f , and the Debye–Waller term M depend on sin θ/λ which changes with λ for fixed scattering angle θ ∼ 45◦ (Fig. 55). The penetration length of X-rays into a sample depends sensitively on wavelength and sample material. Although there are absorption thresholds that dramatically change the penetration length of X-rays, there is an overall atomic number (Z) and wavelength (λ) dependence for the absorption length; absorption length scales roughly as Z −3 –Z −5 and as λ−3 . The penetration length also significantly influences the temperature rise in a sample. The temperature rise anticipated in a sample can be estimated from a simple worst-case scenario of a thin film sample. In this approximation, the absorption/unit thickness is constant and the problem becomes one-dimensional for a round microbeam. The temperature rise is estimated as, R2 P μ 1 . (7.4) + ln T ∼ 2π k 2 R1 Here μ is the absorption coefficient, k is the thermal conductivity, P is the total beam power, R1 is the radius of the beam and R2 is the radius of the heat sink. Because T scales with total power but inverse logarithmically with probe size, often small polychromatic microbeams from even the most intense X-ray source do not significantly heat up samples. The reflectivity and number of reflections are plotted in Fig. 55 as a function of λ. As shown in Fig. 56 the scattering intensity drops by two orders of magnitude from 10 to 20 keV. Correspondingly the number of reflections for a 50% bandpass increases by a

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Fig. 56. A comparison of the on and off axis brilliance an APS type A undulator to the brilliance from the ALS superbend source.

factor of 8 and the penetration depth also improves by around a factor of 8. For small unit-cell crystals, a good compromise is a beam with a bandpass around 10–16 keV. Once the ideal wavelength and bandpass are set, it is important to optimize the average flux from each reflection. For polychromatic microdiffraction the source figure-of-merit is the integrated source brilliance; a brilliant (photons/s/mm2 /mrad2 /eV) X-ray source is essential because of the small phase space that can be accepted for microdiffraction experiments and wide bandpass is essential to collect a sufficient number of reflections. The dedicated station 34-ID-E at the Advanced Photon Source (APS) uses a high-energy ultrahigh brilliance third-generation undulator sources. This source provides on-axis spectra that vary strongly with energy (Fig. 56). By moving slightly off axis, the spectral distribution can be flattened. The dedicated Advanced Light Source (ALS) polychromatic microdiffraction station uses a high-brilliance bend magnet source [108] and operations are moving to a superbend with significantly better brilliance in the crucial 10–20 keV range. Although the on-axis source brilliance is 4 orders of magnitude greater for the APS source than for the ALS source, the integrated brilliance of the ALS spectra is only about two orders of magnitude lower than for the APS undulator and is actually higher than for an APS bend magnet. A comparison of the on and off axis brilliance an APS type A undulator to the brilliance from an ALS superbend source is given in Fig. 56.

7.2. Microbeam monochromator After the source, the next key element is an insertable monochromator that can cycle the beam between polychromatic and monochromatic modes. The ability to change to mono-

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chromatic beams allows the energy of individual reflections to be precisely determined. This is necessary to determine the hydrostatic stress of the local volume. Moreover, monochromator allows the beam energy to be swept over an energy range so that only a few reflections are simultaneously excited, even from a sample with small grains. This allows for interpretation of complicated Laue patterns generated from samples with many small grains or with highly deformed grains. Energy scanning also provides full 3D volumes in reciprocal space for the most demanding measurements of intensity redistribution due to the presence of defects. The 34-ID-E monochromator is a two-crystal monochromator with a small offset that is designed to efficiently pass best source brilliance for monochromatic measurements and to broaden the undulator harmonic spectra for polychromatic measurements [103]. Although the monochromator has a small offset, it uses different parts of the incident beam to bring the monochromatic and polychromatic beams into registry at the exit slit of the monochromator tank. The ALS polychromatic microdiffraction station uses a four-crystal design with no beam displacement [108]. This design inherently keeps the beam fixed when cycled between polychromatic and monochromatic modes. The thermal behaviour of a simple water-cooled incident beam monochromator is acceptable even on a high-thermal density undulator line as described previously [102]. The choice between a simple two-crystal non-dispersive monochromator with small displacement and a four-crystal monochromator with no beam displacement depends on the source properties. For a bend-magnet source with a polychromatic beam, four-crystal geometry has the advantage of simultaneously restricting the source size in the plane of scattering.

7.3. Achromatic focusing The next key element is an achromatic focusing system. Kirkpatrick–Baez total-externalreflection mirrors are the common choice for achromatic focusing to submicron dimensions as they allow for efficient collection of the useable phase space [104,128]. The mirrors focus the polychromatic/monochromatic beam onto a sample that is mounted on a precision 3-axis translation stage. For 3D measurements a differential aperture wire is also mounted on a separate 3-axis stage. The stages typically require reproducibility of less than 100 nm and stages on 34-ID-E are rated to 50 nm bi-directional reproducibility.

7.4. Detector The detector is also a key element for polychromatic microdiffraction. Because elastic strain measurements require absolute angle measurements, it is important to have a detector with uniform inter pixel spacing. Similarly fitted peaks can only be accurately centered if the efficiency of the detector is uniform and if the dynamic range is sufficiently large to accept both weak and strong signals without distortion of the peak shape. The number of pixels also determines the angular precision that can be achieved within a given solid angle. Large solid angle is needed to intercept many reflections, and small pixel size is needed

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to get high angular resolution on detected reflections. Finally, for many experiments, readout time is essential because the data collection time can be small compared to the detector readout time. Detectors with readout speeds better than 1 frame/sec are essential, and readout speeds up to 100 frames/sec are desirable. The X-ray area detector is positioned with a linear encoder to allow for triangulation calibration of the detector-to-sample distances.

7.5. Software DAXM. Methods to recover the depth resolved Laue images from along the penetrating X-ray beam are still developing. The current software was written by Yang and Larson, and is based on fitting each pixel to a step function to precisely locate the position of the wire when its shadow blocks intensity into the pixel. The pixel position and the wire edge position are then used to ray trace back to the incident beam and determine the source of the intensity into the pixel. This method works remarkably well for most samples. Indexing. Beyond measuring normals, it is essential to identify the crystal plane belonging to each reflection (indexing). This important step is a general problem of Laue diffraction [129,130], but most existing methods are difficult or impossible to use with microdiffraction data due to overlapping patterns. Several methods similar to the one described below have been found to work well with simple cubic metals where only a few (4–20) reflections are collected for each grain and where only a few (5–10) overlapping Laue patterns are present. An indexing flowchart is outlined in Fig. 57. In this example, Bragg plane normal and 2θ angles for each reflection are noted and possible indices of each reflection are estimated based on 2θ and the energy bandpass. For a given bandpass and a known crystal structure, there are only a limited number of possible pairs of indices h, k, l’s. The upper bound of h, k, and l is set by the energy bandpass. The angles between the Bragg plane normals of possible pairs of indices are compared to find pairs within the estimated measurement and strain uncertainty. Angles between three or more reflections are then compared to identify all reflections from a single grain. With a reasonable selection of bandpass, a manageable number of possible indices can be obtained. For example, with a 10% bandpass at 20 keV, the number of possible indices for each observed reflection is ∼200–500 for a Si crystal. The list of possible indices increases linearly with bandpass; the algorithm slows for very large bandpass. Once a list of possible indices for each reflection is tabulated, the experimental and theoretical angles between pairs of reflections can be checked. Pairs that do not fall within a specified uncertainty indicate that one or both assumed indices are incorrect, or that the reflections belong to distinct grains all possible combinations of indices can be checked, until only pairs that are consistent with the known crystal structure are determined. When the relative angles between more than three reflections are checked, the number of possible indices converges quickly. The example above is only one possible approach. In a more elegant approach, one calculates the Laue pattern anticipated about each indexed reflection assuming a rotation of the grain about the reflection normal. If the reflection index is correct, there will be

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Fig. 57. Flow-chart for indexing. The possible indices for each reflection are determined from the known bandpass of the radiation, the known unit cell of an unstrained grain, and the 2θ angle of the reflection. The measured angles between reflections are then compared to the theoretical angles for an unstrained grain with all the possible indices. If the angles lie within the expected experimental and strain uncertainty, then the indices are tentatively assigned to the same grain. If no indices can be found which fall within the expected angular uncertainty, then at least one reflection is stored for later evaluation with a new grain. The flow chart does not show the steps required for calibrating the detector system.


597

a simultaneous alignment of a large number of theoretical and experimental reflections when the orientation angle of the model grain is correct. Once the reflections from a single grain are determined, these can be removed from the pattern and the remaining reflections can be indexed to discover a second grain. This process can be continued until virtually all reflections are indexed. In this way several grains can be indexed from a single Laue image. Even grains of two different crystal structures can be identified. For example, a Laue image from a heterostructure of thin films can be analyzed to index the reflections from each layer. Of course the presence of an unsuspected crystal structure can complicate the analysis. Once the indices of each layer are determined, the orientation and strain tensors can be found as described previously. Still another approach utilizes a Hough transform to find poles in reciprocal space. This approach is particularly useful for measurements of highly mosaic thin film samples [131].

8. Concluding remarks X-ray microdiffraction provides a powerful tool for the study of plastic deformation in materials. Plastic deformation introduces geometrically necessary dislocations and dislocation walls that produce elongated diffraction streaks. The streaks can be used to estimate the dislocation density tensor. For single slip deformation the predominant slip system can be identified because of its distinctly different streaking of the Laue patterns. For multiple slip deformation the components of the dislocation density tensor may be determined. The influence of various dislocation structures on the orientation space and white beam diffraction is summarized in Table 3.

Acknowledgements Research is sponsored by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy (DOE), under contract DE-AC0500OR22725 with UT-Battelle, LLC. Experimental work performed on beamlines 34-ID at the Advanced Photon Source, which is supported under DOE contract No. W-31-109ENG-38 with Argonne National Laboratory and on beamline 7.3.3 of the Advanced Light Source, which is supported by DOE Contract No. DE-AC02-05CH11231. We gratefully acknowledge a long-standing collaboration that has made this research possible, with Bennett C. Larson, John D. Budai, Jonathan Z. Tischler, Wenjun Liu and Judy W. Pang.

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CHAPTER 80

X-Ray Imaging of Phonon Interaction with Dislocations DORON SHILO and

EMIL ZOLOTOYABKO Department of Materials Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel



Contents 1. Introduction 605 2. Survey of dislocation interaction with phonons 606 2.1. Dislocation dynamics 606 2.2. Viscous drag of dislocations 610 2.3. Experimental methods for studying phonon interaction with dislocations 612 2.4. Theoretical models describing the scattering of acoustic waves by the dislocation strain field 614 2.5. Theoretical models for the dynamic interaction of acoustic waves with dislocations 615 3. Stroboscopic X-ray imaging of acoustic waves 620 4. Interaction of acoustic waves with individual dislocations in brittle ceramics 624 4.1. Experimental results 624 4.2. Theoretical model and simulations 625 4.3. Dislocation analysis 632 4.4. Extraction of the physical characteristics of vibrating dislocations 634 4.5. Conclusions 636 References 638

1. Introduction Phonon-dislocation interaction has a strong effect on various materials properties. For example, phonon-induced dislocation vibrations significantly influence all characteristics related to the phonon flux, such as thermalsuch as and acoustic attenuation. Besides that, interaction of moving dislocations with phonons determines the viscosity coefficient in the dislocation motion, which in turn controls the dislocation dynamics. The interaction between dislocations and phonons has been extensively studied for decades. Nevertheless, the established experimental methods (internal friction and thermal conductivity measurements) suffer from two principal disadvantages that hamper the studies in this field. First, the existing methods provide only averaged information over an ensemble of dislocations, with neither spatial nor temporal resolution. In other words, we are unable to resolve the interaction of phonons with individual dislocations, which is the subject of theoretical models. Secondly, the existing methods can be applied (and have been applied) only to ductile crystals, such as metals and alkali halides. Ductile crystals are characterized by high viscosity of dislocation motions, which results in the slowing down of dislocation movements. As shown later on, the viscosity coefficients in brittle ceramic crystals, such as LiNbO3 , may be 2–3 orders of magnitude lower than in ductile crystals. In order to overcome these difficulties and limitations we have developed a new imaging technique, called high-frequency stroboscopic X-ray topography. It allows us to visualize, on the same image, the deformation fields of both individual acoustic wave fronts and vibrating dislocations. In order to visualize rapidly changing dynamic deformation fields, we conduct high-frequency stroboscopic measurements in which the acoustic wave propagation is synchronized with X-ray bursts coming from a synchrotron source to the sample position. Comprehensive theoretical analysis enabled us to find optimal experimental conditions for enhanced contrast. We also found a way to apply this technique to non-piezoelectric crystals. Due to these developments, it is now possible to visualize an interaction of short wavelength (down to 6 µm) surface acoustic waves with individual dislocations in a wide variety of crystals. X-ray images, which were taken from LiNbO3 crystals excited by surface acoustic waves, revealed significant distortions of acoustic wave fronts in the vicinity of dislocation lines. We showed that these distortions are related to the dynamic deformation field of the vibrating dislocation string. A theoretical model developed enabled us to simulate the overall dynamic displacement field, which is a combination of the displacement fields of the surface acoustic waves and the vibrating dislocations. By comparing the experimental and simulated images, the amplitudes and velocities of individual vibrating dislocations were determined for the first time. It was found that dislocations can reach nearly “relativistic” velocities (close to the speed of shear bulk waves) under the subtle strains generated by acoustic waves. This is because

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Ch. 80

of the very low viscosity of dislocation motion in LiNbO3 , being at least 2 orders of magnitude lower than any value measured until now in all previously investigated materials. Stroboscopic X-ray imaging in the GHz frequency range, as reviewed in this chapter, significantly expands the ability to investigate dislocation interactions with phonons in single crystals, including brittle ceramics, which were beyond the scope of the established experimental techniques. The chapter is written to encompass all the information required to introduce the reader fully into the subject. Section 2 presents the relevant background of the theory of dislocations (with a focus on their dynamics) and comprises a survey of experimental methods and theoretical models for studying their interaction with phonons. In Section 3, the principles of stroboscopic X-ray imaging of acoustic waves are explained and the experimental technique is described in detail. Section 4 is devoted to the application of this method for studying the interaction of acoustic waves with individual dislocations in brittle ceramic crystals, such as LiNbO3 .

2. Survey of dislocation interaction with phonons 2.1. Dislocation dynamics The motion of dislocations in their gliding plane is generally determined by the following equation [1]: m∗

∂ 2ξ ∂ 2ξ ∂ξ = FExt − FP − FBar , − T +B 2 2 ∂t ∂t ∂l

(1)

where ξ describes the temporally and spatially dependent deviation of the dislocation shape from a straight line situated along an energetically favored lattice direction. The first term on the left side represents the dislocation inertia, where the effective mass per unit length is approximately m∗ ∼ = 0.5ρb2 , with ρ being the material density and b the Burgers vector. The second term on the left side gives the resistance of the dislocation to its elongation, where the line tension, T ≈ 0.5μb2 , approximately equals the energy per unit length of the dislocation line [1–3], with μ being an effective shear modulus. The values of m∗ and T depend on the dislocation character (e.g., edge, screw or mixed dislocation), and the ratio between them,

T = m∗

μ = VT ρ

(2)

defines the shear wave velocity, VT , which also gives the speed of the atomic movements along the dislocation line. The third term on the left side of eq. (1) describes the viscous resistance to dislocation motion. The viscosity coefficient, B, is governed mainly by phonon interactions with the moving dislocation. The viscosity significantly affects not only the dislocation dynamics, but also the phonon propagation within the crystal. Thus, the dislocation interaction with

§2.1

X-ray imaging of phonon interaction with dislocations

607

phonons plays a significant role in such phenomena as heat conductivity [4,5] and attenuation of acoustic waves [6,7]. The terms on the right hand side of eq. (1) describe forces per unit length that act on the dislocation line. The expression FExt is the force exerted by the external stress, σij , on a unit of dislocation length, and it is given by the expression [8] FExt =

(3)

ni σij bj ,

where the vector n with components ni is a unit vector normal to the glide plane. The term FP on the right side of eq. (1) is called the Peierls force [9] and is caused by the periodic lattice potential applied to the dislocation line. The Peierls potential appears as a periodic array of valleys and hills in the glide plane. The Peierls force is typically expressed as [9] FP = σP b sin

2πξ a

(4)

where σP is the external stress required to overcome the Peierls potential barrier (i.e. to move the dislocation out of a potential valley), and a is the distance between adjacent potential valleys, which as a rule is equal to the shortest lattice translation. Note, that for screw and mixed dislocations the vectors a and b are not parallel and hence are not necessarily equal. Nevertheless, since b also tends to follow the shortest lattice translation, in many cases a = b. The ratio σP /μeff is roughly 10−4 in fcc metals, 10−3 in bcc metals and ionic crystals, and 10−2 in covalent crystals such as silicon [1]. The term, FBar , on the right side of eq. (1) represents local forces exerted on the dislocation line by defects, such as precipitates or other dislocations. These defects typically form pinning points which prevent the dislocation motion across a crystal. By using eq. (1), one can predict both the steady-state shape of a dislocation line and the mode of dislocation motion. Let us consider, for example, a single crystal which during its growth or during annealing treatment is subjected to elevated temperature and no external forces (i.e. FExt ∼ = 0). In the steady state, the time derivatives of ξ are equal zero. If the density of pinning points is low, there exist rather long dislocation segments, for which FBar ∼ = 0. These segments obey the following stationary equation 2πξ ∂ 2ξ =0 T 2 − σP b sin a ∂l

(5)

which follows from eq. (1) by setting all parameters mentioned above to zero and using eq. (4) for the Peierls force. The trivial solution of eq. (5), ξ = 0, describes a dislocation line which is located within a single Peierls potential valley. This is extremely rare since the dislocation segments are, generally, pinned with their ends located at different potential valleys. In this case, there is competition between the first term of eq. (5), which acts to straighten the pinned dislocation line, and the second term, which tries to confine the dislocation segment within a certain potential valley. In a crystal, in which the second term is dominant, the dislocation lines

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Fig. 1. Possible shapes of dislocation lines, which are pinned at two points located in different Peierls valleys (indicated by dashed lines): (a) – dislocation line is composed of long segments, which are completely confined to Peierls valleys; (b) – dislocation line is composed of short segments which are located in Peierls valleys and connected by kinks, which pass over potential barriers, (c) – nearly straight dislocation line, which is not confined to Peierls valleys.

are typically confined within Peierls valleys and form shapes as illustrated in Fig. 1(a). A classic example of such a crystal is silicon [10]. If the second term of eq. (5) is not dominant, the dislocation line can follow another solution of eq. (5) that describes a more complicated dislocation shape (see Fig. 1(b)). In this case, it is composed of segments located in adjacent Peierls valleys and connected by small fragments called “kinks” [2]. Thus, the dislocation line can link the two pinning points by forming a set of kinks as illustrated in Fig. 1(b). For geometrical kinks the average distance between two adjacent ones is L = a/δ

(6)

where δ is the angle between the direction of the Peierls valleys and a straight line that connects the pinning points. The width of an individual kink is given by [2]: ∂l πT Wk = a (7) =a ∂ξ ξ =a/2 2abσP and is typically between a few and a few tens of lattice parameters.

§2.1


609

The effect of the Peierls potential on dislocation motion depends on the ratio between L and Wk . If L ≫ Wk , the majority of the dislocation line is confined to Peierls valleys (as illustrated in Fig. 1(b)), and the entire dislocation segment is subjected to the Peierls force which hampers its motion. In contrast, if L Wk , the dislocation line is no longer confined to Peierls valleys (see Fig. 1(c)), and correspondingly, the dislocation motion is no longer subject to Peierls forces. Considering the dislocation motion under the general eq. (1), one can indicate three different regimes corresponding to three specific intervals of the external force. If FExt is larger than both FBar and FP , the dislocation movement in the long term will be governed by viscosity resistance. According to eq. (1), after a short time the dislocation will reach a uniform steady state velocity BVd = FExt ,

(8)

where Vd = ∂ξ/∂t. If the external force is in the range, FP < FExt < FBar , the dislocation line can be pinned as a whole, but segments located far from the pinning points can easily move. If the dislocation can get away the pinning points by thermal activation, its motion will be determined by the combined effect of the activation rate at the pinning points and the viscous motion between the pinning points. In the range where FExt is smaller than both FBar and FP , the dislocation segment can overcome the Peierls potential by a thermal activation process only. This is done by a formation of “bulges” on the dislocation line, which are composed of two kinks and a very short fragment between them (see stage (a) in Fig. 2). After the formation of a bulge, the kinks move in opposite directions along the Peierls valley and thus drag the dislocation line to the neighboring valley (see stages (b) and (c) in Fig. 2). The stress required to move kinks parallel to the potential valleys is very small; h ence, kinks typically move in viscous mode. The kinks that are formed by this mechanism are called “thermal kinks”, in contrast to geometrical kinks which were depicted in Fig. 1(b). It becomes clear from the above analysis that the viscous resistance to dislocation motion plays a significant role in dislocation dynamics. As follows from eq. (1), the viscosity coefficient, B, is the only parameter whose magnitude is not a priori known because of

Fig. 2. Thermally activated motion of a linear dislocation over the Peierls barrier. Stage (a) – formation of a bulge, which consists of a pair of kinks and a short segment already situated in the adjacent Peierls valley. Stage (b) – movement of the kinks in opposite directions along the Peierls valley. Stage (c) – drag of the entire dislocation line into the adjacent Peierls valley.

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the complexity of viscosity mechanisms. Therefore, there is a necessity for experimental studies aimed at obtaining viscosity coefficients to dislocation motion for different materials.

2.2. Viscous drag of dislocations It is possible to divide the mechanisms of viscous drag of dislocations into three categories which are described below. 2.2.1. General viscous behavior The dynamic deformation accompanying the dislocation motion can be considered in terms of elastic waves that create an excess of phonons and disturb the local phonon equilibrium. The dispersion and absorption of the non-equilibrium phonons results in a viscous resistance to dislocation motion. The viscous behavior is a general property of a crystal, which causes an attenuation or drag to any waves, not only to those that are generated by moving dislocations. For example, waves that create a local excess of phonons can be caused by temperature gradients or by an external excitation of acoustic waves. Therefore, the mechanisms that are responsible for general viscous behavior also affect the heat conductivity and attenuation of acoustic waves. There are different mechanisms for the dispersion and absorption of phonons in crystals. The dominant is phonon-phonon interaction, the theory of which was developed by Akhiezer [11]. As was shown later on in Ref. [12], phonon–phonon interactions significantly contribute to the overall viscosity coefficient of dislocation motion. Similar to the phonon interaction with the “phonon gas”, there also exists interaction with the “electron gas” [13], but its effect is much weaker, even in metals [1,14]. Another mechanism that causes the dispersion and absorption of excess phonons is due to the thermoelastic effect, which reveals itself in changing temperatures under stress. Hence, the dislocation motion is accompanied by heat transfer and thereby causes an energy loss. 2.2.2. “Phonon wind” – scattering of phonons due to non-linear elastic behavior In this mechanism, phonons are scattered by the deformation field of the dislocation due to small changes in the material density and elastic modules (the detailed description of this effect is given in Section 2.4). Phonon scattering influences the attenuation of acoustic waves due to their scattering by static dislocations and the resistance to dislocation motion due to the scattering of the crystal phonons by moving dislocations. In the coordinate system of a moving dislocation, the phonons propagate with the same speed as the dislocation moves, but in the opposite direction. This “phonon wind” transfers momentum to the dislocation line opposite to the dislocation motion, with momentum proportional to the dislocation velocity. Calculations [15] showed that this mechanism usually makes an important contribution to the overall viscosity coefficient. 2.2.3. Acoustic emission Time-dependent changes of the dislocation velocity and related kinetic energy losses are a source of acoustic emission [14]. Local velocity changes are due to interactions with

§2.2

611


crystal defects or thermal fluctuations. The latter is called “fluttering” and has a similar effect on thermal conductivity and attenuation of acoustic waves. The Peierls potential also causes small perturbations of the dislocation velocity as the dislocation line moves from valleys to hills and so forth (the radiation friction effect). Calculations [14] showed that the mechanisms of acoustic emission are relatively weak, and their influence is only important under certain conditions such as low temperatures or very high Peierls barriers. 2.2.4. Viscosity coefficients in different materials It can be concluded that the viscosity coefficient of dislocation motion is a result of phonon interactions with moving dislocations. Most of the described mechanisms for viscous drag of dislocations also have a significant effect on the heat conductivity and attenuation of acoustic waves. The room temperature viscosity coefficients, B, of dislocation motion, measured by different methods described in the next section, are summarized in Table 1. The wide ranges of measured B-values (hundreds of percent) are, in part, due to the uncertainty of the dislocation character and partially due to large errors inherent to some of the experimental techniques. Due to limitations of conventional experimental methods, the variety of crystals in which the viscosity coefficients were measured is also very limited and restricted to metals and alkali-halides, which are ductile at room temperature. The typical viscosity coefficients in ductile materials are rather high, ranging from 0.1 to 3 mPoise. In brittle materials, the viscosity coefficients are much smaller and cannot be measured by conventional methods. In order to evaluate them one can assume the proportionality between the viscosity coefficient, B, and total acoustic wave attenuation, α. This assumption is well justified for general viscous behavior (see Section 2.2.1) for which a mechanical viscosity, η, can be defined, and it can be proven that both B ∝ η [12,14] and α ∝ η [7]. Other types of viscosity mechanisms, including the phonon wind and acoustic emission, also affect both the B- and α-values, but not necessarily in the same proportion. The room-temperature values of the acoustic attenuation, α, for different types of materials are summarized in Table 2, which shows remarkable variation among groups containing metals, semiconductors, and insulating ceramics. The typical α-values for semiconductors are an order of magnitude smaller than those for metals, and the typical α-values for insulating ceramic crystals are 2–3 orders of magnitude smaller than in metals. This fact motivates the study of dislocation motion in insulating ceramic crystals such as the ones used in acoustic devices due to their low acoustic attenuation.

Table 1 Measured values of the viscosity coefficient, B, of dislocation motion in single crystals at room temperature (Ref. [14]) Crystal

LiF

NaCl

KCl

KBr

Cu

Al

Pb

Zn

Nb

B (mPoise)

0.3–1.3

0.1–0.3

0.3–0.8

1.7–2.0

0.1–0.9

0.2–3.1

0.4

0.4–2.5

0.2

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D. Shilo and E. Zolotoyabko Table 2 Total acoustic attenuation coefficients (at 1 GHz) in cubic single crystals at room temperature (Ref. [7]) Crystal Metals Al Cu Au Semiconductors Si Ge Ceramic insulators MgO Strontium titanate Yttrium iron garnet Yttrium aluminum garnet

Propagation direction

Wave mode

Attenuation coefficient (dB/m)

110 100 110

Longitudinal Longitudinal Longitudinal

7500 27000 20000

100 111 100 100

Longitudinal Longitudinal Longitudinal Shear

1000 650 2300 1000

100 100 100 100 100 100 100

Longitudinal Shear Longitudinal Longitudinal Shear Longitudinal Shear

330 40 600 200 34 20–32 110

2.3. Experimental methods for studying phonon interaction with dislocations Until our research, there were three conventional experimental methods for studying the phonon interactions with dislocations: measurement of dislocation mobility, internal friction, and measurement of heat conductivity. 2.3.1. Measurements of dislocation mobility This method [14,16] is based on the measurement of the distance, x, that individual dislocation propagates under application of a stress pulse with an amplitude σ , and duration, t. If the stress value is high enough to provide FExt ≫ FBar , FP (see eq. (1)) the whole dislocation line, after a short time, τ0 , reaches the steady state velocity, Vd (σ ), as was explained in Section 2.1. Substituting FExt = σ b (see eq. (3)) in eq. (8), yields: BVd = σ b.

(9)

The dislocation velocity is calculated as Vd = x/t, which is valid until τ0 ≪ t, and after that the B-value is extracted using eq. (9). In order to measure the distance, x, selective etching of dislocation pits before and after the application of the stress pulse is done, followed by the subsequent measurement of the distance between pits by optical or electron microscopy. The accuracy to measure the B-value for an individual dislocation is about 30%. At room temperature, this technique can be used only for ductile materials since brittle materials are subjected to fracture before the dislocation motion starts (i.e. at smaller

§2.3


613

stresses). By using this technique, brittle materials can be investigated only at elevated temperatures (when strictly speaking they are no longer brittle). Unfortunately, the results of these studies cannot be extrapolated to room temperature because the mechanism of dislocation motion is changed with temperature (thermal activation instead of viscous drag (see Section 2.1)). Except the overall value of B, this method does not provide any information concerning the details of phonon interaction with dislocations and its sensitivity to the phonon frequency and structure of the dislocation line. Dislocation vibrations under an excitation of acoustic waves, which play very important role in heat conductivity and attenuation of acoustic waves, are beyond the scope of this technique. 2.3.2. Internal friction In this method [3], the effect of vibrating dislocations on acoustic wave propagation is probed by measuring the changes of the acoustic wave attenuation and velocity as functions of its frequency and amplitude. These measurements provide information, not only on the overall value of B but also on the specific mode of interaction between a dislocation and acoustic wave. The majority of experimental results were explained in the framework of the vibrating string model, developed by Granato and Lücke [3], which describes the vibration of pinned dislocation segments (for details see Section 2.5.2). The B-value is determined by measuring the asymptotic limit of the acoustic wave attenuation, α∞ , at frequencies much higher than the resonance frequencies. According to the vibrating string model, the part of α∞ which is caused by dislocation vibrations, is given by [17]: dis = α∞

4χμb2 N , π 2 BVT

(10)

where N is the density of free dislocation segments, and χ is the Schmid factor. The latter connects the acoustic stress tensor to the effective shear resolved stress that acts on the dis -values from measured α dislocation line in its glide plane. In order to extract the α∞ ∞ coefficients, the acoustic attenuation is taken twice, before and after gamma irradiation. dis Point defects, generated by the radiation, prevent the dislocation motion and hence α∞ is nearly given by the difference between α∞ -values measured before and after sample irradiation. Evaluation of the density of free dislocation segments in eq. (10) is more problematic since the dislocation lines most often have multiple pinning points and cannot vibrate. This complication is overcome by using specimens subjected to plastic deformation at room temperature with no annealing. In such specimens the majority of dislocations are free to move. The experimental procedure limits the implementation of internal friction technique to ductile materials only. 2.3.3. Measurements of heat conductivity The principle of this method is similar to that one used for the internal friction measurements, but now the subject is an effect of vibrating dislocations on heat conductivity. The heat conductivity at low temperatures is determined by measuring the traveling time of a

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Ch. 80

thermal pulse excited by a laser [18]. An improved technique comprises two-dimensional mapping of the thermal pulse propagation [5,19] and allows one to study heat conductivity along different crystal directions. In this method, as for the internal friction technique, one has to separate the net effect of dislocation vibrations from other interactions which influence the heat conductivity. This is accomplished by doing measurements before and after plastic deformation at room temperature, and hence this method is also limited by applications to ductile materials. 2.3.4. Drawbacks of conventional experimental methods We repeat that the described experimental methods suffer from two principal disadvantages that hamper the studies in this field. The first is that the methods for studying the phonon interaction with dislocations (i.e. internal friction and heat conductivity measurements) provide only averaged information on the ensemble of dislocations with neither spatial nor temporal resolution. In other words, they are not able to probe the phonon interaction with individual dislocations, which is the subject of theoretical models (see Section 2.5). There is an attempt to overcome this problem (see Section 2.5.2) by summing the effect of numerous dislocations having different geometrical parameters. This approach is problematic since the strength of interaction changes by several orders of magnitude depending on geometrical factors, which may be very different for individual dislocations. Moreover, the result of the summation depends on some poorly defined parameters such as the average length of free dislocation segments, which can hardly be measured and may vary within several orders of magnitude. Another disadvantage of the existing techniques relates to the fact that they can be applied (and have been applied) only in ductile crystals such as metals and alkali halides. Ductile crystals are characterized by high viscosity coefficients (see Section 2.2.4), which result in sluggish dislocation motion. As is mentioned in Section 2.2.4 and shown recently by us in Ref. [20], the viscosity coefficients in brittle ceramic crystals, such as LiNbO3 , may be 2–3 orders of magnitude lower than in ductile crystals. Correspondingly, the velocities of dislocation segments can reach very high values (close to the speed of sound) which cannot be measured experimentally. In order to overcome these difficulties we developed a fast time-resolved X-ray visualization technique which simultaneously gives the images of the propagating acoustic wave and of the vibrating dislocation (see Section 3). In order to analyze these images, the theoretical models dealing with phonon scattering by dislocations should be revised. This is done in Sections 2.4–2.5.

2.4. Theoretical models describing the scattering of acoustic waves by the dislocation strain field This section deals with scattering of acoustic waves by a static dislocation or by a dislocation which moves with no relation to acoustic waves. In this case, the acoustic waves are scattered due to non-linear elastic behavior in the vicinity of the dislocation line. In other words, the scattering mechanism is based on the changes of the acoustic wave speed

§2.5


615

caused by strain-induced modification of material density and elastic modules. The overall effect of the strain field on the acoustic wave speed can be expressed, as [21]: Vi − Vi0 Vi0

=−

γjik εj k ,

(11)

jk

where Vi and Vi0 are the acoustic wave speeds with and without the strain field, respectively, εj k is the strain tensor, and γjik is the tensor of Gruneisen constants which are of the order of 1. The index i denotes a specific mode of acoustic waves. For example, putting the strain field of screw dislocation in an isotropic material [1], ε = b/4πr (with r being the distance from the dislocation line), into eq. (11), yields Vi Vi0

=−

γ ib . 4πr

(12)

The changes in the acoustic velocity cause refraction of the acoustic wave when it crosses the dislocation line. In order to evaluate the refraction effect, one can use geometrical optics, giving the scattering angle of the order of Vi /Vi . This approach is justified at distances r larger than the acoustic wavelength, λ. Therefore, for r > λ = 5 µm, b = 0.5 nm, and γ i = 1, the scattering angle is of the order of 10−5 rad or smaller. Comprehensive calculations, based on the quantum scattering theory [22–24], also proved that the acoustic wave refraction by static dislocations has negligible effect on the acoustic wave propagation as compared with the effect of the dynamic interaction of acoustic waves and dislocations, which is considered in the next section.

2.5. Theoretical models for the dynamic interaction of acoustic waves with dislocations Dynamic interaction between an acoustic wave and dislocation means that a whole dislocation line or some of its segments are vibrating under the applied acoustic stress field. In general, the dislocation vibrations cause rather strong changes in the attenuation and velocity of the acoustic waves. More specifically, they lead to the wave field distortions of the primary acoustic wave and the emission of secondary acoustic waves. There are different modes of dislocation vibrations which require appropriate modeling. In the subsequent sub-sections, we will distinguish between three types of dislocation vibrations which relate to three different types of dislocation dynamics as mentioned in Section 2.1. First, we will analyze the situation when FExt (i.e. the force applied to the dislocation by the acoustic stress field) is larger than both FBar and FP , and a whole dislocation line moves uniformly. The second type of motion arises when FP < FExt < FBar and corresponds to dislocation segments pinned at both their ends, which vibrate as strings. In the third case, the dislocation line is confined within Peierls potential valleys (i.e. FExt < FP ) and only very limited dislocation motion is permitted.

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Ch. 80

2.5.1. Vibrating dislocation in a uniform motion Eshelby and Nabarro [25,26] developed a model that describes uniform motion of a free dislocation, which is not subjected to pinning barriers, Peierls barriers, or viscous resistance. The solution is given for a screw dislocation under the influence of a bulk shear wave. The model calculations focus on the displacement field, udis z , of the vibrating screw dislocation, which is located parallel to the z-axis and vibrates in the x-direction according to ξ = ξ0 cos(ωt + φ),

(13)

where ω = 2πf is the circular vibration frequency and φ is the phase shift of the dislocation vibration with regard to the acoustic wave. At distances, r ≪ λ, the displacement field follows the local shape modification of the dislocation line, and hence, it can be expressed as a displacement field of static dislocation, udis , located at x = ξ(t). Thus, the dislocation vibration forms a dynamic displacement field, which near the dislocation line is given by: udis = udis (x − ξ, y) − udis (x, y).

(14)

Eq. (14) is valid for any dislocation which vibrates in any mode of motion. In the model described the expression for specific displacement field of screw dislocation in an isotropic material [1,27]: b −1 y tan uz = 2π x

(15)

was substituted to eq. (13). Note that the combination of displacement fields described by eqs (14) and (15) does not satisfy the wave equation, and, hence, it cannot be used to obtain the dynamic displacement field in a whole medium. Moreover, the time which is needed for elastic deformation to arrive at the regions far away from the dislocation line is much larger than the time period of the dislocation vibrations and hence eq. (14) is not valid in the range, r ≫ λ. Eshelby and Nabarro [25,26] looked for a solution which satisfies both a general wave equation at r ≫ λ and eqs (14)–(15) at r ≪ λ. They suggested the following expression for the dynamic displacement field (in cylindrical coordinates (r, ϕ)): udis z = A0 u0 sin ϕ J1 (kr) cos(ωt + φ) + Y1 (kr) sin(ωt + φ) , π A0 = − , ln(4/kb) − γ − 1/2

(16) (17)

where k is the acoustic wave vector, J and Y are the Bessel functions of the first and second kinds, respectively, and γ = 0.5772 is the Euler constant. The displacement amplitude, 0 = 1/2ku . A comparison u0 , relates to the strain amplitude of the acoustic wave by εyz 0

§2.5


617

of the asymptotic behavior of this solution with eqs (14) and (15) provides the vibrating amplitude:

ξ0 =

0 λ2 2A0 εyz

π 2b

.

(18)

Recently, Maurel et al. [28] developed more general approach which allows obtaining solutions for any dislocation character. They applied it to both screw and edge dislocations and showed important distinctions that arise when considering different dislocation character and various polarizations of the acoustic waves. 2.5.2. The vibrating string model This model describes the vibrating dislocation segment, having a length L0 and pinned at both its ends. The appropriate equation of motion is m∗

∂ 2ξ ∂ξ ∂ 2ξ = σ b, −T 2 +B 2 ∂t ∂t ∂z

(19)

which is derived from eq. (1) by neglecting FP and using FBar to describe the pinning properly. The acoustic stress field, σ , is given by σ = σ0 exp −α dis x cos(ωt − x/V )

(20)

where V is the acoustic wave velocity, which is influenced by the vibrating dislocations and hence is a function of ω. The model assumes that partial attenuation coefficient, α dis , due to vibrating dislocations depends on frequency, ω. The solution of eqs (19) and (20) is given by the series [3] ∞ (2n + 1)πz 4σ b 1 Hn (ω, ωn ) cos(ωt − φn ), sin ξ= ∗ m 2n + 1 L

(21)

Hn (ω, ωn ) =

(22)

n=0

=

B , m∗

+ 2 2 ,−1/2 ωn2 − ω2 + ω ,

ωn = (2n + 1)

π 2n + 1 λ VT = ω L0 2 L0

(23) (24)

and tan φn =

ω . ωn2 − ω2

(25)

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Ch. 80

Extraction of the expression ωn2 − ω2 from eq. (25), its substitution to eq. (22), and utilizing some trigonometric transformations yields the simplified expression for Hn : Hn =

| sin φn | . ω

(26)

In a majority of situations, the first term (n = 0) in the expansion series (eq. (21)) dominates [3]. In any case it is possible to learn something about the behavior of the system by considering the first term only. In this approximation, using the relationship, σ0 = με0 , as well as eqs (23) and (26), the dislocation vibrations can be written as ξ = ξ0 sin

πz cos(ωt − φ0 ), L0

(27)

with amplitude ξ0 =

4με0 b | sin φ0 |. Bω

(28)

Further summation over the dislocation ensemble allows obtaining expressions for the acoustic parameters which are measured by internal friction, i.e. the attenuation coefficient and the acoustic wave velocity. Keeping only the first term in the series (21), these parameters can be expressed as: α dis =

ξ0 2χbN 4χμb2 N | sin φ| = 2 sin2 φ0 λ πε0 π BVT

(29)

and χ ′ VT2 N V − VT | sin 2φ0 |, =− VT ω

(30)

where N is the density of dislocation segments that can glide freely and χ ′ is some geometrical factor smaller than 1. Substituting the values of B, typical to metals and alkali-halides from Table 1 into eq. (23), yields ≈ 200 GHz. Hence, in all the internal friction experiments ω ≪ . Moreover, according to eqs (25)–(26), at frequencies much larger than resonant frequencies (i.e. ω ≫ ωn ), the value of | sin φ0 | is approximately 1. Therefore, eq. (29) transforms to eq. (10), which was used for evaluation of the B-values in Section 2.3. The vibrating string model is appropriate if one can neglect the Peierls force. However, usually the Peierls barrier, σP , is larger than the acoustic stress amplitude. Yet the vibrating string model is valid for dislocation lines that are initially not confined to the Peierls potential valleys, as illustrated in Fig. 1(c). It turned out that this situation is very common, at least in fcc metals. If a majority of the dislocation lines is initially located within the Peierls valleys, other models described in the next section should be used.

§2.5


619

2.5.3. Vibrating dislocations located within the Peierls potential valleys Dislocations that are confined to Peierls valleys may exhibit limited vibrating motion in three forms: (1) Dislocation segments which are located at the bottom of potential valleys vibrate within these valleys. Modeling of this problem can be found in Refs [25,29]. (2) The acoustic waves produce bulges on the dislocation line similarly to the case of thermally activated kinks (see Fig. 2). In this case, there exists a resonance frequency which depends on the activation energy for bulge formation and is temperature dependent [1,2]. (3) Kinks which initially exist will vibrate along a direction parallel to Peierls valleys. Such kinks will also interact with other neighboring kinks, this interaction being non-linear. It is possible to obtain a set of linear equations by approximating the kink motion by small amplitude vibrations [2,29]. If the kink density is high enough to allow a collective motion of the dislocation line, the results of this model [2] are similar to those obtained by the vibrating string model. 2.5.4. Concluding remarks The dynamic interaction of acoustic waves with dislocation results in three main effects: (i) attenuation of acoustic waves; (ii) motion of vibrating dislocations, and (iii) emission of secondary acoustic waves. The strength of interaction regarding all the effects mentioned is determined by a single parameter ξ0 /λ, i.e. the ratio of the vibrating amplitude, ξ0 , and the acoustic wavelength, λ. According to eq. (29), partial attenuation of the acoustic wave due to dislocation vibrations is proportional to ξ0 /λ. The overall speed of dislocation motion is determined by the amplitude of its velocity, Vd0 = ξ0 ω, which is also proportional to ξ0 /λ. The strain field amplitude of the emitted acoustic waves at far distances from the disdis location line is determined by udis 0 kf , where u0 is the amplitude of the displacement field udis and kf is the wave vector of the emitted waves. As a rule, udis 0 is proportional to the amplitude of the dislocation motion, ξ0 , and kf is approximately equals to 2π/λ. Hence, the strain field amplitude of the emitted acoustic waves is approximately proportional to ξ0 /λ. Thus, in each effect, the strength of the dynamic interaction is determined by ξ0 /λ ∝ ξ0 ω. According to eq. (28) the product, ξ0 ω, is proportional to 1/B. Hence, the insulating ceramic crystals, which are expected to have very small viscosity coefficients of dislocation motion (see discussion in Section 2.2.4), should also reveal a very strong dynamic interaction between acoustic waves and dislocations. The existing models of dynamic interaction of acoustic waves with dislocations describe only a small part of the possible interaction modes. In particular, it turned out that the specific solution of the equation of motion for individual dislocations is strongly influenced by many geometrical parameters, such as the dislocation character, its orientation with respect to an acoustic wave, the length of the free dislocation segments, etc. Since the conventional experimental methods do not allow resolving the interaction of acoustic waves with individual dislocations and do not provide information about these geometrical parameters, there is an inherent difficulty in formulating a model which can be used to interpret the experimental results in this field (see examples in Refs [4,5]).

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Ch. 80

3. Stroboscopic X-ray imaging of acoustic waves The survey in Section 2 emphasizes the necessity for the development of experimental methods that are sensitive to an interaction of acoustic waves with individual dislocations and provide comprehensive information on the dislocation character and acoustic wave parameters. These requirements are met in the stroboscopic X-ray diffraction imaging (stroboscopic X-ray topography) which, in principle, allows us to visualize the deformation fields of acoustic wave fronts and vibrating dislocations in the same image. Visualization of acoustic wave fronts provides complete information on acoustic wave propagation and scattering processes since the normals to the wave fronts indicate the local direction of energy flow. Note that the acoustic wave fronts can be visualized by other methods, e.g., optically [30], acoustically [31] or by scanning electron microscopy [32]. However, only stroboscopic X-ray diffraction imaging has a unique capability to simultaneously “look” at both the traveling acoustic waves and lattice defects, such as dislocations and, hence, to probe the interaction between them. In order to achieve time-resolved visualization of acoustic waves, one has to first of all avoid time averaging of the dynamic deformation field induced by propagating acoustic waves in the crystal. For this purpose, the investigated samples are illuminated by X-ray pulses that are much shorter than the period of acoustic vibrations (see Fig. 3). However, since the number of X-ray quanta in a single pulse is not enough for the formation of the X-ray image (actually we need about 109 pulses per image even working with modern synchrotron sources), we use the stroboscopic principle of registration, which allows a data accumulation for long periods of time without destroying the phase relations between the acoustic wave and X-ray burst periodicities.

Fig. 3. The principle of stroboscopic measurement. In a given crystal point, all incident X-ray bursts interact with periodic deformation fields having exactly the same magnitude and phase.

§3


621

This is achieved by phase locking the acoustic signal to the X-ray burst periodicity created by the electron bunch pattern within the synchrotron storage ring. As a necessary condition, the acoustic frequency, f = mfx , should be an integer multiple of the X-ray burst frequency, fx . Being synchronized with an acoustic wave, every X-ray burst probes the same instantaneous deformation field during the entire exposure [33]. As a result, the rapidly oscillating deformation field of the acoustic wave, propagating at the speed 3–5 km/s, is revealed in the diffraction images as if frozen in time. The experimental setup, which we use for stroboscopic X-ray topography at the ID19 beam line of the European Synchrotron Radiation Facility (ESRF, Grenoble, France), is schematically illustrated in Fig. 4. It is based on the capability of modern synchrotron X-ray sources to provide very short bursts (of about 50 ps) of high intensity coherent X-rays, which are strictly periodic in time. The electronic signal from the synchrotron storage ring, having exactly the same frequency, fx = 5.68 MHz, as the X-ray bursts, is applied to a frequency synthesizer which produces the output phase-locked signal of a multiple frequency, f = mfx . The value of m is chosen to provide the output frequency, f , to be as close as possible to the resonance frequency of the interdigital transducer (IDT) deposited on top of the sample. The high-frequency output signal is then amplified and applied to the interdigital transducer, which generates surface acoustic waves (SAW). The stroboscopic X-ray diffraction image from the vibrating crystal is collected by using special X-ray photographic film with a spatial resolution down to 0.5 µm. SAW are particularly suitable for the X-ray imaging in the Bragg scattering geometry since the acoustic energy is concentrated within a thin layer (with a thickness of the order of the SAW wavelength) beneath the sample surface, which is comparable to the X-ray penetration depth. Furthermore, SAW can be excited to have shorter wavelengths by an order of magnitude than those characteristic for bulk acoustic waves. Short wavelengths

Fig. 4. Schematic illustration of the experimental setup for the stroboscopic X-ray imaging experiments. The sinusoidal signal inserted to the interdigital transducer (IDT) is phase locked with the X-ray bursts.

622


Ch. 80

Fig. 5. Stroboscopic X-ray diffraction image of a 0.58 GHz SAW propagating in LiNbO3 with the speed of 3.5 km/s. The spacing between individual acoustic wave fronts equals 6 µm.

are especially important for visualizing the acoustic wave interaction with crystal defects of low dimensionality, such as dislocations. The first stroboscopic synchrotron X-ray diffraction experiments aimed at visualizing traveling acoustic waves in crystals were performed in the early 1980s with a 30 MHz SAW [34,35]. Because of the relatively large SAW wavelength (about 100 µm) those experiments did not exhibit phonon interactions with dislocations. An attempt at the stroboscopic X-ray imaging with a 500 MHz SAW was reported in Ref. [36], but those days it was impossible to resolve short individual acoustic wave fronts because of the limited coherence of the X-rays used. A breakthrough in this field has been achieved in experiments that we performed with 0.29 GHz (i.e. 12 µm) SAW devices at ESRF [33,37]. The images obtained exhibited well-resolved individual acoustic wave fronts and their local distortions in the vicinity of dislocation lines. However, because of the non-optimized image contrast in these initial experiments, it was hard to quantitatively investigate the interaction of acoustic waves with dislocations. Detailed analysis of the X-ray focusing and de-focusing process by SAW [38] allowed us to find optimal experimental conditions for enhanced contrast in the stroboscopic X-ray diffraction images. As a result, we succeeded to visualize individual acoustic wave fronts of a 6 µm SAW (f = 0.58 GHz) with excellent contrast allowing quantitative image analysis [39]. An example of such an image is shown in Fig. 5, in which alternating dark and bright lines are individual acoustic wave fronts of the high-frequency SAW propagating through a crystal. This image provides a ‘snapshot’ of the dynamic deformation field which actually changes with a periodicity of 1.6 ns. The experiments mentioned above were performed with SAW devices, which were fabricated on top of specifically oriented highly piezoelectric crystals (LiNbO3 ) (see Fig. 6). A development of a modified technique allowed us to apply the stroboscopic X-ray imaging to weakly- and non-piezoelectric crystals [40]. This has been achieved by transferring SAW from an external piezoelectric transducer to the crystal investigated by generating evanescent acoustic waves via viscous liquid as a coupling medium. The modified tech-

§3


623

Fig. 6. SAW device, comprising an interdigital transducer (IDT) on top of LiNbO3 crystal. Insert shows Cartesian coordinate system (X,Y,Z) related to hexagonal unit cell of LiNbO3 .

nique provided images of a 0.29 GHz SAW propagating in Si and GaAs as is demonstrated in Fig. 7. These technical developments were crucial for the establishment of the stroboscopic X-ray diffraction imaging as a regular method to study acoustic wave (phonon) interaction with dislocations, which is the subject of the next section.

Fig. 7. Stroboscopic X-ray diffraction images of a 0.29 GHz SAW in Si (upper panel), GaAs (lower panel) and LiNbO3 (in the middle). Material-dependent changes in the SAW wavelength, reflecting the differences in the SAW velocity, are clearly seen.

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4. Interaction of acoustic waves with individual dislocations in brittle ceramics The experimental results presented in this section, which helped us to understand the phonon interaction with dislocations in brittle ceramics, were obtained with LiNbO3 -based SAW devices. They were produced on the polished surfaces of the Y-cut LiNbO3 wafers, 76 mm in diameter and 0.5 mm thick. For device fabrication, a system of sectioned interdigital electrodes, consisting of 1.5 µm wide metal fingers separated by the same width blank intervals, was deposited on top of the wafers by using standard lift-off photolithography. The doubled distance between adjacent electrodes, i.e. the SAW wavelength was 6 µm, which corresponded the SAW frequency, f = 0.58 GHz, for Z-propagating SAW. The second type of SAW devices utilized the electrode structure providing the SAW wavelength of 9.8 µm (f = 0.355 GHz). Stroboscopic diffraction images were taken from LiNbO3 single crystals ((030) reflection) at X-ray energy of 10 keV, utilizing the 16-bunch mode of the ESRF storage ring operation (fx = 5.68 MHz). The frequency multiplication coefficient was m = 102 for f = 0.58 GHz and m = 63 for f = 0.355 GHz. 4.1. Experimental results A typical stroboscopic X-ray image revealing the propagation of a 6 µm SAW and its interaction with individual dislocations is shown in Fig. 8(a). Well-resolved acoustic wave fronts form alternating dark and bright lines that pass like a ruler through the entire image area. Besides that, the traces of five dislocation segments are also visible. Dislocations (dark lines) are better resolved in the image shown in Fig. 8(b), which was taken for comparison from exactly the same crystal area, but after switching off the SAW. The two long dislocation segments are almost parallel to the surface since their visible lengths are in the range of tens of microns, whereas the X-ray penetration depth under experimental conditions was only 9 µm. In the vicinity of the dislocation lines, strong wave front deflections occur (see Fig. 8(a)). In particular, in the vicinity of the two long dislocation segments, wave front deflections are clearly visible as black streaks forming large angles with acoustic wave fronts existing far away from the interaction region. We show later on that these deflections arise as a result of the dislocation movements under SAW excitation. Similar features of the deflected wave fronts were found in all images taken from different samples and using SAW excitation of different wavelengths. For example, distortions in a 9.8 µm SAW wave front, when crossing the dislocation line, are clearly seen in Fig. 9. Such strong wave front deflection in a range of several microns around the dislocation line cannot be explained in terms of the acoustic wave scattering by the static strain field of individual dislocation. Following the discussion in Section 2.4, the effect of the static dislocation strain field on the refraction angle at a distance of several microns from the dislocation line should be of the order of 10−5 rad, which is 3–4 orders of magnitude lower than that observed in Figs 8 and 9. Thus, we can unambiguously conclude that the wave front deflections visualized are related to vibrating dislocations (see Section 2.5) which are involved into dynamic interaction with acoustic waves (phonons).

§4.2


625

Fig. 8. (a) Stroboscopic X-ray diffraction image of a 0.58 GHz SAW propagating in LiNbO3 and its interaction with linear dislocations. Remarkable wave front deflections in close vicinity to dislocation lines are clearly seen. The spacing between non-distorted individual acoustic wave fronts equals 6 µm. (b) Stroboscopic X-ray diffraction image taken from exactly the same crystal region as in (a), but with no SAW excitation. Images are taken from Ref. [20].

Additional support of this idea comes from the following observation: deflection of acoustic wave fronts is a common feature for all dislocation lines situated along the crystal surface, except those that are parallel to the Z-axis of LiNbO3 . An example of such behavior is shown in Fig. 10, which reveals pronounced wave front deflections in the left part of the dislocation line only. Wave-front deflections are practically absent in the right part of the dislocation line that is nearly parallel to the z-axis (the acoustic wave fronts far away from the dislocation line are perpendicular to the z-axis). As is explained in Section 4.3, dislocation lines parallel to the z-axis are confined to Peierls potential valleys and can hardly participate in high amplitude vibrations. 4.2. Theoretical model and simulations In order to explain experimental data and extract quantitative information on dislocation vibrations and related parameters, a model for simulating the observed X-ray images was

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Fig. 9. (a) Stroboscopic X-ray diffraction image of a 0.355 GHz SAW propagation in LiNbO3 revealing pronounced wave front distortions in close vicinity to dislocation line. The spacing between non-distorted individual acoustic wave fronts equals 9.8 µm. (b) Stroboscopic X-ray diffraction image taken from exactly the same crystal region as in (a), but with no SAW excitation.

developed by us [20], which is based on the calculation of the dynamic deformation field of a vibrating dislocation. Most of the known models in this field (see Section 2.5), deal with calculating the effect of dislocation vibrations far away from the dislocation line. This approach is not suitable for our purpose since experimentally the wave front deflections are observed in a limited space (less than the one acoustic wavelength, λ) around the dislocation line. Keeping this in mind, we adopted the approach of Eshelby and Nabarro [25,26] (see also Section 2.5.1) for calculating the displacement field of the vibrating dislocations close to the dislocation line. A combination of the dynamic displacement fields of the SAW and of the vibrating dislocation was then used for simulating the local shapes of acoustic wave fronts and comparing them with the observed features in the X-ray images. Note that the X-ray contrast is not always proportional to the amplitude of atomic displacement, so the exact dependence can be more complicated [38]. Nevertheless, the local geometrical shapes of alternating dark and bright lines in X-ray images follow the local shapes of acoustic wave fronts which are determined by the combined displacement field.

§4.2


627

Fig. 10. Stroboscopic X-ray diffraction image showing an interaction of a 0.58 GHz SAW with a dislocation line which is almost parallel to the crystal surface. On the left side of the dislocation segment, remarkable deflections of acoustic wave fronts are clearly visible. On the right part of the dislocation segment, which is parallel to the Z-crystallographic direction of LiNbO3 , the SAW wave fronts remain undistorted. The spacing between nondistorted individual acoustic wave fronts equals 6 µm.

In order to describe the displacement field of the SAW, uS , we used an orthogonal coordinate system (x, y, z) (see Fig. 11), with the y-axis normal to the crystal surface and the z-axis parallel to the SAW wave vector, ks . The Bragg scattering geometry is sensitive only to the uy component of the displacement vector, u. In the case of SAW [41]: uSy = uy0 (y) cos(ks z − ωt).

(31)

An auxiliary coordinate system (x ′ , y ′ , z′ ) is related to the vibrating dislocation, with the z′ axis parallel to the dislocation line (see Fig. 11). The X-ray topographs indicate that the dislocation lines, which strongly interact with the SAW, are almost parallel to

Fig. 11. The coordinate systems related to the propagating SAW (x, y, z) and the vibrating dislocation line (x ′ , y ′ , z′ ).

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Ch. 80

the crystal surface. Hence, we chose the y ′ axis parallel to the y axis and the x ′ axis to be perpendicular to both the y ′ and z′ axes, so that the x ′ –z′ and x–z planes coincide. Since screw dislocations parallel to the crystal surface result in uy = 0, the deformation fields were simulated for edge dislocations with Burgers vector, b, along the x ′ axis, i.e. perpendicular to the dislocation line (the z′ axis). Practically, we have mixed dislocations, but the difference in the displacement component, uy , between mixed and edge dislocation results in a geometrical factor close to one. According to Eshelby and Nabarro (see Section 2.5.1), at distances r ≪ λ (counted outwards from the dislocation line) the displacement field of the dislocation follows the instantaneous location of the dislocation segments. If so, in order to find the displacement field of a vibrating dislocation we can use the known one for a static dislocation, udis y , but taken in the locations, x ′ = ξ(z′ , t), modified by the dislocation vibrations, ξ . Thus, the dislocation vibration creates a dynamic displacement field which, when close enough to the dislocation line, can be expressed as: dis ′ ′ dis ′ ′ udis y = uy (x − ξ, y ) − uy (x , y ).

(32)

As the static displacement field, udis y in eq. (32), we used a well-known solution for an edge dislocation in isotropic medium [27]: ′ ′ udis y (x , y ) = −

x ′2 − y ′2 ′2 b ′2 , (1 − 2υ) ln x + y + ′2 8π(1 − υ) x + y ′2

(33)

where υ is Poisson’s ratio. As is explained in Section 4.4, the solution of the dislocation’s equation of motion, at time intervals much larger than the vibration period, is given by ξ = ξ0 sin(kd z′ − ωt + φ),

(34)

where ξ0 is the vibrating amplitude (real) and φ is the phase shift of dislocation vibration with respect to the phase of the SAW. The wave vector, kd , is defined by the spatial periodicity of the force applied to dislocation by the SAW and is kd = kS cos ,

(35)

where is the angle between the dislocation line and ks (i.e. between the z′ and z axes). The ratio ξ0 /λ manifests the strength of the dynamic interaction between acoustic waves and dislocations (see Section 2.5.4) and determines the amplitude of the dislocation velocity Vd0 = ωξ0 = 2π

ξ0 VR , λ

where VR = ω/kS is the speed of the SAW.

(36)

§4.2


629

Substituting eqs (33) and (34) into eq. (32), yields the expression ′ ′ udis y (x , y )

′ (x − ξ0 sin(kd z′ − ωt + φ))2 + y ′2 b (1 − 2υ) ln =− 8π(1 − υ) x ′2 + y ′2 (x ′ − ξ0 sin(kd z′ − ωt + φ))2 − y ′2 x ′2 − y ′2 . (37) − + ′ (x − ξ0 sin(kd z′ − ωt + φ))2 + y ′2 x ′2 + y ′2

The total dynamic displacement field was calculated in the (x ′ , y ′ , z′ ) coordinate system by adding expressions (31) and (37): uy = uSy + udis y

(38)

after the following coordinate transformation x = x ′ cos − z′ sin ,

z = x ′ sin + z′ cos ,

(39)

applied to eq. (31). The time dependence of the displacement field is formally revealed in both eqs (31) and (37) containing the terms that depend on the product, ωt. However, in the stroboscopic mode of measurement, the X-ray pulses probe the displacement field at certain instants corresponding to ωt = 2πm (where m is an integer) and, hence, all periodic time dependences practically disappear. Additional simplification comes when one considers depth dependent effects in the displacement fields. In this context, both eqs (31) and (37) depend on the coordinates y ′ and y. However, the influence of the y ′ and y values is revealed only in some changes of the am′ plitudes of uSy and udis y . Detailed simulations show that within a reasonable range of y and y-values, which are smaller than the X-ray penetration depth (and also y ′ < λ), the exact choice of the y ′ and y-values does not significantly influence the overall displacement field map in the x ′ –z′ plane. As mentioned above, eq. (37) is valid for r ≪ λ. Beyond this range, the solution is very complicated and can be derived only in specific simple cases [25,26]. We used eq. (37) in the range |x ′ | < λ/2 and neglected the dynamic displacement field of the dislocation line in the range, |x ′ | > λ/2. This neglecting is justified by two findings: (i) the displacement field (37) decays as ξ0 /r with the distance, r, from the dislocation line; (ii) experimentally, the wave front distortions were observed only in the range, |x ′ | < λ/2. As an example of a simulation of the local shapes of acoustic wave fronts, the dynamic displacement map calculated for ξ0 /λ = 0.14 and φ = π/2 is shown in Fig. 12 together with one of the experimental X-ray images. It can be seen that the simulated map reproduces all main features of the wave front distortions that appear in the X-ray image collected. The thin white lines, which were imposed artificially in order to follow the wave front distortions, are practically identical in both images. For a deeper understanding, we show two separate contributions to the dynamic deformation field in Fig. 13. In the middle of the map, the dynamic part of the dislocation deformation field, udis y , is presented ′ for |x | < λ/2. The right and left sides of the map show the SAW’s contribution, uSy , in

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Fig. 12. Stroboscopic X-ray diffraction image (a), showing wave front deflections in the vicinity of a vibrating dislocation in LiNbO3 , as compared with the simulated map (b) of the total dynamic displacement field for ξ0 /λ = 0.14 and φ = π/2. The spacing between non-distorted individual acoustic wave fronts equals 6 µm.

Fig. 13. Displacement fields of SAWs and the vibrating dislocation, shown separately, under the same conditions ′ ′ S as in Fig. 12. The contribution of udis y is drawn within |x | < λ/2, while that of uy – at |x | > λ/2. The displacement field of the vibrating dislocation forms a zigzag pattern near x ′ = 0. Two types of segments that comprise the zig-zag pattern are indicated by S1 and S2. The instantaneous shape, ξ(z′ ), of the vibrating dislocation is shown by the broken white line.

the range of |x ′ | > λ/2. It can be seen that the term, udis y , produces a dark “zigzag” like pattern, which is actually an image of the vibrating dislocation line, ξ(z′ ). The latter is calculated by the means of eq. (34) and indicated by a wavy vertical white line in Fig. 13. The angle between the zigzag segments and the z′ axis becomes smaller as the normalized amplitude, ξ0 /λ, decreases. In the total dynamic deformation field (see Fig. 12), some of the dark segments (indicated by S1 in Fig. 13) that form the zigzag pattern disappear since they are located exactly in the middle of the bright SAW wave fronts. Thus, the observed pattern in the vicinity of

§4.2


631

Fig. 14. Stroboscopic X-ray diffraction image (a) showing wave front distortions in the vicinity of a vibrating dislocation in LiNbO3 , as compared with the simulated map (b) of total dynamic displacement field for ξ0 /λ = 0.14 and φ = −π/2. The spacing between non-distorted individual acoustic wave fronts equals 6 µm.

the dislocation line is actually an image of the second type of zigzag segments (indicated by S2 in Fig. 13), which remain in the image. On the other hand, for φ = −π/2, the S2 segments of the zigzag pattern disappear, and only the S1 segments remain in the image. As a result, the simulated map (see Fig. 14) exhibits dark streaks in the vicinity of the dislocation line, which are tilted by certain angles, as compared with Fig. 12. An example of the observed X-ray image of this type is also shown in Fig. 14. Comparing the simulated maps with experimental images allowed us to extract the values of φ and ξ0 /λ for each individual vibrating dislocation that appeared in the collected X-ray images. It was found that the normalized amplitude (strength of phonondislocation interaction) is in the range of ξ0 /λ = 0.08–0.14 under a 6 µm SAW excitation and ξ0 /λ = 0.02–0.06 under a 9.8 µm SAW excitation. The lower vibration amplitude under an excitation of a 9.8 µm SAW is probably a result of weaker amplitude of SAW generated at a frequency which is shifted a few MHz away from the exact resonance value for a 0.355 GHz SAW transducer. For all vibrating dislocations the φ-values extracted from the displacement maps were found to be either φ = π/2 or φ = −π/2. As is explained in Section 4.4, this means that they vibrate in the resonant mode. The amplitude of the dislocation velocity can be evaluated by substituting the extracted ξ0 /λ values into eq. (36), which yields Vd0 /VR = 0.5–0.88 for samples excited by a 0.58 GHz SAW. Using the relationship VT = 1.08VR , between the Rayleigh velocity, VR , and shear bulk velocity, VT , which is appropriate for YZ-LiNbO3 [41], we obtain Vd0 /VT = 0.46–0.81. The dislocation velocities are 2–3 orders of magnitude higher than those previously measured in metals and alkali halides [3]. This result indicates that the dynamic interaction between acoustic waves and dislocations (see discussion in Section 2.5.4) in LiNbO3 is several orders of magnitude stronger than that of metals and alkali halides. Moreover, a part of dislocations have the velocities not far from the velocity of shear bulk waves, VT , which is considered to be the highest possible value of dislocation motion [1,27]. Until now such high dislocation velocities were observed only under extremely

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high stresses [42]. The physical reasons for high dislocation velocities under subtle strains introduced by SAW will be clarified in the next section.

4.3. Dislocation analysis As mentioned above, dislocations can move fast only if they are not subjected to Peierls barriers. In our case, the magnitude of the SAW-induced stress field (typically, σ/μeff < 10−4 ) is much smaller than the Peierls stress (typically, σP /μeff ≈ 10−3 –10−2 for brittle ceramic crystals) and hence, the applied external force, FExt , is definitely smaller than the Peierls force, FP (see eqs (3) and (4) as well as discussion in Section 2.1). Thus, the collected stroboscopic X-ray images give strong hints that the vibrating dislocations are not confined to Peierls potential valleys. This important point is demonstrated below by direct dislocation analysis in X-ray images. According to Section 2.1, a dislocation does not “feel” the Peierls potential barrier if L Wk , where L is the average distance between adjacent kinks and Wk is the width of an individual kink. Substituting eqs (6) and (7) into this inequality, yields: δ

2abσP , πT

(40)

where δ is the angle between the direction of the Peierls valley and the whole dislocation line (see Fig. 1(c)). By using typical magnitudes of the parameters involved in eq. (40): T ≈ 0.5μeff b2 [1–3], a ∼ = b, and σP /μeff ≈ 10−2 [1], one obtains δ 6.5◦ .

(41)

Important information about the directions of vibrating dislocations can be extracted from the collected X-ray images by measuring the angles, f , that dislocation images form with the zf -axis, which is perpendicular to visible acoustic wave fronts (see Fig. 15). Performing this analysis, one has to keep in mind that the image of the dislocation line is a projection of its direction in a three-dimensional crystal onto the plane of the X-ray photographic film. For the X-ray scattering geometry that we used in our experiments (see Fig. 15), the following relations exist between the real crystal coordinate system (x, y, z) and the coordinate system (xf , zf ) in the plane of the film: xf = x sin θ − y cos θ, zf = z,

(42)

where θ is the Bragg angle for the X-ray reflection used. By using eq. (42), the angle f can be expressed as: lx sin θ − ly cos θ , f = arctan lz

(43)

§4.3

633


Fig. 15. Projection of a dislocation line on the X-ray photographic film.

where the vector [lx , ly , lz ] denotes the direction of the dislocation line in the crystal. Using eq. (43), it is possible to understand whether or not the vibrating dislocation lines are confined to the Peierls potential valleys by calculating the difference between the magnitudes of f and Pf , the latter being related to the directions of the Peierls potential valleys in LiNbO3 . For the reader’s convenience, the glide systems for dislocations in LiNbO3 and their possible Pf -values are summarized in Table 3. The left column specifies the three gliding systems known for LiNbO3 [43]. The most common is the basal plane system (see Fig. 16(a)) with Burgers vectors, |b| = 0.5148 nm, equal to a – translations of the hexagonal unit cell [44,45]. Supplementary gliding systems are the pyramidal and prismatic plane systems (see Figs 16 (b) and (c)). In both of them the smallest Burgers vectors, |b| = 0.5494 nm, are ¯ The study [41] of the high temperature plastic deformation along the directions 31 1101. at 1070 ◦ C > 0.8Tm (Tm stands for melting temperature) showed that the critical resolved Table 3 Directions of Peierls valleys for major glide systems in LiNbO3 . Angles, P f , calculated by means of eq. (42), are placed in last column Glide system

Possible directions of Peierls potential valleys

Cartesian axes [lx , ly , lz ]

Calculated |P f|

Basal

1 ¯ 3 1120

[a, 0, 0] √ [a/2, a 3/2, 0] √ [-a/2, a 3/2, 0] √ [3a/2, a 3/2, c] √ [−3a/2, a 3/2, c] √ [0, a 3, c] √ [3a/2, a 3/2, c] √ [−3a/2, a 3/2, c] √ [0, a 3, c] [0, 0, c]

90◦ 90◦ 90◦

1 ¯ 3 1120(0001)

Pyramidal 1 ¯ ¯ 3 1101{1012}

Prismatic 1 ¯ ¯ 3 1101{1210}

1 1101 ¯ 3

1 ¯ 3 1101

[0001]

3.46◦ 27.67◦ 30.33◦ 3.46◦ 27.67◦ 30.33◦ 0

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Fig. 16. Three major glide systems in LiNbO3 , (a) – in the basal plane, (b) – in the pyramidal plane, and (c) – in the prismatic plane. The glide planes are marked in gray. The shortest Burgers vectors are indicated by arrows.

shear stresses for these gliding systems are very similar. Taking into account also the proximity of Burgers vectors, it is reasonable to assume that dislocations related to all the three gliding systems can be formed during the crystal growth. As a rule, the directions of Peierls potential valleys and Burgers vectors form a two dimensional primitive unit cell within the gliding plane [1]. This consideration allowed us to indicate possible directions of Peierls potential valleys and calculate the Pf -values (the last column of Table 3) for each of the three gliding systems. When analyzing the collected X-ray images, we found out that the values of f , measured for a large number of vibrating dislocations, do not fit the calculated ones. Instead, the measured angles are distributed in the range of f = 6◦ –20◦ with no bias toward any values. Furthermore, for most imaged dislocations, the difference between f and the nearest Pf -value was larger than 10◦ , and hence, it is reasonable to assume that the condition (41) is fulfilled. This result strongly supports the conclusion that the vibrating dislocations are not confined to Peierls potential valleys that is a necessary condition for high speed dislocation movement. However, in a majority of metals and alkali halides, which are frequently investigated by the internal friction technique, the dislocations also are not confined to Peierls potential valleys [3], yet the measured dislocation velocities are much lower. In fact, to obtain such high dislocation velocities as we measured, the viscosity coefficient, B, for dislocation motion should be very small. The appropriate B value for LiNbO3 can be extracted by analyzing the dislocation’s equation of motion.

4.4. Extraction of the physical characteristics of vibrating dislocations The equation of motion for dislocations not subjected to the Peierls force [1,2] can be written as: m∗

∂ 2ξ ∂ 2ξ ∂ξ − T +B = FExt . 2 ′2 ∂t ∂t ∂z

(44)

This equation is very similar to eq. (19), which was used by Lücke and Granato (see Ref [3] and Section 2.5.2). The only difference between their solution and the solution given below stems from the differences in the boundary conditions and a different application of the

§4.4


635

external force, FExt , with regards to the dislocation line. The SAW-induced applied external force can be expressed by means of eq. (3) and Hooke’s law, σij = cij kl εkl , where εkl is the SAW’s strain field, which is calculated by taking spatial derivatives ∂uSi /∂xj , of the SAW’s displacement field, uSi (see, e.g., eq. (31)) and cij kl is the tensor of elastic moduli. The applied external force depends on the dislocation character and, generally can be written as: eff

FExt = με0 b sin(kd z′ − ωt),

(45)

eff

where ε0 is an effective value of the SAW-induced strain amplitude, which actually depends on the dislocation character. In order to derive eq. (45), the equality ks z = kd z′ , i.e. ks = kd cos (see eq. (35)), is used. Substituting eqs (2) and (44) into eq. (43) and dividing by m∗ ∼ = 0.5ρb2 (see discussion in Section 2.1) yields the following normalized equation: 2 VT2 eff ∂ 2ξ ∂ξ 2∂ ξ − V + = 2 ε sin(kd z′ − ωt), T ∂t b 0 ∂t 2 ∂z′2

(46)

in which = B/m∗ according to eq. (23). The solution of eq. (16) depends on the boundary conditions, which in our case can be imposed by using the geometrical parameters of individual vibrating dislocations deduced from the X-ray images. The capability to obtain this information is a significant advantage of the stroboscopic X-ray diffraction techniques over other non-diffraction methods. According to X-ray images (see, e.g., Figs 8 and 9), the vibrating dislocations are almost parallel to the crystal surface but yet have rather long parts propagating in depth. Therefore, the dislocation lines usually end at the surface, and this end is most probably not pinned. Concerning the second end, the X-ray images show that dislocations penetrate much deeper than the X-ray penetration length. Since the latter approximately equals to the SAW’s penetration depth, we can conclude that a part of dislocation line excited by the SAW is free of pinning points. In that situation the dislocation vibrations will attenuate before they arrive at the dislocation end located in the crystal bulk and hence one can neglect the reflected wave from this end, even if it is pinned. Based on these considerations, it is reasonable to choose boundary conditions that correspond to a long dislocation string (much longer than the SAW wavelength), which is free at both its ends. The solution of eq. (46) under the above mentioned boundary conditions and at times much larger than the vibration period is given by eq. (34). Substituting this solution into eq. (45), yields eff

ξ0 =

eff

με b 2VT2 ε0 | sin φ| = 0 | sin φ|, bω Bω

(47)

where tan φ =

ω . − ω2

VT2 kd2

(48)

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Ch. 80

Eq. (48) is equivalent to eq. (25) that appears in the solution of Lücke and Granato [3], if one uses the relationship ω02 = VT2 kd2 , where ω0 is the dislocation’s resonant frequency, which is not equal to ω. Substituting the relationship, ω = VR ks , and eq. (35) into eq. (48), yields: tan φ =

1 . 2 ω [(VT /VR ) cos2 − 1]

(49)

Based on eq. (49), it is easy to understand why the phase shift of the dislocation vibrations, extracted by comparing the experimental images and calculated dynamic deformation maps, was always close to |φ| ∼ = π/2. In fact, setting VT /VR ∼ = 1.08 [41] and ◦ 0.94 < cos < 1 (for < 20 as measured in the collected images), one finds that the denominator in eq. (49) is close to zero (resonance conditions), i.e. |φ| ∼ = π/2. Moreover, since the exact value of VT /VR depends on the direction of the dislocation line, the denominator in eq. (49) can be either slightly positive or negative, providing the phase shift with both signs, φ ∼ = −π/2. As shown in Section 4.2, a change of the sign of = π/2 or φ ∼ φ from −π/2 to +π/2 causes sign changing in wave front deflections when crossing the dislocation line (compare Figs 12 and 14). An image of two dislocation lines located in the same crystal region and producing wave front deflections of both negative and positive signs is given in Fig. 8(a). The viscosity coefficients, B, for dislocation motion in LiNbO3 were evaluated by substituting the obtained values of |φ| ∼ = π/2 and ξ0 /λ = 0.08–0.14 as well as the exeff perimental and material parameters: μ = 60 GPa, ε0 = 5 × 10−5 , b = 0.55 nm, and ω = 2π × 0.58 GHz, into eq. (47). Such evaluated B-values are in the range of B = (5–9) × 10−6 Poise, i.e. 2–3 orders of magnitude lower than those measured in metals and alkali halides [14]. The obtained results correlate well with the very low total acoustic attenuations, α, in LiNbO3 and other brittle ceramic crystals (see discussion in Section 2.2) as compared with ductile materials. A last remark relates to possible effect of SAW on other types of dislocations in LiNbO3 . Note that the right-hand term in eq. (47) is similar to the solution by Lücke and Granato [3], which appears in eq. (28). A comparison between these two equations allowed us to conclude that the vibration amplitude in model [3] is even four times higher than in our model. This fact has an important implication on the subject, since dislocations in the basal gliding system of LiNbO3 have the geometry that was considered in model [3] (i.e. a dislocation line is perpendicular to the acoustic wave propagation). Thus, these dislocations can indeed vibrate with very large amplitudes as follows from the well established model [3]. Unfortunately, we are not able to visualize these specific dislocation vibrations since in the scattering geometry used, the traces of these dislocation lines in the X-ray images are parallel to the acoustic wave fronts. Nevertheless, the phonon-induced vibrations of these dislocations may have a significant effect on the attenuation of acoustic waves in LiNbO3 . 4.5. Conclusions In this chapter we showed that stroboscopic X-ray diffraction imaging provides unique information on the phonon-induced motions of individual dislocations. The key point of the

§4.5


637

technique developed is the phase locking of the periodic high-frequency deformation field (affecting the dislocation movement) to the periodic X-ray illumination. In such a way, fast dislocation dynamics (on a ns to sub-ns scale) can be investigated as if it is frozen in time. It should be emphasized that the X-ray diffraction imaging technique is the only method allowing to “see” in one image the static and dynamic deformation fields with spatial resolution better than 0.5 µm and thus to study the dynamic interaction of dislocations with acoustic waves (phonons). Due to high spatial and temporal resolution, this technique is able to probe an elementary phonon interaction with an individual dislocation, which is a subject of theoretical analysis. By using this technique we succeeded to visualize individual acoustic wave fronts in a GHz range. Due to improved imaging contrast we were also able to resolve the wave front distortions when crossing the dislocation lines. We showed that the observed wave front deflections are caused by dislocation vibrations, as a result of the interaction with phonons. For quantitative image analysis, we developed a model which allowed us to simulate the dynamic displacement maps containing both contributions of SAW and vibrating dislocations. By comparing experimental images and simulated dynamic displacement maps, the amplitude and velocity of vibrating dislocations were extracted. Further solution of the dislocation’s equation of motion, using boundary conditions that were deduced when analyzing experimental X-ray images, allowed us to obtain the dislocation viscosity coefficients, B. The B-values found in LiNbO3 were 2–3 orders of magnitude lower than any value measured before in ductile materials. The results obtained for brittle ceramic crystals, such a LiNbO3 , prove that the technique developed significantly expands our abilities to investigate phonon interaction with dislocations. At the same time, the models that were used rely on several approximations. The dynamic displacement maps were simulated for edge dislocation in isotropic material with the dislocation line parallel to the crystal surface. By taking X-ray images from different reflections, one can find the precise orientation and type of every dislocation in the image and after that perform more precise numerical simulations of the displacement maps using specific elastic constants of LiNbO3 . Using anisotropic models may lower the evaluated amplitudes of dislocation velocity by a factor of 2 or 3. In this case, the dislocation velocities are no longer “relativistic”, but yet remain two orders of magnitude higher than any velocity value measured before in metals and alkali halides under acoustic wave excitation. The extraction of the value of B by substituting the parameters obtained, ξ0 /λ and φ, into eq. (47) also relies on the approximated values of T or m∗ . As pointed out by Seeger [46], the exact value of T can be numerically calculated if the dislocation orientation and character are known. Following Seeger’s analysis [2,46], an increase of B by a factor of 2, due to some uncertainty in the T -values, may be considered. Finally, applying both corrections mentioned can raise the actual value of B by a factor of 5. This corrected value is still two orders of magnitude lower than any value measured until now in all materials investigated. The experimental technique developed paves the route for further research in several directions. First, it is desirable to perform more accurate simulations of the dynamic displacement maps and to achieve more exact solutions for dislocation motion in order to have more accurate data on dislocation velocities and viscosity coefficients in brittle crystals. Secondly, the method developed [40] of applying stroboscopic X-ray imaging to

638


non-piezoelectric crystals opens the route for investigating dislocation interactions with phonons in a wide variety of semiconductor and optical crystals that are beyond the scope of other experimental methods. This can lead to the detection of new modes of dislocation vibration under acoustic wave excitation and can significantly expand the database of the viscosity coefficients, B, in different materials. Stroboscopic X-ray diffraction imaging can also be applied for the very ambitious goal of investigating dislocation motions at relativistic velocities. According to eq. (47), the amplitude of dislocation vibration and, correspondingly, the dislocation velocity depends eff directly on the applied strain amplitude, ε0 . Thus, a few times increase of the SAW amplitude (that is technically possible), in principle, may result in dislocation velocities very close to VT . The question of whether a dislocation can move faster than VT is still open. Five decades ago Eshelby showed [47], basing on the theory of elasticity, that there exist solutions for the dislocation motion faster than VT , that are accompanied by a strong emission of acoustic waves. For a long time, these solutions were considered to be unrealistic since the dislocation’s energy infinitely increases at Vd = VT . Thus, for decades no physical argument to overcome this infinitely high energy barrier was found. However, recent molecular dynamic simulations [48,49] have demonstrated that the dislocation velocity can overcome the barrier of VT and can even be larger than the speed of longitudinal bulk waves. In these simulations, the dislocation velocities jumped over VT within less than one ps time interval, i.e. when passing a distance of about one Burgers vector. Dislocation velocities above VT have never been observed experimentally. Apparently, nearly relativistic velocities were obtained by the application of a pulsed mechanical stress of about 0.01 µ in amplitude during about 1 µs [42]. Measurements of the dislocation positions before and after pulse application allowed us to calculate the average dislocation velocity but failed to provide any information about the dislocation behavior during this motion. Applying the stroboscopic X-ray diffraction imaging to study the dislocation movements at relativistic velocities will provide important information not only on the limiting velocity itself but also on possible modifications of the dislocation strain fields at these velocities. Such experiments will open a new topic of research in dislocation mechanics.

References [1] F.R.N. Nabarro, Theory of Crystal Dislocations (Oxford, University Press, 1967). [2] A. Seeger, J. de Physique 42 (1981) C5-201. [3] A.V. Granato and K. Lücke, in: Physical Acoustics, Vol. 4a, ed. W.P. Mason (Academic Press, London, 1966). [4] G.A. Kneezel and A.V. Granato, Phys. Rev. B 25 (1982) 2581. [5] G.A. Northrop, E.J. Cotts, A.C. Anderson and J.P. Wolfe, Phys. Rev. B 27 (1983) 6395. [6] A.J. Slobodnik, Jr., in: Acoustic Surface Waves, ed. A.A. Oliner (Springer-Verlag, Berlin, 1978). [7] A.A. Auld, Acoustic Fields and Waves in Solids (Krieger Publishing Company, Malabar, 1990). [8] L.D. Landau and E.M. Lifshitz, Theory of Elasticity (Pergamon, Oxford, 1986). [9] R.E. Peierls, Proc. Phys. Soc. London 52 (1940) 34. [10] H. Alexander and P. Haasen, Solid State Physics 22 (1968) 28. [11] A.I. Akhiezer, J. Phys. USSR 1 (1939) 277.

X-ray imaging of phonon interaction with dislocations [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49]

639

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Author Index Italic numbers refer to reference pages. Numbers between brackets are the reference numbers. No distinction is made between first and co-author(s).

Aaronson, H.I. 261 [18]; 261 [47]; 261 [48]; 261 [49] Abarbanel, H.D.I. 218 [20]; 221 [139] Abe, H. 415 [88] Abercombe, R.E. 219 [63] Abu Al-Rub, R.K. 46 [95] Acharya, A. 45 [43] Ackland, G.J. 495 [60] Adams, G.G. 44 [6] Agrait, N. 46 [100] Aifantis, E.C. 219 [46]; 219 [47]; 219 [56]; 219 [61]; 220 [125]; 223 [239]; 223 [240]; 599 [53] Aindow, M. 261 [46] Ajayan, P. 452 [102] Akhiezer, A.I. 638 [11] Albohr, O. 45 [34] Alexander, H. 220 [105]; 363 [75]; 450 [2]; 450 [7]; 450 [23]; 450 [41]; 450 [44]; 451 [45]; 452 [93]; 452 [98]; 452 [112]; 495 [40]; 638 [10] Alexander, P. 451 [56] Alippi, A. 219 [71] Alisivatos, A.P. 497 [107] Allan, P. 362 [16] Almer, J. 219 [83]; 599 [71] Alshits, V.I. 77 [3]; 78 [11]; 78 [12]; 78 [13]; 78 [14]; 78 [38]; 78 [39]; 78 [46]; 78 [49]; 79 [73] Al’shitz, V.A. 639 [14]; 639 [15] Amazit, Y. 414 [63] Ambroise, J.P. 414 [24] Ambrosini, B. 414 [29] Amelinckx, S. 363 [69]; 363 [81]; 363 [103] Ananthakrishna, G. 217 [3]; 218 [15]; 218 [16]; 218 [18]; 219 [50]; 219 [51]; 220 [102]; 220 [104]; 220 [107]; 220 [108]; 221 [133]; 221 [134]; 221 [135]; 221 [136]; 221 [137]; 221 [166]; 222 [172]; 222 [186]; 222 [190]; 222 [191]; 222 [192]; 223 [209]; 223 [210]; 223 [213]; 223 [219]; 223 [226]; 223 [230]

Anderson, A.C. 638 [5]; 639 [18] Anderson, P. 494 [4]; 494 [18]; 495 [47] Anderson, P.W. 221 [164] Anstis, G. 450 [15]; 452 [94] Antipov, S. 451 [58] Antonelli, A. 450 [36] Apih, T. 414 [30] Arai, S. 495 [52] Archard, J.F. 44 [17] Aref, H. 221 [154] Argon, A.S. 450 [24]; 450 [29] Armstrong, R.W. 495 [53] Arsenlis, A. 598 [41]; 598 [42] Art, A. 363 [103] Artz, E. 494 [9] Arvia, A.J. 219 [60] Arzt, E. 600 [121] Asaro, R.J. 46 [74]; 79 [75] Asgari, S. 361 [5] Ashby, M.F. 44 [12]; 45 [35]; 45 [48]; 46 [92]; 223 [241]; 598 [13]; 598 [47] Assali, L. 450 [28] Assmus, W. 415 [68] Assoufid, L. 600 [104] Audier, M. 414 [51]; 415 [86]; 415 [90]; 415 [91] Auld, A.A. 638 [7] Babu, S.S. 599 [87] Bachteler, J. 414 [44]; 414 [58] Bacon, D.J. 78 [30]; 261 [31] Bacroix, B. 598 [40] Baer, S.M. 222 [204]; 222 [207] Bahnck, B.J.D. 451 [52] Bahr, D.F. 494 [23] Bai, J. 415 [88] Bak, P. 219 [73]; 413 [7] Baker, S.P. 494 [9] Bako, B. 219 [86]; 220 [112]; 220 [120] Balakrishnan, V. 220 [93]; 220 [94] Balescu, R. 220 [116] 641

642

Author Index

Balik, J. 222 [179] Balint, D.S. 46 [88] Balough, P. 219 [85] Baluc, N. 416 [124]; 416 [125]; 416 [147]; 416 [150]; 417 [153] Bammann, D.J. 599 [80] Bandyopadhyay, S.K. 221 [170] Bao, Z. 600 [96] Barabash, O.M. 599 [87]; 600 [113]; 600 [122] Barabash, R.I. 598 [7]; 598 [29]; 599 [68]; 599 [73]; 599 [74]; 599 [79]; 599 [87]; 599 [88]; 599 [89]; 599 [90]; 599 [92]; 600 [94]; 600 [112]; 600 [113]; 600 [119]; 600 [120]; 600 [122] Barat, P. 221 [170] Barber, J.R. 45 [32] Barbour, J.C. 600 [96] Barlat, F. 220 [128] Barnett, D.M. 78 [29]; 78 [30]; 78 [35]; 78 [36]; 78 [45]; 78 [47]; 78 [48]; 79 [62]; 79 [70]; 79 [72]; 79 [74]; 79 [75] Barquins, M. 222 [189] Bartsch, M. 413 [15]; 413 [16]; 413 [17]; 415 [102]; 416 [122]; 416 [151]; 417 [154]; 451 [60] Baruchel, J. 639 [33]; 639 [37]; 639 [40] Basinski, S.J. 218 [29]; 361 [1]; 362 [29]; 362 [52]; 362 [58]; 362 [62]; 364 [111] Basinski, Z.S. 218 [29]; 361 [1]; 362 [29]; 362 [52]; 362 [55]; 362 [56]; 362 [58]; 362 [61]; 362 [62]; 363 [71]; 363 [85]; 363 [96]; 364 [110]; 364 [111] Baskes, M.I. 46 [93]; 599 [52] Bassani, J.L. 45 [43]; 46 [75] Bataronov, I. 451 [58] Batson, P. 451 [89] Batterman, B.W. 600 [107]; 600 [108]; 600 [117]; 600 [118] Baufeld, B. 413 [16]; 415 [102] Baumann, I. 639 [45] Baumberger, T. 44 [22] Baumgartel, H. 495 [36] Bay, B. 599 [81] Bay, K. 218 [30]; 219 [79] Bean, J.C. 451 [48] Beaudoin, A.J. 222 [186]; 223 [242] Becker, R. 598 [42] Beckman, S. 450 [32] Begley, M.R. 45 [50]; 599 [54] Bekele, M. 223 [209]; 223 [210] Bell, W.L. 363 [104]; 496 [64] Bellissent, R. 414 [24]; 414 [51]; 414 [52]; 414 [54] Belytschko, T. 46 [109]

Ben-Zion, Y. 44 [20] Bengu, E. 452 [102] Bennetto, J. 450 [27] Benoit, E. 222 [205] Benoit, W. 450 [25] Benson, S.W. 222 [198] Benzerga, A.A. 45 [62]; 45 [63] Berg, P. 218 [19] Bergman, C. 417 [157] Bernardini, J. 417 [157] Bernshteyn, M.L. 362 [26] Bertoud, P. 44 [22] Bertram, R. 222 [203] Berveiller, M. 261 [39] Bever, M.B. 217 [2] Bevis, M. 362 [16]; 362 [37]; 362 [38]; 362 [39]; 362 [40]; 362 [41] Beyss, M. 413 [14]; 413 [15]; 415 [102] Bharathi, M.S. 219 [50]; 219 [51]; 221 [137]; 221 [166]; 222 [186]; 223 [226]; 223 [230] Bhattacharya, K. 260 [4] Bhushan, B. 45 [69]; 46 [104]; 495 [54] Bianchi, A. 415 [72] Bilby, B.A. 218 [34]; 261 [33]; 362 [31]; 363 [88]; 363 [98] Bilello, J.C. 600 [101] Bingert, J.F. 361 [9] Birnbaum, H.K. 496 [80] Birnie, D. 260 [2] Blažková, J. 223 [215] Blech, I. 413 [4] Blech, I.A. 600 [114] Bletry, M. 415 [87] Blewitt, T.H. 362 [60]; 363 [101] Bluher, R. 414 [37] Blumenau, A. 450 [33]; 450 [34]; 451 [87]; 451 [88] Bnagert, U. 451 [88] Boas, W. 362 [50] Boˇcek, M. 219 [54] Bodner, S.R. 223 [227] Bogy, D.B. 45 [28] Bohr, T. 222 [195] Bojar, T. 362 [59] Bonneville, J. 416 [124]; 416 [125]; 416 [147]; 416 [150]; 417 [153] Booth, V. 222 [208] Borgesen, P. 600 [115] Born, E. 600 [109] Bouchaud, E. 44 [14]; 219 [57]; 599 [78] Bouchaud, J. 219 [81]; 450 [39] Boudard, M. 414 [51]; 414 [52]; 414 [54]; 414 [56]

Author Index Boulanger, L. 45 [61]; 496 [63] Bourret, A. 450 [20] Bouvier, S. 46 [89] Bowden, F.P. 44 [1]; 44 [18]; 44 [19] Bower, A.F. 44 [15] Bowles, J.S. 260 [8] Bradby, J.E. 495 [28]; 495 [50]; 495 [51] Bravman, J.C. 494 [12]; 600 [107]; 600 [108]; 600 [117]; 600 [118]; 600 [119]; 600 [120] Bréchet, Y. 45 [62]; 45 [63]; 218 [33]; 221 [163]; 415 [91]; 451 [55] Breghezan, A. 363 [81] Bresson, L. 414 [21]; 414 [22]; 414 [23]; 416 [123] Breuer, D. 598 [37] Briddon, P. 450 [34]; 451 [87]; 451 [88] Bridgman, P.W. 219 [45] Brinck, A. 218 [27]; 218 [28] Brindley, A. 221 [132] Brinson, L.C. 600 [96] Broomhead, D. 221 [148] Bross, H. 639 [24] Brown, L.M. 79 [66]; 218 [8] Brown, P.D. 451 [70] Brown, R. 221 [139] Brown, R.A. 451 [69] Brown, W.L. 600 [107]; 600 [108]; 600 [117] Browning, N.D. 451 [91] Bruls, G. 415 [68] Brunel, M. 639 [32] Brunner, D. 413 [19]; 416 [146] Brunton, J.H. 44 [12] Bryant, P. 221 [139] Buchheit, T. 46 [91] Budai, J.D. 599 [61]; 599 [84]; 599 [86]; 600 [95]; 600 [106] Bujard, M. 450 [25] Bulatov, V.V. 45 [60]; 363 [99]; 363 [100]; 450 [24]; 450 [29]; 450 [35]; 598 [42] Bullough, R. 261 [33] Buras, B. 597 [1] Burgers, J.M. 78 [50] Burkov, S.E. 416 [127] Busing, W.R. 600 [97] Butte, B.M.J. 222 [203] Byerlee, J.D. 219 [64] Cabot, A. 497 [107] Cahill, D.G. 497 [97]; 497 [98]; 497 [99] Cahn, J.W. 362 [45]; 413 [4] Cahn, R.W. 362 [18]; 362 [19] Cai, W. 45 [60]; 450 [24]; 450 [35]

643

Caillard, D. 414 [21]; 414 [22]; 414 [23]; 416 [123]; 450 [42] Caldwell, W.A. 600 [118] Callout, J.L. 222 [205] Calvayrac, Y. 414 [24]; 414 [26]; 414 [27]; 414 [28]; 414 [53]; 417 [157] Canova, G.R. 45 [61]; 413 [18]; 496 [63] Capio, C.D. 639 [44] Capitan, M.J. 414 [53] Carlson, J.M. 222 [188] Carnahan, B. 451 [73] Caroli, C. 46 [98] Carpenter, S.H. 220 [106] Carpick, R.W. 46 [100] Carr, J. 223 [224] Carr, T.W. 222 [208] Carstanjen, H.D. 413 [19]; 416 [146] Carter, C.B. 363 [66]; 363 [77] Castaing, B. 222 [194] Castelnau, O. 598 [40] Caudron, R. 414 [54] Cecot, W. 45 [29] Celestre, R.S. 600 [107]; 600 [108]; 600 [117]; 600 [118] Celli, V. 416 [140]; 450 [3] Celotto, S. 261 [13]; 261 [34] Cerva, H. 639 [35] Chadrasekhar, S. 220 [121] Chai, Y.W. 261 [38] Chaikin, P.M. 220 [90] Chang, R.W. 45 [28] Chang, Y.C. 261 [50] Chateigner, D. 601 [130] Chattopadhyay, K. 415 [74] Chelikowsky, J.R. 450 [19]; 451 [90] Chen, I.-W. 261 [10] Chenbuo 78 [5] Cheng, T.T. 261 [46] Chernikov, M.A. 414 [29]; 415 [72] Cherns, D. 451 [85]; 451 [92] Chevrier, J. 219 [59] Chhabra, A.B. 221 [160] Chi Yuwei 78 [6] Chiang, Y.-M. 260 [2] Chiao, Y.-H. 261 [10] Chiarello, R. 219 [58] Chichili, D.R. 361 [7] Chihab, K. 219 [43] Chin, G.Y. 361 [3]; 362 [43] Chiu, Y.-T. 78 [37] Chmelik, F. 222 [179] Christian, J.W. 260 [5]; 261 [22]; 261 [44]; 362 [21]; 362 [32]; 362 [42]; 363 [98]

644 Chrzan, D.C. 450 [32]; 495 [30]; 497 [107]; 598 [36] Chun, J.S. 497 [94] Chung, J. 452 [103] Chung, J.-S. 598 [5]; 599 [84]; 599 [86]; 600 [103]; 600 [104] Chung, K.-S. 598 [7]; 600 [106] Chung, M.Y. 77 [1]; 77 [2] Ciavarella, M. 45 [32] Ciliberto, S. 221 [138] Clarebrough, L.M. 362 [51]; 363 [68]; 363 [83] Clark, G.F. 639 [34]; 639 [36] Clark, S.J. 495 [60] Clarke, D.R. 45 [36]; 495 [49] Cleveringa, H.H.M. 46 [84] Clifton, R.J. 44 [4] Cochet Muchy, D. 451 [55] Cockayne, D.J.H. 363 [67]; 450 [11] Coddens, G. 414 [24]; 414 [25]; 414 [26]; 414 [27]; 414 [28] Coene, W. 451 [72] Coffman, V.R. 598 [46] Cohen, J.B. 600 [100] Cohen, M. 261 [32] Cohen, M.L. 450 [19]; 495 [30]; 495 [32]; 496 [61] Coker, D. 45 [24] Cole, J.D. 222 [200] Coltman, R.R. 362 [60]; 363 [101] Colton, R.J. 494 [17] Coniglio, A. 221 [155] Constantini, M. 219 [71] Cook, R.F. 495 [49] Corcoran, S.G. 494 [17] Cordier, P. 414 [23]; 416 [124]; 416 [125] Cote, M. 496 [61] Cottrell, A.H. 79 [57]; 218 [34]; 218 [35]; 362 [48]; 363 [88] Cotts, E.J. 638 [5] Coujou, A. 363 [106] Couret, A. 363 [106] Crain, J. 495 [60] Crellin, E.B. 362 [16] Crimp, M.A. 451 [80] Crocker, A.G. 261 [27]; 362 [31]; 362 [33]; 362 [34]; 362 [37]; 362 [38]; 362 [39]; 362 [40]; 362 [41]; 362 [42] Crooks, R.M. 494 [19] Crutchfield, J.P. 221 [144] Csepregi, L. 495 [36] Cuitiño, A.M. 46 [76] Currat, R. 415 [73] Cyrankowski, E. 497 [104] Czjzek, G. 363 [82]

Author Index Dahmen, U. 261 [20]; 495 [33] Dai, M.X. 415 [99] Dai, Y. 363 [79] D’Anna, G. 221 [168] Darinskii, A.N. 78 [46]; 78 [49] Darling, T.W. 415 [72] Dattagupta, S. 220 [93]; 220 [94] David, S.A. 599 [87]; 600 [122] Davies, M.J. 222 [197] Davis, R.F. 600 [111]; 600 [112]; 600 [113] De, P. 416 [109] de Arcangelis, L. 221 [155] de Boissieu, M. 414 [51]; 414 [52]; 414 [54]; 414 [56]; 414 [57]; 414 [63]; 415 [73]; 415 [86]; 415 [87]; 415 [91] de Borst, R. 46 [108] De Fouquet, J. 452 [108] De Koning, M. 450 [24]; 450 [35] de la Rubia, T.D. 45 [57]; 363 [99]; 363 [100]; 450 [24] de Wit, R. 79 [54] Dee, G. 223 [233] deGennes, P.-G. 220 [92] DeHosson, J.Th.M. 496 [75]; 497 [105] Demelio, G. 45 [32] Demenet, J. 450 [12]; 451 [61]; 452 [108] Demler, E. 598 [46] DenBaars, S.P. 600 [113] Deshpande, V.S. 45 [54]; 45 [70]; 46 [87]; 46 [88]; 46 [102]; 46 [103]; 46 [110] Desjardins, P. 497 [97] Devincre, B. 45 [55]; 220 [111]; 598 [45] Diao, D. 495 [54] Diener, F. 222 [205] Diener, M. 222 [205] Dieterich, J.H. 44 [2]; 44 [21] Dietz, V. 450 [44] Differt, K. 220 [127] Dimiduk, D.M. 45 [41]; 494 [14] Ding, D. 414 [43]; 414 [45]; 414 [46]; 414 [47]; 416 [110]; 416 [111]; 416 [114]; 416 [115]; 416 [116] Ding, D.H. 416 [113] Ding, M. 221 [143] Ding Haojiang 78 [5]; 78 [6] Diodati, P. 219 [70] Dmitrienko, V.E. 415 [76] Dobes, F. 220 [98] Doherty, R.D. 361 [5]; 361 [8]; 361 [10]; 364 [112] Dolinsek, J. 414 [29]; 414 [30] Domnich, V. 494 [5]; 494 [20]; 495 [26]

Author Index Donth, H. 639 [29] Dorn, J.E. 416 [141] Dotera, T. 415 [81] Duan, X.Y. 414 [55]; 416 [112] Dub, S.N. 494 [5]; 495 [26] Dubois, J.M. 414 [30]; 415 [91]; 417 [157] Dugain, F. 415 [73] Duneau, M. 414 [38] Dunin-Barkovskii, L. 218 [33] Dunn, M.L. 77 [4] Dunne, D.P. 261 [36] Duquesne, J.-Y. 414 [59]; 415 [66] Duran, J. 220 [92] Durand-Charre, M. 415 [86] Eckhaus, W. 220 [129]; 222 [206] Eckmann, J.P. 221 [138] Edagawa, K. 413 [13]; 414 [20]; 414 [31]; 414 [32]; 414 [33]; 414 [34]; 414 [35]; 414 [36]; 415 [89]; 415 [93]; 415 [94]; 416 [126]; 416 [128]; 416 [133]; 416 [134]; 416 [137]; 417 [155] Edelman, J. 600 [103] Edwards, R.L. 494 [10] Einfeldt, S. 600 [111]; 600 [112]; 600 [113] El-Azab, A. 220 [115] El-Danaf, E. 361 [5]; 361 [8] Elkaim, E. 414 [56] Elser, V. 415 [78] Embury, J.D. 362 [29]; 362 [56]; 362 [61]; 362 [62]; 363 [85]; 363 [96] Engeleke, C. 218 [27]; 218 [28] Entwisle, A.E. 361 [12] Erneux, T. 222 [204]; 222 [207]; 222 [208] Eshelby, J.D. 78 [22]; 79 [69]; 416 [117]; 639 [25]; 639 [47] Espinosa, H. 600 [96] Essmann, U. 219 [77]; 220 [127] Estrin, Y. 46 [85]; 218 [33]; 218 [40]; 219 [43]; 221 [163]; 222 [176]; 222 [177]; 223 [228]; 223 [235] Etison, I. 45 [28]; 45 [31] Eubank, S. 221 [151] Evans, A.G. 45 [37] Eykholt, R. 221 [150] Faivre, G. 78 [21] Fall, C. 450 [33]; 450 [34]; 451 [87]; 451 [88] Falster, R.J. 451 [66] Fan, T.Y. 416 [112] Fanti, F. 639 [17] Farber, B. 452 [111]; 452 [112] Farenc, S. 363 [106]

Faria, L.O. 44 [7] Farmer, J.D. 221 [144]; 221 [151] Fazzio, A. 450 [36] Fdrozhzhin, A. 451 [58] Feder, J. 219 [69] Feldman, E.P. 78 [19] Feltham, P. 220 [97] Feng, J. 415 [100]; 415 [101]; 415 [103] Feuerbacher, M. 413 [15]; 413 [16]; 413 [17]; 415 [102]; 415 [106]; 416 [115]; 416 [122]; 416 [151]; 417 [154] Feuillet, G. 451 [85] Field, J.S. 495 [25]; 495 [44] Field, S. 221 [169] Figge, S. 600 [113] Fivel, M.C. 45 [61]; 496 [63] Fleck, N.A. 45 [35]; 45 [44]; 46 [92]; 46 [94]; 223 [241]; 598 [47] Flinn, P.A. 494 [8] Florando, J.N. 45 [39]; 494 [14] Foiles, S.M. 496 [68] Fordeux, A. 363 [81] Foreman, A.J.E. 416 [118] Forsen, B.H. 218 [17] Forwood, C.T. 363 [68] Foster, K. 415 [67] Franciosi, P. 261 [39] Francoual, S. 414 [57] Frank, F.C. 363 [87] Frank, W. 414 [37] Fraser, S.J. 222 [197]; 222 [198] Frauenheim, T. 450 [33]; 450 [34]; 451 [87]; 451 [88] Freitag, E.H. 44 [18] Fressengeas, C. 217 [3]; 219 [49]; 221 [134]; 221 [135]; 221 [136]; 221 [137]; 221 [166]; 222 [186] Freund, A. 601 [128] Frey, F. 415 [73] Friedel, J. 79 [58] Friedman, L.H. 45 [58]; 46 [83] Fries, E. 639 [43] Frisch, U. 221 [157] Fruhauf, J. 495 [37] Fujimoto, H. 600 [107] Fujita, H. 363 [107] Fujita, K. 451 [51] Fukatsu, S. 451 [51] Furuhara, T. 261 [11]; 261 [47] Gadaud, P. 452 [108] Gaeler, F. 415 [82] Galdrikian, B. 221 [151] Galeckas, A. 451 [59]

645

646 Gambaro, D. 451 [66] Gang Feng 598 [51] Gao, H. 45 [42]; 45 [49]; 598 [48]; 598 [49]; 600 [96]; 639 [48] Gao, H.J. 599 [55] Gao, Y.-F. 44 [15]; 46 [106] Gardner, D.S. 494 [8] Garg, A. 261 [50] Gartner, K. 497 [100] Gas, P. 417 [157] Gastaldi, J. 414 [54] Gavazza, S.D. 79 [70] Gavilano, J.L. 414 [29] Gay, P. 599 [63] Gee, M.L. 46 [99] George, A. 450 [16]; 450 [39]; 451 [53] George, E.P. 45 [38] Gerberich, W.W. 46 [93]; 494 [4]; 494 [17]; 494 [18]; 494 [23]; 495 [47] Gerk, A.P. 452 [107] Gevers, R. 363 [103] Geyer, B. 413 [17]; 416 [122]; 417 [154] Ghoniem, N.M. 45 [59] Giacometti, E. 416 [147]; 417 [153] Giamei, A. 450 [14] Giannakopoulos, A.E. 494 [3] Gibbings, J. 451 [50] Gibbs, J.W. 261 [23] Gibson, M. 452 [96] Gilman, J.J. 220 [100]; 220 [101] Glaisher, R. 452 [99] Glazov, M. 223 [212] Glazov, M.V. 220 [128]; 223 [211] Gleiter, H. 598 [23] Goddard, P.A. 639 [34]; 639 [36] Godfrey, A. 599 [83] Gog, T. 639 [45] Gogotsi, Y. 494 [5]; 494 [20] Gogotsi, Y.G. 495 [26] Golding, B. 414 [60] Goldman, A. 414 [51] Gottschalk, H. 450 [44]; 452 [93]; 452 [112] Gottschalk, H.H. 451 [56] Gottstein, G. 496 [72] Gouldstone, A. 494 [3] Graeff, W. 639 [35] Granato, A.V. 638 [3]; 638 [4]; 639 [17] Granick, S. 600 [96] Grasman, J. 222 [200] Grassberger, P. 219 [76]; 221 [146] Grasset, F.M. 219 [61] Grasso, J.R. 219 [66]; 220 [119] Gratias, D. 413 [4]; 414 [21]; 414 [22]; 414 [27]; 414 [53]; 415 [96]; 416 [123]; 417 [157]

Author Index Gray, G.T., III 361 [4]; 361 [6]; 361 [9]; 363 [97] Grebogi, C. 221 [143] Greene, J.E. 496 [90]; 497 [94]; 497 [96]; 497 [97]; 497 [98]; 497 [99] Greenwood, J.A. 44 [13] Greer, J.R. 45 [40]; 598 [50]; 598 [51] Gregoriy, F. 363 [93] Greiser, J. 416 [121] Gremaud, G. 414 [61]; 450 [25] Griebenow, M. 639 [45] Grigolini, P. 221 [171] Gröller, E. 223 [231] Groma, I. 46 [77]; 46 [78]; 46 [79]; 219 [82]; 219 [84]; 219 [85]; 219 [86]; 220 [112]; 220 [117]; 220 [120] Gruen, D. 496 [69] Grunlan, J.C. 46 [93] Grussbach, G. 221 [165] Gudmundson, P. 45 [46] Guimares, J. 44 [7] Gumbsch, P. 416 [135]; 600 [96]; 639 [48] Gundlach, C. 219 [83] Guo, J.Q. 415 [87]; 417 [155] Guo, Z. 496 [83]; 496 [84] Guo Fenglin 78 [6] Gurtin, M.E. 45 [45] Gusev, V.E. 639 [30] Gutekunst, G. 451 [75] Gutierrezsoza, A. 451 [88] Gutkin, M.Yu. 78 [43] Gutmanas, E. 639 [42] Guyoncourt, D.M.M. 362 [38] Guyot, P. 413 [18]; 415 [90]; 415 [91] Haas, A. 415 [68] Haasen, P. 220 [105]; 451 [57]; 495 [40]; 638 [10] Habenschuss, A. 600 [101] Hafner, J. 415 [75] Hähner, P. 217 [4]; 218 [30]; 219 [44]; 219 [48]; 219 [79]; 219 [80]; 220 [110]; 220 [113]; 220 [114]; 222 [183]; 222 [184]; 222 [185]; 223 [236] Hall, E.O. 362 [17]; 496 [65] Hall, M.G. 261 [18] Halsey, T.C. 221 [158] Ham, R. 450 [14] Hamerský, M. 223 [215]; 223 [217]; 223 [218] Hamilton, J.C. 496 [62] Hammond, C. 261 [30] Han, X. 45 [59] Hanada, S. 451 [82] Hanke, G. 415 [102] Hannemann, B. 495 [37]

Author Index Hanner, P. 598 [27] Hansen, A. 219 [62] Hansen, N. 220 [123]; 598 [30]; 598 [35]; 598 [36]; 598 [38]; 598 [39]; 599 [80]; 599 [81]; 599 [82]; 599 [83] Hara, T. 496 [81] Harding, D.S. 495 [24] Hargreaves, M.E. 362 [51] Härtwig, J. 639 [40] Hashimoto, T. 413 [10]; 417 [152] Hatherly, M. 598 [20] Hatton, P.D. 495 [60] Hazzledine, P.M. 363 [76] He, L.L. 495 [27] Head, A.K. 363 [68] Heckscher, F. 362 [37]; 362 [38] Heggie, M. 450 [31]; 450 [33]; 450 [34]; 450 [38]; 451 [87]; 451 [88] Heidelbach, F. 601 [130] Helliwell, J.R. 598 [8] Hemker, K.J. 361 [7]; 494 [10]; 494 [13]; 599 [75] Henley, C.L. 414 [48]; 414 [50]; 415 [78] Hennion, B. 414 [26]; 414 [51]; 414 [52]; 414 [54] Hentschel, H.G.E. 221 [159] Herrasti, P. 219 [60] Heslot, F. 222 [194] Heuberger, A. 495 [36] Hezemans, A.M.F. 601 [129] Higgens, F.P. 220 [106] Hill, M.J. 495 [43] Hinkel, C. 415 [68] Hiraga, K. 450 [13]; 451 [79]; 451 [82] Hirsch, P.B. 363 [65]; 363 [80]; 415 [95]; 450 [21]; 450 [43]; 452 [94]; 495 [56]; 599 [62]; 599 [63] Hirth, J.P. 45 [57]; 46 [82]; 79 [60]; 79 [75]; 79 [76]; 261 [14]; 261 [16]; 261 [17]; 261 [29]; 261 [34]; 261 [35]; 261 [41]; 261 [42]; 363 [84]; 363 [86]; 416 [142]; 450 [4]; 451 [64]; 451 [73]; 452 [105]; 495 [31]; 598 [43]; 639 [27] Hobert, H. 497 [100] Hobgood, H. 451 [61] Hockey, B.J. 495 [49] Holder, J. 639 [17] Holleck, H. 496 [92] Holmes, S.M. 363 [77] Holt, D.L. 220 [124] Hommel, D. 600 [112]; 600 [113] Homola, H.M. 46 [99] Honeycombe, R.K.W. 600 [126] Hong, S. 494 [12] Hons, A. 450 [11] Hoover, W.G. 78 [41]

647

Hopler, R. 600 [109] Horstemeyer, M.F. 46 [93]; 599 [52] Horsthemke, W. 220 [130] Horton, J.A. 599 [87] Hosford, W.F. 600 [123] Hosono, K. 414 [36] Houghton, D. 451 [49] Houston, J.E. 494 [19] Howard, A.J. 494 [19] Howe, J.M. 261 [9]; 261 [25]; 261 [40]; 261 [47]; 261 [50] Howie, A. 363 [65]; 363 [102]; 415 [95]; 495 [56] Hradil, K. 415 [73] Hu, C. 414 [43]; 414 [45]; 414 [46]; 414 [47]; 416 [110]; 416 [114]; 416 [116] Hu, C.Z. 416 [113] Huang, X. 599 [83]; 600 [125] Huang, Y. 45 [42]; 45 [52]; 450 [30]; 598 [48]; 598 [49] Hughes, D.A. 598 [36]; 599 [80]; 599 [81]; 599 [82] Hugo, G.R. 261 [45] Hull, R. 451 [48]; 451 [52]; 451 [54] Humble, P. 363 [68] Humphreys, C. 451 [70] Humphreys, F.J. 598 [20] Hurtado, J.A. 45 [67]; 45 [68] Huston, A.R. 78 [24] Hutchings, R. 494 [1] Hutchinson, J. 450 [15] Hutchinson, J.W. 45 [35]; 45 [42]; 45 [44]; 45 [50]; 45 [51]; 46 [94]; 223 [241]; 598 [47]; 598 [48]; 598 [49]; 599 [54] Hwang, K.C. 45 [52] Hyun, S. 44 [16]; 45 [33] Ice, G.E. 598 [4]; 598 [5]; 598 [7]; 599 [60]; 599 [61]; 599 [74]; 599 [84]; 599 [85]; 599 [86]; 599 [87]; 599 [88]; 599 [89]; 599 [90]; 599 [91]; 599 [92]; 600 [94]; 600 [95]; 600 [101]; 600 [102]; 600 [103]; 600 [104]; 600 [106]; 600 [112]; 600 [113]; 600 [119]; 600 [120]; 600 [122] Ichihara, M. 415 [89] Ichihashi, T. 452 [102] Iijima, S. 452 [95]; 452 [102] Ilschner, B. 220 [105] Imai, M. 450 [26] Imai, Y. 416 [148] Indenbom, V.L. 78 [31]; 78 [34]; 79 [53]; 79 [63]; 79 [73]; 639 [14] Inoko, F. 600 [125] Inui, H. 451 [81]

648 Inuiz, H. 363 [93] Ishihara, K.N. 414 [40] Ishii, K. 362 [24] Ishii, Y. 414 [42]; 415 [85]; 415 [88]; 415 [92] Ishimasa, T. 414 [56]; 414 [57] Israelachvili, J.N. 46 [99] Ito, A. 223 [222] Ito, H. 497 [96] Iunin, Y.L. 450 [40]; 452 [111]; 452 [112] Jackobsen, B. 219 [83] Jamieson, J.C. 495 [59] Jang, Y.H. 45 [32] Janot, C. 414 [51]; 414 [52]; 415 [91] Janssen, T. 414 [39] Jaric, M.V. 415 [79]; 415 [84] Javaraman, S. 494 [10] Jeanclaude, V. 219 [49] Jeanneret, J.-F. 416 [147] Jendrich, U. 451 [57]; 452 [109] Jensen, D.-J. 598 [35]; 598 [39]; 599 [69] Jensen, H.J. 222 [193] Jensen, M.H. 221 [158]; 222 [195] Jensen, R.V. 221 [160] Jeong, H.-C. 415 [77]; 415 [80] Jergan, E. 78 [44] Jin, M. 496 [76] Jisoon Ihm 496 [89] John, T.M. 222 [192] Johnson, K.L. 46 [97]; 494 [16] Johnston, W.G. 220 [100] Jones, C.K.R.T. 222 [200] Jones, R. 450 [22]; 450 [31]; 450 [33]; 450 [34]; 450 [38]; 451 [67]; 451 [87]; 451 [88]; 452 [113] Jordan-Sweet, J.L. 600 [116] Jossang, T. 363 [84] Joulaud, J.-L. 414 [53]; 417 [157] Jurkschat, K. 451 [66] Justo, J. 450 [24]; 450 [28]; 450 [35]; 450 [36] Jyobe, F. 451 [65] Kabler, M. 416 [140]; 450 [3] Kabutoya, E. 414 [20]; 417 [155] Kadanoff, L.P. 221 [158] Kailer, A. 495 [26] Kajiwara, S. 261 [12] Kajiyama, K. 414 [31]; 414 [32] Kakegawa, K. 416 [149] Kaldor, S.K. 600 [116] Kalidindi, S.R. 361 [5]; 361 [8]; 361 [10]; 364 [112] Kalk, A. 218 [41]; 222 [180]; 222 [181]

Author Index Kalugin, P.A. 413 [9] Kamat, S.V. 451 [73] Kamensky, V.G. 415 [76] Kamimura, Y. 416 [126]; 416 [144] Kamiura, Y. 451 [65] Kamiya, A. 415 [89] Kamphorst, S.O. 221 [138] Kaneko, Y. 414 [57] Kaneyama, T. 415 [98] Kantz, H. 218 [22] Karnthaler, H.P. 363 [70]; 363 [78] Karr, B.W. 497 [97]; 497 [98]; 497 [99] Kaschner, G.C. 361 [9] Kasper, J.S. 495 [58] Kassner, M.E. 600 [96] Kato, K. 415 [71] Katona, T.M. 600 [113] Katz, A. 414 [38] Kawabata, T. 450 [13] Kear, B.H. 450 [14] Keer, L.M. 45 [65]; 45 [66] Kelchner, C.L. 496 [62] Keller, R. 494 [9] Kelly, A. 599 [63] Kelly, P.M. 261 [30] Kelton, K.F. 415 [67]; 415 [74] Kerkovian, J. 222 [200] Khasawinah, S. 495 [35] Kiemel, T. 222 [203] Kiho, H. 362 [24] Kilgore, B.D. 44 [21] Kim, H.J. 496 [78] Kim, J.Y. 415 [67] Kim, K.-S. 45 [67]; 45 [68]; 46 [101]; 46 [106]; 46 [107] Kim, K.S. 600 [96] Kim, K.W. 598 [9] Kim, Y.-W. 497 [96] Kim, Y.H. 496 [78] Kimura, K. 415 [69]; 415 [70]; 415 [71] King, G. 221 [148] Kingery, W.D. 260 [2] Kinsman, K.R. 261 [18] Kirchner, H.O.K. 77 [3]; 78 [38]; 78 [39]; 416 [144]; 416 [145]; 451 [83] Kirchner, P.D. 495 [49] Kirihara, K. 415 [69]; 415 [70]; 415 [71] Kiritani, M. 218 [14] Kisielowski, C. 450 [32]; 451 [86] Kitayev, A.Y. 413 [9] Kittle, M. 450 [1] Kleiser, T. 219 [54] Klimanek, P. 361 [11]; 598 [34]; 598 [37]; 599 [64]; 599 [79]

Author Index Knauss, W. 600 [96] Kneezel, G.A. 638 [4] Koch, C. 452 [100]; 452 [101]; 452 [103]; 452 [104] Koch, C.C. 598 [26] Kocks, U.F. 218 [39]; 361 [2] Koehler, J.S. 79 [71] Koester, U. 413 [14]; 416 [121] Kogut, L. 45 [31] Koh, H.-J. 494 [3] Koiwa, M. 414 [65] Kok, S. 222 [186]; 223 [242] Kolar, H. 451 [45]; 451 [71]; 452 [98] Konstantinidis, A.K. 219 [56] Kopmann, W. 218 [28] Korbel, A. 223 [234]; 362 [28]; 362 [30] Korhonen, M.A. 600 [115] Kortan, A.R. 414 [60]; 414 [62]; 415 [73] Koschella, U. 415 [82]; 415 [83] Kosevich, A.M. 78 [19]; 79 [55] Koslowski, M. 223 [225] Kostlan, E.J. 78 [40] Koutsos, V. 219 [61] Krabbendam, H. 601 [129] Krajci, M. 415 [75] Kramer, D.E. 494 [23] Kramer, M.J. 416 [119]; 416 [120] Kratochvil, J. 220 [126]; 220 [131] Krauss, G. 496 [87] Krenn, C.R. 495 [30]; 495 [32]; 496 [89] Kreuzer, H.G.M. 46 [71]; 46 [72] Kriese, M.D. 494 [23] Krim, J. 219 [58] Krivoglaz, M.A. 598 [11]; 598 [12]; 598 [29]; 599 [73] Kroll, M.C. 495 [49] Kröner, E. 78 [52]; 220 [118] Kronmuller, H. 414 [37] Kroon, J. 601 [129] Ku, C.-H. 600 [116] Kubin, L.P. 45 [55]; 217 [3]; 218 [11]; 218 [12]; 218 [33]; 218 [40]; 219 [43]; 219 [81]; 220 [111]; 221 [135]; 221 [136]; 221 [137]; 221 [163]; 221 [166]; 222 [176]; 222 [177]; 222 [186]; 223 [228]; 598 [45]; 600 [96] Kubota, Y. 415 [69] Kuhlmann-Wilsdorf, D. 45 [64]; 220 [122]; 220 [123]; 598 [38]; 599 [81] Kuksenko, V. 219 [64] Kulik, A. 414 [61] Kumar, K.S. 598 [24] Kumar, N. 218 [21] Kuo, K.H. 414 [41]; 415 [104]

649

Kuramoto, Y. 223 [221] Kurtz, W. 451 [77] Kurzydlowski, K.J. 496 [79] Kusters, K. 450 [23] Kuwabara, M. 451 [74]; 452 [99] Kveder, V. 450 [1] Kycia, S. 414 [51] Lagneborg, R. 218 [17] Laird, C. 218 [9]; 223 [211]; 261 [48] Lakin, E. 639 [39]; 639 [40] Lalli, L.A. 223 [212] Landau, L.D. 599 [76]; 638 [8] Landman, U. 45 [25] Langdon, T. 496 [70]; 496 [71] Langer, J. 600 [96] Langer, J.S. 222 [188]; 223 [233] Langmaack, E. 218 [27] Langsdorf, A. 415 [68] Lapusta, N. 44 [9]; 44 [20] Larson, B.C. 598 [7]; 599 [60]; 599 [61]; 599 [84]; 599 [85]; 599 [86]; 599 [91]; 600 [95]; 600 [96]; 600 [106] Last, H. 494 [13] Latora, V. 221 [171] Lauriat, J.P. 414 [56] Lauridsen, E.M. 599 [69] Lawn, B.R. 495 [42] Le Bolloc’h, D. 415 [88] Le Chatelier, F. 218 [13] Le Thanh, V. 219 [59] Lebyodkin, M. 218 [33]; 221 [137]; 221 [166]; 222 [186]; 223 [242] Lebyodkin, M.A. 221 [163] Lee, B. 451 [79] Lee, C.S. 496 [83]; 496 [84] Lee, T.C. 496 [80] Lefebvre, S. 414 [53] Lefever, R. 220 [130] Lei, J. 414 [47]; 414 [55]; 416 [111]; 416 [114] Leisure, R.G. 415 [67] Lendvai, J. 219 [82] Lenosky, T. 450 [24] Lentzen, M. 451 [72] LeSar, R. 223 [225] Letoublon, A. 414 [52]; 414 [54]; 414 [56] Levade, C. 450 [42] Levine, D. 413 [1]; 413 [6] Levine, L.E. 599 [72] Levinstein, H.J. 639 [44] Levitov, L.S. 413 [9] Levy, H.A. 600 [97] Li, C.-Y. 600 [115]

650

Author Index

Li, G. 222 [199] Li, J.C.M. 496 [67] Li, Q. 46 [101]; 639 [49] Li, X. 495 [54] Li, X.F. 416 [112] Li, Z.C. 495 [27] Liangjian 78 [5] Libchaber, A. 222 [194] Lieberman, D.S. 260 [7] Liebertz, H. 413 [14]; 416 [121] Lienert, U. 219 [83]; 599 [69]; 599 [71] Liew, K.M. 78 [8] Lifshitz, E.M. 599 [76]; 638 [8] Lilleodden, E.T. 494 [4]; 494 [17]; 494 [18]; 495 [47]; 496 [68]; 496 [74]; 497 [102]; 598 [51] Lim, S. 451 [70] Lim, S.C. 44 [12] Ling, X. 221 [169] Liniger, E.G. 600 [116] Linnros, J. 451 [59] Litvinov, Y.M. 600 [105] Liu, L. 495 [27] Liu, Q. 598 [35]; 598 [36] Liu, W. 599 [85]; 600 [95]; 600 [112] Lizzio, R. 78 [42] Lockner, D.A. 219 [64]; 219 [75] Löffelmann, H. 223 [231] Lograsso, T.A. 414 [64]; 417 [156] Lomer, W.M. 362 [47] Long, G.G. 599 [72] Long, N. 452 [93] Longtin, A. 221 [151] Loretto, M.H. 363 [83] Lothe, J. 46 [82]; 78 [29]; 78 [32]; 78 [33]; 78 [34]; 78 [35]; 78 [45]; 78 [46]; 78 [47]; 78 [48]; 79 [60]; 79 [64]; 79 [74]; 79 [75]; 261 [14]; 261 [35]; 363 [86]; 416 [142]; 450 [4]; 495 [31]; 598 [43]; 639 [27] Lou, J. 46 [91] Louat, N. 222 [175] Louchet, A. 451 [47] Louchet, F. 450 [17]; 451 [55]; 452 [110] Louie, S.G. 496 [61] Lowe, W. 600 [103] Lowe, W.P. 599 [84]; 600 [106] Loyalka, S.K. 495 [34] Lubensky, T.C. 220 [90]; 413 [5]; 413 [6]; 413 [8] Lücke, K. 638 [3] Lüders, W. 218 [25] Luedtke, W.D. 45 [25] Lukáˇc, P. 222 [179]; 223 [217]; 223 [218] Lund, F. 639 [28] Lunt, A. 600 [104]

Lykotrafitis, G. 45 [24] Lyonnard, S. 414 [26]; 414 [27]; 414 [28] Ma, Q. 45 [36] Ma, X. 261 [24]; 261 [41]; 261 [43] Ma, X.L. 416 [121] MacDowell, A.A. 600 [107]; 600 [108]; 600 [117]; 600 [118] Macherauch, E. 222 [178] MacKenzie, J.K. 260 [8] Maclean, J.R. 495 [60] Madec, R. 598 [45] Mader, S. 363 [82] Maeda, K. 450 [5]; 451 [46]; 451 [51]; 451 [65] Maeda, N. 452 [97] Magerl, A. 599 [64] Mahadevan, L. 600 [96] Mahajan, S. 261 [9]; 261 [44]; 361 [3]; 362 [21] Mahon, G.J. 261 [9] Maißner, O. 600 [124] Majumdar, A. 600 [96] Maki, T. 261 [11]; 496 [77] Malen, K. 78 [27] Malygin, G.A. 218 [24] Mancini, L. 414 [54] Mandal, P. 414 [35]; 414 [36] Mandelbrot, B.B. 219 [67]; 219 [68]; 221 [154] Mané, R. 221 [142] Maneveau, C. 221 [160] Mann, A.B. 495 [29] Mañosa, L. 219 [65] Marchesoni, F. 219 [70] Marder, A.R. 496 [86]; 496 [87] Marder, J.M. 496 [86] Margulies, L. 599 [70] Marieb, T. 600 [107] Marklund, S. 450 [9] Marks, L. 452 [102] Maroudas, D. 451 [69] Marsan, D. 219 [74] Martins, J.A.C. 44 [7] Mashiyama, H. 223 [222] Mason, T.A. 361 [9] Mason, W.P. 639 [12]; 639 [13]; 639 [21] Mass, U. 222 [198] Materlik, G. 639 [45] Mathewson, C.H. 362 [49] Matsuda, O. 639 [30] Matsuo, Y. 415 [88] Matsuoka, S. 496 [82] Matthews, H. 639 [41] Maugin, G.A. 78 [39] Maugis, D. 222 [189]

Author Index Mauldin, P.J. 361 [9] Maurel, A. 639 [28] May, C. 599 [64] Mayer, J. 451 [75] Maynard, J.D. 414 [62] Mazor, P. 452 [108] McCormick, P.G. 218 [37]; 222 [182]; 223 [235] McGuiggan, P.M. 46 [99] McHargue, C.J. 494 [21]; 495 [46] Mckelvy, M. 452 [99] Mecking, H. 361 [2] Mehregany, M. 452 [103] Mendelson, K.S. 452 [106] Menzel, R. 497 [100] Mera, Y. 451 [51] Mercier, J.F. 639 [28] Mercier, S. 46 [85] Merkle, K.L. 496 [73] Merten, L. 78 [15]; 78 [16]; 78 [17]; 78 [18] Messerschmidt, U. 413 [15]; 413 [16]; 413 [17]; 415 [102]; 416 [122]; 416 [151]; 417 [154]; 451 [60] Metzger, T. 600 [109] Metzmacher, C. 413 [16] Michalske, T.A. 494 [19] Migliori, A. 415 [72] Miguel, M.C. 219 [66]; 220 [117]; 220 [119] Mihalkovic, M. 415 [75] Mikata, Y. 78 [9] Mikhailov, M.A. 600 [105] Mikulla, R. 416 [135]; 416 [136] Milik, A. 223 [231] Millard, D.J. 362 [22] Mills, D.M. 597 [2]; 597 [3] Minor, A.M. 495 [38]; 495 [48]; 495 [55]; 496 [74]; 496 [75]; 496 [76]; 496 [85]; 497 [95]; 497 [102]; 497 [104]; 497 [105]; 497 [106]; 497 [107] Minowa, K. 451 [84] Miroux, A. 598 [40] Mitarai, Y. 414 [65] Mitchell, J.N. 261 [42] Mitchell, T.E. 261 [42]; 362 [53]; 451 [63]; 451 [64] Miyajima, N. 261 [11] Miyaki, K. 416 [126] Mizutani, U. 415 [89] Molénat, G. 218 [27] Molinari, A. 46 [85] Molinari, J.F. 44 [16]; 45 [30]; 45 [33] Moller, H. 452 [109] Moller, H.J. 450 [6] Mompiou, F. 414 [23]; 416 [123] Moon, D.M. 363 [74]

651

Mori, M. 414 [56] Mori, T. 363 [107] Moritani, T. 261 [11] Morris, J.W., Jr. 495 [30]; 495 [32]; 495 [48]; 496 [74]; 496 [75]; 496 [76]; 496 [78]; 496 [83]; 496 [84]; 496 [85]; 496 [89]; 497 [95]; 497 [102] Morton, A.J. 363 [68] Moss, S.C. 415 [88] Moss, W.C. 78 [41] Mott, N.F. 220 [96] Mou, Y. 261 [49] Muddle, B.C. 261 [45] Mügge, O. 219 [52]; 362 [36] Mughrabi, H. 218 [7]; 218 [10]; 218 [23]; 219 [78]; 598 [14]; 598 [15]; 598 [16]; 598 [28]; 598 [33]; 599 [56]; 599 [65]; 599 [66]; 599 [67]; 600 [127] Muhlstein, C.L. 494 [15] Mukherjee, P. 221 [170] Muller, G.M. 45 [35]; 223 [241]; 598 [47] Mullford, R.A. 218 [39] Munroe, P. 495 [50]; 495 [51] Munroe, P.R. 497 [101] Mura, T. 79 [65] Murray, J.D. 222 [196] Murthy, K.P.N. 218 [18]; 221 [134]; 221 [162] Müser, M.H. 45 [26]; 45 [27] Nabarro, F.R.N. 46 [81]; 78 [40]; 78 [51]; 79 [59]; 363 [89]; 363 [90]; 416 [130]; 598 [17]; 598 [18]; 598 [19]; 638 [1]; 639 [22]; 639 [26] Nadgornyi, E. 639 [16] Nagata, T. 415 [71] Nakajima, H. 417 [156] Nakata, T. 415 [69] Narita, N. 362 [20]; 362 [54] Navrátil, V. 223 [215] Needleman, A. 44 [23]; 45 [24]; 45 [53]; 45 [54]; 45 [58]; 45 [62]; 45 [63]; 45 [70]; 46 [73]; 46 [80]; 46 [83]; 46 [84]; 46 [87]; 46 [88]; 46 [89]; 46 [96]; 46 [102]; 46 [103]; 46 [108] Neelakantan, K. 223 [229] Nelson, D.R. 415 [84] Nelson, J.C. 494 [4]; 494 [18]; 495 [47] Nemat-Nasser, S.S. 600 [96] Nembach, E. 218 [27] Nesbitt, E.A. 362 [43] Neuhäuser, H. 218 [6]; 218 [27]; 218 [28]; 219 [44]; 219 [53]; 219 [55] Newell, A.C. 223 [220] Nicholson, R.B. 363 [65]; 495 [56] Nickel, K.G. 495 [26]

652

Author Index

Nie, J.F. 261 [45] Niewczas, M. 362 [29]; 362 [61]; 362 [62]; 362 [63]; 363 [64]; 363 [85] Nikitenki, V.I. 450 [40] Nikitenko, V. 452 [111]; 452 [112] Ninomiya, T. 416 [140]; 450 [3] Nishibori, E. 415 [69]; 415 [71] Nix, W.D. 45 [39]; 45 [40]; 45 [42]; 45 [49]; 362 [27]; 494 [8]; 494 [11]; 494 [12]; 494 [14]; 496 [68]; 496 [88]; 598 [48]; 598 [49]; 598 [50]; 598 [51]; 599 [55]; 600 [118] Nixon, T. 261 [29] Nori, F. 221 [168]; 221 [169] Noronha, S. 221 [152] Noronha, S.J. 221 [134]; 221 [135]; 221 [136] Northrop, G.A. 638 [5]; 639 [19] Northrup, J.E. 450 [19] Nortmann, A. 218 [42]; 222 [181]; 223 [237] Norton, D.P. 599 [86] Nosonovsky, M. 45 [69] Novikov, D.V. 639 [45] Nowacki, J.P. 78 [11]; 78 [12]; 78 [13]; 78 [14] Noyan, I.C. 600 [100]; 600 [116] Nozieres, P. 46 [98] Nunes, R. 450 [27]; 450 [28] Nye, J.F. 45 [47]; 362 [44]; 600 [93] Oberg, S. 450 [31]; 450 [34] Ocon, P. 219 [60] Odagaki, T. 415 [83] Oden, J.T. 45 [29] Ogawa, K. 261 [12] Ogletree, D.F. 46 [100] Ohmura, T. 496 [81]; 496 [82]; 496 [85] Ohta, T. 223 [222] Oliver, W.C. 45 [40]; 494 [1]; 494 [2]; 494 [21]; 494 [22]; 495 [24]; 495 [46]; 598 [50] Olsen, A. 450 [18]; 450 [19] Olson, C.J. 221 [169] Olson, G.B. 260 [1]; 261 [32] Onuki, A. 220 [91] Orlov, A. 450 [10] Orlov, S.S. 78 [31]; 79 [63] Orlov, V. 452 [112] Orlova, A. 220 [131] Ortin, J. 219 [65] Ortiz, M. 45 [30]; 46 [76] Osetskiy, A.I. 223 [216] Ostlund, S. 413 [3]; 413 [6] Ostrom, P. 218 [17] Otsuka, K. 260 [3] Ott, E. 221 [143] Ott, H.R. 414 [29]; 415 [72]

Ourmazd, A. 450 [43]; 452 [94] Overwijk, W. 451 [72] Ovidko, I.A. 598 [25] Owen, W.S. 260 [1] Packard, N.H. 221 [144] Padmore, H.A. 600 [107]; 600 [108]; 600 [117]; 600 [118] Page, T.F. 494 [21]; 495 [46] Paladin, G. 222 [195] Pan, E. 78 [10] Pande, C.S. 363 [76] Pang, J. 599 [88] Pang, J.F. 599 [89] Pang, J.W.L. 600 [94] Pantleon, W. 219 [83]; 598 [31]; 598 [32]; 598 [37]; 599 [71] Paparo, G. 219 [71] Parisi, G. 221 [157] Park, C. 599 [86] Park, J.S. 599 [87] Park, Y.H. 598 [9] Parks, D.M. 598 [41]; 598 [42] Parsey, J.M., Jr. 414 [60] Parthasarathy, T.A. 45 [41] Pashley, D.W. 363 [65]; 495 [56] Pastur, L.A. 78 [19] Patel, J.R. 600 [107]; 600 [108]; 600 [117]; 600 [118]; 600 [119]; 600 [120] Patterson, A.L. 600 [99] Paviot, V.M. 494 [11] Paxton, A.T. 361 [12] Payne, A. 219 [59] Payne, M.C. 495 [57] Peach, M. 79 [71] Pei, L. 44 [16]; 45 [33] Peierls, R.E. 416 [129]; 638 [9] Peissker, E. 220 [105] Pelcovits, P.A. 416 [109] Penning, P. 218 [36] Pennycook, S.J. 451 [91] Pensisson, J. 450 [20] Peralta, P. 451 [64] Perez, R. 495 [57] Pérez-Magrané, R. 219 [65] Pernot, E. 639 [33]; 639 [37] Perrin, B. 414 [59]; 415 [66] Persson, B.N.J. 45 [34]; 222 [187] Persson, P.A. 44 [19] Petch, N.J. 496 [66] Peter, A. 639 [43] Pethica, J.B. 494 [1]; 494 [22]; 495 [29] Peticolas, L. 451 [52] Petri, A. 219 [71]

Author Index Petrov, I. 497 [94]; 497 [95]; 497 [96]; 497 [97]; 497 [98]; 497 [99] Petukhov, B.V. 450 [37] Pfrommer, B.G. 496 [61] Pharr, G.M. 45 [38]; 45 [52]; 218 [31]; 494 [2]; 495 [24]; 598 [7]; 600 [95] Piazza, S. 219 [70] Pielke, R.A. 221 [150] Pilling, M.J. 222 [198] Pilz, R.O. 495 [60] Pin Lu 78 [8] Piobert, A. 218 [26] Pippan, R. 46 [71]; 46 [72] Pirouz, P. 361 [13]; 450 [43]; 451 [59]; 451 [61] Plachke, D. 413 [19]; 416 [146] Planes, A. 219 [65] Plass, R. 452 [102] Plessing, J. 219 [55] Plimpton, S.J. 496 [62]; 599 [52] Polonsky, I.A. 45 [65]; 45 [66] Pomeau, Y. 218 [19] Pond, R.C. 261 [13]; 261 [15]; 261 [16]; 261 [24]; 261 [26]; 261 [28]; 261 [29]; 261 [31]; 261 [34]; 261 [35]; 261 [41]; 261 [43]; 261 [46]; 452 [105] Ponimarev, A. 219 [64] Poole, W.J. 46 [92] Pope, S.B. 222 [198] Popovici, G. 495 [34]; 495 [35] Portevin, A. 218 [13] Potzsch, A. 361 [11] Poulsen, H.F. 219 [83]; 599 [70]; 599 [71] Poulsena, H.F. 599 [69] Povirk, G.L. 44 [23] Prakash, V. 44 [4]; 44 [11] Preble, E.A. 600 [111] Prelas, M.A. 495 [34]; 495 [35] Procaccia, I. 219 [76]; 221 [146]; 221 [154]; 221 [158]; 221 [159]; 223 [223] Proult, A. 416 [124]; 416 [125] Putaux, J. 451 [83] Qin, Q.-H. 78 [7] Qin, Y. 416 [111] Qin, Y.L. 416 [113] Qu, S. 45 [52] Quaouire, L. 221 [134] Quinney, H. 217 [1] Quivy, A. 414 [53] Rabczuk, T. 46 [109] Rabier, J. 416 [124]; 416 [125]; 450 [12]; 451 [53] Rabitz, H. 222 [199]

653

Radovitzky, R. 45 [30] Radowicz, A. 78 [11]; 78 [12]; 78 [13]; 78 [14] Ràfols, I. 219 [65] Raghavan, R.S. 222 [172] Raj, S.V. 218 [31] Rajasekar, S. 221 [162] Rajesh, S. 222 [190]; 223 [213]; 223 [214]; 223 [226] Rajnak, S. 416 [141] Ramaswamy, S. 413 [6]; 413 [8] Ramesh, K.T. 361 [7] Ranganathan, S. 415 [74] Ranjith, K. 44 [8]; 44 [9] Rashkeev, S.N. 220 [128] Räuchle, W. 222 [178] Ravelli, R.B.G. 601 [129] Ray, I.L.F. 363 [67] Read, T.A. 260 [7] Read, W.T. 78 [22]; 79 [56]; 416 [117]; 599 [77] Rebonato, R. 600 [101] Redmann, J.K. 362 [60]; 363 [101] Redner, S. 221 [155] Reichhardt, C. 221 [169] Reisinger, A. 220 [92] Remmers, J.J.C. 46 [108] Renardy, M. 44 [5] Renyi, A. 221 [161] Repetto, E.A. 45 [30] Reppich, B. 220 [105] Reuter, M. 451 [54] Reynolds, G.A.M. 414 [60] Rhee, M. 45 [57] Rice, J.R. 44 [8]; 44 [9]; 44 [10]; 44 [20]; 46 [86] Richmond, O. 223 [212] Richter, P.H. 223 [223] Ricker, M. 414 [58] Riecker, R.E. 361 [15] Ringer, E.M. 45 [25] Rinzel, J. 222 [202]; 222 [204] Risken, H. 219 [88] Ritchie, R.O. 494 [15] Rizzi, E. 219 [44]; 222 [185]; 223 [236] Robbins, M.O. 44 [16]; 45 [26]; 45 [27]; 45 [33] Robertson, C.F. 45 [61]; 496 [63] Robertson, I.M. 496 [80] Robinson, W.H. 363 [74] Rockett, A. 496 [90] Roder, C. 600 [113] Roitburd, A.L. 261 [19] Rollett, J.S. 600 [98] Romanov, A.E. 78 [43]; 598 [21] Ronnpagel, K. 598 [28] Rooney, T.P. 361 [15] Rosakis, A.J. 45 [24]

654 Roschchupkin, D.V. 639 [32] Rosen, A. 223 [227] Rosenfeld, R. 413 [15]; 415 [102] Roshchupkin, A. 451 [58] Rosi, F.D. 362 [49] Roskowski, A.M. 600 [111]; 600 [112] Rösner, H. 218 [27] Ross, F. 451 [54]; 452 [96]; 452 [100] Ross, J. 223 [223] Ross, M. 361 [15] Roth, E.P. 639 [18] Roth, J. 415 [82]; 416 [136] Roucau, C. 414 [22] Roundy, D. 495 [30]; 495 [32] Rousell, M.R. 222 [197]; 222 [198] Rouviere, J. 450 [20] Rowcliffe, D.J. 495 [43] Ruan, J. 46 [104] Rubanov, S. 497 [101] Ruelle, D. 221 [138]; 221 [145] Ruff, A.W. 495 [53] Ruhle, M. 451 [74]; 451 [75]; 451 [76]; 451 [77] Ruina, A.L. 44 [3]; 44 [10] Rumi De 219 [72] Ryaboshapka, K.P. 598 [29]; 599 [59]; 599 [73] Rybin, V. 450 [10] Saada, G. 78 [20]; 78 [21]; 363 [64]; 599 [78] Saha, R. 496 [88] Sahoo, D. 218 [16]; 220 [104]; 220 [107] Sahoo, M. 362 [55] Saimoto, S. 362 [55]; 364 [110] Saito, T. 416 [126] Saka, H. 495 [45]; 495 [52] Sakai, A. 600 [110] Sakata, M. 415 [69]; 415 [71] Salem, A.A. 361 [10]; 364 [112] Salmeron, M. 46 [100] Salvarezza, R.C. 219 [60] Sander, L.M. 221 [156] Sandvik, B.P.J. 261 [37] Sankey, O. 450 [30] Sano, M. 221 [149] Sarkar, A. 221 [170] Sarrazit, F. 261 [28] Sathish, S. 414 [61] Sato, M. 451 [68] Sato, T.J. 415 [73] Sauer, T. 221 [143] Sauer, W. 639 [33]; 639 [37] Sauerland, N. 451 [56] Savart, F. 218 [32] Sawada, Y. 221 [149]

Author Index Scafetta, N. 221 [171] Scattergood, R.O. 78 [30] Schaaf, D. 416 [136] Schall, P. 415 [106]; 416 [151] Scharwaechter, P. 414 [37] Schiller, C. 220 [125] Schiller, P. 450 [8] Schmid, E. 362 [50] Schreiber, J. 600 [124] Schreiber, M. 221 [165] Schreiber, T. 218 [22] Schreuer, J. 414 [64] Schroeter, W. 450 [1] Schwab, A. 600 [124] Schwartz, D.S. 261 [42] Schwarz, K.W. 45 [56] Schwink, Ch. 218 [41]; 218 [42]; 222 [180]; 222 [181]; 223 [237]; 223 [238] Seatons, P.W. 222 [198] Seefeld, M. 598 [22] Seeger, A. 218 [5]; 363 [82]; 450 [8]; 638 [2]; 639 [24]; 639 [46] Segall, R.L. 363 [83] Seidel, H. 495 [36] Semadeni, F. 416 [150] Senkader, K. 451 [66] Sepilo, B. 414 [28] Serra, A. 261 [31] Sethna, J.P. 598 [46] Seung-Hoon Jhi 496 [89] Sevillano, G. 219 [81] Shaklee, J.B. 415 [67] Shan, Z. 497 [105]; 497 [107] Shan, Z.W. 497 [104] Shang, P. 261 [46] Shastri, S.D. 219 [83] Shaw, L.J. 415 [78] Shaw, R.S. 221 [144] Shechtman, D. 413 [4] Shenoy, G.K. 597 [2] Sheremetyev, I.A. 600 [105] Sherman, A. 222 [203] Shi, S. 639 [49] Shibata, K. 415 [73] Shibuya, T. 413 [10]; 416 [148] Shield, J.E. 416 [119]; 416 [120] Shilo, D. 639 [20]; 639 [33]; 639 [37]; 639 [38]; 639 [39]; 639 [40] Shin, H. 495 [53] Shindo, D. 450 [13]; 451 [70]; 451 [79]; 451 [82]; 452 [93] Shinoda, K. 416 [128]; 416 [134]; 416 [149] Shiraki, Y. 451 [51] Shockley, W. 78 [22]; 416 [117]; 599 [77]

Author Index Short, K. 451 [52] Shraiman, B.I. 221 [158] Shrotriya, P. 46 [91]; 46 [106]; 46 [107] Shtremel, M.A. 362 [26] Shvindlerman, L.S. 496 [72] Shyne, J.C. 362 [27] Sidorin, A. 219 [64] Siekluka, Z. 223 [234] Sigga, E.D. 221 [154] Sigle, W. 451 [76]; 451 [77]; 451 [78] Silcock, J. 450 [14] Silcox, J. 363 [80] Simpson, A.D. 495 [57] Simsic, M. 414 [30] Singh, A. 415 [74] Sitch, P. 450 [31] Skodje, K.T. 222 [197] Sleeswyk, A.W. 222 [173]; 362 [23]; 363 [95] Slobodnik, A.J., Jr. 638 [6] Smirnova, I.S. 78 [23] Smith, D. 496 [93] Smith, D.A. 261 [21] Smith, E. 261 [33] So, Y.G. 415 [94] Soboyejo, W.O. 46 [91] Socolar, J.E.S. 413 [2]; 413 [5] Soer, W.A. 496 [75]; 497 [105] Sohler, W. 639 [45] Soldatov, V.P. 223 [216] Solina, D.H. 219 [58] Song, S.G. 361 [4]; 363 [97] Sørensen, H.O. 219 [83] Southgate, P.D. 452 [106] Specht, E.S. 601 [131] Specht, P. 450 [32]; 452 [112] Specht, S. 451 [56] Speck, J.S. 600 [113] Spence, J.C.H. 450 [11]; 450 [18]; 450 [19]; 450 [30]; 451 [45]; 451 [71]; 451 [74]; 452 [93]; 452 [98]; 452 [99]; 452 [100]; 452 [103]; 452 [104] Spolenak, R. 600 [107]; 600 [108]; 600 [117]; 600 [118]; 600 [119]; 600 [120] Spoor, P.S. 414 [62] Sprengel, W. 417 [156] Springer, F. 218 [42]; 222 [180]; 223 [238] Sprušil, B. 223 [217] Sreenivasan, K.R. 221 [160] Stach, E. 451 [54] Stach, E.A. 494 [15]; 495 [48]; 496 [74]; 496 [75]; 496 [76]; 496 [85]; 497 [102]; 497 [104]; 497 [106] Stagni, L. 78 [42] Stanley, H.E. 221 [167]

655

Steeds, J.W. 79 [68]; 363 [72]; 363 [73] Steinhardt, P.J. 413 [1]; 413 [2]; 413 [3]; 413 [5]; 413 [6]; 415 [77]; 415 [80]; 415 [81] Stejskalolvá, V. 223 [217] Sterzel, R. 415 [68] Steurer, W. 414 [25]; 414 [41]; 414 [64] Stoch, E.A. 497 [95] Stölken, J.S. 45 [37] Stout, R.B. 220 [99] Stratonovich, R.L. 219 [89] Stroh, A.N. 78 [25]; 78 [26]; 363 [92] Suck, J.-B. 415 [73] Sugawara, Y. 639 [30] Sumino, K. 450 [26]; 451 [62]; 451 [68]; 451 [84]; 495 [41] Sun, Y.F. 416 [112] Sunakawa, H. 600 [110] Sundgren, J.E. 496 [90] Sung, T. 495 [34]; 495 [35] Suprijadi 495 [52] Suresh, S. 494 [3]; 598 [24] Suzuki, K. 414 [33]; 414 [34]; 414 [35]; 414 [36]; 415 [89]; 451 [51]; 452 [97] Suzuki, M. 220 [109] Suzuki, T. 416 [131]; 416 [139]; 416 [144]; 416 [145] Swadener, F.G. 45 [38] Swain, M.V. 495 [25]; 495 [28]; 495 [44]; 495 [50]; 495 [51] Swanger, L.A. 79 [72] Syed Asif, S.A. 497 [104]; 497 [105]; 497 [106]; 497 [107] Synge, J.L. 79 [61] Szczerba, M. 362 [28] Szczerba, M.S. 362 [29]; 362 [46]; 362 [56]; 362 [57]; 362 [59]; 363 [96] Székely, F. 219 [82] Szmolyan, P. 223 [231] Tabor, D. 44 [1]; 494 [7] Tachi, M. 495 [52] Takakura, H. 415 [87] Takamura, J. 362 [20]; 362 [54] Takasugi, T. 451 [79] Takata, M. 415 [69]; 415 [71] Takens, F. 221 [141] Takeuchi, S. 413 [10]; 413 [11]; 414 [20]; 414 [32]; 414 [33]; 414 [34]; 414 [35]; 414 [36]; 415 [89]; 416 [126]; 416 [128]; 416 [131]; 416 [133]; 416 [134]; 416 [137]; 416 [138]; 416 [139]; 416 [143]; 416 [148]; 416 [149]; 417 [152]; 417 [155]; 451 [46]; 452 [97] Takigahira, M. 639 [30]

656 Tamura, I. 496 [77] Tamura, N. 415 [88]; 416 [115]; 599 [84]; 599 [86]; 600 [107]; 600 [108]; 600 [117]; 600 [118]; 600 [119]; 600 [120] Tamura, R. 414 [20]; 414 [32]; 416 [133]; 416 [137]; 416 [148]; 417 [155] Tamura, S. 639 [30] Tan, M.J. 78 [8] Tanaka, K. 414 [65] Tanaka, M. 415 [98] Tanaka, Y. 639 [30] Tang, C. 219 [73] Tang, L.H. 415 [79] Tangyunyong, P. 494 [19] Tanner, B.K. 639 [34]; 639 [36] Tartaglino, U. 45 [34] Taylor, G.I. 217 [1]; 223 [243] Tazzari, S. 597 [1] Teichler, H. 450 [2]; 450 [7] Teodosiu, C. 78 [28]; 598 [44] Terauchi, M. 415 [98] Texier, M. 416 [124]; 416 [125] Theiler, J. 221 [147]; 221 [151] Thibault-Desseaux, J. 450 [17]; 451 [83] Thomas, G. 363 [104]; 363 [105]; 496 [64] Thomas, R.C. 494 [19] Thompson, L.J. 496 [73] Thompson, N. 362 [22]; 362 [35]; 363 [91] Thomson, R. 223 [225]; 416 [140]; 450 [3]; 599 [72] Thornton, P.R. 362 [53] Thurston, R.N. 362 [43] Thust, A. 451 [72] Ting, T.C.T. 77 [1]; 77 [2] Tischler, J. 599 [85] Tischler, J.Z. 599 [61]; 599 [84]; 599 [86]; 600 [95]; 600 [104]; 600 [106] Titchner, A.L. 217 [2] Tokarski, T. 362 [59] Tomlin, A.S. 222 [198]; 222 [199] Toner, J. 413 [8] Torquato, S. 600 [96] Tortorelli, D.A. 223 [242] Tosatti, E. 45 [34]; 222 [187] Toth, J. 222 [199] Trebin, H.-R. 414 [44]; 414 [58]; 415 [82]; 415 [83]; 416 [135]; 416 [136] Tromp, R. 451 [54] Tsai, A.-P. 415 [73]; 415 [87]; 417 [155] Tsuzaki, K. 496 [77]; 496 [81]; 496 [82] Tsuzuki, T. 223 [221] Tu, K.N. 600 [115] Tucoulou, R. 639 [32] Tuppen, C. 451 [50]

Author Index Turanyi, T. 222 [198] Turbal, A.V. 600 [105] Turner, F.J. 361 [14] Twesten, R. 452 [96] Twozydlo, W.W. 45 [29] Tymiak, N.I. 46 [93]; 494 [23] Uchic, M.D. 45 [41]; 494 [14] Umerski, A. 450 [38]; 451 [67] Umezaki, M. 415 [83] Ungar, T. 218 [23]; 598 [28]; 598 [33]; 600 [127] Unterwald, F. 451 [52] Urbakh, M. 45 [26] Urban, K. 413 [12]; 413 [14]; 413 [15]; 413 [16]; 413 [17]; 415 [96]; 415 [97]; 415 [102]; 415 [106]; 416 [115]; 416 [122]; 416 [151]; 417 [154] Usami, N. 451 [51] Valek, B.C. 600 [107]; 600 [108]; 600 [117]; 600 [118]; 600 [119]; 600 [120] Valsakumar, M.C. 218 [18]; 220 [108]; 221 [134]; 221 [162]; 222 [191]; 223 [219] van den Beukel, A. 218 [38]; 222 [174] Van der Giessen, E. 45 [53]; 45 [54]; 45 [58]; 45 [62]; 45 [63]; 45 [70]; 46 [73]; 46 [78]; 46 [79]; 46 [80]; 46 [83]; 46 [84]; 46 [87]; 46 [88]; 46 [102]; 46 [103]; 46 [110] van der Pol, B. 222 [201] van Heerden, D. 495 [29] van Kampen, N.G. 219 [87] van Saarloos, W. 223 [232] Van Swol, F. 600 [96] Van Swygenhoven, H. 598 [24] Vanderbilt, D. 450 [27] Vanderschaeve, G. 414 [21]; 450 [42] Vara, J.M. 219 [60] Varin, R.A. 496 [79] Vazquez, L. 219 [60] Venables, J.A. 363 [94]; 363 [108]; 363 [109] Venkadesan, S. 221 [134]; 221 [162] Venkataraman, G. 220 [93]; 220 [95]; 223 [229] Verbraak, G.A. 362 [23] Vergnol, J. 219 [43] Vespignani, A. 219 [66]; 219 [71]; 220 [119] Veyssiere, P. 363 [93] Viccaro, P.J. 597 [2] Victoria, M. 363 [79] Vidal, C. 218 [19] Vitek, J.M. 599 [87] Vitek, V. 451 [75] Vives, E. 219 [65] Vlachavas, D.S. 261 [26]

Author Index Vladimirov, V.I. 598 [21] Vlassak, J.J. 494 [11] Vöhringer, O. 222 [178] Volinsky, A.A. 494 [23] Volkert, C. 600 [121] Volokitin, A.I. 45 [34] Vonlanthen, P. 414 [29] Vostrý, P. 223 [217] Voyiadjis, G.Z. 46 [95] Vulpiani, A. 222 [195] Walgraef, D. 220 [125] Walker, F. 599 [74]; 599 [90] Walker, L.R. 78 [24] Wall, M. 495 [33] Wang, J. 46 [107] Wang, J.N. 416 [132] Wang, P.-C. 600 [116] Wang, R. 414 [43]; 414 [45]; 414 [46]; 414 [47]; 414 [55]; 415 [99]; 415 [100]; 415 [101]; 415 [103]; 415 [104]; 415 [105]; 415 [107]; 416 [108]; 416 [110]; 416 [111]; 416 [113]; 416 [114]; 416 [115]; 416 [116] Wang, Z. 45 [59]; 415 [100] Waren, O.L. 497 [107] Warren, B.E. 598 [6] Warren, O.L. 497 [104]; 497 [105]; 497 [106] Wayman, C.M. 260 [3]; 260 [6]; 261 [36]; 261 [37] Weber, E. 451 [60] Weber, E.R. 450 [32] Webster, G.A. 220 [103] Wechsler, M.S. 260 [7] Wei, Y. 45 [51] Weihs, T.P. 494 [12]; 495 [29] Weiland, H. 599 [84] Weiss, J. 219 [66]; 219 [74]; 220 [119] Welzel, G. 219 [55] Wenk, H.R. 601 [130] Wentorf, R.H. 495 [58] Weppelmann, E.R. 495 [25]; 495 [44] Werner, M. 451 [60] Wert, J.A. 600 [125] Wesch, W. 497 [100] Wessel, K. 450 [41] Wessels, E.J.H. 363 [89]; 363 [90] Weygand, D. 45 [58]; 46 [83] Whatmore, R.W. 639 [34]; 639 [36] Whelan, M.J. 363 [65]; 363 [67]; 415 [95]; 495 [56] Whitehead, J.A. 223 [220] Whitney, H. 221 [140] Widjaja, A. 46 [73]; 46 [90]; 46 [110]

Widom, M. 414 [49] Wienecke, H.A. 77 [4] Wiesenfeld, K. 219 [73] Wilkens, M. 598 [28]; 598 [33]; 599 [57]; 599 [58]; 600 [127] Williams, D.B. 363 [66] Williams, D.R. 223 [211] Williams, E. 600 [103] Williams, J.S. 495 [28]; 495 [50]; 495 [51] Williams, W.S. 452 [107]; 496 [91] Williamson, J.B.P. 44 [13] Willis, J.R. 79 [67]; 79 [68] Wilshaw, P.R. 451 [66] Wilson, K.G. 221 [153] Wilson, R.G. 495 [34]; 495 [35] Winning, M. 496 [72] Winther, G. 599 [70]; 599 [83] Wintner, E. 363 [78] Wirth, B.D. 363 [99]; 363 [100] Witt, C. 600 [121] Witt, J. 221 [169] Witten, T.A., Jr. 221 [156] Woirgaard, J. 452 [108] Wolfe, J.P. 638 [5]; 639 [19]; 639 [31] Wollgarten, M. 413 [14]; 413 [15]; 413 [16]; 415 [96]; 415 [97]; 415 [102] Wong-Leung, J. 495 [50]; 495 [51] Woolfson, M.M. 598 [10] Worthington, P.J. 221 [132] Wright, O.B. 639 [30] Wu, D. 414 [64] Wu, K.-C. 78 [37] Wu, X. 495 [27] Wyrobek, J.T. 494 [4]; 494 [18]; 495 [47] Wyrobek, T.J. 497 [104] Xiao, S.Q. 261 [40] Xu, G. 46 [105] Xu, X. 450 [32] Xu, X.-P. 46 [96] Xu, Y.B. 495 [27] Yamaguchi, M. 451 [81] Yamamoto, A. 414 [40] Yamashita, Y. 450 [5]; 451 [51]; 451 [65] Yan, Y. 415 [103]; 415 [105]; 415 [107] Yan, Y.F. 415 [104]; 416 [108] Yang, B. 222 [198] Yang, D. 46 [91] Yang, W. 414 [43]; 414 [46]; 416 [110]; 416 [114]; 416 [115]; 598 [7]; 599 [61]; 599 [84]; 599 [85]; 599 [86]; 600 [95] Yang, W.G. 416 [113]; 600 [106] Yefimov, S. 46 [78]; 46 [79]

657

658 Yeom, H.Y. 598 [9] Yew, C.H. 45 [29] Yin, J. 414 [55] Yip, S. 450 [24]; 450 [29] Yonenaga, I. 450 [42]; 451 [62]; 451 [70]; 451 [84] Yoo, M.H. 362 [25]; 451 [82] Yoon, H.G. 598 [9] Yorke, J.A. 221 [143] Yoshida, M. 451 [79] Yoshida, Y. 416 [149] Yoshinaga, H. 416 [139] Yu, H.H. 46 [106] Yu-Lung Chiu 363 [93] Yuan, F.G. 78 [10] Zagoruyko, L.N. 223 [216] Zaiser, M. 217 [4]; 218 [5]; 218 [30]; 219 [61]; 219 [79]; 219 [80]; 220 [117]; 223 [212]; 598 [27] Zaoui, A. 261 [39] Zapperi, S. 219 [66]; 220 [119]

Author Index Zarembowitch, A. 414 [63] Zasadzinski, J. 223 [234] Zbib, H.M. 45 [57]; 219 [46] Zeng, K.-Y. 494 [3] Zeng, X. 221 [150] Zhan, Z. 415 [107] Zhang, C. 46 [105] Zhang, H. 416 [108] Zhang, M. 451 [61] Zhang, S. 223 [235] Zhang, Z. 415 [96]; 415 [97]; 416 [108]; 451 [76]; 451 [77] Zheng, G. 44 [20] Zhu, W.-J. 414 [50] Ziegenbein, A. 219 [44] Zielinski, W. 496 [79] Ziman, J.M. 639 [22]; 639 [23] Zimmerman, J.A. 496 [68] Zolotoyabko, E. 639 [20]; 639 [33]; 639 [37]; 639 [38]; 639 [39]; 639 [40] Zontone, F. 601 [130] Zorman, C. 452 [103]

Subject Index band 188 – type 161 – velocity 91, 195, 203, 206, 207 band-gap states 437 basic orthogonal system 272 Basinski hardening 359 bent slow manifold 186 bicrystals 436 bifurcation 85, 108, 128, 168, 175, 179, 180 – diagram 179 – point 169 boundary value problem 16, 17 Bragg reflection 439, 503 Burgers vector 514, 606, 628, 633, 634, 638 – of the twinning dislocation 278

η1 direction 268, 269, 273 η2 direction 268, 270 Ŵ-surfaces 402, 403 ab-initio simulations 428 abrupt kink regime 404 absorption correction 546 achromatic focusing 594 acoustic – activity 114 – attenuation 605, 611–613, 636 – emission (AE) 98, 99, 114, 157, 610, 611 activation energy 424, 449 activation enthalpy 430 additive noise 131 adhesion 4 adiabatic conditions 20 AE activity 98 AE signals 98, 100, 115 AE sources 100 AFM 94, 96, 97 AK model 136, 161, 167, 169, 170, 184, 188, 196, 198, 207, 210, 214, 215, 217 Al thin films 470 Amontons–Coulomb description 3 Amontons–Coulomb framework 5 Amontons–Coulomb friction 6 Ananthakrishna model 136, 166, 169 Anderson transition 145, 150 annihilation 432 anomalous fluctuations 116, 117 anti-phase defects 421, 426, 428 antiphase boundary 435 asperity 4, 8, 10, 20, 34, 41, 42 asperity size 40 athermal fluctuations 83, 107, 109 athermal process 107 atomic force microscopy (AFM) 94 atomic surface 369 atomistic calculations 428 autocorrelation 102, 118, 125, 138 – function 138 – time 139 avalanches 100

cathodo-plastic effect 433 cell – formation 105 – patterns 127 – splitting 105 – structure 84, 87, 88, 102, 103, 127 Celli–Thomson theory 422 cells 89, 103, 127 cellular patterns 102 chaos 137, 139, 144, 150, 155, 156, 212 chaotic 180 – B-type 152 – behavior 137 – domain 197 – dynamics 144 – regime 143, 188, 190–192, 195 – signal 137 – state 160, 161, 216 – stress drops 212, 215 – type 146 characterization of – cellular patterns 100 – chaotic behavior 137 – collective behaviour 94 – collective effects 93 – PSBs 100 – stress–strain series 143 civilian transformations 230 climb 436, 449 659

660

Subject Index

– of interfacial line defects 237 – process 367, 399, 405 coefficient of friction 5, 41, 43 coherency strains 243 coherent nanodiffraction 447 cohesive relation 30, 31 cohesive strength 31, 34, 36 collective – behavior 84, 136 – – of dislocations 85–87, 94, 109, 136, 160, 215 – degrees of freedom 170, 215 – dislocation 119 – – dynamics 116 – effects 83, 92–94, 197, 215 – – of dislocations 86, 119 – modes 83, 85, 120, 136, 160, 215 – motion of dislocations 98 colored noise 108 combined electro-elastic line defect 51, 55, 56 compatibility 14 complex order parameter 178 configuration of dislocations 191 conjugate – dislocations 301, 310, 313, 316, 320, 328 – glide plane 278, 281, 282 – plane 266, 279, 281 – slip system 289 – twinning 267, 279, 281, 282 – – direction 268 – – system 279 conservation of energy 19 constitutive – equations 12 – length scale 4 – relations 14, 18 contact 3 – area 4, 5, 7, 8, 10 – length 22, 23, 25, 31 – pressure 23, 25, 26, 28, 35 – size 33, 34, 38, 42 – stress 31 continuum – mechanics 11, 12 – plasticity 35 – slip theory 13 contrast factor 543 conventional continuum plasticity 8 convergence studies 22 coordinate systems 272, 273, 302 coordinate transformation 272 copper single crystal 279, 310, 311 core – defect 434 – reconstruction 449

– structure 439 – – of an a2 [110]M Lomer dislocation 323 – – of dislocation 323 correlated – IDBs 529 – kink motion 428 – noise 108 correlation – dimension 85, 100, 139, 141, 144, 147–149, 156, 180, 212 – function 114, 126, 138 – integral 139 – length 118, 123, 128, 152, 156 – time 128, 195–197 correspondence matrix 271, 274, 275, 295 – analysis 309 – method 314, 329, 331 Cottrell density 113 Cottrell’s type dislocations 170 critical – glide plane 278, 281–284, 295, 296 – plane 296 – slip system 289 – stress for twinning nucleation 278 cross-glide 111 cross glide plane 278, 281–283, 288, 296, 326 – reflection 287 cross plane 360 cross-slip 89, 93, 170, 171 cross-correlation 115 crossover 144–146, 155, 156, 160, 161, 170, 188, 216 crystal approximants to quasicrystals 387 crystal axis system 272, 273 crystallography of deformation twinning 267 Cu-4%Al 279, 289, 292, 294, 295, 327, 328, 338, 357, 358 Cu-8%Al 279, 280, 286, 287, 290, 291, 293, 295, 297–299, 301, 310, 312, 313, 316, 328, 344, 345, 347, 355–357 cube dislocations 299, 301, 343, 349–351, 354, 355 decagonal quasicrystals 374, 379 decorated Cottrell dislocations 170 deep core states 437 deep states 426 defect – clusters 286, 293, 358, 360 – energies 429 – identification 284 – of the 1st kind 506 – of the 2nd type 506

Subject Index – structures 343 deformation – gradient matrix 271, 272, 275 – gradient tensor 275 – matrix 272 – twinning 265, 269 – twins 267, 268, 347 density of kinks 445 detector 594 deterministic dynamics 136, 161 deviatoric strain in stripped GaN layers 560 deviatoric strain tensor 558 dichromatic pattern 234 differential aperture microscopy 535 diffracting conditions 281, 284, 299, 345 diffracting planes 281, 283, 284 dimensional reduction 165, 184 disconnection defect 449 disconnections 234 discrete dislocation plasticity 9–11, 15, 18, 20–22, 26, 42–44 dislocation – boundaries 526 – configurations 185 – core 302, 322, 428, 439 – debris 267, 310, 357, 358, 360 – density 115, 124, 125 – dipoles 327 – dynamics 105, 109, 110, 116, 433, 605, 606, 609, 615, 637 – formation and multiplication 576 – grouping within boundaries 544 – in quasicrystals 367, 390, 392 – loop 427 – mobility 428 – obstacle 21 – results 354 – sources 21 – substructure 267, 277–281, 285–287, 291, 295–301, 309, 310, 328, 347, 348, 355, 358, 360 – velocity 424, 431, 610–612, 628, 631, 637, 638 – vibrations 605, 613–616, 618, 619, 625, 626, 628, 635–637 displacement jumps 19 dissociation 425 – of kinks 428 distorted Thompson tetrahedron 271, 277, 296, 314, 326 distribution function approach 120 distribution theoretic approach 86, 119, 216 dopants 433 double cross-slip 93, 200 double kink mechanism 433

661

double-period (DP) reconstructions 447 DSA, see dynamic strain aging – based model 208 – framework 159, 198 ductile-brittle transition 433 dynamic displacement map 629, 637 dynamic strain aging (DSA) 90, 91, 113, 136, 157–159, 184, 197, 198, 202, 214, 217 dynamical – approach 136, 157, 160, 196, 197, 216 – regime 146 – systems 137, 215 dynamics 144 early stages of plastic deformation in interconnects 569 edge dislocation dipoles 358 edge GN dislocations 523 elastic equations of balance 381 elastic strain 556 electric fields of dislocations 77 electron microdiffraction 449 electrostatic analogue of a dislocation 53, 54 electrostatic line defect 76 embedding dimension 138 energy-loss absorption spectra 439 energy-loss spectroscopy 436, 449 entropy 424 equation of motion 617, 619, 628, 634, 637 Eshelby’s method 393, 412 Ewald sphere diagram 503 experimental attractor 181 extended “elastic moduli” of a piezoelectric 55 extended Indenbom–Orlov formula 62 extension of – the Barnett–Swanger formula 66 – the Burgers solution 60 – the Indenbom–Alshits formulae 67 – the Mura representation 61 – the Peach–Koehler formula 64 extrinsic stacking faults 343, 344, 355 faulted dipole 267, 286, 291–293, 316, 326, 327, 329, 331, 348, 359 faulted dislocation dipole 434 faulted loop 295, 329, 331, 332 Fibonacci lattice 369–371 Fibonacci numbers 396 field-emission electron gun 446 fine debris 357 finite deformation 18, 19 finite rotations 19 Fisher–Kolmogorov equation 194

662

Subject Index

fluctuation induced patterns 127 fluctuation-dissipation 84 – method 107 – relations of plastic flow 118 fluctuations 85, 94, 107–109, 116–119, 128, 129 focused ion beam (FIB) 462 Fokker–Planck approach 86, 216 Fokker–Planck (FP) equation 130, 134 Fokker–Planck like equation 110 forbidden reflection 430, 439, 441 forest densities 158 formation energy 424, 429 forward Hopf bifurcation 179, 193 fractal 86, 96 – characteristics 103 – concept 215 – dimension 94, 97, 99, 100, 103, 126, 132, 142, 162 – nature 99 – object 103, 139 – structure 103, 132, 133 Frank – dipole 291, 326, 329, 330, 334–336, 338, 361 – dislocations 329, 334, 337, 343, 346–351, 353–356, 360 – loop 329 – node equation 302 free energy 109, 169, 179 – function 196 – functional 108, 109 – like function 84, 178, 180 – of kink migration 424 friction coefficient 41 frictional sliding 3, 21 frontal migration 230 full strain tensor 558 g.b values 285–290 GaN 436 generalized – dimensions 142 – Hooke’s law 381 – plastic distortion 56 – Renyi dimensions 142 geometrically necessary – boundaries (GNBs) 508, 511 – dislocations (GNDs) 14, 26, 28, 29, 508, 509 – (unpaired) dislocations 519 Ginzburg–Landau – approach 84, 108 – equation 175, 216 – theory 196 – type equation 128

– type of equation 167 glide – of interfacial line defects 237 – plane 425, 607, 613, 634 – process 367, 399 – set 449 glissile interface 229 gradient cell–wall dislocation structure grain boundaries 477

551

habit plane 246, 268 Hausdorff dimensions 142 heat – capacity 20 – conductivity 607, 610–614 – – matrix 20 Heidenreich–Shockley – dislocation 270, 291, 302 – partial dislocation 269 – partials 351, 355 heterogeneous dislocation network near an indentation 588 Hirth–Lothe theory 422, 423 homogeneous lattice deformation 255 Hopf bifurcation 167–169, 174, 176, 180, 200 hopping – bands 155, 189 – type 155, 197 – type bands 190, 195 Hurst exponent 96, 97, 157 hydrodynamic instability 387 hydrodynamic stability 385 hydrogen 428 hydrogen plasma 434 icosahedral quasicrystal 372, 376 IDB-1s 527 IDB-2s 527 ideal strength 458 image forces 431, 434 images of dislocation cores 445 immobile – components 113, 114 – density 171, 193, 196 – dislocation densities 109 – dislocations 170, 171 impurity atoms 433 in situ nanoindentation 455 incidental dislocation boundaries (IDBs) 508, 511 incoherent twinning front 307, 309, 322, 323, 330, 361 indentation force 23

Subject Index indexing 595 initiation of sliding 8 instrumentation 590 interfacial line-defects 232 intermetallic alloys 425 intermetallic structures 435 intermittent – AE activity 99 – collective motion 100 – loading 433 internal friction 433, 605, 612–614, 618, 634 internal stress fluctuations 156 intrinsic stacking fault 289, 292, 326, 329, 331, 337–339, 355, 356 invariant plane 268, 277 invisibility condition 392 invisibility criterion 284 Ito calculus 130 jerky 94 – behavior 90 – motion 114, 117, 157 – nature 117 – phase 152 – regime 154 jog 421, 436 K1 plane 268, 269, 273, 274, 296 K2 plane 268, 269, 278, 331, 336, 337, 339, 340, 361 Kaplan–Yorke conjecture 148 Kaplan–Yorke dimension 213 kink – collisions 433 – contrast 439 – density 444, 449 – diffusion coefficient 424 – dynamics 423 – energies 428 – formation 430 – – energy 431, 432 – mean free path 424, 433 – migration energy 432 – motion 430 – structure 449 – velocity 424 kink-kink interaction 433 kink-pair – embryos 423 – formation 423 – separation 429, 431 kinks 421, 608, 609, 619, 632 Kirkpatrick–Baez mirrors 590

663

Langevin 85 – approach 85, 86, 107, 108, 114–116, 216, 217 – description 216 – dynamics 215, 216 – equation 84, 107, 108, 115, 116, 129 – type 127 largest Lyapunov exponent 141, 149 lattice – correspondence 270 – image 443 – rotation gradients 583 – rotations 18 – – in stripped GaN layers 560 lattice-invariant deformation 236 Laue diffraction 503 Laue microdiffraction 534 left kinks 428 length scale 12, 512 length-dependent regime 431 limit cycle 167, 171, 174, 175, 200, 201 – solutions 169, 171, 174, 201 LiNbO3 605, 606, 614, 622–627, 630, 631, 633, 634, 636, 637 local isomorphism class (LI class) 369 locked state 384 Lomer – dislocations 278, 288, 291, 309, 317, 320–324 – locks along [011]M 317 ¯ M 322 – locks along [110] Lomer–Cottrell dislocations 267, 278, 289, 290, 295, 360 Lomer–Cottrell network 286, 289, 290, 292, 293, 311, 316 Lüders – band 87, 89 – front 87 – phenomenon 89 Lyapunov – dimension 148 – exponents 85, 140, 141, 144, 183, 213 – spectrum 140, 147–149, 213 manifold 184 martensitic steel 482 martensitic transformations 227 material length scales 28 matrix 129 – structure 88, 102, 127, 134, 135, 216 mechanical twinning 265, 267, 302 mechanical twins 266 metal-ceramics 434

664

Subject Index

microbeam monochromator 593 microdiffraction patterns 448 military transformations 229 Mises solid 13 misfit 244 – dislocations 434 – in stripped GaN layers 560 – relief 247 misorientation vector 538 mobile – components 113 – density 158, 174, 193 – dislocation 170, 171, 194 – – density 109, 113, 169, 183 model attractor 213 modified Hooke’s law 55 Monte Carlo method 115 multifractal 86, 142 – analysis 141, 144–146, 150 – character 146 – formalism 142 – measure 142 – range 151 – spectrum 151 multiple slip 524 multiple time scale 160, 162, 165 – dynamical systems 161 multiple-scattering 447 multiplicative noise 129, 131, 132 multiplicative stochastic differential equation 129 multiscale dislocation structure 507 multiscale model 208 nanodiffraction 446 nanoindentation 455 negative strain rate sensitivity (SRS) 91, 159, 160, 169, 170, 174, 183, 184, 186, 196, 197, 210 Ni grains 583 Nix–Gao relation 29 node conversion mechanism 349, 351 noise 85, 107, 108, 115, 128, 140 – induced drift 130 – induced transition 129, 133, 134 nominal contact length 22, 23 non-local plastic constitutive relations 9 non-planar dissociation 435 noncorrelated IDBs 528 nonlinear dynamical approach 86 nonlinear dynamics 83, 136, 137, 160 nonlinear time series analysis 137 nucleation of double kinks 432

nullcline method 165, 169, 171 nullclines 173, 174, 200, 201, 203, 204 obstacle pining 430 operating reflections 281, 285, 287, 288, 301, 312–314, 345, 348, 357 order parameter 84, 107, 108, 129, 130, 134, 136, 152, 175, 179, 196, 216 order-parameter-like variable 175 orientation relationship 248, 272, 280–282, 295 Orowan loop 341 overshoot 278, 279 oxygen atoms 434 oxygen transport 434 oxygen-deficient core 434 parent axis system 272 parent-martensite interface 246 partial dislocations 265, 292, 307, 316, 322, 333, 338–340, 354, 357 partial kinks 428 partitioning of energy 38 Peach–Koehler force 17, 21 Peierls – barrier 609, 611, 616, 618, 632 – force 607, 609, 618, 632, 634 – potential 401, 402, 430, 607, 609, 611, 615, 618, 619, 625, 632–634 – stress 37, 421 – transition 425 – valley 423, 440, 444 penetration length of X-rays 592 Penrose lattice 372, 391, 399, 400 perfectly ordered model 383 periodic continuation 428 persistent slip bands (PSB) 87, 88, 102, 127, 128, 134 phason – degrees of freedom 367 – distortion 371, 376 – elastic constant 378, 380, 385 – elastic field 367 – elasticity 367, 383 – flip 371, 372, 400 – strain 371, 391, 392 – stress 381 phenomenological continuum plasticity 18 phonon – distortion 376, 392 – elastic constant 380, 382, 383 – elastic field 367 – elasticity 367, 381 – strain 392

Subject Index – stress 381 – wind 610, 611 phonon and phason degrees of freedom 368, 370, 376 phonon–phason coupling 367, 387 – constant 378, 380, 381 phonon-dislocation interaction 605, 631 photoplastic effect 426 pinned segments 432 pinned state 185, 187, 191, 197 pinning 183 – centers 432, 433 plane of shear 268, 270 plasma-enhanced kink motion 433 plastic and elastic strain 559 plastic deformation 8 – of single crystals 278 plastic dissipation 19, 20, 38, 39 plastic strain gradients 14, 26 plastic zone 25, 26 plasticity theory 14 PLC – bands 90 – dynamics 152, 154, 157, 208, 210 – effect, see Portevin–Le Chatelier effect – instability 91, 138, 147, 215 – models 165 – state 180 – time series 157 polar semiconductors 425 polarization stress 17 pole mechanism 304, 306, 307, 321, 322, 329, 331, 332, 336, 346, 349, 351, 355, 360, 361 polychromatic (white-beam) microdiffraction 534 pop-ins 456 pop-out 456 Portevin–Le Chatelier effect 84, 86, 87, 89–91, 110, 113, 136, 140, 146, 152, 154–157, 159–161, 169, 171, 183, 184, 188, 196, 198, 208, 215–217 positive Lyapunov exponent 149, 181 potential-like function 175 power law 98, 99, 132, 144, 150, 152, 155, 182, 183, 191, 195–197, 210 – distribution 99, 103, 133, 145, 149, 152, 182, 183, 210 – regimes 180 – state 144, 149, 160, 195, 216 – statistics 149, 152, 155, 156 – type 146 – – of distribution 132 power-law regime 188 pre-exponential 431

primary dislocation 287, 354 primary glide plane 278, 281–283 primary plane 360 primary slip system 278 propagating – band type 202 – bands 183, 194, 195 – type 191, 197 – type A bands 90, 161 propagative type A bands 210 propagative type of bands 208 PSB cells 216 quasicrystals 367 quasiperiodicity 368 radial direction 515 radiation damage 449 radiation-enhanced dislocation glide 431 random GNDs 544 random tiling 383 randomly distributed dislocations 509 rate and state framework 7 rate and state friction law 6 reciprocal twinning mode 269 reciprocal twinning plane 268 recombination kinetics 421 reconstructed attractors 213 reconstructed strange attractor 213 reductive perturbative approach 174, 196 reductive perturbative method 175 reindexation matrix 274, 275, 281, 295 relaxation oscillation 167 representative volume element 539 resolved shear stress 278, 279 reverse Hopf bifurcation 180, 193, 197 right kinks 428 rotation matrix 273–275 rough surfaces 4 saddle-point structures 428 scalar time series 138 scaling regime 192 Schmid factor 613 Schmid’s law 421 screw GN dislocations 524 second invariant plane 268 second undistorted plane 268, 270 secondary stacking faults 343, 356, 357 self free energy of dislocations 393 self-affine fractal surface 8 self-affine geometry 96 self-affinity 86, 215

665

666

Subject Index

self-energy 434 – of 4D dislocation 63, 70 self-organization 508 self-organized criticality 149, 161 self-similar geometry 96, 103 self-similar pore fractals 132 sensitivity to initial conditions 137, 140, 161 separation of time scales 165, 215 sessile cores 434 sessile intersection of interfacial line defects 241 shear direction 268 shear plane 268 Shockley dislocation 292, 306, 307, 340, 342, 349–351, 354, 355, 357 Shockley partials 315 shuffle core 449 shuffle planes 425 silicon 464 similarity transformation 274 singular value decomposition 139, 213 size dependence 8 size dependent 8 size scale dependence 9 size-dependent plasticity 13 slip systems 276 slow manifold 164, 166–170, 184–188, 191–194, 196, 214, 216 – analysis 165 – parameter 194 slow order parameter 129 slow–fast dynamical system 162, 163, 165, 183, 184 slow–fast dynamics 160, 169 smooth kink regime 404 SOC 149, 155, 156 – dynamics 157 software 595 soliton 427 source limited plasticity 15, 43 spatial configurations 191 spatial coupling 92, 108, 199, 204, 214 spatio-temporal chaos 85 spinel 433 SRS, see strain rate sensitivity stacking fault tetrahedra 267, 286, 293, 339–342, 358, 360, 361 stacking faults 265, 278, 290, 294, 297, 299, 306, 307, 312, 321, 324, 333, 340, 341, 344–346, 348, 354, 356, 357, 359, 361, 439 stair-rod dislocations 278, 291, 292, 326, 331, 339, 340 stair-rods 434 starting-stress anomaly 428 state of locked phason 384

state of unlocked phason 384 static Debye–Waller factor 506 statistical – approach 86, 105, 116 – description 105 – effects 20 – framework 120 – models 109 statistically stored – edge dislocations 517 – (paired) dislocations 516 – screw dislocations 517 STEM 437 STEM ELS 439 stereographic projection 281, 282 stick-slip behavior 90, 169 stick-slip systems 160, 163 stochastic – approach 83, 85, 86, 105, 107, 116, 216 – basis 129 – description 110 – differential equation 130 – framework 109 – method 83, 107 – processes 137 strain gradient 14 – plasticity theory 29 strain rate sensitivity (SRS) 116, 118, 129, 158, 184, 196, 198 strain-gradient plasticity 524 strange attractor 85, 138, 139, 141, 142, 181 Stratonovich calculus 130, 135 strength 359 – of twinned crystal 359 stress fluctuations 128 stress–time series 86, 180 stroboscopic 605, 606, 622 – diffraction images 624 – mode of measurement 629 – principle 620 – synchrotron X-ray diffraction 622 – X-ray imaging 620, 621, 626, 632, 637 – X-ray diffraction 620–623, 625–627, 630, 631, 635, 636, 638 – X-ray topography 605, 620, 621 Stroh-like formalism for piezoelectrics 71 strontium titanate 439 Stuart–Landau equation 175, 178 subcritical bifurcation 179, 200 “super stair-rod” dislocation 331 supercritical bifurcation 179 superdislocations 436 superposition 16, 18

Subject Index surface acoustic waves (SAW) 605, 621–632, 635–638 surface reconstruction 446 surface roughness 4, 11, 439 surrogate data analysis 141 SWL 96 synchrotron 605, 620, 621 TEM 94, 100, 105 TEM imaging 439 TEM observation – weak-beam 293 TEM observations 285, 299, 309, 310, 333, 354, 357 – bright field 291, 296, 300 – dark field 347 – high resolution 294, 338, 344, 358 – weak-beam 292, 294, 298, 301, 311, 313, 314, 316, 327, 328, 348, 357 TEM observations of – cube dislocations 299, 301, 309 – dislocation substructure 296, 297 – faulted dipoles 291, 293 – faulted dipoles in twin 316 – Frank dislocations 346, 347 – Lomer–Cottrell network 289, 292–294, 311 – – in twin 310, 311, 313, 314, 316 – primary dislocations 287, 291 – secondary stacking faults 356, 357 – stacking fault tetrahedra 294 – twin substructure – – in copper 311, 327, 348 – – in Cu-4%Al 328 – – in Cu-8%Al 298, 300, 301, 313, 314, 316, 345, 347, 357 temperature distribution 40 temperature rise in a sample 592 tensile stress-strain characteristics of – Cu 279 – Cu-4%Al 279 – Cu-8%Al 279 termination 439 termination reflections 440 terrace plain 246 theory of deformation twinning 269, 271, 275 thermal conductivity 605, 611 thermal fluctuations 107 thermodynamic instability 386 thermodynamic stability 385 Thompson tetrahedron 267, 271, 276, 277 threading dislocations in stripped GaN layers 560 three-fold dissociation 425, 435

667

TiN 487 tomographic methods 105 transformation – displacement 250 – matrix 272–274 transformation of ¯ M 317 – a2 [011] – a2 [101]M 348 – a2 [101]M primary dislocations 360 – a2 [110]T 322 – a faulted dipole 330 – a LC lock 321 – a Lomer dislocation 323 – a Lomer lock 317, 321, 322 – a stacking fault 333 ¯ M Frank dipole 335 – an a3 [111] – Burgers vectors and glide planes 277 – conjugate LCs 315 – crystallographic planes and directions 270 – faulted dipoles 326, 328, 338 – Lomer dislocations 322 – partial dislocations 361 – stacking fault tetrahedra 339 – the faulted dipole 334 – the stacking fault 331 – vector components 272 transient dynamics 167, 194 transverse plane (orientation space) 515 twin axes 272 twin axis system 274 twin-matrix orientation relationship 280 twinning – direction 268, 272 – elements 268–270 – – in the fcc lattice 269 – front 280, 307–309, 324, 329, 338, 339, 361 – mode 268, 269, 271 – plane 268 – shear 268–270 – system 270, 278 – transformation 267, 270, 271, 277, 280, 310, 312 twinning transformation of – Burgers vectors and lattice planes 276 – dislocations 295 – faulted dipoles 326, 328, 330, 334, 338 – forest dislocation 324 – Frank dipole 335 – Lomer dislocation 322, 323 – Lomer lock 317, 321, 322 – Lomer–Cottrell dislocations 309, 321 – partial dislocations 361 – perfect dislocations 278 – perfect lattice 275

668

Subject Index

– primary dislocations 299, 302 – stacking fault tetrahedra 278, 339 – stacking faults 331, 333 – stair-rod dislocations 278 two-beam bright-field diffracting 345 – conditions 284, 344 two-beam dark-field diffracting conditions 344, 345 type A 214 type A bands 91, 145–147, 149, 155, 156, 197, 201, 206, 207 type B bands 91, 145–147, 150, 155, 161, 191, 195, 197, 206, 212, 213 type B hopping bands 90 type C 90 type C bands 90, 91, 191, 197, 206, 210 type I twins 268, 273 type II twin 268, 269 types of bands 169, 170, 188, 214 UHV TEM 443 ultra-high vacuum electron microscopes uncorrelated 189 – bands 190, 195 undistorted plane 268 unfaulted dipoles 286 unloading calculations 38 unlocked state 384

439

unlocking stress 434 unpinned state 185, 187 unpinning 183, 192 – energy 425 updated Lagrangian formulation

18

vacancy type 292 – faulted dipoles 327, 358 velocity of propagating bands 192 vibrating string model 613, 617–619 viscosity of dislocation 606 – motions 605 viscous drag of dislocations 610, 611 visualization of dislocation configurations

190

weak-beam dark-field diffracting conditions 284 weak-beam observations 292, 294, 298, 301, 311, 313, 327, 328 wedge 21 weld joint 576 white noise 108, 129, 138 X-ray diffraction

637

Z configuration 292, 326 – of a dipole 327 Z dipoles of vacancy type 330

Dislocations in Solids, Volume 13

Dislocations in Solids, Volume 12 (Dislocations in Solids)

Dislocations in Solids, Volume 15

Dislocations in Solids, Volume 11

Dislocations in Solids: Elastic Theory v. 1 (Dislocations in Solids)